Using approximations to Lyapunov exponents to predict changes in dynamical behaviour in numerical solutions to stochastic delay differential equations

Abstract

This book chapter explores the parameter values at which there are changes in qualitative behaviour of the numerical solutions to parameter-dependent linear stochastic delay differential equations with multiplicative noise. A possible tool in this analysis is the calculation of the approximate local Lyapunov exponents. We show that estimates for the maximal local Lyapunov exponent have predictable distributions dependent upon the parameter values and the fixed step length of the numerical method, and that changes in the qualitative behaviour of the solutions occur at parameter values that depend on the step length.