Circuit Experiment Of Stochastic P-Bifurcation In Generalized Van Der Pol Equation

Noise has an important influence on the behavior of nonlinear systems,which may lead to the occurrence of stochastic bifurcation,stochastic resonance,coherent resonance and stochastic synchronization.Although great achievements have been made in theoretical research,less experimental research has been carried out.The main reason is the statistical index of system behavior extracted from experiments,which requires a large number of sample number experiments.This is difficult for most physical and engineering systems to achieve,regardless of time and cost.In this paper,a stochastic dynamic experiment is carried out on the basis of a nonlinear circuit system in order to accumulate experience for the nonlinear stochastic dynamic experiment of the subsequent engineering system.From a dynamic point of view,the generalized Van der Pol equation can be equivalent to a considerable number of physical and engineering systems with self excited vibration.Therefore,the study of its dynamic behavior has an important guiding role in understanding the dynamic mechanism of these complex systems.In view of the generalized Van der Pol equation with tri-stable state,we have studied the deterministic bifurcation and stochastic bifurcation phenomenon from three aspects of theory,numerical and circuit experiments.The main contents are as follows:a)The averaged equations of the system is obtained by the deterministic averaged method,and the singularity analysis is carried out.The bifurcation curves of the transition set and the response are obtained.It is found that the variation of nonlinear damping coefficients can lead to seven different types of bifurcation modes,of which four bifurcations contain up to three stable states,and three bifurcations contain up to two stable states.The numerical analysis shows that the analytical approximation results have high accuracy.b)Under the condition of keeping the topology of the circuit unchanged,the variable damping is used to adjust the nonlinear damping coefficient in a wide range by parameter optimization design,so that four kinds of three stable state bifurcation behaviors can be shown in the circuit.The circuit experiment is verified from two aspects of phase diagram and bifurcation diagram.The results show that the deterministic part of the circuit system is of high precisionc)For a typical three stable state deterministic bifurcation system,we studied the effect of additive Gauss white noise excitation,we got the amplitude of the stable state probability density function by using stochastic averaged method.We got critical parameters condition of stochastic P bifurcation and all kinds of PDF curve by the singularity analysis.The bifurcation diagram shows that the change of the excitation strength and the stability coefficient can lead to the occurrence of stochastic P bifurcation.d)The nine order Van der Pol circuit stochastic dynamic experiment system is constructed.The theoretical results of stochastic P bifurcation are qualitatively verified from the perspective of stochastic response,and the convergence of long time response distribution is discussed.e)Large sample number of stochastic circuit experiments have been carried out initially,which lays a foundation for the establishment of large data of stochastic dynamics in a multi stable state system.