Bifurcation Structure of Dissipative Solitons

Abstract

In this paper we analyse in detail the structure of the phase space of a reversible dynamical system describing the stationary solutions of a model for a nonlinear optical cavity. We compare our results with the general picture described in [P.D. Woods and A.R. Champneys, Physica D {f 129} (1999) 147 ; P. Coullet, C. Riera and C. Tresser, Phys. Rev. Lett. {f 84} (2000) 3069] and find that the stable and unstable manifolds of homogeneous and pattern solutions present a much higher level of complexity than predicted, including the existence of additional localized solutions and fronts. This extra complexity arises due to homoclinic and heteroclinic intersections of the invariant manifolds of low-amplitude periodic solutions, and to the fact that these periodic solutions together with the high-amplitude ones constitute a one-parameter family generating a closed line on the symmetry plane.