July  2003, 9(4): 859-876. doi: 10.3934/dcds.2003.9.859

Ulam's scheme revisited: digital modeling of chaotic attractors via micro-perturbations

1. 

Center for Applied Mathematics and Computational Physics, Budapest University of Technology and Economics, H-1111 Budapest, Műegyetem rkp.3, Hungary, Hungary

Received  May 2002 Revised  December 2002 Published  April 2003

We consider discretizations $f_N$ of expanding maps $f:I \to I$ in the strict sense: i.e. we assume that the only information available on the map is a finite set of integers. Using this definition for computability, we show that by adding a random perturbation of order $1/N$, the invariant measure corresponding to $f$ can be approximated and we can also give estimates of the error term. We prove that the randomized discrete scheme is equivalent to Ulam's scheme applied to the polygonal approximation of $f$, thus providing a new interpretation of Ulam's scheme. We also compare the efficiency of the randomized iterative scheme to the direct solution of the $N \times N$ linear system.
Citation: Gábor Domokos, Domokos Szász. Ulam's scheme revisited: digital modeling of chaotic attractors via micro-perturbations. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 859-876. doi: 10.3934/dcds.2003.9.859
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