Quasi-Periodic Waves Along a Pulsating Propagating Front in a Reaction-Diffusion System

We consider a system of reaction-diffusion equations for which there exists a solution with a uniformly propagating front, and from which solutions describing a pulsating propagating front with periodic travelling and standing waves along the front, bifurcate upercritically. We construct a pulsating propagating solution branch, with quasi-periodic travelling waves along its front, which connects the periodic travelling and standing wave branches. Thus the quasi-periodic branch arises as a secondary bifurcation from the uniformly propagating solution. Our construction involves a perturbation analysis in the neighborhood of a certain degenerate point in parameter space, which we identify. The stability of the various solution branches is investigated.