Title: Singularly Perturbed Problems in Case of Exchange of Stabilities
Abstract: Introduction 0.1.On the development of the theory of singular perturbations.Differential equations whose highest derivatives are multiplied by small parameters are usually called singularly perturbed differential equations.Equations of such type are frequently used for modeling processes with different scales in time and/or in space.We refer to the monographs and surveys [2, 9, 15, 20, 21, 26, 32, 33, 41, 45-48, 63, 65, 66, 71,72,89], where numerous examples from biochemistry, neurophysiology, hydrodynamics, semiconductor physics, etc. can be found.Challenging applications and the aim of using simplified models for describing complex processes and structures have led to an intensive study of singularly perturbed problems in the last fifty years.During this period, different methods have been developed for investigating various classes of singularly perturbed problems.The framework of these approaches is referred to as the "theory of singular perturbations."There is a vast amount literature devoted to this theory, and, since that field is still under active research, the number of relevant papers is increasing very quickly.Among these publications, there are numerous monographs and survey articles from which we would like to quote only a few related to the subject of this survey [16,18,21,30,33,40,42,58,61,62,75,78,[87][88][89][90][91].Some of these publications were published in different issues of the series "Itogi Nauki i Tekhniki."In the process of developing the theory of singular perturbations, the seminal papers of A. N. Tikhonov on the asymptotic behavior of solutions of singularly perturbed problems when the small parameter tends to zero play a fundamental role [85,86].A. B. Vasil'eva substantially extended the approach of A. N. Tikhonov by introducing the method of boundary-layer functions, by which it is possible to construct asymptotic expansions uniformly approximating solution of initial-and boundaryvalue problems for systems of singularly perturbed nonlinear ordinary differential equations (systems of Tikhonov's type).We refer the reader to the monographs [87][88][89].This survey is devoted to a class of singularly perturbed differential equations which cannot be treated in the framework of the standard theory of singularly perturbed problems for both ordinary and partial differential equations.These problems are called singularly perturbed problems in the case of exchange of stabilities.This notion goes back to [56]; the reason for using it will be explained in Sec.0.4.The main results presented in what follows are concerned with the asymptotic behavior of solutions of a rather broad class of singularly perturbed problems in the case of exchange of stabilities as the small parameter tends to zero.This survey is substantially based on papers of the authors and their colleagues published in the last six to seven years in the framework of cooperation between the Department of Mathematics of the Faculty of Physics of the Lomonosov Moscow State University and the Department of Dynamical Systems of the Weierstrass Institute for Applied Analysis and Stochastics in Berlin.In order to classify the type of new problems under consideration in the framework of the theory of singularly perturbed problems, we next present the fundamental theorem of A. N. Tikhonov in its simplest form.