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. 401430. ISSN 12013390

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Abstract
We examine the stabilityinstability behaviour of a polynomial difference equa tion with stateindependent, 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 statedependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the EulerMaruyama dis cretisation of an It^otype 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.
Item type:  Article 

ID code:  29102 
Keywords:  nonlinear stochastic differencial equation, stability, instability, Probabilities. Mathematical statistics, Discrete Mathematics and Combinatorics, Analysis, Applied Mathematics 
Subjects:  Science > Mathematics > Probabilities. Mathematical statistics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Pure Administrator 
Date Deposited:  22 Mar 2011 12:17 
Last modified:  05 May 2016 14:03 
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URI:  http://strathprints.strath.ac.uk/id/eprint/29102 
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