COMPUTER METHOD FOR ANALYZING THE STABILITY OF DIFFERENTIAL EQUATIONS SYSTEMS
Abstract
This article proposes approach to the stability analysis in the sense of Lyapunov for systems
of ordinary differential equations. The approach is based on stability criteria in the form of necessary
and sufficient conditions obtained on the basis of matrix multiplicative transformations of
difference schemes of numerical integration. The matrix, multiplicative form of criteria implies the
possibility of their cyclic program implementation in the form of a cycle by the number of multipliers.
It is mathematically proved that the replacement of an infinite matrix product with a finite
product, which is necessary in the programming process, preserves the certainty of the stability
analysis according to the proposed criteria. The dependence of the certainty of computer stability
analysis on the error of the difference solution of a system of ordinary differential equations is
investigated. In order to improve the accuracy of difference approximations of the solution and
linearization of the system, the method of variable piecewise polynomial approximation of the
solution is used. The method gives continuous and continuously differentiable approximations of
the desired solutions over the entire integration interval. The required approximations are obtained
on the basis of a piecewise-polynomial approximation by Newtonian interpolation polynomials
converted to the form of a polynomial with numerical coefficients. Computer approximation
of integrands increases the accuracy of integral calculation. This increases the accuracy of calculating
expressions in each multiplier of matrix products, and consequently increases the certainty
of analysis using stability criteria. A program and numerical experiment was conducted to analyze
the stability of the Lorentz system under given initial conditions and parameters changes. Based
on the numerical data obtained during the experiment, the stability nature of the system under
study is unambiguously established. In General, the proposed approach makes it possible to perform
a stability analysis arbitrary systems of ordinary differential equations in real time mode
without access to methods of the qualitative theory of differential equations and systems of computer
mathematics.
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