Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds

Author:

Guder Rabbijah1,Kreuzer Edwin1

Affiliation:

1. Technical University of Hamburg-Harburg, Ocean Engineering Section II, Eissendorfer Str. 42, 21073 Hamburg, Germany

Abstract

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modelling and Simulation,Engineering (miscellaneous)

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