http://www.cds.caltech.edu/~macmardg/wiki/api.php?action=feedcontributions&user=Macmardg&feedformat=atomMacMynowski - User contributions [en]2019-07-17T19:13:16ZUser contributionsMediaWiki 1.19.1http://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2016-05-11T04:19:32Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 51<br />
* Number of distinct journals published in: 28<br />
** Ratio: 0.55<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 65 (an extra 13 from a single paper)<br />
** Number of coauthors with 4 or more coauthored papers: 7 (Tziperman, Caldeira, Kravitz, Keith, Andersen, Hall, Rasch)<br />
** Most frequent coauthor: tie; Ben Kravitz, at 8 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-03-15T16:26:38Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Final exam will be due by 9am on Wednesday the 16th. (I fly out on the 17th, so need some time to grade. That means that if a few of you want to hand it in by noon on the 16th thats ok, as long as not everyone does that; let me know if you want til noon.)<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant]<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Feb_2016.pdf Lecture notes]<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6-wi16.pdf hw6-wi16.pdf] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Mar_2016.pdf Lecture notes]<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw7-wi16.pdf hw7-wi16.pdf] <br> Due: 31 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw8-wi16.pdf hw8-wi16.pdf] <br> Due: 38 Feb (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back by 9am on Wednesday, March 16. Open book, 3 hour time limit in one sitting.<br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-03-05T18:35:10Z<p>Macmardg: /* Announcements */</p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Final exam will be due by 9am on Wednesday the 16th. (I fly out on the 17th, so need some time to grade. That means that if a few of you want to hand it in by noon on the 16th thats ok, as long as not everyone does that; let me know if you want til noon.)<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant]<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Feb_2016.pdf Lecture notes]<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6-wi16.pdf hw6-wi16.pdf] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw7-wi16.pdf hw7-wi16.pdf] <br> Due: 31 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw8-wi16.pdf hw8-wi16.pdf] <br> Due: 38 Feb (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back by 9am on Wednesday, March 16. Open book, 3 hour time limit in one sitting.<br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-03-05T18:33:44Z<p>Macmardg: /* Grading */</p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Feb_2016.pdf Lecture notes]<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6-wi16.pdf hw6-wi16.pdf] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw7-wi16.pdf hw7-wi16.pdf] <br> Due: 31 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw8-wi16.pdf hw8-wi16.pdf] <br> Due: 38 Feb (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back by 9am on Wednesday, March 16. Open book, 3 hour time limit in one sitting.<br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-19T17:25:22Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Feb_2016.pdf Lecture notes]<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6-wi16.pdf hw6-wi16.pdf] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw7-wi16.pdf hw7-wi16.pdf] <br> Due: 31 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw8-wi16.pdf hw8-wi16.pdf] <br> Due: 38 Feb (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-19T17:22:02Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Feb_2016.pdf Lecture notes]<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6-wi16.pdf hw6-wi16.pdf] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-13T18:25:49Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6-wi16.pdf hw6-wi16.pdf] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-10T20:35:37Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7, 3.9<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-08T22:12:40Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
* Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week4Notes.pdf SMT] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5Notes.pdf Lyapunov] <br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-Week5NotesCMT.pdf CMT]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140a-wi11-InvManRemark.pdf Invariant] <br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-08T15:31:01Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes]<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-08T15:29:53Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5(2013).pdf L5 2013 notes]<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-08T15:27:25Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-02-01T16:00:26Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 17 Feb <br> 19 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb '''2 hrs''' <br> ''26 Feb''<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 29 Feb* <br> 2 Mar* <br> 4 Mar*<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-25T18:31:27Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-25T18:13:54Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
|<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-17T17:18:17Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Chaos, fractals, and global analysis using SOStools<br />
* Chaotic systems (logistic map, Mandelbrot set)<br />
* SOStools for finding Lyapunov functions<br />
| <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-15T01:20:09Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* '''2 hrs''' <br> ''22 Jan''<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-13T17:42:05Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-13T17:41:39Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-08T20:08:48Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7*, 9* Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-08T20:08:03Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
* HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7, 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-08T20:06:53Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7, 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-06T19:34:18Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W1_2016.pdf Lecture notes]<br />
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7, 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-06T01:05:39Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
* Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan <br> ''8 Jan''<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13 Jan <br> ''15 Jan''<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan* <br> 22 Jan*<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> '''(2 hr?)''' <br> ''Jan 29''<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 1 Feb* <br> 3 Feb* <br> 5 Feb*<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 8 Feb <br> 10 Feb <br> 12 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 15 Feb <br> 17 Feb <br> 19 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 22 Feb <br> 24 Feb* <br> 26 Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 29 Feb <br> 2 Mar <br> 4 Mar<br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 7, 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-04T22:42:07Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-04T22:04:43Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Monday 2-3 pm<br />
|}<br />
<br />
=== Announcements ===<br />
* Homework will be due Wednesdays in class (or by 5pm to Benson). HW 1 is posted below.<br />
* If you have not received an email to sign up for Piazza, please email us!<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-04T16:36:11Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1] [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162016-01-03T18:49:10Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L1-1.pdf L1-1]<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-31T23:47:58Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<br />
<font color='red' size='+1'>(Page currently under construction, not up to date beyond week 1)</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-31T23:47:11Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<br />
<font color='red' size='+1'>(Page currently under construction, not up to date beyond week 1)</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [http://www.cds.caltech.edu/~macmardg/courses/cds140/wi16/hw1-wi16.pdf hw1-wi16.pdf][[CDS 140a Winter 2016 Homework 1|HW 1]] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-30T22:14:56Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: <br />
** DGM: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2016 Homework 1|HW 1]] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-30T22:13:46Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2016 Homework 1|HW 1]] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140a_Winter_2016_Homework_1CDS 140a Winter 2016 Homework 12015-12-30T22:04:01Z<p>Macmardg: Created page with "{{CDS homework | instructor = D. MacMartin and J. Doyle | course = CDS 140 | semester = Winter 2016 | title = Problem Set #1 | issued = 4 Jan 2016 | due = 13 Jan 2015 at..."</p>
<hr />
<div>{{CDS homework<br />
| instructor = D. MacMartin and J. Doyle<br />
| course = CDS 140<br />
| semester = Winter 2016<br />
| title = Problem Set #1<br />
| issued = 4 Jan 2016<br />
| due = 13 Jan 2015 at 5:00 pm<br>In class or to box across 107 STL (?)<br />
| pdf = cds140-wi16_hw1.pdf<br />
}} __MATHJAX__<br />
<br />
<!--<br />
Notes from 2013 (HW #1):<br />
* Many of the students in the class have ''not'' had ACM 125a/CDS 101. They have not done proofs, Cauchy convergence, etc.<br />
* Would be worth having Friday recitation to provide background on relevant ACM 125a topics and proof techniques<br />
* Think about whether we want to require CDS 201 for CDS 140 (but then aero students may never take the class)<br />
--><br />
<br />
<!--<br />
Notes from 2014 (HW #1):<br />
* In #6, there were quite a few questions about how to show convergence of the infinite series and also how to show that the limit was in E. We tried to explain the proof to them, but most of the students did not know anything about closedness or completeness, so we told them not to worry about it.<br />
<br />
* In #7, the question was about what to do when the determinant equals 0 as the theorem in the textbook does not cover the case where the deteminant is 0. We told them to not worry about this case.<br />
--><br />
<br />
<!-- 2014 feedback from TA office hours (HW #2)<br />
TA1: There were a lot of questions on problem 3. Also, quite a few students either had serious issues with linear algebra or even explicitly said that their linear algebra was poor, so it might be a good idea to do that linear algebra recitation.<br />
<br />
