Modern methods for studying nonlinear problems ideas, algorithms, software

Dr. Mark Friedman

Department of Mathematical Sciences
University of Alabama in Huntsville

March 30, 2007
202 Madison Hall
3:00 PM (Refreshments at 2:30 in 201 Madison Hall)

Abstract

Nonlinear equations are a basis for scientific and engineering problems. In these problems it is crucial to detect and classify the qualitative changes in the solution structure as the problem parameters vary. The principal approach of numerical bifurcation analysis is based on continuation of solutions to well-defined operator equations. Such computational results give a deeper understanding of the solution behavior, stability, multiplicity, and bifurcations.

1. Introduction to numerical bifurcation analysis.
   1. Basic nonlinear phenomena: classification of bifurcations.
   2. Numerical bifurcation analysis: continuation and related issues. 
2. My work: Software for Bifurcation analysis of large equilibrium systems in Matlaba.
   1. The Continuation of Invariant Subspaces (CIS) algorithm produces a smoothly varying basis for an invariant subspace R(s) of a parameter-dependent matrix A(s). In the case when A(s) is the Jacobian matrix for a system that comes from a spatial discretization of a partial differential equation, it will typically be large and sparse.
   2. Cl_matcont is a user-friendly MATLAB package for the study of dynamical systems and their bifurcations. We incorporate the CIS algorithm into Cl_matcont to extend its functionality to large scale bifurcation computations via subspace reduction. 

Joint work with D. Bindel, J. Demmel, W. Govaerts, J. Hughes, Yu. A. Kuznetsov, I Savin, and W. Qiu.