Summary.
Computer simulation of dynamical systems involves a phase space which is the finite set of machine arithmetic. Rounding state values of the continuous system to this grid yields a spatially discrete dynamical system, often with different dynamical behaviour. Discretization of an invertible smooth system gives a system with set-valued negative semitrajectories. As the grid is refined, asymptotic behaviour of the semitrajectories follows probabilistic laws which correspond to a set-valued Markov chain, whose transition probabilities can be explicitly calculated. The results are illustrated for two-dimensional dynamical systems obtained by discretization of fractional linear transformations of the unit disc in the complex plane.
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Received January 9, 2001; accepted January 2, 2002 Online publication April 8, 2002 Communicated by E. Doedel
Communicated by E. Doedel
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Diamond, P., Vladimirov, I. Set-Valued Markov Chains and Negative Semitrajectories of Discretized Dynamical Systems. J. Nonlinear Sci. 12, 113–141 (2002). https://doi.org/10.1007/s00332-001-0450-4
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DOI: https://doi.org/10.1007/s00332-001-0450-4