TA2: I also had a lot of questions on problem 3, and also a lot of questions on problem 1a. Two or three persons had some questions on how to compute generalized eigenvectors, and why there are different methods. I gave a brief explanation of how to compute generalized eigenvectors according to the method of Perko (around page 42, depending on the copy of Perko), as one of the students had questions about that, but didn't get into all of the details. That method I believe is a good method and better than other methods I've seen. An example of computing generalized eigenvectors for a 3x3 matrix seemed to be helpful to show some of the students. A lot of people (some in office hours) are not familiar with how to compute e^{Jt} for a Jordan form with off diagonal elements (ones).<br />
--><br />
<br />
<!--<br />
Notes for 2015:<br />
* This homework set is a combination of HW #1 and HW #2 from previous years. The course now requires ACM 104 as a pre-requisite, so the proofs should be OK for the students.<br />
* Left the number of problems at 6, with a combination of theory and numerical examples. <br />
--><br />
<br />
'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on<br />
this homework set (including reading).<br />
<br />
<ol><br />
<!-- Commented out for 2015 <br />
<li> '''Perko, Section 1.1, Exercise 3''': Find the general solution of the linear system<br />
\begin{align}<br />
\dot x_1 &= x_1 \\ x_2 &= a x_2<br />
\end{align}<br />
where $a$ is a constant. Sketch the phase portraits for $a = -1$, $a = 0$ and $a = 1$ and notice the qualitative structure of the phase portrait is the same for all $a < 0$ as well as for all $a > 0$, but that it changes at the parameter value $a = 0$, called a ''bifurcation value''.<br />
</li><br />
<br />
<li> '''Perko, Section 1.1, Exercise 6:'''<br />
<br><br />
(a) If $u(t)$ and $v(t)$ are sollutions of the linear system<br />
\begin{align}<br />
\dot x = A x,<br />
\end{align}<br />
prove that for any constants $a$ and $b$, $w(t) = a u(t) + b v(t)$ is a solution.<br />
<br />
<br><br />
(b) For<br />
\begin{align}<br />
A = \begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix},<br />
\end{align}<br />
find solutions $u(t)$ and $v(t)$ of $\dot x = A x$ such that every solution is a linear combination of $u(t)$ and $v(t)$.<br />
</li><br />
--><br />
<br />
<!-- Comments, RMM 19 Jan 2015: this problem is too simple to be very interesting. Delete next time around. --><br />
<li> '''Perko, Section 1.2, Exercise 6:''' Let the $n \times n$ matrix $A$ have real, distinct eigenvalues. Let $\phi(t, x)$ the the solution of the initial value problem<br />
\begin{align}<br />
\dot x &= A x &\qquad x(0) &= x_0.<br />
\end{align}<br />
Show that for each fixed $t \in {\mathbb R}$,<br />
\begin{align}<br />
\lim_{y_0 \to x_0} \phi(t, y_0) = \phi(t, x_0).<br />
\end{align}<br />
This shows that the solution $\phi(t, x_0)$ is a continuous function of the initial condition.<br />
</li><br />
<br />
<!-- Comments, RMM 19 Jan 2015: Make more explicit that students should compute e^A, not just the eigenvalues and eigenvectors. Otherwise, students just use part (b) and they don't explore *how* to compute e^A --><br />
<li> (Based on Perko, Section 1.3, Exercise 5, 6) <br />
<br> (a) For each matrix below, find the eigenvalues and eigenvectors of $A$ and $e^A$:<br />
{| style="margin: 1em auto 1em auto"<br />
|-<br />
| (i) $\begin{bmatrix} a & 0 \\ 0 & -b \end{bmatrix}$<br />
| width=10% |<br />
| (ii) $\begin{bmatrix} 1 & 0 \\ a & 1 \end{bmatrix}$<br />
| width=10% |<br />
| (iii) $\begin{bmatrix} 5 & -6 \\ 3 & -4 \end{bmatrix}$<br />
|}<br />
(remember to show the steps required for these [simple!] computations, don't just plug in values from MATLAB or Mathematica; see notes at the bottom of the page).<br />
<br />
<!-- RMM, 19 Jan 2015: One student only showed this for the matrices above. Make more clear next time --><br />
(b) Show that if $x$ is the eigenvector of $A$ corresponding to the eigenvalue $\lambda$, then $x$ is also an eigenvector of $e^A$ corresponding to the eigenvalue $e^\lambda$.<br />
<!-- Deleted in Jan 2014<br />
<br> <br />
(c) If $A = P \text{diag} [\lambda_j] P^{-1}$, use Corollary 1 in Section 1.3 to show that<br />
\begin{align}<br />
\det\, e^A = e^{\text{trace}\, A}<br />
\end{align}<br />
</li><br />
<br />
<!-- Deleted in Jan 2015<br />
<li> '''Perko, Section 1.4, Exercise 4:''' Given<br />
\begin{align}<br />
A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 1 & 0 & -1 \end{bmatrix},<br />
\end{align}<br />
compute the $3 \times 3$ matrix $e^{At}$ and solve $\dot x = A x$.<br />
</li><br />
--><br />
<br />
<li> (Based on Perko, Section 1.4, Exercise 6) Let $A:{\mathbb R}^n \to {\mathbb R}^n$ be a linear transformation that leaves a subspace $E \subset {\mathbb R}^n$ invariant (i.e., for all $x \in E$, $A x \in E$). Show that if $x(t)$ is the solution of the initial value problem<br />
\begin{align}<br />
\dot x &= A x &\qquad x(0) &= x_0<br />
\end{align}<br />
with $x_0 \in E$, then $x(t) \in E$ for all $t \in {\mathbb R}$.<br />
</li><br />
<br />
<!-- Deleted in Jan 2015<br />
<li> '''Perko, Section 1.5, Exercise 1:''' Use the theorem in Section 1.5 to determine if the linear system $\dot x = A x$ has a saddle, node, focus or center at the origin and determine the stability of each node or focus:<br />
{| style="margin: 1em auto 1em auto"<br />
|-<br />
| (a) $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$<br />
| width=10% |<br />
| (b) $A = \begin{bmatrix} \lambda & -2 \\ 1 & \lambda \end{bmatrix}$<br />
| width=10% |<br />
| (c) $A = \begin{bmatrix} \lambda & 2 \\ 1 & \lambda \end{bmatrix}$<br />
|}<br />
(If your answer depends on the value of a parameter, make sure to describe all possible cases.)<br />
</li><br />
--><br />
<br />
<li> '''Perko, Section 1.6, Exercise 2:''' Solve the initial value problem <br />
\begin{align}<br />
\dot x &= A x &\qquad x(0) &= x_0<br />
\end{align}<br />
with<br />
\begin{align}<br />
A &= \begin{bmatrix} 0 & -2 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & -2 \end{bmatrix}.<br />
\end{align}<br />
Determine the stable and unstable subspaces and sketch the (3D) phase portrait. (Hint: see Figure 1 in Section 1.6 for an example of a 3D phase portrait.)<br />
</li><br />
<br />
<!-- Commented out for 2014<br />
<li> '''Perko, Section 1.7, problem 3, parts (a) and (d)''': Solve the initial value problem (1) with the matrix<br />
<center><amsmath><br />
\text{(a)}\quad A = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 \end{bmatrix}, \qquad<br />
\text{(d)}\quad A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & 1 \end{bmatrix}, \qquad<br />
</amsmath></center><br />
</li><br />
--><br />
<!-- Deleted in Jan 2015<br />
<li> '''Perko, Section 1.8, problem 10'''<br />
Suppose that the elementary blocks <amsmath>B</amsmath> in the Jordan form of the matrix <amsmath>A</amsmath>, have no ones or <amsmath>I_2</amsmath> blocks off the diagonal, so that they are of the form<br />
<center><amsmath><br />
B = \begin{bmatrix}<br />
\lambda & 0 & 0 & \dots & 0 \\<br />
0 & \lambda & 0 & \dots & 0 \\<br />
\dots & & & & & \\<br />
0 & \dots & & \lambda & 0 \\<br />
0 & \dots & & 0 & \lambda <br />
\end{bmatrix}<br />
\qquad\text{or}\qquad<br />
B = \begin{bmatrix}<br />
D & 0 & 0 & \dots & 0 \\<br />
0 & D & 0 & \dots & 0 \\<br />
\dots & & & & & \\<br />
0 & \dots & & D & 0 \\<br />
0 & \dots & & 0 & D<br />
\end{bmatrix},<br />
\quad<br />
D = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.<br />
</amsmath></center><br />
(a) Show that if all of the eigenvalues of <amsmath>A</amsmath> have nonpositive real parts, then for each <amsmath>x_0 \in {\mathbb R}^n</amsmath> there is a positive constant <amsmath>M</amsmath> such that <amsmath>|x(t)| \leq M</amsmath> for all <amsmath>t \geq 0</amsmath> where <amsmath>x(t)</amsmath> is the solution of the initial value problem.<br />
<br />
(b) Show via a simple counterexample that this is not true if the Jordan blocks have non-zero off diagonal entries (with the same constraint on the eigenvalues).<br />
</li><br />
--><br />
<!-- Removed in Jan 2015<br />
<li> '''Perko, Section 1.9, problem 3''' (modified): Consider the linear system<br />
<center><amsmath><br />
\dot x = \begin{bmatrix} a & 0 & 0 & 0\\ a & 0 & -b & 0 \\ a & 0 & -b & 0 \\ a & a & 0 & -b \end{bmatrix} x,<br />
</amsmath></center><br />
where <amsmath>a,\,b > 0</amsmath>.<br />
<br />
(a) Compute the solutions to the differential equation. You should provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).<br />
:* Note: you should show the various (regular and generalized) eigenvectors associated with each eigenvalue. OK to check your answer with MATLAB, but be sure to show that you know how to solve it by hand.<br />
<br />
(b) Find the stable, unstable and center subspaces for this system (in the original coordinates).<br />
</li><br />
--><br />
<br />
<li> '''Perko, Section 1.9, problem 5, parts (c), (d2)''': Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>. Show that<br><br />
<br />
(c) If <amsmath>x_0 \in E^c</amsmath>, <amsmath>x_0 \neq 0</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq |x(t)| \leq M</amsmath>;<br />
:* Note: in the book, Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath> in the book is the set <amsmath>E</amsmath> minus the point 0. <br />
<br />
(d2) If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0</amsmath> has nonzero components in both <amsmath>E^s</amsmath> and <amsmath>E^u</amsmath>, then <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;<br />
</li><br />
<br />
<!-- Replaced in 2014 with problem below <br />
<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 in Section 1.10 to solve the nonhomogeneous linear system<br />
<center><amsmath><br />
\dot x = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix} x + \begin{bmatrix} t \\ 1 \end{bmatrix}<br />
</amsmath></center><br />
with the initial condition<br />
<center><amsmath><br />
x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.<br />
</amsmath></center><br />
</li><br />
--><br />
<br />
<li><br />
Consider the system<br />
<center><amsmath><br />
\frac{dx}{dt} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ 1 \end{bmatrix}u, \qquad<br />
y = \begin{bmatrix} 1 & 0 \end{bmatrix} x.<br />
</amsmath></center><br />
(a) Show that the unforced system (<amsmath>u = 0</amsmath>) is stable but not asymptotically stable.<br />
<br />
(b) Given <amsmath>x(0) = x_0</amsmath> and <amsmath>u(t) = \cos(\omega*t)</amsmath>, solve for the output <amsmath>y(t)</amsmath>. Show that when <amsmath>\omega=1</amsmath> the output is unbounded.<br />
</li><br />
</ol><br />
<br />
<hr><br />
Notes:<br />
* The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko (third edition) as the definitive source of the problem statement.<br />
* There are a number of problems that can be solved using MATLAB, Mathematica or a similar program. If you just give the answer with no explanation (or say "via MATLAB"), the TAs will take off points. Instead, you should show how the solutions can be worked out by hand, along the lines of what is done in the text book. It is fine to check everything with MATLAB or your favorite software tool.<br />
