View source for ACM 101/AM 125b/CDS 140a, Winter 2013

{{warning|This page is out of date.  Please refer to the [[CDS 140a]] page for the latest information}}
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<font color='blue' size='+2'>Differential Equations and Dynamical Systems</font>__NOTOC__
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'''Instructors'''
* Richard Murray (CDS/BE), murray@cds.caltech.edu
* Doug MacMartin (CDS), macmardg@cds.caltech.edu
* Lectures: Tu/Th, 9-10:30, 105 ANB
* Office hours: Wed 2-3 pm (please e-mail to confirm)
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'''Teaching Assistants'''
* Katheryn Broersma (CDS), katheryn@caltech.edu
* Vanessa Jönsson (CDS), vjonsson@caltech.edu
* Office hours: Fridays, 11-12 in 314 ANB; Mondays, 12-1 in 243 ANB
* [https://piazza.com/caltech/winter2013/cds140 Piazza]
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=== 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.

===Announcements ===

* 16 Dec 2012: set up [https://piazza.com/caltech/winter2013/cds140 Piazza] page + added additional course info
* 23 Aug 2012: web page creation

=== Lecture Schedule ===

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| '''Date'''
| '''Topic'''
| '''Reading'''
| '''Homework'''
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| 8 Jan <br> 10 Jan <br> RMM
| Linear Differential Equations I
* Course overview and administration
* Linear differential equations
* Matrix exponential, diagonalization
* Stable and unstable spaces
* Planar systems, behavior of solutions
| 
Perko, 1.1-1.6<br>
Optional:
* J&S, Ch 1; Sec 2.1-2.5
* Ver, Sec 5.1-5.2; 6.1-6.2
| [[CDS 140a Winter 2013 Homework 1|HW 1]] <br> Due: 15 Jan (Tue)
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| 15 Jan <br> 17 Jan <br> RMM
| Linear Differential Equations II
* S + N decomposition, Jordan form
* Stability theory
* Linear systems with inputs (nonhomogeneous systems)
* Boundary value problems (if time)
| Perko, 1.7-1.10 + notes
| [[CDS 140a Winter 2013 Homework 2|HW 2]] <br> Due: 22 Jan (Tue)
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| 22 Jan <br> 24 Jan <br> RMM
| Nonlinear differential equations
* Existence and uniqueness
* Flow of a differential equation
* Linearization
| Perko, 2.1-2.6
| [[CDS 140a Winter 2013 Homework 3|HW 3]] <br> Due: 29 Jan (Tue)
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| 29 Jan* <br> 31 Jan <br> DGM
| Behavior of differential equations
* Stable and unstable manifolds  
* Stability of equilibrium points
| Perko, 2.7-2.10 
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* [[Media:cds140a-wi11-Week4Notes.pdf|Some notes]]
* [[Media:cds140a-wi11-InvManRemark.pdf|Remark on invariant manifolds]]
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| | [[CDS 140a Winter 2013 Homework 4|HW 4]] <br> Due: 5 Feb (Tue)
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| 5 Feb* <br> 7 Feb <br> DGM
| Non-hyperbolic differential equations
* Lyapunov functions
* Center manifold theorem
| Perko, 2.11-2.13
<!--
* [[Media:cds140a-wi11-Week5Notes.pdf|Notes on Lyapunov]]
* [[Media:cds140a-wi11-Week5NotesCMT.pdf|Notes on Center Manifold]]
* [[Media:cds140a-wi11-CMTLimitCycleExample.pdf|An Example (Center Manifold / Limit Cycle)]]
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| [[CDS 140a Winter 2013 Homework 5|HW 5]] <br> Due: 12 Feb (Tue)
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| 12 Feb <br> 14 Feb* <br> DGM
| Hamiltonian systems
* Gradient and Hamiltonian systems
* Energy based stability methods
* Applications
| Perko 2.14 + notes
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/L6-Hamiltonian.pdf Scanned lecture notes]
* [http://www.cds.caltech.edu/~macmardg/courses/cds140a/MarsdenMechSystems.pdf Marsden, Mechanical systems]
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* [[Media:cds140a-wi11-Week6NotesHamGrad.pdf|Gradient and Hamiltonian Systems]]
* [[Media:cds140a-wi11-Week6NotesLagHamSum.pdf|Lagrangian, Summary]]
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| [[CDS 140a Winter 2013 Homework 6|HW 6]] <br> Due: 19 Feb (Tue)
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| 19&nbsp;Feb <br> 21 Feb* <br> 26 Feb <br> RMM
| Limit cycles
* Limit sets and attractors
* Periodic orbits and limit cycles
* Poincare' map
* Bendixson criterion for limit cycles in the plane
| Perko, 3.1-3.5, 3.7, 3.9
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* [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L7-1_orbits-15Feb11.pdf Lecture notes on orbits and attractors]
* [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L7-2_limitcycles-17Feb11.pdf Lecture notes on limit cycles]
* [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L7-3_bfs_oscillators.pdf BFS notes on oscillators]
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| [[CDS 140a Winter 2013 Homework 7|HW 7]] <br> Due: 5 Mar (Tue)
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| 28 Feb <br> 5 Mar <br> 7 Mar* <br> RMM
| Bifurcations
* Structural stability
* Bifurcation of equilibrium points
* Hopf bifurcation
| Perko 4.1-4.4 + notes
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* [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L8-1_parmsens-24Feb11.pdf Lecture notes on structural stability, parameter sensitivity]
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* [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L8-1_bfs_sensitivity.pdf BFS notes on parameter sensitivity]
<!--
* [http://www.cds.caltech.edu/~murray/courses/cds140/wi11/caltech/L8-2_bifurcations-01Mar11.pdf Lecture notes on bifurcations]
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| [[CDS 140a Winter 2013 Homework 8|HW 8]] <br> Due: 12 Mar (Tue)
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| 12 Mar <br> 
| Course review
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=== Textbook ===

The primary text for the course (available via the online bookstore) is
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|- valign=top
| 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 (on reserve in SFL):
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|- 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.
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| align=right | &nbsp;[Ver]&nbsp;
| F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition.  Springer, 2006.
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=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 8 one-week 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 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.

No collaboration is allowed on the ﬁnal exam.

== Old Announcements ==
[[Category:Courses]]