||Ling Hong and J. X. Xu (2012) “Crises in Chaotic Systems” In Global Analysis of Nonlinear Dynamics, Edited by Jian-Qiao Sun and Albert C. J. Luo. Springer, New York, Pages 75-106.
Abstract: In this chapter, following the pioneering work of Professor C. S. Hsu (1995), a generalized cell mapping digraph (GCMD) method is presented on the basis of a correspondence between generalized cell mapping dynamical systems and digraphs. The correspondence is theoretically proved with the help of set theory in the cell state space. State cells are afresh classified, and self-cycling sets, persistent self-cycling sets, and transient self-cycling sets are defined. The algorithms of digraphs are adopted for the purpose of determining the global evolution properties of the systems. After all the self-cycling sets are condensed by using digraphic condensation method, a topological sorting of the global transient state cells can be efficiently achieved. Based on the different treatments, the global properties can be divided into qualitative and quantitative properties. In the analysis of the qualitative properties, only Boolean operations are used. As a result, the complicated behavior of nonlinear dynamical systems can be efficiently studied in a new way. Here crises in chaotic systems are investigated by means of the GCMD method. Three examples include a boundary crisis, a double crisis at a vertex of two-parameter space, and a hyperchaotic crisis in a high-dimensional chaotic system with two positive Lyapunov exponents.