Symmetry and bifurcation of periodic solutions in Neumann boundary value problems

Abstract

We study a vector-valued reaction-diffusion equation with Neumann boundary conditions $(u:[0,\pi ]\to R^2$). Unlike what is observed for scalar equations, where no heteroclinic connections involving periodic solutions occur, we find that steady-state/Hopf and Hopf/Hopf mode interactions produce heteroclinic solutions connecting at least one solution of standing wave type. This is achieved by restricting a problem with periodic boundary conditions and equivariant under $\mathrm{O}(2)$ symmetry to a suitable fixed-point space.

For completeness, we include a description of the solutions for Hopf bifurcation and mode interactions involving Hopf bifurcation, namely, steady-state/Hopf and Hopf/Hopf.