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GÓRA, PAWEŁ, BOYARSKY, ABRAHAM and ESLAMI, PEYMAN (2012) METASTABLE SYSTEMS AS RANDOM MAPS. International Journal of Bifurcation and Chaos, 22 (11). p. 1250279. ISSN 0218-1274

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Official URL: http://dx.doi.org/10.1142/S0218127412502793


Metastable dynamical systems were recently studied [González-Tokman et al., 2011] in the framework of one-dimensional piecewise expanding maps on two disjoint invariant sets, each possessing its own ergodic absolutely continuous invariant measure (acim). Under small deterministic perturbations, holes between the two disjoint systems are created, and the two ergodic systems merge into one. The long term dynamics of the newly formed metastable system is defined by the unique acim on the combined ergodic sets. The main result of [González-Tokman et al., 2011] proves that this combined acim can be approximated by a convex combination of the disjoint acims with weights depending on the ratio of the respective measures of the holes. In this note we present an entirely different approach to metastable systems. We consider two piecewise expanding maps: one is the original map, τ1, defined on two disjoint invariant sets of ℝN and the other is a deterministically perturbed version of τ1, τ2, which allows passage between the two disjoint invariant sets of τ1. We model this system by a position dependent random map based on τ1 and τ2, to which we associate position dependent probabilities that reflect the switching between the maps. A typical orbit spends a long time in one of the ergodic sets but eventually switches to the other. Such behavior can be attributed to physical holes as between adjoining billiard tables or more abstract situations where balls can "leap" from one table to the other. Using results for random maps, a result similar to the one-dimensional main result of [González-Tokman et al., 2011] is proved in N dimensions. We also consider holes in more than two invariant sets. A number of examples are presented.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Journal or Publication:International Journal of Bifurcation and Chaos
Digital Object Identifier (DOI):10.1142/S0218127412502793
Keywords:Metastable systems; absolutely continuous invariant measures; random maps
ID Code:976821
Deposited By: Danielle Dennie
Deposited On:29 Jan 2013 13:43
Last Modified:18 Jan 2018 17:43


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