Math 557B -- Dynamical Systems and Chaos

Announcements

(Last revised on January 11, 2018.)

* Nothing to see here.

Syllabus

These are the topics I'm planning to cover. It may get updated as we go along.

(Last revised on January 12, 2018.)

- More ergodic theory (~3 weeks): invariant measures for
  continuous maps; unique ergodicity; spectral properties of
  transfer operators; applications and examples.

- Elements of hyperbolic dynamics (2-3 weeks): hyperbolic
  sets and their stability; stable and unstable foliations;
  horseshoes & transverse homoclinic intersections.

- Selected topics in chaotic dynamics (~3 weeks): Lyapunov
  exponents; measures of dynamical complexity; symbolic &
  topological dynamics; delay coordinates and nonlinear time
  series analysis; period-doubling route to chaos.

- Dimension reduction (~3 weeks): timescale separation;
  transfer and Koopman operators.

- There may be additional topics if time permits.  These
  will be chosen based on instructor and student interests.

Grading. Your grade in this course will be based on (i) problem sets and (ii) a project. You are encouraged to work together on problems, 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 topic is up to you, but needs my approval first. The abstract will be made public after you have had a chance to make revisions.

Project abstracts from Spring 2018

• Robert Banks
• Hannah Biegel
• Spence Lunderman
• Jessica Pillow

Project abstracts from Fall 2017

• Armando Albornoz-Basto
• Robert Banks
• Hannah Biegel
• Kyle Gwirtz
• Spence Lunderman
• Jessica Pillow
• Jonathan Ramalheira-Tsu
• Brandon Tippings

Main references

1. Brin and Stuck, Introduction to Dynamical Systems

2. Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Additional general references

1. Guillemin and Pollack, Differential Topology

2. Devaney, Hirsch, and Smale, Differential Equations, Dynamical Systems, and An Introduction to Chaos

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

4. Strogatz, Nonlinear Dynamics and Chaos

References with a focus on Hamiltonian systems

1. Sussman and Wisdom, >Structure and Interpretation of Classical Mechanics,

2. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction,

The contents on this site, including but not limited to all documents and source code, are licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License.

This page was last updated on September 16, 2018.