Hopf bifurcation near a double singular point with Z2-symmetry and X0-breaking

This paper deals with nonlinear equations f(x,λ,α)=0 and the corresponding ODEs xt=f(x,λ,α) satisfying f(0,λ,α)=0 and a Z2-symmetry. In particular, we are interested in Hopf points, which indicate the bifurcation of periodic solutions of xt=f(x,λ,α) from (steady-state) solutions of f(x,λ,α)=0. It is shown that under suitable nondegeneracy conditions, there bifurcate two paths of Hopf points from a double singular point, where x=0 and fx(0,λ,α) has a double zero eigenvalue with one eigenvector symmetric and one anti-symmetric. This result gives a new example of finding Hopf points through local singular points. Our main tools for analysis are some extended systems, which also provide easily implemented algorithms for the numerical computation of the bifurcating Hopf points. A supporting numerical example for a Brusselator model is also presented.