LEVEL LINE METHOD IN THE THEORY OF SINGULARLY PERTURBED EQUATIONS

Abstract

In this article, we briefly analyze previous studies on non-autonomous and autonomous singularly perturbed ordinary differential equations. It is noted that the main results are obtained using the stability of rest points for non-autonomous equations and equilibrium positions for autonomous equations. The question is raised of the study of the asymptotic behavior of solutions of singularly perturbed equations without invoking stability conditions. To solve the problem, a method has been developed using the level lines of conjugate harmonic functions generated by given analytic functions. The demonstration of the method was carried out for simplicity of presentation, for a linear scalar equation of the first order. The right side is only required to be analytic in some open, simply connected domain of the complex plane. The method consists of two parts: topological and analytical. In the first part, using the level lines of the conjugate harmonic functions, the covered and division of the region is produced. In the second part, with the choice of integration paths, the study of the asymptotic behavior of the solution of a given equation is carried out. The study was carried out without involving stability conditions. Next, an analysis of possible directions for the development of the method is carried out.