Limiting phase trajectories and emergence of autoresonance in nonlinear oscillators.

Existence of stable autoresonance (AR) with continuously growing energy is directly connected with the inherent property of nonlinear systems to remain in resonance when the driving frequency varies in time. However, the physical mechanism underlying the transformation of bounded oscillations into AR remains unclear. As this paper demonstrates, the emergence of AR from stable bounded oscillations is basically analogous to the transition from quasilinear to nonlinear oscillations in the time-invariant oscillator driven by an external harmonic excitation with constant frequency, and AR can occur as a result of the loss of stability of the so-called limiting phase trajectory. We obtain the parametric threshold, which determines the transition from bounded oscillations to AR in the time-dependent system. The accuracy of the obtained approximations is confirmed by numerical simulations.