Numerical stability analysis of linear stochastic delay differential equations using Chebyshev spectral continuous time approximation

In this paper, a numerical approach is developed for the stability analysis of linear stochastic delay differential equations (SDDEs) in the parameter space based on the Chebyshev Spectral Continuous Time Approximation (CSCTA) technique. The CSCTA method is used to approximate an infinite-dimensional linear SDDE as a set of linear stochastic differential equations (SDEs). The mean and mean-square stability concepts are employed for the stochastic stability analysis of the resulting SDE. For this purpose, a set of linear deterministic differential equations for both the first and second moments are obtained using the Ito differential rule. Two examples are provided: a first order SDDE with multiplicative stochastic excitation and a second order SDDE with both additive and multiplicative stochastic excitation. In both examples the stability charts obtained from the proposed approach match those obtained using the stochastic semi-discretization method as described by Elbeyli et al. (Commun Nonlinear Sci Numer Simul 10(1):85–94, 2005). In the first example the stability results obtained from both numerical approaches are found to be less conservative than the Lyapunov-based stability region obtained by Samiei et al. (Int J Dyn Control 1(1):64–80, 2013).