Point Mapping under Cell Reference - A Two Scaled Numerical Method for Global Analysis

Abstract

In this chapter, a two-scaled method for global analysis is presented and is dedicated to the 90th birthday of Professor C.S. Hsu. First, the existing numerical global analysis methods are classified into two classes according to their different ways in treating the state space and in detecting and depicting the global structure of a dynamical system. One class of methods treats the state space as continuum and uses points to describe the invariant sets. The other regards the state space as discrete one and uses subsets (cells) to depict the invariant sets. These two classes of methods for global analysis are all single-scaled methods. On the basis of this understanding, it is then pointed out that the previously proposed method of point mapping under cell reference, or PMUCR in short, has laid a framework for the development of a two-scaled numerical method for global analysis that can, to a large extent, take the advantages of both classes of single-scaled methods but will release the difficulties induced by the disadvantages of them. Moreover, the two-scaled method makes it feasible to conduct global analysis of high-dimensional nonlinear systems. The basic ideas and the main implementation steps as well as numerical techniques for the capability enhancement of the two-scaled method for global analysis are elaborated in the chapter. Examples are presented to illustrate the basic ideas and to demonstrate the capabilities of the proposed method.