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Generalized Cell Mapping for Randomly Perturbed Dynamical Systems
Author(s) -
Fischer J.,
Kreuzer E.
Publication year - 2001
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/1521-4001(200111)81:11<769::aid-zamm769>3.0.co;2-9
Subject(s) - randomness , perturbation (astronomy) , attractor , dynamical systems theory , statistical physics , operator (biology) , mathematics , perturbation theory (quantum mechanics) , physics , mathematical analysis , quantum mechanics , statistics , biochemistry , chemistry , repressor , transcription factor , gene
Abstract We present a method to analyze dynamical systems undergoing random perturbations based on the cell mapping approach. Analytical expressions are derived for the transition probabilities from the evolution operator of the system. Thus there is no need for simulation of randomness and the numerical approximations are safe, i.e., we approximate attractors and their basins from within and give lower bounds for exit times. For additively perturbed systems the transition probabilities can be expressed in terms of the transition probabilities of the unperturbed system and the properties of the perturbation. The numerical details concerning the perturbation terms are discussed and their application is shown with an example.