MULTIPERIODIC OSCILLATIONS IN SECOND-ORDER LYAPUNOV'S SYSTEMS WITH A DIFFERENTIATION OPERATOR IN THE DIRECTION OF A QUASI-PERIODIC VECTOR FIELD

Abstract

Abstract. The work is devoted to studying oscillatory solutions of Hamiltonian systems of two equations with a differentiation operator in the directions of a vector field with a quasi-periodically varying velocity, in which the frequencies satisfy the condition of strong incommensurability. The system is quasi–linear, has a pair of conjugate pure imaginary eigenvalues, admits the first integral, and is real analytical in time variables in the neighbourhood of the real axis of the complex plane, and space variables on the real plane. It can be called the Lyapunov system. Each equation of a vector field is represented by an independent quasi-periodic scalar function. According to one study by J. Moser the integrals of such equations, under the condition of strong incommensurability, are described by the sum of linear and quasi-periodic functions. To investigate the existence of oscillatory solutions in the system under consideration, Lyapunov's method is extended to this system, considering the result of J. Moser. The differentiation operator with a quasi-periodic property is considered for the first time, and periodic or constant coefficients are often considered in connection with various issues of the theory of multi-frequency oscillations. Thus, the result of the conducted research is an analogue of the result on Lyapunov's periodic systems in the case of quasi-periodic systems with a special differentiation operator.