Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points

Abstract

We show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations. We extend the approach by Werner (1988) by presenting methods for the detection of bifurcation points and the computation of (multiple) Hopf points. The essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group Γ. The group theoretical tool is an a priori knowledge of the interaction of certain subgroups Σ 0 and Σ of Γ at (possibly multiple) steady state or Hopf bifurcation points (minimal Σ 0-Σ-breaking bifurcation). We introduce a bifurcation graph which shows graphically this a priori information — its edges represent possible symmetry breaking bifurcations. Our analysis follows the lines of Golubitsky, Stewart and Schaeffer (1988) but it is aimed to numerical applications. We have chosen a 4-box-Brusselator model in order to explain our notions and ideas and to discuss the numerical procedure.