Math 557 -- Dynamical Systems and Chaos

Fall 2017

When: MW 3-4:15pm
Where: Saguaro 223 Math 514
Instructor: Kevin K Lin

Office: Math 606
Phone: 626-6628
E-mail: klin@math.arizona.edu

Course web page (this page):
http://math.arizona.edu/~klin/557

Office hours:
see here

Announcements

(Last revised on December 22, 2017.)
* <2017-10-25 Wed> Tim Sauer guest lecture
* <2017-10-16 Mon> Problem Set #1 due
* <2017-10-11 Wed> Projects
1) 15 minute in-class presentation, dates TBA
2) 4--5 page extended abstract
* <2017-08-22 Tue> Starting tomorrow, we will meet in Math 514.

Syllabus

This is a rough list of topics I plan to cover. It will get updated as we go along. 
(Last revised on August 18, 2017.)
Part I: Fixed points, periodic orbits, and invariant manifolds
- Crash course on manifolds
- Fixed points, periodic orbits, and their stability
- Hyperbolic fixed points and periodic orbits
- Stable, unstable, and center manifolds
- Hyperbolic invariant sets and genericity

Part II: Bifurcations of fixed points and periodic orbits
- Normal forms
- Local bifurcations of flows: saddle-node, pitchfork,
  transcritical, Hopf
- Some global bifurcations of flows
- Local bifurcations of maps

Part III: Additional topics (time permitting of course)
- Geometric singular perturbations and slow-fast systems
- Nonautonomous flows?
- Symmetries?

If the second semester course (557B) runs, I plan to
introduce some of the modern mathematical tools useful for
studying chaotic and random dynamics.  I may also cover some
additional topics, depending in part on how much time we
have and on interest (yours and mine).

Grading. Your grade in this course will be based on

  1. Problem sets
  2. Project
I will assign problems from time to time. These will be due a few times over the course of the semester. You are encouraged to work together, but you must write up your own solution. Since this is a graduate course, I expect everyone to make their best effort to solve every problem.

As for projects, you will

  • give a short (about 15 minutes) talk in class at the end of the term, and
  • write an "extended abstract" (a few pages, complete with references).
The project is to be on a topic of your choosing but with instructor approval. The abstract will be made public after you have had a chance to make revisions.

Project abstracts

Main references

  1. J Guckenheimer and P Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer

  2. D Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press

  3. S Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer

#1 is our main text; I will not follow it closely, but everything I cover will be in there. #2 will be closer to Parts I and II of the course, but is unfortunately out of print as far as I know. #3 covers a superset of #1 and can be a useful reference. I'll also hand out notes (or copies of papers) when useful.

Additional general references

  1. V Guillemin and A Pollack, Differential Topology

  2. R Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press

  3. R Devaney, M Hirsch, S Smale, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Springer

  4. S Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley

References with a focus on Hamiltonian systems

  1. G J Sussman and J Wisdom, with M Mayer, Structure and Interpretation of Classical Mechanics, MIT Press

  2. M Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Addison-Wesley

This page was last updated on September 16, 2018.