Reducibility to multiperiodic linear systems with a diagonal differentiation operator and its application to conditionally periodic systems

Abstract

In this paper, the main theorem is proved by establishing the reducibility to an equivalent multiperiodic linear system with a differentiation operator directed along the diagonal of the independent variables space. It is shown that helical lines on a circular cylindrical surface form periodic characteristics of the operator. The reducibility of a multiperiodic system is examined near a helix starting from the initial point of the phase circle, following the classical approach used for periodic systems.

A monodromy matrix is introduced, which remains constant along the first integrals of the characteristic equations and possesses the properties of smoothness and multiperiodicity. The existence of localised positive eigenvalues consistent with the properties of this matrix is demonstrated. It is assumed that at the initial point of the phase circle, the monodromy matrix attains the maximal number of distinct eigenvalues. Their localisation on the cylindrical surface is established using the Gershgorin method.