On the stability of "nonlinear center" under quasiperiodic perturbations

The problem of the stability of the zero solution of a system with critical point of the "center" type at the origin, is considered. Such problem for autonomous systems was investigated by Liapunov. Investigations of Liapunov were continued by the authers for systems periodic in time. In the present paper systems with quasi-periodic dependence on time, are considered. It is supposed that the basic frequencies of quasi-periodic functions sutisfy the standard condition of diophantine type. The problem under consideration can be intepreted as the problem of the stability of the state of equilibrium of the oscillator $\ddot{x} + x^{2n-1} = 0$, $n$ is a integer, $n \geqslant 2$, under "small" quasiperiodic pertubations.