This short survey does not pretend to be absolutely complete and, as with all surveys, is to a certain degree subjective because most attention is paid to scientific work closest to the interests of the author. Nevertheless we make every effort to give an adequate picture of the development of this scientific field.The systematic study of singularly perturbed systems began in the Soviet Union forty years ago. The basic articles are by A. N. Tikhonov [8]-[10]. Before then, some sporadic papers of foreign authors, for instance [1 ], were known. In his first paper [8], Tikhonov considered the initial value problem dz dy (1)where the scalar z and vector y are unknown functions, and/z > 0, is a small positive parameter.Formally setting/z 0, we obtain the so-called degenerate (or reduced) system; its order is less than that of (1) and its solution cannot satisfy all the supplementary conditions (2).Tikhonov investigated the passage to the limit as # 0 of the solution of the problem (1), (2) to some solution of the degenerate system. The following result was established. Let the equation F(z, y, t) 0 have isolated solutions (roots) z Pi(Y, t) in some domain D(y, t). Let one of the roots z 991(y, t) (with q93 < 1 < q92) be stable. The stability condition Tikhonov suggested was in the form of the inequality (3) (z qg)F(z, y, t) < 0.The simplest case F F(z, t) can be demonstrated in Fig. 1: Suppose the function F is positive if z < qgl and negative if z > p, and suppose further that the initial point z belongs to the so-called domain of attraction of the root z qg (y, t). This is the domain q9 < z < 02.In this case (Fig. 1), it is geometrically clear that the integral curve will tend to the curve z 0 (t), > 0 as # --+ 0. In the general case we have the following limiting relations:where f(qg (33, t), 33, t), )3It=0 for 0 < < T. dt The passage to the limit in (5) is uniform with respect to t, while in (4) it is not uniform because z -Ol (y0, 0). In the neighborhood of 0, there is a zone where z(t, #) differs considerably from 5(t)(q0 (t) in Fig. 1). This zone was called the boundary layer.
