Jeremy Gray

THE HISTORY OF THE CONCEPT OF A FINITE-DIMEHSIONAL VECTOR SPACE

The purpose of this note is to draw attention to an interesting problem in the history of mathematics which the pressure of work will prevent me from taking up, and to make a suggestion about a possibly neglected aspect of the problem that should be explored. The problem is: to describe the origins of the concept of a vector space and its recognition as a central topic in mathematics. Moves towards the explicit treatment of topics, and their axiomatization, are always of interest; in addition, it seems that vector spaces have had a curious and convoluted development.

My specific suggestion, drawn from my research into the history of the monodromy group and differential equations in the complex domain, is that the theory of ordinary linear differential equations and geometric complex function theory provided a rich source of finite-dimensional vector spaces throughout the late 18th and 19th centuries. The existence of a finite-dimensional basis for the solutions of an nth-order linear ordinary differential equation with constant coefficients is described, for example, in [START_REF] Euler | De integratione aequationum differentialium altiorum gradium[END_REF], and the theory is oeveloped by many subsequent writers, culminating in [START_REF] Fuchs | Zur Theorie der linearen Differentialgleichungen mit veranderlichen Coefficienten[END_REF]. Fourier, in a study of heat diffusion among n bodies in a circle, derived n simultaneous linear ordinary differential equations, which he solved to obtain finite "Fourier Series" [Grattan- [START_REF] Grattan-Guinness | Joseph Fourier[END_REF]. He was well aware, as were his contemporaries, of the linear character of his problems and of the orthogonality properties of his basis of solutions, which are expressed by the familiar properties of integrals of the form

1 271 0 sin mx sin nx.
Fourier's work is also important for the study of infinite matrices and function spaces which began at the end of the 19th century.

The matrices which appear explicitly in the study of the monodromy (and the monodromy group) of an algebraic function and in the related study of the singular points of a differential equation operate implicitly on vectors (the n-tuples of possible values or independent solutions); see, for example, [START_REF] Hermite | Sur les fonctions algebriques[END_REF], [START_REF] Riemann | Theorie der Abel'schen Functionen[END_REF], Jordan [1872), or most writers on complex functtion theory throughout the century. The sheer size of the literature on complex functions is daunting, and I wish to draw attention to an important source of ideas about vectors which might, perhaps, otherwise be ignored.

It does not seem that much has been written on the history of vector spaces, with the notable exceptions of [START_REF] Bernkopf | The development of function spaces[END_REF], on the origin of function spaces; [START_REF] Bourbaki | Elements d'histoire des mathematiques[END_REF], which examines number-theoretic roots; [START_REF] Bourbaki | Elements d'histoire des mathematiques[END_REF], which considers rather more topics; and Hawkins' considerable reconstruction of the theory of matrices [1977a, 1977b]. In a methodological way, Hawkins' papers are also germane to the topic raised in this note; they suggest a sequence of stages for the theory of matrices, from their origins in the study of systems of differential equations by Lagrange and Laplace, through an explicit but untheoretical period, to a generic and finally rigorous theoretical treatment by Weierstrass and Frobenius.

It is likely that the study of vector spaces passed through similar reformulations.

Hawkins suggests that the modern pedagogic approach, which puts vector spaces first and matrices or, better, linear maps second, dates only from the 1930's [Hawkins 1977a, 109].

Two problem areas are important if one looks for early explicit formulations of the concept of a vector space.

In the course of a study of systems of n linear first-order ordinary differential equations, [START_REF] Peano | Integration par ser~es des equations differentielles[END_REF] gave an explicit axiomatic definition of an n-dimensional vector space over the reals. He called his n-tuples "number complexes of order n," which suggests affinity with the study of hyper-complex numbers. Significantly, he had just read [START_REF] Grassmann | Die lineare Ausdehnungslehre[END_REF], which contains a long-neglected treatment of vector spaces and exterior algebras. The study of differential equations with irregular singular points led directly to the study of infinite vector spaces, or rather, as it was then expressed, the study of linear equations in infinitely many unknowns and infinite determinants [START_REF] Hill | On the part of the motion of the lunar perigee which is a function of the motions of the sun and moon[END_REF][START_REF] Appell | Sur une methode elementaire pour obtenir les developpements en serie trigonometrique des fonctions elliptiques[END_REF][START_REF] Poincare | Remarques sur l'emploi de la methode precedent[END_REF]Poincare , 1886)). These topics are treated in [START_REF] Bernkopf | The development of function spaces[END_REF], for they merged into the study of integral equations and thence function spaces, so decisively established by Hilbert from 1904 to 1910. However, as Bernkopf points out, Hilbert was more interested in integral equations and their kernels (generalizations, accordingly, of matrices) than in the spaces of functions.

The study of the topological aspects of these spaces is due to [START_REF] Frechet | Sur quelques points du calcul fonctionnel[END_REF], and the geometric aspects to [START_REF] Schmidt | Uber die Auflosung lineare Gleichungen unendlich vielen Unbekannten[END_REF].

It may well be that the unexpected richness and complexity of infinite-dimensional vector spaces (which need not be isomorphic to their duals, for example) first inclined mathematicians to study the finitedimensional spaces explicitly. This interesting possibility merits further study.

The concept of a finite-dimensional vector space also appears explicitly in the development of Galois theory. Kronecker's constructive idea of a "RationaliUitsbereich" [1882] consists typically of finite linear combinations of algebraic numbers taken over the rationals. The totality of these linear combinations is an algebraic extension of the rationals and therefore a vector space over the rationals. Unhappily, Kronecker's ideas are notoriously hard to date (see [START_REF] Kiernan | The development of Galois t~eory from Lagrange to Artin[END_REF]). The same idea of a field extension is quite explicit in Dedekind and ~Ieber [1881], where it is called a "Modul." Here the module is over a field, usually the field of rational functions. Dedekind had long been interested in fields because of his interest in number theory, and in the paper he wrote with Weber he sought a common language to describe the similar domains of algebraic number theory and algebraic function theory. The first abstract treatment of fields is given in [START_REF] Weber | Allgemeinen Grundlagen der Galois'schen Gleichungentheorie[END_REF], but here vector spaces do not appear explicitly: algebraic extensions of a given field are called "congruence fields" by Weber and treated as fields. The concept of a vector space was, of course, of considerable importance for Artin's modern reformulation of Galois theory [START_REF] Artin | Foundations of Galois theory[END_REF][Artin , 1959] ] (and see also [START_REF] Van Der Waerden | Die Galois-Theorie von Heinrich Weber bis Emil Artin[END_REF]). Hilbert (1894] took up the concepts of fields and vector spaces in number theory, as did [START_REF] Hensel | Theorie der algebraischen Funktionen einer Variabeln[END_REF] in function theory. It is apparently from number theory that Hilbert extracted the theory of finite-dimensional vector spaces, which he said he was generalizing in his work in integral equations. (It is interesting to note that Dedekind and Peano, the two mathematicians of the period who are most prominently identified with the question of what a number is, should be among the first to identify vector spaces as objects worthy of interest.)

Finite-dimensional vector spaces also appeared in the work of S. Lie, F. Engel, W. Killing, and E. Cartan in the theory of continuous transformation groups. The infinitesimal action on a "Lie" group by its Lie algebra was the first object of study, and the successful classification of the complex semisimple algebras is described in [START_REF] Bourbaki | Elements d'histoire des mathematiques[END_REF] and [START_REF] Helgason | Invariant differential equations on homogenous manifolds[END_REF].

