Hopf bifurcation near a double singular point

Abstract

The nonlinear equation f( x, λ, α) = 0, f: X × R 2→ X, where X is a Banach space and f satisfies a Z 2-symmetry relation is considered. Interest centres on a certain type of double singular point, where the solution x is symmetric and f x has a double zero eigenvalue, with one eigenvector symmetric and one antisymmetric. We show that under certain nondegeneracy conditions, which are stated both algebraically and geometrically, there exists a path of Hopf bifurcations or imaginary Hopf bifurcations passing through the double singular point, and for which x is not symmetric except at the double singular point. An easy geometrical test is found to decide which type of phenomenon occurs. A biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths. A pseudo-spectral method was used to obtain numerical results for the Brusselator equations to illustrate the theory.