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

The principal response of a strongly Van der Pol-Duffing oscillator subjected to parametric random narrow-band excitation is investigated. The technique of modified Lindstedt Poincare (MLP) method is used to transform the strongly non-linear system to a weak one 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 behaviors 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. The excellent agreement between theoretical results and numerical ones can be found immediately, and so the present method in this paper is applicable to solve strongly non-linear problems.