Sufficient conditions for sustainability of solutions of systems of ordinary differential equations with time delay. Part III. Nonlinear equations

Background. The paper is devoted to the analysis of stability in the sense of Lyapunov steady-state solutions of systems of nonlinear differential equations with coefficients and with time delays. The cases of continuous and impulsive perturbations are considered. Materials and methods. The study is based on the use of the relationship between the stability of the initial systems of nonlinear differential equations and the stability of specially constructed systems of linear differential equations. When analyzing systems of linear differential equations constructed this way, the properties of logarithmic norms are used. Results. Algorithms are proposed that allow one to obtain sufficient conditions for the stability of solutions of finite systems of nonlinear differential equations with coefficients and with time delays. Sufficient conditions are presented in the form of inequalities connecting the coefficients of linearized systems of equations. The algorithms are effective both in the case of continuous and in the case of impulsive perturbations. Conclusions. The proposed method can be used in the study of nonstationary dynamic systems described by systems of ordinary linear differential equations with time delays.