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 Dynamical Systems
    2. Orbits and Phase Portraits
2 01/30     No class!
02/01     No class!
3 02/06     3. Invariant Sets
    4. Differential Equations as Dynamical Systems
02/08     5. Poincare Maps
II. Topological Equivalence, Bifurcations, and Structural Stability
    1. Equivalence of Dynamical Systems
    2. Topological Classification of Equilibria and Fixed Points
    3. Bifurcations and Bifurcation Diagrams
    4. Topological Normal Forms
    5. Structural Stability
III. Local Bifurcations in Continuous Dynamical Systems
    1. Simplest Bifurcation Conditions
    2. The Fold Bifurcation
    3. The Hopf Bifurcation
    4. Center Manifolds
    5. The Lyapunov-Schmidt Reduction
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
VI. Global Bifurcations
    1. To be announced..
VII. An Introduction to Computer-Assisted Proofs
    1. To be announced..

For the course, I will draw material from the following books:


Thomas Wanner, January 23, 2018.