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On the local dynamics of polynomial difference equations with fading stochastic perturbations

Appleby, John A.D. and Kelly, C. and Mao, Xuerong and Rodkina, A. (2010) On the local dynamics of polynomial difference equations with fading stochastic perturbations. Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematical Analysis, 17. pp. 401-430. ISSN 1201-3390

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Abstract

We examine the stability-instability behaviour of a polynomial difference equa- tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis- cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region.