The asymptotic stability of the equilibrium of parametrically perturbed systems

Quasi-linear systems with many degrees of freedom are investigated for low dissipation and periodic perturbation using Lyapunov's second method. The periodic perturbation can be of small or large amplitude. Criteria of the asymptotic stability of the systems investigated are derived, which can be characterized as the sufficient conditions for parametric imperturbability of the latter when there is a weak dissipative background. The proposed approach enables limiting cases of periodic perturbation to be considered, when the corresponding frequency may approach both zero and infinity. Extensions to the case of non-periodic perturbations which vary very slowly or very rapidly with time are possible.