Stability And Bifurcation Analysis Of Van Der Pol-Duffing Oscillator Model Excited By Stochastic Noise

As a classic nonlinear system,Van Der Pol-Duffing oscillator system has very rich dynamic behavior,and the research results on this system are relatively mature.However,in the actual environment,considering the unevenness of the material medium and the influence of small external disturbances,the stability of the system is likely to change,which will affect the stable operation of the system.In order to restore the actual working conditions more realistically,we regard the influence of these external or internal factors as noise.Considering the Van Der Pol-Duffing vibrator model under the excitation of Stochastic noise,it is necessary to analyze its Stochastic dynamic behavior.Based on the basic theory of stochastic dynamics,this paper establishes the Van Der Pol-Duffing oscillator system under Gaussian white noise excitation and Gaussian colored noise excitation.The Stochastic stability and Stochastic bifurcation of the system are analyzed,and the critical conditions for the occurrence of bifurcation are obtained,and further verified its correctness through numerical simulation.The main research contents of this article:1.Firstly,explain the research status of Van Der Pol-Duffing system and the purpose and significance of the research in detail;and introduce the research status of stochastic dynamics;and introduce the basic knowledge of stochastic dynamics;2.Considering the Van Der Pol-Duffing oscillator model under Gaussian white noise excitation,using the Central manifold theorem and polar coordinate transformation to reduce the dimensionality of the original system,and then use the stochastic averaging method to obtain the stochastic It(?) differential equation of the model,and pass The Lyapunov exponent method and the singular boundary theory analyze the local and global stability of the system,and obtain the stability conditions of the system.Then,the conditions for the occurrence of D-bifurcation of the system are obtained through the Lyapunov exponents,and the FPK equations of the system are solved to obtain Stationary probability density function and joint probability density function;and select appropriate parameters to perform numerical simulation to verify their correctness;3.Establish the Van Der Pol-Duffing vibrator model under Gaussian color noise excitation,use the unified color noise principle to perform noise whitening processing on the system,and obtain the equivalent Van Der Pol-Duffing vibrator model under the excitation of white noise.The central popular theorem and polar coordinate transformation are used to reduce the dimensionality of the system,and the It(?) stochastic differential equation of the model is obtained by using the stochastic averaging method,and then the local and global stability of the system are analyzed,and the stochastic D-bifurcation and stochastic are obtained.The critical condition for the occurrence of P-bifurcation;the FPK equation of the system is solved,the stationary probability density function and the joint probability density function of the system are obtained,and appropriate parameters are selected,and their effectiveness is verified by numerical simulation;4.Summarize the content described in this article and make a further outlook.