Attraction rates, robustness and discretization of attractors

L. Grüne: Attraction rates, robustness and discretization of attractors
SIAM Journal on Numerical Analysis 41, 2096 - 2113, 2003

DOI: 10.1137/S003614290139411X
Keywords: numerical one-step approximation; attractor; attraction rate; robustness; convergence; discrete time dynamical systems
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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.

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