On the stability of the equilibrium under periodic perturbations of an oscillator with a power-law restoring force with a rational exponent

Small time-periodic perturbations of the oscillatorwhere p and q are odd numbers, p > q, are considered. The stability of the equilibrium x = 0 is investigated. The problem is distinguished by the fact that the frequency of unperturbed oscillations is an infinitesimal function of the amplitude. It is shown that in the case of a general equilibrium, for fixed value of q, the Lyapunov constant for values of p that are equal modulo 4q is calculated by the same algorithms, i.e., the problem reduces to a consideration of a finite number (equal to 2q − 2 if q > 1, and equal to 2 if q = 1) of values of p. An estimate, depending on q, of the number of terms of the transformation required for the calculation of the Lyapunov constant for values of p that are equal modulo 4q is given. Particular cases are considered.