Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

Abstract

A linear system with a differentiation operator D with respect to the directions of vector fields of the form of the Lyapunov's system with respect to space independent variables and a multiperiodic toroidal form with respect to time variables is considered. All input data of the system multiperiodic depend on time variables or do not depend on them. The autonomous case of the system was considered in our early work. In this case, some input data received perturbations depending on time variables. We study the question of representing the required motion described by the system in the form of a superposition of individual periodic motions of rationally incommensurable frequencies. The initial problems and the problems of multiperiodicity of motions are studied. It is known that when determining solutions to problems, the system integrates along the characteristics outgoing from the initial points, and then, the initial data is replaced by the first integrals of the characteristic systems. Thus, the required solution consists of the following components: characteristics and first integrals of the characteristic systems of operator D, the matricant and the free term of the system itself. These components, in turn, have periodic and non-periodic structural components, which are essential in revealing the multiperiodic nature of the movements described by the system under study. The representation of a solution with the selected multiperiodic components is called the multiperiodic structure of the solution. It is realized on the basis of the well-known Bohr's theorem on the connection of a periodic function of many variables and a quasiperiodic function of one variable. Thus, more specifically, the multiperiodic structures of general and multiperiodic solutions of homogeneous and inhomogeneous systems with perturbed input data are investigated. In this spirit, the zeros of the operator D and the matricant of the system are studied. The conditions for the absence and existence of multiperiodic solutions of both homogeneous and inhomogeneous systems are established.