FURTHER STUDY OF INTERIOR CRISES IN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Abstract

Crises in systems of ordinary differential equations are investigated by means o f Generalized Cell Mapping Digraph (GCMD) method. We show that a boundary crisis results from a collision between a chaotic attractor and a periodic saddle on i ts basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the origin al chaotic attractor in phase space. We focus here on a sudden change in the siz e of a chaotic attractor, namely an interior crisis. We demonstrate that at an i nterior crisis the chaotic attractor collides with a chaotic saddle within its b asin of attraction. This chaotic saddle is an invariant and nonattracting set an d resembles the new portion of the larger chaotic attractor just after the inter ior crisis. We also investigate the origin and evolution of the chaotic saddle. The local refining procedures of persistent and transient self-cycling sets are given.