The Bifurcation Of The BVDP System Under The Regulation Of Noise And Time Delay

It is generally believed that random fluctuations exist in any real dynamical system.Therefore,we can observe lots of interesting dynamical phenomena in ran-dom systems,such as resonance,stochastic bifurcation,the mean first passage time and so on.Most of all,stochastic bifurcation,as one of the kinetic responses of systems under noise excitations,has gradually become one of the hottest study subjects for biochemistry,physics,biology and other disciplines,and it is usually marked by a qualitative conversion of the steady-state probability density function(SPDF),for example:a transformation from single distribution to bimodal distri-bution.Moreover,it has been shown that the characteristics of noise can become effective bifurcation parameters in the control system.Therefore,we can use the statistical properties of noise as a strategy to regulate the dynamic behavior of the system,for example:noise intensity.At the same time,due to a certain signal transmission time,there will be the system's transmission rate and memory ca-pacity in a real natural system,so the phenomenon of time delay is accordingly widespread.Therefore,it is more valuable for considering the stochastic nonlinear dynamic system with time delay.There has been a growing interest in birhythmic van der Pol(BVDP)system due to its wide range of applications such as cell rhythm,enzymatic reaction,laser and cardiac dynamics.In this paper,based on the actual research background,we use the knowledge of nonlinear dynamics to reasonably add the noise and time delay to pave the way for further exploration of the extremely abundant dynamic behaviors of BVDP systems disturbed by the outside world.This research mainly explores the bifurcation phenomenon of the random B-VDP system which is controlled by the delay self-control feedback.Through the establishment for a stochastic mathematical model,the stochastic averaging method is used to discuss the steady-state response and bifurcation problem of a stochastic BVDP system with delay self-control feedback.Firstly,the harmonic function is utilized for variables to get the amplitude and phase of the stochastic differential equation.Secondly,the corresponding Ito differential equation and stationary FPK equation are obtained after adding the correction term and averaging.The SPDF which can describe the system amplitude can give a deeper analysis of time delay,noise and other external forces on the system response and bifurcation.Finally,the effectiveness of selected method is verified by comparing the analytical solution with the numerical simulation.Compared with the deterministic case,the results show that both the noise and the delay self-control feedback will have profound impacts on the nonlinear system.Furthermore,the noise intensity,the correlation time,the time delay and the feedback strength will influence the system's limit cycle distri-bution in varying degree,such as large correlation time,feedback strength and time delay will suppress the outer limit cycle,while the larger noise intensity will promote the emergence of the outer limit cycle,which means that the proposed strategy for adjusting the system bifurcation is feasible.And the inner limit cycle will be never destroyed due to noise.At the same time,the strength of the noise,the correlation time,the time delay,and the feedback strength can be seen as the bifurcation pa-rameters.In addition,the validity of the results in theoretical analysis is verified through the use of Monte Carlo simulation of the dynamics for the original system.