CDS 140, Winter 2016
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=== Announcements ===  === Announcements ===  
+  * 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.)  
* Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.  * 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!  * [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  * 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.  * 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.  
+  * Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11Week4Notes.pdf SMT]  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11Week5Notes.pdf Lyapunov]  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11Week5NotesCMT.pdf CMT]  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11InvManRemark.pdf Invariant]  
=== Course Description ===  === Course Description ===  
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 20 Jan* '''2 hrs''' <br> ''22 Jan''   20 Jan* '''2 hrs''' <br> ''22 Jan''  
−   Behavior of differential equations  +   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  * Stable and unstable manifolds  
* Stability of equilibrium points for planar systems  * Stability of equilibrium points for planar systems  
 Perko, 2.72.10   Perko, 2.72.10  
−   [  +   [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw3wi16.pdf hw3wi16.pdf] <br> Due: 3 Feb (Wed) 
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−    +   1 Feb* <br> 3 Feb* <br> 5 Feb* 
−   Nonhyperbolic differential equations  +   Nonhyperbolic differential equations [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/L5_2013.pdf L5 2013 notes] 
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11Week4Notes.pdf SMT]  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11Week5Notes.pdf Lyapunov]  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11Week5NotesCMT.pdf CMT]  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/Cds140awi11InvManRemark.pdf Invariant]  
* Lyapunov functions  * Lyapunov functions  
* Center manifold theorem  * Center manifold theorem  
 Perko, 2.112.13   Perko, 2.112.13  
−   [  +   [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw4wi16.pdf hw4wi16.pdf] <br> Due: 10 Feb (Wed) 
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−    +   8 Feb <br> 10 Feb <br> 12 Feb 
 Global behavior   Global behavior  
* Limit sets and attractors  * Limit sets and attractors  
−  * KrasovskiiLasalle invariance principle  +  * KrasovskiiLasalle invariance principle 
* Periodic orbits and limit cycles  * Periodic orbits and limit cycles  
 Perko, 3.13.3   Perko, 3.13.3  
−   [  +   [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw5wi16.pdf hw5wi16.pdf] <br> Due: 17 Feb (Wed) 
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−    +   17 Feb <br> 19 Feb 
 Limit cycles   Limit cycles  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Feb_2016.pdf Lecture notes]  
* Poincare' map  * Poincare' map  
* Bendixson criterion for limit cycles in the plane  * Bendixson criterion for limit cycles in the plane  
−  +   Perko, 3.43.5, 3.7, 3.9  
−   Perko, 3.43.5, 3.7  +   [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw6wi16.pdf hw6wi16.pdf] <br> Due: 24 Feb (Wed) 
−   [  +  
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−    +   22 Feb <br> '''2 hrs''' <br> ''26 Feb'' 
 Bifurcations   Bifurcations  
+  [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/LectureNotes_Mar_2016.pdf Lecture notes]  
* Sensitivity analysis  * Sensitivity analysis  
* Structural stability  * Structural stability  
* Bifurcation of equilibrium points  * Bifurcation of equilibrium points  
 Perko 4.14.2 <br>   Perko 4.14.2 <br>  
−  +   [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw7wi16.pdf hw7wi16.pdf] <br> Due: 31 Feb (Wed)  
−  +  
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−    +   29 Feb* <br> 2 Mar* <br> 4 Mar* 
 Bifurcations   Bifurcations  
* Hopf bifurcation  * Hopf bifurcation  
−  * Application example  +  * Application example 
 Perko 4.34.5 + notes   Perko 4.34.5 + notes  
−   [  +   [http://www.cds.caltech.edu/~macmardg/courses/cds140a/wi16/hw8wi16.pdf hw8wi16.pdf] <br> Due: 38 Feb (Wed) 
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 7*, 9* Mar <br>   7*, 9* Mar <br>  
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The ﬁnal grade will be based on homework and a ﬁnal exam:  The ﬁnal grade will be based on homework and a ﬁnal exam:  
* Homework (75%)  There will be 9 oneweek 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.''  * Homework (75%)  There will be 9 oneweek 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  +  * 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. 
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.  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. 
Latest revision as of 16:26, 15 March 2016
Introduction to Dynamics  
Instructors

Teaching Assistants

[edit] Announcements
 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.)
 Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.
 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.
 Some good quality typed up lecture notes from 2011 on manifolds, Lyapunov, and centre manifold (also posted below)
[edit] 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, PoincaréBendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.
[edit] Lecture Schedule
Date  Topic  Reading  Homework 
4 Jan 6 Jan 8 Jan 
Linear Differential Equations L11 Lecture notes

Perko, 1.11.10 
hw1wi16.pdf Due: 13 Jan (Wed) 
11 Jan 13 Jan 15 Jan 
Nonlinear differential equations

Perko, 2.12.6  hw2wi16.pdf Due: 20 Jan (Wed) 
20 Jan* 2 hrs 22 Jan 
Chaos, fractals, and global analysis using SOStools


25 Jan (2 hrs) Jan 29 
Behavior of differential equations L4

Perko, 2.72.10  hw3wi16.pdf Due: 3 Feb (Wed) 
1 Feb* 3 Feb* 5 Feb* 
Nonhyperbolic differential equations L5 2013 notes

Perko, 2.112.13  hw4wi16.pdf Due: 10 Feb (Wed) 
8 Feb 10 Feb 12 Feb 
Global behavior

Perko, 3.13.3  hw5wi16.pdf Due: 17 Feb (Wed) 
17 Feb 19 Feb 
Limit cycles

Perko, 3.43.5, 3.7, 3.9  hw6wi16.pdf Due: 24 Feb (Wed) 
22 Feb 2 hrs 26 Feb 
Bifurcations

Perko 4.14.2 
hw7wi16.pdf Due: 31 Feb (Wed) 
29 Feb* 2 Mar* 4 Mar* 
Bifurcations

Perko 4.34.5 + notes  hw8wi16.pdf Due: 38 Feb (Wed) 
7*, 9* Mar 
Course review  Final exam Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab 
[edit] Textbook
The primary text for the course (available via the online bookstore) is
[Perko]  L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006. 
The following additional texts may be useful for some students:
[G&H]  J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. SpringerVerlag, 1990. 
[H&S]  M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. SpringerVerlag, 1990. 
[J&S]  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) 
[Ver]  F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL) 
[edit] Grading
The ﬁnal grade will be based on homework and a ﬁnal exam:
 Homework (75%)  There will be 9 oneweek problem sets, due in class approximately one week after they are assigned. Late homework will not be accepted without prior permission from the instructor.
 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.
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.
[edit] 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.