Approximation of Attractors by a Variable Time-Stepping Algorithm for Runge-Kutta Methods

Abstract

Numerical simulations are frequently used in applying continuous dynamical systems associated with systems of ordinary differential equations (o.d.e.'s) to concrete problems as fluid flow, chemical reactions, economic markets, etc. From dynamical point of view numerical methods are associated with discrete dynamical systems. In numerical analysis of initial-value problems the convergence is of fundamental importance. It is natural to ask, for example if the limit sets from the discrete case approximate the corresponding limit sets from the continuous case. These aspects are very important in the case of a dynamical system depending on many parameters which presents an alternance of regular, stationary, periodic or chaotic regimes according to the parameter variation. The theoretical treatments of attractors alternance are very difficult and they are accompanied by or reduced to numerical computations. In the field of numerical analysis in dynamical systems is included our paper. We prove that a variable time-stepping algorithm for Runge-Kutta methods generates a discrete dynamical system. We show that for the invariant sets, in particular attractors, of a dynamical system associated with a autonomous system of o.d.e.'s invariant sets of the numerical method correspond. Key­Words: Discrete and continuous dynamical systems, autonomous systems of ordinary differential equations, Runge-Kutta method, variable time-stepping algorithm, orbits, positively invariant sets, attractors.