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Random and deterministic perturbation of a class of skew-product systems

Broomhead, David S., Hadjiloucas, Demetris and Nicol, Matthew (1999) Random and deterministic perturbation of a class of skew-product systems Dynamics and Stability of Systems, 14 (2). pp. 115-128.


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<p>This paper is concerned with the stability properties of skew-products <i>T</i> (,i>x</i>,<i>y</i>) = (<i>f</i>(<i>x</i>), <i>g</i>(<i>x</i>,<i>y</i>)) in which (<i>f</i>,<i>X</i>,<i>mu</i>) is an ergodic map of a compact metric space <i>X</i> and <i>g</i>: <i>X</i> x <b>R</b><i><sup>n</i></sup> → <b>R</b><i><sup>n</i></sup> is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. </p> <p>We study the orbit stability and stability of mixing of <i>T</i> (,i>x</i>,<i>y</i>) = (<i>f</i>(<i>x</i>), <i>g</i>(<i>x</i>,<i>y</i>)) under deterministic and random perturbation of <i>g</i>. We show that such systems are stable in the sense that for any <i>epsilon</i> > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance <i>epsilon</i> of each other except for a fraction <i>epsilon</i> of the time. </p> <p>Furthermore, we show that the invariant measure for the perturbed system is continuous (in the Hutchinson metric) as a function of the size of the perturbation to g (Lipschitz topology) and the noise distribution. Our results have applications to the stability of Iterated Function Systems which 'contract on average'.</p>

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Broomhead, David S.
Hadjiloucas, Demetris
Nicol, Matthew
Date : 20 August 1999
DOI : 10.1080/026811199282029
Additional Information : This is a pre-copy-editing, author-prepared, <b>peer-reviewed</b> PDF of an article published in <i>Dynamics 115-128. Click <a href=";id=doi:10.1080/026811199282029" >here</a> to access the publisher's version. © 1999 Taylor and Francis Group.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:40
Last Modified : 31 Oct 2017 14:01

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