Applied Dynamical Systems

Math 2219

Fall 1998

Bard Ermentrout

Lectures:

MWF 1:00-1:50 PM Thackeray 704

Instructor:

Bard Ermentrout

Textbook:

Elements of Applied Bifurcation Theory. Y. Kuznetsov, Springer NY, 1998 (2nd Edition)

Optional Reading

Introduction to Applied Nonlinear Dynamical Systems and Chaos. S. Wiggins, Springer-Verlag, NY, 1990

Abstract

The course offers an introduction to Dynamical Systems from an applied and practical point of view. The goal in this course will be to teach the student how to compute the behavior of differential equations as parameters varies. Techniques that will be used include bifurcation analysis and computatation of normal forms, geometric methods, and the method of averaging. There will also be a computer component; numerical techniques required will be discussed.

Prerequisites


Syllabus

  1. Review of differential equations
    1. Fixed points and stability
    2. Planar systems
    3. Homework 1
      1. Nullclines
      2. Index theory
      3. Bendixson Theorem
      4. Poincare-Bendixson Theory
    4. Lyapunov Functions
  2. Introduction to Dynamical Systems (Chapter 1, K)
    1. Definitions
    2. Orbits
    3. Invariant sets -- Smale horseshoe
    4. Poincare maps
    5. Extra homework
  3. Topological equivalence (Chapter 2, K)
    1. Equilibria and fixed points
    2. Bifurcation diagrams
    3. Topological normal forms
  4. Codimension-1 bifurcations in ODEs (Chapter 3,5)
    1. Fold
    2. Hopf
  5. Codimension-1 bifurcations of maps (Chapt 4)
    1. Fold
    2. Flip
    3. Hopf (Neimark-Sacker)
  6. Homoclinic and heteroclinic bifurcations (Chapt 6)
  7. Codim-2 bifurcations of flows (Chapt 8)

Grading Policy

Grades will be based on homework (33%), a midterm exam (33%) and a final exam (33%). Exams will be take-home with both written and computer components. Homework will generally be due a week after it is assigned. No late homework will be accepted. Homework must be in hardcopy only.

Computer test

This is to test whether you have your browser configured correctly to run ODE files interactively with the free software that comes with the course.