STABILITY ANALYSIS OF RIGID SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Abstract

A method for analyzing stability in the sense of Lyapunov for systems of ordinary differential
equations is proposed. The method is based on stability criteria in the form of necessary and sufficient
conditions obtained on the basis of vector-matrix transformations of difference numerical
integration 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 increase
the reliability of the stability analysis, the approximations of the solution included in the
construction of the criteria are based on piecewise interpolation approximation by Lagrange polynomials
converted to a form with numerical coefficients. A programming and numerical experiment
is 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 carried
out on the basis of the presented criteria and the results of the program clearly determine the nature
of the stability in real time. Based on the results of the experiment, it can be argued that replacing
the difference approximations of the solution with piecewise interpolation approximations
increases the reliability of the stability analysis, reduces the study time, and makes it possible to
determine the asymptotic properties of the solution. In general, the proposed approach is an alternative
to the methods of the qualitative theory of differential equations and makes it possible to
reliably determine the stability of rigid systems of ordinary differential equations in real time.