Numerical determination of an emanating branch of Hopf bifurcation points in a two-parameter problem

Abstract

In a two-parameter problem a branch of Hopf bifurcation points can bifurcate from a branch of simple turning points of the steady state problem, at a point for which the Frechet derivative has a double eigenvalue zero with a one-dimensional nullspace. It is indicated how the origin of a branch of Hopf points can be detected during the continuation of a branch of simple turning points.Further, an augmented system of equations is presented, for which this “origin for Hopf bifurcation” is an isolated solution. If the steady state problem is described by a system of algebraic equations, Newton's method for the solution of the augmented system can be implemented very efficiently. The authors also discuss switching to the branch of Hopf points.Results are given for the one-dimensional “Brusselator” model, a system of four partial differential equations.