乘性色噪声激励下广义分数阶van der Pol-Duffing振子的随机分岔分析

本文研究了在乘性色噪声激励下含分数阶导数项的广义Duffing振子的随机分岔.首先，利用一种回复力和阻尼力的线性组合等效替换系统中的分数阶导数项；其次，对系统中的三次项进行线性化处理，利用最小均方误差原理，将系统转变成整数阶系统，由随机平均法求得系统的稳态概率密度函数；最后，通过拟不可积Hamilton系统随机平均法得到系统不变测度的最大Lyapunov指数，并对系统进行随机D-分岔和P-分岔分析.研究发现，分数阶导数阶数、噪声的自相关时间等参数的改变可以诱发系统发生随机P-分岔.

Stochastic bifurcation analysis of generalized fractional order van der Pol-Duffing oscillators under multiplicative color noise excitation

This article investigates the stochastic bifurcation of a generalized Duffing oscillator with fractional derivative term under multiplicative colored noise excitation. Firstly, a linear combination of restoring force and damping force is used to equivalently replace the fractional order derivative term in the system. Secondly, the cubic term in the system is linearized, the system is transformed into an integer order system by using the principle of minimum mean square error, the steady-state probability density function of the system is obtained by the stochastic averaging method. Finally, the maximum Lyapunov exponent of the invariant measure of the system is obtained by the stochastic averaging method of quasi-non-integrable Hamilton system, and the stochastic D-bifurcation and P-bifurcation of the system are analyzed. It is found that the changes of fractional derivative order, autocorrelation time of noise and noise intensity can induce stochastic P- bifurcation of the system.