A Numerical Liapunov-Schmidt Method with Applications to Hopf Bifurcation on a Square
Peter Ashwin, Klaus Böhmer, Zhen Mei · Mathematics of Computation · 1995
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.