On the stability of the zero solution of an essentially non-linear differential equation of the second order

Abstract

Small periodic in time perturbations of an essentially non-linear differential equation of the second order, are studied. It is supposed that the restoring force of the unperturbed equation consists of both conservative and dissipative parts, the first one being a root function. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. Peculiarity of the problem under consideration is that the frequency of oscillations is an infinitely small function of the amplitude. A problem of zero-solution stability is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method of its calculation. Liapunov’s approach can not be applied to nonautonomous, in particular to periodic perturbations because it is based on the possibility to exclude the time variable from the system. Through modification of Lyapunov’s method the following results are obtained. “Action angle” variables are introduced. A polynomial transformation of the action variable that permits to calculate Lyapunov’s constant is presented. In general case the structure of the polynomial transformation is studied. It appears that the “length” of the polynomial is a periodic function of the exponent of a root function in the conservative part of the restoring force in unperturbed equation. The least period is equal to 4. Refs 6.