Composite Asymptotic Expansions

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris

For further volumes: http://www.springer.com/series/304

2066

•

Augustin Fruchard



Reinhard Sch¨afke

Composite Asymptotic Expansions

123

Augustin Fruchard Laboratoire de Math´ematiques, Informatique et Applications Universit´e de Haute Alsace Mulhouse France

Reinhard Sch¨afke Institut de Recherche Math´ematique Avanc´ee Universit´e de Strasbourg Strasbourg France

ISBN 978-3-642-34034-5 ISBN 978-3-642-34035-2 (eBook) DOI 10.1007/978-3-642-34035-2 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012953999 Mathematics Subject Classification (2010): 41A60, 34E, 34M30, 34M60 c Springer-Verlag Berlin Heidelberg 2013  This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The purpose of this memoir is to present a theory of asymptotic expansions for functions of two variables, using at the same time functions of one variable and functions of the quotient of these two variables. These composite asymptotic expansions (CAsEs for short) are particularly well suited to the description of solutions of singularly perturbed ordinary differential equations. Their use is classical for boundary layers, but less familiar near turning points. Let us describe the context in a few words. Consider an equation of the form "

dy D ˚.x; y; "/ dx

(1)

where ˚ is infinitely differentiable, x and y are real or complex variables, and " is a small parameter, real positive or in a sector of the complex plane. Such an equation might be obtained from an autonomous slow–fast system du D "f .u; v; "/ dt dv D g.u; v; "/ dt

(2)

by elimination of the variable t, i.e. by considering the function y D v ı u1 . The slow set of (1) is the subset L of C2 determined by ˚.x; y; 0/ D 0. A point .x; y/ of L is called regular if @˚ .x; y; 0/ ¤ 0; otherwise it is called a turning @y point . In a neighborhood of a regular x; e y / 2 L , one easily sees that (1) has a P point .e unique formal solution b y D n0 yn .x/"n and it is known that certain solutions of (1) have b y as asymptotic expansion in appropriate domains. The classical theory of composite expansions (see e.g. Vasi’leva/Butuzov [59]) is useful to describe the boundary layer (also called inner layer) of a solution which takes at e x an initial value close enough to e y . For example, in the real framework, if the point .e x; e y / is

v

vi

Preface

< 0, then an approximation of the solution y D y.x; "/  P can be given, uniformly on some interval Œe x ;e x C ı, of the form n0 yn .x/ C  x  n " , containing the formal solution b z xe y as well as functions z exponentially

attracting, i.e.

n

@˚ .e x; e y ; 0/ @y

n

"

decreasing at infinity. See also Benoˆıt/El Hamidi/Fruchard [4] for details. At a turning point .x  ; y  /, the coefficients of the formal solution may have poles or ramified singularities and the classical theory of composite expansions no longer applies. The most common method to obtain an approximation of the solutions is the matching of the so-called inner and outer expansions; in our memoir we would like to propose an alternative. We present an extensive study of infinite CAsEs of the form  X   an .x/ Cgn xx n ; 

(3)

n0

 a certain root of ", that are uniformly valid for x in some sector with vertex x  satisfying K jj  jx  x  j  L with some K; L > 0, and sometimes even in a full neighborhoodof x  of size O .jj/. Such CAsEs provide outer expansions  with respect to  and inner expansions by replacing by re-expanding gn xx  x D X and re-expanding an .X /. One advantage of composite expansions is that they provide approximations also in “intermediate” ranges K jj˛  jxj  L jjˇ , where 0 < ˇ  ˛ < 1. Composite expansions and even more composite approximations are not new. In the context of matched asymptotic expansions, they are often found as the sum of inner and outer expansions, less the terms common to both see e.g. Kevorkian/Cole [35], p.13, or Skinner [54], p.4. We propose to work directly with composite expansions using the many properties and theorems we present and compare with inner and outer expansions in a second step. More remarks concerning the relations with previous work will be given in Chap. 7. There is a vast literature on singular perturbation methods for ordinary differential equations. The presentation of another book on this subject should be well motivated. We hope to shed some new light on composite expansions with the following features. • We work with infinite asymptotic expansions, discuss their algebraic and analytic properties, and use them to obtain new results. • The expansions are mainly presented in the complex domain. Their regions of validity are modified sectors with vertex at a turning point. • We also present a Gevrey theory of CAsEs—so far the Gevrey theory was confined to classical uniform asymptotic expansions where the coefficients depend upon one or more variables different from the expansion variable. In the literature on singularly perturbed ordinary differential equations the Gevrey theory was mainly used in full neighborhoods of a turning point

Preface

vii

(e.g. Canalis-Durand/Ramis/Sch¨afke/Sibuya [9]) whereas here we have (quasi-) sectors with vertex at a turning point. • We exhibit three applications, in two of which new results are obtained, thanks to the Gevrey theory of CAsEs. The plan of the memoir is as follows. In the first chapter, we present simple examples of linear differential equations to convince the reader that composite asymptotic expansions are natural, even unavoidable, near turning points. The second chapter presents the definition of CAsEs and their compatibility with algebraic and analytic operations and discusses the relation with the method of matched asymptotic expansions. The third chapter contains the Gevrey version of the theory of CAsEs. The relation with exponentially small terms is an important feature. Chapter 4 states and proves our most important theorem on the existence of Gevrey CAsEs. It essentially states, for a family of holomorphic functions of .x; / that are bounded on a so-called consistent good covering of a punctured neighborhood of .0; 0/, that each of them has a Gevrey CAsE provided their differences are exponentially small. We emphasize that here, unlike in the classical Gevrey theory, two kinds of exponential smallness appear: the differences in the p p x-plane are O e cjxj =jj and hence only small away from the origin whereas the   p differences in the -plane are O e c=jj as usual. Chapter 5 applies the theory of CAsEs to singularly perturbed nonlinear ordinary differential equations of first-order near turning points. We present two methods for obtaining CAsEs. The first is direct, following the classical lines: study formal solutions of the form (3), then prove the existence of an analytic solution having this formal solution as a CAsE. The second method is more indirect and uses the main result of Chap. 4, embedding a single solution in a family of solutions on a good covering of a neighborhood of the turning point and showing that their differences satisfy exponential estimates. A reader familiar with differential equations in the real domain might find this method too “complex,” but it is, to our knowledge, the simplest one in the context of singular perturbation. We first treat a type of equations allowing analytic continuation of the solutions to a small neighborhood of the turning point and then consider more general equations. The most powerful result in this section is Theorem 5.17. In the sixth chapter, we apply the previous results and use the general properties of CAsEs for several well-known problems of singular perturbation: canards solutions near multiple turning points, non-smooth canards, and Ackerberg–O’Malley resonance. The linear differential equation of second order appearing in the third problem is reduced to a first-order Riccati equation to which our results are applied. For the first and third problems, we obtain new results, and for the second problem, we can add Gevrey properties and Gevrey CAsEs to the known results. The memoir focuses on ordinary differential equations and only considers firstorder nonlinear and second-order linear equations, but we are convinced that CAsEs

viii

Preface

can be very useful for equations of higher order and for other types of functional equations, especially difference equations. We have included a few exercises for the benefit of the reader—in most of them, the results of the corresponding section are applied to examples. Acknowledgements The work of the second author was supported in part by grants of the French National Research Agency (ref. ANR-10-BLAN 0102 and ANR-11-BS01-0009).

Contents

1

Four Introductory Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Extensions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Second Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Third Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Fourth Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6 Composite Expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 6 9 11 13 14

2 Composite Asymptotic Expansions: General Study .. . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Composite Formal Series . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Composite Expansions: Definition and Basic Properties . . . . . . . . . . . . . 2.4 Composite Expansions and Matching .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Continuation of Composite Expansions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Quotients of CAsEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 Multiple CAsEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

17 17 19 22 29 34 37 40

3 Composite Asymptotic Expansions: Gevrey Theory .. . . . . . . . . . . . . . . . . . . . 3.1 Composite Gevrey Formal Series . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Composite Gevrey Asymptotic Expansions . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Flat Gevrey Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Theorems of Borel–Ritt Type . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Consistent Good Coverings.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

43 43 47 53 55 59

4 A Theorem of Ramis–Sibuya Type . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Statement of the Theorem and Beginning of the Proof . . . . . . . . . . . . . . . 4.2 The Slow Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 The Fast Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . j 4.4 Study of the Functions gn . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 The Case  Negative.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Composition of a Gevrey CAsE and an Analytic Function .. . . . . . . . . . .

63 63 67 70 71 74 76

ix

x

Contents

5 Composite Expansions and Singularly Perturbed Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 5.1 Classical CAsEs at a Regular Point . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 82 5.2 CAsEs at a Turning Point: The Quasi-linear Case .. . . . . . . . . . . . . . . . . . . . 89 5.2.1 Composite Formal Solutions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 90 5.2.2 Analytic Solutions and CAsEs . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 5.2.3 The Gevrey Character of CAsEs . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 5.2.4 Remarks and Extensions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 5.3 CAsE at a Turning Point: A Generalization . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106 6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Canards in a Multiple Turning Point . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Non-smooth Canards.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Equations of “Union Jack” Type . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Non-smooth Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Ackerberg–O’Malley Resonance .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

119 120 126 126 132 136

7 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 151 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 155 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 159

Chapter 1

Four Introductory Examples

Here we present simple examples, showing that solutions of singularly perturbed differential equations naturally have CAsEs near turning points. All examples are linear equations of first order. We emphasize that these are only illustrative examples. Our theory of CAsEs goes far beyond these examples. The first example is among the simplest ones having a turning point. The second one contains a control parameter for “duck hunting” or “canard hunting”. The third example also contains a control parameter, but the turning point is no longer simple; this implies that the canard solutions are no longer overstable solutions in the sense of Wallet [60]. Finally the fourth example relates to “fake ducks” or “fake canard solutions”: the slow curve is first repelling and then attracting. In this situation, any solution with bounded initial condition at the turning point is defined and bounded on an interval containing this turning point, but this solution can have a CAsE only if the initial condition has an expansion in powers of . We will see that this necessary condition is also sufficient.

1.1 First Example Let us begin with the equation "

dy D 2xy C "g.x/; dx

(1.1)

where " > 0 is a small parameter and x; y are real variables. We assume that the function g W R ! R is of class C 1 and bounded, as well as all its derivatives. These assumptions are made to simplify the presentation, but most of the following results are valid with weaker assumptions. For example the existence of the solution y  below only uses the continuity of g; in the same way, the proof that this solution locally has an asymptotic expansion of order N only needs that g is of class C N .

A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 1, © Springer-Verlag Berlin Heidelberg 2013

1

2

1 Four Introductory Examples

Fig. 1.1 Some solutions of (1.1) with " D 0:02, jxj  1, jyj  1 and g.x/ D x C 1. On the right, the solutions y  and y C

With minor modifications, we can also treat the case of a finite interval or of unbounded functions. Moreover, we could present this example in the complex domain; this would correspond better to the approach of the memoir. We wish to present, however, these examples in the real framework: this illustrates the fact that CA sE s are also very useful in this context. By the way, the applications presented in Chap. 6 have their origin in the real context. Since (1.1) is linear, its solutions (Fig. 1.1) are defined on R and can be expressed by variation of constants. For arbitrary x0 , C , the formula   Z x x 2 =" t 2 =" CC y.x; "/ D e e g.t/dt : (1.2) x0

gives a solution of (1.1). The slow curve mentioned in the preface is y D 0; it is attracting for x < 0 and repelling for x > 0. Equation (1.1) has a simple turning point at x D 0, i.e. the coefficient of y has a simple zero at x D 0. In order to structure the set of solutions, it is convenient to consider solutions having a nice behavior at infinity. Here we have two such distinguished solutions (Fig. 1.1). In this example, it is easy to verify that for each fixed " > 0, a unique function y  .  ; "/ exists, bounded on R , which satisfies (1.1) for that value " of the small parameter. This solution is given by (1.2) with x0 D 1 and C D 0, namely Z x 2 2 e t =" g.t/dt: (1.3) y  .x; "/ D e x =" 1

As " is a variable in this memoir, what we call “solution” is actually a family of solutions depending on ". Formula (1.3) defines a family of solutions of (1.1) that are not only bounded on R , but moreover uniformly with respect to " (in other words, it is a function of both variables x and " which is bounded on R  0; "0  for some fixed "0 > 0). In the sequel, we sometimes use the term “bounded” for “bounded uniformly with respect to "”.

1.1 First Example

3

We want to have a description of the solution, not only for x < 0 but also for x near 0 and for x > 0. We begin with the case x < 0. By successive integrations by parts, we easily show that, for any fixed positive ı, the solution y  also admits a uniform asymptotic expansion in the sense of P Poincar´e1 on   1; ıŒ, of the form b y D n0 yn .x/"n : for all integer N > 0, there exists a constant CN such that for all x < ı and all " 2 0; "0  ˇ ˇ N 1 X ˇ ˇ  nˇ ˇ y .x; "/  yn .x/" ˇ  CN "N : ˇ nD0

Actually this expansion is the unique formal solution of (1.1), given recursively by y0 .x/ D 0;

y1 .x/ D 

1 g.x/; 2x

(1.4)

and

1 0 y .x/; n  1: (1.5) 2x n To see that b y is indeedPthe expansion of y  , one can for example write y D 1 .N / N n y Cz" with y .N / D N nD0 yn " and check that z satisfies an equation analogous to (1.1), hence is bounded on the ray   1; ıŒ. If one replaces 1 by C1, the same formula (1.3) also yields a unique solution y C bounded on RC , which has an asymptotic expansion on ı; C1Œ for all ı > 0. Since this expansion is the unique formal solution of (1.1), it is the same for y  and for y C . ynC1 .x/ D

• In the very special case where g is odd, then on the one hand we have y  D y C and on the other hand formulas (1.4) and (1.5) imply that for all n 2 N the function yn is even and without pole at x D 0; so the formal solution b y is defined at x D 0. It is therefore natural to ask whether the common expansion P n n0 yn .x/" , valid for fixed x non-zero, remains valid near 0. In this example it is the case and this can be shown using y .N / as before. For analogous equations, e.g. with 4x 3 instead of 2x in (1.1), cf. 1.2.2, the result is no longer true. We still have y  D y C , hence a solution bounded on all R, but the coefficients of the formal solution generally have poles at x D 0 and the partial sums y .N / of b y cannot be uniform approximations of y. Equation (1.1) is one of the simplest examples where the theory of overstability applies [60]. We do not pursue the discussion further in this direction because we want to present CAsEs and not overstability. • If g is not odd, however, then the expansion of y C yields an approximation of 2 y  on ı; C1Œ. Indeed, one has y  .x; "/ D y C .x; "/ C I."/e x =" with I."/ D Z C1 2 e t =" g.t/dt. If g is not odd, then the function I is non-zero. To see this, 1

1

In this memoir, we will use the notation y   b y for the Poincar´e asymptotic.

4

1 Four Introductory Examples

 p R C1 p  1 one can write I."/ D 0 e s=" g. s / C g. s / 2p ds and use the fact that s the Laplace transform is one-to-one. Especially, if the even part of g—given by  g C .x/ D 12 g.x/ C g.x/ —is non-flat, then it satisfies g C .x/  C x 2N with C ¤ 0 and N 2 N, and one obtains I."/  C 0 "N C1=2 . As a consequence, at any given point x > 0, y  .x; "/ has an exponentially large value with respect   to ": there exist c; a; "0 > 0 (depending on x) such that jy  .x; "/j  c exp a" for all " 2 0; "0 Œ. It is possible to describe precisely the domain where y  is bounded and the domain where y  tends to infinity in terms of N , but we do not pursue this further. p Instead we describe y  .x; "/ for x-values of the order of  D ", that is to say, when x and " tend to 0 such that x is bounded. It is easily seen that for such values, y  ispbounded, and even tends to 0. Indeed the change of variable x D X (with  D ") in (1.3) gives for all real K y  .X; "/ D 

Z

X

eX

2 T 2

1

g.T /d T D O./

(1.6)

as  tends to 0, uniformly for X  K. We will see that y  admits an expansion in powers of , involving both functions of the slow variable x and of the fast variable X D x . This can be seen by a succession of integrations by parts. Indeed, let S denote the operator defined by g.x/ D g.0/ C xSg.x/:

(1.7)

Since g is C 1 and bounded on R, so is Sg (and even tends to 0 at infinity). A first integration by parts gives y  .x; "/ D e x

2 ="

 Z g.0/

x

e t

2 ="

Z

1

D

 

g.0/ U  x

with U  .X / D e X has, for all real K,

2

RX 1

x

dt C

te t

1

 Sg.x/ C e " 2

" 2

x 2 ="

Z

2 ="

x

 Sg.t/dt

e t

2 ="

.Sg/0 .t/dt:

1

e T d T: Applying (1.6) to .Sg/0 instead of g, one then 2

y  .x; "/ D g.0/U  as  ! 0, uniformly over all x with

x 

x  

   2" Sg.x/ C O 3

 K. Repeating the integration by parts, we

obtain, with the operator S given by (1.7) and the operator SD D

d dx

1.1 First Example

5 N 1 X

y  .x; "/ D

 1

  n  DS g .0/2nC1 U  x  2

(1.8)

nD0 N 1   1 X   1 n  S 2 DS g .x/2nC2 C O 2N C1 2 nD0

as  ! 0, uniformly for all x with x  K. This is an example of a composite    P expansion of the form n0 an .x/ C gn x n where  n1  a0 D 0; a2n D  12 S 12 DS g ; a2nC1 D 0 and gn D cn U  with c2n D 0; c2nC1 D

(1.9)

 1

n  DS g .0/: 2

Moreover, the function U  has a asymptotic expansion at infinity, given by U  .X / 

X .1/nC1 1:3 : : : .2n  1/2n1 X 2n1

(1.10)

n0

D

1 1 3 C  C :::; 3 2X 4X 8X 5

X ! 1:

There is an analogous formula for the solution y C which is bounded on RC : y C .x; "/ D

N 1 X

n    DS g .0/2nC1 U C x  2

 1

nD0 N 1   1 X  1 n  S 2 DS g .x/2nC2 C O 2N C1 2 nD0

uniformly for all x with

x 

C

 K, with U .X / D e

Z X2

X

e T d T D 2

C1

U  .X /. In the intersection jxj  K, we can compare the CAsEs of y  and y C . The result is y  .x; /  y C .x; / D

N 1  X

  n    DS g .0/2nC1 .U   U C / x C O 2N C1 : 2

1

nD0

6

1 Four Introductory Examples

p X2 e , this corresponds to the formula deduced from the Z 1 2 2 y  .x; /  y C .x; / D e x =" e t =" g.t/ dt: (1.11)

As .U   U C /.X / D definition

1

In the case where g is odd, .DS/n g is odd for all n 2 N, so the first part of the expansion (1.8) is identically zero. We thus find that y  has a classical asymptotic expansion in powers of 2 D ", whose coefficients are functions of x only. By the way, let us notice that half of the terms an and gn vanish. Therefore it would be possible to rewrite (1.8) as a series in powers of ", but this is particular to the example.

1.2 Extensions 1.2.1. Before presenting the second example, we would like to investigate generalizations and extensions of the first example. The first generalization concerns the local nature of the result. First, any solution of (1.1), with initial value y.x0 ; "/ bounded (for " 2 0; "0 ) at a given point x0 < 0, has a CAsE on Œx0 C ı; 0 for all ı 2 0; jx0 jŒ ; moreover this CAsE is the same as that of y  since the two solutions are exponentially close to one another on Œx0 C ı; 0. For similar reasons, the result is valid only with a local hypothesis on g: if g is of class C 1 on some interval   r; rŒ, then for all x0 2  r; 0Œ, all ı 2 0; jx0 jŒ and any function c D c."/ bounded on 0; "0 , the solution of (1.1) with initial condition y.x0 ; "/ D c."/ has a CAsE of the form (1.8) on Œx0 C ı; 0. The analogous statement holds for positive x0 . These CA sE s do not depend on initial conditions. As before, they are a priori different if x0 is positive respectively negative. 1.2.2. The second generalization is to replace the term 2x in (1.1) with p x p1 , where p is an even integer.2 We still have a unique solution y  bounded on R and aZunique solution y C bounded on RC . They are given now by y ˙ .x; "/ D x

e t

p ="

g.t/dt. The condition y  D y C is still equivalent to g odd. P Seeking a formal solution b y D n0 yn .x/"n leads to

ex

p ="

˙1

y0 .x/ D 0; y1 .x/ D 

1 1 g.x/; then ynC1 .x/ D y 0 .x/: p1 px p x p1 n

In general, this formal solution is not defined at x D 0, even when g is odd. In the case where g is odd, the expansion of y  is valid for both positive and negative x, but cannot be valid in the neighborhood of 0. The same method of successive

2

Here we are interested only in the case p even, but the case p odd also has its interest, cf. Eq. (6.15) in Sect. 6.2.

1.2 Extensions

7

integrations by parts shows (with g odd or not) that y  has a CAsE, combining both functions of x and functions of the fast variable X D x , with  D "1=p . The calculations are more lengthy and complicated but not more difficult. The following p  1 “special functions” appear Uk .X / D e X

p

Z

X

e T T k1 d T; p

k D 1; : : : ; p  1:

1

   P We finally obtain for y  a CAsE of the form n0 an .x/Cgn x n with  D "1=p  and gn of the form gn D cn1 U1 C  Ccn;p1 Up1 . As above for U  , these functions  Uk also have an asymptotic expansion when X tends to 1. 1.2.3. A third extension concerns equations where the function g depends on ". If g—and all its derivatives with respect to x—have an asymptotic expansion in powers of ", then one can show that the functions y ˙ still have CAsEs with functions gn˙ proportional to U ˙ ; the proportionality factor is the same for both signs. More precisely these CAsEs are given as before by y ˙ .x; "/ D

N 1 X

AN n; n .0; "/2nC1 U ˙

x 



(1.12)

nD0 N 1   1 X BN n; n .x; "/2nC2 C O 2N C1 2 nD0

where Amn W .x; "/ 7!

m X

Amnk .x/"k is the jet of order m with respect to " of the

kD0  n function 12 DS g and Bmn is the jet of order m with respect to " of the function n   S 12 DS g . The only change required would be the necessary and sufficient condition for y  D y C . Instead of “g odd”, this condition becomes

Z

1

e t

p ="

g.t; "/ dt D 0:

1

When y  D y C , we obtain again that the solution bounded on R has a classical expansion in the case p D 2 because the factors of U ˙ in (1.12) vanish. In contrast, if p  4 this is no longer necessarily the case. 1.2.4. In our fourth extension we replace p x p1 by f .x/, where f is a function of class C 1 such that xf .x/ > 0 if x ¤ 0 and f .x/ D ax p1 C O.x p /; x ! 0 with a ¤ 0. The solution y  is then written

8

1 Four Introductory Examples iR d

R

−d

Fig. 1.2 In dashed lines the boundary of V  , in solid lines that of V  , in gray the intersection V  \ .V  /

Z



y .x; "/ D e

x

e F .t /=" g.t/dt

F .x/=" 1

with F .x/ D

Rx 0

Z

0

f .t/dt, if we add the assumption that

e F .t /=" dt converges

1

for " > 0 small enough. A diffeomorphism x D './ reduces to an equation of the form "

dz D p  p1 z C "h./; d

which provides for y  a CAsE of the form X  1  an .x/ C gn ' .x/ n : n0

Our general theory of CAsEs shows that such a CAsE can also be transformed into a CAsE of the variable X D x , i.e. of the form X

bn .x/ C hn

 x  

n ;

n0

cf. Proposition 2.6(c). Note that, for this extension, we need other “fast” functions besides U  and U C . 1.2.5. Our last extension is to use the complex framework. If the function g is analytic and bounded in a horizontal strip S D fx 2 C I jIm xj < d g, then one  by can show ˚ that the function y defined  (1.3) is analytic in S and bounded in  V D x 2 S I j arg x  j < 3  ı [ D.0; K/, for all ı; K > 0 (Fig. 1.2). 4 Analogously, the function y C is analytic and bounded in V  . In the intersection

1.3 Second Example

9

 V  \ .V  / D fx 2 C I jRe xj < jIm xj < d g [ D.0; K/, their difference is 2 O e x =" as shown by formula (1.11), and hence exponentially small as " ! 0 for every (fixed) x with jRe xj < jIm xj < d .

By the way, these five extensions can be combined.

1.3 Second Example Now consider an equation already studied in a similar form by Lobry in the introductory chapter [41]. This is the equation "

dy D 2xy C "g.x/ C "˛; dx

(1.13)

where " > 0 is a small parameter, ˛ 2 R is a control parameter, and where the function g W R ! R is of class C 1 and bounded as well as all its derivatives. The question is: Are there any values of ˛ for which a bounded solution exists3 on all R? The answer is “yes” and is well-known, see [41]. To see this, recall that y  defined by Z x  2 2  (1.14) e t =" ˛ C g.t/ dt: y  .x; "/ D e x =" 1 

is the unique solution bounded on R , for any given ˛. Replacing 1 by C1, we obtain the unique solution y C bounded on RC . We deduce that there exists a solution y bounded on all R if and only if y C D y  ; this yields an equation for the parameter ˛, whose solution is Z ˛."/ D 

C1 1

e t

2 ="

  Z

C1

g.t/dt

e t

2 ="

 dt :

(1.15)

1

Since we assumed g of class C 1 , we deduce that ˛ has an asymptotic expansion as " tends to 0. The solutions y ˙ corresponding to this value of ˛ are studied using the third extension (with the function .x; "/ 7! g.x/ C ˛."/ playing the role of g) and we have y  D y C by the choice of ˛. The solution y has therefore also an asymptotic expansion, whose coefficients are functions of class C 1 , including at x D 0.

3

Recall that “bounded” means uniformly with respect to " in some interval 0; "0 . In the present context, it turns out that, for all fixed ", there is a unique value ˛ D ˛."/ for which (1.13) has a solution y D y.x; "/ bounded on R in the classical sense, and that the function y so defined is also bounded on R 0; "0 .

10

1 Four Introductory Examples

We directly compute the first terms y0 .x/ D 0;

˛0 D g.0/;

y1 .x/ D 

 1  g.x/ C ˛0 : 2x

(1.16)

For the subsequent terms, it suffices to note that ˛n and yn .x/ are determined uniquely by the fact that yn has no pole at x D 0. They are calculated recursively by  1  0 yn .x/  ˛n ; ˛n D yn0 .0/: ynC1 .x/ D (1.17) 2x Extension. In the complex domain, assume g analytic and satisfying jg.x/j  2 M e M jxj in some horizontal strip S D fx 2 C I jIm xj < d g for some d > 0. Equation (1.13) has a landscape consisting of level curves of the function R W x 7! Re .x 2 /. Using Extensions 1.2.3 and 1.2.5, we can show that, C for any bounded value of ˛˚ D ˛."/ and for all fixed ı 2  0; d Œ, the solution y 3 is bounded in the domain x 2 S I j arg xj < 4  ı . If moreover ˛ has an asymptotic expansion in powers of ", then y C also has an asymptotic expansion P C C y .x; "/  n0 yn .x/"n in the domain ˚ DıC D x 2 S I j arg xj <

3 4

 ı; jxj > ı



for all ı > 0. Precisely, for all N 2 N, there exists M D M.ı; N / > 0 such that N 1 ˇ ˇ X ˇ ˇ C .x; "/  ynC .x/"n ˇ  M "N ˇy

(1.18)

nD0

for all x 2 DıC . Notice that this domain DıC includes most of the east “mountain” and the north and south “valleys” in the strip S , cf. Fig. 1.3. It is the same for y  in Dı D fx 2 S I j arg x  j < 3 2  ı; jxj > ıg. In the case where ˛ is given by (1.15), i.e. if y C D y  , then the expansion (1.18) on DıC and the analogous one for y  on Dı are valid. Since both expansions concern the same function on the intersection Dı \ DıC , one has ynC D yn for all n; first on the intersection, then on the whole strip by identity theorem for holomorphic functions. The coefficients ynC D yn DW yn are given by (1.16) and (1.17). Since y C is analytic in all S and (1.18) is satisfied for x 2 Dı [ DıC D S n D 0 .0; ı/, where D 0 .0; ı/ is the closed disk of center 0 and radius ı, the maximum modulus principle yields that the inequality is satisfied for all x in S . Briefly, as the turning point x D 0 is simple, the landscape has only two mountains, so a solution defined and bounded on some parts of both mountains is automatically bounded in a neighborhood of the turning point. Such a solution is called overstable, following the terminology adopted by Wallet [60].

1.4 Third Example

11

iR

R

Fig. 1.3 The level curves Re .x 2 / D constant; in gray the domain DıC

1.4 Third Example Let us now consider an example with a multiple turning point: we replace the term 2x by 4x 3 ; in other words, consider equation "

dy D 4x 3 y C "g.x/ C "˛; dx

(1.19)

For all ˛ D ˛."/, there still exists a unique solution y C bounded on RC and a unique solution y  bounded on R . There is also a unique value of ˛ for which these two solutions coincide and yield a solution y bounded on R (Fig. 1.4). This solution is given by y.x; "/ D e x

4 ="

Z

x

e t

4 ="

  ˛ C g.t/ dt

(1.20)

1

R C1 t 4 =" e g.t/dt with ˛ D ˛."/ D  1 : R C1 4 =" t e dt 1 Again we can show that ˛."/ has an asymptotic expansion of the form ˛."/  1 X ˛n "n=2 . In general, however, the solution y has no uniform asymptotic expansion nD0

12

1 Four Introductory Examples y

y

1

1

1

x

1

x

Fig. 1.4 The solutions y C and y  for g.x/ D 3x 2 C 3x. On the left ˛ D 0:507,pon the right    2 ˛ D 0:35. Here jxj  5, 5  y  3 and " D 14 . The exact value is ˛ 14 D  34 2  34 0:506984

in powers of "1=2 in a realPneighborhood of 0. Indeed, an expansion would be P such n=2 n=2 a formal solution b y D y .x/" ; b a D ˛ " satisfying y0 .x/ D n0 n0   n  0 n  1 1 0, y2 .x/ D  4x 3 g.x/ C ˛0 and y2nC2 .x/ D 4x 3 y2n .x/  ˛2n . If the Taylor expansion of some coefficient y2n contains a nonzero term in x 2 or x 3 , then the following coefficient has a pole at x D 0, whatever the choice of ˛2n . We will see that it is still possible to give an approximation of y valid in a real neighborhood of 0, using functions of the variable x , where  D "1=4 . This idea was the basis of Thomas Forget’s thesis [20] and his subsequent articles [21, 22]. Indeed, isolate the first terms of the expansion of g writing g.x/ D g0 C g1 x C g2 x 2 C 4x 3 h.x/. The first formula of (1.20) becomes       y.x; "/ D ˛."/ C g0 U0 x C g1 U1 x C Z x   4 2  x x 4 =" g U e t =" 4t 3 h.t/dt  2 2  Ce

(1.21)

1

with Uj .X /

D e

X4

Z

X

e T T j d T: 4

1

Integration by parts gives e

x 4 ="

Z

x

e 1

t 4 ="

4t h.t/dt D "h.x/  "e 3

x 4 ="

Z

x

e t

4 ="

h0 .t/dt

1

which is of the same form as in Formula (1.21) with an additional factor ", hence can be repeated. In the same manner as in 1.2.2, we finally obtain a CAsE. The fact that y C is equal to y  implies that it is a CAsE on the whole real axis. More precisely, there are functions an 2 C 1 .R/ and linear combinations gn of the

1.5 Fourth Example

13

 x  P 1  a C O.N / functions U0 ; U1 ; U2 such that y ˙ .x; "/ D N .x/ C g n n nD0  uniformly on R. In the complex domain, the landscape associated to (1.19), given by the real part of x 4 , has four mountains where Re .x 4 / > 0. The solution being studied is close to the slow curve on two mountains, east and west, but not necessarily on the other two mountains, north and south; therefore it is not necessarily overstable.

1.5 Fourth Example Let us replace the term 2x by 2x in the first example (1.1), i.e. consider "

dy D 2xy C "g.x/ dx

(1.22)

with g W R ! R of class C 1 and bounded as well as all its derivatives. Contrary to (1.1), the slow curve y D 0 is repelling for x < 0 and attracting for x > 0. Fix an initial value c D c."/, bounded as 0 < " ! 0. The solution with initial condition y.0; c."/; "/ D c."/ is then bounded on R as " ! 0. It is given by the formula Z x 2 2 2 y.x; c."/; "/ D e x =" e t =" g.t/ dt C c."/e x =" : (1.23) 0

By successive integrations by parts, one shows as in Sect. 1.1 that y.x; c."/; "/ has X an asymptotic expansion yn .x/"n for x far from 0. More precisely, for all ı > 0, n1

one has y.x; c."/; "/ 

X

yn .x/"n

n1

as 0 < " ! 0, uniformly on   1; ı [ Œı; C1Œ. The coefficients yn .x/ are deter1 1 0 mined similarly to (1.4) and (1.5) by y1 .x/ D 2x g.x/ and ynC1 .x/ D  2x yn .x/. This expansion is independent of the initial condition c."/; the exponential term in (1.23) does not contribute outside of neighborhoods of 0. As in the previous sections, the question of the behavior of these solutions near x D 0 arises. We proceed again by integration by parts using g.x/ D g.0/ C x Sg.x/; we obtain  y.x; c."/; "/D g.0/U.X / C 2" Sg.x/ C c."/  Z  x 2 =" " x 2 =" x t 2 =" " Sg.0/ e  e e .Sg/0 .t/ dt 2 2 0

RX p 2 2 with  D " and U.X / D 0 e X CT d T . Repeating the integration by parts, we obtain as in Sect. 1.1

14

1 Four Introductory Examples

y  .x; "/ D

N 1 X

.Vn g/.0/2nC1 U

x  

C

nD0

1 2

N 1 X

.S Vn g/.x/2nC2 C

nD0

c."/ 

1 2

N 1 X

n

!

