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 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.

Course Outline

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:

Homework assignments will be posted here and updated regularly.

Assignments 1 and 2

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Notes:

Fold bifurcation and Maple code

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Maple code to compute the flip bifurcation normal form

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Hopf Bifurcation and Normal Form for the van der Pol Equation

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Notes on a center manifold calculation

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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 October 1.

  2. Progress Report (1-2 pages); Due November 1.

  3. Final Report; Due Dec. 8.

An important part of this course is learning to compute bifurcation diagrams using AUTO. A good way to get started with XPPAUT is to run a few of the DEMO problems, although many of these will be describe in class. Also, use the book B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems, SIAM, 2002.

For more information contact J. Keener, 1-6089

E-mail: keener@math.utah.edu