  The phase reduction method for a limit cycle oscillator subjected to a strong
amplitude-modulated high-frequency force is developed. An equation for the
phase dynamics is derived by introducing a new, effective phase response curve.
We show that if the effective phase response curve is everywhere positive
(negative), then an entrainment of the oscillator to an envelope frequency is
possible only when this frequency is higher (lower) than the natural frequency
of the oscillator. Also, by using the Pontryagin maximum principle, we have
derived an optimal waveform of the perturbation that ensures an entrainment of
the oscillator with minimal power. The theoretical results are demonstrated
with the Stuart-Landau oscillator and model neurons.