* For numerical calculations, it is OK to use MATLAB to invert a matrix. But you should not use it to compute the matrix exponential and just put down the answer. Instead, show how to get the matrix exponential into a form in which the calculation can be done by hand and then carry out the computation.<br />
* For phase portraits, you should generate the diagram by hand and make sure to label any important features. Describe why the portrait looks as it does based on the relevant properties of the dynamical system (eg, eigenvalues of the A matrix).<br />
* For the final exam, you will ''not'' be allowed to use MATLAB, Mathematica or similar programs, so make sure you understand what you are computing and drawing!</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-30T22:02:58Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2016 Homework 1|HW 1]] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2016 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2016 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2016 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=Template:CDS_homeworkTemplate:CDS homework2015-12-30T22:01:15Z<p>Macmardg: Created page with "{| width=100% |- | align=left | {{{instructor}}} | align=right | Issued: {{{issued}}} |- valign=top | {{{course}}}, {{{semester}}} {{#if: {{{pdf|}}} | [[Media:{{{pdf}}}|(P..."</p>
<hr />
<div>{| width=100%<br />
|-<br />
| align=left | {{{instructor}}}<br />
| align=right | Issued: {{{issued}}}<br />
|- valign=top<br />
| [[{{{course}}}, {{{semester}}}]]<br />
{{#if: {{{pdf|}}} | [[Media:{{{pdf}}}|(PDF)]] }}<br />
| align=right | Due: {{{due}}}<br />
|}</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-30T21:57:49Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 4 Jan <br> 6 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2015 Homework 1|HW 1]] <br> Due: 13 Jan (Wed)<br />
|- valign=top<br />
| 11 Jan <br> 13&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2015 Homework 2|HW 2]] <br> Due: 20 Jan (Wed)<br />
|- valign=top<br />
| 20 Jan <br> 22 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2015 Homework 3|HW 3]] <br> Due: 27 Jan (Wed)<br />
|- valign=top<br />
| 25 Jan <br> TBD&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2015 Homework 4|HW 4]] <br> Due: 3 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2015 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2015 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2015 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2015 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 9 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=Main_PageMain Page2015-12-30T21:54:04Z<p>Macmardg: </p>
<hr />
<div><!-- #REDIRECT [[CDS 140, Winter 2016]] !--><br />
== Courses on this site==<br />
<br />
* [[CDS 140, Winter 2016]]<br />
* [[CDS 140b, Spring 2012]]<br />
* [[CDS 101/110a, Fall 2014]]<br />
* [[CDS 101/110a, Fall 2013]]<br />
* [[CDS 101/110a, Fall 2012]]<br />
* [[CDS 101/110a, Fall 2011]]<br />
* [[CDS 101/110a, Fall 2010]]<br />
* [[CDS 101/110a, Fall 2009]]<br />
* [[CDS 110b, Winter 2011]]<br />
* [[CDS 110b, Winter 2010]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-30T20:42:40Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 1-1:55, 213 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: TBD<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 5 Jan <br> 7 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2015 Homework 1|HW 1]] <br> Due: 14 Jan (Wed)<br />
|- valign=top<br />
| 12 Jan <br> 16&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2015 Homework 2|HW 2]] <br> Due: 21 Jan (Wed)<br />
|- valign=top<br />
| 21 Jan <br> 23 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2015 Homework 3|HW 3]] <br> Due: 28 Jan (Wed)<br />
|- valign=top<br />
| 26 Jan <br> 30&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2015 Homework 4|HW 4]] <br> Due: 4 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2015 Homework 5|HW 5]] <br> Due: 11 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2015 Homework 6|HW 6]] <br> Due: 18 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2015 Homework 7|HW 7]] <br> Due: 25 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2015 Homework 8|HW 8]] <br> Due: 4 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 11 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2015CDS 140, Winter 20152015-12-30T20:41:07Z<p>Macmardg: Macmardg moved page CDS 140, Winter 2015 to CDS 140, Winter 2016: original error</p>
<hr />
<div>#REDIRECT [[CDS 140, Winter 2016]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-30T20:41:07Z<p>Macmardg: Macmardg moved page CDS 140, Winter 2015 to CDS 140, Winter 2016: original error</p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 11-12:30, 312 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Tue, 4-5 pm, Ann 106<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 5 Jan <br> 7 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2015 Homework 1|HW 1]] <br> Due: 14 Jan (Wed)<br />
|- valign=top<br />
| 12 Jan <br> 16&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2015 Homework 2|HW 2]] <br> Due: 21 Jan (Wed)<br />
|- valign=top<br />
| 21 Jan <br> 23 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2015 Homework 3|HW 3]] <br> Due: 28 Jan (Wed)<br />
|- valign=top<br />
| 26 Jan <br> 30&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2015 Homework 4|HW 4]] <br> Due: 4 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2015 Homework 5|HW 5]] <br> Due: 11 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2015 Homework 6|HW 6]] <br> Due: 18 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2015 Homework 7|HW 7]] <br> Due: 25 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2015 Homework 8|HW 8]] <br> Due: 4 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 11 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2015-12-28T21:39:59Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 49<br />
* Number of distinct journals published in: 27<br />
** Ratio: 0.55<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 64 (an extra 13 from a single paper)<br />
** Number of coauthors with 4 or more coauthored papers: 7 (Tziperman, Caldeira, Kravitz, Keith, Andersen, Hall, Rasch)<br />
** Most frequent coauthor: tie; Ben Kravitz and David Keith, at 7 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-04T15:13:29Z<p>Macmardg: </p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Douglas MacMartin, macmardg@cds.caltech.edu<br />
* John Doyle, doyle@cds.caltech.edu<br />
* Lectures: MWF, 11-12:30, 312 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Benson Christalin (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Tue, 4-5 pm, Ann 106<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 5 Jan <br> 7 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2015 Homework 1|HW 1]] <br> Due: 14 Jan (Wed)<br />
|- valign=top<br />
| 12 Jan <br> 16&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2015 Homework 2|HW 2]] <br> Due: 21 Jan (Wed)<br />
|- valign=top<br />
| 21 Jan <br> 23 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2015 Homework 3|HW 3]] <br> Due: 28 Jan (Wed)<br />
|- valign=top<br />
| 26 Jan <br> 30&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2015 Homework 4|HW 4]] <br> Due: 4 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2015 Homework 5|HW 5]] <br> Due: 11 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2015 Homework 6|HW 6]] <br> Due: 18 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2015 Homework 7|HW 7]] <br> Due: 25 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2015 Homework 8|HW 8]] <br> Due: 4 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 11 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_140,_Winter_2016CDS 140, Winter 20162015-12-04T01:24:57Z<p>Macmardg: Created page with "{| width=100% |- | colspan=2 align=center | <font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__ <font color='red' size='+3'>Page currently under construction..."</p>
<hr />
<div>{| width=100%<br />
|-<br />
| colspan=2 align=center |<br />
<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__<br />
<font color='red' size='+3'>Page currently under construction</font>__NOTOC__<br />
|- valign=top<br />
| width=50% |<br />
'''Instructors'''<br />
* Richard Murray (CDS/BE), murray@cds.caltech.edu<br />
* Lectures: MWF, 11-12:30, 312 ANB<br />
* Office hours: Wed 2-3 pm (please e-mail to confirm)<br />
| width=50% |<br />
'''Teaching Assistants'''<br />
* Vanessa Jonsson (CDS)<br />
* Contact: cds140-tas@cds.caltech.edu<br />
* Office hours: Tue, 4-5 pm, Ann 106<br />
|}<br />
<br />
=== Course Description ===<br />
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.<br />
<br />
=== Lecture Schedule ===<br />
<br />
{| class="mw-collapsible " width=100% border=1 cellpadding=5<br />
|-<br />
| '''Date'''<br />
| '''Topic'''<br />
| '''Reading'''<br />
| '''Homework'''<br />
|- valign=top<br />
|- valign=top<br />
| 5 Jan <br> 7 Jan<br />
| Linear Differential Equations<br />
* Course overview and administration<br />
* Linear differential equations<br />
* Matrix exponential, diagonalization<br />
* Stable and unstable spaces<br />
* S + N decomposition, Jordan form<br />
| <br />
Perko, 1.1-1.10<br><br />
| [[CDS 140a Winter 2015 Homework 1|HW 1]] <br> Due: 14 Jan (Wed)<br />
|- valign=top<br />
| 12 Jan <br> 16&nbsp;Jan<br />
| Nonlinear differential equations<br />
* Existence and uniqueness<br />
* Flow of a differential equation<br />
* Linearization<br />
| Perko, 2.1-2.6<br />
| [[CDS 140a Winter 2015 Homework 2|HW 2]] <br> Due: 21 Jan (Wed)<br />
|- valign=top<br />
| 21 Jan <br> 23 Jan<br />
| Behavior of differential equations<br />
* Stable and unstable manifolds <br />
* Stability of equilibrium points for planar systems<br />
| Perko, 2.7-2.10 <br />
| [[CDS 140a Winter 2015 Homework 3|HW 3]] <br> Due: 28 Jan (Wed)<br />
|- valign=top<br />
| 26 Jan <br> 30&nbsp;Jan<br />
| Non-hyperbolic differential equations<br />
* Lyapunov functions<br />
* Center manifold theorem<br />
| Perko, 2.11-2.13<br />
| [[CDS 140a Winter 2015 Homework 4|HW 4]] <br> Due: 4 Feb (Wed)<br />
|- valign=top<br />
| 4 Feb <br> 6 Feb<br />
| Global behavior<br />
* Limit sets and attractors<br />
* Krasovskii-Lasalle invariance principle (if time)<br />
* Periodic orbits and limit cycles<br />
| Perko, 3.1-3.3<br />
| [[CDS 140a Winter 2015 Homework 5|HW 5]] <br> Due: 11 Feb (Wed)<br />
|- valign=top<br />
| 9 Feb <br> 11 Feb<br />
| Limit cycles<br />
* Poincare' map<br />
* Bendixson criterion for limit cycles in the plane<br />
* Describing functions (if time)<br />
| Perko, 3.4-3.5, 3.7<br />
| [[CDS 140a Winter 2015 Homework 6|HW 6]] <br> Due: 18 Feb (Wed)<br />
|- valign=top<br />
| 18 Feb <br> 20 Feb<br />
| Bifurcations<br />
* Sensitivity analysis<br />
* Structural stability<br />
* Bifurcation of equilibrium points<br />
| Perko 4.1-4.2 <br><br />
[[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_Page|BFS]] 3.2 and 5.4 ([[Media:cds140-wi15_bfs-sensitivity.pdf|PDF]])<br />
