Syllabus
| Lectures | Brief Description | Sections in Text |
|---|---|---|
| 9/11 | Introduction: Discrete and Continuous-time Dynamical Systems, fixed points Here is a roughly annotated and extended version of the file we used during class (I removed the cobweb plot because that's your assignment). |
Chapter 1.1, 1.2 |
| 9/13 | Attracting fixed points (proof here!) and periodic points. | Chapter 1.3, 1.4 |
| 9/15 | Stability, doubling map, eventually periodic. | Chapter 1.5 |
| 9/18 | Bifurcations. (Mathematica Bifurcations) Bifurcation diagrams: full and chaotic region. |
|
| 9/19 | Proof Writing X-Hour, 12:15 pm | |
| 9/20 | Chaos! Sensitivity to initial conditions, chaotic orbits. | |
| 9/22 | Two dimensional linear maps. | 2.2, 2.3, 2.4 |
| 9/25 | Nonlinear maps and the Jacobian, Henon map. | 2.5 |
| 9/27 | Stable/unstable manifolds, Henon map. Henon Map Animations. |
2.6 |
| 9/29 | Lyapunov exponents. Logistic Lyapunov Animation. |
3.1, 3.2 |
| 10/2 | Itineraries, transition diagrams. | 1.8, 3.3 |
| 10/4 | Lyapunov exponents, flat-topped tent map. Lyapunov and Cobweb for Logistic Flat-topped Tent Map |
|
| 10/5 | Midterm Exam | 7 pm, Kemeny 201. |
| 10/6 | Sarkovskii's Theorem. Three Implies Chaos. | |
| 10/9 | Fractals arising from maps, Cantor set, fractal dimension. | |
| 10/11 | Probabilistic construction of fractals, Koch Snowflake and others. Mathematica Fractal File Fractal Fern |
|
| 10/13 | Box-counting dimension. Henon Strange Attractor |
|
| 10/16 | Julia and Mandelbrot sets; Basins of Infinity. Mandelbrot and Julia Sets |
|
| 10/18 | Crash Course in Differential Equations (Review). | |
| 10/20 |
Limit Cycles: Poincare-Bendixson Theorem. Stable Limit Cycles |
|
| 10/23 |
Stable Manifolds, Limit Sets. Monday Worksheet Solutions |
|
| 10/25 | Chaos in Higher Dimensions. Strange Attractors. | |
| 10/26 | Midterm Exam | 7-9pm |
| 10/30 | Attractors in Non-linear Systems Lorenz Attractor and Delay Embeddings |
|
| 11/1 | Delay Embeddings and Shannon Entropy | |
| 11/3 | Permutation Entropy, Generating Partitions, KL Divergence Methods | |
| 11/6 | Recurrence Plots and Estimating Lyapunov Exponents Sample Data Example Recurrence Plots |
|
| 11/8 | Local Rules and Cellular Automata, Spread of Information on Graphs. First Examples |
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| 11/10 | More Cellular Automata Langton's Ant, Game of Life. |
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| 11/13 | Last day: talking about chaos. | |
| 11/17 | Final Exam | 8:00 am - 11:00 am |
