Picture of a black hole

Strathclyde Open Access research that creates ripples...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of research papers by University of Strathclyde researchers, including by Strathclyde physicists involved in observing gravitational waves and black hole mergers as part of the Laser Interferometer Gravitational-Wave Observatory (LIGO) - but also other internationally significant research from the Department of Physics. Discover why Strathclyde's physics research is making ripples...

Strathprints also exposes world leading research from the Faculties of Science, Engineering, Humanities & Social Sciences, and from the Strathclyde Business School.

Discover more...

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

[img]
Preview
PDF
199_AP_CK_AR_XM.pdf - Preprint

Download (426kB) | Preview

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.