Bifurcation Theory

Math 689-001

Spring 2018


The following table contains the schedule for the course. This page will be updated regularly throughout the semester.

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:


Thomas Wanner, April 3, 2018.