2nC2

.SV g/.0/

e x

2 ="

  C O 2N C1 ;

nD0 1 where V D P 2 DS. Now suppose that c."/ has an asymptotics expansion in powers of : c."/  n0 cn n . Then we obtain again a composite expansion

y.x; c."/; "/ D

X   an .x/ C gn x n n0

with a0 D 0;

 n a2nC2 D 12 S  12 DS g;

a2nC1 D 0;

n D 0; 1; 2; : : : as well as g0 .X / D c0 e X and for n  1 g2n .X / D dn e X 2

2

 n1  dn D c2n  12 S  12 DS g .0/

with and for n  0

g2nC1 .X / D bn U.X / C c2nC1 e X with 2

bn D



n   12 DS g .0/:

Unlike Sect. 1.1, this expansion depends on the initial condition, but is the same for positive or negative x. One can generalize and extend this example, but this investigation, which is similar to Sect. 1.2, is left to the reader.

1.6 Composite Expansions If one wants to generalize the method of the previous sections to nonlinear equations, it is necessary to use a larger family of coefficient functions. In particular, one has to take into account the products of functions Uj˙ Uk˙ , solutions of ordinary   differential equations "y 0 D px p1 y C U ˙ x , and products of functions of x and functions of x . One possible strategy is to construct an algebra A containing the functions X 7! X n and X 7! Uj˙ .X / that is stable by the operators J ˙ defined by

1.6 Composite Expansions

J ˙ v.X / D e X

15

p

Z

X

e T v.T /d T: p

˙1

The construction of the smallest algebra with these properties leads to define a large number of “special functions”; this was the strategy adopted by Forget in his thesis for approximating canard solutions as in Sect. 1.4. An additional complication comes from the non-uniqueness of the writing. For example, the word x can be considered both as a function of x of order 0 in  and as a function of x of order 1 in , if it is written x . This required a modification of the algebra: only functions in A tending to 0 at infinity can be allowed. The strategy we have adopted is to consider from the beginning larger algebras: all functions having an asymptotic expansion at infinity without constant term are allowed. An advantage of this strategy is its simplicity; a drawback is to have a priori less information about the coefficients.

Chapter 2

Composite Asymptotic Expansions: General Study

In this chapter, we present the general theory of CAsEs: their definition and their behavior with respect to the basic operations of addition, multiplication, division, differentiation, integration, composition and analytic continuation. In Sect. 2.4, we also link our CAsEs to the inner and outer expansions of the classical method of matching. Using these inner and outer expansions is also a good method for determining the coefficients of a composite expansion in practice, provided one can show the existence of a composite expansion independently. Many problems solved using CAsEs have their origin in the real variable, so a purely “real” presentation might seem enough. However, an essential element in solving some problems is the Gevrey character of the CAsEs, which will be developed in Chap. 3. In order to obtain Gevrey properties, the only method known so far is to apply our “key-theorem” 4.1 of Ramis–Sibuya type, for which the complex framework is essential. Therefore the presentation here uses the complex variable; a presentation of the results in the real domain can be found in Fruchard/Sch¨afke [27, 28].

2.1 Notation The notation N refers to the set of all natural numbers, including 0. The open disk of center 0 and radius r is denoted by D.0; r/. Given ˛ < ˇ  ˛C2 and 0 < r  1, S.˛; ˇ; r/ is the sector S.˛; ˇ; r/ D fx 2 C I 0 < jxj < r; ˛ < arg x < ˇg: f . A sector is usually considered as part of the Riemann surface of the logarithm C Since our sectors will always have an opening less than 2, however, we consider them as subsets of C D C n f0g.

A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 2, © Springer-Verlag Berlin Heidelberg 2013

17

18

2 Composite Asymptotic Expansions: General Study

We say that a function f holomorphic and bounded on a sector S has an asymptotic P expansion at x D 0 (in the sense of Poincar´e) if there exists a formal series 0 a x  and for all N 2 N there is some constant CN such that N 1 ˇ ˇ X ˇ ˇ jxjN ˇf .x/  a x  ˇ  CN D0

for all x 2 S . In that case, we write X f .x/  a x  ; S 3 x ! 0: 0

We say that a function g holomorphic and bounded on an infinite sector S has an asymptotic expansion at X D 1 if the function f W x 7! g.1=x/ has an asymptotic expansion at x D 0. Given a sector S D S.˛; ˇ; r/ and  > 0, V .˛; ˇ; r; / denotes the union of the sector S and the disk D.0; /: V .˛; ˇ; r; / D fx 2 C I .jxj < r and ˛ < arg x < ˇ/ or jxj < g:

(2.1)

For  < 0, we define V .˛; ˇ; r; / D fx 2 C I  < jxj < r and ˛ < arg x < ˇg:

(2.2)

In the sequel, we call these sets quasi-sectors , for  positive or negative (Fig. 2.1). For simplicity, we often only consider the case  > 0. The necessary changes in the case  < 0 are minor and will be indicated. Given an infinite quasi-sector V D V .˛; ˇ; 1; /, G .V / denotes the vector space of holomorphic functions g bounded in V and having an asymptotic expansion in P the Poincar´e sense at infinity without constant term g.X /  1 g X  ; V 3 X ! 1, i.e. 8N 2 N

9CN > 0

8X 2 V;

N 1 ˇ ˇ X ˇ ˇ jX jN ˇg.X /  g X  ˇ  CN : D1

Let T W G .V / ! G .V / denote the operator which, to a function g, associates the function Tg given by Tg.X / D Xg.X /  g1 (2.3) where g1 X 1 is the first term of the asymptotic expansion of g at infinity. Sometimes V will be an annulus: V D A.r; 1/ D fx 2 C I r < jxj < 1g. In that case, the Banach space G .V / is therefore the space of functions holomorphic and bounded on V , tending to 0 as X ! 1.

2.2 Composite Formal Series Fig. 2.1 Two examples of quasi-sectors, on the left  < 0, on the right  > 0

19 ¹

r

¯

0

®

Given a number r0 > 0, H .r0 / denotes the vector space of functions a holomorphic and bounded in the disk D.0; r0 / of radius r0 centered in 0. Similarly to T, let S W H .r0 / ! H .r0 / be the operator which, to a function a, associates the function Sa given by a.x/  a.0/ : (2.4) Sa.x/ D x On the expansions, the operators S and T act as a shift to the left: if g.X /  1 X X X g X  , then Tg.X /  gC1 X  and if a.x/ D a x  , then Sa.x/ D 1

1 X

1

D0

aC1 x  .

D0

2.2 Composite Formal Series Definition 2.1. Let V D V .˛; ˇ; 1; / be an infinite quasi-sector (with  positive or negative) and r0 > 0. A composite formal series associated to V and D.0; r0 / is an expression of the form b y .x; / D

X   an .x/ C gn x n ;

(2.5)

n0

where an 2 H .r0 / and gn 2 G .V /. The functions an form the slow part of the composite formal series, and the gn the fast part.  Remarks. 1. More precisely, a composite formal series is an element of H .r0 /  N G .V / . As for classical formal series, we could therefore represent a comX  an .x/ C gn .X / n —or in the form posite formal series in the form X n0

n0

 an .x/; gn .X / n —using three variables. We will however not have to

20

2 Composite Asymptotic Expansions: General Study

consider functions of three variables asymptotic to a composite formal series: the three variables are always related by x D X . 2. In this section, the symbol  introduced in the above definition is arbitrary, because we are just dealing with formal series. Later on, however,  will be a new independent variable and we will consider the behavior of functions as  tends to 0. In the context of singularly perturbed differential equations, as was the case in the introductory examples,  will be connected to " by p D " with a suitable integer p. 3. The names “slow part” and “fast part” are motivated by their behavior with respect to differentiation d=dx. In general, differentiation does not change the -order of a slow term. For a fast term, however, differentiation introduces a (large) factor 1=. For details, see below (Definition 2.3). b .r0 ; V /. Let C b .r0 ; V / denote the vector space The “Differential” Algebra C of composite formal series associated with V and D.0; r0 /, endowed with the canonical addition and multiplication by constants, the ultrametric distance y 1 b y2/ d.b y1; b y 2 / D 2val.b ; where val.b y / D minfn  0 I an or gn 6 0g

(2.6)

and the topology induced by this distance. b .r0 ; V / and, to simplify the Let I denote the canonical inclusion of H .r0 / in C b .r0 ; V /. The symbol notation, let the same letter denote the inclusion of G .V / in C  denotes the real number, the function of .x; / with value  and the composite formal series with a single term a1 D 1. Due to the fact that gn .X / has an asymptotic expansion when X ! 1, the b .r0 ; V / with a structure of algebra as follows. operators S and T endow C In order to define the product of two composite b y andb z, we expand  formal series  x  n P the product term by term: if b y .x; / D n0 an .x/ C gn   and b z.x; / D   x  n P  , then we put n0 bn .x/ C hn  b y b z.x; / D

X

n  X

n0

D0

 I.a /.x; / C I.g /.x; / 

!    I.bn /.x; / C I.hn /.x; / n :

b 0 ; V / and it remains This is a convergent series with respect to the topology  of C .r   to define products of images by I. The sets I H .r0 / and I G .V / are naturally equipped with a structure of algebra, hence we just have to define the product of an element I.a/.x; /, a 2 H .r0 /, and an element I.g/.x; /, g 2 G .V /. For this 1 X X a x  and g.X /  g X  , we observe purpose, with the notation a.x/ D D0

>0

2.2 Composite Formal Series

21

  x  x   x  D a and xg D g . In other C xSa.x/ g C Tg 0 1       words, a composite asymptotic expansion of the product of functions a.x/ g x with respect to  can be obtained by

that a.x/g

x

a.x/ g

x 

D a0 g

x  

C g1 Sa.x/ C Sa.x/ Tg

x 

:

(2.7)

By iterating this formula, we define with the convention g0 D 0 I.a/.x; / I.g/.x; / D

X

g .S a/.x/ C a .T g/

 x  

 :

(2.8)

0

Remarks. 1. The above formula implies that the product of composite formal series is again a composite formal series. It also shows the need to have an asymptotic expansion of g; otherwise we could not define T g. This is the main reason why we require the functions gn in Definition 2.1 to have asymptotic expansions as X ! 1. 2. Classical composite series [4,59] are a special case of our CAsEs: the functions gn decay exponentially. Recall that a function g W J D ; C1Œ! R has an exponential decay if there exist C; A > 0 such that jg.X /j  C exp.AX / for all X 2 J: A function g with exponential decay satisfies g.X / D O.X N /; X ! C1 for all integer N , so is flat : it admits the zero series as asymptotic expansion. 3. In the case of classical composite series, the slow part of a product depends only on the slow parts of the factors. This can be seen for example on (2.8): when all the g are zero, the product of a slow term and a fast term generates only fast terms; cf. also Remark 2, p. 11 of [4]. In contrast, for our composite series, formula (2.8) shows that the product of a slow term and a fast term yields many slow terms, so that everything is intertwined. Composite series are also compatible with the left composition. b .r0 ; V / be a composite formal series without constant Lemma 2.2. Let b y 2 C b .r0 ; V /ŒŒy;  be a b 2 C term, i.e. with a0 .x/ 0 and g0 .X / 0. Let P formal series in two variables whose coefficients are composite formal series: X b .r0 ; V /. Then the expression b .x; y; / D P pj;k .x; /y j k with pj;k 2 C j;k0

b y /.x; / D P b .x; b Q.b y .x; /; / WD

X

pj;k .x; /b y .x; /j k

j;k0

b W C b .r0 ; V / ! defines a composite formal series. Moreover, the application Q b C .r0 ; V / is well defined and 1-Lipschitz.

22

2 Composite Asymptotic Expansions: General Study

b y /.x; / The proof is immediate, thanks to the convergence of the series defining Q.b for the ultrametric topology induced by the distance (2.6). Let us now define the derivative of a composite formal series. Since derivatives of functions in H .r0 / are not necessarily bounded and those of functions in G .V / have no longer necessarily an asymptotic expansion, this differentiation is somewhat more difficult to treat, although the formula is simple. In particular, it is required to reduce slightly the definition domains of the functions. In the real framework, it is not always possible to define the derivative of a composite formal series.   b r0 ; V .˛; ˇ; 1; / is a composite formal series given by Definition 2.3. If b y2C b y .x; / D

X   an .x/ C gn x n ; n0

such that the first fast term g0 is identically zero, then its derivative with respect to y x, db , is given by dx

X  x  n db y 0 an0 .x/ C gnC1 .x; / D   dx n0   be This formula defines an element of C r 0 ; V .e ˛; e ˇ; 1; e / for any e r 0 2 0; r0 Œ, ˛ < e ˛ <e ˇ < ˇ and any e  < . Remark. A priori, the derivative of a composite formal series is only defined if the d d first fast term g0 is identically zero. The operator  dx D dx .  /, however, is defined without condition on g0 . Exercise 2.4. Give a detailed proof of Lemma 2.2.

2.3 Composite Expansions: Definition and Basic Properties Until now, the objects considered were formal expressions. We now want to define the composite expansion of a function of two variables x and . The simplest and most natural way would be to consider functions defined on a product of sectors in x and . For some applications, however, it will be convenient that the x-domain contains a neighborhood of 0 of size proportional to jj. For other applications, it will be necessary to remove a neighborhood of 0. That is why we introduced the quasi-sectors (2.1) and (2.2). Definition 2.5. Let V D V .˛; ˇ; 1; / denote an infinite quasi-sector, let S2 D S.˛2 ; ˇ2 ; 0 / denote a finite sector and let ˛1 < ˇ1 be such that ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ. Let y.x; / be a holomorphic function defined for  2 S2 and x 2

2.3 Composite Expansions: Definition and Basic Properties

V .˛1 ; ˇ1 ; r0 ;  jj/. Finally, let b y .x; / D y as CAsE and we write say that y has b

23

X   b .r0 ; V /. We an .x/ C gn x n 2 C n0

y.x; /  b y .x; /; as S2 3  ! 0; x 2 V .˛1 ; ˇ1 ; r0 ;  jj/; if, for any integer N , there exists a constant KN such that for all  2 S2 and all x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ ˇ ˇ N 1  ˇ X  x  n ˇˇ ˇ an .x/ C gn   ˇ  KN jjN : ˇy.x; /  ˇ ˇ

(2.9)

nD0

Again, the functions an are the slow part of the CAsE and the gn are its fast part. The conditions on the angles ˛j ; ˇj ensure the implication: if  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ then x= 2 V . Remarks. 1. In the case of an annulus V D A.r; 1/, r > 0, there  is no P condition on the angles. A composite expansion y.x; /  n0 an .x/ C  x  n gn   is then another form of a monomial expansion introduced in CanalisDurand/Mozo/Sch¨afke [8]. If we put u D =x, then  D xu and the function z.x; u/ D y.x; xu/ is defined on a sector in xu, defined in  [8] after Definition 3.4, i.e. the set of .x; u/ such that jxj < r0 ; juj < min 1r ; r00 and  P arg.xu/ 2 ˛1 ; ˇ1 Œ, and admits the monomial expansion z.x; u/  n0 an .x/  C bn .u/ .xu/n defined in [8], Definition 3.6, with bn .u/ D gn .1=u/. 2. For the sake of simplicity, we ask the functions an to be holomorphic in the whole disk D.0; r0 /, while y itself is only defined for x 2 V .˛1 ; ˇ1 ; r0 ;  jj/. In Sect. 2.4 we shall have to generalize the definition of CAsE to a situation where the functions an are holomorphic on a more general domain containing 0, cf. the remark after Proposition 2.20. 3. A function y.x; / cannot have two different CAsEs as S2 3  ! 0 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/. Indeed, one has lim y.x; / D a0 .x/ for x 2 S.˛1 ; ˇ1 ; r0 /, !0

hence the holomorphic function a0 2 H .r0 / is uniquely determined, therefore also a0 .0/. We continue with lim y.X; / D a0 .0/ C g0 .X /, and so on. It !0

should be noted that, to prove this uniqueness, only the property that gn .X / tends to 0 as X ! 1 was used; this will be useful in Sect. 4.1. It is immediate that CAsEs are compatible with addition and scalar multiplication. For compatibility with the multiplication of expansions, the only less obvious point   is to show that a product a.x/g x , a 2 H .r0 /, g 2 G .V / has a CAsE. This is a consequence of Formula (2.7) and of the fact that   S and T are endomorphisms. Definition (2.8) was made so that we have a.x/ g x  I.a/.x; / I.g/.x; /.

24

2 Composite Asymptotic Expansions: General Study

Composition. The CAsEs are also compatible with the left and right composition by a holomorphic function, as expressed in the following proposition. Statement (a) concerns left composition by a function of three variables, but in the case of a CA sE without term in 0 . Statement (b) treats the case of left composition without this restriction, but by a function of one variable only. These two statements are complementary. For the right composition, we have considered only functions of one variable x for the sake of simplicity, but it is possible to generalize the result to the case of a function ' of the two variables x and , such that '.0; 0/ D 0 and @' @x .0; 0/ D 1. The Gevrey version of this generalization is given in Sect. 4.6, Theorem 4.7. We have not formulated any statement concerning a change of the variable  because we do not need it. Proposition 2.6. (a) Let P .x; z; / be a holomorphic function defined when jzj < V .˛1 ; ˇ1 ; r0 ; jj/ such that all coefficients r,  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 P Pn of the expansion P .x; z; / D n0 Pn .x; /zn have a CAsE Pn .x; /  b n .x; / as S2 3  ! 0, x 2 V .˛1 ; ˇ1 ; r0 ; jj/. Let y.x; / D O./ be a P function having a CAsE b y .x; / as S2 3  ! 0 and x 2 V .˛1 ; ˇ1 ; r0 ; jj/ without terms in 0 . Suppose that supx; jy.x; /j < r. Then the function u W .x; / 7! P .x; y.x; /; / has the CAsE b y /.x; / D Q.b

X

b n .x; /b P y .x; /n :

n0

as S2 3  ! 0; x 2 V . (b) Consider a holomorphic function y defined for  2 S2 and x 2 V where V D V .˛1 ; ˇ1 ; r0 ;  jj/, with range in a bounded set W C and having a CA sE as  ! 0. Let f be a holomorphic function in a neighborhood of the closure of W . Then the function z D f ı y has a CAsE as S2 3  ! 0, x 2 V . (c) Let ' be a holomorphic function defined for jxj < x1 such that '.0/D 0 and P ' 0 .0/ D 1 and let z D z.u; / be a function with a CAsE n0 an .u/ C    gn u n as S2 3  ! 0 and u 2 V .˛1 ; ˇ1 ; r0 ; jj/, with an 2 H .r0 / and ˛1; e ˇ 1 with ˛1 < e ˛1 < e ˇ 1 < ˇ1 andall Q <  there gn 2 G .V /. Then for all e are e r;e 0 > 0 such that the function y W .x; / 7! z '.x/;  has a CAsE as   0 / 3  ! 0 and x 2 V e ˛1; e ˇ 1 ;e r; e jj . S.˛2 ; ˇ2 ;e Remark. In (a), the assumption “y bounded by r” is not essential: simply reduce the -domain if it is not satisfied. X Proof. (a) For all N 2 N , the finite sum Pn .x; /y.x; /n has a CAsE 0nN 1

(compatibility with product and sum). It remains to verify that a constant L D L.N / exists such that the remainder is bounded by L jjN . This is evident from the assumptions.

2.3 Composite Expansions: Definition and Basic Properties

25

(b) By modifying f and y if necessary, we may  assume that a0 .0/ D 0. Using a Taylor expansion, it suffices to prove that f a0 .x/ C g0 . x / has a CAsE.   Set h.u; v/ D f .u C v/. It suffices to show that h a0 .x/; g0 . x / has a CAsE as  tends to 0. To show this, we write h.x; y/ D h.x; 0/ C h.0; y/  h.0; 0/ C xy k.x; y/ with some holomorphic function k of two variables x; y; therefore     h a0 .x/; g0 . x / D h.a0 .x/; 0/ C h 0; g0 . x /  h.0; 0/   Ca0 .x/g0 . x / k a0 .x/; g0 . x / :   Since a0 .0/ D 0, the product a0 .x/g0 x is of the form O./; we obtain a CAsE   for h a0 .x/; g0 . x / by iterating this procedure.   Note that the leading to  term  of the CAsE of f y.x; / (without  the  reduction  a0 .0/ D 0) has f a0 .x/ as slow part and f a0 .0/ C g0 . x /  f a0 .0/ as fast part.   (c) If e r;e 0 are small enough, then ' V .e ˛1; e ˇ 1 ;e r; e jj/ V .˛1 ; ˇ1 ; r0 ; jj/ if  '.x/   2 S2 ; jj < e 0 . It suffices to show that b  has a CAsE, if b is in G .V /. 1 For that purpose, we introduce the functions h and defined by '.x/  x1 D h.x/   and .x; t/ D x= 1 C txh.x/ . The function h can be analytically continued to a function defined for jxj < x1 , still denoted h by abuse and  .x;1/  ofnotation, D b .x; 0/ D x; .x; 1/ D '.x/. The Taylor expansion of b '.x/   with respect to t gives for all N 2 N b

 '.x/  

D

N 1 X

1 @n b nŠ @t n



.x;t /  1 jt D0 C .N 1/Š 

Z

nD0

1 0

@N @t N

  /  Using the fact that @t@ f .x;t D h.x/. f /  defined by . f /.X / D X 2 f 0 .X /, we obtain b

 '.x/  

D

N 1 X

n nŠ

h.x/n . n b/

.x;t /  jt D 

.x;t /  

.1 /n1 d :

with the operator



Z h.x/



x 

nD0 N C .N1/Š

b

1

N

N

. b/ 0



.x; /  .1 

(2.10)  /

n1

d

and one can verify that the last term is O.N /. The compatibility of CA  sEs with t addition and multiplication then yields the existence of a CAsE for b '.x/  . u

26

2 Composite Asymptotic Expansions: General Study

Differentiation. As was the case for composite formal series, CAsEs are compatible with differentiation if the domains are slightly reduced and if the first fast term is identically zero. Recall and complete the notation of Definition 2.5: let ˛; ˛1 ; ˛2 ; ˇ; ˇ1 ; ˇ2 2 R with ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ and ˛2 < ˇ2 , let 0 ; r0 > 0 and let  2 R. Let V D V .˛; ˇ; 1; /, S2 D S.˛2 ; ˇ2 ; 0 / and V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/. Moreover, let e r 0 2 0; r0 Œ, e  < , e ˛1 ; e ˇ 1 be such that ˛1 < e ˛1 < e ˇ 1 < ˇ1 and e e ˇ2  ˛2 < ˇ 1  e ˛ 1 and e ˛ ; ˇ such that ˛ <e ˛ e a 1  ˇ2 < e ˇ 1  ˛2  e ˇ < ˇ: e D V .e e1 ./ D V .e ˛1; e ˇ 1 ;e r 0; e  jj/. Let V ˛; e ˇ; 1; e / and V Lemma 2.7. Let y.x; / be a function defined for  2 S2 and x 2 V1 such that  x  n P b .r0 ; V / as S2 3  ! 0.  DW b y .x; / 2 C y.x; /  n0 an .x/ C gn  Assume that g0 .X / 0. Then one has X  x  n db y dy 0 .x; /  .x; / D an0 .x/ C gnC1   dx dx n0 db y b .e e/. .x; / 2 C r 0; V dx Proof. Let ı D min.jj.  e /; r0 e r 0 / and, for N 2 N arbitrary,

e 1 ./, where as S2 3  ! 0 and x 2 V

RN .x; / D y.x; / 

X

an .x/ C gn

 x  

n :

(2.11)

n
One has

1 ı

DO



1 jj



. The Cauchy formula for the derivative gives

ˇ ˇ ˇ ˇ Z ˇ ˇ 1 ˇ dRN C1 RN C1 .u; / ˇˇ 1 ˇDˇ ˇ .x; / max jRN C1 .u; /j; d u ˇ ˇ 2 i ˇ  ı juxjDı ˇ dx 2 juxjDı .u  x/ dRN C1 .x; / D O.jjN /. Since by the Cauchy formula the functions dx 0 e, we deduce that dRN .x; / D aN and gN0 C1 are bounded in D.0;e r 0 /, resp. in V dx   O jjN . t u which yields

Remark. Lemma 2.7 is not valid in the real framework. For example, it is wellknown that there are small functions with unbounded derivatives. We have the following result in real framework, however: if the derivative of a function with CA sE also has a CA sE , then formula (2.12) below implies that the CA sE of the derivative can be obtained by differentiating the CAsE of the function term by term.

2.3 Composite Expansions: Definition and Basic Properties

27

Integration. Integration of a CAsE does not always yield a CAsE because of possible terms X1 in the expansions of functions in G .V /. If all these terms are absent, integration poses no problem, as the following statement shows.    P Proposition 2.8. Consider a CAsE y.x; /  n0 an .x/ C gn x n defined for  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/, such that all functions gn satisfy gn .X / D O.X 2 / as X ! 1. Let r 2 S.˛ Z 1 ; ˇ1 ; r0 /. x

Then the function .x; / 7!

y.t; / dt has a CAsE. More precisely, one has r

Z

x r

b.x; /  Y b.r; /; where y.t; / dt  Y 1  X   b.x; / D A0 .x/ C An .x/ C Gn1 x n Y

(2.12)

nD1

Z

x

with An .x/ D

Z an .t/ dt and Gn .X / D 

r

1

gn .T / d T . X

b.r; / with the formal series in which Gn1 . r / has been Here we have identified Y  replaced by its asymptotic expansion as  ! 0. The proof is immediate: one has An 2 H .r0 / and, by hypothesis, Gn 2 G .V /. Before stating the result in the general case, we introduce some notation. Let ` be an analytic function in the quasisector V D V .˛; ˇ; 1; / such that its derivative `0 has an asymptotic expansion at infinity starting with X1 : `0 .X / 

X

cn X n with c1 D 1:

n1

One can choose e.g. `.X / D log.X   / with  … V . If one wants a function having real values on the real axis, one may use `.X / D 12 log.X 2 C L2 / with L   large enough. Observe that the expression ` x will not be bounded for  2 S2 and   x in some quasi-sector V .˛; ˇ; r0 ;  jj/, but we have ` x D O .jlog.jj/j/ there,   because `.X / D log X C C C O X 1 as V 3 X ! 1 with some constant C . The statement in the general case is as follows.    P b denote Proposition 2.9. Given a CAsE y.x; /  n0 an .x/ C gn x n , let R P n b the series of residues of the gn .X /: R./ D n0 gn1  . Let r 2 S.˛1 ; ˇ1 ; r0 /. Z x b.x; /  Y b.r; /, with Then one has y.t; / dt  Y r

28

2 Composite Asymptotic Expansions: General Study

     b.x; / D R./ b Y ` x  ` r C A0 .x/C 1  X   An .x/ C Hn1 x n ;

(2.13)

nD1

Z

Z

x

where An .x/ D

an .t/ dt and Hn .X / D  r

1



 gn .T /  gn1 `0 .T / d T .

X

b Again,  r identified Y .r; / with the formal expression obtained by replacing  r  we have `  and Hn  by their expansions as  tends to 0. Proof. The classical Borel–Ritt theorem (see below) provides a function   R./ with b R./ as asymptotic expansion. The difference y.x; /  R./`0 x satisfies the condition of Proposition 2.8, so its integral has a CAsE. The “generalized CAsE” for y follows. t u We also have a statement similar to the classical Borel–Ritt theorem. The statement of the classical theorem is as follows: given any sequence .an /n2N of complex numbers and any sector S.˛; ˇ; 0 /, there exists a function a D P n a./ defined and holomorphic on S and having the formal series 1 a  as n nD0 asymptotic expansion. The result is also true when .an /n2N is a sequence of bounded analytic functions of a complex variable x. In the case of our CAsEs, the statement is as follows. Lemma 2.10. (Borel–Ritt) Let V D V .˛; ˇ; 1; / be an infinite quasi-sector ( > 0 or < 0), S2 D S.˛2 ; ˇ2 ; 0 / a finite sector, r0 > 0 and let ˛1 < ˇ1 be such that y .x; / D  ˛  ˛1 ˇ2< ˇ1  ˛2  ˇ. Given a composite formal series b P x n b a  .x/ C g 2 C .r ; V /, there exists a holomorphic function y.x; / n n  0 n0 defined for  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ such that y.x; /  b y .x; / as  ! 0. Proof. Simply use the P Borel–Ritt theoremPfor classical uniform asymptotic expant u sion twice: once for an .x/n , once for gn .X /n . Exercise 2.11. (a) Prove that the equation y C y" D 2x C 2x 2 in the complex domain has a unique solution y D y.x; "/ holomorphic on the annulus 2 j"j1=2  jxj  12 satisfying y.x; "/ D 2x C 2x 2 C o.1/ as " ! 0 uniformly on this annulus. (b) Using the properties of CAsEs discussed in this section, show that z.x; / D y.x; 2 / has a CAsE in the annulus 2 jj < jxj < 12 , as  ! 0. Exercise 2.12. Let f D f .x; / be a holomorphic function defined when  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V .˛1 ; ˇ1 ; r0 ; jj/ and having a CAsE f .x; / 

X   b .r0 ; V /: an .x/ C gn x n 2 C n0

2.4 Composite Expansions and Matching

29

(a) Suppose that a0 .0/ ¤ 0 and g0 D 0 identically. Prove that the function 1=f has a CAsE,  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V .˛1 ; ˇ1 ; r0 ; jj/ (with the same ). (b) Assume only a0 .0/ ¤ 0 and g0 arbitrary. Prove that 1=f has a CAsE,  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V .˛1 ; ˇ1 ; r0 ; jj/ if a0 .0/ C g0 .X / does not vanish on the closure of V .˛1  ˇ2 ; ˇ1  ˛2 ; ; 1/. Exercise 2.13. Give a detailed proof of Lemma 2.7. Exercise 2.14. This example comes from Skinner’s book [54]. Prove that the function z given by z.x; / D xC2x 3 C has a CAsE for  > 0 and x 2; C1Œ for any  > 1. Compute an asymptotic expansion (containing a term in ln ) for Z

1

F ./ D

z.x; /dx: 0

Exercise 2.15. Suppose that y.x; / is a function holomorphic and bounded on the set of all complex .x; / with jj < 0 , K jj < jxj < L, where 0 ; K; L are some positive numbers. Using the Laurent decomposition of y, prove that y has a CAsE as  ! 0, uniformly on the given annulus and that this CAsE is actually convergent. Using this result, solve again the Exercises 2.11(b) and 2.14, except for the value of .

2.4 Composite Expansions and Matching Our concept of composite expansion combines P the classical asymptotic expansion n in the sense of Poincar´ e of the form y.x; /  n0 cn .x/ and an expansion of P n the form y.X; /  n0 hn .X / . The former expansions are called “outer”, the latter are called “inner” expansions. These inner and outer expansions are central in the method of matched asymptotic expansion. Although CAsEs are different from both, there are close links with inner and outer expansions. On the one hand, we show that a function with a CAsE also has an inner and an outer expansion, and that these two expansions have a common region of validity. In other words, a proof of existence of a CAsE can provide a solid foundation for the method of matching. On the other hand, the converse is true: if the method of matching is valid, i.e. if a function has inner and outer expansions with a common region of validity, and if moreover such expansions satisfy an additional property, then the function also has a CAsE. We emphasize that the results of this section, especially Proposition 2.17, are not new. They present the classical relations between inner, outer and uniform expansions adapted to our framework (see Chap. 7 for some more details). The first result is the following.

30

2 Composite Asymptotic Expansions: General Study

Proposition 2.16. Let .an /n2N be a family of functions of H .r0 / and .gn /n2N a 1 X anm x m and family of functions of G .V / with V D V .˛; ˇ; 1; /. Let an .x/ D gn .X / 

X

mD0

gnm X m denote their expansions. Suppose that

m>0

y.x; / 

X

an .x/ C gn

 x  

n

n0

as S2 3  ! 0 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ in the sense of Definition 2.5. Then, for fixed x 2 S.˛1 ; ˇ1 ; r0 /, one has X

y.x; / 

cn .x/n as S2 3  ! 0;

(2.14)

n0

X

where cn .x/ D an .x/ C

gl;nl x ln . Moreover, for all r > 0, this expansion

0ln1

is uniform with respect to x on all x 2 S.˛1 ; ˇ1 ; r0 / such that jxj > r. Similarly, if X 2 V and ˛3 ; ˇ3 ; 3 are such that  2 S.˛3 ; ˇ3 ; 3 / implies  2 S2 and X 2 V .˛1 ; ˇ1 ; r0 ;  jj/, then one has y.X; / 

X

hn .X /n as S.˛3 ; ˇ3 ; 3 / 3  ! 0;

(2.15)

n0

where hn .X / D gn .X / C

X

anl;l X l . The expansion is uniform with respect to

0ln

X on compact subsets of V satisfying the above condition. Remarks. 1. According to the literature, we will call the first expansion (2.14) outer expansion and the second (2.15) inner expansion. Each function cn of the outer expansion may have a singularity at x D 0 but only a pole of order at most n; similarly each function hn of the inner expansion has polynomial growth of order at most n as X ! 1. Thus the restraint index in the sense of Wasow [62], Chap. VIII equals 1. 2. One can show that for every 2 0; 1Œ, the outer expansion (2.14) is uniform on jxj > jj , and that the inner expansion (2.15) is uniform on jX j < jj , which justifies the method of matched asymptotic expansions when a CAsE exists. N In both cases, we need to use 1 terms in order to obtain an approximation  N with remainder O  . It is often preferable, however, to have uniform approximations throughout the domain instead of two different expansions on overlapping regions. Such uniform approximations seem indispensable if we want to obtain estimates of Gevrey kind.