Geometry, with its n-tuples of points in ann-dimensional space, is another obvious sorce of the concept. Although interest is not usually centred on the ambient space but upon configurations within it, linear combinations of figures do occur. For example, if S = 0 and S' :: 0 define two conics, so does "AS + "AS' ::0 for scalars A, A'. The investigations of matrix representations by Frobenius and others, the related study of hyper-complex numbers, and researches into finite-dimensional algebras would also have to be considered in any study of the origins of finite-dimensional vector spaces (see [START_REF] Hawkins | Hypercomplex numbers, Lie groups and the creation of group representation theory[END_REF]Navy 1973, Chap. 4]). Finally, there is the physicist's vector algebra (see [START_REF] Crowe | A history of vector analysis[END_REF]) and its generalization to tensors (see [START_REF] Weyl | Raum[END_REF]).

It is perhaps worth raising the related questions: when did the concept of a vector space with additional structure, such as an inner product, first emerge, and how and when did it gain general acceptance? The concept may first have emerged with the Killing form of a Lie algebra [START_REF] Killing | Die Zusammensetzung der stetigen endlichen Transformationsgruppen, I. Mathematische Annalen[END_REF][START_REF] Killing | Die Zusammensetzung der stetigen endlichen Transformationsgruppen, I. Mathematische Annalen[END_REF], its final acceptance may have come with von Neumann's work on unitary operators in the late 1920's (see, e.g., [START_REF] Von Neumann | Ober die Grundlagen der Quanten-mechanik[END_REF]).

It is possible that the concept of a vector space is one of many which mathematicians have found they have been using for many years without knowing it, or, perhaps one should say, needing to know it. Other examples taken at random are semigroups, long used in the theory of integral equations [START_REF] Hille | What is a semigroup?[END_REF]], and even groups (see [START_REF] Wussing | Die Genesis des abstrakten Gruppenbegriffes[END_REF]).

If this is so, then subtle attention paid to the ideas in their historical context may prove to be instructive about the nature of mathematical activity and the necessity, or otherwise, of abstraction. For the problem is not merely to assess the neglect of Grassmann by mathematicians, but to respond to the fact that for 50 years mathematicians worked confidently without feeling the need to elaborate an abstract structure, when it is now so often assumed that making such abstractions is essential to mathematical work.

ACKNOWLEDGMENT

I should like to thank Ivor Grattan-Guinness for his helpful advice during the preparation of this note.


A L Kelley

The Stable, Center-Stable, Center, Center-Unstable, Unstable Manifolds

INTRODUCTION

Our purpose is to give a proof of the existence and smoothness of the invariant manifolds in the title for a system of ordinary differential equations defined in a neighborhood of a critical point, periodic orbit, or periodic surface. In system (1) below when the center equation (y-equation) is absent, the existence, and to some extent the smoothness, of stable and unstable manifolds is well known (see Theorem 4.1 on page 330 and Theorem 4.2 on page 333 of Coddington and Levinson [1], for example). For the associated perturbed system with no center equation [system (45) below with no y-equation] the existence of an invariant manifold called a periodic surface or integral manifold is well known [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF]- [START_REF] Sacker | A new approach to the perturbation theory of invariant surfaces[END_REF]. For smoothness of this manifold see [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF], p. 480; [START_REF] Diliberto | Perturbation theorems for periodic surfaces[END_REF], [START_REF] Sacker | A new approach to the perturbation theory of invariant surfaces[END_REF]. Sections 6 and 7 of this paper are in part related to the idea of the classical periodic surface. See the remark at the end of Section 6 and Theorem 5 in Section 7. A discussion of the work of Krylov-Bogoliubov-Mitropolsky [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF], [START_REF] Krylov | The application of methods of nonlinear mechanics to the theory of stationary oscillations[END_REF], Diliberto [START_REF] Diliberto | Perturbation theorems for periodic surfaces[END_REF], and Levinson [START_REF] Levinson | Small periodic perturbations of an autonomous system with a stable orbit[END_REF] occurs in Hale [START_REF] Hale | On the method of Krylov-Bogoliubov-Mitropolski for the existence of integral manifolds of perturbed differential systems[END_REF]. The techniques we use in this paper are closely related to those of Krylov-Bogoliubov-Mitropolsky.

For Hamiltonian systems or systems of differential equations with an integral, the existence of two-dimensional subcenter manifolds (invariant manifolds related to pairs of imaginary eigenvalues) is known; see, for example, [START_REF] Kelley | On the Liapounov subcenter manifold[END_REF]- [START_REF] Siegel | Vorlesungen uber Himmelsmechanik[END_REF]. Lykova [START_REF] Lykova | On the question of stability of solutions of systems of nonlinear differential equations[END_REF], [START_REF] Lykova | Investigation of the solutions of nonlinear systems close to integrable systems by using the method of integral manifolds[END_REF] seems to have been the first to consider systems from the manifold standpoint with a center equation present but without an integral. However, she considered only the two-dimensional center case. Chen [START_REF] Chen | On nonelementary hyperbolic fixed points of diffeomorphisms[END_REF] has considered a corresponding problem for diffeomorphisms in the one-dimensional and two-dimensional center case. With no restriction on dimensions the center-stable, center, center-unstable manifolds have occurred only recently in the work of Pliss [START_REF] Pl!ss | Principal reduction in the theory of the stability of motion[END_REF] and Kelley [/7], [START_REF] Kelley | Stability of the center-stable manifold[END_REF]. The theorem of Pliss states (without going into detail) that the stability of the center-stable manifold is completely determined by the stability of the center manifold. Although Pliss only proved the theorem for systems of ordinary differential equations in a neighborhood of a critical point, the same theorem is true for systems in a neighborhood of a periodic orbit or periodic surface. A proof of this extension to the Pliss theorem is given in [START_REF] Kelley | Stability of the center-stable manifold[END_REF]. In [START_REF] Kelley | The center manifold and integral manifolds for Hamiltonian systems[END_REF] is found an elementary application of the concept of the center manifold to Hamiltonian systems of equations.

Professor J. Hale has brought to our attention many of the references for this paper-in particular, [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF], [START_REF] Hale | On the method of Krylov-Bogoliubov-Mitropolski for the existence of integral manifolds of perturbed differential systems[END_REF], [START_REF] Krylov | The application of methods of nonlinear mechanics to the theory of stationary oscillations[END_REF], [START_REF] Levinson | Small periodic perturbations of an autonomous system with a stable orbit[END_REF], [START_REF] Lykova | On the question of stability of solutions of systems of nonlinear differential equations[END_REF], and [START_REF] Lykova | Investigation of the solutions of nonlinear systems close to integrable systems by using the method of integral manifolds[END_REF]. The referee has pointed out that Lemma 2 below is a direct consequence of Gronwall's inequality since the right-hand derivative of the absolute value is less than or equal to the absolute value of the derivative. Section 4 of the present paper is due to D. V. Anasov.

NOTATION

The norm I • I will represent the Euclidean norm on vectors and the operator norm on matrices, and ( •, •) will represent the usual scalar product on pairs of vectors. If F = F(p) is a smooth vector valued function of the vector p, then F '1) will represent the Jacobian matrix of partial derivatives. D~ will designate the usual partial differential operator; D~ = (}IPijop'I_t ... op~", where p = (p 1 , ... , Pn) is ann-tuple of nonnegative integers, I pI= p 1 + ... + Pn, and n = dimp. In the proof of Lemma 3 below, it will be convenient to use the notation p = p 1 + p 2 , where plj = 1, 2) designates an n-tuple rather than a component of an n-tuple. The meaning will be clear from the context.

