Course Announcement
Math 6740 - Bifurcation Theory
Time: T,TH 09:10-10:30 am
Place: AEB 360
Because of my travel schedule, we will not meet on 1/17, 19.
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, 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.
D. G. Schaeffer and J. W. Cain, Ordinary Differential Equations,Basics and Beyond, Springer, 2016.
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.)
- Introduction: Continuation and homotopy, What is a bifurcation?, the implicit function
theorem
- Examples of bifurcations; algebraic equations, discrete maps, Hopf. Use of XPPAUT and/or MATCONT to compute bifurcation curves.
- Steady state bifurcations; Sturm Liouville problems, Turing, global
continuation theorems
- Bifurcation of dynamical systems (Kuznetsov)
- Bifurcation in PDE's; Ginzberg-Landau equations, Turing revisited
- 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
Fold Normal Form
MATLAB files:
Matlab codes will be posted here:
XPP files:
XPPAUT files will be posted here:
Euler column
Bratu's BVP
Maple Codes:
Maple codes are posted here:
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:
- Project Proposal (1 page); Due February 23.
- Progress Report (1-2 pages); Due March 23.
- Final Report; Due April 23
An important part of this course is learning how to compute bifurcation
diagramsusing 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 matcont5p2 - I'm still working on this.
For more information contact J. Keener, 1-6089
E-mail: keener@math.utah.edu