Modern methods for studying nonlinear problems: numerical bifurcation analysis

Dr. Mark Friedman

Department of Mathematical Sciences
University of Alabama in Huntsville

November 2, 2001

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.

I. Introduction to numerical bifurcation analysis.

1. Basic nonlinear phenomena: classification of bifurcations.
2. Examples of bifurcation analysis, a model optimization problem.
3. Numerical bifurcation analysis: continuation and related issues.
4. Bifurcation software under development.

II. My work (in progress): Continuation of Invariant Subspaces for large bifurcations problems.

The Continuation of Invariant Subspaces (CIS) algorithm [Demmel, Dieci, Friedman 2001] and [Dieci, Friedman 2000], produces a smooth orthogonal similarity transformation to block triangular form of a parameter dependent matrix A(s). We extend the CIS algorithm to the case of large sparse matrices using an approach in [Brandts 2001] and develop reliable procedures for updating the invariant subspace of interest when the eigenvalue set from it coalesces and/or overlaps with the eigenvalue set from its complement during the continuation process. This allows for a robust subspace reduction to be used in bifurcation computations.