Multiperiodic solutions of linear systems with a differentiation operator in the directions of the diagonal of the space of independent variables

Abstract.

In the paper, we generalize the methods of studying periodic solutions of linear systems of ordinary differential equations to linear multiperiodic systems with a differentiation operator along the main diagonal of the space of independent variables based on the study of the characteristic systems of partial differential operator on a multidimensional cylindrical manifold. A point on this manifold with time coordinates moves along the helix of a circular cylindrical surface , which is a periodic characteristic of the diagonal differentiation operator. In conclusion, we established the conditions for the multiperiodicity of solutions of linear systems with such an operator; justified the criteria for the multiperiodicity of solutions of systems associated with the matrix of those systems; determined the connections between periodic solutions of homogeneous and inhomogeneous systems based on the concept of conjugate systems; and provided integral representations of multiperiodic solutions using the Green function.