Hopf bifurcation and symmetry: travelling and standing waves on the circle

Synopsis In this paper we consider Hopf bifurcation in the presence of O (2) symmetry. The system of reaction diffusion equations u t , = D (µ) u xx + f (µ, u ) provided with periodic boundary conditions may serve as a model problem. We prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability (with asymptotic phase) of these periodic solutions, giving explicit formulae. Finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation.