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