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.