INVARIANT MANIFOLDS

Consider the real, Ck( 1 ~ k < oo) system of ordinary differential equations e =a+ B(e, x,y, z), x =Ax+ X(O, x,y, z),

(1) y = By + "?(0, x, y, z), z = Cz + 2(0, x,y, z), where A, B, C are constant square matrices in real canonical form; A has eigenvalues with negative real parts; B has eigenvalues with zero real parts (B ="' 0 is allowed); C has eigenvalues with positive real parts; 8, x, etc., are vectors; a is a constant vector; e, X, r, 2 are defined and Ck in N 8 = {(8, x, y, x)l8 arbitrary, I x I + I y I + I z I < 8}, and have multiple period w in 8; e, x, f, Z, (X, :f, Z)(x,y,z) = o when (x,y, z) = 0. Equation (I) represents a system of ordinary differential equations in a neighborhood of a critical point, periodic orbit, or periodic surface, depending on whether 8 is absent from (I), dim() = I, or dim 8 > I, respectively.

In the last two cases the condition a cj= 0 would also hold, but for our purposes one need not assume anything about a except that it is constant. THEOREM 1. For system (1) with 3 ~ k < oo, there exists invariant manifolds M+ = {(8, x, y, z)l() arbitrary, I x I < 8 1 , y = v+ (8, x)

, z = w+((), x)}, M-= {(8, x, y, z)l() arbitrary, x = u-((), z), y = v-((), z), I z I < 8 1 },
where v+, w+, u-, v-are real vector-valued functions defined and Ck-Z in some neighborhood N 8 for 8 1 sufficiently small; v+, w+, u-, v-have multiple period w 1 in (); v+, w+, u-, v-, (v+, w+, u-, v-)(x.z) = 0 when (x, z) = 0; M+, M-are (locally) unique.

For system (I) The invariant manifolds M+, M*+, M*, M*-, M-are called, respectively, the stable manifold, the center-stable manifold, the center manifold, the center-unstable manifold, and the unstable manifold.

Proof Introducing the scalar change of variables (x, y, z) ---+ (.\x, .\y, ,\z)

and multiplying e, X, r, 2 by r/>(1 X 1 2 + I y 1 2 + I z 1 2 + K.\ 2 ) where K is a sufficiently large positive constant and r/>(r) is a C 00 real-valued function satisfying rp(r) === I for 0 ~ r ~ t and ,P(r) === 0 for I ~ r < oo, we obtain 0 = a + E> (8, x, y, z, ,\), x = Ax + X(8, x, y, z, .\), [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF] y = By + Y(8, x, y, z, .\), z = Cz + Z(8, x, y, z, .\), where e(e, X, y, z, A) = c/>(1 X 1 2 + I y 1 2 + I z 1 2 + KA 2 )B(e, Ax, Ay, .\z), X(e, x, y, z, A) = 4>(1 X 1 2 + I y 1 2 + I z 1 2 + KA 2 )A-1 X'(e, Ax, Ay, .\z), etc., and the following conditions hold.

(2i) e, X, Y, Z exist and are continuous for all (e, x, y, z, A) and for each A fixed are Ck in (e, x, y, z).

(2ii) e, X, Y, Z have multiple p.eriod w in e.

(2iii) e, X, Y, Z, (X, Y, Z)c,, 11 ,z> == 0 when (x, y, z) = 0.

(2iv) e, X, Y, z == 0 for I X 1 2 + I y 1 2 + I z 1 2 ~ 1.

(2v) D~e.x,y,z)(B, X, Y, Z)--* 0 uniformly in (e, x, y, z) as A--* 0 for o ~I pI~ k.

If A ::1= 0, then systems (1) and (2) are locally (near {(e, x, y, z)le arbitrary, (x, y, z) = 0}) related by a scalar change of variables. Therefore it is sufficient to prove Theorem 1 for system (2). More precisely, however, we will prove Theorem 2 which will imply Theorem 1. in (e, x, z).

(3 ii) v+, w+, u-, v-have multiple period win e.

(3 iii) v+, w+, u-, v-, (v+, w+, u-, v-)(x,z) -0 when (x, z) = 0.

For system (2) with 2 ~ k < oo there exist invariant manifolds M[+ = {(e, x, y, z)l(e, x, y) arbitrary, z = w*+(e, x, y, A), I A I < S}, Mt = {(e, x, y, z)l(e, y) arbitrary, x = u*(e, y, A), z = w*(e, y, A), I A I < S}, Mt-= {(e, x, y, z)l(e, y, z) arbitrary, x = u*-(e, y, z, A), I A I < S}, where the following conditions hold.

(3 iv) w*+, u*, w*, u*-are real vector-valued functions defined and continuous in N 6 = {(0, x, y, z, A.)I(O, X, y, z) arbitrary, I A I < o} for some 8 > 0 sufficiently small, and for each A fixed these functions are Ck-I in (0, x, y, z).

(3 v) w*+, u*, w*, u*-have multiple period win 0.

(3 vi) w*+, u*, w*, u*-, (w*+, u*, w*, u*-)<x.y,z) == 0 when (x, y, z) = 0.

Moreover, MA+, MA-are (locally) unique (but M'f+, M'f, Mfneed not be).

Proof. Let (tfo, t, YJ, ~) where tfo = t/J(t) = !f(t, 0, x, y, z, A.), g = t(t) = t(t, 0, x, y, z, A.), etc., represent the unique solution of (2) with initial condition (0, x, y, z) at t = 0. From (2 i, iv) the solution exists and is continuous for all (t, 0, x, y, z, A.) and for each A fixed is Ck in (t, 0, x, y, z).

The functions v+, w+ which determine MA +will now be constructed as the unique solution to the differential-integral system

() = a + 6J(O, x, v+(O, x, A.), w+(O, x, A.), A.), (3a) x = Ax + X(O,
where (tfo+, g+) with tfo+ = !f+(t) = !f+(t, 0, x, v+, w+, A.), g+ = g+(t) = g+(t, 0, x, v+, w+, A.) represents the unique solution to (3a) with initial condition (0, x) at t = 0. To explicitly designate the functional dependence of the solution of (3a) on v+, w+, these functions are included in the arguments of tfo+, g+. In (3b) the functions tfo+, g+ occurring in the integrand are understood to be !f+(a) = !f+(a, 0, x, v+, w+, A.), g+(a) = g+(a, 0, x, v+, w+, A.).

Assuming (3) has a unique solution (v+, w+), v+ =v+(O, x, A.), w+ = w+(O, x, A), which satisfies conditions (3 i-iii), we can easily show that MA+ is an invariant manifold for system [START_REF] Bogoliubov | Asymptotic Methods in the Theory of Non-linear Oscillations[END_REF]. Since (3a) is an autonomous system, !f+(a, !f+(t, 0, x, v+, w+, A), g+(t, 0, x, v+, w+, A), A.)

= !f+(t + a, 0, x, v+, w+, A.), g+(a, !f+(t, 0, x, v+, w+, A), g+(t, 0, x, v+, w+, A), v+, w+, A.)

= g+(t + a, 0, x, v+, w+, A.).

Replacing (8, x) and MA + is an invariant manifold for (2).

To solve (3), inequalities involving the matrices A, B, Care basic.

LEMMA l. There exists fl. > 0, y;;;:;, 0, ifl. > y, such that, for all x, y, z, and these inequalities imply

(Ax, x) ~ -2/l.l x 12, I (By,y) I~ riY 1 2 , <Cz, z) ;;;:;, 2!1.1 z 1 2 1 t!Jt 1 ~ e;viti (-OJ< t <OJ), (4) (5)
This lemma is well known and the proof is omitted. These inequalities are introduced as a lemma because they are used extensively below.

