UDC 517.925.51 The Algorithm of Reducibility of Inhomogeneous Systems with Polynomially Periodic Matrix on the Basis of Spectral Method Yu. <...> A. Konyaev∗, A. F. Salimova†, Nguyen Viet Khoa∗ ∗ Department of Higher Mathematics Peoples’ Friendship University of Russia Street Mikluho Maklay 6, 117198, Moscow, Russia † Department of Higher Mathematics National Research University “Higher School of Economics” Street Myasnitskaya 20, 101000, Moscow, Russia The paper is devoted to investigation of the class of linear and quasi-linear systems of ordinary differential equations, the matrix of which can be characterized as polynomially periodic. <...> The main aim of this article is to generate a new algorithm of their splitting in order to create equivalent sets with almost diagonal matrix that are simpler to analyze. <...> Another objective is formulating and proving of sufficient stability conditions or asymptotic stability of their trivial decision. <...> The authors of the paper develop an analytical method which appears to be a summary of known classical theorems. <...> At the heart of the offered algorithm of reducibility lies one of options of splitting method, which is conducted by a spectrum of a defining matrix in studied non-autonomous system (taking into account its splitting on diagonal and non-diagonal part) lies. <...> The present article shows possibilities of reducibility of sets of the specified class depending on structure of a matrix spectrum. <...> Theorems of stability or asymptotic stability of the trivial decision of the transformed equivalent systems and the relevant initial systems that is development and generalization of a spectral method of research of stability for the class of non-autonomous systems considered in work are proved. <...> Key words and phrases: quasi-linear system, spectral method, polynomially periodic matrix, splitting method, stability. 1. <...> Introduction A big class of ordinary differential equations (ODE) sets with a polynomially periodic matrix, that arise at the description of various physical systems or technical processes, is discovered in the article. <...> A0(t)0 is imposed, by which with the help of a certain algorithm it is possible to transform the system od ODE (1) to a system of differential matrix equation of a Received 8th August, 2013. (1) for which matrix A(t) is polynomially periodic and can be presented in the form of series <...>