Stability of the Equilibrium of an Oscillator with an Infinitely High Natural Oscillation Frequency

We consider the stability of the equilibrium state of an oscillator with an infinitely high natural oscillation frequency under time-periodic perturbations of the oscillator. It is shown that the problem of stability in the case of general equilibrium can be solved by considering only a linear approximation of the perturbation. In the singular case, a procedure is proposed to construct a nonzero constant, if it exists, whose sign specifies whether the state of equilibrium is asymptotically stable or unstable.