| [[CDS 140a Winter 2015 Homework 7|HW 7]] <br> Due: 25 Feb (Wed)<br />
|- valign=top<br />
| 23 Feb <br> 25&nbsp;Feb? <br> 27&nbsp;Feb?<br />
| Bifurcations<br />
* Hopf bifurcation<br />
* Application example: rotating stall and surge in turbomachinery<br />
| Perko 4.3-4.5 + notes<br />
| [[CDS 140a Winter 2015 Homework 8|HW 8]] <br> Due: 4 Mar (Wed)<br />
|- valign=top<br />
| 2 Mar <br> 6 Mar <br />
| Nonlinear control systems<br />
| {{obc08|OBC}}, Chapter 1<br />
| OBC 1.3, 1.4ab, 1.5<br> Due: 11 Mar (Wed)<br />
|- valign=top<br />
| 9 Mar <br> <br />
| Course review<br />
| <!-- Reading --><br />
| Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab<br />
|}<br />
<br />
=== Textbook ===<br />
<br />
The primary text for the course (available via the online bookstore) is<br />
{|<br />
|- valign=top<br />
| align=right | &nbsp;[Perko]&nbsp;<br />
| L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.<br />
|}<br />
<br />
The following additional texts may be useful for some students:<br />
{|<br />
|- valign=top<br />
| align=right| &nbsp;[G&H]&nbsp;<br />
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right| &nbsp;[H&S]&nbsp;<br />
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.<br />
|- valign=top<br />
| align=right | &nbsp;[J&S]&nbsp;<br />
| D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)<br />
|- valign=top<br />
| align=right | &nbsp;[Ver]&nbsp;<br />
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)<br />
|}<br />
<br />
=== Grading ===<br />
The ﬁnal grade will be based on homework and a ﬁnal exam:<br />
* Homework (75%) - There will be 9 one-week problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.''<br />
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period). <br />
<br />
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.<br />
<br />
=== Collaboration Policy ===<br />
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.<br />
<br />
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).<br />
<br />
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.<br />
<br />
[[Category:Courses]]</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2015-08-04T23:06:11Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 48<br />
* Number of distinct journals published in: 26<br />
** Ratio: 0.54<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 64 (an extra 13 from a single paper)<br />
** Number of coauthors with 5 or more coauthored papers: 4 (Tziperman, Caldeira, Kravitz, Keith)<br />
** Most frequent coauthor: David Keith, at 7 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2015-08-04T22:56:52Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 48<br />
* Number of distinct journals published in: 26<br />
** Ratio: 0.54<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 63 (an extra 13 from a single paper)<br />
** Number of coauthors with 5 or more coauthored papers: 4 (Tziperman, Caldeira, Kravitz, Keith)<br />
** Most frequent coauthor: David Keith, at 7 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2015-03-27T15:55:29Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 46<br />
* Number of distinct journals published in: 25<br />
** Ratio: 0.54<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 63 (an extra 13 from a single paper)<br />
** Number of coauthors with 5 or more coauthored papers: 4 (Tziperman, Caldeira, Kravitz, Keith)<br />
** Most frequent coauthor: David Keith, at 7 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2015-03-27T15:09:06Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 46<br />
* Number of distinct journals published in: 25<br />
** Ratio: 0.54<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 62 (an extra 13 from a single paper)<br />
** Number of coauthors with 5 or more coauthored papers: 4 (Tziperman, Caldeira, Kravitz, Keith)<br />
** Most frequent coauthor: David Keith, at 7 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=UselessStatisticsUselessStatistics2015-01-14T17:49:45Z<p>Macmardg: </p>
<hr />
<div>'''Irrelevant Useless Statistics'''<br />
<br />
* Number of journal articles (accepted or in print): 45<br />
* Number of distinct journals published in: 25<br />
** Ratio: 0.56<br />
** Most frequent journal: Applied Optics (7)<br />
* Number of distinct coauthors: 61 (an extra 13 from a single paper)<br />
** Number of coauthors with 5 or more coauthored papers: 4 (Tziperman, Caldeira, Kravitz, Keith)<br />
** Most frequent coauthor: David Keith, at 7 <br />
* Erdos number: 4 (Lall, Boyd, Diaconis)<br />
* According to The Mathematics Genealogy Project, 18 academic-generations back on my family tree is Isaac Newton, and 21 is Galileo.</div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_101/110a,_Fall_2014CDS 101/110a, Fall 20142014-12-17T22:28:44Z<p>Macmardg: </p>
<hr />
<div>{{cds101-fa14}}<br />
This is the homepage for CDS 101 (Analysis and Design of Feedback Systems) and CDS 110a (Introduction to Control Theory) for Fall 2014. __NOTOC__<br />
----<br />
<br />
<table width=100%><br />
<tr valign=top><br />
<td><br />
'''Instructor'''<br />
* Doug MacMartin, macmardg@cds.caltech.edu<br />
* Lectures: MW, 2-3 pm, 105 Annenberg; lecture/recitation F 2-3 pm (also 105 ANB)<br />
** Friday Lecture or Recitation is optional for CDS 101<br />
* Office hours: Mondays, 1-2 pm <br />
* Prior years: [http://www.cds.caltech.edu/~murray/wiki/CDS_101/110a%2C_Fall_2006 FA06], [http://www.cds.caltech.edu/~murray/wiki/CDS_101/110a%2C_Fall_2007 FA07],[http://www.cds.caltech.edu/~murray/wiki/CDS_101/110a%2C_Fall_2008 FA08], [[CDS 101/110a, Fall 2009 | FA09]], [[CDS 101/110a, Fall 2010 | FA10]], [[CDS 101/110a, Fall 2011 | FA11]], [[CDS 101/110a, Fall 2012 | FA12]], [[CDS 101/110a, Fall 2013 | FA13]]<br />
</td><td><br />
'''Teaching Assistants''' [mailto:cds110-tas@cds.caltech.edu cds110-tas@cds.caltech.edu]<br />
* Ivan Papusha<br />
* David Flicker<br />
* Jerry Cruz<br />
* Ioannis Filippidis<br />
* Office hours:<br />
** Mon 3-5 pm in 107 ANB (243 ANB on 10/20 only)<br />
** Tue 7-9 pm in 107 ANB<br />
''' Course Ombuds:''' [mailto:efeldman@caltech.edu Ellen Feldman] <br />
<br />
</td></tr><br />
</table><br />
<br />
== Announcements ==<br />
* 17 Dec: Final exam statistics: CDS 110 mean = 38.6/60, std = 11.3. CDS 101 mean = 28/40, std = 6.7<br />
* 7 Dec: Plots of lead/lag frequency response with a parameterization different from what I used in class: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa14/pdf/LeadLag-freqresp.pdf LeadLag.pdf]. This may or may not be helpful.<br />
* 5 Dec: Extra copies of the final are available outside 107 Steele Lab. Please return to Nikki Fauntleroy by '''Thu Dec 11 5pm''' (Fri Dec 12 8am at the latest). In addition please fill out TQFR surveys on Caltech Access.<br />
* 3 Dec: The TAs will hold additional office hours on Thu 4 Dec, 7--9pm (107 ANB).<br />
* 21 Nov: NOTE: for today only, class is in 102 Steele<br />
* 18 Nov: HW #7 is available. <br />
** Problem 3: compare your design to C(s)=41(1+s/4)/(1+s/100) if you do not have access to another classmate's controller<br />
** Problem 4/(CDS101 Problem 2): see [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa13/pdf/MacMartin-Kravitz-Keith-Jarvis-2013.pdf MacMartin-etal-2013.pdf]<br />
* 18 Nov: Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
<br />
<table style="float: right" border=0><br />
<tr><br />
<td><br />
[[#Old Announcements|Archive]]<br />
</td></tr><br />
</table><br />
<br />
<!--<br />
== Homework/Exam Statistics ==<br />
<table align=right border=0><br />
<tr><br />
<td><br />
</td></tr><br />
</table><br />
Note that the solutions can also be obtained from the [[CDS 101/110a, Fall 2014 - Course Schedule|course schedule]] page after the homework for that week has been handed in. Solutions are only available from within the caltech.edu domain.<br />
* Oct. 11: HW1 -- CDS101 [ Avg. Grade = xx/20 | Avg. Time Spent = x hrs. ], CDS110a [ Avg. Grade = xx/60 | Avg. Time Spent = x hrs. ]<br />
--><br />
= Course Syllabus =<br />
<table style="float: right" border=1 width=20% cellpadding=6><br />
<tr><br />
<td><br />
<center>'''Contents'''</center><br />
<ul><br />
<li> [[#Grading|Grading]] </li><br />
<li> [[#Lectures and Recitations|Lectures/Recitations]] </li><br />
<li> [[#Collaboration Policy|Collaboration Policy]] </li><br />
<li> [[#Software|Software]] </li><br />
<li> [[#Course Text and References|Course Text]] </li><br />
<li> [[#Course_Schedule|Course Schedule]]</li><br />
</ul><br />
</td></tr><br />
</table><br />
CDS 101/110 provides an introduction to feedback and control in physical, biological, engineering, and information sciences. Basic principles of<br />
feedback and its use as a tool for altering the dynamics of systems and managing uncertainty. Key themes throughout the course will include<br />
input/output response, modeling and model reduction, linear versus nonlinear models, and local versus global behavior. The course has several variants:<br />
<br />
* CDS 101 is a 6 unit (2-0-4) class intended for advanced students in science and engineering who are interested in the principles and tools of feedback control, but not the analytical techniques for design and synthesis of control systems. <br />
<br />
* CDS 110 is a 12 unit class (3-0-9) that provides a traditional first course in control for engineers and applied scientists. It assumes a stronger mathematical background, including working knowledge of linear algebra and ODEs. Familiarity with complex variables (Laplace transforms, residue theory) is helpful but not required. <br />
<br />
=== Lectures and Recitations ===<br />
The main course lectures are on MW from 2-3 pm in 105 Annenberg. Friday 2-3 will either be an additional lecture or a problem solving (recitation) session run by the course teaching assistants. CDS 101 students are not required to attend the Friday lecture or recitation, although they are welcome to do so (CDS 101 homework will typically require much less use of Matlab or Python). <br />
<br />
=== Grading ===<br />
The final grade will be based on homework sets, a midterm exam, and a final exam: <br />
<br />
*''Homework (50%):'' Homework sets will be handed out weekly and due on Wednesdays by 5 pm to the locked box outside of 102 Steele. Students are allowed three grace periods of two days each that can be used at any time (but no more than 1 grace period per homework set). Late homework beyond the grace period will not be accepted without a note from the health center or the Dean; please plan accordingly. MATLAB or Python code are considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).<br />
<br />
* ''Midterm exam (20%):'' A midterm exam will be handed out at the beginning of midterms period (29 Oct) and due at the end of the midterm examination period (4 Nov). The midterm exam will be open book and computers will be allowed (though not required). <br />
<br />
* ''Final exam (30%):'' The final exam will be handed out on the last day of class (5 Dec) and due on Friday of finals week (Dec 12). It will be an open book exam and computers will be allowed (though not required).<br />
<br />
