ACM 101/AM 125b/CDS 140a, Winter 2011

Differential Equations and Dynamical Systems
Instructors Richard Murray (CDS/BE) Gentian Buzi (CDS) Lectures: Tu/Th, 9-10:30, 105 ANB	Teaching Assistants Molei Tao (CDS) Zhaoqian (Steven) Xie (CDS) Office hours: Mon 2-3pm, 314 ANB; Wed 12-2pm, 107 ANB

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

• 02 Mar 2011: For those interested, we have posted an example illustrating the use of CMT and Lyapunov functions.
• 08 Feb 2011: HW #6 is now posted; due 17 Feb 2011
• 01 Feb 2011: HW #5 is now posted; due 10 Feb 2011
• 25 Jan 2011: HW #4 is now posted; due 3 Feb 2011
• 18 Jan 2011: HW #3 is now posted; due 27 Jan 2011
• 11 Jan 2011: HW #2 is now posted; due 20 Jan 2011
• 6 Jan 2011: Office hours for HW #1 will be on 10 Jan (Mon) from 4-6 pm in 235 ANB; office hours for future weeks will be on Wednesdays from 12-2 in 235 ANB
• 6 Jan 2011: Course ombuds: Jeff Amelang and Matanya Horowitz
• 3 Jan 2011: HW #1 is now posted; due 11 Jan 2011
• 14 Nov 2010: web page creation

Lecture Schedule

Date	Topic	Reading	Homework
4 Jan 6 Jan	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 J&S, 2.1-2.5	HW 1
11 Jan 13 Jan	Linear Differential Equations II S + N decomposition, Jordan form Stability theory Linear systems with inputs (nonhomogeneous systems) Boundary value problems	Perko, 1.7-1.10 + notes	HW 2
18 Jan 20 Jan	Nonlinear differential equations Existence and uniqueness Flow of a differential equation Linearization	Perko, 2.1-2.6	HW 3
25 Jan 27 Jan	Behavior of differential equations Stable and unstable manifolds Stability of equilibrium points	Perko, 2.7-2.10 Some notes Remark on invariant manifolds	HW 4
1 Feb* 3 Feb*	Non-hyperbolic differential equations Lyapunov functions Center manifold theorem	Perko, 2.11-2.13 Notes on Lyapunov Notes on Center Manifold An Example (Center Manifold / Limit Cycle)	HW 5
8 Feb* 10 Feb	Hamiltonian systems Gradient and Hamiltonian systems Energy based stability methods Applications	Perko 2.14 + notes Gradient and Hamiltonian Systems Lagrangian, Summary	HW 6
15 Feb 17 Feb 22 Feb	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, 3.9 Lecture notes on orbits and attractors Lecture notes on limit cycles BFS notes on oscillators	HW 7
24 Feb 1 Mar 3 Mar	Bifurcations Structural stability Bifurcation of equilibrium points Hopf bifurcation	Perko 4.1-4.4 + notes Lecture notes on structural stability, parameter sensitivity BFS notes on parameter sensitivity Lecture notes on bifurcations	HW 8
8 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.

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 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.