Computer-Oriented Lyapunov Stability Criteria for Systems of Nonlinear Ordinary Differential Equations

An approach to Lyapunov stability analysis of nonlinear ordinary differential equations systems is being developed in order to obtain stability criteria involving software implementation. The approach is based on recurrent multiplicative transformations of numerical integration difference schemes under general constraints. The magnitude of the solution perturbation is determined as an infinite vector product multiplied by the initial data perturbation. Consequently, the magnitude of the disturbance is proportional to the vector product, which determines the nature of the system stability. From this follow the criteria of stability and asymptotic stability in multiplicative form. The resulting criteria are converted to additive and logarithmic form. Further, with additional constraints, stability criteria are constructed based on the right-side behavior of nonlinear system and its derivatives. The mathematical construction of the criteria entails their software implementation possibility in the course of the system approximate solution. The results of numerical experiment to analyze the stability of nonlinear systems based on the obtained criteria are presented. The experiment involves estimating the value from the left side of the criteria. Its limited change corresponds to stability, monotonous striving towards zero characterizes asymptotic stability, unlimited growth is a sign of instability. The nature of the stability of the studied systems is unambiguously established on the basis of the experimental results. The obtained criteria differ from the known ones by the construction. The criteria are programmatically implemented and allow real-time analysis of the stability of nonlinear ordinary differential equations systems.