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 TBD TBD

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

• 14 Nov 2010: web page creation

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.

Lecture Schedule

Week	Date	Topic	Reading	Homework
1	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
2	11 Jan 13 Jan	Linear Differential Equations II S + N decomposition, Jordan form Stability theory Linear systems with inputs (nonhomogeneous systems)	Perko, 1.7-1.10
3	18 Jan 20 Jan	Nonlinear differential equations Existence and uniqueness Flow of a differential equation Linearization	Perko, 2.1-2.6
4	25 Jan 27 Jan	Behavior of differential equations Stable and unstable manifolds Stability and Lyapunov functions Planar systems, phase diagrams	Perko, 2.6-2.10
5	1 Feb* 3 Feb	Non-hyperbolic differential equations Center manifold theorem Normal forms	Perko, 2.11-2.13
6	8 Feb 10 Feb	Hamiltonian systems Gradient and Hamiltonian systems Energy based stability methods Applications	Perko 2.14 + notes
7	15 Feb 17 Feb	Limit cycles Limit sets and attractors Periodic orbits and limit cycles Poincare' map Poincare'-Bendeixson criterion	Perko, 3.1-3.9
8	22 Feb 24 Feb	Bifurcations Structural stability Bifurcation of equilibrium points Hopf bifurcation	Perko 4.1-4.4
9	1 Mar 3 Mar	Advanced topics Higher co-dimension bifurcations Homoclinic bifurcation Introduction to chaos	Perko, 4-3, 4.5-4.8, notes
10	8 Mar	Course review

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 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 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.