Double crises in fuzzy chaotic systems

Abstract

Double crises of chaotic oscillators in the presence of fuzzy uncertainty are studied by means of the fuzzy generalized cell mapping method. A fuzzy chaotic attractor is characterized by its global topology and membership distribution. A fuzzy crisis implies a simultaneous sudden change both in the topology of the chaotic attractor and in its membership distribution. It happens when a fuzzy chaotic attractor collides with a regular or a chaotic saddle. By increasing a small constant bias in the forcing and considering both the fuzzy noise intensity and the bias together as controls, a double crisis vertex is identified in a two-parameter space, where four curves of crisis meet and four distinct crises coincide. The crises involve three different basic sets of fuzzy chaos: a chaotic attractor, a chaotic set on a fractal basin boundary, and a chaotic set in the interior of a basin and disjoint from the attractor. Two examples are presented including two boundary crises and two interior crises for two distinct fuzzy chaotic attractors. Here we concentrate on a fuzzy double crisis vertex involving the coincidence of four distinct crisis events, each of which involves a simultaneous sudden change in a fuzzy chaotic set. It is shown that the dynamics of the fuzzy chaotic systems is extremely rich at the vertex.