Mathematical Science Seminar: Spatial discretisation of dynamical systems

Events

Speaker

Dr P.E. Kloeden
Huazhong University of Science & Technology

Abstract

Numerical simulations of dynamical systems essentially require the system to be replaced by another one on a finite computer number field. This can have some profound effects on the resulting dynamical behaviour such as the collapse of chaotic behaviour. Spatial discretisation will be discussed in this talk for more general discretised domains. The issue is how to construct such a discretisation to ensure that certain dynamical features are preserved. It will be shown that invariant measures are the most appropriate properties the compare. The talk will focus on discrete time dynamical systems, i.e., difference equations which involve possible discontinuous functions. Since these functions need not map the spatial grand onto itself, here are two ways it can be approximated, by a set-valued mapping or by a Markov chain. It will be shown that the semi-invariant measures (which reduce to invariant measures for continuous mappings) are the ones that can be approximated and that the set-valued and stochastic approaches give the same results. The proof uses interval stochastic matrices.