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

• Published in: Dynamics and stability of systems 1999-06, Vol.14 (2), p.115-128
• Main Authors: Broomhead, David, Hadjiloucas, Demetris, Nicol, Matthew
• Format: Article
• Language: English
• ISSN: 0268-1110; 1468-9367; 1465-3389; 1468-9375
• DOI: 10.1080/026811199282029

This paper is concerned with the stability properties of skew-products T (x,y) = (f(x), g(x,y)) in which (f,X,mu) is an ergodic map of a compact metric space X and g: Xx Rn Rn is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. We study the orbit stability and stability of mixing of T(x,y) = (f(x), g(x,y)) under deterministic and random perturbation of g. We show that such systems are stable in the sense that for any > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance of each other except for a fraction of the time. 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'.