Differential systems stability analysis based on matrix multiplicative criteria

Abstract

This article proposes a method of stability analysis in the sense of Liapunov for systems of ordinary differential equations. The method is based on matrix multiplicative stability criteria obtained from transformations of numerical integration difference schemes. Criteria are formed in the form of necessary and sufficient conditions. A distinctive feature of the criteria is that they do not employ the methods of the qualitative theory of differential equations. Specifically, for the case of linear systems, the evaluation of characteristic numbers and characteristic indicators is not necessary. When analyzing the stability of nonlinear systems, the construction of Liapunov functions is not required. Matrix multiplicative form of the criteria allows computerization of stability analysis. The software implementation of the criteria is performed in real time and, based on its results, allows to make an unambiguous conclusion about the nature of stability of the system under study.