Attraction Rates, Robustness, and Discretization of Attractors

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Grüne, Lars:
Attraction Rates, Robustness, and Discretization of Attractors.
Bayreuth , 2001

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Abstract

We investigate necessary and sufficient conditions for the convergence of attractors of discrete time dynamical systems induced by numerical one-step approximations of ordinary differential equations (ODEs) to an attractor for the approximated ODE. We show that both the existence of uniform attraction rates (i.e., uniform speed of convergence towards the attractors) and uniform robustness with respect to perturbations of the numerical attractors are necessary and sufficient for this convergence property. In addition, we can conclude estimates for the rate of convergence in the Hausdorff metric.