2.4 Composite Expansions and Matching

31

3. In cases where the existence of a composite expansion for a function y.x; / can be shown indirectly, but the functions an and gn are not yet known, one method for determining them is to apply the preceding proposition. For fixed nonP zero x, one computes the outer expansion y.x; /  n0 cn .x/n , then one eliminates the terms with negative powers of x to obtain the slow P parts an .x/.n Analogously, one computes the inner expansion y.X; /  n0 hn .X / and throws away the terms with non-negative powers of X , which gives gn .X /. In practice, the calculation of inner and outer expansions often leads to recurrence equations for their coefficients. This allows to compute an ; gn without having to use the cumbersome formulas for multiplication of composite formal series. In the case of singularly perturbed differential equations, as noted by Gautheron/Isambert [29], the computation of the inner expansion is more involved than the outer one. The latter only needs algebraic operations (if the Taylor expansions of the coefficients of the equation are known). The former, however, requires solving linear differential equations and choosing the constant of integration such that the solution has a certain asymptotic behavior; this introduces transcendence. For this reason, Isambert [32] calls these outer and inner expansions algebraic and transcendental expansions, respectively. Proof of Proposition 2.16: Let N 2 N be fixed and recall the notation (2.11). X Furthermore, set glm X m : rlk .X / D gl .X /  0<m
By hypothesis, there are positive constants CN ; Ak n and Clk such that 8 2 S2 8x 2 V1; WD V .˛1 ; ˇ1 ; r0 ; jj/ 8x 2 V1;

jRN .x; /j  CN jjN ;

ˇ ˇ X ˇ ˇ akl x l ˇ  Ak n jxjn ˇak .x/ 

(2.16)

l
and

8X 2 V

jrlk .X /j  Clk jX jk :

(2.17)

An elementary calculation gives y.x; / 

X

cn .x/ D RN .x; / C n

n
X

 X X x  l ln l gl    gl nlx 0
l
D RN .x; / C

X

l
  rl Nl x l ;

(2.18)

l
hence, as jxj > r,  ˇ ˇ  X X ˇ nˇ lN jjN : cn .x/ ˇ  CN C Cl Nl r ˇy.x; /  n
l
(2.19)

32

2 Composite Asymptotic Expansions: General Study

Similarly, one has y.X; / 

P n
 X X an .X /  anl l X l n n
hn .X /n D RN .X; / C

k
l
therefore, for all R > 0 and for jX j  R,  ˇ ˇ  X X ˇ ˇ hn .X /n ˇ  CN C AkNk RN k jjN : ˇy.X; /  n
(2.20)

k
t u Conversely, one has the following statement. Proposition 2.17. Let y be a function defined for  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V ./ D V .˛1 ; ˇ1 ; r0 ;  jj/. Assume that there are real numbers a; b;   with 0 < a < b and 0 < < 1, and for each n 2 N a function cn , cn .x/ D Pn x1 C an .x/, Pn polynomial without constant term, an 2 H .r0 / and a function hn D Qn C gn , Qn polynomial and gn 2 G .V /, V D V .˛; ˇ; 1; /, ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ, with the following properties. Assumption 1. For all N 2 N, there is a constant C > 0 such that ˇ ˇ N 1 ˇ ˇ X ˇ nˇ cn .x/ ˇ  C jjN.1 / ˇy.x; /  ˇ ˇ

(2.21)

nD0

for all  2 S2 and all x 2 V ./ with jxj > ajj and ˇ ˇ N 1 ˇ ˇ X ˇ nˇ hn .X / ˇ  C jjN ˇy.X; /  ˇ ˇ

(2.22)

nD0

for all  2 S2 and all X 2 V such that X 2 V ./ with jX j < bjj 1 . Assumption 2. For any n 2 N, the polynomials Pn and Qn have degree less than n C 1. Then y has a CAsE for  2 S2 and x 2 V ./; precisely y.x; / 

1  X   an .x/ C gn x n : nD0

2.4 Composite Expansions and Matching

33

Remarks. 1. As there is a common region for expansions (2.21) and (2.22), these expansions are necessarily consistent, as shown in the proof, cf. (2.25). 2. In general we cannot get better than jjN.1 / in the remainder of (2.21) and jjN in that of (2.22), as the first neglected terms have this size when PN and QN are of degree N . 3. This statement is a special case of a general theorem of Eckhaus’ book [17]. In the classical method of matched asymptotic expansions, one first establishes inner and outer expansions on growing domains as  ! 0 having a nonempty intersection. Then one constructs so-called “composite” expansions of which our CAsEs are an example, cf. also Chap. 7. PC1 m Proof of Proposition 2.17: Let cn .x/ D mDn cnm x and MnN be the largest integer M such that M C n  N.1  /. Then (2.21) implies that for any N 2 N there exists C2 > 0 such that ˇ ˇ N 1 M nN ˇ ˇ X X ˇ ˇ cnm x m n ˇ  C2 jjN.1 / ˇy.x; /  ˇ ˇ nD0 mDn

as  2 S2 and x 2 V ./, a jj < jxj < b jj . For any integer S , we can find a constant C3 such that ˇ ˇ ˇ ˇ X ˇ ˇ m nˇ ˇy.x; /  cnm x  ˇ  C3 jjS (2.23) ˇ ˇ ˇ n0;mn;m Cn<S as  2 S2 and x 2 V ./, a jj P < jxj < b jj . m Similarly, noting hn .X /  C1 , one can find replacing X by mDn znm X for any integer S there exists a constant C4 such that ˇ ˇ ˇ ˇ X ˇ ˇ S q pCq ˇ ˇy.x; /  zpq x  ˇ  C4 jj ˇ ˇ ˇ p0;qp;q. 1/Cp<S

x 

that

(2.24)

as  2 S2 and x 2 V ./, a jj < jxj < b jj . As (2.23) and (2.24) uniquely determine the coefficients cnm and zpq , they must coincide, i.e. cnm D znCm;m for any n 2 N and m 2 Z, m  n. One thus has the formal equality 1 1 X X   b b c n .x/n ; (2.25) hn x n D nD0

nD0

where b hn and b c n denote the series associated with hn and cn .

34

2 Composite Asymptotic Expansions: General Study

N  X   an .x/ C gn x n . When a jj < jxj, Now consider the sum YN .x; / D nD0 ˇ ˇ ˇ ˇ n1 ˇ ˇ  x  NX znq x q q ˇˇ  C5 jj.N n/.1 / and hence with znq D we find with ˇˇgn   ˇ ˇ qD1 cnCq;q

ˇ ! ˇ N 1 n ˇ X X ˇ ˇ ˇ ˇ m ˇy.x; /  YN .x; /ˇ  ˇˇy.x; /  an .x/ C n ˇ C cn;m x ˇ ˇ nD0

mD1

C6 jjN.1 / :

This implies ˇ ˇ ˇ ˇ P ˇy.x; /  YN .x; /ˇ  ˇˇy.x; /  N 1 cn .x/n ˇˇ C C6 jjN.1 / nD0

 C7 jj

N.1 /

(2.26)

as  2 S2 , x 2 V ./, a jj < jxj. Using the expansions of the an , we similarly find that jy.x; /  YN .x; /j  C8 jjN also when  2 S2 , x 2 V ./, jxj < b jj . Together with (2.26), this shows that for all N , there is a constant C9 such that for all  2 S2 and x 2 V ./ one has jy.x; /  YN .x; /j  C9 jjN with D min. ; 1  /. The statement to be proven corresponds to jjN instead of jjN in this last inequality. It is obtained in two steps. On the one hand, this last assertion can also be N written: there is C10 with jy.x; /  YS .x; /j  x  C10 jj , if S > N . On the other hand the fact that all functions an .x/ and gn  are bounded on all x;  in question implies that there is a constant C11 such that jYS .x; /  YN .x; /j  C11 jjN .

t u

Exercise 2.18. Add details for Remark 2 after Proposition 2.16.

2.5 Continuation of Composite Expansions In connection with the inner and outer expansions of the method of matching, we also have two results of continuation of CAsEs, which will be very useful for solutions of differential equations. The first result says essentially that a function with a CAsE for x in a quasi-sector, whose inner expansion exists on a larger quasi-sector, admits the CAsE also on the larger quasi-sector. The precise result is as follows.

2.5 Continuation of Composite Expansions

35

Fig. 2.2 Some domains V and ˝ with different signs of , jj   and . Bold line: the boundary of V , thin line: that of ˝, their intersection in dark gray

Proposition 2.19. Let y be a function defined for  2 S2 D S.˛2 ; ˇ2 ; 0 / and   P x 2 V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/ and having a CAsE n0 an .x/ C gn x n , as S2 3  ! 0 and x 2 V1 ./, with an 2 H .r0 / and gn 2 G .V /, where V D V .˛; ˇ; 1; /, ˛ D ˛1  ˇ2 and ˇ D ˇ1  ˛2 . Let  > . In the case where  > jj, set ˝ D D.0; /, otherwise set ˝ D V .˛; ˇ;  C ; / with  > 0 arbitrarily small (Fig. 2.2). Assume that the function Y W .X; / 7! y.X; / can be analytically continued P on ˝  S2 and that it has an asymptotic expansion Y .X; /  1 h .X /n as  nD0 n tends to 0, uniformly on ˝. Then y can be analytically continued to the set of all .x; / with  2 S2 and with x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ and has a CAsE there as  ! 0. Remarks. 1. The domain ˝ has been chosen bounded, with ˝ \ V ¤ ; and V [ ˝ D V .˛; ˇ; 1; /. Signs of  and  are arbitrary. We will use this result particularly in the case  < 0 < . 2. The assumption on the domain and the asymptotic expansion of Y may be slightly weakened (the domain with respect to X may depend on the argument of ), but the version presented is sufficient for our applications to differential equations. 3. It is possible to show this result using Propositions 2.16 and 2.17, but we prefer to present an independent proof. One reason for this choice is that this proof will serve for the Gevrey analog Proposition 3.8. In contrast to this, we have no Gevrey analog of Proposition 2.17.

36

2 Composite Asymptotic Expansions: General Study

Proof of Proposition 2.19: The expansion of Y in the assumption and the inner expansion corresponding to the CAsE of y given by Proposition 2.16 coexist on some open region, hence coincide by the uniqueness of an asymptotic expansion. Thus, the functions hn .X / of the assumption are necessarily the analytic continuations of the coefficients of this inner expansion. The hypothesis implies that y.x; / can be analytically continued to the set of e1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/, precisely by all .x; / such that  2 S2 and x 2 V x putting y.x; / D Y  ;  . We now use again the notation introduced in the proof of Proposition 2.16. We have to show that the remainder RN .x; / given by (2.11) e1 ./ n V1 ./. The assumption is bounded by a constant times jjN , also for x 2 V e1 ./ n V1 ./, one has x= 2 ˝. By on ˝ ensures that for all  2 S2 and all x 2 V hypothesis, there exists DN such that ˇ X   ˇˇ ˇ hn x n ˇ  DN jjN : ˇy.x; /  n
Now, the equality above (2.20) can be written RN .x; / D y.x; / 

X

hn

  X x  n X l a k :   .x/  a x k kl 

n
k
l
Moreover, modifying the constants Ak n if necessary, inequality (2.16) is valid for e 1 ./ n V1 ./. This shows that for all all x 2 D.0; r0 /, hence in particular for x 2 V e  2 S2 and for all x 2 V 1 ./ n V1 ./   X N k jRN .x; /j  DN C jjN Ak Nk M

(2.27)

k
with M D supX 2˝ jX j D  C ı or . The second result concerns outward continuation.

t u

r 0 and let y be a function defined for  2 S2 D Proposition 2.20. Let 0 < r0 < e e1 ./ D V .˛1 ; ˇ1 ;e S.˛2 ; ˇ2 ; 0 / and x 2 V r 0 ;  jj/. Suppose that y has a CAsE x n P  , as S2 3  ! 0 and x 2 V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/, n0 an .x/ C gn  with an 2 H .r0 / and gn 2 G .V /, V D V .˛; ˇ; 1; / such that ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ. P n Assume moreover that y has an asymptotic expansion y.x; /  1 nD0 cn .x/ as  tends to 0, uniformly for x 2 V .˛1 ; ˇ1 ; r0  ;e r 0 / with  > 0 arbitrarily small. e 1 ./. Then (2.9) is satisfied for all  2 S2 and all x 2 V e 1 ./, Remark. By abuse of notation, we say that y has a CAsE for  2 S2 and x 2 V although the functions an are not necessarily defined on the whole disk D.0;e r 0 /. Proof. First, we can use Proposition 2.16 on the quasi-sector V .˛1 ; ˇ1 ; r0  ; r0 /, and by comparing (2.14) with the second hypothesis, we obtain that the functions cn

2.6 Quotients of CA sEs

37

of the hypothesis are analytic continuations of those of the proposition. Therefore we can also continue the functions an analytically on D.0; r0 /[V .˛1 ; ˇ1 ; r0 ;e r 0 /. e 1 ./ n V1 ./. By (2.18), one has It remains to estimate RN for x 2 V RN .x; / D y.x; / 

X

cn .x/n 

n
X

rl Nl

x  l : 

(2.28)

l
and by (2.17) jrl Nl .X /j  Cl Nl jX jlN . By hypothesis, there are AN > 0 such that X jy.x; /  cn .x/n j  AN jjN n
Exercise 2.21. Use Propositions 2.16 and 2.17 to prove Proposition 2.19.

2.6 Quotients of CAsEs Here we investigate under which conditions the multiplicative inverse of a function with a CAsE has a CAsE. If the first slow term a0 is non zero at x D 0, then this inverse has a CAsE thanks to composition with the function f 7! 1=f , see Exercise 2.12. Here we investigate a more general situation. Let y D y.x; / be a function defined and analytic for  2 S D S.ı; ı;0 / x n P  and x 2 V .˛; ˇ; r0 ; jj/, having a CAsE y.x; /  n0 an .x/ C gn  as  ! 0. We propose a slightly more general statement, more useful in practice: it establishes conditions under which there exists k 2 N such that the function .x; / 7! k =y.x; / has a CAsE. By Proposition 2.16, y has an inner expansion X hn .X /n as S2 3  ! 0; y.X; /  n0

uniformly with respect to X on compact subsets of S.˛1 ; ˇ1 ; 1/ where ˛1 D ˛  ı; ˇ1 D ˇ C ı and X al;nl X nl : (2.29) hn .X / D gn .X / C 0ln

For all n, let

C1 X mDn

hnm X m denote the asymptotic expansion of hn at infinity and

let vn D val.hn / denote the least integer m  n such that hnm ¤ 0. If hn is flat, we

38

2 Composite Asymptotic Expansions: General Study

put vn D val.hn / D C1. We say that y is degenerate, if it is flat or if there exists N 2 N such that h0 D : : : D hN 1 D 0 and hN ¤ 0 is flat. If y is nondegenerate, let C.y/ denote the pair .N; M / with N 2 N such that h0 D : : : D hN 1 D 0, hN ¤ 0 and M D val.hN /  N . Proposition 2.22. With the previous notation, the following three conditions are equivalent. r 0 < r0 and e    such that the function .x; / 7! (a) There exist k 2 N, e 0 < 0 , e k =y.x; / has a CAsE as  ! 0 in e S D S.ı; ı;e 0 / and x 2 V .˛; ˇ;e r 0; e /. (b) y is non-degenerate and, if C.y/ D .N; M /, one has val.hn /  M  n C N for all n  N . (c) There exist k 2 N, ` 2 Z such that the function .x; / 7! k x ` y.x; / has a CA sE whose first slow coefficient e a0 satisfies e a0 .0/ ¤ 0. Remarks. 1. The second condition in (b) can also be written in terms of the outer expansion and the expansions of its coefficients. This is a consequence of the relation (2.25). 2. Graphically, the second condition in (b) means that the points with coordinates .n; m/ such that hnm ¤ 0 (the “support” of the inner expansion) are all in the quadrant on the right of the vertical line and above the line of slope 1 passing through C.y/, see Fig. 2.3. Since hn has a polynomial part of degree at most n, we already know that this support is in the quadrant on the right of the axis and above the second bisector. The change of variable y ! z W .x; / 7! k x ` y.x; / induces a shift of just C.y/ D .N; M / on the supports, with N D k  ` and M D `. 3. The proof also provides a procedure to calculate the CAsE for k =y.x; /. Using the above shift, the situation is reduced to the case where y has a first slow term a0 non-zero at x D 0. Thus we obtain the CAsE by left composition with the function u 7! 1=u. Proof of Proposition 2.22: We show the implications (b))(c))(a))(b). Suppose that condition (b) is satisfied.  M If M < 0, first consider z.x; / D x y.x; /. As a product of two functions having CAsEs, z has a CAsE on the same domain as y. The corresponding inner expansion is that of X M y.X; / and satisfies therefore a condition similar to (b) with .N; 0/ instead of .N; M /. If M > 0, consider z.x; / D x M y.x; /. As before, z has a CAsE and the corresponding inner expansion is that of M X M y.X; /, thus satisfies a condition similar to (b) with .N C M; 0/ instead of .N; M /. Therefore both cases M > 0 and M < 0 can be reduced to the case M D 0. If M D 0, then formula (2.29) and the condition on the hn show that asm D 0 for 0  s < N and m  0, and that aN 0 ¤ 0. Since the functions as are analytic, this implies as D 0 for s D 0; : : : ; N  1. Then the function e y W .x; / 7! N y.x; / has also a CAsE on the same domain as y and satisfies the condition (c). To sum up, in the three cases y satisfies condition (c) with k D N C M and ` D M .

2.6 Quotients of CA sEs

39

M 0

C(y) N

slope q Fig. 2.3 In gray, the part of the plane containing the support of the inner expansion of y. In bold, the boundary of the analogous part for z

Now suppose the condition (c) satisfied and set z.x; / D k x ` y.x; /. For e r 0 and e 0 small enough and e    suitable, the function z does not vanish when 2e S D S.ı; ı;e 0 / and x 2 V .˛; ˇ;e r 0; e /. Proposition 2.6(b) applies with the function f W u 7! 1=u and we deduce that 1z has a CAsE. In the case where `  0, the function .x; / 7! x ` =z.x; / has therefore a CAsE, which proves (a). In the case  ` ` < 0, the function .x; / 7! x =z.x; / has a CAsE, which gives (a) with k  ` instead of k. Finally, suppose that condition (a) is satisfied, let e y .x; / D k =y.x; / and let e hn denote the coefficients of the inner expansion of e y . Since .ye y /.x; / D k , there is a first term hr which is not identically zero in the inner expansion of y and a first term e hs for e y , with r C s D k and hre hs D 1. Each of these functions is of polynomial growth as X ! 1, so none can be flat. Thus the two functions y and e y e; M f/ D C.e are nondegenerate. Set .N; M / D C.y/ and .N y /. For a proof by contradiction, suppose there ˚  exists n > N such that val.hn / < s /M M  n C N . Let q D min val.h I s > N < 1 and M D fs  N I val.hs /  sN sq D M  N qg; it is a set of cardinal at least 2 (containing at least N and some s for which the minimum q is attained) and finite (since val.hs /  s). If e y satisfies condition (b), we set e q D 1, otherwise e q is the analog of q for y. Switching e y and y if necessary, we can assume without loss of generality that e g and N denote the finite and q e q . Let then K D minfval.e hs /  sq I s > N e g if nonempty set of all s 2 N such that val.e hs /  sq D K. Note that N D fN e q > q and that the cardinal of N is at least 2 if e q D q. Recall that the minimum of M is n1 D N , and let n2 D max M ; let e n1 and e n2 denote the minimum P and the maximum of N . Consider P the inner expansion of p D ye y : p.X; /  n0 pn .X /n . Thus one has pn D rCsDn hre hs for all e n1 then hn1 he n1 CK D M N q Cnq CK; n  0. If n D n1 Ce n1 has valuation M Ce if r C s D n D n1 C e n1 with r ¤ n1 , then the valuation of hre hs is greater than that number because of the choice of n1 and e n1 . So we obtain pn1 Ce n1 ¤ 0. Similarly, we

40

2 Composite Asymptotic Expansions: General Study

also get pn2 Ce n1  e n2 , this contradicts the assumption n2 ¤ 0. Since n1 < n2 and e t u that the product ye y is reduced to the monomial k .

2.7 Multiple CAsEs In this section we call vertex of a function y a point near which y has a CAsE which is not a classical asymptotic expansion, i.e. at least one of the fast terms gn is non zero. Here we would like to discuss the case of a domain with two vertices on its boundary. The following statement shows that it not necessary to generalize the concept of CAsE for uniform expansions on domains that have several vertices on their boundary, because we can reduce this situation to the case of a single vertex. For simplicity, we only study the case of a real interval. Proposition 2.23. Let a < b < c < d be four real numbers and let y W a; d Œ0; 1  ! R be a function having a CAsE y.x; / 

1  X   n an .x/ C gn xa   nD0

as  ! 0, uniformly on a; cŒ, with an holomorphic in a neighborhood of Œa; c and gn 2 G .S /, S D S.ı; ı; 1/ with some ı > 0. Assume that y also has a CAsE y.x; / 

1  X   n bn .x/ C hn d x   nD0

as  ! 0, uniformly on b; d Œ, with bn holomorphic in a neighborhood of Œb; d  and hn 2 G .S /. Then y has an asymptotic expansion y.x; / 

1  X     n cn .x/ C gn xa C hn d x   

(2.30)

nD0

as  ! 0, uniform on a; d Œ with the gn ; hn of the previous formulas and with functions cn holomorphic in a neighborhood of Œa; d . P P m More precisely, if gn .X /  1 gnm X m and hn .X /  1 as mD1 mD1 hnm X Pn1 `n X ! 1, then cn .x/ D an .x/ `D0 h` n` .d x/ when x is in a neighborhood P `n of Œa; c and cn .x/ D bn .x/  n1 when x is in a neighborhood `D0 g` n` .x  a/ of Œb; d . Remark. The functions cn are the non-polar parts of the functions bn at the point x D a and the non-polar parts of the functions an at the point x D d .

2.7 Multiple CA sEs

41

Proof. By P the classical Borel–Ritt theorem, P we construct two functions g; h with g.X; /  n0 gn .X /n and h.X; /  n0 hn .X /n uniformly on S . Then we     consider the difference z.x; / D y.x; /g xa ;  h d x ;  . Proposition 2.16,   applied to h respectively g, shows that z has two slow expansions uniform on Œa; c and on Œb; d . By the uniqueness of asymptotic expansions, they must coincide on Œb; c. This implies that their coefficients must be continuations of each other and we obtain the statement. t u

Chapter 3

Composite Asymptotic Expansions: Gevrey Theory

In the following chapters (Chaps. 5 and 6) we will apply CAsEs to singularly perturbed differential equations. We will see at this occasion that the notions of composite formal series and composite expansions of Gevrey kind play a key-role. The notion of Gevrey asymptotics, which was introduced by Ramis [48] for asymptotics in the independent variable, has played a key-role in the classical theory of asymptotic expansions uniform in a full neighborhood of a turning point (see e.g. [9]). In the present chapter, we generalize this concept to composite expansions and thus to expansions valid on quasi-sectors with vertex at a turning point. From now on, p denotes a positive integer. In general (except in Sect. 5.1) this integer is at least 2.

3.1 Composite Gevrey Formal Series Definition 3.1. Let r0 > 0,  2 R , V D V .˛; ˇ; 1; / be given by (2.1) or (2.2) X and let   b .r0 ; V /; an .x/ C gn x n 2 C b y .x; / D where gn .X / 

X

n0

gnm X

m

as V 3 X ! 1. We say that b y is Gevrey of order

m>0

and type .L1 ; L2 /, if there is a constant C > 0 such that for all n 2 N one has   sup jan .x/j  CLn1  pn C 1 and for all n; M 2 N and all X 2 V one has

1 p

jxj
ˇ ˇ M 1 X ˇ ˇ  M Cn  m ˇ ˇ gnm X ˇ  CLn1 LM jX j ˇgn .X /  2  p C1 : M

(3.1)

mD1

Here the sum is meant to be 0 if M D 0 or M D 1. We say that functions gn .X /; n D 0; 1; : : :, holomorphic in V have consistent asymptotic expansions of Gevrey order p1 (and type .L1 ; L2 /) if there is a constant C > 0 such that (3.1) is satisfied for all n; M 2 N and X 2 V . A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 3, © Springer-Verlag Berlin Heidelberg 2013

43

44

3 Composite Asymptotic Expansions: Gevrey Theory

We have defined above the notion of type for Gevrey composite formal series. Observe that we cannot talk of “the” type of a series, because any pair of numbers larger than L1 ; L2 will also be a type of the series. We do not want to discuss the optimality of the constants L1 and L2 for composite formal seriesin this memoir.  It would be possible to formulate (3.1) with products  pn C 1  M p C 1 using the inequalities  .a C 1/ .b C 1/   .a C b C 1/  2aCb  .a C 1/ .b C 1/; if a; b > 0 (3.2) but we prefer to write (3.1). This also simplifies some proofs. We deduce easily from (3.1) that  mCn  C1 (3.3) jgnm j  CLn1 Lm 2 p for all n; m 2 N, ˇm  1, see Exercise 3.3. ForM  1, the ˇ  inequalities (3.1) are M Cn  C 1 with the operator T of equivalent to jX j ˇ.TM 1 gn /.X /ˇ  CLn1 LM 2 p (2.3). It will be helpful to avoid having to prove the inequalities (3.1) for X near 0. This is the purpose of the following proposition. Proposition 3.2. The functions gn .X /; n D 0; 1; : : :, have consistent asymptotic expansions of Gevrey order p1 and type .L1 ; L2 / if and only if there are two constants C; T > 0 such that   sup jgn .X /j  CLn1  pn C 1 (3.4) X 2V

for all n 2 N and such that ˇ ˇ M 1 X ˇ ˇ  M Cn  m ˇ ˇ gnm X ˇ  CLn1 LM jX j ˇgn .X /  2  p C1 M

(3.5)

mD1

for all n; M 2 N and all X 2 V with jX j > T . Proof. The necessity of the conditions is obvious (for (3.4), choose M D 0). For their sufficiency, one has to show, from (3.4) and (3.5), that (3.1) is also true for X 2 V with jX j  T . First (3.5) implies (3.3). Using (3.2) we obtain for jX j  T ˇ ˇ M 1 X ˇ ˇ   gnm X m ˇˇ  T M CLn1  pn C 1 C jX jM ˇˇgn .X /  mD1

M 1 X

 mCn  CLn1 Lm C 1 T M m 2 p mD1   e Ln LM  M Cn C 1 C 1 2 p eDC with C



P k1

T L2

k .    pk C 1 .

t u

3.1 Composite Gevrey Formal Series

45

The inequalities (3.1) are also essential for the compatibility of the new concept with elementary operations. We leave the case of addition and differentiation as exercises. Below we show the compatibility with multiplication. Consider first the classical b c.x; / D b a1 .x; /b a 2 .x; ˇ / of two ˇ P case of a product 1 n ˇaj n .x/ˇ  formal series b aj .x; / D a .x/ of Gevrey order . If sup j n n0 p jxj
jxj
n X



 p

   C 1  n C1 p

D0

and we obtain the Gevrey character of b c from the inequality n X



 n p

     C 1  p C 1  Kp  pn C 1

(3.6)

D0

     with the constant Kp D 2 1 C  p1 C 1 C : : : : C  p1 C 1 C 2p. p b Now consider h.x; / D b g 1 .x; /b g 2 .x; / of two formal series P a xproduct n b g j .x; / D gjn .  / of Gevrey order p1 and suppose, according to Proposiˇ ˇ   tion 3.2, that sup ˇgjn .X /ˇ  Cj Lnj  pn C 1 for all n. As before, one shows that h satisfy the coefficients hn of b sup jhn .X /j  Kp C1 C2 max.L1 ; L2 /n 

n p

 C1 :

We must also show that the hn .X /, n 2 N, have consistent Gevrey asymptotic expansions, i.e. (3.5). We therefore assume (without loss of generality with the same constants) ˇ ˇ  M Cn  jX j ˇTM 1 gj n .X /ˇ  CLn1 LM 2  p C1 for all same estimate for the gj nm . Then we estimate ˇ n; M withˇ M  1 and the P e/ D jX j ˇTM 1 hn .X /ˇ using hn D nD0 g1 g2;n . Applying the formula T.f f e e T.f /f C f1 f several times for each term of this sum, we obtain M n X X ˇ ˇ M 1    nCM m  2 n M ˇ ˇ  C T h .X / L L  Cm C1 : jX j n 1 2 p C1  p D0 mD1

e such that for all n; M Similarly to (3.6), we show the existence of a constant K M n X X



 Cm p

     m e M Cn C 1 : C 1  nCM C 1  K p p

D0 mD0

ˇ ˇ This completes the estimate of the terms jX j ˇTM 1 hn .X /ˇ.

46

3 Composite Asymptotic Expansions: Gevrey Theory

P It remains to treat the mixed case. Let b y .x; / D n0 an .x/n and b z.x; / D x  n 1  g be two composite formal series of Gevrey order and of type n0 n  p P m to Definition 3.1. Let an .x/ D m0 anm x when jxj < r0 and .L1 ; L2 / according P gn .X /  m1 gnm X m as X ! 1. According to the definition of the product of two composite formal series, one has P

.b y b z/.x; / D

X

n

n0

n X 

X   I.a /I.gn / .x; / D bn .x/ C hn . x / n ; (3.7)

D0

n0

where bn .x/ D

X

gl;m .Sm a /.x/ and hn .X / D

ClCmDn

X

a m .Tm gl /.X /:

ClCmDn

According to the hypothesis, we have ja m j  CL1

  1 m    p C1 ; r0

sup jSm a j  CL1

  2 m    p C1 ; r0

and jglm j  CLl1 Lm 2

 mCl p

  mCl  C 1 and sup jTm gl .X /j  CLl1 Lm 2 p C1 :

This implies that X

sup jbn .x/j  C 2

 mCl

Ll1 Lm 2

p

  m   C 1 L1 r20  p C 1 :

ClCmDn

Using a larger L1 if necessary, we may assume that q WD X

sup jbn .x/j  C 2 Ln1

qm

 mCl p

2L2 r0 L1

< 1 and we obtain

   C 1  p C 1

ClCmDn



n    C 2 Ln1 X  n  p C 1  p C 1 : 1  q D0

The desired estimate for bn is again a consequence of inequality (3.6). The estimate for hn .X / is the same; it remains to verify that the hn .X / have consistent expansions of Gevrey order p1 . As this is equivalent to estimating X TM 1 hn .X / D

X ClCmDn

it is again analogous to the previous case.

a m X TmCM 1 gl .X /;

3.2 Composite Gevrey Asymptotic Expansions

47

Unlike the case of addition, multiplication and differentiation, the compatibility of the notion of composite Gevrey formal series with the composition on the left or right by an analytic function would be very tedious and difficult to show using only the definition. It will be proven in Sect. 4.6 using Proposition 3.20 and our key-theorem 4.1. Exercise 3.3. Deduce (3.3) from (3.1) with the same constants C , L1 and L2 . Hint: use (3.1) for M D m and M D m C 1. Exercise 3.4. Show that the functions gn , n D 0; 1; : : :, holomorphic in V have consistent asymptotic expansions of Gevrey order p1 iff there is a constant A > 0 such that the remainders of the Gevrey estimates for the functions An X n gn .X / are the same for jX j > R, i.e. iff there exist constants K; L > 0 such that for all m; n 2 N and all X 2 V with jX j > R:   Tm An X n gn .X /  K Lm mŠI here T is given by (2.3). Prove that one may choose L D L1 =L2 if L1 ; L2 are the constants in (3.1). Exercise 3.5. Prove that the derivative of a composite formal series b y .x; / in b C.r0 ; V .˛; ˇ; 1:// of Gevrey order p1 defined in Definition 2.3 is also a composite formal series of Gevrey order p1 if considered as an element of every C.e r 0 ; V .e ˛; e ˇ; 1:e  // if 0 < e r 0 < r0 , 0 < ˛ < e ˛ <e ˇ < ˇ and e  < .

3.2 Composite Gevrey Asymptotic Expansions The definition of a Gevrey CAsE is close to Definition 2.5; only the constant KN of (2.9) is specified. Definition 3.6. Let V D V .˛; ˇ; 1; / be an infinite quasi-sector ( positive or negative), S2 D S.˛2 ; ˇ2 ; 0 / a finite sector and ˛1 < ˇ1 such that ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ. Let y.x; / be a holomorphic for  2 S2 and x 2  function defined  P b .r0 ; V /. Then we y .x; / D n0 an .x/ C gn x n 2 C V .˛1 ; ˇ1 ; r0 ;  jj/ and b y as CAsE of Gevrey order say that y has b

1 p

and we write y.x; /  1 b y .x; / as p

S2 3  ! 0 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/, if b y .x; / is of Gevrey order p1 and type .L1 ; L2 / in the sense of Definition 3.1 for some L1 ; L2 > 0 and if there exists a constant C such that, for all N , for all  2 S2 and all x 2 V .˛1 ; ˇ1 ; r0 ;  jj/, ˇ ˇ N 1  ˇ X  x  n ˇˇ N  N ˇ an .x/ C gn   ˇ  CLN ˇy.x; /  1  p C 1 jj : ˇ ˇ nD0

(3.8)

48

3 Composite Asymptotic Expansions: Gevrey Theory

Remark. In the case of an annulus, this definition of Gevrey CAsE is equivalent to that of monomial Gevrey asymptotic expansion; cf. [8], Definition 3.6. To see this, one proceeds in the same manner as in Remark 1 after our Definition 2.5. Condition (3.5) is automatically satisfied in this case because the gn .X / are holomorphic. The notion of monomial summability of [8], however, has not been generalized for CAsEs in our memoir. A first step in this direction is Proposition 3.14. The compatibility of Definition 3.6 with addition, differentiation and integration are left as exercises, cf. Exercises 3.10, 3.11. Below we detail the compatibility with multiplication. We show it only for the P case of a mixed product of y.x; /  1 b y .x; / D n0 an .x/n with z.x; /  1 p p   P zO.x; / D n0 gn x n ; this is the most interesting case. Let us write N 1 X y.x; / D an .x/n C PN .x; / nD0

and z.x; / D

N 1 X

gn

x  n  C QN .x; / 

nD0

N  N where the two remainders are bounded by CLN 1  p C 1 jj . The first estimate below is classical; by the way, it is the same for the two cases y1 y2 and z1 z2 that are not detailed. ˇ ˇ N 1 n ˇ X X  x ˇˇ ˇ n  a .x/gn  ˇ jRN .x; /j WD ˇ.y  z/.x; /  ˇ ˇ nD0 D0 ˇN 1 ˇ ˇX ˇ  x  N n ˇ ˇ D ˇ Pn .x; /gN n   C y.x; /QN .x; /ˇ ˇ ˇ nD0

N  C 2 LN 1 jj

N X



n p

   C 1  Npn C 1

nD0

N  and (3.6) implies jRN .x; /j  Kp C 2 LN 1  p C 1 with the constant Kp indicated there. With the notation of (3.7), we have to estimate N 1 X nD0

n

n X

a .x/gn

D0

x  



N 1  X

bn .x/ C hn

 x  n  : 

nD0

By (2.7) applied m times we have a.x/g

x 

D

m1 X lD0

  al .Tl g/ C gl .Sl a/ .x; /l C .Sm a/.Tm g/.x; /m :

3.2 Composite Gevrey Asymptotic Expansions

Thus it remains to estimate

X

49

.Sm a /.Tm gl /.x; /. This is analogous to the

ClCmDN

case of the product of Gevrey formal series dealt with in Sect. 3.1. Finally we obtain as desired ˇ ˇ N 1  ˇ X  x  n ˇˇ N  N ˇ bn .x/ C hn   ˇ  Kp C 2 LN ˇ.y  z/.x; /  1  p C 1 jj ˇ ˇ nD0

t u

with the constant Kp indicated in (3.6).

The result for quotients of Gevrey CAsEs and its proof are identical to Proposition 2.22 and its proof for “ordinary” CAsEs if we add the word “Gevrey”. Simply use Theorem 4.7 (b) of Sect. 4.6 instead of Proposition 2.6 (b) to obtain the CAsE of the composition of a CAsE with the function y 7! 1=y. The analog of Proposition 2.16 for Gevrey asymptotics is also true. X X anm x m 2 H .r0 / and gn .X /  gnm X m 2 Proposition 3.7. Let an .x/ D m0

G .V /. Suppose that

y.x; /  1

p

m>0

X   an .x/ C gn x n n0

as S2 3  ! 0 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ in the sense of Definition 2.5. Then, for all r > 0, one has X y.x; /  1 cn .x/n as S2 3  ! 0 p

n0

uniformly with X respect to x on all x 2 S.˛1 ; ˇ1 ; r0 / such that jxj > r, with cn .x/ D an .x/ C gl;nl x ln . 0ln1

Similarly, given a compact subset K of V and ˛3 ; ˇ3 ; 3 such that X 2 K and  2 S.˛3 ; ˇ3 ; 3 / imply  2 S2 and X 2 V .˛1 ; ˇ1 ; r0 ;  jj/, one has y.X; /  1

1 X

p

hn .X /n as S.˛3 ; ˇ3 ; 3 / 3  ! 0

nD0

uniformly for X 2 K, where hn .X / D

X

anl;l X l C gn .X /.

0ln

Proof. We modify the proof of Proposition 2.16 by specifying the constants in a form adapted to the  NGevrey  asymptotics. From  kCldefinition,  particularly (3.1), one has l k CN D CLN C 1 . We also have 1  p C 1 and Clk D CL1 L2  p jjak jj WD sup jak .x/j  CLk1  jxj
k p

 C1 :

50

3 Composite Asymptotic Expansions: Gevrey Theory

l Furthermore, using the Cauchy XInequality, one has jakl j  jjak jj jr0 j . The akl x l is bounded by .n C 1/jjak jj on the disk function b W x 7! ak .x/  l
jxj < r0 . Thus, the (analytic) function x 7! x n b.x/ is bounded by .nC1/jjak jjr n on the circle jxj D r for all r < r0 . By the maximum modulus principle, we deduce that 1/jjak jjr0n if jxj  r0 . Thus one can choose  k jb.x/j   .n C k n Ak n D CL1  p C 1 .n C 1/r0 in (2.16). For the outer expansion, we simply estimate in (2.19) X   X l N l lN   N  CN C Cl;N l r lN  C Np C1 L1 L2 r  C L1 C Lr2  Np C1 : l
lN

For the inner expansion, one estimates in (2.20) using (3.2) CN C

X

Ak;N k RN k  C

k
X kN

e LN C 1  where

eDC C

k

Lk1  N p

p

 N k  C 1 .N  k C 1/ rR0

C1



 .   X   p C 1 : . C 1/ r0RL1

(3.9) (3.10)

0

t u Concerning Remark 2 below Proposition 2.16, it is possible to derive Gevrey estimates for jxj > jj or jX j < jj , 2 0; 1Œ. We do not detail these estimates, however. We now present Gevrey analogs for Propositions 2.19 and 2.20: under similar conditions on inner resp. outer expansions, the domain of validity of a Gevrey CAsE can be extended. Proposition 3.8. Let y be a function defined for  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/ and having a CAsE of Gevrey order p1 in the sense of Definition 3.6: X   y.x; /  1 an .x/ C gn x n p

n0

as S2 3  ! 0 and x 2 V1 ./, with an 2 H .r0 / and gn 2 G .V /, V D V .˛; ˇ; 1; / such that ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ. Let  > . In the case where  > jj, set ˝ D D.0; /, otherwise set ˝ D V .˛; ˇ;  C ; / with  > 0 arbitrarily small. Assume that the function Y W .X; / 7! y.X; / can be analytically continued on ˝  S2 and that it has a Gevrey asymptotic expansion Y .X; /  1

1 X

p

nD0

as  tends to 0; uniformly on ˝.

hn .X /n

3.2 Composite Gevrey Asymptotic Expansions

51

Then y can be analytically continued to the set of all .x; / such that  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/. Moreover y has a CAsE of Gevrey order p1 as  ! 0 in this set. Proof. With the notation of the former proofs, one first has to give a Gevrey estimate e1 ./ n V1 ./, V e1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/ for RN .x; / given by (2.11), for x in V and secondly to show that the functions gn can be analytically continued and have consistent Gevrey expansions on all V .˛1  ˇ2 ; ˇ1  ˛2 ; 1; /. One estimates RN starting from (2.27) and proceeding N  as in (3.9). By assumption, one can choose DN of the form DN D CLN  C1 . With the same constant 1 p k  k n Ak n D CL1  p C 1 .n C 1/r0 as before, inequality (2.16) is valid for jxj < r0 , hence   X N  N k e LN jjN  C Ak Nk M jRN .x; /j  DN C 1  p C1 k
e given by (3.10) and M D supfjxj I x 2 ˝g. with C In order to prove that the functions gn , n 2 N, have consistent P Gevrey expansions, we use Proposition 3.2. By assumption, the formal series n0 hn .X /n is of Gevrey order p1 on ˝ , i.e. there are constants H; L1 such that for all n 2 N sup jhn .X /j  HLn1 

jX j2˝

n p

 C1 :

(3.11)

P By definition of a Gevrey CAsE, the formal series ak .x/k is also of Gevrey 1 order p ; reducing L1 if needed, one deduces that a constant A exists such that   sup jak .x/j  ALk1  pk C 1 for all k. By the Cauchy Inequality, this implies jxj
implies that jgn .X /j  HLn1 

n p

X   nl  l l   Q n n C1 C ALnl 1  p C 1 r0 M  C L1  p C 1 0ln

l .   P  l  for X 2 ˝ and n 2 N, with CQ D H C l0 LM C 1 . Therefore r p 1 0 the gn satisfy (3.4) for X 2 ˝, and also for X 2 V by assumption, hence for X 2 V .˛1  ˇ2 ; ˇ1  ˛2 ; 1; /. Finally (3.5) is also satisfied by assumption. The assumptions of Proposition 3.2 are met and the conclusion follows. t u Proposition 3.9. Let 0 < r0 < e r 0 and y be a function defined for  2 S2 D e 1 ./ D V .˛1 ; ˇ1 ;e S.˛2 ; ˇ2 ; 0 / and x 2 V r 0 ;  jj/. Assume that    P y .x; / D n0 an .x/ C gn x n ; y.x; /  1 b p

as S2 3  ! 0 and x 2 V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/, in the sense of Definition 3.6.

52

3 Composite Asymptotic Expansions: Gevrey Theory

P n Suppose that y has an asymptotic expansion 1 nD0 cn .x/ of Gevrey order as  tends to 0, uniformly for x 2 V .˛1 ; ˇ1 ;e r 0 ; r0   / for some  > 0. e1 ./. Then y.x; /  1 b y .x; / as S2 3  ! 0 and x 2 V

1 p

p

Remark. As in Proposition 2.20, we say by abuse of language that y has a Gevrey e 1 ./, even if the functions an are not defined on the full CA sE for  2 S2 and x 2 V disk D.0;e r 0 /. Proof. By Proposition 2.20, the functions an may be analytically continued on the union D.0; r0 / [ V .˛1 ; ˇ1 ;e r 0 ; r0   /. We also have to verify (3.8) for x 2 V1 ./ with jxj  r0 , i.e. to estimate RN given by (2.28). By (3.1), there are C; L1 ; L2 > 0 such that for all l < N 2 N l  jrl N l .X /j  CLl1 LN 2

N p

 C 1 jX jN Cl :

Increasing the constants C and L1 if necessary, one has by assumption ˇ ˇ X N  N ˇ ˇ cn .x/n ˇ  CLN ˇy.x; /  1  p C 1 jj : n
We deduce 0 jRN .x; /j  C @LN 1 C

X

1 l lN A Ll1 LN r0  2

N p

 C 1 jjN

0l
  L2 N  N  C L1 C  p C 1 jjN : r0 t u Exercise 3.10. State and prove the analog of Lemma 2.7 (which concerns differentiation of CAsEs) for Gevrey CAsEs. Exercise 3.11. (a) Show that Proposition 2.8 (which concerns integration of CA sE s in the absence of residues) and its proof can be carried to Gevrey CA sE s. Additionally, it has to be shown here that the functions Gn1 of the proposition have consistent asymptotic expansion of Gevrey order p1 . (b) Carry Proposition 2.9 and its proof over to Gevrey CAsEs. First, prove that the b series R./ in the proposition is Gevrey of order p1 . Then use the classical b Borel–Ritt–Gevrey theorem to obtain a function R./ with R./  1 R./ p

(see e.g. the beginning of the proof of Lemma 3.15 below). Finally reduce the statement to part (a) as in the proof of Proposition 2.9.

3.3 Flat Gevrey Functions

53

3.3 Flat Gevrey Functions As for the classical asymptotic, flat functions are of particular interest. Here two concepts of flatness can be defined, depending on whether one asks the functions an and gn to be identically zero, or only their coefficients anm and gnm . Since an analytic function is uniquely determined by the coefficients of its Taylor series, the situation for gn and for an is not symmetrical. Definition 3.12. With the notation of Definition 3.6, we say that a function y.x; / is flat in the strong sense if it has a CAsE and if all functions an and gn of the corresponding formal series are identically zero. We say that y is flat in the weak sense if it has a CAsE and if all functions an are zero and all the coefficients gnm of the asymptotic expansions (3.1) of functions gn vanish. One of the key-points of the classical theory of Gevrey asymptotic expansions is the relationship between flat Gevrey functions and exponentially small terms. Below we present an analog for CAsEs. Proposition 3.13. Let y be a function defined and analytic for  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/. (a) If y is flat in the strong sense, then there are A; C > 0 such that jy.x; /j  C exp.A= jjp /

(3.12)

for  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/. Conversely, if y satisfies (3.12), then y is flat in the strong sense. (b) If y is flat in the weak sense, then there are B; C > 0 such that jy.x; /j  C exp.B jxjp = jjp /

(3.13)

for  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/. Conversely, if a function y satisfies (3.13), and if moreover y has a CAsE, then y is flat in the weak sense. Remark. The assumption that y has a CAsE is essential ˚for the in the weak  converse p . case, as shown e.g. by the function y.x; / D 1=2 exp  x Proof. Item (a) is classical. If y is flat in the strong sense, then we have for all N the estimate N  N (3.14) jy.x; /j  CLN 1  p C 1 jj : p  1 . We then obtain (3.12) Now choose N close to its optimal value N  p jjL 1 with A D L11 . Conversely, a function satisfying (3.12) satisfies (3.14) for all L1 > it is flat in the strong sense.

1 , A

therefore

54

3 Composite Asymptotic Expansions: Gevrey Theory

(b) As y is bounded because of Definition 3.6, it suffices to prove the inequality when x 2 V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/ satisfies jxj  K jj, where K > 0 is large enough. Enlarging L2 if needed, we may assume that r0 C 0  L2 =L1 , so jxj  L2 =L1 for all x 2 V1 ./. According to the hypothesis, we have for any positive integer N jy.x; /j 

ˇ N    ˇ  x ˇ n ˇgn  ˇ j j C C L1 jj  Np C 1

N 1 ˇ X nD0

and, for all n and all positive integers M (cf. (3.1)) ˇ  ˇ M M Cn  M Cn   ˇ ˇ  p C1 : jj ˇgn x ˇ jjn  CLn1 L2 = jxj The inequality L1  L2 = jxj and the choice of M C n D N lead to N    jy.x; /j  C.N C 1/ L2 j=xj  Np C 1 for any positive integer N . By choosing N close to its optimal value N  p jxj (here we use that jxj  K jj with K large) we obtain the existence of a p jjL 2 ˇ ˇp   p e such that jy.x; /j  C e ˇˇ x ˇˇ exp  jxjp p . This implies the statement constant C  jj L p

2

for all 0 < B < L2 . Conversely, if y has a CAsE and satisfies (3.13), then not only the slow part of its CA sE is zero, but also its outer expansion. The first part of Proposition 2.16 shows that the functions of its fast part also have an asymptotic expansion identically zero, cf. (2.14). t u In the framework of CAsEs, we also have a statement analogous to the classical Watson lemma: if a function is flat on sufficiently large sectors in  and in x, then the function is zero. Proposition 3.14. Let 0 ; r0 > 0,  2 R, ! >

C =p and ı >  C =p. Set

D D f.x; / I  2 S. ; !; 0 /; x 2 V . C arg ; ı C arg ; r0 ; jj/g: Let y.x; / be a holomorphic function defined on D such that, for all  ˛2 < ˇ2  ! and ˛1 < ˇ1 satisfying  C ˇ2  ˛1 < ˇ1  ı C ˛2 , its restrictions to  2 S.˛2 ; ˇ2 ; 0 / and to x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ have a CAsE of Gevrey order p1 flat in the weak sense. Then y D 0. Proof. From the previous proposition, there are constants C; B > 0 such that the p p function y satisfies jy.x; /j  C e Bjxj =jj for all .x; / 2 D. A priori, the constants depend on the selected domains for the restrictions, but as we can cover D by a finite number of such domains, it suffices to take the maximum of the constants.

3.4 Theorems of Borel–Ritt Type

55

In the sequel, we can assume without loss of generality that ı   > !  . Put Q D min.0; /. We choose ˛2 D ; ˇ2 D !, ˛1 D  C ! and ˇ1 D ı C ; the restriction e y of y to  2 S.˛2 ; ˇ2 ; 0 /, x 2 V .˛1 ; ˇ1 ; r0 ;  Q 0 / therefore p p satisfies je y .x; /j  C e Bjxj =jj . For all fixed x 2 V .˛1 ; ˇ1 ; r0 ;  Q 0 /, this function is holomorphic and exponentially small on the sector S.˛2 ; ˇ2 ; 0 / of opening angle > =p. By the classical Watson lemma (see [2], Proposition 11 p. 75), e y D 0. The identity theorem for analytic functions yields the statement. t u P 1 n Given a formal series an .x/ of Gevrey order p and coefficients gnm satisfying the necessary condition (3.3), there is at most one function analytic on the domain D as described in Proposition 3.14 with a CAsE of Gevrey order p1 corresponding to these data. In a summability theory of CAsEs which remains to be established, sucha function, if it exists, might be called the sum of the doubly   P P g n x n , b g n .X / D gnm X m on D. formal series n0 an .x/ C b

3.4 Theorems of Borel–Ritt Type As in the classical theory of Gevrey asymptotics, it will be useful to construct functions with a prescribed Gevrey CAsE. As in the classical Gevrey theory, this will only be possible if the size of the sectors is less than =p. We present two results of this type. In the first one, Lemma 3.15, the functions an ; gn are given. In the second one, Corollary 3.17, the functions an and the coefficients gnm of the gn are given. Lemma 3.15. Let r0 > 0, ˛; ˇ; ˛1 ; ˇ1 ; ˛2 ; ˇ2 ;  2 R satisfying ˛1 < ˇ1 , ˛2 < ˇ2 and ˛  ˛1  ˇ2 < ˇ1  ˛2  ˇ. Let V D V .˛; ˇ; 1; / be an infinite quasi-sector and S2 D S.˛2 ; ˇ2 ; 0 / a finite sector. Finally, let b y .x; / D

X   b .r0 ; V / an .x/ C gn x n 2 C n0

be a composite formal series of Gevrey order p1 as in Definition 3.1. Assume that ˇ2  ˛2 < =p. Then there exists a holomorphic function y defined for  2 S2 and x 2 V .˛1 ; ˇ1 ; r0 ;  jj/ which has b y as CAsE of Gevrey order p1 . Proof. According to the assumption, one has for all n 2 N, sup jan .x/j      jxj
Let us first briefly recall the proof of the Borel–Ritt–Gevrey theorem (cf. [2], Proposition 10, p. 73), i.e. the classical construction of a function with a prescribed 1 X a .x; / D an .x/n asymptotic expansion of Gevrey order p1 . Given a series b nD0

56

3 Composite Asymptotic Expansions: Gevrey Theory 1 X

an .x/ n  t n ; it is called the  C 1 p nD0 formal Borel transform of b a.x; "/. We have slightly modified it compared to most common presentations: we do not introduce a shift in the exponent. This simplifies our presentation. The above series has a radius of convergence at least L11 . Let ' D 1 D 12 .ˇ2  ˛2 /, 0 < % < 1=L1 and T D %e i' . Now we set 2 .˛2 C ˇ2 /, 1 p

of Gevrey order

and type L1 , put a.x; M t/ D

e a.x; / D LT;p .a/.x; M / WD p

Z

T

e t

p =p

a.x; M t/ d .t p / I

(3.15)

0

it is called the truncated Laplace transform of aM (also a bit modified). Note that it is defined for all  2 C and x 2 D.0; r0 /. The formal Borel and truncated Laplace transforms are important in the Gevrey theory because they transform a Gevrey formal series into an actual function admitting it as Gevrey asymptotic expansion. This is based on the formula n

L1ei' ;p .t n / D 

p

 C 1 n as jarg   'j <

 p:

In order to show that e a.x; / has b a .x; / as asymptotic expansion of Gevrey order 1 , we split a.x; M t/ D P .x; t/ C r N N .x; t/, where p PN .x; t/ D

N 1 X nD0

1 X an .x/ an .x/ n  t n and rN .x; t/ D n  tn  p C1  C 1 p nDN

and N is any positive integer. Then, for  2 S2 and for jxj  r0 , one has e a .x; / 

N 1 X

an .x/n D I C J

nD0

p

Z

1ei'

with I D  Since

e

t p =p

p

Z

T

PN .x; t/ d.t / and J D p

T

e t

p =p

rN .x; t/ d.t p /.

0 N

jPN .x; t/j  %

jtj C N

N 1 X

.L1 %/n as jtj  % ;

nD0

C one has first, with the constant K D , 1  L1 %   p s cos.p / N s d.s p / exp  p jj 0   .% cos.p /1=p /N  Np C 1 jjN :

jI j  K jjp %N D

K cos.p /

Z

C1

3.4 Theorems of Borel–Ritt Type

57

With the same constant K, we have the inequality jrN .x; t/j  K%N jtjN as t 2 Œ0; T . We therefore obtain similarly   p Z % s cos.p / %N s N d.s p / exp  jJ j  K jjp jjp 0   1=p N  N  K % cos.p /  p C 1 jjN :  cos.p / ˇ ˇ N 1 ˇ ˇ X  N  ˇ nˇ ee L  Np C 1 jjN with the constants a.x; /  This proves ˇe an .x/ ˇ  C ˇ ˇ nD0   1=p 1 2C 2K e D D and e L D % cos.p / C , i.e. the cos.p / .1  L1 %/ cos.p / asymptotic of Gevrey order p1 of e a, uniformly for x 2 D.0; r0 /. 1 X Similarly, we construct e g.X; / having b gD gn .X /n as asymptotic expannD1

sion of Gevrey order p1 uniformly for X 2 V .˛; ˇ; 1; /. The function given by   y.x; / D e a.x; / C e g x ;  finally has the desired CAsE. u t Regarding the second result, where only the coefficients gnm are given with the condition (3.3), we must first construct functions gn .X / satisfying (3.1). This is done in the statement below. Lemma 3.16. Let gnm , n; m 2 N be complex numbers satisfying (3.3). Let ˛ < ˇ < ˛ C =p and  2 R. Then there are C 0 ; L01 ; L02 > 0 and a sequence of functions gn defined on V .˛; ˇ; 1; / such that for all X 2 V .˛; ˇ; 1; / ˇ ˇ M 1 X ˇ ˇ   gnm X m ˇˇ  C 0 L01 n L02 M  MpCn C 1 ; jX jM ˇˇgn .X /  mD1

i.e.

x  n n0 gn   is of Gevrey order

P

1 p

in the sense of Definition 3.1.

Corollary 3.17. Let gnm , n; m 2 N be complex numbers satisfying (3.3) and an 2 H .r0 / be holomorphic functions   such that there exist constants C; L1 satisfying supjxj
as CAsE of Gevrey order p1 . The corollary is an immediate consequence of the two previous lemmas.

58

3 Composite Asymptotic Expansions: Gevrey Theory

Proof of Lemma 3.16: The series b g n .X / D

1 X

gn;mn X m are of Gevrey order 

mDnC1 1 p;

their coefficients satisfy precisely jgn;mn j  C

L1 L2

n

Lm 2

m p

 C1 for all n; m.

The method presented in the proof of Lemma g n , replacing the  applies to each b  3.15

variable  by X 1 and the constant C by C   % 2 0; 1=L2 Œ. Put c D cos p.ˇ˛/ . 2 Then we define the functions e g n by Z e g n .X / D X

%

p 0

e t

pXp

L1 L2

n

. More precisely, let T > jj and

1 X

gn;mn t m d .t p / m  . C 1/ p mDnC1

on V D V .˛; ˇ; 1; /. For all M  n and all X 2 V , jX j  T , the estimates in the proof of Lemma 3.15 imply ˇ ˇ  n M 1 ˇ ˇ X   2C L1 ˇ M m ˇ g n .X /  gn;mn X ˇ  .%c 1=p /M  M : ˇe p C1 jX j ˇ ˇ .1  L2 %/c L2 mDnC1 With eD C

2C ; .1  L2 %/c

e % D %c 1=p ;

(3.16)

replacing M by M C n and multiplying by jX jn , this implies that the functions gn W X 7! e g n .X /X n satisfy ˇ ˇ n  M 1 ˇ ˇ X   ˇ m ˇ e L1 gnm X ˇ  C e %M  MpCn C 1 jX jM ; ˇgn .X /  ˇ ˇ L2e % mD1

e of (3.16), e i.e. (3.5) is satisfied for jX j > T (with the constants C L1 D 1 e L2 D instead of C , L1 and L2 ).

L1 L2 %Q

and

%Q

In order to apply Proposition 3.2, it remains to estimate gn .X / for X 2 V .˛; ˇ; 1; /, jX j  T . An estimate analogous to that of J in the former proof shows that  n  n Z % L1 s 2C g n .X /j  jX jp exp.s p jX jp / d.s p / je 1  L2 % L2 % 0  n e exp.%p T p / L1 ; C L2 p p e e where  L1 T nC is the constant defined in (3.16). Thus we have jgn .X /j  C exp.% T / for jX j  T . This finally implies (3.4) and thus completes the proof of the L2 lemma. t u

3.5 Consistent Good Coverings

59

Exercise 3.18. With the notation of Lemma 3.15, show that, if b y has type .L1 ; L2 /, L1 then there exists y having b y as CAsE of type e L1 for all e L1 > cos.p , where D / 1 .ˇ2  ˛2 /. 2

3.5 Consistent Good Coverings Now we will use the results of Sect. 3.4 to associate, to a function y.x; / having a CAsE of Gevrey order p1 , a family of functions defined on other sectors in the variable  and on other quasi-sectors in x with the following properties. First, P they all have CAsEs of Gevrey order p1 so that the formal series n an .x/n and P g n .X /n are the same for all these functions. Second, the union of their domains nb contains the set of all .x; / satisfying 0 < jj < 0 and x 2 A. jj ; r0 /. Here and in the sequel we use the annulus A.; %/ D fx j  < jxj < %g; if % > 0; % > : By Proposition 3.13, the difference of any two functions of such a family is exponentially small on the intersection of their domains; here the cases of strong resp. weak flatness have to be distinguished. Let V D V .˛; ˇ; 1; / be an infinite quasi-sector, S D S. ; !; 0 / a finite sector with 0 < !  < ˇ  ˛, < such that ˛ <  ! <  < ˇ and y.x; / a holomorphic function defined for  2 S and x 2 V . ; ; r0 ;  jj/ such that 1  X   an .x/ C gn x n as S 3  ! 0 y.x; /  1 (3.17) p

nD0

where g0 ; g1 ; : : : are functions of G .V / satisfying (3.1) with coefficients gnm . In order to associate a family of functions to y, we first construct their domains of definition. Firstly we choose infinite sectors V j D V .˛ j ; ˇ j ; 1; /, j D 2; : : : ; J , where J is an integer at least 2, such that their opening angles are less than =p and such that, completing them with V 1 WD V , they form a good covering of C if  > 0, or of the infinite annulus S A.jj ; 1/ D fX 2 C I jj < jX j < 1g if   0. This means that we have Jj D1 V j D C, resp. A.jj ; 1/, as well as V j \ V m bounded if j 62 fm  1; m; m C 1g, i.e. ˛ j < ˇ j 1 < ˛ j C1 . By convention, superscripts are mod J and angles are mod 2. In order to obtain large opening angles for the quasi-sectors in x, we now choose small sectors in , so that X D x= is in V j for x in some sector only slightly smaller than V j . Therefore the given -sector S is not necessarily included in the family of -sectors we will choose. More precisely, let 2ı > 0 be less than the minimum of the opening angles of the intersections V j \ V j C1 , i.e. 2ı < minfˇ j  ˛ j C1 ; j D 1; : : : ; J g (with the convention ˇ J  ˛ J C1 D ˇ J  ˛ 1  2). Then we choose a family of finite

60

3 Composite Asymptotic Expansions: Gevrey Theory

 j  Fig. 3.1 Examples of good coverings by quasi-sectors Vl ./ 0<j J for l D 1. For clarity, the radii were chosen slightly different

sectors Sl D S.˛l ; ˇl ; 0 /, l D 1; : : : ; L such that they form a good covering of the punctured disk D.0; 0 / , i.e. their union is D.0; 0 / and Sl \ Sm D ; if l 62 fm  1; m; m C 1g, and such that the opening angles satisfy ˇl  ˛l  ı. Here and in the sequel, indices are taken modulo L. Let 'l D .˛l C ˇl /=2 denote the j directions bisecting the sectors Sl . Finally, we choose the quasi-sectors Vl ./ D j j j j V .˛l ; ˇl ; r0 ;  jj/ with ˛l WD ˛ j C 'l C ı and ˇl WD ˇ j C 'l  ı. It is easily j verified that for every  2 Sl , the families Vl ./, j D 1; : : : ; J form a good covering of D.0; r0 / if  > 0, of the annulus A. jj ; r0 / if   0, and that j  2 Sl and x 2 Vl ./ imply x= 2 V j . j The collection of V j ; Sl ; Vl ./ constructed in this way is a consistent good covering in the sense of the following definition (Fig. 3.1). j

Definition 3.19. We call the collection of V j ; Sl ; Vl ./, j D 1; : : : ; J , l D 1; : : : ; L a consistent good covering of A .0 ; r0 ; / D f.x; / I 0 < jj < 0 ;  jj < jxj < r0 g if V j D V .˛ j ; ˇ j ; 1; /, j D 1; : : : ; J , form a good covering of A.; 1/, if Sl D S.˛l ; ˇl ; 0 /, l D 1; : : : ; L, form a good covering of D.0; 0 / , if there is ı > 0 such that maxfˇl  ˛l ; l D 1; : : : ; Lg  ı < j

j

j

1 2 j

minfˇ j  ˛ j C1 ; j D 1; : : : ; J g j

and if Vl ./ D V .˛l ; ˇl ; r0 ;  jj/ with ˛l D ˛ j C 'l C ı and ˇl D ˇ j C 'l  ı, where 'l D .˛l C ˇl /=2. We call the number max.ˇl  ˛l ; l D 1; : : : ; L/ the resolution of the covering.

3.5 Consistent Good Coverings

61

1 Lemma 3.15 establishes the existence for  2 Sl and  n P1 of functions ylx holomorphic 1 1 x 2 Vl ./ such that yl .x; /  1 .x/Cg . /  with the functions an .x/ a n n  nD0 p and gn .X / of (3.17). Moreover, reducing  and 0 if necessary, Corollary 3.17  j implies the existence of sequences of functions gn .X / n2N , j D 2; : : : ; J , holomorphic in V j satisfying (3.1) with the gnm of (3.17) and some constants j j C; L1 ; L2 and functions yl .x; / holomorphic on all  2 Sl , x 2 Vl ./ such that

j

yl .x; /  1

p

1  X   an .x/ C gnj x n :

(3.18)

nD0

We complete the family of functions by setting gn1 D gn for all n. j Thus we have a family fyl glD1;:::;LIj D1;:::;J of functions defined on a consistent good covering having CAsEs to the same formal series as for the given function y. Proposition 3.13 implies that there are constants A; B; C > 0 such that ˇ ˇ p ˇ j ˇ j ˇylC1 .x; /  yl .x; /ˇ  C e A=jj ; j

(3.19) j

as j D 1; : : : ; J; l D 1; : : : ; L;  2 SlC1 \ Sl ; x 2 VlC1 ./ \ Vl ./; ˇ ˇ p p ˇ ˇ j C1 j ˇyl .x; /  yl .x; /ˇ  C e Bjxj =jj ; as j D 1; : : : ; J; l D 1; : : : ; L;  2 Sl ; x 2 Vl

j C1

(3.20)

j

./ \ Vl ./ as well as

ˇ ˇ 1 ˇy .x; /  y.x; /ˇ  C e A=jjp ; l

(3.21)

as l D 1; : : : ; L;  2 Sl \ S; if this intersection is non-empty, and x 2 Vl1 ./ \ V . ; ; r0 ;  jj/. We have therefore proved the following. Proposition 3.20. Let V D V .˛; ˇ; 1; / be an infinite quasi-sector, S D S. ; !; 0 / a finite sector with ! < ˇ˛, < such that ˛ < ! <  < ˇ and y.x; / a holomorphic function defined as  2 S and x 2 V . ; ; r0 ;  jj/ admitting a CAsE of Gevrey order p1 . j

Then there exists a consistent good covering V j ; Sl ; Vl ./, j D 1; : : : ; J , l D j 1; : : : ; L, of A .0 ; r0 ; / and a family of holomorphic bounded functions yl .x; / j defined as  2 Sl and x 2 Vl ./ such that the differences are exponentially small, i.e. there are constants A; B; C > 0 such that (3.19)–(3.21) are satisfied. The next chapter is devoted to proving the converse of this result.

Chapter 4

A Theorem of Ramis–Sibuya Type

j

In this chapter, we present a key-result of this memoir: functions yl defined and analytic on domains forming a consistent good covering in the sense of Definition 3.19 and satisfying (3.19), (3.20) have CAsEs of Gevrey order p1 and P P g n .X /n are the same. The result the associated formal series n an .x/n and n b is valid for  positive or negative, but in order to simplify the presentation, we only allow positive  in Sects. 4.2–4.4. The modifications for negative  are indicated in Sect. 4.5. In the last Sect. 4.6, the key-result is used to show the compatibility of Gevrey CAsEs and composition with an analytic function.

4.1 Statement of the Theorem and Beginning of the Proof Theorem 4.1. Let J; L be two nonzero integers,  a real number, and let V j D j j j V .˛ j ; ˇ j ; 1; /; Sl D S.˛l ; ˇl ; 0 /; Vl ./ D V .˛l ; ˇl ; r0 ;  jj/, j D 1; : : : ; J , l D 1; : : : ; L, be a consistent good covering of resolution ı of A .0 ; r0 ; / ej ./ D in the sense of Definition 3.19. Let e r 0 > r0 and e  > . Put V l j j V .˛l 2ı; ˇl C2ı;e r0 ; e  jj/. Assume that there are holomorphic bounded functions j ej ./ and positive constants A; B; C such that yl .x; / defined for  2 Sl and x 2 V l the differences satisfy the following estimates: ˇ ˇ  ˇ j ˇ j ˇylC1 .x; /  yl .x; /ˇ  C exp 

A jjp



(4.1)

e j ./ \ V ej ./ and if  2 Sl \ SlC1 and x 2 V l lC1 ˇ ˇ ˇp  ˇ j C1  j ˇy .x; /  yl .x; /ˇ  C exp  B ˇ x ˇ l

A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 4, © Springer-Verlag Berlin Heidelberg 2013

(4.2)

63

64

4 A Theorem of Ramis–Sibuya Type

ej ./ \ V ej C1 ./. Then the restrictions of the functions y j to the if  2 Sl and x 2 V l l l j set of all .x; / such that  2 Sl , x 2 Vl ./ have CAsEs of Gevrey order p1 in the sense of Definition 3.6. More precisely, there exists a sequence .an /n2N ; an 2 H .r0 /, for each j D j j e such that for any pair 1; : : : ; J a sequence .gn /n2N ; gn 2 G .V j / and a constant C j of indices .j; l/, any  2 Sl , any x 2 Vl ./ and any integer N > 0 N 1  ˇ X   ˇˇ   ˇ j ee AN=p  Np C 1 jjN an .x/ C gnj x n ˇ  C ˇyl .x; / 

(4.3)

nD0

where e A D min.A; Br0 / with the constants A; B of (4.1), (4.2). Moreover, the j sequences of functions .gn /n2N have consistent asymptotic expansions of Gevrey order p1 and type .e A1=p ; B 1=p / with coefficients independent of j , i.e. there is a e such that for all double sequence .gnm /n;m2N of complex numbers and a constant C M; n 2 N; j 2 f1; : : : ; J g and all X 2 V j p

M 1 ˇ ˇ X  ˇ ˇ e en=p M=p  M Cn A gnm X m ˇ  C B  p C1 : jX jM ˇgnj .X / 

(4.4)

mD1 j

Remark. The asymptotics of the functions yl can only be valid in quasi-sectors j Vl ./ whose opening angle is smaller than that of their quasi-sectors of definition ej ./. Indeed, the functions gnj are constructed from y j using Cauchy type V l l integrals, so the quasi-sectors V j on which they are defined must already have an j e ./; then the condition that the quotients x= opening angle smaller than that of V l j j must be in the domain V of gn implies that the asymptotics can be valid only in j quasi-sectors Vl ./ of opening angle still smaller than V j . By construction, the ej ./ is ˇ j  ˛ j C 2ı, that of V j is ˇ j  ˛ j and that of V j ./ opening angle of V l l is ˇ j  ˛ j  2ı and we have the following implications j

if  2 Sl and x 2 Vl ./; then x= 2 V j

(4.5)

ej ./: r 0 ; then X 2 V if X 2 V j and  2 Sl with jX j < e l

(4.6)

and conversely

e ./ can be arbitrarily close to r0 and The constants e r 0 and e  in the definition of V l , respectively, but not equal. j

j

Principle of the proof. The idea is to write each function yl as the sum of two j int functions ylext and yl each of which has an asymptotic expansion in powers of 1  of Gevrey order p such that the first function ylext is independent of j and has P as its asymptotic expansion the slow part an .x/n , whereas the second function

4.1 Statement of the Theorem and Beginning of the Proof

65

xl2

xl2 °3

xl1

xl3

°2

xl3

°4

xl1

°1

x4l

xl4

 j  Fig. 4.1 The covering e V l ./ 0<j J and the paths  j

P j  j int yl has as its asymptotic expansion the fast part gn x n . In order to construct these two functions, we use the Cauchy–Heine formula with respect to the variable ej;j C1 ./ WD V e j ./ \ V e j C1 ./. In the x by choosing points in the intersections V l ˇ l l ˇ ˇ jˇ j case   0, we simply select points xl with ˇxl ˇ D e r 0 ; the functions are defined in formulas (4.9) and (4.10) below. In the case  < 0, we assume without ˇ lossˇ of ˇ j ˇ j j x l ./ˇ D x l ./ with ˇe generality that  < e  < 0 and, besides xl , we choose points e j; here the functions are defined in formulas (4.24) and (4.25) in Sect. 4.5. The je j functions we define depend on the choice of the points xl , but are independent of j j int the e x l ./. Moreover, these functions ylext and yl have the desired behavior only j 1 j j in sectors excluding the points xl and xl . As neither the given functions yl , nor j j the coefficient functions an and gn we construct depend on the points xl , modifying j j them yields the desired behavior of yl on the entire quasi-sector Vl ./. e j;j C1./ will be used throughout the proof. Notations similar to V l We may assume without loss of generality that p

A  Br0 :

(4.7)

In the sequel, we only treat the case   0; the changes needed for the case  < 0 are given in Sect. 4.5. For each j 2 f1; : : : ; J g, let j 2 ˛ j C1  ı; ˇ j C ıŒ. For j j j each .j; l/ 2 f1; : : : ; J g  f1; : : : ; Lg put xl D e r 0 e i. C'l / . Thus xl is in the ej;j C1 ./ for all jj < 0 . Let  j denote the closed path consisting of closure of V l j 1 j 1 j to xl arbitrarily close to the circular arc the segment Œ0; xl , a path from xl j ej ./ with of radius r0 and the segment Œxl ; 0 (Fig. 4.1). For  2 Sl and x 2 V l j 1 j < arg x < arg xl , the Cauchy formula yields arg xl Z

j

j

yl .u; / j du D 2 iyl .x; / and ux

Z k

ylk .u; / du D 0 if k ¤ j; ux

(4.8)

66

4 A Theorem of Ramis–Sibuya Type

because only  j contains x in its interior. Rearranging the sum of these integrals, we obtain the Cauchy–Heine formula j

j int

yl .x; / D ylext .x; / C yl with ylext .x; / D

.x; /

J Z k 1 X xl ylk .u; / du 2 i xlk1 u  x

(4.9)

kD1

where integration is along paths arbitrarily close to arcs of the circle of radius e r 0, and J Z k 1 X xl ylkC1 .u; /  ylk .u; / j int yl .x; / D du: (4.10) 2 i ux 0 kD1

By Definition (4.9), the function ylext .x; / is holomorphic and bounded on j int j j int D.0; r0 /Sl . From the relation yl .x; / D yl .x; /ylext .x; /, the function yl ej ./, jxj  r0 ; formula (4.10) extends is holomorphic and bounded as  2 Sl , x 2 V l j int j 1 j yl in the infinite x-sector S.arg xl ; arg xl ; 1/. Concerning the functions ylext , we will prove in Sect. 4.2 the following result. Lemma 4.2. There exists a sequence .an /n2N ; an 2 H .r0 / and a constant C > 0 such that, for any integer N > 0, any index l 2 f1; : : : ; Lg and any pair .x; / 2 D.0; r0 /  Sl N 1 ˇ ˇ X   ˇ ext ˇ an .x/n ˇ  CAN=p  Np C 1 jjN : ˇyl .x; /  nD0

j

Given the dependence of xl with respect to j , it is convenient to introduce the functions j j int Yl .X; / WD yl .X; / on the set of .X; / such that  2 Sl and, either arg.X / 2  j 1 C ı; j  ıŒ, or j (jX j  r0 and X 2 V j ). Let El denote this set; it depends on the choice of the j . For these functions, we have the following result. Lemma 4.3. There is a constant C > 0 and, for each j D 1; : : : ; J , a sequence j .gn /n2N of functions holomorphic in V j , tending to 0 as arg X 2  j 1 Cı; j ıŒ; jX j ! 1 such that for all l 2 f1; : : : ; Lg, all N 2 N and all .X; / with  2 Sl and arg X 2  j 1 C ı; j  ıŒ one has N 1 ˇ ˇ X   ˇ j ˇ gnj .X /n ˇ  CAN=p  Np C 1 jjN : ˇYl .X; / 

(4.11)

nD0

Moreover, for any compact subset K of V j , there exists 0 > 0 such that for all l 2 f1; : : : ; Lg, all N 2 N , all  2 Sl with jj < 0 and all X 2 K, inequality (4.11) holds.

4.2 The Slow Part

67

A priori the functions and the constant in the statement depend on j ; j D 1; : : : ; J . This lemma is proved in Sect. 4.3.  j j As yl .x; / D ylext .x; / C Yl x ;  , we almost have a CAsE. More precisely, e such that for all j; l; N and all  2 Sl , x 2 S. j 1 C 'l C there is a constant C j 2ı; C 'l  2ı; r0 / N 1  ˇ X   ˇˇ   ˇ j e AN=p  N C 1 jjN : an .x/ C gnj x n ˇ  C ˇyl .x; /  p

(4.12)

nD0 j

Since the functions gn tend to 0 as jX j ! 1, they, as well as the an , are uniquely determined by the inequalities (4.12), according to Remark 3 after Definition 2.5. j As a consequence, the an and the gn do not depend on the choice of the j , i.e. of j the xl , and the inequalities (4.12) are therefore valid in any finite union of sectors in x we can obtain by different choices of these points. Using the last statement of the lemma, one can complete these inequalities for quasi-sectors in the case  > 0. This proves (4.3); one still cannot talk about CAsE, because one has not yet j the required property that the gn .X / have asymptotic expansions as X ! 1. j Inequalities of the form (4.4), i.e. the fact that the gn .X /; n 2 N have consistent 1 asymptotic expansions of Gevrey order p , will be proved in Sect. 4.4. This will complete the proof of the theorem.

4.2 The Slow Part Formula (4.9) defines, for each index l, a function analytic and bounded in D.0; r0 /  Sl , and the differences satisfy, for all x 2 D.0; r0 / and all  2 Sl;lC1 WD Sl \ SlC1 J  ext  X 2 i ylC1 .x; /  ylext .x; / D kD1

Z

xlk k1 xlC1

k ylC1 .u; /  ylk .u; / du C ux

J Z X kD1

k xlC1

xlk

k ylC1 .u; /  ylkC1 .u; / du: ux

k1 By assumption, on the arc from xlC1 to xlk one has

ˇ ˇ k  ˇy .u; /  y k .u; /ˇ  C exp  lC1 l

A jjp



:

For the second integral, we use the fact that, depending on the values of u on k the arc from xlk to xlC1 , one (at least) of the following two identities holds:   k   kC1 k k either ylC1 .u/  yl .u/ D ylC1 .u/  ylk .u/ C ylk .u/  ylkC1 .u/ or ylC1 .u/

68

4 A Theorem of Ramis–Sibuya Type

    kC1 kC1 k  ylkC1 .u/ D ylC1 .u/  ylC1 .u/ C ylC1 .u/  ylkC1 .u/ . Using (4.1) and (4.2), and with paths of integration "-close to the arc of circle, " arbitrarily small, we obtain ˇ ˇ  p     Be ˇ ˇ k r  ˇylC1 .u; /  ylkC1 .u; /ˇ  C exp  jjAp C exp  jjp0    2C exp  jjAp ; p

r 0 . On the paths of integration, one has juxj  e r 0 r0 ". since A  Br0 and r0 < e Hence, since " is arbitrary, one has for all .x; / 2 D.0; r0 /  Sl;lC1 ˇ ˇ ext ˇy .x; /  y ext .x; /ˇ  lC1 l

2C

e r 0 r0

 exp 

A jjp



:

(4.13)

Using the classical Ramis–Sibuya theorem [49–51] (see Lemma 4.4 below), we obtain that the functions ylext have a common asymptotic expansion of Gevrey order 1 p uniformly in D.0; r0 /. For the completeness of the memoir, we give a proof of this classical result below. Lemma 4.4. Assume that Sl D S.˛l ; ˇl ; 0 /; l D 1; : : : ; L form a good covering of the punctured disk D.0; 0 / . Let fl W Sl ! C,l D 1; : : : ; L, be holomorphic functions bounded by some constant C > 0. We assume that there are constants p D; A > 0 such that jflC1 ./  fl ./j  De A=jj for all l D 1; : : : ; L and all  2 Sl;lC1 . Then the functions f1 ; : : : fL have a common asymptotic expansion of e and a sequence .cn /n2N such that Gevrey order p1 , i.e. there are a constant C ˇ ˇ N 1 ˇ ˇ X  ˇ ˇ e N=p  N cn n ˇ  C  p C 1 jjN A ˇfl ./  ˇ ˇ nD0

for all N 2 N , all l and all  2 Sl . e depends only on the data A; C; D; 0 ; ˛l ; ˇl . The coeffiRemark. The constant C cients cn can be expressed by integrals, see formula (4.15) below. Therefore, if the functions depend on parameters analytically, resp. continuously, if the functions are uniformly bounded and if the constants D and A do not depend on the parameters, then the cn are analytic, resp. continuous with respect to these parameters and the estimates for the remainders in the lemma are uniform. This will be essential in the sequel. Proof. Fix  > 0 such that ˇl  ˛lC1  2 for all l 2 f1; : : : ; Lg. Set Tl D 0 e i l with l 2 Œ˛lC1 ; ˇl  to be chosen. Applying the Cauchy formula to each fk , k D 1; : : : ; L, and rearranging the sum of these equations, we obtain the classical Cauchy–Heine formula  Z Tk L Z Tk fkC1 .v/  fk .v/ fk .v/ 1 X dv C dv ; fl ./ D 2 i v 0 Tk1 v   kD1

(4.14)

4.2 The Slow Part

69

for 0 < jj < 0 with l1 < arg  < l , where, as previously, integration from Tk1 to Tk is along a path arbitrarily close to the arc of the circle of radius 0 . For n 2 N set  Z Tk L Z Tk   dv 1 X dv fkC1 .v/  fk .v/ nC1 C cn D fk .v/ nC1 : 2 i v v 0 Tk1

(4.15)

kD1

Now fix N 2 N and put r./ D fl ./ 

N 1 X

cn n :

nD0 N 1 X n 1 N by in (4.14), one obtains Replacing C v vnC1 vN .v  / nD0

r./ D r1 ./ C r2 ./ with

(4.16)

L Z N X Tk fkC1 .v/  fk .v/ dv r1 ./ D 2 i vN .v  / 0 kD1

and r2 ./ D

L Z N X Tk fk .v/ d v: N .v  / 2 i v Tk1 kD1

First assume ˛l C   arg   ˇl   and jj  .1   /0 . Then, using l1 D ˛l and l D ˇl , we obtain jv  j  jvj sin  on every segment Œ0; Tk . Taking the hypothesis into account, this implies 1 jr1 ./j  2

Z

0 0

  p D Np De A=t dt jjN jjN  sin  t N C1 2pAN=p sin 

For the other part of the remainder, one has jr2 ./j 

C 2

Z

2 0

0 d C1 N  0

jjN D

C jjN : N  0

e Combining the two estimates withıthestatement  gives a constant C , depending on N N N=p C; D; 0 ;  and on supN 0 A  p C1 . In the case where arg   ˛l <  , one has ˇl1  arg    and using another choice of angle k , we obtain a similar estimate of the remainder   for fl1 : As p p jfl1 ./  fl ./j  De A=jj and jjN e A=jj  AN=p  Np C1 , we conclude that the estimate of the remainder also holds for fl .

70

4 A Theorem of Ramis–Sibuya Type

In the case ˇl  arg  <  , we proceed similarly with flC1 instead of fl ; in the case 0 .1   /  jj  0 , the estimate for the remainder is done simply by using e large enough. the estimates for cn and fl , if we choose C t u

4.3 The Fast Part Consider now, for each l D 1; : : : ; L and each j D 1; : : : ; J , the function j

j int

Yl .X; / D yl

.X; / D

J Z k 1 X xl ylkC1 .u; /  ylk .u; / du 2 i u  X 0 kD1

D

j yl .X; /

 ylext .X; /

(4.17)

defined on the set El of all .X; / such that  2 Sl and, either arg.X / 2  j 1 C j ı; j  ıŒ, or jX j < r0 , X 2 V j . Recall that the set El depends on the choice of the j and that we use the notation of Sect. 4.1. In order to prove Lemma 4.3, we first prove j

e > 0 such that for all .j; l/ 2 f1; : : : ; J g  Lemma 4.5. There exists a constant C j f1; : : : ; Lg and all .X; / 2 El;lC1 one has ˇ ˇ  ˇ j ˇ e j exp  ˇYlC1 .X; /  Yl .X; /ˇ  C

A jjp



:

(4.18)

e Proof. If X 2 V j and jX j < r0 , then X 2 V l;lC1 ./. We use in this case j

  j j j j ext YlC1 .X; /  Yl .X; / D ylC1  yl C ylext  ylC1 .X; / and the estimates (4.1) and (4.13). If arg.X / 2  j 1 C ı; j  ıŒ and jX j  r0 , j int we use again the integral formula defining yl , which gives j

j

YlC1 .X; /  Yl .X; / ! Z x k kC1 Z xk J kC1 k l .y lC1 .y  ylk /.u; / 1 X lC1  ylC1 /.u; / l du  du D 2 i u  X u  X 0 0 kD1 Z  k kC1 J kC1 k l .y C ylk  ylC1 /.u; / 1 X lC1  yl du C D 2 i u  X 0 kD1 ! Z  k kC1 Z xk kC1 k l .y lC1 .y  ylk /.u; / lC1  ylC1 /.u; / l du C du u  X u  X lk xlk where lk D e r 0 e i. Cl / , 'l < l < 'lC1 denotes a point in the intersection k;kC1 e k;kC1 ./. On each path, the denominator u  X is bounded below in e ./ \ V V j

l

lC1

j

4.4 Study of the Functions gn

71

kC1 k modulus by r0 sin.ı=2/, the numerator ylC1  ylkC1 C ylk  ylC1 in the first integral   A is bounded by 2C exp  jjp by (4.1), and the numerators of the second and third  Br p    integrals are bounded by C exp  jjp0  C exp  jjAp by (4.2) and (4.7). t u

Lemma 4.3 results directly from the previous lemma by the Ramis–Sibuya j Theorem 4.4; the analytic dependence of the functions gn .X / thereby obtained follows from formula (4.15) defining these coefficients, as indicated in the remark following Lemma 4.4. Similarly, we first deduce from their definitions by integrals j that all the Yl .X; / tend to 0 as X ! 1, arg.X / 2  j 1 C ı; j  ıŒ, if  is in j a compact set not containing 0 and then that the gn .X / tend to 0 as X ! 1. j

4.4 Study of the Functions gn j

j

Recall that the functions gn of the asymptotic expansion of the function Yl of Lemma 4.3 are obtained by application of Lemma 4.4. Hence they are defined by gnj .X / D

Z

L 1 X 2 i

k k1

kD1

Z 0

j

Yk .X; v/

k



dv C vnC1

j YkC1 .X; v/



(4.19)

j Yk .X; v/

 dv vnC1

!

where k is in the closure of Sk;kC1 , k D 1; : : : ; L satisfy j1 j D : : : D jL j DW r  j 0 . Recall also that the functions Yl are defined by (4.17). Therefore formula (4.19) is valid for all X 2 V j such that, either r jX j < r0 , or j 1 < arg X < j . Recall j also that the asymptotic expansion (4.12) uniquely determines the functions gn and j that these are thus independent of the choice of the xl in (4.17) and of the choice of the k in (4.19). Finally, note the following estimate of these functions, which is a j consequence of (4.11) and the choice of xl : there exists a constant C such that ˇ ˇ j   ˇg .X /ˇ  CAn=p  n C 1 n p

(4.20)

for all n and X 2 V j . j It remains to show that the functions gn ; n 2 N have consistent asymptotic 1 expansions of Gevrey order p in the sense of Definition 3.1. This will be done by an j C1

j

adaptation of the proof of Lemma 4.4: one first estimates the differences Yl Yl , j C1 j then gn  gn and finally the remainders of the asymptotic expansions. First from (4.6), if  2 Sl and X 2 V j;j C1 are such that jX j < r0 , then ej;j C1 ./. Moreover, since y ext do not depend on j , one has in this case X 2 V l l j C1

Yl

j

j C1

.X; /  Yl .X; / D yl

j

.X; /  yl .X; /

72

4 A Theorem of Ramis–Sibuya Type

and (4.2) applies. We obtain, for X 2 V j;j C1 and  2 Sl such that jX j < r0 , the estimate ˇ j C1 ˇ j ˇY .X; /  Yl .X; /ˇ  C exp .BjX jp / : (4.21) l j C1

j

Now we use (4.19) to estimate gn  gn . Contrary to Lemma 4.3, we want here an asymptotic result as X tends to infinity in the sector V j;j C1 . For that purpose, j C1 j we will use the exponential estimate for the differences Yl  Yl given by (4.21). From now on, the modulus r of the points l is no longer fixed, but must necessarily depend on X . One has gnj C1 .X /



gnj .X /

L 1 X D 2 i lD1

Z

l 0

Z

  dv j C1 j  Yl .X; v/ nC1 C Yl v l1 l

  dv j C1 j C1 j j YlC1  Yl  YlC1 C Yl .X; v/ nC1 v

!

where the integrals from l1 to l are arbitrarily close to the arcs of the circle of ej;j C1 ./. We choose r D r0 = jX j, when radius r. On these arcs, one has X 2 V l e exists such jX j  L WD 0 =r0 C 1. Then using (4.21), we obtain that a constant C that for all jX j  L and all .n; j /, the sum of the integrals from l1 to l is less e r n jX jn exp.B jX jp /. than C 0 For the integrals from 0 to l , one can apply (4.18) if the numerator is written j C1 j C1 j j j C1 .YlC1  Yl /  .YlC1  Yl / and one can apply (4.21) if it is written .YlC1  j j C1 j YlC1 /  .Yl  Yl /. This implies that there is a constant CN such that the sum of the integrals from 0 to l is bounded by CN

Z

r

  p p min e A=v ; e BjX j vn1 d v :

0

Recall that jX j  L and r D r0 = jX j. With q > 0 given by A=q p D B, this becomes ! Z q=jX j Z r0 =jX j A=vp n1 BjX jp n1 N C e v dv C e v dv ; q=jX j

0

where jX j  L  1: With a change of variable in the first integral, we deduce that this sum of integrals from 0 to l is bounded by CN



1 n=p A p



 p n BjX jp 1 n C ; B q e jX j jX j p n

n

R1 here  .a; u/ D u e t t a1 dt, Re a > 0, is the incomplete Gamma function. It is preferable to use the incomplete Gamma function because simple estimates are different if n is small, respectively large.

j

4.4 Study of the Functions gn

73

Combining the two estimates, we deduce the existence of a constant C such that   ˇ j C1 ˇ   p ˇg .X /  gnj .X /ˇ  C An=p  pn ; B jX jp C q n jX jn e BjX j n

(4.22)

for all n; j and X 2 V j;j C1 , jX j  L. Now we use the Cauchy–Heine formula in the variable Z D X1 . Consider the sector V j and first only the X 2 V j with jX j  LC1 and ˛ j Cı < arg X < ˇ j ı. For each k D 1; : : : ; J , let Xk 2 V k;kC1 be such that jXk j D L; in particular arg Xj 1 D ˛ j and arg Xj D ˇ j . Let Zk D X1k . With gmn D

 Z ZkC1 J Z Zk      d    d 1 X k 1 gnkC1 1  gnk 1 ; C g n  2 i  mC1  mC1 0 Zk kD1

we thus obtain, by a calculation similar to that of r1 and r2 in the proof of Lemma 4.4 X

M

gnj .X /



M 1 X

! gmn X

m

mD0

Z Zk g kC1  1   g k  1  J 1 X n n     d C D 1 M 2 i  X 0 kD1   ! Z Zk gnk 1   d M   1 Zk1  X

j j for X 2 V j , jX j  L ˇ C 1,1 ˇ˛ C ı < argX < ˇ  ı. Using the estimate (4.22) ˇ ˇ and the lower bound   X  jj sin ı on the paths of integration, one estimates the absolute value of the sum of integrals from 0 to Zk by

JC 2 sin ı

 Z n=p A

C1



0

n

p;B

t

p

 dt C q n t M C1

Z

C1

e

B t p

0

dt



t M CnC1

and, after the change of variable s D Bt p and using formula Z

1

s a1  .b; s/ds D

0

1  .a C b/; a

one estimates this sum of integrals by    JC p : C 1 An=p B M=p  nCM p 2p sin ı M The estimate for integrals from Zk1 to Zk follows in a simpler manner from (4.20). Their sum is bounded by J C An=p

1 L



1 1  LC1

n p

 C 1 LM :

74

4 A Theorem of Ramis–Sibuya Type

    Since there is a constant D such that  pn C 1 LM  DB M=p  nCM p C 1 for all n; M 2 N, we deduce the existence of a constant C such that ˇ  ˇˇ M 1 ˇ X   ˇ M j m ˇ gn .X /  gmn X C1 ˇ  CAn=p B M=p  nCM ˇX p ˇ ˇ

(4.23)

mD0

for all n; M and X 2 V j with jX j  L C 1 and ˛ j C ı < argX < ˇ j  ı. For X 2 V j with argX close to ˛ j or argX close to ˇ j , we proceed analogously to the end of the proof of Lemma 4.4. As the second case is analogous, we only j 1 treat the first case. One first considers the estimates for gn and then one adds j j 1 terms estimating the differences gn  g n . For that purpose, we  use the fact that a constant D exists such that T M  pn ; BT p  DB M=p  nCM C 1 and p  nCM  nCM BT p M=p T e  DB  p C 1 for all n; M; T; B > 0 (use the fact that for ˛; ˇ > 0, the function f .t/ D t ˛  .ˇ; t/ reaches its maximum at a point t such that f .t/ D ˛1 t ˛Cˇ e t ). Thus, we finally have proved that there is a constant C such that (4.23) holds for all n; j; M and all X 2 V j with jX j  L C 1. Thanks to Proposition 3.2 j and (4.20), this is enough to show that the sequence of gn , n D 0; 1; : : : has 1 consistent expansions of Gevrey order p in the sense of Definition 3.1. The proof of Theorem 4.1 in the case   0 is now complete. t u

4.5 The Case  Negative We indicate in this section how the proof has to be modified in the case  < e  < 0. As before, for each j 2 f1; : : : ; J g, choose j 2 ˛ j C1  ı; ˇ j C ıŒ. For each j j .j; l/ 2 f1; : : : ; J g  f1; : : : ; Lg we set xl D e r 0 e i. C'l / and for all  ¤ 0 we j j j j j x l ./ D je j e i. C'l / . Therefore xl and e x l are in the closure of set e xl D e e j;j C1./, consisting ej;j C1./ for all jj < 0 . Let  j denote a closed path in V V l l j 1 j 1 j 1 j to xl arbitrarily close to the of the segment from e x l to xl , a path from xl j j arc of the circle of radius r0 , the segment from xl to e x l and finally a path from j j 1 x l arbitrarily close to the arc of the circle of radius je j. For  2 Sl and e x l to e j j 1 j e < argx < argxl , we obtain as before x 2 V l ./ with argxl Z

j

j

yl .u; / j du D 2 iyl .x; / and ux

Z k

ylk .u; / du D 0 for all k ¤ j ux

and rearranging the sum of these integrals, we obtain the Cauchy–Heine formula j

j int

yl .x; / D ylext .x; / C yl

.x; /

4.5 The Case  Negative

75

xl2

xl3

xl2 γ3

xl1

xl3

j ξl

γ2

xl1

j

ξl γ4

γ1

xl4

xl4 Fig. 4.2 The quasi-sectors and the paths of integration for  < 0

with ylext .x; /

J Z k 1 X xl ylk .u; / du D 2 i xlk1 u  x

(4.24)

kD1

where the integration is done along a path arbitrarily close to the arc of circle of radius e r 0 , and j int yl .x; /

J Z k 1 X xl ylkC1 .u; /  ylk .u; / du  D 2 i ux x kl kD1 e

(4.25)

J Z xk l ylk .u; / 1 X e du 2 i ux x k1 l kD1 e

where the integration is done along a path arbitrarily close to the arc of circle of j radius je j, cf. Fig. 4.2. Since neither yl nor ylext depend on the points e x kl , the j int functions yl do not depend on them either. The treatment of the functions ylext is identical to the case e  > 0; for the function j int k1 k x l to e x l , which were not in the yl , one has to check that the integrals from e formula (4.10) in the case e  > 0, cause no problem. j j int First, the functions Yl .X; / D yl .X; / are defined on all .X; / such that j 1  2 Sl and, either (arg.X / 2  C ı; j  ıŒ and jX j > e ) or (jX j > e , j j jX j  r0 and X 2 V ). Let El denote this set; it depends on the choice of the j and on jj. Lemmas 4.5 and 4.3 still hold. Indeed, the formula expressing the difference of the functions in the case jX j  r0 becomes

76

4 A Theorem of Ramis–Sibuya Type j

j

YlC1 .X; /  Yl .X; / D Z  k kC1 J kC1 k l .y C ylk  ylC1 /.u; / 1 X lC1  yl duC 2 i u  X e  kl kD1 Z  k kC1 Z xk kC1 k l .y lC1 .y  ylk /.u; / lC1  ylC1 /.u; / l du C du u  X u  X lk xlk ! Z e k  kl .y k lC1  yl /.u; / du u  X e  k1 l k k where lk D e r 0 e i. Cl / and e  kl D je j e j. Cl / , 'l < l < 'lC1 denote the points ek;kC1 ./ \ V ek;kC1 ./. Here the points e in the intersection V x kl respectively e x klC1 l lC1 j j in formula (4.25) for Yl respectively YlC1 have been changed into e  kl before the subtraction. Only the last integral did not appear before; it is exponentially small by j assumption (4.1). In the study of the functions gn , nothing changes.

