宽带噪声激励下分数阶黏弹性碰撞系统的稳定性分析和随机分岔

Stability Analysis and Stochastic Bifurcation of Fractional-OrderViscoelastic Collision System under Broadband Noise Excitation

研究基于分数阶黏弹性材料构造的Van der pol减振系统在外部宽带噪声激励下的随机稳定性和随机分岔行为.考虑约束条件的影响,引入非平滑Zhuravlev变换,将碰撞系统转化为无碰撞的动力学系统.利用一组拟周期函数近似替换分数阶微分,通过随机平均法得到系统的It8随机微分方程,根据最大Lyapunov指数法和奇异边界理论分类讨论系统的随机稳定性,利用拟Hamilton系统随机平均法分析系统在线性It8方程下的随机分岔行为,得到D-分岔的临界条件,进一步求出与系统幅值相关的稳态概率密度函数.使用MATLAB绘制稳态概率密度曲线,直观展现系统发生的稳态变化.结果表明,当分数阶阶次和噪声强度在一定阈值内变化时,可诱导系统产生P-分岔行为.

The stochastic stability and stochastic bifurcation behavior of Van der pol vibration damping system constructed based on fractional-order viscoelastic material was studied under external broadband noise excitation.Considering the influence of constraints condition,a non-smooth Zhuravlev transforma tion was introduced to transform the collision system into a collision-free dynamic system.A set of quasi-periodic functions was used to replace the fraction al-order differential element approximately,the stochastic average method was used to obtain the It stochastic differential equation of the system.The stochastic stability of the system was classified and discussed based on the maximum Lyapunov exponent method and singular boundary theory.The stochastic b ifurcation behavior of the system under the linear It equation was analyzed by using the pseudo Halmiton system stochastic average method,and the critical condition for D-bifurcation was obtained.Furthermore,the stationary probability density function related to the amplitude of the syst em was obtained.Using the steady-state probability density curves drawn by MATLAB to visually display the changes of steady state that occurred in the system.The results show that the system can generate P-bifurcation behavior when the fractional-order and noise intensity change within a certain threshold.