Numerical investigations of bifurcations in stochastic delay differential equations

Stewart J. Norton, Neville J. Ford

Department of Mathematics, Chester College, Parkgate Road, Chester, CH1 4BJ, UK

s.norton@chester.ac.uk

Contributed talk

We are interested in changes in qualitative behaviour of solutions to stochastic delay differential equations. As a starting point we consider the stochastic delay logistic equation with multiplicative noise of the form

$$dY(t)=\lambda Y(t)[1-Y(t-1)] dt + \mu Y(t-1) dW(t)$$

where $W(t)$ is white noise. When $\mu=0$ this equation reduces to the familiar deterministic delay logistic equation and it is known to have a Hopf-bifurcation when the parameter $\lambda= \pi/2$.

For simple numerical schemes with fixed step lengths we know (see [1,2,3]) that the actual bifurcation value of the parameter is perturbed by an amount that depends both on the choice of numerical method and on the step length $h>0$.

We consider now the corresponding question for the case where $\mu \neq 0$. Here the stochastic term may be expected to influence the qualitative behaviour of solution trajectories. We discuss the practical implications of using numerical methods to detect bifurcation values of $\lambda$ and present some numerical results.