The Algorithm of Reducibility of Inhomogeneous Systems with Polynomially Periodic Matrix on the Basis of Spectral Method
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 question is topical since the analysis of a considered class of non-autonomous systems with the use of known methods (for example, the method of functions of Lyapunov) is complicated. In addition, the usage of spectral and other methods while solving non-uniform sets might cause extra difficulties. 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. This simplifies the analysis of questions of stability. 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.