A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square

Author:

Ashwin Peter,Böhmer Klaus,Mei Zhen

Abstract

We discuss an iterative method for calculating the reduced bifurcation equation of the Liapunov-Schmidt method and its numerical approximation. Using appropriate genericity assumptions (with symmetry), we derive a Taylor series for the reduced equation, where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation. This method is used to calculate reduced equations at Hopf bifurcation of the two-dimensional Brusselator equations on a square with Neumann and Dirichlet boundary conditions. We examine several Hopf bifurcations within the three-parameter space. There are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

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