On the stability of the zero solution of a periodic reversible second-order differential equation

Abstract

The problem of the stability of the zero solution of the second-order differential equation describing the periodic perturbations of an oscillator with a nonlinear reducing force is studied. The problem in the autonomous case was solved by A.M. Lyapunov. The so called transcendental case when all members of the decomposition of the right part of the differential equation into series are to be taken into account, is considered. This case takes place for reversible differential equations, i. e. equations that do not change when time is replaced by the opposite value. The problem is solved by the methods of the KAM theory, according to which in any neighborhood of the equilibrium position at the origin of the phase plane there are periodic invariant two-dimensional tori that separate the three-dimensional configuration space. These tori are considered as two-dimensional periodic invariant surfaces covering the time axis from where the stability (non-asymptotic) of the zero solution followed. The problem to be solved is characterized by the fact that the unperturbed part of the equation contains a dissipative term (a term dependent on velocity) which has the same order of smallness as the restoring force. It is established that if a dissipative part of the perturbation is small enough then the unperturbed movement is stable according to Lyapunov.