SINGULARLY PERTURBED EQUATION WITH TWO SADDLE POINTS

Abstract

In this paper, we consider a linear singularly perturbed equation in the presence of two saddle points. Previously, such classes of equations have not been studied. The problem of an asymptotic study of solutions of the considered equation is posed. To solve this problem, the level lines of conjugate harmonic functions are used. Using level lines, the necessary geometric constructions were carried out, which includes covered and division areas. Further, integration paths are chosen that ensure the boundedness of solutions. The possibility of passing from one zero to another and the dependence of the asymptotic behavior of solutions on the initial value of the independent variable are shown. The existence of boundary-layer lines, boundary-layer and regular regions is proved.