We now develop a useful generalization of an inequality used by Hale [START_REF] Hale | Integral manifolds of perturbed differential systems[END_REF].

LEMMA 2. Let a be a nonnegative constant, and let b(t) be a continuous real-valued function defined on a finite or infinite interval I which contains the origin. If cp(t) is a C 1 vector or matrix which satisfies

I g;(t) I ~ al cp(t) I + b(t), tEl (6) then Proof. From (6) fort~ 0, I cp(t) I -I cp(O) I ~ I cp(t) -cp(O) I = If: g;(-r) dr I ~ It I g;(r) I dr ~ r {al cp(r) I + b(r)} dr. 0 0
Therefore,

I cp(t) I ~ I cp(O) I + ( {al cp(r) I + b(-r)} d-r. ( 7
)
Consider the scalar function rp(t) defined by

r/;(t) = arp(t) + b(t), rfo(O) = I cp(O) I• Thus tfo(t) = lcp(O) I + r {atfo(r) + b(r)} dr. 0 (8)
Subtracting ( 8) from ( 7), we obtain

I cp(t) I -tfo(t) ~ f a{l cp(-r) I -t/J(r)} dr 0
and it now follows from the Gronwall inequality ([1], problem I, Chapter 1) that, fort~ 0, tEl,

The proof for t < 0 is similar. This completes the proof of Lemma 2.

We now proceed to solve (3) by means of a contraction mapping in a Banach space. For l, m positive integers define

Im! = {x = x(8, x) satisfying (9i-v)}. ( 9
i) X is a real vector-valued function defined and C! for all 8 and (8, x, v(8, x), w(8, x), A.), [START_REF] Lavita | Concerning a theorem of Liapounov[END_REF] x = Ax + X(8, x, v(B, x), w(8, x), A.), where tj/V,W) = .p = tjl(t) = tjl(t, 8, X, V, W, A), [START_REF] Liapounov | Prob!eme General de Ia Stabilite du Mouvement[END_REF] g(v,w) = g = g(!) = g{t, 8, X, V, w, A.) represents the unique solution of [START_REF] Lavita | Concerning a theorem of Liapounov[END_REF] with initial condition (8, x) at t = 0. LEMMA 3. For 0 "( t < oo, all 8, I xI < 1, v E l:~-1, wE l:!-I, I A I < S 0 with S 0 > 0 chosen sufficiently small, the solution (t/J, g) of [START_REF] Lavita | Concerning a theorem of Liapounov[END_REF] [given explicitly in (11)] exists and satisfies

I X I< 1. (9 ii) dim x = m. (9 iii) x has multiple period w in 8. (9 iv) x, Xx = 0 when x = 0. (9 v) II X II = max sup sup I 1Y(9,o:)X(8, x) I < 00. o,; [PI,;; l 6 [x[ <I With the norm II •II given in (9 v), Im! is a Banach space. Define Xm 1 to be the closed unit ball in Im 1 • Let Iv! = I~im Y , etc. For v E 1:~-1 , wE l:~-I, 2 "( k < oo, consider the system 0 = a + f9
(11 i) I g(t) I "( r~'tl x J, (11 ii) I D~e.xlifl(t) I "( ex(t)ell(~) t
(1 "( I PI "( k -1), (11 iii) I D~e.x)g(t) I "( ex(t)e(-1'+/l(~)) t

( 1 "( I p I "( k -1 ),

where ex(t) is a polynomial in t with positive coefficients, f3(A.) ~ 0 is continuous in A., f3(A.)---->-0 as A---->-0; these inequalities are uniform in 8, v E l:=-1 , w E l:~-1, and (11 ii, iii) hold uniformly in I x I < 1 also.

Proof. The proof is accomplished in a finite number of steps wherein we find a succession of ex's and f3's. In (11 ii, iii) we take ex and f3 to be the largest of the ex's and f3's constructed, respectively.

IfF = F(B, x) is any smooth vector-valued function of (8, x), then

F(B, x') -F(B, x") = r Fx(B, sx' + (1 -s)x") ds • {x'-x"}. 0 In particular if F(B, 0) -0, then F(B, x) = r Fx(O, sx) ds. X. 0
Thus from (2 iii), (9 iv), for v E X 11 1, wE X/, I xI < 1, we have

X(B, x, v(B, x), w(B, x), ,\) = r {Xx(B, sx, v(B, sx), w(B, sx), ,\) 0 + Xy(B, sx, ... )vx(B, sx) + X.(B, sx, ... )wx(B, sx)} ds • x.
Hence from (2 v),

I X(B, x, v(B, x), w(B, x), ,\) I < fLI x I, (I X I < 1, I,\ I < 8 0 ) (12)
for o 0 > 0 sufficiently small. From (4), ( 10)-( 12) with I x I < I, I,\ I < o 0 , t = Ag + X(rp, g, v(rp, g), w(rp, g),,\),

(djdt)l g 1 2 = 2(Ag, D + 2(X(rp, g, ... ), t), (dfdt)l g 1 2 < -4~LI g 1 2 + 21 X II g I < -2~LI g 1 2 , which implies inequality (11 i) and the existence of (rp, g) as stated. From [START_REF] Lavita | Concerning a theorem of Liapounov[END_REF] as,\--+ 0, it follows from (4) for I,\ I < o 0 , o 0 restricted further if necessary, [START_REF] Lykova | Investigation of the solutions of nonlinear systems close to integrable systems by using the method of integral manifolds[END_REF] for all real vectors p, dim p = dim x. The procedure used to obtain (12) also yields I Xe + Xuve + X.wel < .81(,\)l g I < .81(,\)e-,.t [START_REF] Chen | On nonelementary hyperbolic fixed points of diffeomorphisms[END_REF] for I xI < 1 where Xe = X 6 (1/l, g, ... ), etc.; {3 1 (>..) ~ 0, {3 1 (>..)--+ 0 as A--+ 0.

From (2 v) and ( 13)-( 15), I f<e.x) I ,::;; /32(>..) 11/J<e.x) I + /32(>..) lg(e,x) I, (dfdt) lg<e.x) 1 2 ,::;; -2fLI g(e,x) 1 2 + 2{32(..\)e-l'tl!fl<e.x) II g(e,x) I, [START_REF] Pl!ss | Principal reduction in the theory of the stability of motion[END_REF] 11/J<e.x)(O) I = I g(e.x)(O) I = 1, I X I < I, I,.\ I <Do, where {3 2 (>..) ~ 0, {3 2 (>..)--+ 0 as)..-+ 0. From Lemma 2 for 0 ,::;; t < oo, so that (dfdt)l g(e,x) 1 2 ,::;; -2~--tl g(e x) 1 2 + 2{32(/..)e-~'tefl•<:t>t X l1 + S: e-fl.<.\>~1 g(e.x)('r)[ dr!l g(e.x) I• [START_REF] Kelley | Stability of the center-stable manifold[END_REF] By restricting D 0 further if necessary, we may assume {3 2 (>..) < f.t for I).. I ~ D 0 so that near t = 0, I g(e,x)(t) I is a decreasing function of t; I g(e,x)(t) I ,::;; I for 0 ,::;; t < E, E > 0 sufficiently small. Now compare [START_REF] Kelley | Stability of the center-stable manifold[END_REF] Since f (0) = I, the inequality j ~ -f.tf implies f ( t) > 0 for 0 ,::;; t < oo. Thus J,::;; ( -~--t + /32(/..)) f + {32(>-.)e<-l'+fl.<mt{l + 2t}, f (t) ,::;; {1 + f3 2 (A)(t + t2)}e<-1'+fl2<mt, and by restricting D 0 further if necessary we have If (t) I ,::;; 2 for 0 ,::;; t < oo so that (20) holds for all 0 ~ t < oo. Hence we have proved (11 iii) for I p I = 1. The crude inequality I g<e.x>(t) I ~ 2 in (17) yields so that (11 ii) holds for I p I = I. We now proceed inductively. Let DfB,x)rf = rfp, D(e,x)g = qP, and consider the case p = p 1 + p 2 , I p 1 I = I P2l =I, I pI = 2. Assuming k;? 3 (otherwise we are done), we have from (

