Week | Date | ||
---|---|---|---|
1 | 01/23 | Motivation: The Tacoma Narrows Bridge | |
I. Introduction to Dynamical Systems | |||
01/25 | 1. Continuous- and Discrete-Time Dynamical Systems | ||
2 | 01/30 & 02/01 | No class! | |
3 | 02/06 | 2. Orbits and Phase Portraits | |
02/08 | 3. Invariant Sets | ||
4 | 02/13 | 4. Differential Equations as Dynamical Systems | |
02/15 | 5. Links between Continuous-Time and Discrete-Time | ||
II. Topological Equivalence, Bifurcations, and Structural Stability | |||
5 | 02/20 | 1. Equivalence of Dynamical Systems | |
02/22 | 2. Topological Classification of Equilibria and Fixed Points | ||
6 | 02/27 | 3. Bifurcations and Bifurcation Diagrams | |
03/01 | 4. Topological Normal Forms | ||
7 | 03/06 | 5. Local Bifurcations Near Equilibria | |
03/08 | 6. Structural Stability | ||
03/13 & 03/15 | Spring Break! | ||
III. Local Bifurcations in Continuous Dynamical Systems | |||
8 | 03/20 | 1. Simplest Bifurcation Conditions | |
2. The Normal Form of the Fold Bifurcation | |||
03/22 | 3. Generic Fold Bifurcation | ||
9 | 03/27 | 4. The Normal Form of Hopf Bifurcation | |
03/29 | 5. Generic Hopf Bifurcation | ||
10 | 04/03 | 6. A Model from Population Dynamics | |
04/05 | 7. Center Manifold Theorems | ||
11 | 04/10 & 04/11 | No class! | |
12 | 04/17 | 8. Fold Bifurcations in Arbitrary Dimensions | |
04/19 | 9. Hopf Bifurcations in Arbitrary Dimensions | ||
IV. Local Bifurcations in Discrete Dynamical Systems | |||
1. Simplest Bifurcation Conditions | |||
2. The Fold Bifurcation | |||
3. The Flip Bifurcation | |||
4. The Neimark-Sacker Bifurcation | |||
5. Computation of Center Manifolds | |||
V. Bifurcations in Symmetric Systems | |||
1. Equivariant Dynamical Systems | |||
2. Equivariant Lyapunov-Schmidt Reduction | |||
3. Symmetry-Breaking Pitchfork Bifurcations | |||
16 | 05/15 | Student Presentations! (10:30am - 1:15pm) |
For the course, I will draw material from the following books: