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, fractals, Julia sets and the Mandelbrot set, 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.
Discrete Chaos, by Saber Elaydi. We will cover chapters 1, 2, 3, 6, 7 and parts of chapters 4, 5. The lectures will differ somewhat from the text however and some extra topics will be covered.
| Homework | 20 % |
| Midterm Exams (20 % each) | 40 % |
| Final Exam | 40 % |
| Midterm I | Monday, February 19 |
| Midterm II | Monday, April 2 |
| Final Exam | Friday, May 11, 10:30-12:30 |
| Week | Topic | Reading |
| 1/17-1/19 | Iteration. Fixed points. Cobweb plots. | 1.1-1.5 |
| 1/22-1/26 | Periodic points. Stability. | 1.6,1.7 |
| 1/29-2/2 | Changes of coords. Bifurcations. | 1.8,2.5 |
| 2/5-2/9 | More bifurcations. Cascades. Itineraries. | 2.5,2.1 |
| 2/12-2/16 | Sharkovski theorem. Singer theorem. | 2.1,2.2,2.4 |
| 2/19-2/23 | Midterm I. Cantor Sets. | 3.2 |
| 2/26-3/2 | Sequence spaces. Symbolic dynamics. | 3.6,3.7 |
| 3/5-3/9 | Chaos. | 3.3-3.5 |
| 3/12-3/16 | Fractals. Iterated functions systems. | 6.1, 6.3 |
| 3/19-3/23 | Fractal dimensions. Hutchinson theorem. | 6.4, 6.2 |
| 3/26-3/30 | Spring Break. | |
| 4/2-4/6 | Midterm II. Complex dynamics. | 7.1,7.2 |
| 4/9-4/13 | Julia sets. Mandelbrot set. | 7.3,7.4 |
| 4/16-4/20 | Maps of the plane. Fixed points. Stability. | 4.2,4.5-4.7 |
| 4/23-4/27 | Stable manifolds. Horseshoes. | 4.10,5.2,5.3 |
| 4/30-5/4 | Forced pendulum. Other applications. | extra topics |