+ [X8 + Xyv8 + X.wa]!f" + X.
(djdt)f2 = -21Lf2 + 4,BiA)rx 3 (t)e<-!'+fJ,<;>.>>tl, 1(0) = t, j = -~L I+ 2,Bi.\)rxa(t)e<-!'+fl,u))t, I (0) = t,
where exa(t) is a polynomial in t with positive coefficients, exa(t) ~ cx 2 (t), exa(t) for 0 ~ t < oo; {3-t(>..) ~ {3 3 (>..), [f.Ja(>..)]2 ~ 0; f.Ji>..)-+ 0 as >..-+ 0; it follows that (25) fort ~ 0 as long as I gP(t)

I ~ 1. Since j ~ -fl-f, f(O) = t,
we havef(t) > 0 for 0 ~ t < oo, so that j ~ ( -~L + f.Ji>..))f + 2,Bi>..)exa(t)e<-"+ll,<-'llt,

f(t) ~ e<-1'+/l,<-'llt ~~ + J: 2,Bi>..)cxa(r) drl. ( 26
)
Thus by restricting 8 0 further, if necessary, it follows that f (t) ~ 1 for 0 ~ t < oo and hence ( 23)-( 25) are valid for all 0 ~ t < oo. Inequalities (24)- (26) show that (11 ii, iii) hold for 1 ~ I p I ~ 2. By continuing in this manner (a finite number of steps), one proves (11 ii, iii) valid for all1 ~ I p I ~ k -1. This completes the proof of Lemma 3.

For v E X~-

and (!fo<v,wl, g<v,wl) is the solution of (10) which is given explicitly in [START_REF] Liapounov | Prob!eme General de Ia Stabilite du Mouvement[END_REF]. Proof. The fact that T maps .I~-1 X .I~-1 into itself for 1 A I ~ 8 1 , 8 1 sufficiently small, is an immediate consequence of (2 iii, v), [START_REF] Hale | Integral manifolds of perturbed differential systems[END_REF], and Lemma 3.

For (v, w) E X~-1 X X~-1 define T(v, w) = (T 1 v, T 2 w).
To show that T is a contraction in the Ck-2 topology, it is sufficient to show that T 1 is a contraction on .I~-1 in the Ck-2 topology uniformly in w E .I!-I, and similarly that T 2 is a contraction on .I~-1 in the Ck-2 topology uniformly in v E .I~-1 • We will give the argument for T 1 ; the argument for T 2 is completely analogous. To show T 1 a contraction, it is sufficient to show uniformly in e, I X I < I, I A I ~ 81' wE x;-1 , 0 ~ I pI ~ k-2, that the inequality (28) holds for vi, v2 E .{~-1.

Let (yi, gi) = (tjJ(vi,w>, g(vi,w>), (j = 1, 2), and let P = y(v;,w) = Y(tjJi, g1, ... ),(j = I, 2). To prove(28)we will show that uniformly in{}, I xI <I, I A I ::;:;; 8 1 , wE .I~-1 , 0 ~ I p I ~ k -2, the inequalities

I D~e.xM1 -tjJ2)1 ~ cx(t)eflWt II vl -v211, (29)
I D~e.x)W -g 2 )1 ::;:;; cx(t)eh•+J'I(,\))t II v 1 -v 2 ll
hold where cx(t) is a polynomial in t with positive coefficients, {3(A) ;:?: 0 is continuous in I A I ~ 8 1 , {3(A) --* 0 as A -+ 0. If we suppose (29) valid, then, by restricting 8 1 to be sufficiently small, inequality (28} is immediate. One computes from (27)

D{e.xl(T 1 v 1 -T 1 v 2 ) = r e-B"D(e,xl(YI -Y 2
) da +oo and then uses (2 iii, v), (5), Lemma 3, and (29) to verify that the interchange of differentiation and integration is valid and that {28) holds. The mean-value theorem as presented at the beginning of the proof of Lemma 3 is used repeatedly.

Hence it remains to prove (29). From [START_REF] Lavita | Concerning a theorem of Liapounov[END_REF],

(dfdt)(tjJ1 -tjJ2) = (91 -(92, (djdt)(e -g 2 ) = A(e -g 2 ) + X 1 -X 2 , ( 30
)
I (dfdt)(tjJl -ifJ2) I ~ I f91 -f92
I + II v 1 - v 2 II, (31)
Thus where ~2(A) = 3~1(A) ~ 0, (3 2 (A)-+ 0 as A-+ 0. By using properties (2 iii, v) and applying the mean-value theorem, we obtain I XI -X 2 I ,s;; /3 3 (A){I g 1 I + I gz I} •{I t/J 1 -t/J 2 I + I g1 _ gz 1 + I vl(!fl, g1) _ v2(!f2, e) 1 + I w(t/J\ e) -w(t/J 2 , e) 1},

where {3 3 (A) ~ 0, (3 3 (>..)-+ 0 as A-+ 0. From (11 i) for I xI < 1, I e(t) I + I g 2 (t) I ,s;; 2e-~'t, so that (31) and ( 33) in (32) yields

I X 1 -X 2 I ,s;; (3 4 (>..)r~'t{l tf1 -t/J 2 I + I g 1 -gz I + II v 1 -v 2 ll}, (33)
where (3 4 (>..) ~ {3 3 (A) + f3lA) ~ 0, (3 4 (>..) -+ 0 as A-->-0. By restricting <\ to be sufficiently small we may assume (3 4 (A) ,s;; JL for ) >.. ) ,s;; 8 1 ; then from (30)

we finally obtain I (d/dt)(t/1 1 -t/1 2 ) I ,s;; f3i.\){l t/1 1 -t/1 2 I + I g 1 -e 2 I + II v 1 -v 2 1\}, (34a) (dfdt)l eg 2 1 2 ,s;; -2fLI gl -e 2 1 2 Since rftl(O) -rp 2 (0) = 0, we have from (34a) and Lemma 2, for t ~ 0, Since g 1 (0) -gz(O) = 0, the inequality

(35)
is valid in some neighborhood of t = 0, and as long as (35) remains valid 29) is proved for the case I p I = 0.

(t ~ 0), I rp 1 (t) -rp 2 (t) 1 ~ 2,8
Assuming that k ? 3 (otherwise we are finished), we now want to show that ( 29) is valid for 1 pI = 1. Let D~e.x)tf 1 = t/J/, etc. Then from [START_REF] Lavita | Concerning a theorem of Liapounov[END_REF],

(dfdt)(tfi/ -tfp 2 ) = Be 1 tP/ -Be 2 tfp 2 + Bx 1 t/ -e, 2 t/ + •••, (dfdt)(t/ -tp 2 ) = A(t/ -tp 2 ) + Xitfi/ -X 8
= •••, 1)*+ =•••,represents
the unique solution of (40a) with initial condition (8, x,y) at t = 0. This system is also solved by iteration, but with the following modification. For l, m positive integers define

Xm 1 = {x = x(B, x,y) satisfying (41 i-v) below}.
(41 i) xis a real vector-valued function defined and C 1 for all (B, x,y).

