Steady-state Response And Stochastic Bifurcation Of Fractional-order Nonlinear Systems Excited By Gaussian White Noise

Dynamic behaviors of the stochastic nonlinear system with fractional derivative have been a very attractive and promising research area.The previous results showed that traditional integer order model can’t simulate the actual system with memory factors.However,the researchers found that fractional stochastic nonlinear systems have wider practical value.It can not only simulate the internal properties of the real system more accurate than the integer order nonlinear model,but also give a more reasonable explanation.Especially when simulating viscoelastic material,the advantage of fractional order model is more prominent.Therefore,it is worthwhile to study the nonlinear system with fractional derivative.There are many theoretical methods to analyze the above systems.In this paper,the steady-state response of the Duffing-Van der Pol system with fractional derivative and the bifurcation problem of the Van der Pol system are considered via the stochastic averaging method.Firstly,the generalized Van der Pol transformation is applied to obtain the stochastic differential equations of amplitude and phase.Secondly,the Ito averaging differential equation is obtained by adding the correction term.Then,establish the corresponding stationary FPK equation.The steady-state probability density function of the system is obtained by solving the stationary FPK equation.Finally,analysing the influence of fractional derivative term on the response of the system based on the steady-state probability density function.The results show that:1.In fractional Duffing-Van der Pol systems,the fractional derivative term can significantly affect the steady-state response of the system.It is found that the peak of the steady-state probability density curve shiftes left gradually and decreases when the order or coefficient of the fractional order increases.This phenomenon shows that the steady-state response of the system is gradually weakened.Therefore,we can change the fractional damping term to control system response.In addition,the study found that changing other parameters or noise intensity also has a negligible effect on the response of fractional order systems.Last,Monte Carlo numerical simulation method is used to verify the effectiveness of the stochastic averaging method for fractional stochastic nonlinear problems.2.For the deterministic fractional Van der Pol oscillator,the phase diagram evolves from a limit cycle to a point when increasing fractional order.This indicates that the change of fractional order will cause the phase transition bifurcation of the system and the introduction of fractional order leads to more complex system behavior.Then studying stochastic fractional Van der Pol oscillator,we found that increasing fractional order steady-state probability density curve evolves from two peak to a single peak,and the peak of joint probability density surface evolves from concave into convex.This indicates that changing fractional order can cause stochastic bifurcation.