On multi-periodic solutions of quasilinear autonomous systems with an operator of differentiation on the Lyapunov’s vector field

Abstract

On multi-periodic solutions of quasilinear autonomous systems with an operator of differentiation on the Lyapunov's vector field A quasilinear autonomous system with an operator of differentiation with respect to the characteristic directions of time and space variables associated with a Lyapunov's vector field is considered. The question of the existence of multi-periodic solutions on time variables is investigated, when the matrix of a linear system along characteristics has the property of exponential stability. And the non-linear part of the system is sufficiently smooth. In the note, on the basis of Lyapunov's method, the necessary properties of the characteristics of the system with the specified differentiation operator were substantiated; theorems on the existence and uniqueness of multi-periodic solutions of linear homogeneous and nonhomogeneous systems were proved; sufficient conditions for the existence of a unique multi-periodic solution of a quasilinear system were established. In the study of a nonlinear system, the method of contraction mapping was used.

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

“…A method for studying the multiperiodic structure of oscillatory solutions of perturbed linear autonomous systems of the form (1.1) -(1.2) was developed. The main essence of the method for studying the multiperiodic structures of solution of the system under consideration is a combination of the known methods [1][2][3] with the methods used in [11,12] for the autonomous systems. In conclusion, the sufficient conditions for the existence of the multiperiodic solutions of linear systems (1.…”

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10] and applied to the study of solutions different problems in the partial differential equations [11,12]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12], in particular, problems on multi-frequency solutions of equations from control theory [13]. The methods of research for multiperiodic solutions are successfully combined by methods for studying solutions of boundary value problems for equations of mathematical physics.…”

“…As noted above, the considered system of partial differential equations along with multidimensional time contains space independent variables, according to which differentiation is carried out to the directions of the different vector fields. The autonomous case of this system was considered in [11,12], where differentiation with respect to time variables was carried out in the direction of the main diagonal of space, and the free term of the system was independent of time variables. In this case, these parameters of the systems received perturbations depending on time variables.…”

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

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10][11][12][13][14] and applied to the study of solutions different problems in the partial differential equations [15,16]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], in particular, problems on multi-frequency solutions of equations from control theory [17]. Many oscillatory phenomena are described by systems with a differentiation operator with respect to toroidal vector fields, and new methods based on the ideas of the Fourier [18], Poincaré-Lyapunov and Hamilton-Jacobi methods [19,20] appear to establish their periodic oscillatory solutions.…”

“…As noted above, the considered system of partial differential equations along with multidimensional time contains space independent variables, according to which differentiation is carried out to the directions of the different vector fields. The autonomous case of this system was considered in [15,16], where differentiation with respect to time variables was carried out in the direction of the main diagonal of space, and the free term of the system was independent of time variables. In this case, these parameters of the systems received perturbations depending on time variables.…”