Limiting phase trajectories and energy exchange between anharmonic oscillator and external force

We present a simple analytical description of transient vibrations of forced anharmonic oscillator in terms of periodic non-smooth functions. Such a description (in chosen variables) is similar to that of a vibro-impact process. The main attention is paid to properties of the limiting phase trajectories (LPTs) describing intensive energy exchange between the oscillator and external force. We show that two dynamical transitions occur in the undamped system under consideration with increasing parameter of nonlinearity. The first transition consists in fast change of slow modulation period and drastic change of amplitude. This transition is characterized by appearance of dynamics resembling vibro-impact behavior in appropriate coordinates (amplitude and phase shift between the oscillator and external force). It corresponds to qualitative transformation of the LPT. The second transition corresponds to annihilation of two stationary points. Numerical study confirms our analytical results.