On the Development of Singular Perturbation Theory at Moscow State University and Elsewhere

Abstract

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.

Hierarchical Selective Recruitment in Linear-Threshold Brain Networks Part II: Multilayer Dynamics and Top-Down Recruitment

“…The classical result on singularly perturbed ODEs goes back to Tikhonov [44], [45,Thm 11.1] and has since inspired an extensive literature, see, e.g. [46]- [49]. Tikhonov's result, however, requires smoothness of the vector fields, which is not satisfied by linear-threshold dynamics.…”

H∞optimal control of vehicle active suspension systems in two time scales

“…An excellent survey of the historical development of singular perturbations is found in a book by O'Malley [11]. Other historical surveys concerning the research activity in singular perturbation theory at Moscow State University and elsewhere can are found [12,13]. The curse of dimensionality coupled with stiffness poses formidable computational complexities for the analysis and design of multiple time scale systems.…”

The initial jumps of solutions and integral terms in singular BVP of linear higher order integro-differential equations

“…Development of different asymptotic methods can be found, e.g., in O'Malley [15] and Vasil'eva [20].…”