View source for CDS 140, Winter 2016

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<font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__
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'''Instructors'''
* Douglas MacMartin, macmardg@cds.caltech.edu
* John Doyle, doyle@cds.caltech.edu
* Lectures: MWF, 1-1:55, 213 ANB
* Office hours: 
** DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)
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'''Teaching Assistants'''
* Benson Christalin (CDS)
* Contact: cds140-tas@cds.caltech.edu
* Office hours: Monday 2-3 pm
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=== Announcements ===
* Homework will be due Wednesdays in class (or by 5pm to Benson).  Homework assignments are posted below.
* [https://piazza.com/caltech/winter2016/cds140/home Piazza] If you have not received an email to sign up for Piazza, please email us!
* Dates for recitation are shown in italics in schedule.  Dates where JCD is lecturing given with asterisk
* 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.

=== Course Description ===
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.

=== Lecture Schedule ===

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| '''Date'''
| '''Topic'''
| '''Reading'''
| '''Homework'''
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| 4 Jan <br>  6 Jan <br> ''8 Jan''
| 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]
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LinearSystemsNotes.pdf LinSys notes]
* Course overview and administration
* Linear differential equations
* Matrix exponential, diagonalization
* Stable and unstable spaces
* S + N decomposition, Jordan form
| 
Perko, 1.1-1.10<br>
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw1-wi16.pdf hw1-wi16.pdf] <br> Due: 13 Jan (Wed)
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| 11 Jan <br> 13 Jan <br> ''15 Jan''
| Nonlinear differential equations
[http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_W2_2016.pdf Lecture notes]
* Existence and uniqueness
* Flow of a differential equation
* Linearization
| Perko, 2.1-2.6
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw2-wi16.pdf hw2-wi16.pdf] <br> Due: 20 Jan (Wed)
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| 20 Jan* '''2 hrs''' <br> ''22 Jan''
| Chaos, fractals, and global analysis using SOStools
* Chaotic systems (logistic map, Mandelbrot set)
* SOStools for finding Lyapunov functions
|  
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| 25 Jan <br> '''(2 hrs)''' <br> ''Jan 29''
| Behavior of differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L4.pdf L4]
* Stable and unstable manifolds  
* Stability of equilibrium points for planar systems
| Perko, 2.7-2.10 
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3-wi16.pdf hw3-wi16.pdf] <br> Due: 3 Feb (Wed)
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| 1 Feb* <br> 3 Feb* <br> 5 Feb*
| Non-hyperbolic differential equations
* Lyapunov functions
* Center manifold theorem
| Perko, 2.11-2.13
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4-wi16.pdf hw4-wi16.pdf] <br> Due: 10 Feb (Wed)
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| 8 Feb <br> 10 Feb <br> 12 Feb
| Global behavior
* Limit sets and attractors
* Krasovskii-Lasalle invariance principle (if time)
* Periodic orbits and limit cycles
| Perko, 3.1-3.3
| [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5-wi16.pdf hw5-wi16.pdf] <br> Due: 17 Feb (Wed)
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| 17 Feb <br> 19 Feb
| Limit cycles
* Poincare' map
* Bendixson criterion for limit cycles in the plane
* Describing functions (if time)
| Perko, 3.4-3.5, 3.7
| [[CDS 140a Winter 2016 Homework 6|HW 6]] <br> Due: 17 Feb (Wed)
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| 22 Feb '''2 hrs''' <br> ''26 Feb''
| Bifurcations
* Sensitivity analysis
* Structural stability
* Bifurcation of equilibrium points
| Perko 4.1-4.2 <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]])
| [[CDS 140a Winter 2016 Homework 7|HW 7]] <br> Due: 24 Feb (Wed)
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| 29 Feb* <br> 2 Mar* <br> 4 Mar*
| Bifurcations
* Hopf bifurcation
* Application example
| Perko 4.3-4.5 + notes
| [[CDS 140a Winter 2016 Homework 8|HW 8]] <br> Due: 2 Mar (Wed)
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| 7*, 9* Mar <br> 
| Course review
| <!-- Reading -->
| Final exam  <br> Due: 18 Mar (Wed).  Pick up from Nikki Fountleroy, 107 Steele Lab
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=== Textbook ===

The primary text for the course (available via the online bookstore) is
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| align=right | &nbsp;[Perko]&nbsp;
| L. Perko, Differential Equations and Dynamical Systems, Third Edition.  Springer, 2006.
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The following additional texts may be useful for some students:
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|- valign=top
| align=right| &nbsp;[G&H]&nbsp;
| J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields.  Springer-Verlag, 1990.
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| align=right| &nbsp;[H&S]&nbsp;
| M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.
|- valign=top
| align=right | &nbsp;[J&S]&nbsp;
| 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)
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| align=right | &nbsp;[Ver]&nbsp;
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition.  Springer, 2006. (On reserve in SFL)
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=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* 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.''
* 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). 

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.

=== Collaboration Policy ===
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.

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).

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.

[[Category:Courses]]