4.6 Composition of a Gevrey CAsE and an Analytic Function Theorem 4.1 and its converse Proposition 3.20 yield a quick proof of a generalization of Proposition 2.6 to Gevrey CAsEs: the composition—left or right—of a function with a Gevrey CAsE and an analytic function has a Gevrey CAsE. First, we recall the analogous result for the classical Gevrey expansions. On the one hand this allows to present the proof using the classical Ramis–Sibuya theorem; on the other hand this result will serve in the sequel. Proposition 4.6 (classical). Consider a holomorphic function y defined on D  S , where D is a domain and where S is the sector S.˛; ˇ; 0 /, with range in a bounded set W C and having an asymptotic expansion of Gevrey order p1 in , uniform with respect to x 2 D. (a) Let f be a holomorphic function on a neighborhood of the closure of W . Then the function z D f ı y has an expansion of Gevrey order p1 in  2 S , uniform with respect to x 2 D. (b) Let ' W D.0; x1 /  D.0; 0 / ! C be a holomorphic bounded function such that '.0; 0/ D 0 and @' .0; 0/ D 1. Let K be a compact subset of D and let @x e e Then the function D C be such that '.x; / 2 K if  2 S and x 2 D. z W .x; / 7! y.'.x; /; / has an expansion of Gevrey order p1 in , uniform e with respect to x 2 D. P1 n Proof. Let y.x; /  1 nD0 yn .x/ be the asymptotic expansion of y. We p choose sectors Sl , l D 2; : : : ; L of opening angles less than p , such that, with S1 WD S , we obtain a good covering of D.0; 0 / . As indicated in the proof of Lemma 3.15, there exist functions yl on D  Sl , l D 2; : : : ; L, having

4.6 Composition of a Gevrey CA sE and an Analytic Function

77

P1

as asymptotic expansion of Gevrey order p1 , uniformly on D. With y1 D y we obtain that the differences have the 0 series as expansion of Gevrey order p1 ; thus they are exponentially small: there are constants C; A > 0 such that jyl .x; /  yl1 .x; /j  C exp.A= jjp / as  2 Sl1 \ Sl and x 2 D. Under the assumption of item (a), if jj is small enough, one can define zl D f ı yl and one has n nD0 yn .x/

jzl .x; /  zl1 .x; /j  CM exp.A= jjp / where M is the maximum of jf 0 j on the closure of W . By Lemma 4.4, this implies (a). One can find a more detailed proof in [9]. Under the assumption of (b), we set analogously zl .x; / D yl .'.x; /; / and e and  2 Sl is small enough. Then we easily check that they are defined if x 2 D one obviously has jzl .x; /  zl1 .x; /j  C exp.A= jjp / as  2 Sl1 \ Sl and e and the conclusion follows as before. x2D t u Theorem 4.7. Consider a holomorphic function y defined for  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V1 ./ D V .˛1 ; ˇ1 ; r0 ;  jj/, with range in a bounded set W C and having a CAsE of Gevrey order p1 . (a) Let P .x; z; / be a holomorphic function defined for jzj < r,  2 S2 D S.˛2 ; ˇ2 ; 0 / and x 2 V1 ./ having a CAsE of Gevrey order p1 : P .x; z; /  1 p   x  n P b P .x; z; / D  as S2 3  ! 0, x 2 V1 ./, n0 An .x; z/ C Gn  ; z uniformly for jzj < r, i.e. the constants of Definitions 3.6 and 3.1 are independent of z with jzj < r. Assume furthermore that y.x; / D O./. Then the function u W .x; / 7! P .x; y.x; /; / has a CAsE of Gevrey order p1 . (b) Let f be a function holomorphic on a neighborhood of the closure of W . Then the function z D f ı y has a CAsE of Gevrey order p1 for  2 S2 and x 2 V1 . (c) Let ' W D.0; x1 /  D.0; 0 / ! C be a bounded holomorphic function such that '.0; 0/ D 0 and @' r  x1 , e  2 R and e ˛1; e ˇ 1 with @x .0; 0/ D 1. Let 0 < e ˛1 < e ˛1 < e ˇ 1 < ˇ1 such that '.x; / 2 V1 ./ as  2 S.˛2 ; ˇ2 ; 0 / and e 1 ./ D V .e x2V ˛1; e ˇ 1 ;e r; e  jj/. Then the function z W .x; / 7! y.'.x; /; / e 1 ./. has a CAsE of Gevrey order p1 for  2 S.˛2 ; ˇ2 ; 0 / and x 2 V Remarks. 1. We will use the result of (c) mainly in two cases: the case where ' is independent of  and the case of the shift '.x; / D .x C T; / with T 2 C fixed. 2. The proof below is very indirect and does not show how to compute in practice the CAsE of the composition, left or right, from the CAsE of y. This was done in the proof of Proposition 2.6 for cases (a) and (b), and also for case (c) when ' is independent of . 3. In practice, suitable values for e r;e ˛1; e ˇ 1 and e  can be determined from ' in the case (c). For example in the proof of Corollary 5.3 we use the shift '.x; / D x C R R > 0. In this case, we can choosee r D r  R0 and, for any

 with 1some ,e ˛ 1 D ˛1 C ı, e ı 2 0; ˇ1 ˛ ˇ 1 D ˇ1  ı and e  D minf  R; R= sin ıg. 2

78

4 A Theorem of Ramis–Sibuya Type

When ' depends only on x, the function of the proof below is identically zero and we can choose e  arbitrarily close to , if e r is reduced. j Proof of Theorem 4.7: (a) Using Proposition 3.20, we associate to y a family .yl / on a consistent good covering whose differences satisfy (3.19)–(3.21). Similarly, j we associate to P a family .Pl / on the same covering with differences satisfying (3.19)–(3.21) uniformly with respect to z. To do this, simply replace coefficients and scalar functions of scalar values by coefficients resp. values in the Banach space of bounded holomorphic functions  in D.0; r/.  j j j We put ul .x; / D Pl x; yl .x; /;  . We find that the estimates (3.19)– j

(3.21) are satisfied for u and the ul , if necessary increasing the constant C . By j Theorem 4.1, the functions ul have a common CAsE of Gevrey order p1 . By j

Proposition 3.13 (a) applied to u  ul on each intersection, the function u has the same CAsE of Gevrey order p1 . Comparing with Proposition 2.6, we obtain the formal series of the CAsE. The proof of (b) is completely analogous. (c) Set .x; / D 2 '.x; 0; /, where 2 ' is the finite difference of ' with respect to the second variable, defined by '.x; /  '.x; / D 2 '.x; ; /.  /:

(4.26)

Thus one has '.x; / D '.x; 0/ C  .x; /: Reducing 0 if needed, the Cauchy inequalities show that the function is bounded by some constant K. j j As in the proof of (a), associate to y a family of functions yl defined on Vl ./ D j j V .˛l ; ˇl ; r0 ; jj/, satisfying (3.19)–(3.21). First let rN > 0, N 0 > 0 and ı > 0 be j small enough and let N <   K so that the the composition of all yl with ' is j j j possible: Set zl .x; / D yl .'.x; /; / for  2 S.˛2 ; ˇ2 ; N 0 / and x 2 VNl ./ D j j j V .˛l C ı; ˇl  ı; rN ; jj/. N According to the conditions on ', the functions zl are j e ./ form a consistent good well defined and we check that the sets Sl , V j and V l covering if rN > 0 and N 0 > 0 are small enough. j e C e , thus they have a The zl satisfy (3.19)–(3.21) with other constants e A; B; 1 common CAsE of Gevrey order p by Theorem 4.1. The function z has the same expansion for  2 S.˛2 ; ˇ2 ; N 0 / and x 2 V .˛1 C ı; ˇ1  ı; rN ; jj/, N as in (a). Now e1 ./. we have to prove that this CAsE remains valid for  2 S.˛2 ; ˇ2 ; 0 / and u 2 V The function ˚ W .X; / 7! '.X; / satisfies ˚.X; / D X C  .0; 0/ C O.2 / hence, from Proposition 4.6 (b), the function Z W .X; / 7! z.X; / can be continued analytically and has a Gevrey expansion, uniform for  2 S.˛2 ; ˇ2 ; N 0 / and X 2 V .e ˛1; e ˇ 1 ; .jj C ı/ jj ; e jj/. Using Propositions 3.7 and 3.8, we can extend the Gevrey CAsE of z for  2 S.˛2 ; ˇ2 ; N 0 / and u 2 V .e ˛1; e ˇ 1 ; rN ; e jj/. Similarly, with Propositions 4.6(b), 3.7 and 3.9, we extend this Gevrey CAsE for  2 S.˛2 ; ˇ2 ; N 0 / and u 2 V .e ˛1; e ˇ 1 ;e r; e jj/. Finally, changing the constants C; L1 and L2 in (3.8) if necessary, we obtain the statement. t u

4.6 Composition of a Gevrey CA sE and an Analytic Function

79

Exercise 4.8. This exercise concerns simultaneous integration of a family of functions on a consistent good covering. Let J; L 2 N;  a real number,  < 0 and let V j D V .˛ j ; ˇ j ; 1; /; Sl D j j j S.˛l ; ˇl ; 0 /; Vl ./ D V .˛l ; ˇl ; r0 ;  jj/, j D 1; : : : ; J , l D 1; : : : ; L, be a consistent good covering of A .0 ; r0 ; / of resolution ı in the sense of Definition 3.19. ej ./ D V .˛ j 2ı; ˇ j C2ı;e Lete r 0 > r0 and e  >  and put V r 0; e  jj/. Consider l l l j ej ./ such holomorphic bounded functions yl .x; / defined for  2 Sl and x 2 V l that their differences satisfy the estimates (4.1) and (4.2). j Finally, let 'l D .˛l C ˇl /=2, choose j 2˛ j C1  ı; ˇ j C ıŒ and let xl D j r0 e i . C'l / similar to the beginning of the proof of Theorem 4.1. J Z k 1 X xl k (a) Put Rl ./ D yl .u; / du for  2 Sl , l D 1; : : : ; L. Show that k1 2 i kD1 xl p A=jjp e b /, A D min.A; Br /, and that R ./  1 R./ R ./  R ./ D O.e e lC1

l

0

l

p

b where R./ is defined as in Proposition 2.9 for the CAsEs associated to the j family yl by Theorem 4.1. Let `.X / denote some function such that `.X /  log.X / is holomorphic and single valued on the annulus jX j > jj. Here we use on V j the branch of the logarithm such that arg X 2˛ j ; ˇ j Œ and assume, without loss in generality, that ˛ 1 < ˛ 2 < ˇ 1 < ˛ 3 < ˇ 2 < : : : < ˛ J < ˇ J 1 < ˛ 1 C 2 < ˇ J : In intersections V j;j C1 , the two possible determinations of log X coincide for j D 1; : : : ; J  1, but not in V J \ V 1 . j Define zl recursively by Z z1l .x; /

x

WD xlJ

   e1l ./; yl1 .u; /  Rl ./`0 u du for  2 Sl ; x 2 V

and then for j D 1; : : : ; J  1 j C1

zl

Z .x; / WD

x j

xl

   j C1 j j yl .u; /  Rl ./`0 u du C zl xl

ej C1 ./: if  2 Sl ; x 2 V l j (b) Prove that the differences of the functions zl satisfy estimates analogous to (4.2) and Z xJ  lC1    A=jjp  j j yl1 .u; /  Rl ./`0 u du D O e e zlC1 .x; /  zl .x; /  xlJ

ej ./ \ V ej ./. if  2 Sl \ SlC1 and x 2 V l lC1

80

4 A Theorem of Ramis–Sibuya Type

(c) Fix any x  with jx  j D r0 . For all l 2 f1; : : : ; Lg choose ml 2 f1; : : : ; J g such e ml ./ and define wj by that x  2 V l l  l e wl .x; / D zl .x; /  zm l .x ; / for  2 Sl ; x 2 V l ./: j

j

j

j

Prove that the differences of the functions wl satisfy estimates analogous to (4.1) and (4.2). More precisely the value of B is the same as in the estimates j for the functions yl , the value for A has to be replaced by e A. Conclude that the j functions wl admit CAsEs of Gevrey order p1 .   j j j (d) Clearly, Yl .x; / WD wl .x; / C Rl ./` x defines antiderivatives of yl . j b.x; /  Y b.x  ; / as Sl 3  ! 0, x 2 V j ./, Show that Y .x; /  1 Y l

p

b was defined in Proposition 2.9. where Y

l

Chapter 5

Composite Expansions and Singularly Perturbed Differential Equations

Our notion of composite expansion has been motivated by the study of singularly perturbed ordinary differential equations of the form (1), rewritten below for convenience "y 0 D ˚.x; y; "/: (5.1) Contrary to the equations of Chap. 1, the variables x and y are now complex and the small parameter " is in a sector of the complex plane S D S.ı; ı; "0 / containing some part of the positive real axis, with ı; "0 > 0 fixed small enough. Even though most applications only need real values of ", we will need to consider " in a sector. We assume ˚ analytic with respect to x and y in a domain D C2 and of Gevrey order 1 with respect to " in S ; this also allows to treat equations containing a control parameter. This is useful for equations where canards solutions might occur for certain values of the parameter. Recall that the slow set L is the subset of D of equation ˚.x; y; 0/ D 0 and that a point .x  ; y  / of L is called regular if @˚ .x  ; y  ; 0/ ¤ 0. In a @y neighborhood of a regular point, by the implicit function theorem, L is the graph   of some analytic  function y0 , called slow function, satisfying y0 .x / D y . Set @˚ f .x/ D @y x; y0 .x/; 0 . If we continue such a function y0 analytically, two situations may occur: either we reach the boundary of D, or we reach a turning point, i.e. a point where f vanishes. A priori the function y0 has a singularity at a turning point. In this memoir, we consider only the very particular situation where y0 is defined and analytic in a neighborhood of the turning point. In Sect. 5.1, we detail the use of CAsEs in a neighborhood of a regular point, where we slightly improve the classical composite expansions. The case of a turning point is presented in the later sections, first in a simple situation, called quasi-linear in Sect. 5.2, then in a more general situation in Sect. 5.3.

A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 5, © Springer-Verlag Berlin Heidelberg 2013

81

82

5 Composite Expansions and Singularly Perturbed Differential Equations

5.1 Classical CAsEs at a Regular Point We consider an initial value problem of the form "y 0 D ˚.x; y; "/ ;

y.x  ; "/ D v."/

(5.2)

with the previous notation and assumptions. We assume that v is of Gevrey order 1 with respectPto " 2 S . It is easily verified that there exists a unique formal solution b y .x; "/ D n0 an .x/"n of (5.1) without the initial condition, whose coefficients an are analytic in a neighborhood of x  and such that a0 D y0 , the slow function, cf. formula (5.3) below. It is known [6,9] that this formal solution satisfies estimates of Gevrey order 1 in ". Furthermore, any solution of (5.1), which is bounded in a domain of the form D1  S with D1 D D.x  ; r0 / and S D S.ı; ı; "0 /, is e1  e e 1 D D.x  ;e asymptotic of Gevrey order 1 to b y in any subdomain D S (i.e. D r 0/ e e e and e S D S.ı; ı;e "0 / with e r 0 < r0 , ı < ı and e "0 < "0 ). It is also known [4, 59] that the solution of the initial value problem (5.2) is  defined and bounded for " 2 S and x in some sector this  that P of vertex x and  solution has a composite expansion of the form n0 an .x/ C gn xx "n , " where the an are the coefficients of the formal solution and the gn are exponentially decreasing. Theorem 5.1 and Corollary 5.3 confirm this result and generalize it to the complex framework. They also provide Gevrey estimates for these expansions. More precisely, we show that, if the initial condition v is sufficiently close to y  D y0 .x  /, then the solution of problem (5.2) has a Gevrey CAsE for " 2 S and for x in an appropriate domain. The domain in x may be chosen rather large, provided that v is close enough to y  . To avoid having a too complicated statement, and also to make proofs not too long, we have chosen to write two separate statements. Before presenting these statements, we make the following reductions. Firstly, changing y in y  y0 we reduce to y0 0. Secondly, a shift and a rotation of the variable x allow to assume x  D 0 and f .0/ < 0. This permits us to detail the formal solution to some extent. Since ˚.x; 0; 0/ D 0, the function ˚ is of the form ˚.x; y; "/ D y .x; y/C"P .x; y; "/, with  and P analytic (and  .x; 0/ D f .x/).  P Thus the coefficient of "nC1 in the Taylor expansion of ˚ x; k1 ak .x/"k ; " is of the form f:anC1 C n , where n depends only on the terms a1 ; : : : ; an and 0 .x/ D P .x; 0; 0/. Therefore the formal solution can be obtained recursively by a1 D 

0 ; f

anC1 D

1 0 .a  n /: f n

(5.3)

Our first statement concerns initial conditions close to the slow curve. Theorem 5.1. Let ı; "0 ; r0 ; r2 > 0 with ı < 6 , set S D S.ı; ı; "0 / and consider problem (5.2) where x  D 0, ˚ is analytic for jxj < r0 , jyj < r2 , of Gevrey order 1 for " 2 S and satisfies ˚.x; 0; 0/ D 0, and where v is Gevrey-1 for " 2 S and satisfies jv.0/j < r2 .

5.1 Classical CA sEs at a Regular Point

83

Set f .x/ D @˚ .x; 0; 0/ and assume that f .0/ < 0 and for all jxj < r0 , jyj < r2 @y and " 2 S.ı; ı; "0 / ˇ ˇ ˇ ˇ ˇ ˇ .x; y; "/   .x; y; "/ˇ  ˇ < ı and ˇ @˚ ˇ arg @˚ @y @y

1 2

jf .0/j:

(5.4)

Then, for any r < r0 and for "0 sufficiently small, the solution y of (5.2) has a  CA sE of Gevrey order 1 for " ! 0 in S and x 2 S 0 D S  2 C 3ı; 2  3ı; r . P The slow part of this expansion is the formal solution n1 an .x/"n determined by (5.3) and  the fast part consistsof functions gn of exponential decay in the sector S 00 D S  2 C 2ı; 2  2ı; 1 . More precisely, there exist a sequence .gn /n2N , gn 2 G .S 00 / and constants C1 ; C2 > 0 such that, for all x 2 S 0 , all " 2 S and all integer N > 0 N 1  ˇ X   ˇˇ ˇ an .x/ C gn x" "n ˇ  C1 C2N N Š j"jN : ˇy.x; / 

(5.5)

nD0

Moreover, there are C3 ; C4 ; C5 > 0 such that, for all X 2 S 00 and all n 2 N  nC1=2 C5 jX j e if C5 jX j  n ; jgn .X /j  C3 C4n C5 jX j jgn .X /j 

C3 C4n

(5.6)

nŠ otherwise:

Remarks. 1. By continuity of the partial derivative of ˚, condition (5.4) is satisfied if r0 ; r2 and "0 are small enough. 2. The existence of composite expansions at a regular point is classical in the real framework. The real proofs can easily be adapted to the complex framework. The Gevrey estimates, however, are new up to our knowledge. We believe that these estimates may be useful for certain problems. 3. The coefficients gn can be calculated from the formal solution of the inner equation and using Proposition 2.16. The inner equation is obtained by the change of variables x D "X; y.x/ D Y .X /. Recall the notation ˚.x; y; "/ D y .x; y/ C "P .x; y; "/, where  and P are analytic and  .x; 0/ D f .x/. Then the inner equation can be written dY D Y  ."X; Y / C "P ."X; Y; "/; dX

Y .0; "/ D v."/:

P b.X; "/ D n0 Yn .X /"n of this initial value problem is The formal solution Y calculated by successively solving differential equations with initial conditions; these equations are linear, except for the first one. P For the first term, we obtain using the notation .Y / D  .0; Y / and v."/ D n0 vn "n dY 0 D Y0 .Y0 /; dX

Y0 .0/ D v0 :

84

5 Composite Expansions and Singularly Perturbed Differential Equations

Z

Y0 .X /

This determines Y0 implicitly by the formula v0

subsequent terms, we obtain Yn0 D A.X /Yn C Bn .X / ; where A D Y0 We obtain

0

.Y0 / C

du D X . For the u .u/

Yn .0/ D vn

.Y0 / and Bn only depends on the terms Y0 ; : : : ; Yn1 .

Yn .X / D vn e

RX 0

Z A

C

X

e

RX u

A

Bn .u/du:

0

Our Theorem 5.1 and Proposition 2.16 yield that Yn can be written as Yn D Pn C gn where Pn is a polynomial of degree at most n and gn is the coefficient of the fast part of the CAsE. Thus gn is obtained by discarding the polynomial part of Yn . Proof of Theorem 5.1: We first show the existence of y for x 2 S 0 and " 2 S . The proof follows the lines of Lemma 7 in [25]. For this purpose, we fix x 2 S 0 and " 2 S and we show that the function u W t 7! y.tx; "/ is defined on Œ0; 1 and satisfies ju.t/j < r2 . Since ju.0/j D jv."/j < r2 , it is sufficient to show that the set fjuj < e r 2 g is invariant under (5.1) for some e r 2 with d ju.0/j < e r 2 < r2 , i.e. dt ju.t/j < 0 if ju.t/j D e r 2 . We have d dt u.t/u.t/

    D 2Re y 0 .tx; "/ x y.tx; "/ D 2Re 1" ˚.tx; u.t/; "/ x u.t/ :

We now write Z ˚.tx; u.t/; "/ D ˚.tx; 0; 0/ C " 0

1

@˚ .tx; 0; s"/ds C u.t/ @"

Z 0

1

@˚ .tx; su.t/; "/ds @y

ˇ ˇ and we use that ˚.tx; 0; 0/ D 0, ˇ @˚ .tx; 0; s"/ˇ is bounded by some constant M > @" R1 0, and the complex number ' D 0 @˚ @y .tx; su.t/; "/ds satisfies j arg '  j < ı and 1 j'j  2 jf .0/j. As a consequence, we obtain jarg ˚.tx; u.t/; "/  arg u.t/  j < 2ı if ju.t/j D r2 e e r 2 and "0 is small enough (more exactly, ˇ  if "0 < 2M jf .0/j sin ı). Since ˇ j arg xj < ˇ ˇ  1 1  3ı and j arg " j < ı, this gives ˇarg " ˚.tx; u.t/; "/ x u.t/   ˇ < 2 , hence 2 d dt u.t/u.t/

< 0. This completes the proof of existence of y. To show that y has a CAsE, we compare it to a family of solutions of ODEs close to (5.1) and we apply our key-theorem 4.1. Let N 2 N with N  2 ı and, for l D 0; : : : ; N  1, let Sl D S.˛l ; ˇl ; 0 / with ˛l D

2.l  1/ 2.l C 1/ and ˇl D : N N

5.1 Classical CA sEs at a Regular Point

85

In this way, N is large enough so that P S0 S . Let vl W Sl ! C be a function asymptotic of Gevrey order 1 to b v D n0 vn "n , whose existence is ensured by the Borel–Ritt–Gevrey theorem (e.g. obtained by Borel transform of b v and truncated Laplace transform, see the beginning of the proof of Lemma 3.15). Reducing "0 if required, we can assume that these functions vl are bounded by r2 . To fix ideas one can choose for v0 the restriction of v to S0 , but this is not essential. Similarly, let ˚l W D.0; r0 /  D.0; r2 /  Sl ! C be asymptotic of Gevrey order 1 in " to the same series as ˚. Since the functions v0 ; : : : ; vN 1 and v are Gevrey-1 asymptotic to the same series, and similarly for ˚0 ; : : : ; ˚N 1 and ˚, there exist C; A > 0 such that for all l 2 f0; : : : ; N  1g, all x 2 D.0; r0 /, all y 2 D.0; r2 / and all " 2 Sl;lC1 D Sl \ SlC1 jvlC1 ."/  vl ."/j  C e A=j"j and j˚lC1 .x; y; "/  ˚l .x; y; "/j  C e A=j"j (5.7) as well as jvl ."/  v."/j  C e A=j"j and j˚l .x; y; "/  ˚.x; y; "/j  C e A=j"j for x 2 D.0; r0 /, y 2 D.0; r2 / and " 2 S \ Sl if this intersection is not empty. In order to apply Theorem 4.1, we will consider two families .yl1 /0l
 2l  2l  C 3ı and ˇl1 D C  3ı: N 2 N 2

1 Lemma 5.2. For all x in the intersection Sl;1 lC1 D Sl1 \ SlC1 and all " in Sl; lC1 D Sl \ SlC1 , one has 1 jylC1 .x; "/  yl1 .x; "/j  C.1 C r0 /e A=j"j ;

where A and C are given in (5.7). Similarly, with the notation S and S 0 of Theorem 5.1, for all x in Sl1 \ S 0 and all " in Sl \ S one has jyl1 .x; "/  y.x; "/j  C.1 C r0 /e A=j"j : 1 Proof. Put zl D ylC1  yl1 , let

b.x; "/ D ˚lC1 .x; yl .x; "/; "/  ˚l .x; yl .x; "/; "/;

86

5 Composite Expansions and Singularly Perturbed Differential Equations

recall the notation 2 introduced in (4.26), defined by ˚lC1 .x; z; "/  ˚lC1 .x; y; "/ D 2 ˚lC1 .x; y; z; "/.z  y/ and let d.x; "/ D 2 ˚lC1 .x; yl .x; "/; ylC1 .x; "/; "/. Since Z

1

2 ˚.x; y; z; "/ D 0

@˚ .x; y C t.z  y/; "/dt; @y

ˇ ˇ assumption (5.4) implies ˇ arg d.x; "/   ˇ < ı. Then zl is the solution of the initial value problem "z0l D d.x; "/zl C "b.x; "/;

zl .0; "/ D vlC1 ."/  vl ."/:

Given x 2 Sl;1 lC1 , we choose as path of integration the segment from 0 to x. Then variation of constants gives zl .x; "/ D zl .0; "/ exp

 R 1 x "

0

 Z d.u; "/du C

x

b.u/ exp 0

 R 1 x "

u

 d.v; "/ du:

For  2 Sl; lC1 and u 2 Œ0; x, one has Re

1

" d.u; "/e

i arg u



< 0;

which implies Z

jxj

jzl .x; "/j  jzl .0; "/j C

jb.se i arg x /jds  C.1 C r0 /e A=j"j :

0

The second estimate of the lemma is proved similarly.

t u

Continuation of the proof of Theorem 5.1: Now we construct the second family. Fix e r 0 2 0; r0 Œ. For each l 2 f0; : : : ; N  1g, let xl D e r 0 e 2l i=N and let yl2 be the 2 solution with initial condition yl .xl ; "/ D 0. For "0 small enough, yl2 is defined on Dl  Sl , where Dl is the intersection of the disk D.0; r0 / and the sector of vertex xl and angles ˛l1 and ˇl1 , cf. Fig. 5.1. By an argument analogous to the proof of Lemma 5.2, we show that for x 2 Dl \ DlC1 and " 2 Sl;lC1 , one has 2 jylC1 .x; "/  yl2 .x; "/j  C.1 C 2r0 /e A=j"j :

To apply Theorem 4.1, it remains to be shown that the solutions yl1 and yl2 are exponentially close in the intersection of their domain of existence Dl \ Sl1 D Sl1 . Their difference w D yl2  yl1 satisfies the equation "w0 D D.x; "/w

5.1 Classical CA sEs at a Regular Point

87

Fig. 5.1 In bold, the boundary of Dl

where D.x; "/ D 2 ˚.x; yl1 .x; "/; yl2 .x; "/; "/, and the initial condition w.0; "/ D yl2 .0; "/ vl ."/. One has jw.0; "/j  2r2 ; by (5.4), D.x; "/j < ı ˇ one also has ˇ j arg ˇ <   ı and " 2 Sl , and jD.x; "/j  12 jf .0/j. For x 2 Sl1 such that ˇ arg x  2l N 3 we obtain jw.x; "/j  2r2 e Bjxj=j"j with B D 14 jf .0/j. We have shown above that the conditions of Theorem 4.1 are satisfied. We obtain j that each function yl ; j D 1 or 2, 0  l < N , has a composite expansion of Gevrey order 1. Since the solution y is exponentially close to yl1 on .S 0 \ Sl1 /  .S \ Sl /, it has a Gevrey expansion, too. Moreover, since the formal solution does not contain poles, by Proposition 2.16 it coincides with the slow expansion and all the coefficients gmn are zero. Therefore all the functions of the fast expansion gn are flat of Gevrey order 1 as the variable X tends to infinity; thus they decrease exponentially. Since the functions gn have consistent Gevrey asymptotic expansions in the sense of Definition 3.1, formula (3.1) gives (5.6) with a suitable choice of the integer M depending on n and jX j. More precisely, if C; L1 ; L2 > 0 are constants (from Definition 3.1) such that for all n; M and X (here p D 1) ˇ ˇ jX jM ˇgn .X /ˇ  CLn1 LM 2  .M C n C 1/; we obtain (5.6) using the estimate kŠ  k kC1=2 e 1k for k D M C n. We choose C3 D e C; C4 D L1 ; C5 D 1=L2 and for the integer M : M D 0 if n  jX j =L2 t u and M D jX j =L2  n otherwise. Theorem 5.1 only applies to initial conditions close to the slow curve, but our results on the extension of CAsEs allow to treat initial conditions far from the slow curve. Corollary 5.3. With the notation of Theorem 5.1, assume that v.0/ is in the basin of attraction of 0 for the fast equation Y 0 D ˚.0; Y; 0/:

(5.8)

88

5 Composite Expansions and Singularly Perturbed Differential Equations

Fig. 5.2 The sector S.˛; ˇ; r0 / and the shift S 0 of the  sector  S  2 C ı; 2  ı; r0  "R

Then there exist ı; "1 > 0 such that the solution y of (5.2) has a CAsE of Gevrey order 1 as " ! 0 in S1 D S.ı; ı; "1 / and x 2 S.ı; ı; r0 /. More generally, if ˛ < 0 < ˇ are such that v.0/ is in the basin of attraction of 0 for the equation Y 0 D e d i ˚.0; Y; 0/ for all d 2 Œ˛; ˇ, then the solution y of (5.2) has a CAsE of Gevrey order 1 as S1 3 " ! 0 and x 2 S.˛; ˇ; r0 /. Proof. First of all, if v.0/ is in the basin of attraction of 0 for the fast equation (5.8) and if ı is sufficiently small, then v.0/ is also in the basin of attraction for the equation e id Y 0 D ˚.0; Y; 0/ for all jd j < ı. Thus it suffices to prove the generalization. Since v.0/ is in the basin of attraction of 0, there is R > 0 such that w."/ D y."R; "/ satisfies the condition of the previous Theorem 5.1, i.e. jw."/j < r2 for all " 2 S . First we show that w has a Gevrey-1 expansion. Let 'R D 'R .v; "/ be the flow at time R of the inner equation, obtained by putting x D "X dY D ˚."X; Y; "/: dX

(5.9)

The function 'R is defined by 'R .v; "/ D Yv .R; "/ if Yv is the solution of (5.9) with initial condition Yv .0; "/ D v. By a classical theorem, 'R is analytic with respect to v and ". By assumption, y.0; "/ D v."/ has a Gevrey-1 expansion. Since the composition of a Gevrey-1 function and an analytic function is Gevrey-1 (cf. the beginning of Sect. 4.6), the function w W " 7! 'R .v."/; "/ is Gevrey-1. Now we apply Theorem 5.1 to the problem "

dz e z; "/ ; D ˚.u; du

z.0; "/ D w."/

(5.10)

e z; "/ D ˚.u C "R; z; "/. Thus the solution z, which is linked to y where we set ˚.u;    P by z.u; "/ D y.u C "R; "/, has a Gevrey-1 CAsE n0 bn .u/ C hn u" "n , for u in   S 0 D S  2 C 2ı; 2  2ı; r0 and " in S D S.ı; ı; "0 / (Fig. 5.2). By Theorem 4.7 (cf. also Remark 3 which follows this theorem) the solution y of (5.2) also has

5.2 CA sEs at a Turning Point: The Quasi-linear Case

89

  a CAsE for " 2 S and x 2 V D V  2 C 3ı; 2  3ı; j"j; r0  R"0 with  D R= sin ı, if "0 is small enough. From the assumptions, the function y is defined for x in the sector S.˛; ˇ; r0 /. Using once again the inner equation (5.9), the analyticity of the flow, and the fact that the composition of a Gevrey function and an analytic function is Gevrey, we deduce that the function Y defined by Y .X; "/ D y."X; "/ is Gevrey-1 for " 2 S , uniformly for X 2 S.˛; ˇ; R/. By Proposition 3.8, y has a CA sE on S.˛; ˇ; r0 /. t u

5.2

CAsEs

at a Turning Point: The Quasi-linear Case

In this section, r0 ; r2 ; "0 ; ı > 0 are fixed, ı > 0 is sufficiently small, D1 and D2 are the disks of center 0 and radius r0 and r2 respectively, and ˙ denotes the sector ˙ D S.ı; ı; "0 /. Consider an equation of the form "y 0 D px p1 y C "P .x; y; "/;

(5.11)

where x; y 2 C, " is a small parameter and P is analytic and bounded on D1  D2  ˙ and of Gevrey order 1 as ˙ 3 " ! 0. Let  D "1=p . In the sequel, we sometimes use " and  at the same time, being understood that we always have the relation " D p . Equation (5.11) is called quasi-linear , because the inner equation obtained by dY x D X , Y .X / D y.X /, i.e. dX D pX p1 Y C P .X; Y; p / reduces to a linear homogeneous equation if  is replaced by 0. The general form of the quasi-linear equation corresponds to the case when px p1 is replaced by a function f .x/ holomorphic in a neighborhood of 0 and with a zero of order p  1 at this point. This form can be reduced to the normal form (5.11) by a change of variable (see Remark 8 in Sect. 5.2.4). Alternatively, one can modify the statements and proofs of this part to fit this general form. As the description of the domains with respect to x is easier for (5.11), we have chosen to present the theory only for this equation in the sequel. We chose a simple equation depending on ", although we can handle more general equations (see Remark 2 in Sect. 5.2.4 and a generalization in Sect. 5.3). For this equation, two proofs of the existence of solutions having CAsEs are possible. The first one, described in Sects. 5.2.1 and 5.2.2, is based on a classical model: one first determines a composite formal solution and then shows the existence of solutions having it as CAsE. The second proof uses the key-theorem 4.1 and shows the existence of a Gevrey CAsE without giving its coefficients. Although the second proof is shorter, we have chosen to present also the first one because it can deal with equations that are not within the framework of the second proof (e.g. real equations that are only C 1 ) and where the CAsEs are not necessarily Gevrey.

