DISCONTINUOUS BIFURCATIONS OF CHAOTIC ATTRACTORS IN FORCED OSCILLATORS BY GENERALIZED CELL MAPPING DIGRAPH (GCMD) METHOD

Abstract

Discontinuous bifurcations of chaotic attractors are investigated by means of Generalized Cell Mapping Digraph (GCMD) method. The two categories of discontinuous bifurcations of chaotic attractors are found in typical systems described by dissipative ordinary differential equations, namely catastrophic and explosive bifurcations. For the case of a catastrophe, the two different types, namely a regular saddle and a chaotic saddle catastrophe, are studied, in particular, the subtle distinction between them can be explicitly observed in phase portrait. We show that a regular saddle catastrophe results from a collision between a chaotic attractor and a regular saddle on its smooth basin boundary. In such a case the chaotic attractor is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the original chaotic attractor in phase space. We demonstrate that a chaotic saddle catastrophe results from a collision between a chaotic attractor and a chaotic saddle in its fractal basin boundary. In such a case the chaotic attractor is suddenly destroyed as the parameter passes through a critical value, simultaneously the chaotic saddle also undergoes an abrupt enlargement in its size, namely, the chaotic attractor is converted into a new incremental portion of the chaotic saddle after the collision. For the case of an explosion, a chaotic saddle explosion is studied, in which there is a sudden increase in the size of a chaotic attractor as the parameter passes through a critical value. We demonstrate that at the chaotic saddle explosion the chaotic attractor collides with a chaotic saddle already existent when the explosion occurs. This chaotic saddle is an invariant and nonattracting set. We also investigate the origin and evolution of the chaotic saddle. A locally refining process of Generalized Cell Mapping (GCM) method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space.