单边碰撞分数阶Rayleigh振子的随机P-分岔

Random P-bifurcation of fractional-order Rayleigh oscillator with unilateral collision

基于非光滑变换和随机平均法分析了随机激励下含有分数阶微分的Rayleigh振子碰撞振动系统的随机P-分岔问题。基于Caputo定义计算了分数阶导数,将分数阶微分等效为相应的阻尼力与恢复力,并用非光滑变换将原系统等效为一个新的不含速度跳的系统;基于随机平均法建立了随机伊藤方程,得到了随机响应的Markovian近似,进而计算出系统的概率密度函数及其稳态解;引入突变理论推导出随机P-分岔的临界参数条件表达式,并分析了分数阶系数、分数阶阶次、恢复系数等主要参数对分数阶Rayleigh振子碰撞系统发生分岔的影响。

Here, based on non-smooth transformation and the stochastic average method, random p-bifurcation problems of Rayleigh oscillator collision system with fractional-order differential under random excitation were analyzed. Firstly, the fractional-order derivative was calculated based on Caputo’s definition, and fractional derivative was equivalent to the corresponding damping force and restoring force. The non-smooth transformation was used to make the original system be equivalent to a new system without velocity jump. Secondly, the random ITO equation was established based on the stochastic average method to obtain Markovian approximation of the random response, and furthermore the system’s probability density function and steady-state solution were calculated. Finally, the catastrophe theory was introduced to derive the critical parametric condition expression of random p-bifurcation, and effects of main parameters, such as, fractional order coefficient, fractional order number and recovery coefficient on fractional-order Rayleigh oscillator collision system having bifurcations were analyzed.