90

5 Composite Expansions and Singularly Perturbed Differential Equations

5.2.1 Composite Formal Solutions Let V D V .˛; ˇ; 1; /, where  is a real number, positive or negative, and where 3 b ˛ and ˇ satisfy  3 2p < ˛ < 0 < ˇ < 2p . We recall that C .r0 ; V / denotes the set of e .V / composite formal series associated to V and the disk D1 . We also introduce C the set of composite formal series associated to V and D1 where the coefficients an are not necessarily bounded on the disk. The motivation for this modification e .V /; we lies in the fact that the derivative of the slow part of a CAsE is still in C e .V / embedded with the will not have to differentiate fast parts. The vector space C f denote the vector ultrametric distance (cf. (2.6)) is a Banach space. Similarly, let H space of functions holomorphic in D1 , not necessarily bounded. Here is the main result. Theorem 5.4. With the notation above, Eq. (5.11) has a unique composite formal e .V /. solution b y in C In Sect. 5.2.3 we will see that the existence of this formal solution is a direct consequence of the existence of analytic solutions of (5.11) with composite expansion. We find it instructive, however, to give an independent proof for the existence of this formal solution and of a solution having it as a CAsE, avoiding the techniques developed in Chap. 4. The idea of proof is as follows. We start by solving the linear non-homogeneous equation "y 0 D px p1 y C "h.x; / (5.12) e .V /. In Lemma 5.7 we prove that (5.12) has a unique solution in where h 2 C e b denotes the operator which, to a formal series b b .h/. If Q C .V /, denoted by ˚ y, b y /.x; / D associates the formal series obtained by substituting b y in P , i.e. Q.b P .x; b y .x; /; p /, we then check that the equation   b b Q.y/ yD˚ e .V /. satisfies all the conditions of the Banach fixed point theorem in C By linearity, Eq. (5.12) is solved by developing h in a composite formal series and adding the solutions of each equation containing only one term of the series. Therefore this amounts to solving (5.12) in two cases: when the function h depends   f , and when h depends only on X D x , i.e. h.x; / D k x only on x, i.e. h 2 H   with k 2 G .V /. We start with the second case. Lemma 5.5. For all k 2 G .V /, the equation "y 0 D px p1 y C "k has a unique solution in G .V /.

x  

(5.13)

5.2 CA sEs at a Turning Point: The Quasi-linear Case

Proof. The change of unknown y.x; / D Y

91

x



;

leads to equation

dY D pX p1 Y C k.X /: dX Since V satisfies  3 2p < ˛ < 0 < ˇ < bounded in G .V /: Z Y .X / D exp.X p /

3 2p ,

this equation has a unique solution

X

exp.t p /k.t/dt 1

where the path of integration is in V and its part far enough from the origin is a ray t u fX1 C t I t 2 RC g with some X1 2 V . f , we will use the following lemma. In order to solve (5.12) in the first case h 2 H Although it is more or less classical (similar results can be found for example in [5, 9]), we join a proof for the completeness of the memoir. f , there is a unique family .b Lemma 5.6. For all h 2 H a0 ; : : : ;b ap2 / of p1 formal series in powers of " without constant term (i.e. b al 2 "CŒŒ") such that the equation "y 0 D px p1 y C "h.x/ C

p2 X

b al x l

(5.14)

lD0

has a formal solution u without constant term and with coefficients analytic in the Pb f . Moreover, b disk D1 : b u.x; "/ D >0 u .x/" ; u 2 H u is uniquely determined. P P Proof. Let us plug b u D >0 u .x/" and b a l D >0 al " in (5.14), taking into account the constraint that u has no pole at x D 0; we obtain for the term of order 1 in ": p2 X 0 D px p1 u1 .x/ C h.x/ C al1 x l : P

lD0

Setting h.x/ D 0 h x , this gives al1 D hl for l D 0; : : : ; p  2 and u1 .x/ D X  p1 hCp1 x  , i.e. u1 D  p1 Sp1 h. Concerning the term of order n in ", n  2, 

0

we obtain u0n1 .x/ D px p1 un .x/ C

p2 X

aln x l :

lD0

The condition that un has no pole at x D 0 is satisfied if and only if aln is the coefficient of the term of degree l in the series expansion of u0n1 , which is equivalent to un D p1 Sp1 u0n1 . t u

92

5 Composite Expansions and Singularly Perturbed Differential Equations

f . The change of unknown y D b Let us return to Eq. (5.12) with h 2 H u C z, where b u is given by Lemma 5.6, yields equation "z0 D px p1 z 

p2 X

b al x l :

(5.15)

lD0

By linearity, it suffices to show that for all l 2 f0; : : : ; p  2g there is a unique e .V / of equation solution in C "z0 D px p1 z  "x l : The change of unknown z.x; / D lC1 Z

x



;

gives

Z 0 D pX p1 Z  X l whose unique solution in G .V / is given by Z Z.X / D exp.X p /

1

t l exp.t p /dt:

X

f , (5.12) has a unique solution in C e .V /. Combining with Hence, when h 2 H Lemma 5.5, we thus obtain the following result. e .V / there exists a unique solution of (5.12) in C e .V /. Lemma 5.7. For all h 2 C e .V / ! C e .V /; h 7! ˚.h/ b .h/. The map ˚ bWC b It is denoted ˚ defined in this way is contracting with ratio 12 . e .V / and to the Proof of Theorem 5.4: According to the variant of Lemma 2.2 for C b e e b lemma above, the map ˚ ı Q W C .V / ! C .V / is contracting with ratio 12 . Since e .V / is complete, this map has thus a unique fixed point in C e .V /. C t u Remarks. 1. Another way to present Lemma 5.7 is to say that the formula b h.x; / D ˚

Z

x

exp 0

n  x p 



 t p o 

h.t; /dt

e .V / into C e .V /. b from C defines an operator ˚ 2. The proof of the theorem shows that, as for Lemma 2.2, the statement still holds if one only assumes that P .x; y; "/ is a formal series in " and y, with coefficients e .V /. in C 3. In practice, one does not compute the composite formal solution of (5.11) as suggested by the proof of Theorem 5.4. Since we know that it exists, we can compute it as indicated in Remark 3 after Proposition 2.16: one first computes the inner and outer expansions, then one rejects the polar part of the outer expansion

5.2 CA sEs at a Turning Point: The Quasi-linear Case

93

to obtain the slow expansion and the polynomial part of the inner expansion to obtain the fast expansion. P The outer expansion is given by the formal solution n1 vn .x/"n of (5.11). This solution has coefficients regular except at the turning point 0; it is given recursively by v1 .x/ D 

P .x; 0; 0/ ; px p1

vnC1 .x/ D

.v0n  qn /.x/ px p1

where qn is the coefficient (depending on v1 ; : : : ; vn )of P the term of order n in " obtained by Taylor expansion of the function " 7! P x; 1n v .x/" ; " . To compute the inner expansion, one plugs x D X , y.x/ D Y .X / (with p D ") in (5.11), which gives the inner equation dY D pX p1 Y C G.X; Y; / dX

(5.16)

with G.X; Y; /PD P .X; Y; p /. One shows that there is a unique formal n b D solution Y n1 Vn .X / such that all Vn .X / are of polynomial growth as V 3 X ! 1. This formal solution is determined recursively by calculating the unique solution of polynomial growth on V of the equation dV n D pX p1 Vn C Gn .X / dX

(5.17)

where G1 .X / D G.X; 0; 0/ and Gn is the coefficient (depending on V1 ; : : : ; Vn1 ) of the term of order n  1 in  obtained by Taylor expansion  P of the function  7! G X; 1
X 1

exp.X p  s p / Gn .s/ds:

5.2.2 Analytic Solutions and CAsEs The purpose of this section is to show that Eq. (5.11) admits solutions having CAsEs; see Theorem 5.8 below. The notations are those of the beginning of Sect. 5.2: the numbers r0 ; r2 ; "0 ; ı > 0 are given, with "0 ; ı sufficiently small, the variable " D p is in the sector ˙ D 2ı S.ı; ı; "0 /, D1 is the disk D.0; r0 / and D2 the disk D.0; r2 /. We set ˛ D  3 2p C p ,  

 1=p 2ı . ˇ D 3 2p  p and we fix r1 2 0; r0 cos.2ı/ Theorem 5.8. Consider equation (5.11) with P analytic and bounded in D1 D2  ˙ and having a uniform asymptotic expansion as ˙ 3 " ! 0.

94

5 Composite Expansions and Singularly Perturbed Differential Equations

Then, for all > 0, there  exist 1 > 0 and a solution y.x; / of (5.11) defined for  2 S1 D S  pı ; pı ; 1 and x 2 V ./ D V ˛; ˇ; r1 ; jj . Furthermore y has a CAsE as S1 3  ! 0 and x 2 V ./. Remark. In this statement, the equation is quasi-linear and solutions can be extended to a neighborhood of 0; therefore quasi-sectors can be chosen with  > 0. In Sect. 5.3, this will not be the case and we need to consider quasi-sectors with  < 0. This is one reason why we defined our CAsEs on quasi-sectors with  positive or negative, and not on sectors. For reasons of convenience, we postpone the proof of Theorem 5.8 at the end of this Sect. 5.2.2 and we first deal with preliminary notions and results. Given jd j < ı, x0 2 C and a domain ˝ containing 0 and x0 on its boundary, we say that ˝ is ı -descending from x0 with respect to the landscape Rd W x 7! Re .x p e id /

(5.18)

if for all x 2 ˝, there is a piecewise C 1 path x , of length denoted by `, joining x0 to x in ˝ [ f0; x0 g, parametrized by its arc length, i.e. such that jx0 .t/j D 1 for all t 2 Œ0; `, and such that for all t 2 Œ0; ` and all d 2   ı; ıŒ   Re x .t/p1 x0 .t/e id  jx .t/p1 j

(5.19)

with  D sin ı. This means that at every point of x , the angle between the tangent to x and the direction of steepest descent for Rd is at most 2  ı. For such paths, we have the following useful estimates. Lemma 5.9. Let x be a piecewise C 1 path of length ` from x0 to x in ˝ [ f0; x0 g, parametrized by its arc length and satisfying (5.19). Then for all j 2 f0; 1; : : : ; p  1g, there is a constant Lj > 0 such that ˇ p pˇ Z ˇ p pˇ ˇ ˇ ˇ  = ˇ j j C1 Ij .x; / WD ˇe x = ˇ ˇe ˇ jj jd j  Lj jj x

for all x 2 ˝,  2 S1 . Moreover, if jx .t/j  L jj for all t 2 Œ0; `, then Ij .x; / 

1 1Cj p j C1 L : jj p

R ˇ p pˇ R`ˇ p pˇ Here x ˇe  = ˇ jjj jd j is a notation for 0 ˇe x .t / = ˇ jx .t/jj dt; recall that jx0 .t/j D 1. Proof. We have to estimate ˇ p pˇ Z ˇ ˇ Ij .x; / D ˇe x = ˇ

` 0

ˇ ˇ ˇ x .t /p =p ˇ ˇe ˇ jx .t/jj dt:

5.2 CA sEs at a Turning Point: The Quasi-linear Case

95

  We denote d D p arg  and we first introduce the variable s D Re x .t/p e id . By (5.19), we have   ds D p Re x .t/p1 x0 .t/e id  p  jx .t/jp1 dt with  D sin ı; moreover, one has jx .t/j  jsj1=p . With  D Re .x p e id /= jjp , we obtain Z Ij .x; /  e 

Re .x p eid / p Re .x0 eid /

Z Finally we estimate e 

 j C1  p

p

e s=jj

1



e

j C1 1 jjj C1  1 ds  e jsj p p p

u

juj

j C1 p 1

Z



if  is positive and by 1 C

Z

1

du D

C1

Z

1

e u juj

j C1 p 1

du



e  j C j

j C1 p 1

d by

0

j C j

max.0;1/

j C1 p 1

d  1 C

2p j C1

if

 is negative. For the second assertion, we use jx .t/j  LR jj instead of jx .t/j  s 1=p and we 1 1 obtain it with Ij .x; /  p Lj C1p jjj C1 e   e u du. u t The following lemma will be used repeatedly in the proof of Theorem 5.8. Lemma 5.10. Consider equation p y 0 D px p1 y C p Q.x; y; /

(5.20)

with Q.x; u; / analytic and  bounded on the set of all .x; u; / such  that u 2 D2 ,  2 S0 WD S  pı ; pı ; 0 and x 2 V0 ./ WD V ˛; ˇ; r0 ; jj , where  > 0,  1=p

 2ı 3 2ı . ˛ D  3 2p C p and ˇ D 2p  p . Let r1 2 0; r0 cos.2ı/ Then there are 1 > 0 and a solution y of (5.20) defined on the set     of all ; x such that  2 S1 WD S  pı ; pı ; 1 and x 2 V1 ./ WD V ˛; ˇ; r1 ; jj . Moreover, y.x; / D O./ uniformly in the previous set. Proof. We first construct y for  2 S0 small enough and x in a part ˝ of V0 ./, consisting of the sector S.˛; ˇ; r1 / and a small “curvilinear triangle” tangent to this sector. Then we show that y can be continued analytically to V1 ./ n ˝ if  is small enough.  1=p . The condition of the statement on r1 implies that x0 Set x0 D r1 cos.2ı/ is in V0 ./ for all  2 S0 . Let ˝ denote the union of S.˛; ˇ; r1 / and the interior of the “curvilinear triangle” T .r1 ; ı/ whose image by F W C ! C;

x 7! x p

(5.21)

96

5 Composite Expansions and Singularly Perturbed Differential Equations

is the triangle of vertices x0 ; r1 e i 2ı and r1 e 2ıi . This triangle is equilateral at x0 and has the angles   4ı and twice 2ı. The choice of x0 is done so that the parts of @˝ joining x0 to the points r1 e 2ıi=p and r1 e 2ıi=p are sent by F to segments p tangent to the circle of radius r1 . By construction, V1 ./ n ˝ is a part of the disk D.0; jj/ (Fig. 5.3). The reason for adding this little triangle lies in that ˝ is ı-descending from x0 with respect to the landscape (5.18) for all jd in the case where   j p< ı. For example, j arg xj  p , we can choose x such that arg x0 x .t/p is constant and in the case where j arg xj > p we can choose as path x the union of the segments Œx0 ; 0 and p Œ0; x. In other words, we can choose x such that x .Œ0; `/p is either Œx0 ; 0[Œ0; x p  p p or Œx0 ; x , but other choices of x are possible. Let E be the Banach space of functions z holomorphic on ˝  S1 such that there is a constant Z  0 satisfying jz.x; /j  Z jj for all .x; /. For z 2 E , let jjzjj denote the smallest of these constants Z. With  to be chosen properly, let M be the (non-empty) set of all z 2 E with norm jjzjj  . For z 2 M , we put p

p

.T z/.x; / D e x

p =p

p

Z

e 

p

p =p

Q.; z.; /; p / d :

(5.22)

x

A fixed point y of T is a solution of (5.11) defined on ˝  S1 which satisfies y.x0 ; / D 0. We want to show using the Banach fixed point theorem that T admits a fixed point in M , if 1 is small enough. We choose  D L0 sup jQj in the definition of M . Note first that T is well defined for all z 2 M , if 1 is small enough: it suffices that 1  < r2 . Then, using Lemma 5.9, we obtain j.T z/.x; /j  L0 jj sup jQj as z 2 M , .x; / 2 ˝  S1 . This implies T .M / M . Now let Z q.x; z1 ; z2 ; "/ D 2 Q.x; z1 ; z2 ; "/ D 0

1

@Q  @z x; z1

 C .z2  z1 / d :

(5.23)

Then Q.x; z2 ; "/  Q.x; z1 ; "/ D q.x; z1 ; z2 ; "/.z2  z1 / for all x 2 D1 , z1 ; z2 2 D2 and " 2 ˙. Reducing r slightly if necessary and using the Cauchy inequalities, we can assume that q is bounded, say by K. We obtain ˇ p pˇ Z ˇ p pˇ ˇ ˇ ˇ  = ˇ j.T z2  T z1 /.x; /j  ˇe x = ˇ ˇe ˇ K jz2 .; /  z1 .; /j jd j x

and thus using Lemma 5.9 again jjT z2  T z1 jj  L0 K1 jjz2  z1 jj for all z1 ; z2 2 M . If 1 satisfies also L0 K1 < 1, then T is a contraction in M and has therefore a (unique) fixed point in M . This proves the existence of the solution y on ˝ S1 and that its values are in the closure of D.0; jj/. For the analytic continuation of y to V1 ./ n ˝, we introduce Y .X; / D y.X; /. It satisfies the inner differential equation

5.2 CA sEs at a Turning Point: The Quasi-linear Case

97

Fig. 5.3 Left in the x-plane, right the images by F W x 7! x p ; here p D 2. In bold, the boundary of ˝. Right, the sector in dark gray is covered by two sheets of the image of ˝

dY D pX p1 Y C Q.X; Y; p / dX and takes a value jY .X0 ; "/j D O.jj/ for some small arbitrary X0 > 0. The theorem of dependence with respect to initial conditions and parameters then implies that Y is defined and takes values O.jj/ on the disk jX j  . This allows to continue y analytically to the set of all .x; / such that  2 S1 and x 2 V1 ./; we still have y.x; / D O./. t u Our last preliminary result is that two solutions of (5.11) on some domain are exponentially close one to each other on a smaller domain. e./ WD r 1 and set V Lemma 5.11. With the above notation, let r1 < e    V ˛; ˇ;e r 1 ; jj , V ./ D V ˛; ˇ; r1 ; jj . let u1 and u2 be two solutions of (5.11) satisfying u1 .x; /; u2 .x; / D O./ e ./. Then u1 and u2 are uniformly on the set of all .x; / such that  2 S1 ; x 2 V exponentially close on V ./. More precisely, there exists C > 0 such that for all  2 S and all x 2 V ./ ju1 .x; /  u2 .x; /j  C e  =jj : p

 p p with D e r 1  r1 cos.3ı/.

ˇ ˇ Proof. Let K denote a constant such that ˇuj .x; /ˇ  K jj for j D 1; 2 and for those .x; / in the assumption of the lemma. Put z D u1  u2 . Then z is a solution of equation   p z0 D px p1 C p g.x; / z (5.24)   p with g.x; / D 2 P x; u1 .x; /; u2 .x; /;  , where 2 P is determined by (5.23). Since the function P is bounded on D1  D2  S , the Cauchy inequalities show that

2 P is bounded on D1  D.0; Ke /  D.0; Ke /  S if e  is small enough, thus

98

5 Composite Expansions and Singularly Perturbed Differential Equations

  e./. g.x; / is bounded on all .x; / such that  2 e S D S  pı ; pı ;e  and x 2 V i arg x e Given  2 S and x 2 V ./, choose e x De r 1e if j arg xj < 2ı and e x De r1 otherwise. Put Z x G.x; / D exp

g.u; /du: e x A priori G is noncontinuous but since g is bounded, G is bounded. Equation (5.24) e./ gives, for  2 e S and x 2 V ˚  z.x; / D z.e x ; /G.x; / exp 1p .x p  e xp / : e./ and e By the choice of V ./; V x , one has for all  2 e S and all x 2 V ./,  ip arg  p    p Re e .x  e x p / < r1  je x jp cos.3ı/ D  if j arg xj < 2ı and   p x p / < r1 cos ı  je x jp cos.j arg xj  ı/ Re e ip arg  .x p  e   p x jp cos ı <  otherwise: < r1  je Thus there is a constant C such that jz.x; /j  C exp. =jjp / for these values  of .x; /. Increasing the constant if needed, this estimate is still valid as  2 S   ı ı . t u p ; p ; N and x 2 V ./, i.e. for jj  e Proof of Theorem 5.8: Now we are in a position to prove our main result of this section. By Lemma 5.10, there is a solution y.x; / D O./ of (5.11) as  2 S1  1=p   e./ WD V ˛; ˇ;e and x 2 V r 1 ; jj , if r1 < e r 1 < r0 cos.2ı/  . By Theorem 5.4,  x  n P there is also a composite formal solution of (5.11) b y D n0 an .x/Cgn   2   b r0 ; V .˛  ı ; ˇ C ı ; 1; e C / , with some e  > . p p  1=p Now fix N 2 N and let us choose rN0 < r0 such that e r1 < rN0 cos.2ı/  . By x n P 1  , the Lemma 2.7 applied to the partial sum b y N .x; / D N nD1 an .x/ C gn    p1 p p p 0 y N .x; / C  P x; b y N .x; /;    b y N .x; / has a function rN .x; / WD px b   CA sE as S1 3  ! 0 and x 2 VN ./ WD V ˛; ˇ; rN0 ; jj . As b y N is a partial sum of a formal solution, the coefficients of j vanish for j D 0; : : : ; N  1. In particular rN .x; / D O.N / uniformly for x 2 VN ./. Then we make the change of variable y D b y N .x; / C N p z and we obtain for z p 0 p1 p an equation of the form  z D px z C  QN .x; z; /, with QN .x; z; / bounded as jzj < Z,  2 S1 and x 2 VN ./ with some Z > 0. We apply Lemma 5.10 to this e./ equation: it has a solution zN .x; / D O./ as  2 SN WD S. pı ; pı ; N /, x 2 V for some N > 0. Hence we have shown that for all N 2 N, there are N and a solution yN .x; / e./ such that .yN  b of (5.11) defined for  2 S. pı ; pı ; N / and x 2 V y N /.x; / D N pC1 e O. / uniformly on V ./.

5.2 CA sEs at a Turning Point: The Quasi-linear Case

99

By Lemma 5.11, the solution y D y0 is exponentially close to each of the solutions yN , if  is small enough. This implies that it has a CAsE for  2  S  pı ; pı ; 1 and x 2 V0 ./. Changing the constants if necessary, the radius of the sector in  is inessential for a CAsE. t u

5.2.3 The Gevrey Character of

CAsEs

We show that the CAsE of the solution of Theorem 5.8 is of Gevrey order p1 in the sense of Definition 3.6. The key-theorem 4.1 is particularly well suited to achieve this result. The idea of proof is as follows: we first construct a family of solutions, one for each element of a certain consistent covering, then we show that these solutions are exponentially close to each other (cf. Lemma 5.13). More precisely, if the sector in " changes, the two solutions are on the same mountain,1 thus their difference is exponentially small of the form exp.˛=j"j/. If the sector in x changes, however, then the solutions are defined on two adjacent mountains and they have to descend into the valley separating them to become exponentially close. Their difference is thus of the form exp.˛jx p ="j/. It turns out that the estimates obtained correspond exactly to the conditions of Theorem 4.1. It seems difficult to prove this result directly, i.e. in the spirit of the former Sects. 5.2.1 and 5.2.2. Theorem 5.12. Consider Eq. (5.11). Then, for each k D 0; : : : ; p  1, there  are ; 1 > 0 and a solution y of (5.11) defined for  2 S1 WD S  pı ; pı ; 1 and   2ı 2k 3 2ı 2k x 2 Vk ./ D V ˛k ; ˇk ; r1 ;  jj , ˛k D  3 2p C p C p and ˇk D 2p  p C p . Moreover y has a CAsE of Gevrey order p1 . j

Proof. In a first step, we construct a family of functions .yl /, solutions of equations similar to (5.11), where the function P is replaced by functions Pm having the same Gevrey-1 expansion as P for " in a covering .˙m /0m<M . We then check that the j differences of the functions yl satisfy the exponential estimates needed to apply Theorem 4.1. During the proof, we will see that the functions Pm can be constructed j as functions of x and ", but that the solutions yl will be defined as functions of x and the variable  D "1=p . First consider a covering .˙m /0m<M of the punctured disk D.0; "1 / D D.0; "1 / n f0g. Let M 2 N sufficiently in particular  we must have 4  M ı.  large; 2.mC1/ . By Borel and truncated For 0  m < M , let ˙m D S 2.m1/ ; ; " 1 M M

Consider the landscape Rd given by (5.18) with d D p arg  D arg ". We call mountain a connected component of the set Rd > 0. A valley is a connected component of the set Rd < 0. See the lines above Corollary 5.16 for a description and a numbering of these mountains and valleys.

1

100

5 Composite Expansions and Singularly Perturbed Differential Equations

Laplace transforms (cf. the proof of Lemma 3.15) we construct on each sector ˙m a function Pm bounded and having the same Gevrey-1 asymptotic as P . We need to make one complete counterclockwise turn around  D 0, which corresponds to p turns in ". Thus, to the covering .˙m /0m<M of D.0; "1 / 1=p corresponds a covering .Sl / of D.0; 1 / , with 1 D "1 , as follows. 2 Let L D pM . For 0  l < L, we set 'l D l L ; ˛l D 'l  2 .D 'l1 /; ˇl D L 2 'l C L .D 'lC1 / and   ; .l C 1/ 2 ; 1 : Sl D S.˛l ; ˇl ; 1 / D S .l  1/ 2 L L Thus the image of Sl by the mapping F W  7! p is the sector ˙l mod M . We extend the family .Pm / by “cloning”: for l 2 fM; : : : ; L  1g, we  set Pl D Pl mod M . 3 For j 2 f0; : : : ; p  1g, let V j D V .˛ j ; ˇ j ; 1;  with ˛ j D j 2 p  2p C 3ı p;

3 3ı ˇ j D j 2 p C 2p  p and  > 0 small enough. Thus, as soon as ı <

ˇl  ˛l D

 8,

one has

ı   6ı 1 4  < D .ˇ j  ˛ j C1 /: L p 2p 2

From these sectors Sl and infinite quasi-sectors V j , we construct a consistent  j jgood j covering of resolution  pı by considering the quasi-sector Vl ./ D V ˛l ; ˇl ; r1 ;  jj with j

˛l D ˛ j C'l C pı D

2j 2l 3 4ı p C L  2p C p ;

j

ˇl D ˇ j C'l  pı D

2j 2l 3 4ı p C L C 2p  p :

To simplify the sequel, we consider the variables X and x only in sectors. The modifications for quasi-sectors will be pointed out at the end of the proof. More precisely, consider X in the sector S j D S.˛ j ; ˇ j ; 1/ and x in the sector   we j j j e Sl D S e ˛l ; e ˇ l ; r1 with j

j

e ˛ l D ˛l  2ı p D

2j p

3 2ı C 2l L  2p C p ;

j j e ˇ l D ˇl C 2ı p D

2j p

3 2ı C 2l L C 2p  p :

  j 2ı The sector e S l is bisected by 2 pj C Ll and has half-opening 3 2p  p . Let Sl;lC1 D j;j C1 j j Sl \ SlC1 and define e Sl and e S l;lC1 similarly. For  2 Sl , the sector e S l contains essentially the part at distance < r1 of the j th mountain of the landscape Rd given by (5.18) with d D arg " D p arg , and almost the whole two adjacent valleys. j;j C1 The intersection e Sl contains, in turn, much of the valley between the mountains j;j C1 and all  2 Sl , j and j C 1. In particular, one has for all x 2 e Sl ˇ ˇ ˇ.2j C 1/  p arg x ˇ < 

 2

 2ı:

(5.25)

5.2 CA sEs at a Turning Point: The Quasi-linear Case

101

 1=p  ˚  j j Let xl D r1 cos.2ı/ exp 2 i pj C Ll . It turns out that xl is on the bisector j j of e S l . Choose as yl the solution of the equation p y 0 D px p1 y C p Pl .x; y; p /: j

(5.26)

j

with initial condition yl .xl ; / D 0. As we can reduce this case to the one treated in the proof of Lemma 5.10 using j 1=p rotations of  and x, we obtain that yl is defined and bounded by "1 with some j j j  < r in ˝l  Sl , where ˝l contains the sector e S l and a small curvilinear triangle j j tangent to e S l and having a vertex at xl . Lemma 5.13. There exist constants C; A; B > 0 such that, for each .j; l/ 2 f1; ::; J g  f1; ::; Lg, the following inequalities hold ˇ ˇ  ˇ ˇ j j ˇylC1 .x; /  yl .x; /ˇ  C exp 



on e S l;lC1  Sl;lC1 ;

(5.27)

ˇ ˇp  ˇ ˇ j C1  j j;j C1 ˇy Sl .x; /  yl .x; /ˇ  C exp  B ˇ x ˇ on e  Sl : l

(5.28)

A jjp

j

j C1

j

Proof. Let us begin with the estimate (5.28) and put z D yl  yl . Then z satisfies the equation   p z0 D px p1 C p g.x; / z (5.29)   j j C1 with g.x; / D 2 Pl x; yl .x; /; yl .x; /; p , 2 defined in (5.23). Since the function Pl is bounded on D1  D2  S , the Cauchy inequalities show that 2 Pl is 1=p 1=p bounded R x on D1 D.0; "1 /D.0; "1 /S if "1 is small enough. Let G.x; / D exp 0 g.u; /du. It is a function on D1  S bounded by some C1 > 0. Equation j;j C1 (5.29) gives, for x 2 Sl and  2 Sl z.x; / D z.0; /G.x; / exp

˚ x p  :  1=p

Using (5.25), we obtain (5.28) with B D sin.2ı/ and C  2"1 C1 . j j Regarding the estimate (5.27), we set w D ylC1  yl . Then w is solution of the equation   p w0 D px p1 C p h.x; / w C p Q.x; / (5.30) with and

  j j h.x; / D 2 PlC1 x; yl .x; /; ylC1 .x; /; p

    j j Q.x; / D PlC1 x; yl .x; /; p  Pl x; yl .x; /; p : Rx Let H.x; / D exp 0 h.u; /du. It is a function bounded above by K and below by j j 1 for some K > 0. Let  2 @˝l;lC1 with arg  bisecting Sl;lC1 ; thus  is of the form K

102

5 Composite Expansions and Singularly Perturbed Differential Equations

Fig. 5.4 An exaggerated j sketch of the domains ˝l and j ˝lC1 and the point 

xjl+1

0 ξ xjl

     1=p   D r3 exp 2 i pj C lC1=2 ) with r3 > r1 (one finds r3 D r1 cos 2ı  M L j

(Fig. 5.4). The variation of the constant formula gives, for all x 2 Sl;lC1 ˚ p  p  H.x; / C w.x; / D w.; / exp x   H.; / Z x ˚ p  p  H.x; / Q.s; /ds: exp x  s H.s; / 

(5.31)

where the path of integration is chosen descending for the landscape Rd given by 2.lC1/

. (5.18) for all d 2 2l ; M M ˇ ˇ ˇ ˇ 2 Concerning the first term of (5.31), one has jw.; /j < 2r2 and ˇ H.x;/ H.;/ ˇ  K .

   j Moreover, for all x 2 Sl;lC1 and all d 2 2l , one has Re x p e id  ; 2.lC1/ M M ˇ ˚ p  p ˇˇ   p id  ˇ  A1 with A1 D r3 cos M  r1 , thus ˇ exp x   ˇ  Re  e   j A1 exp  jj for all x 2 Sl;lC1 and all  2 Sl;lC1 . The condition A1 > 0 is ensured p by the fact that 4  M ı. Concerning the second term of (5.31), the functions Pl and PlC1 are Gevrey1 asymptotic to the same series, thus there areC2 ; A2 > 0 such that for all x 2 j A2 Sl;lC1 and all  2 Sl;lC1 , jQ.x; /j  C2 exp  jj p . Since the path is chosen

 descending, we have, for all s on this path, all  2 Sl;lC1 and all d 2 2l , ; 2.lC1/ M M ˚ x p  s p   0. Thus we obtain Re     jw.x; /j  2r2 K 2 exp 

A1 jjp



 C .r1 C r3 /K 2 C2 exp 

A2 jjp

 ;

t which gives (5.27) with A D min.A1 ; A2 / and all C  .2r2 C .r1 C r3 /C2 /K 2 . u

5.2 CA sEs at a Turning Point: The Quasi-linear Case

103

End of proof of Theorem 5.12: With  chosen smaller if necessary, the solutions j j yl extend for x in the disk D.0; jj/, thus for x in Vl . Increasing the constant C j if necessary, inequalities (5.27) and (5.28) remain valid when x is in Vl . Now all j the conditions are met to apply Theorem 4.1; this implies that all the functions yl have a Gevrey CAsE. In particular, for each k 2 f0; : : : ; p  1g, the solution y D y0k has the wanted properties. t u

5.2.4 Remarks and Extensions 1. An immediate corollary of Theorem 5.12 is that there is a formal composite series solution of (5.11), and that this formal solution is of Gevrey order p1 in . In other terms, this yields another proof of Theorem 5.4. 2. Theorem 5.12 is valid in the slightly more general case where P is a function of the variable , of Gevrey order p1 in , instead of the variable ". 3. The landscape of equation (5.11) for arg  D 0 consists of a series of p    2k  mountains and p valleys corresponding to the sectors S 2k  ; p C 2p ; r0 , p 2p    2k respectively S 2k C 2p ; p C 3 ; r , k D 1; : : : ; p. We have shown that p 2p 0 a solution with a CAsE exists on a mountain and two adjacent valleys. One can deduce that any solution y of (5.11) with initial condition y.x1 / D y1 sufficiently small, at a point x1 on a mountain, is defined and has a CAsE for  with small argument and for x in the part of the mountain below x1 , in most of the two adjacent valleys and in a neighborhood of order  of the turning point 0.  More precisely, given such a point x1 ¤ 0 with a D 2k  2p < arg.x1 / < p  b D 2k p C 2p , let ı > 0 be such that 2ı < minfarg.x1 /  a; b  arg.x1 /g and let ˝ be the set of all x 2 D.0; x0 / accessible from x1 by a ı-descending path for the landscape Rd given by (5.18), for all d 2   ı; ıŒ. The same proof as in Lemma 5.10 shows that y is defined for  2 S.ı; ı; 0 / and x 2 ˝[D.0; Kjj/ if 0 and K are small enough, and an argument similar to that for the proof of (5.28) shows that y is exponentially close to the solution, here denoted by e y, given by Theorem 5.12, at any point x of ˝ sufficiently far from x1 , with a coefficient in the exponential given by minjd jı fRd .x1 /  Rd .x/g. Then with Proposition 3.13 (a), it is immediate that y has the same CAsE as e y. 4. The region ˝ contains only parts of one mountain and two adjacent valleys, but in fact the solution y extends also to large parts of all the other valleys. Indeed, a modification of the proof of Theorem 5.8 shows that y extends in each sector    2ı 2j 3 2ı Sj D S 2j C C ; C  ; r . This raises the question whether p 2p p p 2p p these continuations have CAsEs; we show below that this is true and that the fast coefficients of these CAsEs are the analytic continuations of those of Theorem 5.8 to other valleys.

104

5 Composite Expansions and Singularly Perturbed Differential Equations

5. More generally, let now e y be a solution of (5.11) with initial condition e y .0; / D u0 ./, where u0 W S D S.; ; 0 / ! C,  > 0 small enough,2 satisfies supS ju0 j  e r < r and has an asymptotic expansion b u0 ./ of Gevrey order p1 as  ! 0. As in the former remark and asSin the proof of Theorem 5.8, we show that e y .x; / extends in D.0; K jj/ [ j Sj with K > 0 and the former sectors Sj , as  2 S.˛; ˇ; 1 /, if 1 > 0 is small enough. This solution e y differs in Sj from two solutions given by Theorem 5.8 continued analytically p p from the adjacent mountains by a quantity of order O.e Bjxj =jj /, but this does not suffice to prove that e y has a CAsE. This was discussed in the remark below Proposition 3.13. We are led to use Theorem 4.1 again. We do this concisely following the model of the proof of Theorem 5.12. First we use sectors Sl ; l D 1; : : : ; L in  of this proof and construct on each Sl functions Pl and u0l with the same Gevrey asymptotics as P , resp. u0 . Let e yl denote the solution of p y 0 D px p1 y C p Pl .x; y; p / with initial condition e y l .0; / D u0l ./ as  2 Sl . We reduce the V j almost to the mountains: V j D  4ı 2  4ı j V .˛ j ; ˇ j ; 1; / with ˛ j D j 2 p  2p  p ; ˇ D j p C 2p C p and  > 0 small enough. We complete these V j in a covering of C with the quasi-sectors   ej D V .e V ˛j ; e ˇ j ; 1; /, with e ˛ j D j 2 C 2p C pı and e ˇ j D .j C1/ 2  2p  pı . p p ej has a non-empty intersection with V j and V j C1 . As in the proof of thus V j ej to these Sl , V j ; V ej Theorem 5.12, one associates the quasi-sectors Vl and V l and thus obtains a consistent good covering. ej , j D Similarly to e y , the solution e y l extends analytically in the union of V l j ej . In V j , we consider the (restrictions 1; : : : p; let e y l denote its restriction to V l l j of) solutions yl of the proof of Theorem 5.12. We show in the same manner as before that estimates similar to Lemma 5.13 are satisfied and thus 4.1  P Theorem a can be applied. We deduce that e y .x; /  1 b .x/ C e y .x; / WD n n0 p  x  n e g n   has a CAsE of Gevrey order p1 in each Sj . Here a0 .x/ D 0 and b e y .x; / is the composite formal solution of (5.11) with initial condition b e y .0; / D b u0 ./, where b u0 ./ is the series associated to u0 ./. In particular, e g 0 .X / is the solution p of Y 0 D pX p1 Y with Y .0/ D u0 .0/, i.e. e g 0 .X / D u0 .0/e X . For the continuation of a solution y in the valleys discussed in item 4, we thus obtain that they have the same CAsE, i.e. that their coefficients are the analytic continuations of the coefficients of the CAsE of y in ˝. 6. It was not our intention to present the best possible constants A and B in the proof of Lemma 5.13. The constant B D sin.2ı/ can be greatly improved if we consider the solutions on smaller quasi-sectors; if we reduce the opening angle to 2=p C ı=p, we can obtain B D cos ı and thus a constant as closed to Other arguments for  can be reduced to this case by rotations x D e i' . If we want a sector of greater opening in , we cover it by small sectors in  and consider the intersection of the corresponding sectors in x . One can consider an initial condition at a point x D L, L 2 C instead of x D 0, in which case we first study the existence and the asymptotics of the corresponding solution at x D 0 using the inner equation.

