Indexed on: 24 May '11Published on: 24 May '11Published in: Physical review. E, Statistical, nonlinear, and soft matter physics
An analytical investigation of nonstationary processes in a Duffing oscillator subject to biharmonic forcing, under conditions of primary resonance, is carried out. The earlier developed methodology of limiting phase trajectories (LPTs) for studying highly nonstationary regimes, characterized by intense energy exchanges between the different degrees of freedom, is successfully applied to the system under investigation. Two distinct types of LPT trajectories are described in the first part of the study. Conditions for the recurrent transitions in time from one type of LPT to another were derived in the first part of the analysis corresponding to the undamped case. An approximation of the LPT related to the higher amplitude of oscillations was derived using nonsmooth transformations. An analysis carried out in the study has revealed the necessary and sufficient conditions for excitation of relaxation oscillations exhibited by a lightly damped system. It was also demonstrated that the mechanism of relaxations may be approximated and explained by the methodology of LPTs, characterized by strong energy exchanges between the coupled oscillators or, alternatively, a single oscillator and an external source of energy. The results of analytical approximations and numerical simulations are observed to be in quite satisfactory agreement.