Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations

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June 2006, 5(2): 367-382. doi: 10.3934/cpaa.2006.5.367

Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations

Neville J. Ford ^1, and Stewart J. Norton ^1,

1.	Department of Mathematics, University of Chester, Parkgate Road, Chester CH1 4BJ, United Kingdom, United Kingdom

Received February 2005 Revised June 2005 Published March 2006

Abstract
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We consider numerical approximations to parameter-dependent linear and logistic stochastic delay differential equations with multiplicative noise. The aim of the investigation is to explore the parameter values at which there are changes in qualitative behaviour of the solutions. One may use a phenomenological approach but a more analytical approach would be attractive. A possible tool in this analysis is the calculation of the approximate local Lyapunov exponents. In this paper we show that the phenomenological approach can be used effectively to estimate bifurcation parameters for deterministic linear equations but one needs to use the dynamical approach for stochastic equations.

Keywords: bifurcations, Stochastic delay differential equations, Lyapunov exponents..

Mathematics Subject Classification: Primary: 34K18, 34K28; Secondary: 34K50, 65C30, 65P3.

Citation: Neville J. Ford, Stewart J. Norton. Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 367-382. doi: 10.3934/cpaa.2006.5.367

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