Difference between revisions of "ACM 101b/AM 125b/CDS 140a, Winter 2014"
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'''Teaching Assistants'''  '''Teaching Assistants'''  
−  * Katie Broersma (CDS),  +  * Katie Broersma (CDS), Anandh Swaminathan (CDS) 
* Contact: cds140tas@cds.caltech.edu  * Contact: cds140tas@cds.caltech.edu  
−  * Office hours: Tue, 3:  +  * Office hours: Tue, 3:004:30, 101 STH 
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=== Course Description ===  === 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.  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.  
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=== Lecture Schedule ===  === Lecture Schedule ===  
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 Behavior of differential equations   Behavior of differential equations  
* Stable and unstable manifolds  * Stable and unstable manifolds  
−  * Stability of equilibrium points  +  * Stability of equilibrium points for planar systems 
 Perko, 2.72.10   Perko, 2.72.10  
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  [[CDS 140a Winter 2014 Homework 4HW 4]] <br> Due: 5 Feb (Wed)    [[CDS 140a Winter 2014 Homework 4HW 4]] <br> Due: 5 Feb (Wed)  
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* Center manifold theorem  * Center manifold theorem  
 Perko, 2.112.13   Perko, 2.112.13  
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 [[CDS 140a Winter 2014 Homework 5HW 5]] <br> Due: 12 Feb (Wed)   [[CDS 140a Winter 2014 Homework 5HW 5]] <br> Due: 12 Feb (Wed)  
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 11 Feb <br> 13 Feb   11 Feb <br> 13 Feb  
−    +   Global behavior 
−  *  +  * Limit sets and attractors 
−  *  +  * KrasovskiiLasalle invariance principle (if time) 
−  *  +  * Periodic orbits and limit cycles 
−   Perko  +   Perko, 3.13.3 
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 [[CDS 140a Winter 2014 Homework 6HW 6]] <br> Due: 19 Feb (Wed)   [[CDS 140a Winter 2014 Homework 6HW 6]] <br> Due: 19 Feb (Wed)  
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−   18 Feb* <br> 20 Feb*  +   18 Feb* <br> 20 Feb* 
 Limit cycles   Limit cycles  
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* Poincare' map  * Poincare' map  
* Bendixson criterion for limit cycles in the plane  * Bendixson criterion for limit cycles in the plane  
−   Perko, 3.  +   Perko, 3.43.5, 3.7 
−  +   [[CDS 140a Winter 2014 Homework 7HW 7]] <br> Due: 26 Feb (Wed)  
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−   [[CDS 140a Winter 2014 Homework 7HW 7]] <br> Due:  
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−    +   25 Feb <br> 27 Feb 
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 Bifurcations   Bifurcations  
+  * Sensitivity analysis  
* Structural stability  * Structural stability  
* Bifurcation of equilibrium points  * Bifurcation of equilibrium points  
+   Perko 4.14.2 + notes  
+  * [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L81_bfs_sensitivity.pdf BFS notes on parameter sensitivity]  
+   [[CDS 140a Winter 2014 Homework 8HW 8]] <br> Due: 5 Mar (Wed)  
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+   4 Mar <br> 6 Mar  
+   Bifurcations  
* Hopf bifurcation  * Hopf bifurcation  
−   Perko 4.  +  * Application example: rotating stall and surge in turbomachinery 
−  +   Perko 4.34.5 + notes  
−  +   [[CDS 140a Winter 2014 Homework 9HW 9]] <br> Due: 12 Mar (Wed)  
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−   [[CDS 140a Winter 2014 Homework  
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 11 Mar* <br>   11 Mar* <br>  
 Course review   Course review  
 <! Reading >   <! Reading >  
−   Final exam <br> Due: 19 Mar (Wed)  +   Final exam <br> Due: 19 Mar (Wed). Pick up from Diane Goodfellow, 246 ANB 
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Latest revision as of 16:25, 30 March 2014
Differential Equations and Dynamical Systems  
Instructors

Teaching Assistants

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.
Lecture Schedule
Date  Topic  Reading  Homework 
7 Jan 9 Jan 
Linear Differential Equations I

Perko, 1.11.6 
HW 1 Due: 15 Jan (Wed) 
14 Jan 16 Jan 
Linear Differential Equations II

Perko, 1.71.10  HW 2 Due: 22 Jan (Wed) 
21 Jan 23 Jan 
Nonlinear differential equations

Perko, 2.12.6  HW 3 Due: 29 Jan (Wed) 
28 Jan 30 Jan 
Behavior of differential equations

Perko, 2.72.10  HW 4 Due: 5 Feb (Wed) 
4 Feb 6 Feb 
Nonhyperbolic differential equations

Perko, 2.112.13  HW 5 Due: 12 Feb (Wed) 
11 Feb 13 Feb 
Global behavior

Perko, 3.13.3  HW 6 Due: 19 Feb (Wed) 
18 Feb* 20 Feb* 
Limit cycles

Perko, 3.43.5, 3.7  HW 7 Due: 26 Feb (Wed) 
25 Feb 27 Feb 
Bifurcations

Perko 4.14.2 + notes  HW 8 Due: 5 Mar (Wed) 
4 Mar 6 Mar 
Bifurcations

Perko 4.34.5 + notes  HW 9 Due: 12 Mar (Wed) 
11 Mar* 
Course review  Final exam Due: 19 Mar (Wed). Pick up from Diane Goodfellow, 246 ANB 
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 (on reserve in SFL):
[J&S]  D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. 
[Ver]  F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. 
Grading
The ﬁnal grade will be based on homework and a ﬁnal exam:
 Homework (75%)  There will be 8 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 at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 48N 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.