This is a one-semester in dynamical systems theory devoted mostly to the study of iteration of mappings of dimension one and two. Most of the basic ideas of dynamical systems theory can be introduced in this setting. Topics to be covered include fixed points, periodic points, stability, bifurcations, stable manifold theorem, Smale's horseshoe map, homoclinic chaos, strange attractors and Poincaré maps. Many ideas from topology and analysis will be introduced along the way. In addition there will be computer labs to illustrate the theory.
Chaos, An Introduction to Dynamical Systems, by Alligood, Sauer and Yorke. We will cover most of chapters 1-6 and 10-13. The lectures will contain topics and examples not treated in the book.
|Midterm Exams (25 % each)||50 %|
|Final Project||25 %|
You will submit a project on a topic of your choice related to dynamical systems or its applications. Some suggestions will be given later. The project is due on Friday, December 17 (the first day of exams).
|Midterm I||Monday, October 4|
|Midterm II||Monday, November 15|
|9/8-9/10||Iteration. Periodic points and stability.||1.1-1.4|
|9/13-9/17||Symbolic dynamics. Sensitive dependence.||1.6-1.8|
|9/20-9/24||Lab I. Lyapunov exponents and chaos.||3.1-3.3|
|9/27-10/1||Invariant Cantor sets andthe shift map.||3.4, 4.1, 4.3|
|10/4-10/8||MT I. 2D maps. Per. points and stab.||2.1-2.4|
|10/11-10/14||Linearization. Stable manifold theorem.||2.5, 2.6, 10.1, 10.4|
|10/18-10/22||Symbolic dynamics and chaos in 2D.||5.1, 5.2, 5.4, 5.5|
|10/25-10/29||Horseshoe map and homoclinic chaos. Lab II.||5.6, 10.2|
|11/1-11/5||Strange attractors. Fractals.||6.1, 6.2, 4.2|
|11/8-11/12||Fractal dimensions, invariant measures||4.5-4.7, 6.4, 6.5|
|11/15-11/19||MT II. Bifurcations.||11.1-11.4|
|11/22-11/14||2D bifurcations, Hopf bifurcation.||11.5-11.8|
|11/29-12/3||Cascades of bifurcations.||12.1-12.4|