Noise-induced dynamics of time-delayed stochastic systems

The noise-induced dynamical behaviors and phenomena in time-delayed stochastic systems have been important topics in many scientific fields, such as biology, physics, neural networks and engineering over the past few decades. In this chapter, we first give a brief introduction to several key concepts and approaches, such as the method of multiple scales, the stochastic averaging method, the delayed Fokker-Planck equations and the two-state model, related to time-delayed stochastic systems. Then, we present a review on some of our recent theoretical and numerical results about the dynamical behaviors and the noise-induced resonances in time-delayed stochastic systems. For instance, the principal resonance and the moment stability of stochastic systems with delayed feedback control are studied under the excitations of narrow-band random noise and Gaussian white noise, respectively. The coherence resonance can be demonstrated through the power-spectrum of a delayed bistable system driven by additive and multiplicative Gaussian white noises. When a weak harmonic signal is applied to this system, the expression of linear spectrum amplification is derived to qualify the stochastic resonance. Finally, we present an overview of the established results of noise-induced dynamics in time-delayed stochastic systems, future challenges and expected prospects.