A COMPUTER-ORIENTED APPROACH TO THE ANALYSIS OF THE LYAPUNOV STABILITY OF NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Abstract

An approach to the analysis of Lyapunov stability of systems of nonlinear ordinary differential equations is developed. The approach is based on vector multiplicative transformations of numerical integration difference schemes under general constraints. In the course of transformations, the magnitude of the perturbation of the solution is determined as an infinite vector product multiplied by the perturbation of the initial data. Consequently, the infinite vector product determines the nature of the stability of the system. This implies criteria for stability and asymptotic stability of a nonlinear system of ordinary differential equations in multiplicative form. The mathematical construction of the criteria entails the possibility of their software implementation, which serves as the basis for computerization of the stability analysis according to Lyapunov. The replacement of an infinite vector product by a finite product, which is necessary in the process of software implementation, preserves the reliability of the stability analysis according to the proposed criteria. Further, varieties of stability criteria are constructed in additive and logarithmic form, equivalent to the previously obtained criteria. Under additional restrictions, stability criteria are constructed according to the nature of the behavior of the right side of the nonlinear system of ODEs and its derivatives. A software and numerical experiment is presented to analyze the stability of systems of nonlinear ODEs based on the obtained criteria. The experiment is reduced to estimating the value from the left side of the criteria. Its limited change corresponds to stability, the monotonous tendency to zero characterizes asymptotic stability, and unlimited growth is a sign of instability. According to the results of the experiment, the nature of the stability of the systems under study was unambiguously established. The proposed approach makes it possible in practice to perform a Lyapunov stability analysis of a nonlinear system of ordinary differential equations in real time.