TRANSITION OF SOLUTIONS OF SINGULARLY PERTURBED EQUATIONS FROM ONE SOLUTION OF THE UNPERTURBED EQUATION TO ANOTHER

Abstract

         The object of study is a singularly perturbed ordinary differential equation in a complex domain, the unperturbed equation of which has two solutions. The subject of the study is to elucidate the possibility of passing the solution of a singularly perturbed equation to another one. The purpose of the study is to prove the existence of areas of attraction of solutions of singularly perturbed equations to solutions of an unperturbed equation and to establish their relationship. To implement the goals and objectives set, the differential equation is replaced by integral ones, using the level line of harmonic functions, the area under consideration is divided into several parts. Further, taking into account the analyticity of the functions, integration paths are chosen and, using asymptotic methods, asymptotic representations in each of the parts are obtained for the solution. It is proved that each of the considered parts is the domain of attraction of a solution of a singularly perturbed equation to one solution of an unperturbed equation. In the theory of autonomous singularly perturbed equations, one of the main provisions is relaxation oscillations. Relaxation oscillations occur with successive alternation of the transition of solutions of autonomous singularly perturbed equations from one equilibrium position, which loses stability, to another stable equilibrium position.The results obtained can be used for a wide class of singularly perturbed equations.