REDUCIBILITY TO MULTIPERIODIC LINEAR SYSTEMS WITH A DIAGONAL DIFFERENTIATION OPERATOR AND ITS APPLICATION TO CONDITIONALLY PERIODIC SYSTEMS

Abstract. The main theorem is proven by substantiating the reducibility of an equivalent multiperiodic linear system, with a differentiation operator in the direction of diagonal of independent variable space. It is shown that helical lines with an elevation angle of  / 4, when applied to a P-circular cylindrical surface, are  -periodic characteristics of the differentiation operator.The reducibility of a multiperiodic linear system is investigated near the helix line which starts from the initial point of the phase circle, following the scheme of substantiating the reducibility of linear periodic systems. The concept of a monodromy matrix is introduced, which is constant along the first integrals of the characteristic equations of the differentiation operator and has the properties of multiperiodicity and smoothness. The existence of localised positive eigenvalues with all the properties of the monodromy matrix is proved. It is assumed that the monodromy matrix accepts the maximum number of different eigenvalues at the initial point of the phase circle compared to other points. In the neighbourhood of these eigenvalues, the localisation of eigenvalues of the monodromy matrix, defined on a cylindrical surface, is realized in its entirety. The Gershgorin method is employed for this purpose. The non-singularity of the monodromy matrix allows the existence of a curve of the complex plane that does not cover zero and surrounds the spectrum of the monodromy matrix. This proves the existence of the logarithm of the monodromy matrix, and therefore the reducibility of a multiperiodic linear system. Hence, a proof of the main theorem for a conditionally periodic system generating a multiperiodic system is obtained as a consequence.