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On the approximation of complicated dynamical behavior. (English) Zbl 0916.58021

This paper deals with a numerical method for the approximation of physically relevant invariant measures based on a discretization of the Frobenius-Perron operators. Since invariant measures under considerations are fixed points of this operator, the authors approximate it by a Galerkin projection and then compute eigenvectors of the discretized operator corresponding to the eigenvalue 1. Based on spectral properties of the Perron-Frobenius operator, the authors determine the dynamical behaviour of the system. Here two essentially different mathematical concepts are used: The authors combine classical convergence results for finite-dimensional approximation of compact operators with results from ergodic theory concerning the approximation of SRB-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples.

MSC:

37A10 Dynamical systems involving one-parameter continuous families of measure-preserving transformations
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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