  We investigate parametric autoresonance: a persisting phase locking which
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 which enables one to use the adiabatic invariant of
the perturbed nonlinear motion.