(41 ii) dim x = m.
{41 iii) X has multiple period win B.

{41 iv) x, X<x.v) == 0 when (x, y) = 0. {41 v)
• II X II = max sup I D1e.x.vJx(B, x, y)l < oo.

0~ jpj ~! (B,x,y)
With the norm in (41 v), Xm 1 is a Banach space. Define Xm 1 to be the closed unit ball in Xm 1 • For conciseness let 1:. 1 = x~imz. Now, for wE x.k consider the system

B = a + B(B, x, y, w(B, x, y), A.), x = Ax + X(8, x, y, w(8, x, y), A.), y = By + Y(B, x, y, w(8, x, y), A.). (42)
Let ,pw = ,P(t) = ,P(t, 8, x, w, A.), tw = ... , 7Jw = ... , represent the unique solution of (42) with initial condition (8, x,y) at t = 0. Now define the transformation T acting on Xzk as follows. For wE X/, (Tw)(B, X, y) = r e-cazc,pw, tw, 7)w, w(,pw, tw, 7Jw), ,\) da.

+OO

Corresponding to Lemma 4, we can now prove LEMMA 5. For I ~ k < oo and I,\ I ~ 8 2 , 8 2 > 0 sufficiently small, the transformation T maps Xzk into itself and is a contraction in the Ck-l topology:

II •11 = II •llk-1 = max sup I D1e.x,y) • 1. o~ IPI ~ k-l <e,x,y)
The proof of Lemma 5 is analogous to the proof of Lemma 4. In fact, the details are even easier to carry out. The function w*+ = w*+(8, x, y, A.) is the unique fixed point of T.

The proof of the existence and smoothness of the center-unstable manifold is similar to that for the center-stable manifold. Once we have both Mf+ and M[-, then However, we can also construct the center manifold MI directly by constructing the functions u* = u*(8, y, .\), w* = w*(8, y, .\) as the unique solution of the following differential-integral system:

(j = a + 0(8, u*(8, y, A), y, w*(8, y, A), A), where tf;* = tf;*(t) = tf;*(t, 8, u*, y, w*, ,\), "'* = ... , represents the unique solution of (43a, b) with initial condition (8,y) at t = 0. The procedure followed here is similar to the procedure used to solve (40).

If another invariant manifold

M~ = {(8, x, y, z)l8 arbitrary, I x I < 1, y = v'(8, x, ,\), z = w'(8, x, ,\)}
satisfies all the properties of MA+, then M; is composed of solutions of (2) which we designate tf;', g', YJ 1 = v'(.j;', g', ,\), ~~ = w'(tf;', f, A.). Since I g' I goes exponentially to zero as t ____, . . o, so do 1 v' ( tf;', g', A.) 1 and 1 w' (if;', r, .\) 1• Therefore v', w' must satisfy (3); but since the solution of (3) is unique, (v', w') = (v+, w+) and M; = MA+. A similar argument shows that M 11 -is also unique. An example of non-uniqueness for the center manifold is given in Section 4 below, and this same counter-example can be used to show non-uniqueness for the center-stable and center-unstable manifolds also. This completes the proof of Theorems 1 and 2.

Let us point out what should already be obvious. Namely, the reason that the center-stable manifold has one more derivative than the stable manifold is because the factor e-Ca occurring in the integrand in ( 40b) is an exponently converging factor whereas the factor e-Ba occurring in the integrand in (27a) is not. (See (5).) However, all the manifolds have one more derivative. This will be discussed in Section 5 below. which can be integrated to yield where c is the constant of integration. This give us the following phase portrait in the (y, x) plane for system (44). ---------~------~--- i' = z to (44) we see that the center-stable and center-unstable manifolds are also non-unique. If, however, the center-stable manifold is stable [the origin is (Lyapounov) stable with respect to the center-manifold], then it is not difficult to show that the center-stable manifold is unique. With respect to -t, the same is true of the center-unstable manifold. If both the centerstable and center-unstable manifolds are unique, then the center-manifold is also unique.

Let X \ \ --~-

ADDITIONAL SMOOTHNESS

In our construction of M~+ and MA-, the procedure was to construct a mapping T on the closed unit ball ik-l of the appropriate Banach space with a Ck-1 topology. Then we proved

T : ik-1 ~ ik-l
and T is a contraction in the Ck-2 topology. Thus the fixed point (say, p) of T has k -2 derivatives. But also p is the limit in the Ck-2 topology of elements in ik-1 • Thus the (k -2)th derivatives of p are uniformly Lipschitzian. Using this fact and the proof method of Theorem 4.2. on page 333 of [1], one can show that, for ,\ sufficiently small, p E Ck-1 , and even more, p E ik-1 . (The details of this program are quite laborious so we do not present them here.) Thus the manifolds M~+, M~-E Ck-1 where system (2) is Ck, 2 , . ;:;_; k < oo. An analogous argument shows that MJ+, MJ, Mf-E Ck where system (2) is Ck, I , . ;:;_; k < oo.

When the y-equation in (1) is absent, then M+ = M*+, M-= M*-.

Therefore in this case M+, Mare as differentiable as system (1). This fact will be used in Section 7 below.

Finally, we remark that multiple periodicity in 8 for system (l) is not essential in the proof of Theorem I. Rather, one needs only to be able to put the original system in a form similar to (2), but without multiple periodicity in 8. Of course, the invariant manifolds also will not exhibit multiple periodicity in 8.

PERTURBATION THEORY FOR M*+, M*, M*-

For simplicity we will not discuss the perturbation theory of M+, Min the general case. But notice that when the center equation (y-equation) is absent from (1), then M*+ = M+, M*-= Mso that the results of this section apply toM+, M-in that special case.

Consider the real Ck, 1 , . ;:;_; k < oo, system of ordinary differential equations e =a+ 8 (8, x,y, z, E), X= Ax+ X(8, x,y, z, E), y = By + "? (8, x, y, z, E), z = Cz + 2(8, x, y, z, E), (45) where a, A, B, Care as in (1); 8, x, etc., are vectors; Eisa real (perturbation) scalar; 8, X, ?, 2 are defined and Ck in N 8 = {(8, x, y, z, E)l8 arbitrary, I x I + I y I + I z I + I E I < 8} for some 8 positive and have multiple period w in 8; e, x, ?, 2, (X, ?, 2)(m, 11 ,z) -0 when (x, y, z, E) = 0. Thus when E = 0, (45) reduces to a system of the form (1). The proof of this theorem is essentially a copy of the proof of Theorem 1. One merely introduces a scalar change of variables (x, y, z, E)-+ (.\x, .\y, ,\z, ,\ 2 E) and then changes the system outside a neighborhood of the (x, y, z, E) origin similar as in the proof of Theorem 1. The essential property of the transformed system will be the analog of (2 v), namely, D'/e.x.v.z,,)(e, X, Y, Z)-+ 0 uniformly in (0, x, y, z, E) as ,\-+ 0 for 0 ~ I p I ~ k, where analogous to the procedure in the proof of Theorem 1, 8(0, X, y, z, E, ,\) = .P(I X 1 2 + I y 1 2 + I z 1 2 + E 2 + K,\ 2 )B(O, AX, ,\y, AZ, A 2 E), etc. With this property there is no difficulty in solving the appropriate differential-integral system for w*+, etc., provided ,\ is sufficiently small.

