Bifurcation Theory Home Page

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Course Announcement

Math 6740 - Bifurcation Theory

Time: T,TH 2:00-3:20 pm.

Place: LCB 322

Text

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, third edition, Springer, 2004.

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems, SIAM, 2002.

The course will begin with an introduction to computations of bifurcation curves using XPPAUT. In addition to the topics in the text, we will cover the Lyapunov-Schmidt method, global bifurcation theorems for Sturm-Liouville eigenvalue problems, the global Hopf bifurcation theorem, bifurcations in pde's, the Ginzberg-Landau equation, the Turing instability and bifurcation (pattern formation), bifurcations such as the Taylor-Couette vortices and Benard instabilities (and maybe thermoacoustic engines.)

1. Introduction: Continuation and homotopy, What is a bifurcation?, the implicit function theorem

2. Examples of bifurcations; algebraic equations, discrete maps, Hopf. Use of XPPAUT to compute bifurcation curves.

3. Steady state bifurcations; Sturm Liouville problems, Turing, global continuation theorems

4. Bifurcation of dynamical systems (Kuznetsov)

5. Bifurcation in PDE's; Ginzberg-Landau equations, Turing revisited

6. Other important examples; Taylor-Couette, Benard, thermoacoustic engines

Homework:

Notes:

Hopf Bifurcation and Normal Form for the van der Pol Equation

Notes on a center manifold calculation

Branching Theory paper

For more information contact J. Keener, 1-6089