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

Gábor Domokos; Domokos Szász

doi:10.3934/dcds.2003.9.859

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2003, Volume 9, Issue 4: 859-876. Doi: 10.3934/dcds.2003.9.859

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

Gábor Domokos^1, and
Domokos Szász^1,

1.
Center for Applied Mathematics and Computational Physics, Budapest University of Technology and Economics, H-1111 Budapest, Műegyetem rkp.3

Received: April 30, 2002

Revised: November 30, 2002

Published: June 2003

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Abstract

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.

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Mathematics Subject Classification: 37M25, 68W20, 37E05.

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