A DISSIPATIVE MAP OF THE PLANE--A MODEL FOR OPTICAL BISTABILITY (DYNAMICAL SYSTEMS).

Abstract

We analyze a dissipative map of the plane. The map was initially defined by Ikeda as a model for bistable behavior in an optical ring cavity. Our analysis is based upon an examination of attracting sets and basins of attraction. The primary tools utilized in the analysis are stable and unstable manifolds of fixed and periodic saddle points. These manifolds determine boundaries of basins of attraction, and the extent and evolution of attracting sets. We perform extensive numerical iterations of the map with a central focus on sudden changes in the topological nature of attractors and basins. Our analysis concentrates on the destruction of the lower branch attractor as a prominent example of attractor/basin interaction. This involves an examination of a possible link between two fixed points L and M, namely the heteroclinic connection Wᵘ(L) ∩ Wˢ(M) ≠ 0. We use two different methods to approach this question. Although the Ikeda map is used as the working model throughout, both of the techniques apply to a more general class of dissipative maps satisfying certain hypotheses. The first of these techniques analyzes Wˢ(M) when Wᵘ(M) ∩ Wˢ(M) ≠ 0, with the result that Wˢ(M) is found to invade some minimum limiting region for Wᵘ(M) ∩ Wˢ(M) ≠ 0 arbitrarily close to tangency. The second approach is more topological in nature. We define a mesh of subregions to bridge the spatial gap between the points L and M, and concentrate on the occurrence of Wᵘ(L) ∩ Wˢ(M) ≠ 0 (destruction of the attractor). The first main result is a necessary condition for the heteroclinic connection in terms of the behavior of the map on these subregions. The second result is a sequence of sufficient conditions for this link. There remains a gap between these two conditions, and in the final sections we present numerical investigations indicating that the concept of intersection links between subregions is useful to resolve cases near the boundary of the destruction region.