Phase bunching of oscillators in parametric resonance

Phase bunching in an ensemble of oscillators is considered by solving the Cauchy problem for the Mathieu equation using the asymptotic method of averaging in the third approximation for the zeroth resonance zone and in the fourth approximation for the first, second, third, and fourth zones in instability regions, as well as in stability regions near the boundaries with instability regions. It is shown that the existence and regularities of bunching follow from analysis of the well-known physical phenomenon, viz., beats of two oscillations. By way of example, parametric oscillations of charges at a node of the electric field of a standing wave are considered.