=== Collaboration Policy ===<br />
<br />
Collaboration on homework assignments is encouraged. You may consult<br />
outside reference materials, other students, the TA, or the<br />
instructor, but you cannot consult homework solutions from<br />
prior years and you must cite any use of material from outside<br />
references. All solutions that are handed in should be written up<br />
individually and should reflect your own understanding of the subject<br />
matter at the time of writing. Python or MATLAB scripts and plots are<br />
considered part of your writeup and should be done individually (you<br />
can share ideas, but not code).<br />
<br />
''No collaboration is allowed on the midterm or final exams.''<br />
<br />
=== Software ===<br />
Computer exercises will be assigned as part of the regular homeworks. The<br />
exercises are designed to be done either in MATLAB, using the Control Toolbox, or in Python.<br />
SIMULINK may be useful but is not required. <br />
Caltech has a site license for Matlab and it may be obtained<br />
from [http://software.caltech.edu IMSS] (Caltech students only). An online tutorial is available at<br />
<center><br />
http://www.engin.umich.edu/group/ctm/basic/basic.html<br />
</center><br />
<br />
=== Course Text and References ===<br />
<br />
The primary course text is [http://www.cds.caltech.edu/~murray/amwiki/Main_Page ''Feedback Systems: An Introduction for Scientists and Engineers''] by {{Astrom}} and Murray (2008). This book is available in the Caltech bookstore and via download from the [http://www.cds.caltech.edu/~murray/amwiki/Main_Page companion web site]. The following additional references may also be useful:<br />
<br />
* A. D. Lewis, ''A Mathematical Approach to Classical Control'', 2003. [http://www.mast.queensu.ca/~andrew/teaching/math332/notes.shtml Online access].<br />
<br />
In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.<br />
<br />
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', McGraw-Hill, 1986.<br />
* G. F. Franklin, J. D. Powell, and A. Emami-Naeni, ''Feedback Control of Dynamic Systems'', Addison-Wesley, 2002.<br />
<br />
=== Course Schedule ===<br />
A detailed course schedule is available on the [[CDS 101/110a, Fall 2014 - Course Schedule|course schedule]] page (also shown on the "menu bar" at the top of each course page).<br />
<br />
== Old Announcements ==<br />
* Old announcements will appear here when they are archived.<br />
* Oct 30: Extra midterms are available on top of the homework dropbox in Steele.<br />
* Oct 23: Matlab and Python files for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/bike_linmod.m bike_linmod.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/bike_linmod.py bike_linmod.py]. For further details on bicycle dynamics, see [http://www.cds.caltech.edu/~murray/amwiki/index.php/Bicycle_dynamics Bicycle dynamics], and the IEEE article [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1499389&isnumber=32202 K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4):26-47, August 2005].<br />
* Oct 17: Simulation files for HW #3: In python: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/cartpend.py cartpend.py], and in Matlab: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 15: Monday's (10/20) office hours will be in 243 ANB. Office hours will be in the usual room after Monday.<br />
* Oct 9: Tuesday night office hours will be from 7-9 pm from now on (in the same room).<br />
* Oct 6: HW #2 files for plotting phase portrait: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/boxgrid.m boxgrid.m] for neater looking plots.<br />
* Sep 30: Everyone in the class should fill out this [https://docs.google.com/spreadsheet/viewform?usp=drive_web&formkey=dE8zMUtqYlFhMjdJa0JKNEhZYmJwQXc6MA#gid=0 survey]. It will help us better gauge the level of background knowledge of the class.<br />
* Sep 29: Everyone should by now have received an invite from Piazza. If you are taking the class and have not received an invite, please email the TAs at [mailto:cds110-tas@cds.caltech.edu cds110-tas@cds.caltech.edu] or sign up [http://piazza.com/caltech/fall2014/cds110 here].<br />
* Sep 27: You will need to know MATLAB for this class. If you are unfamiliar with MATLAB here is a good starting tutorial: [http://www.cyclismo.org/tutorial/matlab/ Basics Tutorial]. Here is a tutorial created by a TA from a previous year: [https://www.dropbox.com/s/uxk94uwbc2rcw60/cds110-matlab-tutorial-pack.zip?dl=0 Tutorial-Pack.zip]. It has some code that you can test out and an example homework problem done in MATLAB. <br />
* Sep 15: If you have not been signed up for Piazza by the first week of class, sign up [http://piazza.com/caltech/fall2014/cds110 here].<br />
* Sep 10: Updated links and created course schedule.<br />
* Aug 9: Website created, currently under construction<br />
<br />
<!-- leaving these as templates <br />
* 20 Nov: HW #7 is available. For the final problem see [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa13/pdf/MacMartin-Kravitz-Keith-Jarvis-2013.pdf MacMartin-etal-2013.pdf],<br />
* 12 Nov: HW #6 is available. Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
* Oct 28: Midterm will be handed out on Wednesday Oct 30. The midterm will not be available online; if you are not in lecture please pick up a midterm from Anissa Scott in 107 Steele during normal work hours. The midterm will be due the following Tuesday Nov 5 at 5pm. We will permit you to use one of your homework grace periods and hand it in ''no later than 8am Nov 6''. If you choose to do so, (a) please send us an email to confirm and (b) hand it in in the usual homework inbox in Steele.<br />
** There will be no office hours on the week of the midterm (Mon-Tue Nov 4-5)<br />
* Oct 27: Matlab and Python files for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/bike_linmod.m bike_linmod.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/bike_linmod.py bike_linmod.py]. For further details on bicycle dynamics, see [http://www.cds.caltech.edu/~murray/amwiki/index.php/Bicycle_dynamics Bicycle dynamics], and the IEEE article [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1499389&isnumber=32202 K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4):26-47, August 2005].<br />
* Oct 22: HW #4 posted. Note that if you downloaded this last night, then the link pointed to the HW assignment from 2012.<br />
* Oct 18: Simulation files for HW #3: In python: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/cartpend.py cartpend.py], and in Matlab: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 6: HW #2 files: Any required files will typically be posted here, under announcements; for this week you need [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/boxgrid.m boxgrid.m] for neater looking plots.<br />
* Oct 2: Friday's recitation will contain a brief Matlab tutorial. There will be an hour long Matlab tutorial on Sunday at 1pm in 328 SFL for those who want more.<br />
* Oct 2: Note, some information for those who are interested in using Python for homework assignments can be found here: [[CDS_101/110_-_Python_Tutorial|Python info]]<br />
* Sep 30: <br />
** Please fill out background survey here: [https://docs.google.com/spreadsheet/viewform?formkey=dHlfYUpKblNOWE5LMEpBSEZ2M2pBVnc6MA#gid=0 Survey]<br />
** Lecture notes and recording for today are available by following the link to the [[CDS 101/110a, Fall 2013 - Course Schedule|course schedule]] page and looking for L1-1. <br />
** Everyone should by now have received an invite from Piazza. If you are taking the class and have not received an invite, please email the TAs at [mailto:cds110-tas@cds.caltech.edu cds110-tas@cds.caltech.edu].<br />
** No TA office hours first week<br />
* Sept 11: Schedule for first week: regular lectures MW in 105 Annenberg. First recitation (F 2--3pm in ANB 105) will be a Matlab tutorial.<br />
<br />
<br />
<br />
* 20 Dec 10: Final course grade histogram [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/pdf/final_hist.pdf final_hist.pdf]. (The odd choice of boundaries between grades is chosen to ensure that students who have almost identical numerical grades receive identical letter grades.)<br />
* 3 Dec 10: Extra copies of the final are located next to the HW boxes in Steele. Stability and controller design portion of the final review can be found [https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B6AUPAlXzdCIYjE0YjBkYWQtYzNmZC00Y2RiLTkwNDMtZjM3NDI5NDhkYzc3&hl=en&authkey=CLe6xtsD here] .<br />
* 27 Nov 10: NO OH THIS WEEKEND -- new times are 4-6 PM on Monday and 8-10 PM on Tuesday in STL 214. Also, the Wednesday deadline for the HW is final. No additional extensions unless you have a note from the dean or the health center.<br />
* 24 Nov 10: Due to a technical error, today's lecture wasn't recorded, so you missed my description of how to choose P or PI or PID and gains...<br />
* 24 Nov 10: Office hours for next week will be Monday 4-6pm and Tuesday 8-10pm (no office hours this weekend).<br />
* 20 Nov 10: Note errata in text re eq 6.24; [http://www.cds.caltech.edu/~murray/amwiki/index.php/Errata:_In_equation_%286.24%29,_the_sign_of_the_sin%28omega_t%29_term_is_incorrect_for_zeta_less_than_one]<br />
* 17 Nov 10: HW #7 is available. You may find the following useful for the final problem: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/L8_2_maglev.m maglev.m].<br />
* 17 Nov 10: there will be no recitation on Friday Nov 26th.<br />
* 15 Nov 10: FYI, HW #8 will be due on *Wednesday* Dec 1 rather than Monday Nov 29, however, to allow time for grading, we can't extend this with grace periods (i.e. consider this a free grace period for everyone.)<br />
* 12 Nov 10: For HW #6: I defined gain and phase margin, but not the "stability margin" asked for in problem 1; you can ignore that and just compute gain and phase margins. (The stability margin is the closest point to -1, that is, the minimum value of 1+L(iw)<br />
* 9 Nov 10: Michelle's modification of amnyquist to plot the unit circle: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/mjnyquist.m mjnyquist.m]<br />
* 9 Nov 09: HW #6 is available. Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
* Nov 8: Midterm histogram: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/pdf/Midterm_hist.pdf Midterm hist.pdf]<br />
* Oct 29: If you did not pick up a midterm in class on Wednesday, copies are in Steele next to where you hand homework in.<br />
* Oct 27: THIS WEEK ONLY: Friday recitation will be in 070 Moore (midterm review).<br />
* Oct 27: Note change in office hours and location.<br />
* Oct 20: NOTE: if you have trouble accessing solutions, they are only accessible from a .caltech.edu domain<br />
* Oct 20: For HW #4, problem 6.10 (proof of Cayley-Hamilton), you can assume that A is diagonalizable (the proof is highly non-trivial if you want to consider the general case of non-trivial Jordan form).<br />
* Oct 18: Matlab file for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/bike_linmod.m bike_linmod.m]<br />
