NUMERICAL MODELING OF LYAPUNOV STABILITY
Stability in the sense of Lyapunov is analyzed using approximate methods for solving the Cauchy problem for ordinary differential equations. The methods used include the Euler, Runge-Kutta, Butcher, and Dormand-Prince methods as well as the piecewise interpolation method with iterative refinement based on the Lagrange interpolation polynomial. The methods are implemented programmatically and are compared with each other for the effectiveness of computer stability analysis. The comparison is performed in terms of reliability and time complexity of the analysis results, specifically, in terms of the length of the segment of the approximate solution, on which the reliable stability estimate is preserved, and in terms of the time to perform the analysis on equal segments. For the difference methods the same step of solving the same differential system is chosen. The piecewise-interpolation method for the same system is chosen with fixed parameters. Stability estimates and comparison parameters are based on a priori proposed criteria. The analyzed differential systems include linear systems with constant and variable coefficients, autonomous and nonlinear systems of general form. The comparison is performed by numerical modeling of the proposed stability criteria and is implemented in the course of solving the Cauchy problem. The results of the numerical experiment show the applicability of each of the considered methods for stability modeling. At the same time, the piecewise interpolation method with iterative refinement based on the Lagrange interpolation polynomial is preferred for real-time computer stability analysis in the course of problem solving. The paper also presents the results of numerical modeling of stability based on the signs of the solution approximation and its two derivatives, in which case the Euler method is applied without comparison with other methods.