GENERALIZED CELL MAPPING DIGRAPH METHOD FORGLOBAL ANALYSIS
全局分析的广义胞映射图论方法

In this paper, according to Hsu's idea that posets and digraphs are introduced intogeneralized cell mapping, a generalized cell mapping digraph method is presented by using thetheory of generalized cell mapping discretizing the continuous state space into the cell state spaceand the theories of set and digraph to achieve the task of global analysis of nonlinear dynamicalsystems. In the cell state space, we make the correspondence between generalized cell mappingdynamical systems and digraphs. The demonstrations of the two theorems of existence of self-cycling set and persistent self-cycling set are given. State cells are classified, and self-cycling sets,persistent self-cycling sets and transient self-cycling sets are defined. The persistent self-cycling setsrepresent the attractors of the systems, while the transient self-cycling sets are usually associatedwith the unstable fixed points and periodic solutions. Digraphs are introduced into generalizedcell mapping systems by defining binary relations in the cell state space, thus, the rich theoriesand the very powerful algorithms in the field of graphs and digraphs are adopted for the purposeof determing the global evolution properties of the systems. After all the self-cycling sets arecondensed by using digraph condensation method, the number of the state cells involved can beefficiently decreased in the global transient analysis, and a topological sorting of the global transientstate cells can be efficiently achieved by digraph algorithms, simultaneously, after transient cellsare classified to transient cell sets according to the number of the domiciles that they have, domainsof attraction and boundary regions can also be determined. Based on the different treatments,the global properties can be divided into qualitative (topological) and quantitative properties. Inthe whole analysis of the qualitative properties, only Boolean operations are used. The Booleanoperations are absolutely accurate, reliable, and time-saving. It is believed that the generalized cellmapping digraph method offers us a new way to examine the complicated behavior of nonlineardynamical systems.