ACM 101b/AM 125b/CDS 140a, Winter 2014

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Differential Equations and Dynamical Systems

Instructors

  • Richard Murray (CDS/BE), murray@cds.caltech.edu
  • Lectures: Tu/Th, 9-10:30, 105 ANB
  • Office hours: Wed 2-3 pm (please e-mail to confirm)

Teaching Assistants

  • Katie Broersma (CDS), Anandh Swaminathan (CDS)
  • Contact: cds140-tas@cds.caltech.edu
  • Office hours: Tue, 3:00-4:30, 101 STH

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.

Lecture Schedule

Date Topic Reading Homework
7 Jan
9 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

HW 1
Due: 15 Jan (Wed)
14 Jan
16 Jan
Linear Differential Equations II
  • S + N decomposition, Jordan form
  • Stability theory
  • Linear systems with inputs (nonhomogeneous systems)
Perko, 1.7-1.10 HW 2
Due: 22 Jan (Wed)
21 Jan
23 Jan
Nonlinear differential equations
  • Existence and uniqueness
  • Flow of a differential equation
  • Linearization
Perko, 2.1-2.6 HW 3
Due: 29 Jan (Wed)
28 Jan
30 Jan
Behavior of differential equations
  • Stable and unstable manifolds
  • Stability of equilibrium points for planar systems
Perko, 2.7-2.10 HW 4
Due: 5 Feb (Wed)
4 Feb
6 Feb
Non-hyperbolic differential equations
  • Lyapunov functions
  • Center manifold theorem
Perko, 2.11-2.13 HW 5
Due: 12 Feb (Wed)
11 Feb
13 Feb
Global behavior
  • Limit sets and attractors
  • Krasovskii-Lasalle invariance principle (if time)
  • Periodic orbits and limit cycles
Perko, 3.1-3.3 HW 6
Due: 19 Feb (Wed)
18 Feb*
20 Feb*
Limit cycles
  • Poincare' map
  • Bendixson criterion for limit cycles in the plane
Perko, 3.4-3.5, 3.7 HW 7
Due: 26 Feb (Wed)
25 Feb
27 Feb
Bifurcations
  • Sensitivity analysis
  • Structural stability
  • Bifurcation of equilibrium points
Perko 4.1-4.2 + notes HW 8
Due: 5 Mar (Wed)
4 Mar
6 Mar
Bifurcations
  • Hopf bifurcation
  • Application example: rotating stall and surge in turbomachinery
Perko 4.3-4.5 + notes HW 9
Due: 12 Mar (Wed)
11 Mar*
Course review Final exam
Due: 19 Mar (Wed). Pick up from Diane Goodfellow, 246 ANB

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.

You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).

No collaboration is allowed on the final exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.

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