Stochastic delay differential equations: Analysis and simulation studies

• Published in: Chaos, solitons and fractals Vol. 165; p. 112819
• Main Authors: Chendur Kumaran, R., Venkatesh, T.G., Swarup, K.S.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2022
• Subjects: Bistable potential; Coloured noise; Fokker–Planck equations; Mean First Passage Time; Path integral; Stochastic delay differential equations; Fokker–Planck equations; Coloured noise; Stochastic delay differential equations; Bistable potential; Path integral; Mean First Passage Time
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2022.112819

Stochastic delay differential equations play an important role in modelling scientific and engineering systems. In this paper the partial differential equation satisfying the time evolution of the probability density function of the state variable governed by the stochastic delay differential equation (SDDE) with additive white noise and coloured noise perturbation is obtained under small time delay and small correlation time approximation using path integral formalism. This partial differential equation reduces to a Fokker–Planck equation for particular linear and non-linear SDDE. Fokker–Planck equation is then solved with reflecting boundary conditions to get the analytical solutions for Stationary Probability Density Function (SPDF). The analytical solutions for SPDF is compared with the SPDF obtained using simulation. Further the Mean First Passage Time (MFPT) of the bistable system is calculated for various values of time delay and noise strength and compared with that of the simulation. The MFPT for the time delayed system for the case of cubic potential driven by Gaussian and Levy noise as well as asymmetric bistable potential driven by a noise and periodic driving force has been investigated. •We consider stochastic delay differential equations (SDDE) with white , colored noise.•A Fokker-Plank equation for SDDE has been derived using the path integral formalism.•Got Stationary probability density function(SPDF) and mean first passage time (MFPT).•Studied symmetric, asymmetric bistable and cubic potential with periodic driving.•SPDF and MFPT got through analysis has been validated against simulation.