Crises and chaotic transients studied by the generalized cell mapping digraph method

Abstract

In this Letter, a generalized cell mapping digraph method is presented on the basis of a correspondence between the generalized cell mapping of dynamical systems and digraphs, and this correspondence is theoretically proved on the basis of set theory in the cell state space. State cells are classified afresh, 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 the 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. A boundary crisis is studied by means of the generalized cell mapping digraph method. Attractors, basins, basin boundaries and unstable solutions are obtained once through a global analysis at low computational cost. Moreover, the approach of a chaotic attractor to an unstable periodic orbit at its basin boundary before a boundary crisis, the collision of the chaotic attractor with the unstable periodic orbit when the crisis occurs, and a chaotic transient after the crisis, are explicitly shown. The limiting probability distribution of the chaotic attractor is calculated.