Applied Dynamical Systems Math 2219 Fall 1998

Textbook:

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

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

• Linear algebra. Solving linear equations, finding nullspaces, eigenvalues, and eigenvectors
• Elementary analysis. Implicit function theorem, some minimal knowledge of metrics, multivariate calculus.
• Differential equations. Solutions to linear ODEs, finding fixed points, determining stability. (There will be a review of this.)
• Computer Ability to use UNIX or Windows to the extent of editing a text file.

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)

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.

• Step 1. Download either the X-windows/Unix source xppaut and compile it on your UNIX platform. Or download the Windows 95/NT binary to run on your windows platform. ( Note If you run off of the math server, euler, XPP is already available On the university UNIX machines, create a link to ~phase/xppaut for the latest Solaris version)
• Step 2. Configure your browser. (I run Netscape, so that is the one I will configure).
• Click on Edit Preferences
• Click on Navigator and then Applications
• Click on Add to add a new application. Fill in as follows:
• Description: differential equations
• MIMEType: application/ode
• Suffixes: ode
• Handled By: Click on Application and type in xppaut %s
• Click OK
• Click OK
• Click on the following to see if the software pops up: sample.ode If it doesn't ask me and we I will help set you up. Note If all you get is text on the screen, you can still use the software. Just copy the text to some file (a good choice is the filename given in the text file) and then run xpp or winpp from the command line. I don't think this is a problem in Windows.