* Oct 17: Problem 2b is officially declared as an extra credit problem. You will get 5 pts. extra credit if you use the SIMULINK tutorial ([[Media:MJ-CDS110a-HW2P2b-SIMULINK.pdf|click here for the SIMULINK tutorial]]) or if you can fix the M-files to generate the proper graphs.<br />
* Oct 11: Simulation files for HW #3: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 7: Due to technical difficulties, yesterday's lecture wasn't recorded... there seems to be a curse on that lecture; the most recent one I could find was from 2007, so you get to hear Prof. Murray's version.<br />
* Oct 05: HW #2 files: Any required files will typically be posted here, under announcements; for this week you need [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~murray/courses/cds101/fa08/matlab/boxgrid.m boxgrid.m] for neater looking plots.<br />
* Sep 29: Recitations in 110 Steele (for CDS101), 206 Thomas (110a, "Mech-E" focus) or 214 Steele (110a, "EE/info/bio/..." focus)<br />
* Sep 29: We will conduct regular recitation sections this week on Friday, and an optional Matlab tutorial on Sunday 2-4, SFL 328.<br />
* Sep 29: Next Monday/Wednesday class in 070 Moore<br />
* Sep 26: CDS 101 students may find this week's Wednesday lecture on modeling useful <br />
* Sep 26: Website should be current, let me know if any links are broken! <br />
* Aug 20, website created, currently under construction<br />
<br />
* 11 Dec 12: Note final is due Friday 14th, not Thursday 13th as (previously) stated below.<br />
* 8 Dec 12: Bug fixed in solution to HW #7.<br />
* 12 Nov 12: HW #6 is available. Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
* Oct 22: Matlab and Python files for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/bike_linmod.m bike_linmod.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/bike_linmod.py bike_linmod.py]. For further details on bicycle dynamics, see [http://www.cds.caltech.edu/~murray/amwiki/index.php/Bicycle_dynamics Bicycle dynamics], and the IEEE article [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1499389&isnumber=32202 K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4):26-47, August 2005].<br />
* Oct 15: Simulation files for HW #3: In python: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/cartpend.py cartpend.py], and in Matlab: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 14: HW #2 files: For this week's homework, you can also use [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/boxgrid.m boxgrid.m] for neater looking plots. <br />
* Oct 14: Note *this week only*, the new Monday office hour will be in SFL 231, group study room 2-4. Stay tuned for next week location.<br />
* Oct 10: Note recitation rooms: 110 Steele for more conceptual focus (including CDS 101), 206 Thomas for Matlab-focused, and 213 Annenberg for Python-focused.<br />
--></div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_101/110a,_Fall_2014CDS 101/110a, Fall 20142014-12-07T22:27:41Z<p>Macmardg: </p>
<hr />
<div>{{cds101-fa14}}<br />
This is the homepage for CDS 101 (Analysis and Design of Feedback Systems) and CDS 110a (Introduction to Control Theory) for Fall 2014. __NOTOC__<br />
----<br />
<br />
<table width=100%><br />
<tr valign=top><br />
<td><br />
'''Instructor'''<br />
* Doug MacMartin, macmardg@cds.caltech.edu<br />
* Lectures: MW, 2-3 pm, 105 Annenberg; lecture/recitation F 2-3 pm (also 105 ANB)<br />
** Friday Lecture or Recitation is optional for CDS 101<br />
* Office hours: Mondays, 1-2 pm <br />
* Prior years: [http://www.cds.caltech.edu/~murray/wiki/CDS_101/110a%2C_Fall_2006 FA06], [http://www.cds.caltech.edu/~murray/wiki/CDS_101/110a%2C_Fall_2007 FA07],[http://www.cds.caltech.edu/~murray/wiki/CDS_101/110a%2C_Fall_2008 FA08], [[CDS 101/110a, Fall 2009 | FA09]], [[CDS 101/110a, Fall 2010 | FA10]], [[CDS 101/110a, Fall 2011 | FA11]], [[CDS 101/110a, Fall 2012 | FA12]], [[CDS 101/110a, Fall 2013 | FA13]]<br />
</td><td><br />
'''Teaching Assistants''' [mailto:cds110-tas@cds.caltech.edu cds110-tas@cds.caltech.edu]<br />
* Ivan Papusha<br />
* David Flicker<br />
* Jerry Cruz<br />
* Ioannis Filippidis<br />
* Office hours:<br />
** Mon 3-5 pm in 107 ANB (243 ANB on 10/20 only)<br />
** Tue 7-9 pm in 107 ANB<br />
''' Course Ombuds:''' [mailto:efeldman@caltech.edu Ellen Feldman] <br />
<br />
</td></tr><br />
</table><br />
<br />
== Announcements ==<br />
* 7 Dec: Plots of lead/lag frequency response with a parameterization different from what I used in class: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa14/pdf/LeadLag-freqresp.pdf LeadLag.pdf]. This may or may not be helpful.<br />
* 5 Dec: Extra copies of the final are available outside 107 Steele Lab. Please return to Nikki Fauntleroy by '''Thu Dec 11 5pm''' (Fri Dec 12 8am at the latest). In addition please fill out TQFR surveys on Caltech Access.<br />
* 3 Dec: The TAs will hold additional office hours on Thu 4 Dec, 7--9pm (107 ANB).<br />
* 21 Nov: NOTE: for today only, class is in 102 Steele<br />
* 18 Nov: HW #7 is available. <br />
** Problem 3: compare your design to C(s)=41(1+s/4)/(1+s/100) if you do not have access to another classmate's controller<br />
** Problem 4/(CDS101 Problem 2): see [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa13/pdf/MacMartin-Kravitz-Keith-Jarvis-2013.pdf MacMartin-etal-2013.pdf]<br />
* 18 Nov: Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
<br />
<table style="float: right" border=0><br />
<tr><br />
<td><br />
[[#Old Announcements|Archive]]<br />
</td></tr><br />
</table><br />
<br />
<!--<br />
== Homework/Exam Statistics ==<br />
<table align=right border=0><br />
<tr><br />
<td><br />
</td></tr><br />
</table><br />
Note that the solutions can also be obtained from the [[CDS 101/110a, Fall 2014 - Course Schedule|course schedule]] page after the homework for that week has been handed in. Solutions are only available from within the caltech.edu domain.<br />
* Oct. 11: HW1 -- CDS101 [ Avg. Grade = xx/20 | Avg. Time Spent = x hrs. ], CDS110a [ Avg. Grade = xx/60 | Avg. Time Spent = x hrs. ]<br />
--><br />
= Course Syllabus =<br />
<table style="float: right" border=1 width=20% cellpadding=6><br />
<tr><br />
<td><br />
<center>'''Contents'''</center><br />
<ul><br />
<li> [[#Grading|Grading]] </li><br />
<li> [[#Lectures and Recitations|Lectures/Recitations]] </li><br />
<li> [[#Collaboration Policy|Collaboration Policy]] </li><br />
<li> [[#Software|Software]] </li><br />
<li> [[#Course Text and References|Course Text]] </li><br />
<li> [[#Course_Schedule|Course Schedule]]</li><br />
</ul><br />
</td></tr><br />
</table><br />
CDS 101/110 provides an introduction to feedback and control in physical, biological, engineering, and information sciences. Basic principles of<br />
feedback and its use as a tool for altering the dynamics of systems and managing uncertainty. Key themes throughout the course will include<br />
input/output response, modeling and model reduction, linear versus nonlinear models, and local versus global behavior. The course has several variants:<br />
<br />
* CDS 101 is a 6 unit (2-0-4) class intended for advanced students in science and engineering who are interested in the principles and tools of feedback control, but not the analytical techniques for design and synthesis of control systems. <br />
<br />
* CDS 110 is a 12 unit class (3-0-9) that provides a traditional first course in control for engineers and applied scientists. It assumes a stronger mathematical background, including working knowledge of linear algebra and ODEs. Familiarity with complex variables (Laplace transforms, residue theory) is helpful but not required. <br />
<br />
=== Lectures and Recitations ===<br />
The main course lectures are on MW from 2-3 pm in 105 Annenberg. Friday 2-3 will either be an additional lecture or a problem solving (recitation) session run by the course teaching assistants. CDS 101 students are not required to attend the Friday lecture or recitation, although they are welcome to do so (CDS 101 homework will typically require much less use of Matlab or Python). <br />
<br />
=== Grading ===<br />
The final grade will be based on homework sets, a midterm exam, and a final exam: <br />
<br />
*''Homework (50%):'' Homework sets will be handed out weekly and due on Wednesdays by 5 pm to the locked box outside of 102 Steele. Students are allowed three grace periods of two days each that can be used at any time (but no more than 1 grace period per homework set). Late homework beyond the grace period will not be accepted without a note from the health center or the Dean; please plan accordingly. MATLAB or Python code are considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).<br />
<br />
* ''Midterm exam (20%):'' A midterm exam will be handed out at the beginning of midterms period (29 Oct) and due at the end of the midterm examination period (4 Nov). The midterm exam will be open book and computers will be allowed (though not required). <br />
<br />
* ''Final exam (30%):'' The final exam will be handed out on the last day of class (5 Dec) and due on Friday of finals week (Dec 12). It will be an open book exam and computers will be allowed (though not required).<br />
<br />
=== Collaboration Policy ===<br />
<br />
Collaboration on homework assignments is encouraged. You may consult<br />
outside reference materials, other students, the TA, or the<br />
instructor, but you cannot consult homework solutions from<br />
prior years and you must cite any use of material from outside<br />
references. All solutions that are handed in should be written up<br />
individually and should reflect your own understanding of the subject<br />
matter at the time of writing. Python or MATLAB scripts and plots are<br />
considered part of your writeup and should be done individually (you<br />
can share ideas, but not code).<br />
<br />
''No collaboration is allowed on the midterm or final exams.''<br />
<br />
=== Software ===<br />
Computer exercises will be assigned as part of the regular homeworks. The<br />
exercises are designed to be done either in MATLAB, using the Control Toolbox, or in Python.<br />
SIMULINK may be useful but is not required. <br />
Caltech has a site license for Matlab and it may be obtained<br />
from [http://software.caltech.edu IMSS] (Caltech students only). An online tutorial is available at<br />
<center><br />
http://www.engin.umich.edu/group/ctm/basic/basic.html<br />
</center><br />
<br />
=== Course Text and References ===<br />
<br />
The primary course text is [http://www.cds.caltech.edu/~murray/amwiki/Main_Page ''Feedback Systems: An Introduction for Scientists and Engineers''] by {{Astrom}} and Murray (2008). This book is available in the Caltech bookstore and via download from the [http://www.cds.caltech.edu/~murray/amwiki/Main_Page companion web site]. The following additional references may also be useful:<br />
<br />
* A. D. Lewis, ''A Mathematical Approach to Classical Control'', 2003. [http://www.mast.queensu.ca/~andrew/teaching/math332/notes.shtml Online access].<br />
<br />
In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.<br />
<br />
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', McGraw-Hill, 1986.<br />
