Time-delayed Feedback Control Of Tri-stable Van Der Pol System Under Noise Excitation

In recent years,more and more attention has been paid to the dynamic behavior and control of multistable system under noise excitation.Noise excitation can cause a series of dynamic phenomena,such as coherence resonance,stochastic P-bifurcation,first passage and so on.To solve these problems,several methods are introduced,such as time-delayed feedback control,fractional derivative and so on.Because of the lack of general design method,the influence of control parameters is usually discussed to provide reference for parameters selection.So far,although the theoretical research in this field has achieved rich results,the experimental research is still less.In addition,the dynamic behavior of the system studied is relatively simple,the control of the system with more complex dynamic behavior has less research results at present.Feedback control is a common control method,Time delay is inevitable in the process of feedback control.The time delay in the feedback control will not only influence the effects of the control,but also change the stability of the system.Therefore,it is of great theoretical and engineering significance to study the effects of time-delayed feedback on the dynamic behavior of the controlled system.In this paper,the effects of time-delayed feedback control on the response of tri-stable van der Pol system are studied from three aspects: theoretical analysis,numerical simulation and experiment(1)The deterministic averaging method is used to solve the theoretical approximate solution of the tri-stable van der Pol system with time-delayed feedback.The effects of time-delayed feedback on the bifurcation behavior of the deterministic system are discussed.Based on the stochastic averaging method and singularity theory,the transition set of stochastic system with time-delayed feedback is solved.The effects of time-delay and feedback strength on the stationary probability density of stochastic system are discussed respectively.It is found that both the feedback strength and the time delay can influence the stationary response distribution of the system under noise excitation,resulting in stochastic P-bifurcation.At the same time,the above results are verified by Monte Carlo numerical simulation.(2)The equation of the tri-stable van der Pol circuit system with time-delayed feedback is derived,and the parameters of the circuit system are designed.Timedelayed feedback control based on STM32 single chip microcomputer is realized,and the fractional Gaussian noise and Gaussian white noise excitation modules are designed with the mixed programming technology of C# and Matlab.(3)Time-delayed feedback control experiments are carried out for deterministic and stochastic systems.The results of theoretical and numerical simulation are verified from the perspective of system response.The feasibility of the time-delayed feedback control of the tri-stable van der Pol system with STM32 single chip microcomputer is proved.(4)The influences of time scale and noise intensity on joint probability density of classical van der Pol system is studied.The results show that the increase of time scale and noise intensity will make the ring structure of joint probability density disappear and induce random P-bifurcation in the sense of joint probability density.