Noise-induced changes to the behaviour of semi-implicit Euler methods for stochastic delay differential equations undergoing bifurcation

Authors:
Neville J. Ford;Stewart J. Norton
Affiliations:
Mathematics Department, University of Chester, Parkgate Road, Chester, CH1 4BJ, UK;Mathematics Department, University of Chester, Parkgate Road, Chester, CH1 4BJ, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

We are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.