Some Relative Problems And Its Applications In Non-linear Delay Fokker-Planck Equations

There are various examples of stochastic systems with time-delay feedback in the inanimate and animate world. In many cases, nonlinear delay Fokker-Planck equations in according with stochastic delay differential equations (SDDE) are used to study the dynamical behaviors of stochastic delay dynamical system. Therefore, more researchers are interested in FPK equations, which describe the evolution of transition probability of stochastic system.This article mainly discusses two aspects of Fokker-Planck equation: on the one hand, the SDDE only have been studied, where delay is considered in the drift coefficient, and is ignored in the diffused coefficient. However, in general stochastic delay dynamical systems, the drift coefficient and diffused coefficient both have delay. Hence, we deduce the delay FPK equation in according with the general nonlinear stochastic delay dynamical systems using perturbation theory. And then, we detain the approximate stationary solution of nonlinear delay FPK equation. On the other hand, the multivariate delay FPK equation in according with multivariate nonlinear stochastic delay dynamical systems, mostly have not been investigated deeply. We deduce the delay FPK equation of multivariate nonlinear stochastic delay dynamical systems using variational theory. Then, the stochastic phase model of neuronal oscillator population with delay coupling has been investigated using the multivariate nonlinear delay FPK equation. Finally, numerical simulation has shown that a number density, which is used to describe the activities of neuronal population, is dynamical evolution with time. Numerical simulation also has shown that the time-delay has significantly impact the synchronous firing patterns in neuronal population.