If there is no center equation (y-equation) in (45), then M*+ = M+, M*-= M-. Since we haven't defined M+, M-in the general perturbation case, we can take this as a definition. Also, with no center equation in ( 45), the center manifold

M* = {(0, x, z)IO arbitrary, x = u*(O, E),
is the same as what is known as the periodic surface. There is an extensive literature concerning this invariant manifold. See, for example, [2], [START_REF] Diliberto | Perturbation theorems for periodic surfaces[END_REF], [START_REF] Hale | Integral manifolds of perturbed differential systems[END_REF], and [START_REF] Sacker | A new approach to the perturbation theory of invariant surfaces[END_REF]. As a corollary to Theorem 3 we have that the periodic surface is as differentiable as the system of differential equations.

PERTURBATION THEORY WITHOUT CENTER

In this section we want to discuss how M+, M-vary with respect to the perturbing function.

Consider the real Ck, 1 :s;; k < oo, system of ordinary differential equations 8 = a + e(B, x, y) + Ef?J(B, x, y), x =Ax+ X(8, x, y) + EX (8, x, y), y =By+ Y(8, x,y) + EY(IJ, x,y), (46) where A, B are constant matrices in real canonical form; A has eigenvalues with negative real parts; B has eigenvalues with positive real parts; 6, x, etc. are vectors; a is a constant vector; € is a perturbation parameter; e, e, X, X, Y, Y are defined and Ck in N 8 = {(8, x, y)IIJ arbitrary, I xI + I y I < 8} and have multiple period w in 8; e, X, Y, (X, Y)<x.ul = 0 when (x, y) = 0.

Let P (for perturbation) represent the triple (EJ, X, Y). From Theorem 4

we know that locally (for (x, y, E) sufficiently small) there exists stable and unstable manifolds M± = M±(P, E). Since M±(P, 0) are independent of P, let M±(P, 0) = M 0 ±. Define Xk,l to be the set of all triples P = (EJ, X, Y) where 8 0 is a sufficiently small positive constant which is independent of P. Moreover, max sup sup I D~a.x)(v+((}, x, P, E) -v+(6, x))l --+ 0 0~\Pi<;;l a txl<5 0 as E--+ 0 uniformly in P E Xk,l, where v+ (8, x) ~ v+((}, x, P, 0).

which
The proof of Theorem 4 is also essentially the same as the proof of Theorem 1. After introducing a scalar change of variables and changing the system outside a neighborhood of the origin, the assertions are readily proved. But the assertions are logically equivalent for both the original system (46) and the transformed system. A similar theorem holds forM-. Proof It is sufficient to prove this theorem for a system which has been transformed from ( 46). Let ()=a+ 8 (8, x,y, e),

x =Ax+ X(8, x,y, e), y =By+ Y(8, x,y, e) be a transformed system. If we introduce the change of variables p = x-u-(8,y, e), q = y -v+ (8, x, e), then, in these new coordinates, (47) has the form ()=a+ B(8,p, q, e), p = Ap + P(8,p, q, e), q = Bq + Q(8,p, q, e), where I P(8,p, q, e) I::;;; fLIP I, I Q(8,p, q, e) I ::;;; fLI q I, Thus we see that p E Ck-1 and that p satisfies (50) provided e, X, y have sufficiently small first-order derivatives. From ( 5), ( 49), (50), (djdt)Jp \ 2 = 2(Ap,p) + 2(P,p) ~ -4JLIP \ 2 + 2JLIP \ 2 ~ -2JLIP \ 2 , (dfdt)J q \ 2 = 2(Bq, q) + 2(Q, q) ~ 4{-t[ q J 2 -2{-t[ q J 2 ~ 2fLI q [ 2 •

Our theorem follows immediately from these differential inequalities. Finally we state a theorem which is closely related to Theorem 4. We want to show that M+(P, E) --4-M+(P 0 , E) in the C 1 -1 topology as P --4-P 0 in the C 1 topology where E is small but fixed. A similar statement will hold relative toM-. THEOREM 6. For P, P 0 E Xk,! if P --4-P 0 in the C 1 topology, max sup sup [ IY( 9 ,.,,y)(P -P 0 )\-4-0, o,;; IPI,;;t B i"'l+lvl <6 then v+(B, x, P, E) --4-v+(B, x, P 0 , E) in the C 1 -1 topology, The proof of Theorem 6 is obtained by introducing the change of variables (48) relative to P 0 , p = xu-(B,y, P 0 , E), q = y -v+(B, x, P 0 , E). System ( 49) will now only be Ck-1 , but otherwise Theorem 6 reduces to Theorem 4 with only minor modifications.

  with 2 ~ k < oo there exist invariant manifolds M*+ = {(8, x, y, z)IB arbitrary, I x I + I y I < 8 1 , z = w*+((), x, y)}, M* = {(8, X, y, z)l8 arbitrary, X = u*((), y), I y I < 81 ' z = w*((), y)}, M*-= {((), x, y, z)l() arbitrary, x = u*-(8, y, z), I y I + I z I < 8 1 }, where w*+, u*, w*, u*-are real vector-valued funct£ons defined and Ck-l in some neighborhood N 81 for 8 1 sufficiently small; w*+, u*, w*, u*-have multiple period win 8; w*+, u*, w*, u*-, (w*+, u*, w*, u*-)(x,y,z) == 0 when (x,y, z)= 0 (M*+, M*, M*-need not be unique).

THEOREM 2 .

 2 For system (2) with 3 ~ k < oo there exists invariant manifolds M;.+ = {(e, x,y, z)le arbitrary, I xI < 1, y = v+(e, x, A), z = w+(e, x, A), I A I < S}, M~.-= {(e, x, y, z)le arbitrary, x = u-(e, z, A), y = v-(e, z, A), I z I < 1, I A I < S}, where the following conditions hold. (3 i) v+, w+, u-, v-are real vector-valued functions defined and continuous in N 1 8 = {(e, x, z, A)le arbitrary, I xI + I z I < 1, I A I < S} for some S > 0 sufficiently small, and for each A fixed these functions are Ck-2

  with the real scalar equation with the initial condition f (0) = 1. As long as If (t) I ,::;; 2 holds, it follows that I g(e,x)(t) I ,::;; f (t).

  j = -fLf + {32(/..)e<-l'+fl.(A)}t ! 1 + r e-fl.(.\h2dr!.

0

  ,e!fpip, + .... The notation Beerfp 2 rfp 1 , etc., is defined by writing out (21) in complete detail. Since (II ii, iii) hold for I p 1 I = I p 2 l = I, we obtain from (21)I ~PI ~ .Sa(A)I!ftP I + ,8 3 (A)I qP I + ,8 3 (A)rx 1 (t)efl,Wt (dfdt)l gp 1 2 ~ -2/LI gp 1 2 + 2,8 3 (A)e-~' 1 l!ftp II gp I + 2,8 3 (.\)rxl(t)e!-t.•+fl.w>tl gp I, l!ftp(O) I = I gp(O) I = 0,(22)where rx 1 (t) is a polynomial with positive coefficients and ,8 3 (A) ;? ,8 2 (,\) ;? 0, ,8 3 (A) ~ 0 as A ~ 0. From Lemma 2 Since I qP(O) I = 0, it follows that near t = 0(23) and as long as this inequality holds, it follows that (24) where rxlt) = r {I + rx!(T)} dr.Using inequality (24) in (22) we haveComparing this differential inequality with the scalar equation

LEMMA 4 .

