Differential Equations: Dynamical Systems and Chaos
Professor Sutherland, Spring 1997
Lorenz attractor applet by John Franks
|
Math 351 Differential Equations: Dynamical Systems and Chaos Professor Sutherland, Spring 1997 |
Lorenz attractor applet by John Franks |
| week | sections | topics |
|---|---|---|
| 1/15 | 1.1-1.4 | Introduction. Begin one-dimensional maps |
| 1/20 | 1.4-1.8 | The logistic family; sensitive dependence; itineraries If your browser is Java-enabled, you might try a logistic map simulator and bifurcation diagram generator |
| 1/27 | 1.8; 2.1-2.2; Devaney: sect 1.6 | The logistic family for a=4 and a>4; Introduction to 2 dimensional maps (especially the Henon map). |
| 2/3 | 2.3-2.5, Appendix A | sinks, sources, and saddles; Linear maps,
eigenvalues
and eigenvectors; nonlinear maps and linearization. |
| 2/10 | 2.5-2.7, 3.1-3.3 | stable and unstable manifolds;matrix times circle equals ellipse; Lyapunov exponents. |
| 2/17 | 3.3-3.5 | conjugacy, transition graphs, fixed points, basins of attraction, Sharkovskii's theorem. |
| 2/24 | Challenge 3, 3.5 | Sharkovskii's theorem; the Schwarzian derivative; (no class 2/28) |
| 3/3 | 4.1-4.4 | Fractals: Cantor Sets, the Sierpinski gasket, basin boundaries. Exam on 3/7, covers Chapters 1-3 |
| 3/10 | 4.5,4.6, 5.1, 5.2 | Fractal dimension, box-counting dimension and correlation dimension. Lyapunov exponents in higher dimensions; Lyapunov dimension. |
| 3/17 | 5.3-5.6 | Fixed point theorems. Markov partitions. The Horseshoe map and subshifts of finite type. |
| 3/24 | Spring Break | |
| 3/31 | 6.1-6.6 | Forward limit sets, chaotic attractors, measure. |
| 4/7 | 7.1-7.6 | Differential equations: linear and nonlinear. potentials. Lyapunov functions. |
| Homework | |
| date due | problems |
|---|---|
| 2/5 | do any 10 T1.2, T1.3, T1.9, T1.12, T1.13, T1.14; 1.2, 1.3, 1.6, 1.8, 1.11, 1.12, 1.16, Challenge 1 (counts as 8 problems). |
| 2/19 | do any 12 T2.5, T2.6, T2.7, T2.9; 2.1, 2.3, 2.4, 2.7, 2.9, Challenge 2 (counts as 8 problems). T3.1, T3.2, T3.3, T3.6, T3.8; 3.1, 3.3, 3.4, 3.7, 3.12. |
| 3/7 | do any 10 T3.9, T3.10, T3.11, T3.13, T4.2, T4.3, T4.6, T4.8, T4.9, T4.10, T4.11. 3.5, 3.9, 3.11, 4.1, 4.2, 4.3, 4.5, 4.7. |
| 3/31 | Redo your exam as a take-home, carefully writing up any problems you didn't attempt or did incorrectly. You needn't redo problems that were done correctly. Do all 6 problems. |
| 3/31 | Do any 8 4.10, 4.12, 4.14. T5.5, T5.6, 5.1, 5.4, 5.5, 5.6, 5.7, 5.8. |
| 4/18 | Do any 10 T6.2, T6.3, T6.6, T6.10, T6.13, T6.14, Challenge 6 (Counts as 5--try it). 6.1, 6.5, 6.6, 6.8. T7.5, T7.6, T7.11. 7.2, 7.3, 7.6. |