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On the oscillatory behaviour of stochastic delay equations


Kelly, Cónall;

Institution: Dublin City University. School of Mathematical Sciences
Subject Keywords: Differential equations; Mathematics; Stochastic analysis; Real-world processes; perturbations; feedback noise;

This is an investigation of the causes of oscillatory behaviour in solutions of stochastic delay differential equations. Delay equations are used to study phenomena in which some part of the history of the system determines its evolution. Real-world interactions are often characterised by inefficiency and such equations are therefore widely used in applications. Real-world processes are also subject to interference in the form of random external perturbations or feedback noise. This interference can have a dramatic effect on the qualitative behaviour of these processes and so should be included in the mathematical analysis.

Specifically, we consider the roles played by delayed feedback and noise perturbation in the onset of oscillation around an equilibrium solution. To this end, we consider a nonlinear equation with fixed delay, and a linear equation with asymptotically vanishing delay. Where necessary, results guaranteeing the global existence and uniqueness of solutions are presented. To facilitate the analysis of the linear equation, we present two difference schemes that are designed to mimic the oscillatory behaviour of its solutions. The first, a discretisation on a uniform mesh, is unsuccessful. We determine the reasons for this failure, and design a successful scheme based on this analysis.

These choices allow the empirical manipulation of the relative involvement of the delay in the behaviour of solutions. In this way, and by comparison with the known qualitative behaviour of the corresponding deterministic delay differential equation, a picture of the mechanisms underlying oscillatory behaviour can be developed.



Suggested citation:

Kelly, Cónall; . () On the oscillatory behaviour of stochastic delay equations [Online]. Available from: [Accessed: 16th September 2019].


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