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

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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 HW 4
1 Feb*
3 Feb*
Non-hyperbolic differential equations
  • Lyapunov functions
  • Center manifold theorem
Perko, 2.11-2.13 HW 5
8 Feb*
10 Feb
Hamiltonian systems
  • Gradient and Hamiltonian systems
  • Energy based stability methods
  • Applications
Perko 2.14 + notes 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 HW 7
24 Feb
1 Mar
3 Mar
Bifurcations
  • Structural stability
  • Bifurcation of equilibrium points
  • Hopf bifurcation
Perko 4.1-4.4 + notes 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.

Old Announcements