A Generalized Theory of Cell-to-Cell Mapping for Nonlinear Dynamical Systems

Abstract

The simple theory of cell mapping and the associated algorithm presented in [1, 2] have been found to form a very effective tool for the global analysis of nonlinear systems. In this paper we generalize the theory by allowing the mapping of a cell to have multiple image cells with appropriate individual mapping probabilities. This generalized theory will be able to deal with very fine and complicated global behavior patterns, if they exist, in a more attractive way without having to utilize extremely small cell sizes. It is found that such a generalized cell mapping can be identified with a Markov chain and the well-developed mathematical theory of Markov chains can be immediately applied. Similar to the simple theory of [1], the generalized cell mapping theory is also eminently suited as a theoretic base for computer alogorithms which will be needed when dealing with systems involving a very large number of cells.