 4 For 2 ~ k < oo and for I A I ~ 8 1 ~ 8 0 , 8 1 > 0 sufficiently small, the transformation T maps X~-1 X X~-1 into itself and is a contraction in the Ck-z topology: ll(v, w)ll = max (II vII, II w Jl), II •11 =II •lik-2 = max sup sup I D'(e.a:l • j.

  o<;: IPI<;:k-2 e 1"' 1 <1

+ 2 (

 2 3iA)e-~'t{l !flt/12 I + \1 vi-v211}. I gl-g2(. (34b)

  8, y, A) = r e-AaX(.j;*, u*(if;*, YJ*, A), YJ*, w*(if;*, YJ*, A),,\) du, -oo y =By+ Y(8, u*(8, y, ,\),y, w*(8, y, ,\), ,\), (43b) 0 w*(8, y, A) = J e-caz(if;*, u*(.j;*, YJ*, A), YJ*, w*(tf;*, YJ*, ,\),A) du, +co

4 .

 4 NoN-UNIQUENESS OF THE CENTER MANIFOLD Consider the pair of real scalar equations x = -x, (44) Dividing x by y we obtain dxfdy = -(xfy 2 )

  , c)= IO ( for y < 0, for y?: 0. Clearly, u(O, c) = u 11 (0, c) = 0 so that M(c) = {(x,y)[x = u(y, c), y arbitrary} is a center manifold for each real constant c. By adding the scalar equation

THEOREM 3 .

 3 For system (45) there exists invariant manifolds M*+ = {(8, x, y, z)l8 arbitrary, I x I + I y I + I E I < 8 1 , z = w*+(8, x, y, E)}, M* = {(0, x, y, z)IO arbitrary, x = u*(O, y, E), I y I + I E I < 8 1 , z = w*(O, y, E)}, M*-= {(0, x, y, z)JO arbitrary, x = u*-(0, y, z, E), I y I + I z I + I E I < 81}, where w*+, u*, w*, u*-are real vector-valued functions defined and Ck in some neighborhood N 8 for 8 1 sufficiently small; w*+, u*, w*, u*-have multiple 1 period w in 0; w*+, u*, w*, u*-, (w*+, u*, w*, u*-)<x.v,z) = 0 when (x,y, z, E)= 0.

  Since A, B in (46) have eigenvalues with nonzero real parts, M* = M+nM-. THEOREM 5. M+, M-have the following characterization for e sufficiently small: M+ = {(8, x,y)l(.f, ~. 'TJ)-+ M* as t-+ +oo}, M-= {(8, x,y)l(.f, ~. 'TJ)-+ M* as t-+ -oo}, where if= .f(t, 8, x,y, e),~= •••, ' TJ = •••, represents the unique solution of (46) with initial condition (8, x, y) at t = 0.

  fL is the positive constant given in[START_REF] Hale | Integral manifolds of perturbed differential systems[END_REF]. [B in (49) is C in[START_REF] Hale | Integral manifolds of perturbed differential systems[END_REF].] Let us compute P, for example, to see that inequality (50) is true. From (48),p = x -v-= Ax +X -v 8 -{a + 8}v., -{Ax +X} = Ax -v 8 -{a + 8(8, v-, y, e)} -v., -{Ax + X(8, v-, y, e)} + X(8, p + v-, y, e) + v 8 -{8(8, v-, y, e) -8(8, p + v-, y, e)} + v.,-{X(8, v-,y, e)-X(8,p + v-,y, e)} = Ax -Av--X(8, v-, y, e) + X(8, p + v-, y, e) + v 8 -{8(8, v-, y, e) -8(8, p + v-, y, e)} + v., -{X(8, v-, y, e) -X(8, p + v-, y, e)}= Ap + P(8,p, q, e).

  10) ~p = [618 + Buve + e.we]fp + [61., + Buvx + e.w.,]qp + 61 88 fp.fp, + ... , gp = [A+ X.,+ Xuvx + X.w.,]qp

	(21)

  4 (,.\)tefl,CAltiJ v 1 -v21J.

	0 1 that
	f(t):s;;;llv1-v21J
	holds for all 0 :s;;; t < oo. Thus (35)-(37) and (39) are valid for all 0 :s;;; t < oo
	and (
	(36)
	Using (36) in (34b) we obtain
	(dfdt)l e _ g2 1 2 ~ _ 2 ,., 1 g1 _ g2 1 2 + 2,8 4 (-\)a: 1 (t)eh•+fJ 4 WHIJ vi -v2 11 • 1 g1 _ g2 I,
	(37)
	(39)
	f(t) ~ el-i•+f3,WJt g II v 1 -v211 + ( ,8 4 (/..)cxt(r) drl/ v 1 -v 2 II I,
	f(t) ~ j~ + ,84(A)a:2(t)leh+!l,WJtll vi -v2 II,

where a: 1 (t) = 1 + 2,8 4 (-\)t.

Comparing (37)

with the scalar equation with f (0) = ill v 1 -v 2 II > 0, we see that holds so long as (35) remains valid. From (38), j = -p.f + ,Bi-\)a:t(t)eh<+llplltjl vi -v211 and since, for 0 ~ t < oo, is valid, it follows thatf(t) > 0 for 0 ~ t < oo and therefore where j ~ ( _,_, +,Bit..))/+ ,84(/..)o:l(t)eh+fJ,O.Jltll vl -v2JI, Therefore, by restricting o 1 further if necessary, we have for I ,\ I :s;;;

  2 t/lp 2 +. •••,

	x = Ax + X(8, x, y, w*+(8, x, y, .\), .\),	(40a)
	y = By + Y(8, x, y, w*+(8, x, y, .\), .\),
	w*+(8, x, y, .\) = r e-CaZ(tfi*+, t*+, 17*+, w*+(tfJ*+, t*+, 17*+, .\), .\) da, (40b)

and, by means of (2 iii, v), Lemma 1, Lemma 3, inequality (29) for the case I p I = 0, and the mean-value theorem, one achieves I (dfdt)(t/J/t/Jp 2 I :s;;; .Bs(.\){1 t/J/ -t/Jp 2 1 + I t/ -tp 2 1} + ,85(.\)<Xa(t)e-86 1 mll V 1 -V 2 II, (dfdt)l t/ -gp2 12 :s;;; -2fLI e -g2 12 + 2,85(.\)<Xa(t

)eh'+!!,Wlt • •{I t/J/t/Jp 2 I + II v 1 -v 2 ll} • I e -e I,

where <X 3 (t) is a polynomial in t with positive coefficients, ,8 5 (.\) ? 0, ,8 5 (.\)-+ 0 as ,\ -+ 0. By restricting o 1 further if necessary, one now readily establishes that (29) is valid for all I p I :s;;; 1. In an analogous manner one proceeds inductively (a finite number of steps) to establish(29) for all 0 :s;;; I p I :s;;; k-2.

This completes the proof of Lemma 4.

The fixed point of the transformation T is designated v+ = v+(8, x, .\), w+ = w+(8, x, .\), and these functions define the stable manifold MA+• The unstable manifold MAis constructed in an analogous manner.

The function w*+ = w*+(8, x, y, ,\) which defines the center-stable manifold M[+ is taken to be the unique solution to the differential-integral system e = a + 8(8, x, y, w*+(8, X, y, ,\), ,\),

+"'

where tfi*+ = tfi*+(t) = tfi*+(t, 8, x,y, w*+, .\), t*+

  Uniformly in P E Xk,l, v+ is defined and Ck on N 8 o = {(8, x, E)l8 arbitrary, I x I < 8 0 , I E I < 8 0 },

are defined and Ck in N 8 , have multiple period w in 8, and satisfy max sup sup I D~e.x,y)P I :s;; 1, O~IPI~l a txl+lul<8

where l < k is a positive integer. For P E Xlc,l let M+(P, E) = {(8, x, y)l8 arbitrary, y = v+(IJ, x, P, E)}, where P in the argument of v+ denotes a functional dependence, THEOREM 4.