Bifurcation Theory Home Page

Send e-mail to : Professor Keener Links: Professor Keener's home page Math Biology Program Department of Mathematics College of Science University of Utah

Course Announcement

Math 6740 - Bifurcation Theory - Spring 2023

Time: MWF 10:45-11:35-am

Place: CSC (Crocker Science Center) 25 and on zoom https://utah.zoom.us/j/98018825769 Passcode: 819555

Lectures will be recorded and posted on Canvas

Prerequisites

Texts

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

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

Other References

H. Kielhofer, Bifurcation Theory, An Introduction with Applications to Partial Differential Equations, Springer, 2012. edition, Springer, 2004.

W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, 2000.

R. Howe, Pattern Formation, An Introduction to Methods, Cambridge University Press, 2006.

The course will begin with an introduction to computations of bifurcation curves using XPPAUT (and MATCONT). 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 (and/or MATCONT) 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:

MATLAB files:

Delayed logistic map

XPP files:

quadratic equation

cubic equation

quartic equation

discretized Bratu's equation

Predator-Prey system w/ Holling II dynamics

Morris-Lecar equations

Maple Codes:

An important part of this course is learning how to compute bifurcation diagrams using either AUTO or MATCONT. A good way to get started with AUTO is with XPPAUT and run a few of the DEMO problems, although many of these will be described in class. Also, use the book B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems, SIAM, 2002. MATCONT is convenient for people familiar with Matlab: I have successfully used matcont4p2, but not later versions (currently at matcont7p3) - I'm still working on this.

For more information contact J. Keener, 1-6089