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• COMPUTER METHOD FOR ANALYZING THE STABILITY OF DIFFERENTIAL EQUATIONS SYSTEMS

  S.G. Bulanov

  2021-02-13

  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.

• STABILITY ANALYSIS OF RIGID SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

  S. G. Bulanov

  2021-08-11

  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.

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

  S.G. Bulanov

  2023-10-23

  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.