Spline Continuation of Equilibrium Solutions to Parameter Dependent Dynamical Systems and its Application to Numerical Bifurcation Analysis

Mr. Jeremy Hughes

Department of Mathematical Sciences
UAHuntsville

January 15, 2010
218 Shelby Center
3:00 (Refreshements at 2:30)

Abstract

Nonlinear parameter dependent systems of the form ∂u / ∂t = f(u, a) are of interest in many disciplines. Bifurcation analysis of these systems provides an important understanding of the change in behavior of these solutions with respect to the dependent parameters. Numerical continuation of equilibrium solutions is a tool that allows for this analysis. Traditional methods of continuation require many iterations of Newton based correction steps. Spline continuation allows accurate calculation of these solutions and their bifurcations while avoiding many costly Newton steps. It also avoids potentially problematic effects of more traditional techniques when applied to large systems. In addition, this method opens possibilities for developing accurate, super-linear convergent methods for location of higher order singularities.