窄带随机噪声参激作用下强非线性系统的响应

将改进的L-P(ModifiedLindstedt-Poincare)方法和随机多尺度法结合起来,研究了窄带随机噪声参激作用下强非线性VanderPol-Duffing系统的响应、稳定性和分叉问题。文中首先由改进的L-P方法引入变换参数,然后用多尺度法分离了系统的快变量,求出了最大Lyapunov指数的解析表达式,得到了系统几乎必然稳定的充分必要条件,讨论了系统的参数对稳定性的影响,分析了系统在1/2亚谐共振区的性态。理论分析与数值计算表明:在一定条件下系统将发生随机跳跃,且随着随机激励带宽的增大,扩散的极限环的宽度将逐渐增大。数值模拟结果表明:MLP方法结合随机多尺度法研究强非线性随机系统的响应、稳定性和分叉问题是有效的。

Responses of strongly non-linear oscillator parametrically excited by narrow-band random noise

The principal response of a Van der Pol-Duffing oscillator subjected to parametric narrow-band random excitation is investigated. The technique of Modified Lindstedt Poincare (MLP) method is used to transform the strongly nonlinear system to a small parameter system by introducing a new expansion parameter, and then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response of the system. The effect of damping, detuning, and bandwidth on the dynamic behaviours such as stability, bifurcation are examined by computing the maximum Lyapunov exponent analytically. Also the numerical simulation is carried out to verify the analytical results, and random jump phenomenon may be observed in the region of the parameters of the system. An excellent agreement between theoretical and numerical results is obtained.It is show that the method of this paper is applicable to strongly non-linear problems. The results obtained for strongly non-linear system complement previous results of weakly non-linear system in the literature.