Investigation On Two-dimensional Stochastic Dynamics Of Multistable VDP Systems

Stochastic bifurcation is an important issue in the study of nonlinear stochastic dynamics.It is a phenomenon that causes behavior to change intrinsically due to the uncertain internal or external factors of the system.As a typical nonlinear self-excited system,van der Pol equation's nonlinear characteristics are widely representative,and the in-depth study of its nonlinear characteristics is helpful for people to understand and analyze the behavior mechanism of the same kind of nonlinear systems.At present,the research is mainly carried out by stochastic averaging method and the theory of solving FPK equation,and carrying out by analytical expression of steady-state probability density.However,this method ignores the influence of phase.It is difficult to fully reflect the stochastic dynamic behavior of the system.Therefore,it is of great significance to study the two-dimensional stochastic dynamic behavior of nonlinear systems.As a common type of noise,Gauss white noise is widely used in circuit,control and other fields,and is often regarded as random excitatio.Among them,the integer order Gauss white noise has independent irrelevance,but the fractional Gauss white noise has long correlation.In view of the fact that fractional noise is a new type of noise concerned in recent years,most of the research is focused on numerical analysis,and there is no experimental report.For several kinds of van der Pol systems with different nonlinear times,such as classical van der Pol systems,fifth-order and ninth-order generalized van der Pol systems,this paper studies the system in Gaussian white noise by means of theoretical,numerical and circuit experiments.In addition to the amplitude probability density,the change of joint probability density is also taken into account in the response of fractional Gaussian white noise excited by noise.The main tasks are as follows:1.For the classical van der Pol equation,the bifurcation behavior of the system is obtained by the average method,and the stochastic P bifurcation results of some joint probability densities are qualitatively verified.The stochastic dynamic behavior of the system excited by integer order and fractional order Gaussian white noise is studied.The influence of fractional order Gaussian white noise on the system is obtained by comparing the integer order Gaussian white noise.2.While proving the correctness and reliability of the circuit experiment system,the existing circuit experiment system is improved,the continuous adjustment of bifurcation parameters of the system is realized,and the automation of bifurcation experiment is realized.The fractional Gaussian white noise generation module is developed and used in the circuit experiment under fractional noise excitation.3.For the van der Pol equation with bistable state,it is found that when the fixed stability coefficient is fixed and the noise intensity is increased.There are three ways to qualitatively change the joint probability density of:(a)single peak ?three peak?bimodal(b)basin peak ? triple peak ?bimodal(c)ring bimodal4.The theoretical and experimental research on the ninth-order van der Pol system with tristable behavior is carried out.Based on the existing research results,a simple theoretical and experimental explanation is carried out in this chapter,and the influence of strong excitation on the stochastic dynamic behavior of the system is mainly studied.It is found that strong random excitation can inhibit the random P bifurcation caused by the change of stability coefficient.