Stochastic Bifurcation Of Nonlinear System With Fractional Derivative Damping

Noise and time delay have important effects in various research fields.For dynamic systems with noise and time delay,there are often many uncertainties.The fractional order controller has become a research hotspot in the control field because of its good genetic and memory characteristics.The existing research results also show that the control effect of the fractional order controller is better than the traditional integer order controller.Based on the research status of fractional calculus,stochastic bifurcation,time delay and fractional order control problems,the stochastic bifurcation of stochastic systems with fractional derivative damping under Gaussian white noise excitation and the stochastic bifurcation of typical nonlinear oscillators with time delay fractional order PD controller are investigated.The first part describes the research background,research status,research content and innovation of this paper,and briefly introduces some definitions and formulas.In the second part,the stochastic P-bifurcation of the Morse oscillator with fractional derivative damping under the excitation of Gaussian white noise is investigated.First,by using the generalized harmonic function,we develop the stochastic averaging method of the strongly nonlinear system under the excitation of Gaussian white noise,and obtain the averaged (?) differential equation about the amplitude and phase difference,then the steadystate probability density of response amplitude is obtained by solving the stationary FokkerPlanck-Kolmogorov(FPK)equation corresponding to the averaged (?) equation.Taking the Morse oscillator as an example,the effects of fractional order,noise intensity and damping coefficient on the system response are discussed.Based on the singularity theory,the critical parameter conditions of stochastic P-bifurcation are obtained,and the effects of system parameters on stochastic bifurcation are studied.In the third part,the stochastic bifurcations of typical nonlinear oscillators with delay fractional order PD controller under Gaussian white noise excitation are studied.The averaged (?) stochastic differential equation of the controlled quasi-integrable Hamiltonian system with fractional derivative damping is obtained by using the stochastic averaging method of the quasi-integrable Hamiltonian system.The steady-state probability density of the response amplitude is obtained by solving the stationary FPK equation corresponding to the averaged (?) equation.Based on the obtained stationary probability density,based on the obtained stationary probability density,the influence of fractional order and time-delay on the system response and the stochastic bifurcation are discussedThe fourth part summarizes the important conclusions of this paper and prospects for future research.