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

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Differential Equations and Dynamical Systems
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)	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 Piazza

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 Piazza page + added additional course info
• 23 Aug 2012: web page creation

Lecture Schedule

Date	Topic	Reading	Homework
8 Jan 10 Jan 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 Optional: J&S, Ch 1; Sec 2.1-2.5 Ver, Sec 5.1-5.2; 6.1-6.2	HW 1 Due: 15 Jan (Tue)
15 Jan 17 Jan 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	HW 2 Due: 22 Jan (Tue)
22 Jan 24 Jan RMM	Nonlinear differential equations Existence and uniqueness Flow of a differential equation Linearization	Perko, 2.1-2.6	HW 3 Due: 29 Jan (Tue)
29 Jan* 31 Jan DGM	Behavior of differential equations Stable and unstable manifolds Stability of equilibrium points	Perko, 2.7-2.10	HW 4 Due: 5 Feb (Tue)
5 Feb* 7 Feb DGM	Non-hyperbolic differential equations Lyapunov functions Center manifold theorem	Perko, 2.11-2.13	HW 5 Due: 12 Feb (Tue)
12 Feb 14 Feb* DGM	Hamiltonian systems Gradient and Hamiltonian systems Energy based stability methods Applications	Perko 2.14 + notes Scanned lecture notes Marsden, Mechanical systems	HW 6 Due: 19 Feb (Tue)
19 Feb 21 Feb* 26 Feb 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	HW 7 Due: 5 Mar (Tue)
28 Feb 5 Mar 7 Mar* RMM	Bifurcations Structural stability Bifurcation of equilibrium points Hopf bifurcation	Perko 4.1-4.4 + notes BFS notes on parameter sensitivity	HW 8 Due: 12 Mar (Tue)
12 Mar	Course review

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 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.