АНАЛИЗ УСТОЙЧИВОСТИ ЖЕСТКИХ СИСТЕМ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

Abstract

A method for analyzing stability in the sense of Lyapunov for systems of ordinary differentialequations is proposed. The method is based on stability criteria in the form of necessary and sufficientconditions obtained on the basis of vector-matrix transformations of difference numericalintegration schemes. The varieties of criteria in multiplicative, additive and matrix form are presented.The design of the criteria implies the possibility of their programmatic realization. To increasethe reliability of the stability analysis, the approximations of the solution included in theconstruction of the criteria are based on piecewise interpolation approximation by Lagrange polynomialsconverted to a form with numerical coefficients. A programming and numerical experimentis carried out to analyze the stability of the Belousov-Jabotinsky periodic reaction model,which belongs to the class of rigid systems, under given initial conditions. The analysis is carriedout on the basis of the presented criteria and the results of the program clearly determine the natureof the stability in real time. Based on the results of the experiment, it can be argued that replacingthe difference approximations of the solution with piecewise interpolation approximationsincreases the reliability of the stability analysis, reduces the study time, and makes it possible todetermine the asymptotic properties of the solution. In general, the proposed approach is an alternativeto the methods of the qualitative theory of differential equations and makes it possible toreliably determine the stability of rigid systems of ordinary differential equations in real time.