Numerical Modeling of Toroidal Dynamical Systems with Invariant Lebesgue Measure
Phil Diamond
Department of Mathematics, University of Queensland,
Brisbane, Queensland 4072, Australia
Peter Kloeden
Department of Computing and Mathematics, Deakin University,
Geelong, Victoria 3217, Australia
Alexei Pokrovskii
Permanent address: Institute of Information Transmission Problems, Russian Academy of Science, Moscow
Department of Mathematics, University of Queensland,
Brisbane, Queensland 4072, Australia
Abstract
Computer simulations of dynamical systems can contain both discretizations, in which finite machine arithmetic replaces continuum state spaces, and realizations, in which a continuous system is replaced by some approximation such as a computational method. In some circumstances, complicated theoretical behavior collapses to trivial and degenerate behavior. In others, the computation may bear little resemblance to the underlying theoretical model. We show here that systems that preserve Lebesgue measure do not suffer such undesirable features. In fact, they can be approximated by such simple maps as permutations of computer state space.
