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

Gábor Domokos, Domokos Szász

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We consider discretizations fN of expanding maps f : I → 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 × N linear system.

Original languageEnglish
Pages (from-to)859-876
Number of pages18
JournalDiscrete and Continuous Dynamical Systems
Volume9
Issue number4
DOIs
Publication statusPublished - Jul 2003

Keywords

  • Computation
  • Digital arithmetic
  • Ergodic maps
  • Expanding maps
  • Minimal perturbation
  • Random perturbation
  • Ulam's scheme

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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