* G. F. Franklin, J. D. Powell, and A. Emami-Naeni, ''Feedback Control of Dynamic Systems'', Addison-Wesley, 2002.<br />
<br />
=== Course Schedule ===<br />
A detailed course schedule is available on the [[CDS 101/110a, Fall 2014 - Course Schedule|course schedule]] page (also shown on the "menu bar" at the top of each course page).<br />
<br />
== Old Announcements ==<br />
* Old announcements will appear here when they are archived.<br />
* Oct 30: Extra midterms are available on top of the homework dropbox in Steele.<br />
* Oct 23: Matlab and Python files for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/bike_linmod.m bike_linmod.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/bike_linmod.py bike_linmod.py]. For further details on bicycle dynamics, see [http://www.cds.caltech.edu/~murray/amwiki/index.php/Bicycle_dynamics Bicycle dynamics], and the IEEE article [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1499389&isnumber=32202 K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4):26-47, August 2005].<br />
* Oct 17: Simulation files for HW #3: In python: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/cartpend.py cartpend.py], and in Matlab: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 15: Monday's (10/20) office hours will be in 243 ANB. Office hours will be in the usual room after Monday.<br />
* Oct 9: Tuesday night office hours will be from 7-9 pm from now on (in the same room).<br />
* Oct 6: HW #2 files for plotting phase portrait: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/boxgrid.m boxgrid.m] for neater looking plots.<br />
* Sep 30: Everyone in the class should fill out this [https://docs.google.com/spreadsheet/viewform?usp=drive_web&formkey=dE8zMUtqYlFhMjdJa0JKNEhZYmJwQXc6MA#gid=0 survey]. It will help us better gauge the level of background knowledge of the class.<br />
* Sep 29: Everyone should by now have received an invite from Piazza. If you are taking the class and have not received an invite, please email the TAs at [mailto:cds110-tas@cds.caltech.edu cds110-tas@cds.caltech.edu] or sign up [http://piazza.com/caltech/fall2014/cds110 here].<br />
* Sep 27: You will need to know MATLAB for this class. If you are unfamiliar with MATLAB here is a good starting tutorial: [http://www.cyclismo.org/tutorial/matlab/ Basics Tutorial]. Here is a tutorial created by a TA from a previous year: [https://www.dropbox.com/s/uxk94uwbc2rcw60/cds110-matlab-tutorial-pack.zip?dl=0 Tutorial-Pack.zip]. It has some code that you can test out and an example homework problem done in MATLAB. <br />
* Sep 15: If you have not been signed up for Piazza by the first week of class, sign up [http://piazza.com/caltech/fall2014/cds110 here].<br />
* Sep 10: Updated links and created course schedule.<br />
* Aug 9: Website created, currently under construction<br />
<br />
<!-- leaving these as templates <br />
* 20 Nov: HW #7 is available. For the final problem see [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa13/pdf/MacMartin-Kravitz-Keith-Jarvis-2013.pdf MacMartin-etal-2013.pdf],<br />
* 12 Nov: HW #6 is available. Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
* Oct 28: Midterm will be handed out on Wednesday Oct 30. The midterm will not be available online; if you are not in lecture please pick up a midterm from Anissa Scott in 107 Steele during normal work hours. The midterm will be due the following Tuesday Nov 5 at 5pm. We will permit you to use one of your homework grace periods and hand it in ''no later than 8am Nov 6''. If you choose to do so, (a) please send us an email to confirm and (b) hand it in in the usual homework inbox in Steele.<br />
** There will be no office hours on the week of the midterm (Mon-Tue Nov 4-5)<br />
* Oct 27: Matlab and Python files for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/bike_linmod.m bike_linmod.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/bike_linmod.py bike_linmod.py]. For further details on bicycle dynamics, see [http://www.cds.caltech.edu/~murray/amwiki/index.php/Bicycle_dynamics Bicycle dynamics], and the IEEE article [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1499389&isnumber=32202 K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4):26-47, August 2005].<br />
* Oct 22: HW #4 posted. Note that if you downloaded this last night, then the link pointed to the HW assignment from 2012.<br />
* Oct 18: Simulation files for HW #3: In python: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/cartpend.py cartpend.py], and in Matlab: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 6: HW #2 files: Any required files will typically be posted here, under announcements; for this week you need [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/boxgrid.m boxgrid.m] for neater looking plots.<br />
* Oct 2: Friday's recitation will contain a brief Matlab tutorial. There will be an hour long Matlab tutorial on Sunday at 1pm in 328 SFL for those who want more.<br />
* Oct 2: Note, some information for those who are interested in using Python for homework assignments can be found here: [[CDS_101/110_-_Python_Tutorial|Python info]]<br />
* Sep 30: <br />
** Please fill out background survey here: [https://docs.google.com/spreadsheet/viewform?formkey=dHlfYUpKblNOWE5LMEpBSEZ2M2pBVnc6MA#gid=0 Survey]<br />
** Lecture notes and recording for today are available by following the link to the [[CDS 101/110a, Fall 2013 - Course Schedule|course schedule]] page and looking for L1-1. <br />
** Everyone should by now have received an invite from Piazza. If you are taking the class and have not received an invite, please email the TAs at [mailto:cds110-tas@cds.caltech.edu cds110-tas@cds.caltech.edu].<br />
** No TA office hours first week<br />
* Sept 11: Schedule for first week: regular lectures MW in 105 Annenberg. First recitation (F 2--3pm in ANB 105) will be a Matlab tutorial.<br />
<br />
<br />
<br />
* 20 Dec 10: Final course grade histogram [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/pdf/final_hist.pdf final_hist.pdf]. (The odd choice of boundaries between grades is chosen to ensure that students who have almost identical numerical grades receive identical letter grades.)<br />
* 3 Dec 10: Extra copies of the final are located next to the HW boxes in Steele. Stability and controller design portion of the final review can be found [https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B6AUPAlXzdCIYjE0YjBkYWQtYzNmZC00Y2RiLTkwNDMtZjM3NDI5NDhkYzc3&hl=en&authkey=CLe6xtsD here] .<br />
* 27 Nov 10: NO OH THIS WEEKEND -- new times are 4-6 PM on Monday and 8-10 PM on Tuesday in STL 214. Also, the Wednesday deadline for the HW is final. No additional extensions unless you have a note from the dean or the health center.<br />
* 24 Nov 10: Due to a technical error, today's lecture wasn't recorded, so you missed my description of how to choose P or PI or PID and gains...<br />
* 24 Nov 10: Office hours for next week will be Monday 4-6pm and Tuesday 8-10pm (no office hours this weekend).<br />
* 20 Nov 10: Note errata in text re eq 6.24; [http://www.cds.caltech.edu/~murray/amwiki/index.php/Errata:_In_equation_%286.24%29,_the_sign_of_the_sin%28omega_t%29_term_is_incorrect_for_zeta_less_than_one]<br />
* 17 Nov 10: HW #7 is available. You may find the following useful for the final problem: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/L8_2_maglev.m maglev.m].<br />
* 17 Nov 10: there will be no recitation on Friday Nov 26th.<br />
* 15 Nov 10: FYI, HW #8 will be due on *Wednesday* Dec 1 rather than Monday Nov 29, however, to allow time for grading, we can't extend this with grace periods (i.e. consider this a free grace period for everyone.)<br />
* 12 Nov 10: For HW #6: I defined gain and phase margin, but not the "stability margin" asked for in problem 1; you can ignore that and just compute gain and phase margins. (The stability margin is the closest point to -1, that is, the minimum value of 1+L(iw)<br />
* 9 Nov 10: Michelle's modification of amnyquist to plot the unit circle: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/mjnyquist.m mjnyquist.m]<br />
* 9 Nov 09: HW #6 is available. Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
* Nov 8: Midterm histogram: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/pdf/Midterm_hist.pdf Midterm hist.pdf]<br />
* Oct 29: If you did not pick up a midterm in class on Wednesday, copies are in Steele next to where you hand homework in.<br />
* Oct 27: THIS WEEK ONLY: Friday recitation will be in 070 Moore (midterm review).<br />
* Oct 27: Note change in office hours and location.<br />
* Oct 20: NOTE: if you have trouble accessing solutions, they are only accessible from a .caltech.edu domain<br />
* Oct 20: For HW #4, problem 6.10 (proof of Cayley-Hamilton), you can assume that A is diagonalizable (the proof is highly non-trivial if you want to consider the general case of non-trivial Jordan form).<br />
* Oct 18: Matlab file for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/bike_linmod.m bike_linmod.m]<br />
* Oct 17: Problem 2b is officially declared as an extra credit problem. You will get 5 pts. extra credit if you use the SIMULINK tutorial ([[Media:MJ-CDS110a-HW2P2b-SIMULINK.pdf|click here for the SIMULINK tutorial]]) or if you can fix the M-files to generate the proper graphs.<br />
* Oct 11: Simulation files for HW #3: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 7: Due to technical difficulties, yesterday's lecture wasn't recorded... there seems to be a curse on that lecture; the most recent one I could find was from 2007, so you get to hear Prof. Murray's version.<br />
* Oct 05: HW #2 files: Any required files will typically be posted here, under announcements; for this week you need [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~murray/courses/cds101/fa08/matlab/boxgrid.m boxgrid.m] for neater looking plots.<br />
* Sep 29: Recitations in 110 Steele (for CDS101), 206 Thomas (110a, "Mech-E" focus) or 214 Steele (110a, "EE/info/bio/..." focus)<br />
* Sep 29: We will conduct regular recitation sections this week on Friday, and an optional Matlab tutorial on Sunday 2-4, SFL 328.<br />
* Sep 29: Next Monday/Wednesday class in 070 Moore<br />
* Sep 26: CDS 101 students may find this week's Wednesday lecture on modeling useful <br />
* Sep 26: Website should be current, let me know if any links are broken! <br />
* Aug 20, website created, currently under construction<br />
<br />
* 11 Dec 12: Note final is due Friday 14th, not Thursday 13th as (previously) stated below.<br />
* 8 Dec 12: Bug fixed in solution to HW #7.<br />
* 12 Nov 12: HW #6 is available. Some useful files: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/ambode.m ambode.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/amnyquist.m amnyquist.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/arrow.m arrow.m]<br />
