Bifurcation Theory Home Page

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Math Biology Program
Department of Mathematics
College of Science
University of Utah



Course Announcement

Math 6740 - Bifurcation Theory

Time: T,TH 09:10-10:30 am

Place: CSC (Crocker Science Center) 13

Texts

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

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

D. G. Schaeffer and J. W. Cain, Ordinary Differential Equations,Basics and Beyond, Springer, 2016.

Other References

H. Kielhofer, Bifurcatiion 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 Formtion, An Introduction to Methods, Cambridge University Press, 2006.

Course Outline

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:

Homework assignments will be posted

here

 and updated regularly. .

Notes:

Additional class notes will be posted

here

Stakgold SIAM Review paper (1971)

Resultant Notes

Fold Normal Form

Normal Form and Hopf bifurcation for van der Pol equation

MATLAB files:

Matlab codes will be posted here:

Delayed logistic map

XPP files:

XPPAUT files will be posted here:

quadratic equation

cubic equation

quartic equation

discretized Bratu's equation

Predator-Prey system w/ Holling II dynamics

Morris-Lecar equations

Euler column

Bratu's BVP

Maple Codes:

Maple codes are posted here:

Hopf bifurcation for Bazykin model (Predator-Prey system w/ Holling II dynamics)

Final Project:

All registered students will be required to do a project that involves numerical computation of bifurcations. The written description of the project must describe the physical or biological problem, describe the mathematical model, and then explore the bifurcations of the solutions. The numerical computations may be done using AUTO, or some other satisfactory package (XPP has an AUTO interface that may be useful). The physical or biological problem may come from another class, from the research literature, or from ones own research.

The schedule for the projects is as follows:

  1. Project Proposal (1 page); Due February 23.

  2. Progress Report (1-2 pages); Due March 23.

  3. Final Report; Due April 23

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 matcont6p11) - I'm still working on this.

For more information contact J. Keener, 1-6089

E-mail: keener@math.utah.edu