Double crises in fuzzy chaotic systems

Double crises of chaotic oscillators in the presence of fuzzy uncertainty are studied by means of the fuzzy generalized cellmappingmethod.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 happenswhen 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 ofwhich 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.