* Oct 22: Matlab and Python files for HW #4: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/bike_linmod.m bike_linmod.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/bike_linmod.py bike_linmod.py]. For further details on bicycle dynamics, see [http://www.cds.caltech.edu/~murray/amwiki/index.php/Bicycle_dynamics Bicycle dynamics], and the IEEE article [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1499389&isnumber=32202 K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4):26-47, August 2005].<br />
* Oct 15: Simulation files for HW #3: In python: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/cartpend.py cartpend.py], and in Matlab: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend.m cartpend.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/cartpend_model.m cartpend_model.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/balance_simple.mdl balance_simple.mdl]<br />
* Oct 14: HW #2 files: For this week's homework, you can also use [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/phaseplot.m phaseplot.m], try it with [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa11/matlab/boxgrid.m boxgrid.m] for neater looking plots. <br />
* Oct 14: Note *this week only*, the new Monday office hour will be in SFL 231, group study room 2-4. Stay tuned for next week location.<br />
* Oct 10: Note recitation rooms: 110 Steele for more conceptual focus (including CDS 101), 206 Thomas for Matlab-focused, and 213 Annenberg for Python-focused.<br />
--></div>Macmardghttp://www.cds.caltech.edu/~macmardg/wiki/index.php?title=CDS_101/110a,_Fall_2014_-_Course_ScheduleCDS 101/110a, Fall 2014 - Course Schedule2014-12-04T15:49:24Z<p>Macmardg: </p>
<hr />
<div>{{cds101-fa14}} __NOTOC__<br />
This page contains the course schedule for CDS 101/110a. Lecture notes, slides, and audio recordings will be posted usually on the same day as class. <br />
<br />
{| border=1 width=100%<br />
|-<br />
| Week || Date || Topic || Reading || Homework<br />
|-<br />
| align=center rowspan=4 | 1 <br />
| colspan=4 |<br />
====={{cds110 topic|1|Introduction and Review}} =====<br />
|-<br />
| 29 Sep (M)<br />
| Introduction to Feedback and Control ({{cds101-fa14 lectures|L1-1.pdf|L1-1}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 1.1-1.2, 1.4-1.5]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|1}}<br />
|-<br />
| 1 Oct (W)<br />
| System Modeling. ({{cds101-fa14 lectures|L1-2.pdf|L1-2}})<br />
* Video: [[Media:Demo_springmass.mp4| coupled mass demo]]<br />
* MATLAB code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/L1_3_modeling.m L1_3_modeling.m], [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/springmass.m springmass.m]<br />
* Python code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/python/L1-3_modeling.py L1-3_modeling.py]<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 2.1-2.4 (review)]<br />
|-<br />
| 3 Oct (F)<br />
| Introduction to Feedback ({{cds101-fa14 lectures|L1-3.pdf|L1-3}}) <br />
* MATLAB code: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa14/matlab/L3_1b.m L3_modeling.m], [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa14/matlab/clodefun.m clodefun.m]<br />
| <br />
|-<br />
| align=center rowspan=4 | 2<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|2|Dynamic Behavior}} =====<br />
|-<br />
| 6 Oct (M)<br />
| Qualitative Analysis and Stability {{cds101-fa14 lectures|L2-1.pdf|L2-1}}; <br />
* Python code for phase portrait: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/python/L2-1.py L2-1.py]<br />
* Predator-prey matlab code: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/predprey.m predprey.m]<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 4.1-4.3]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|2}}<br />
|-<br />
| 8 Oct (W)<br />
| Stability Analysis ({{cds101-fa14 lectures|L2-2.pdf|L2-2}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 4.4]<br />
|-<br />
| 10 Oct (F)<br />
| Lecture on Lyapunov stability ([http://www.cds.caltech.edu/~ifilippi/temp/lyapunov_notes.pdf lecture notes (part 1)], [http://www.cds.caltech.edu/~ifilippi/temp/L2-3.pdf L2-3 (part 2)])<br />
| <br />
|-<br />
| align=center rowspan=4 | 3<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|3|Linear Systems}} =====<br />
|-<br />
| 13 Oct (M)<br />
| Linear Time-Invariant Systems ({{cds101-fa14 lectures|L3-1.pdf|L3-1}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 5.1-5.2]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|3}}<br />
|-<br />
| 15 Oct (W)<br />
| Linear Systems Analysis ({{cds101-fa14 lectures|L3-2.pdf|L3-2}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 5.4]<br />
|-<br />
| 17 Oct (F)<br />
| Recitation <br />
| <br />
|-<br />
| align=center rowspan=4 | 4<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|4|State and Output Feedback}} =====<br />
|-<br />
| 20 Oct (M)<br />
| State Feedback ({{cds101-fa14 lectures|L4-1.pdf|L4-1}}; (sorry, recording incomplete))<br />
* MATLAB code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/L4_1_statefbk.m L4_1_statefbk.m], [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/predprey.m predprey.m], [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/predprey_rh.m predprey_rh.m]<br />
* Python code: [http://www.cds.caltech.edu/~slivings/TA/CDS110ab/L4_1_statefbk.py L4_1_statefbk.py], [http://www.cds.caltech.edu/~slivings/TA/CDS110ab/predprey.py predprey.py]<br />
<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 6.1-6.3]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|4}}<br />
|-<br />
| 22 Oct (W)<br />
| Observability and State Estimation ({{cds101-fa14 lectures|L4-2.pdf|L4-2}})<br />
* MATLAB code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/L4_2_predprey.m L4_2_predprey.m], [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/predprey.m predprey.m], [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/predprey_rh.m predprey_rh.m]<br />
* Python code: [http://www.cds.caltech.edu/~slivings/TA/CDS110ab/L4_2_predprey.py L4_2_predprey.py], [http://www.cds.caltech.edu/~slivings/TA/CDS110ab/predprey.py predprey.py]<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 7.1--7.3]<br />
|-<br />
| 24 Oct (F)<br />
| No class; institute academic holiday for new president ceremony<br />
| <br />
|-<br />
| align=center rowspan=4 | 5<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|5|Reference Tracking and intro to Frequency Domain}} =====<br />
|-<br />
| 27 Oct (M)<br />
| State Estimation (cont'd) and Reference Tracking ({{cds101-fa14 lectures|L5-1.pdf|L5-1}})<br />
* MATLAB code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/dfan_est.m dfan_est.m], [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/dfan_lqr.m dfan_lqr.m], <br />
<!-- * Python code: [http://www.cds.caltech.edu/~slivings/TA/CDS110ab/L4_1_statefbk.py L4_1_statefbk.py], [http://www.cds.caltech.edu/~slivings/TA/CDS110ab/predprey.py predprey.py] --><br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 6.4-6.5]<br />
| rowspan=3 align=center | {{cds101 exam|Midterm guidelines|midterm-guideline}}<br />
|-<br />
| 29 Oct (W)<br />
| State Space Design and Laplace Transforms ({{cds101-fa14 lectures|L5-2.pdf|L5-2}})<br />
* [http://lpsa.swarthmore.edu/LaplaceZTable/Common%20Laplace%20Transform%20Pairs.pdf Laplace transform table]<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 7.5, 8.5]<br />
|-<br />
| 31 Oct (F)<br />
| Midterm review ''In Annenberg''({{cds101-fa14 lectures|cds110_midterm_review.pdf|L5-3}}), [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa12/pdf/MidtermReviewCDS101.pdf cds101_review]<br />
| <br />
|-<br />
| align=center rowspan=4 | 6<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|6|Transfer Functions}} =====<br />
|-<br />
| 3 Nov (M)<br />
| Transfer Functions ({{cds101-fa14 lectures|L6-1.pdf|L6-1}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 8.1-8.2, 8.4]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|5}}<br />
|-<br />
| 5 Nov (W)<br />
| Frequency Response and Bode Plots ({{cds101-fa14 lectures|L6-2.pdf|L6-2}})<br />
* Python code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/L6-2_cruise.py L6-2_cruise.py]<br />
<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 8.3-8.4]<br />
|-<br />
| 7 Nov (F)<br />
| More on transfer functions and block diagrams ({{cds101-fa14 lectures|L6-3.pdf|L6-3}})<br />
| <br />
|-<br />
| align=center rowspan=4 | 7<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|7|Loop Analysis}} =====<br />
|-<br />
| 10 Nov (M)<br />
| Stability of Feedback Systems ({{cds101-fa14 lectures|L7-1.pdf|L7-1}})[http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/L7_1_loopanal.m L7_1_loopanal.m],<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 9.1-9.2]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|6}}<br />
|-<br />
| 12 Nov (W)<br />
| Nyquist Criterion ({{cds101-fa14 lectures|L7-2.pdf|L7-2}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 9.3-9.4]<br />
|-<br />
| 14 Nov (F)<br />
| Recitation<br />
| <br />
|-<br />
| align=center rowspan=4 | 8<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|8|Loop Shaping}} =====<br />
|-<br />
| 17 Nov (M)<br />
| Loop shaping and lead/lag Controller ({{cds101-fa14 lectures|L8-1.pdf|L8-1}})<br />
* [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa10/matlab/L8_1_dfan.m L8_1_dfan.m]<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 11.1-11.3]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|7}}<br />
|-<br />
| 19 Nov (W)<br />
| Loop shaping, continued ({{cds101-fa14 lectures|L8-2.pdf|L8-2}})<br />
* MATLAB code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/pvtol_nested.m pvtol_nested.m]<br />
* Python code: [http://www.cds.caltech.edu/~murray/courses/cds101/fa12/matlab/pvtol-nested.py pvtol-nested.py]<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 11.6] (Examples 2.9, 11.6, 11.12)<br />
|-<br />
| 21 Nov (F)<br />
| Examples ({{cds101-fa14 lectures|L8-3.pdf|L8-3}})<br />
| <br />
|-<br />
| align=center rowspan=4 | 10<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|9|Loop Shaping and Control Design, cont'd}} =====<br />
|-<br />
| 24 Nov (M)<br />
| Control Design using PID ({{cds101-fa14 lectures|L9-1.pdf|L9-1}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 10.1-10.3]<br />
| rowspan=3 align=center | {{cds101-fa14 homework|8}}<br />
|-<br />
| 26 Nov (W)<br />
| Design Example ({{cds101-fa14 lectures|L9-2.pdf|L9-2}})<br />
| [http://www.cds.caltech.edu/~murray/amwiki/Main_Page AM 10.3, 10.5] <br />
|-<br />
| 28 Nov (F)<br />
| No class (Thanksgiving)<br />
| <br />
|-<br />
| align=center rowspan=4 | 9<br />
| colspan=4 |<br />
<br />
===== {{cds110 topic|10|Control Design}} =====<br />
|-<br />
| 1 Dec (M)<br />
| Limits of Performance ({{cds101-fa14 lectures|L10-1.pdf|L10-1}}); [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa09/matlab/L8_2_maglev.m L8_2_maglev.m]<br />
| <br />
| rowspan=3 align=center | Please pick up hardcopy of final outside 102 Steele<br />
|-<br />
| 3 Dec (W)<br />
| Control system implementation ({{cds101-fa14 lectures|L10-2.pdf|L10-2}});<br />
* Additional implementation notes: [http://www.cds.caltech.edu/~macmardg/courses/cds101/fa13/pdf/L10-2.pdf notes.pdf]<br />
|<br />
|- <br />
| 5 Dec (F)<br />
| Recitation, [[CDS_101/110a,_Fall_2012_-_Recitation_Schedule#7_Dec|Final review (previous years)]]<br />
|<br />
|}</div>Macmardg