Parametric autoresonance.

We investigate parametric autoresonance: a persisting phase locking that occurs when the driving frequency of a parametrically excited nonlinear oscillator slowly varies with time. In this regime, the resonant excitation is continuous and unarrested by the oscillator nonlinearity. The system has three characteristic time scales, the fastest one corresponding to the natural frequency of the oscillator. We perform averaging over the fastest time scale and analyze the reduced set of equations analytically and numerically. Analytical results are obtained by exploiting the scale separation between the two remaining time scales that enables one to use the adiabatic invariant of the perturbed nonlinear motion.