2

5.2 CA sEs at a Turning Point: The Quasi-linear Case

105

the optimum 1 as desired. The constant A2 depends on the Gevrey type of the function P and cannot be improved. The constant A1 depends on the point x. According to the data in the proof, this choice becomes worse if r1 tends to r0 , j but for fixed r0 it is possible to choose the points xl with modulus close to r0 , which allows to recover a constant A1 as close to r0  r1 as possible. The optimal constant A depends on the point x and on global properties of the differential equation, outside the local setting we chose in this memoir. 7. We have chosen to present the theory for a small "-sector S.ı; ı; "0 /, because this is close to some applications where we need only positive values of ". Of course, the result remains valid for any other "-sector of small opening: just change arg " and arg x using a rotation. Moreover, it is possible to deduce results on large sectors from Theorem 5.12. For example, if P .x; y; "/ is real for real x; y; ", one can prove, for all ı > 0 small, the existence of the  solution of (5.11) with initial condition 0 at x D r0 for x 2 Œ0; r0  and " 2 S  2 C ı; 2  ı; "1 , if "1 > 0 is small enough. Choose r1 2 0; r0 Œ; since this solution is exponentially close on Œ0; r1 to a solution of the

theorem for a small subsector jarg "  'j < ı, whatever ' 2  2 C 2ı; 2  2ı ,    it also admits a CAsE for x 2 Œ0; r1 , " 2 S  2 C ı; 2  ı; "1 . This will be useful for the study of global canards in Sect. 6.1. 8. The general form of an equation called quasi-linear is "v0 D f .x/v C "P .x; v; "/

(5.32)

with P analytic in D  D.0; /  S for all  > 0, if "0 is small enough. We can apply Theorem 5.12 by making 4.7.  R xa change in the variable x and using Theorem Indeed, the function F .x/ D 0 f .s/ ds satisfies F .x/ D x p 1CO.x/ ; x ! 0, so there is an analytic function h with h.0/ D 0 and h0 .0/ D 1, such that F .x/ D h.x/p . Let ' D h1 be the inverse diffeomorphism; thus we have F '.t/ D t p . The change of variable x D '.t/ then leads to the equation (setting w.t; "/ D v.'.t/; "/) dw e .t; w; "/ D pt p1 w C "P (5.33) " dt e .t; w; "/ D ' 0 .t/P .'.t/; w; "/. By Theorem 5.12 and Remark 3, a solution with P w of (5.33) with initial condition w.t1 / D w1 sufficiently small at a point t1 on a mountain, is defined and has a CAsE for  with small argument and for t in the part of the mountain below t1 , in most of the two adjacent valleys and in a neighborhood of order jj of t D 0. Theorem 4.7 (c) then yields the following result, given without proof. The description and the numbering of the mountains and valleys are detailed above Corollary 5.16. Proposition 5.14. Let j 2 f1; : : : ; pg and x1 be a point of the j -th mountain Mj . Let ı > 0 be arbitrarily small (in particular such that j arg F .x1 /j < 2  2ı).

106

5 Composite Expansions and Singularly Perturbed Differential Equations

Let .x1 / denote the part of Vj 1 [ Mj [ Vj bounded by curves of equation j arg F j D 3  2ı and j arg F  F .x1 / j D 2 C 2ı, cf. Fig. 5.5. Let v be a solution 2 of (5.32) with initial condition v.x1 / D v1 sufficiently small. Then, for ˛; 0 > 0 small enough, v is defined for  2 S D S.ı; ı; 0 / and x 2 .x1 / [ D.0; ˛jj/. Moreover v has a Gevrey CAsE for  2 S and x 2 .x1 /[D.0; ˛jj/; jx x1 j > ı. As in Remarks 4 and 5, v extends to other valleys; it also has a CAsE in these valleys, which is the same as the one calculated there.

5.3

CAsE

at a Turning Point: A Generalization

We want to apply the CAsEs to equations more general than (5.11). Consider again Eq. (5.1) "y 0 D ˚.x; y; "/ with ˚ analytic with respect to x and y in a domain D C2 and of Gevrey order 1 with respect to " in the sector S D S.ı; ı; 0 / It is assumed that the slow set L contains the graph of a slow function y0 , analytic in a simply connected domain D. Thus one has for all x 2 D, x; y0 .x/ 2 D and ˚.x; y0 .x/; 0/ D 0. Recall the notation f .x/ D @˚ @y .x; y0 .x/; 0/. The important assumption we make at present is that D contains a turning point x  , i.e. that f vanishes at x  . It is a restrictive assumption because usually the slow function y0 has a branch point at a turning point; other singularities of y0 are also possible, e.g. poles. Here we assume that y0 remains regular. Using a shift, a rotation and a scaling of the variable x, we can reduce to x  D 0  p1 1 C O.x/ as x ! 0, for some p 2 N; p  2. Let F denote and to f .x/ D px the antiderivative of f vanishing at 0 and let R denote the real part of F . Replacing y by y0 C y, we are lead to the equation   "y 0 D f .x/ C "g.x; "/ y C "h.x; "/ C y 2 P .x; y; "/

(5.34)

  with h.x; "/ D 1" ˚.x; y0 .x/; "/˚.x; y0 .x/; 0/ y00 .x/ where g and P are given by

2 ˚.x; y0 .x/; y0 .x/ C y; "/ D f .x/ C "g.x; "/ C yP .x; y; "/I Recall the notation 2 ˚: ˚.x; y2 ; "/  ˚.x; y1 ; "/ D 2 ˚.x; y1 ; y2 ; "/ .y2  y1 /: The change of unknown y D "u yields the equation   "u0 D f .x/ C "g.x; "/ u C h.x; "/ C "u2 P .x; "u; "/

(5.35)

5.3 CA sE at a Turning Point: A Generalization

0

107

0

x1

F (x1 )

Fig. 5.5 The domain .x1 / and its image by F ; this image is a sector of opening 3  4ı except a sector with vertex F .x1 / and opening  C 4ı. Here F .x/ D x 2 C 1C0;7i x3 4

The new slow function u0 is obtained by putting " D 0 in (5.35), i.e. u0 D h0 =f , where h0 W x 7! h.x; 0/. In general, this slow function has a pole at x D 0, but in the particular case where h0 has a zero of order at least equal to that of f at 0, u0 is again regular at x D 0. The change of unknown u D u0 C v then leads to the quasi-linear equation (5.32). For the case where h0 =f has a pole at x D 0; we present the following theorem. The reduced inner equation corresponding to (5.34) will no longer be linear. The price to pay for this generalization is that the solution is a priori not defined in a disk of radius proportional to  around 0. This is the main reason why we introduced the quasi-sectors with  < 0. As before, we first assume that f .x/ D px p1 and generalize later. We recall the notation of the beginning of Sect. 5.2, Lemma 5.10 and Theorem 5.8: " D p , r0 ; r2 ; "0 ; ı > 0, ˙ D S.ı; ı; "0 /, D1 D D.0; r0 /, D2 D D.0; r2 /, ˛ D  3 C 2ı , 2p p  

 1=p 3 2ı ˇ D 2p  p , r1 2 0; r0 cos.2ı/ . The form of equation (5.34) is slightly modified below: the term P contains also g.x; "/. Theorem 5.15. Consider equation "y 0 D px p1 y C "h.x; "/ C y P .x; y; "/

(5.36)

with h and P analytic and bounded in D1  ˙, resp. D1  D2  ˙ having a uniform asymptotic expansion of Gevrey order 1 as ˙ 3 " ! 0. Assume that there exists r 2 f1; : : : ; p  1g such that, on the one hand h.x; 0/ D O.x r1 /; x ! 0 and on the other hand one has an expansion P .x; y; 0/ D

X

pkl x k y l :

k0;l1; kCrlp1 r Then there exist  2 R, 1 > 0 and a solution y.x;  / D O. / of (5.36) defined ı ı for  2 S1 WD S  p ; p ; 1 and x 2 V ./ D V ˛; ˇ; r1 ; jj . Moreover, y has a CAsE of Gevrey order p1 as S1 3  ! 0 and x 2 V ./.

108

5 Composite Expansions and Singularly Perturbed Differential Equations

Remarks. 1. The condition on P is equivalent to the condition that P .x; 0; 0/ D 0 and the -valuation of the series P . X; r Y; 0/ is at least p  1. Another equivalent condition is that we can write yP .x; y; "/ D

q1 X

x p1r` P` .x/y `C1 C y qC1 Q0 .x; y/ C "yQ1 .x; y; "/ (5.37)

`D1

with holomorphic functions P` .x/, Q0 .x; y/; Q1 .x; y; "/ on D1 , resp. D1  D2 and D1  D2  ˙, where q denotes the smallest integer such that qr  p  1. The inner equation, obtained by putting x D X , y D r Y , is e .X; Y; / Y 0 D pX p1 Y C P with

e .X; Y; / D 1r h.X; / C 1p Y P .X; r Y; p /: P

In the limit  ! 0, we obtain thus the nonlinear equation Y 0 D pX p1 Y C r1 cXX CY Q.X; Y /, where c is the coefficient of x r1 in h.x; 0/ and Q.X; Y / D pkl X k Y l . kCrlDp1

2. The example below shows that the condition imposed on P is necessary and natural. This is the equation "y 0 D 4x 3 y  4"  xy 2 :

(5.38)

Thus we have p D 4, r D 1 and p11 ¤ 0. We show at the end of Chap. 5 that this equation cannot have a solution admitting a CAsE. Proof of Theorem 5.15: It is a modification of that of Theorem 5.8—we only present the proof using the key-theorem 4.1. We first show the existence of a solution y, j then that of a family yl on a good covering and finally that their differences are   exponentially small. We conclude using Theorem 4.1. Let S1 D S  pı ; pı ; 1 with a small 1 > 0 to be determined. The domain in x is different from that of Lemma 5.10, because it does not a priori contain x D 0. Let x0 D r1 cos.2ı/1=p . For m > 0 small enough, we construct a domain ˝.m/ as follows. We consider the quasi-sector V .˛; ˇ; r1 ; m/ and we add the interior of the curvilinear triangle T .r1 ; ı/ of the proof of Lemma 5.10.3 Then we remove from this union two curvilinear triangles: one whose image by the map F W x 7! x p is mp i. 3 2 3ı/ , and another whose image by F is the triangle of vertices 0; mp e 2ı ; sin ı e p m i. 3 2 3ı/ . Let ˝.m/ denote the result; see the triangle of vertices 0; mp e C2ı ; sin ı e Fig. 5.6. As in the proof of Lemma 5.10, the choice of the domain ˝.m/ is motivated

3

p

p

p

i.e. the set whose image by F is the triangle with vertices x0 ; r1 e 2ıi and r1 e 2ıi .

5.3 CA sE at a Turning Point: A Generalization

109

p

0

r1

r1

x0

m

p

x0

mp

Fig. 5.6 The domain ˝.m/ and its image by F W x 7! x p ; here p D 2. As in Fig. 5.3, the part in dark gray is covered by two sheets of the image of ˝.m/

by the fact that it contains V .˛; ˇ; r1 ; m.sin ı/1=p / and that it is ı-descending from x0 with respect to the landscape Rd given by (5.18) for all jd j < ı. The second property can be expressed as follows: for all x 2 ˝.m/ there is a path  W Œ0; ` ! ˝.m/ [ fx0 g, j 0 .t/j D 1 for all t, from x0 to x such that for all t 2 Œ0; ` and all d 2   ı; ıŒ inequality (5.19), i.e.   Re .t/p1  0 .t/e id  j.t/p1 j is satisfied with  D sin ı. Let x denote such a path; Lemma 5.9 can be applied. Let D  1 be a real number to be chosen later. Let E be the Banach space of functions z holomorphic on all .x; / with  2 S1 , x 2 ˝.Djj/ for which there is a constant Z such that jz.x; /j  Z jjr for all .x; /. For z 2 E , we define jjzjj as the smallest of these constants Z. With  to be chosen properly, let M be the (non-empty) set of all z in E of norm jjzjj  . For z 2 M , we set Z p p 1 x p =p .T z/.x; / D p e e  = p h.; p /C (5.39) x  z.; /P .; z.; /; p / d : A fixed point y of T is the unique solution of (5.36) satisfying limx!x0 y.x; / D 0 for all . Formula (5.37) implies the existence of constants C0 ; : : : ; Cq such that jp h.; p / C z.; /P .; z.; /; p /j  C0 jjr1 jjp C q1 X `D1

C` `C1 jjp1r` jjr.`C1/ C Cq jjp1Cr . C qC1 /;

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5 Composite Expansions and Singularly Perturbed Differential Equations

because z 2 M . Since jx .t/j  D jj, we deduce using Lemma 5.9 that j.T z/.x; /j 

K./ r jj pD

Pq1 with K./ D C0 C `D1 C` `C1 C Cq . C qC1 /, if D  1. We therefore choose  > 0 arbitrary and D  1 large enough such that K./  pD. Our estimate then ensures that T W M ! M . Now we show that T is a contraction, under an additional condition for D. Let q.x; z1 ; z2 ; "/ be the holomorphic function such that z2 P .x; z2 ; "/  z1 P .x; z1 ; "/ D q.x; z1 ; z2 ; "/.z2  z1 /, i.e. the 2 -image of the mapping .x; z; "/ 7! zP .x; z; "/. Then q.x; 0; 0; 0/ D 0 for all x and one can expand q.x; z1 ; z2 ; 0/ similarly to P : X

q.x; z1 ; z2 ; 0/ D

qk;`1 ;`2 x k z`11 z`22 :

(5.40)

kCr.`1 C`2 /p1

This implies the estimate jq.x; z1 .x; /; z2 .x; /;  /j  p

q1 X

e ` ` jxjp1r` jjr` C C e q jjp1 .1 C q / C

`D1

e ` , if z1 ; z2 2 M . As before, this implies with some constants C jjT z2  T z1 jj 

(5.41)

e K./ jjz2  z1 jj D

Pq1 q e e e e ` with K./ D `D1 C `  C C q .1 C  /. If D also satisfies K./ < D, then T is a contraction on M . This yields the existence of the solution y stated in the theorem on the set of all .x; / with  2 S1 , x 2 ˝.Djj/ and thus also for x 2 V .˛; ˇ; r1 ;  jj/ with  D D= sin ı. In the next step, we use the same numbers L; M , the same angles 'l ; ˛l ; ˇl ; ˛ j ; j j j j j ˇ ; ˛l ; ˇl ;e ˛l ; e ˇ l and the same sectors Sl , l D 1; : : : ; L, j D 0; : : : ; p 1 as in the proof of Theorem 5.12. By Borel and truncated Laplace transforms, we construct functions hl .x; /, Pl .x; / as in that proof having the same Gevrey asymptotic expansion as the given h and P . Instead of the x-sectors of the previous proof, we use the quasi-sectors V j D j j ej ./ D V e V .˛ j ; ˇ j ; 1; / and V ˛l ; e ˇ l ; r1 ; Djj.sin ı/1=p and corresponding l j sets ˝l .Djj/ (cf. Fig. 5.6), with D  1 to be determined. For each of the differential equations "y 0 D px p1 yC"hl .x; /CyPl .x; y; "/—satisfying the same conditions as (5.36) in the assumption of the theorem — and all j 2 f0; : : : ; p  1g, j j we construct the solution yl on all .x; / with  2 Sl ; x 2 ˝l .Djj/ in the same way as y above. The number D is chosen such that this is possible for all .j; l/.

5.3 CA sE at a Turning Point: A Generalization

111

Now we show analogously to Lemma 5.13 that their differences are exponentially small.4 We have to prove that there are constants C; A; B > 0 such that for each .j; l/ one has ˇ ˇ   ˇ j ˇ j ej ./; (5.42) ˇylC1 .x; /  yl .x; /ˇ  C exp  jjAp if  2 Sl;lC1 ; x 2 V l;lC1 ˇ ˇp  ˇ ˇ j C1  j ˇy ej;j C1 ./; .x; /  yl .x; /ˇ  C exp  B ˇ x ˇ if  2 Sl ; x 2 V l l where, as previously,

ej V l;lC1

and

ej;j C1 V l

(5.43)

denote the corresponding intersections. j C1

j

 yl . Then z satisfies the Let us start with the estimate (5.43) and let z D yl equation   p z0 D px p1 C g.x; / z (5.44) j

j C1

with g.x; / D ql .x; yl .x; /; yl .x; /; p /, where ql is the analogue of q, satisfying z2 Pl .x; z2 ; "/  z1 Pl .x; z1 ; "/ D ql .x; z1 ; z2 ; "/.z2  z1 /. Analogously to (5.41), we obtain jg.x; /j 

q1 X

ˇ ˇ CN ` ` jxjp1r` jjr` C CN q jjp1 .1 C q / DW jjp1 h ˇ x ˇ (5.45)

`D1

with some constants CN ` and hence with some polynomial h of degree at most p  2. This implies ˚  p ˇ ˇ jz.x; /j  K exp Re x C H ˇ x ˇ where H is the antiderivative of h vanishing at 0 and where ˇ ˇK > 0 is a constant. As the degree of H is at most p  1, this implies (5.43), if ˇ x ˇ is large enough, i.e. if  is large enough. Concerning the estimate (5.42), we proceed analogously to the proof of (5.27) j j and we set w D ylC1  yl . Then w satisfies the equation   p w0 D px p1 C e g .x; / w C Q.x; / with

(5.46)

  j j e g .x; / D qlC1 x; yl .x; /; ylC1 .x; /; p

and      j j j Q.x; / D p .hlC1  hl /.x; / C yl .x; / PlC1 x; yl .x; /; p  Pl x; yl .x; /; p :

R   e / D exp p x e Let G.x; to (5.45), we show the exis0 g .u; /du . Similarly ˇ ˇ   1 G.x; /ˇ  tence of constants L and M such that L exp  M jj1p  ˇe   j L exp M jj1p . Again let  2 @˝l;lC1 .Djj/ at a maximal distance from 0 and

4

Instead of integration from 0 to x, however, we must use integration from some point  to x.

112

5 Composite Expansions and Singularly Perturbed Differential Equations

 ˚  j with arg  bisecting Vl;lC1 ./; thus  is of the form  D r2 exp 2 i pj C lC1=2 L    1=p with r2 > r1 (one finds r2 D r1 cos 2ı  M ). Variation of constants gives, for j all x 2 Vl;lC1 ./ w.x; / D w.; / exp

˚ x p

p



Z



e /   p  G.x; C  e / G.;

x

exp

˚ x p







(5.47)

e /  s p  G.x; Q.s; /ds:  e / G.s;

where the path of integration is chosen such that it descends the landscape Rd given  2.lC1/

. by (5.18) for all d 2 2l ; M M Concerning the first term of (5.47), one has ˇ G.x; ˇ ˇ e / ˇ 1p jw.; /j < 2r and ˇ ˇ  L2 exp.2M jj /: e / G.; j

Moreover, for all x 2 Vl;lC1 ./ and all d 2

 2l M

; 2.lC1/ , one has M

    Re x p e id  Re  p e id  A1 with A1 D r2 cos

 M

 r1 , thus ˇ  ˚ p   p ˇˇ ˇ ˇ exp x   ˇ  exp  j

A1 jjp



for all  2 Sl;lC1 and all x 2 Vl;lC1 ./. This implies that this first term is exponentially small. Concerning the second term of (5.47), the functions hl and hlC1 , respectively Pl and PlC1 , are Gevrey-1 asymptotic to the same series, thus there are C2; A2 > 0 j A2 such that for all  2 Sl;lC1 and all x 2 Vl;lC1 ./, jQ.x; /j  C2 exp  jj p .  2l 2.lC1/

Since the path is chosen descending for all d 2 M ; M , we have, for all s ˚ x p  s p   0. We obtain that the second on this path and all  2 Sl;lC1 , Re      2 p term is bounded by .r1 C r2 /C2 L exp  A2 jj C 2M jj1p ; and thus is also exponentially small if 1 is small enough. This finally proves (5.42). We conclude this proof as that of Theorem 5.12 by applying the key-theorem 4.1. t u Let us go back to the general equation (5.34) and recall that F is the antiderivative vanishing at 0 of the coefficient f and that R is its real part. The associated landscape, i.e. the graph of R W C ' R2 ! R, defines a series of p mountains Mj ; j D 0; : : : ; p1 where R > 0 and p valleys Vj where R < 0, delimited by the separatrices of the saddle point R D 0. More precisely, these mountains and valleys are subsets of C and thus the projections of the parts of the landscape usually bearing

5.3 CA sE at a Turning Point: A Generalization

113

these names. Thus we have for example M0 [    [ Mp1 D fx 2 D I R.x/ > 0g. We number these mountains and valleys so that each is connected, that M0 contains a part of the real positive axis, and that they alternate Mj ; Vj ; Mj C1 (modulo p). Thus in a neighborhood of x D 0, Mj is “tangent” to the sector      2j   2j S 2j  2p ; p C 2p ; 1 , and Vj is tangent to the sector S 2j C 2p ; p C 3 ;1 . p p 2p As a corollary of Theorem 5.15, we prove that for each j D 1; : : : ; p, there is a solution of (5.34) having a Gevrey CAsE for any quasi-sector included in Mj [ Vj [ Vj 1 . The meaning of r0 ; r1 ; r2 ; "0 ; ı; D1 ; D2 and ˙ is the same as in Theorem 5.15. Because of the change of the variable x in the proof, the x-sector V ./ has to be modified. Again we incorporate the term g.x; "/ into P . Corollary 5.16. Consider equation "y 0 D f .x/y C "h.x; "/ C y P .x; y; "/

(5.48)

with f analytic in D1 , f .x/ D px p1 CO.x p / as x ! 0 and with h and P bounded analytic in D1  ˙, resp. D1  D2  ˙ and each having a uniform asymptotic expansion of Gevrey order 1 as ˙ 3 " ! 0. Assume that there is r 2 f1; : : : ; p  1g such that, on the one hand h.x; 0/ D O.x r1 /; x ! 0 and on the other hand one has an expansion on D1  D2 X

P .x; y; 0/ D

pkl x k y l :

k0;l1; kCrlp1

Finally, assume that ˛ < ˇ, j and r3 > 0 are such that S.˛; ˇ; r3 / .Vj 1 [ Mj [ Vj / \ D.0; r1 /, and that ı < p6 .ˇ  ˛/. Then, there are  2 R, 1 > 0 and a solution y.x; / D O.r / of (5.48) defined  for  2 S1 WD S. pı ; pı ; 1 / and x 2 V ./ D V ˛ C 3ı ; ˇ  3ı ; r3  ı; jj . p p Moreover y has a CAsE of Gevrey order p1 as S1 3  ! 0 and x 2 V ./. Proof. There exists a function ' with '.u/ D u e 2 ij=p C O.u2 / such that   F '.u/ D '.up /. For  > 0 small enough, ' is a diffeomorphism from D.0; / onto its image. The change of variable x D '.u/ then transforms (5.48) into (5.36) (with the independent variable denoted u). We apply Theorem 5.15  the  and obtain existence of a solution z.u; / having a CAsE for  2 S1 WD S  pı ; pı ; 1 and   u2V e ˛; e ˇ; r1 ; e jj where e ˛ D  3 C 2ı , e ˇ D 3  2ı and 1 > 0, e  2 R. 2p

p

2p

p

Now we apply Theorem 4.7 (c) to the change of variable u D ' 1 .x/. We use  > 0 small enough such that the image of V ˛ C 3ı ; ˇ  3ı ;  ; jj by ' 1 p p 2   is contained in V e ˛; e ˇ; ; e jj , for some  2 R. This is possible because the 2j 3 3 condition of the statement implies 2j p   2p  ˛ < ˇ  p C 2p . Thus we obtain  3ı 3ı  a CAsE in V ˛ C p ; ˇ  p ; 2 ; jj . To extend this CAsE to x 2 V ./ such that  t u 2 < jxj < r3  ı, we use Proposition 3.9.

114

5 Composite Expansions and Singularly Perturbed Differential Equations

Remarks. 1. We compute the composite formal series for the CAsE of the corollary similarly to the end of Sect. 5.2.1. For the slow part, we determine as before the non-polar part at x D 0 of the formal outer solution of (5.48). To compute the fast part, we go over to the inner equation by putting x D X and y D r Y , as was done for Eq. (5.36) in the remark following Theorem 5.15. With the notation f .x/ D px p1 C x p f1 .x/ and h.x; "/ D cx r1 C x r h1 .x/ C "k.x; "/, one has   dY D pX p1 C X p f1 .X / Y C cX r1 C dX M.X; / C 1p Y P .X; r Y; p /

(5.49)

with M.X; / D X r h1 .X / C pr k.X; p /. The condition on P in the corollary implies that the last term of (5.49), 1p YP .X; r Y; p /, is bounded as X and Y are in compacts, uniformly with respect to . From this assumption, defining X Y Q.X; Y / WD pk;l X k Y l (5.50) kCrlDp1

as the quasi-homogeneous part of lowest degree of P , we also obtain 1p YP .X; r Y; p / D Y 2 Q.X; Y / C YP1 .X; Y; /: The inner equation has the following limit as  ! 0 dY D pX p1 Y C cX r1 C Y 2 Q.X; Y /: dX

(5.51)

In the theory of irregular singular points, it is shown that (5.51) hasa unique solution Y0 .X /   pc X rp as jX j ! 1 in a quasi-sector V D V  3 2p C  3 ; 2p  ; 1;  with  > 0 arbitrary and  < 0, jj large enough. The function Y0 has an expansion of Gevrey order p1 Y0 .X / 

X

dl X rpl as V 3 X ! 1:

l1

b.X; / D The inner solution (on V ) is computed by plugging Y P1complete formal n nD0 Yn .X / in (5.49). Thus, Yn .X / for n  1 is determined as the solution of n polynomial growth of a linear non-homogeneous equation of the form dY D dX p1 pX Yn C hn .X /, where hn contains terms of f1 ; M and P as well as Y fast part of the CAsE of Corollary 5.16 on V , i.e. for j D 1, is P0 ;1: : : ; Yn1 . The nCr g .X / with the non-polynomial parts gn of Yn . In order to compute n nD0 the fast parts of the CAsEs for other values of j , use a rotation X Xe 2j i=p .

5.3 CA sE at a Turning Point: A Generalization

115

2. Theorem 5.15 and Corollary 5.16 have the disadvantage of not containing information on the domain of validity of the CAsE, in particular the number  is unknown. The following statement closes this gap using information about the solution Y0 of (5.51). As it is one of our most important results, we present it as a theorem, even though it is a corollary of Theorem 5.15 (and a few other results). Theorem 5.17. Under the conditions of Corollary 5.16, assume that the solution Y0 of the reduced inner equation (5.51) satisfying Y0 .X /   pc X rp as V 3 X ! 1  i i  pı ; e ˇ  2j C can be continued on a neighborhood of the closure of V e ˛  2j p p  ı e e  with some e ˛ ; ˇ such that ˛ < e ˛ < ˇ < ˇ and e  2 R. Then the solution y p ; 1; e of Corollary 5.16 can be continued analytically and has a CAsE of Gevrey order p1   ˛; e ˇ;e r 1; e on the set of all .x; / such that  2 S  ı ; ı ; 1 and x 2 V .e  jj/. p

p

We present a particular case separately. Corollary 5.18. Under the conditions of Corollary 5.16, assume that pkl D 0 if k C rl D p  1. Then, for all e  2 R, there exist 1 > 0 and a solution y.x; / of (5.48) defined for  2 S1 WD S. pı ; pı ; 1 / and x 2 V ./ D V ˛; ˇ;e r 1; e  jj . 1 Moreover y has a CAsE of Gevrey order p as S1 3  ! 0 and x 2 V ./. Proof of Theorem 5.17: According to Corollary 5.16, the solutions y exist and   r 1 ;  jj/ with some have Gevrey CAsEs as  2 S  pı ; pı ; 1 and x 2 V .˛; ˇ;e  2 R, possibly negative, and such that jj is large. Let us fix such a solution y. By Proposition 3.7, Y .X; / WD y.X; / has P a uniform asymptotic expansion n of Gevrey order p1 of the form Y .X; /  1 n0 Zn .X / on compacts of p 2j i     V ˛  pı ; ˇ C pı ; 1;  . Here in particular Z0 .X / D Y0 Xe  p .   Choose X0 2 V e ˛  ı ;e ˇ C ı ; L;  with L > . Then in particular Y0 ./ WD p

p

Y .X0 ; / has an asymptotic expansion of Gevrey order p1 . e of the inner equation (5.49) with initial condition Now consider the solution Y  e Y .X0 ; / D Y0 ./. On the one hand, it coincides with Y on the compacts of V ˛   ı ; ˇ C pı ; 1;  . On the other hand, it reduces to the solution Z0 of (5.51) as  p tends to 0 since Y0 .0/ D Z0 .X0 /. Now from the assumption of the theorem, Y0 can be continued analytically and thus also Z0 can be continued analytically to a neighborhood of the closure of     ˝DV e ˛  pı ; e ˇ C pı ; 1; e  n V ˛  pı ; ˇ C pı ; 1; L : As (5.49) is regularly perturbed (on compacts), the theorem of analytic dependence e with respect to parameters and initial conditions  that Y .X; / is holomor ı ıimplies 1 > 0 is small enough. 1 , if e phic for X in the closure of ˝ and  2 S  p ; p ;e e has a uniform asymptotic expansion of Gevrey order 1 Moreover, we obtain that Y p on this closure.

116

5 Composite Expansions and Singularly Perturbed Differential Equations

e coincide on an open subset of C2 , each one is an analytic As Y and Y continuation of the other. The assumptions of Proposition 3.8 about continuation of CA sE s by using an extended inner expansion are thus satisfied and we can conclude. t u Proof of Corollary 5.18: In this case, Eq. (5.51) is linear and the condition of Theorem 5.17 is obviously satisfied. t u Remarks. 1. In his work [42–44], Matzinger proves the existence of solutions with inner and outer expansions for a more general differential equation, which in particular may contain poles at x D 0, cf. e.g. Theorem 1 of [44]. By our Proposition 2.17, we obtain the existence of CAsEs for these solutions. This naturally raises the question of Gevrey character of these CAsEs, and if we can prove this character analogously to the proof of our Theorem 5.15. 2. In the same way as (5.36), we can treat the following equation, in which the right hand side has an expansion in powers of  instead of ". It is p y 0 D f .x/y C h.x; / C yP .x; y; /;

(5.52)

where f .x/ D px p1 C O.x p / as before, h and P have Gevrey- p1 asymptotic expansions and there exists r 2 f1; : : : p  1g such that their coefficients hj .x/ and Pj .x; y/ of j satisfy the conditions hj .x/ D

X

hj l x lCpCr1j

l0

P0 .x; 0; 0/ D 0 and for j D 0; : : : ; p  2 X

Pj .x; y; 0/ D

pj kl x k y l :

k0;l0; kCrlp1j

If jh00 j is small enough, then the equation p C h00 C

X

p0;k;l l D 0

kCrlDp1

has a simple root P

small in absolute value. Then there exists a unique j r rC1 transformation y D jr1 / reducing D0 aj .x/ C z with a0 .x/ D x C O.x (5.52) to an equation of the same form, satisfying the same conditions and satisfying moreover h0 D : : : D hr1 D 0. For the latter equation, we show the existence of a solution z.x; / D O.r / for  2 S1 and x 2 V ./ with S1 ; V ./ as in Corollary 5.16; this immediately implies the same statement for y. Notice however that the formal solution in this case can be very different from that of Corollary 5.16; for instance each power of  may contain a “slow” factor.

5.3 CA sE at a Turning Point: A Generalization

117

3. All equations in this Chap. 5 could depend analytically (or even only continuously) on additional parameters. The proofs using the fixed point theorem and (in Theorem 4.1) using integral formulas then show that the solutions depend analytically (resp. continuously) on these parameters and their Gevrey CAsEs are uniform with respect to these parameters. 4. We come back to example (5.38). On the positive real axis, one can see that there exists a unique solution y C tending to 0 as x ! C1. This solution, however, cannot be extended to x-values of order  D "1=4 . We can even see that it has singularities on the real axis for some x of order  D "1=5 . Indeed, the change of variables and unknowns x D X; y D 2 Y; " D 5 yields the equation 

dY D 4X 3 Y  4  X Y 2 : dX

(5.53)

This is a singularly perturbed equation (with  taking the role of "). The slow curve X 7! Y0 .X / to be used here is the branch of the algebraic curve 4X 3 Y  4  X Y 2 D 0 asymptotic to 0 as X ! C1; it is not defined if 0  X < 1. The solution Y C corresponding to y C , i.e. Y C .X; / D 2 y C .X; 5 /, satisfies Y C .X; / ! Y0 .X / as  ! 0, for all X 2 1; C1Œ. For all ı > 0, a solution of (5.53) bounded on Œ1ı; 1 cannot exist since the right hand side of the equation, 4X 3 Y  4  X Y 2 , is bounded by a negative constant on Œ1  ı; 1  ı=2 for all Y 2 R. One can also show that Y C has poles in any interval Œ1  ı; 1 if  is small enough, see Exercise 5.19. Also note that the pole orders of the coefficients of the formal solution of (5.38) are too high, incompatible with a CAsE corresponding to p D 4 according to Remark 1 after Proposition 2.16. Indeed, seeking for a formal solution b y .x; "/ D P n y .x/" leads to the recursion formulas n1 n y1 .x/ D

1 ; x3

 X 1  0 y .x/ C x y .x/y .x/ : k nk n1 4x 3 n1

yn .x/ D

kD1

It follows that yn has a pole  of order 5n  2 at x D 0. More precisely yn is of the form yn .x/ D x 5nC2 an C xPn .x/ where Pn is polynomial and where the n1 X ak ank , hence are positive. numbers an satisfy a1 D 1 and for n  2; an D kD1

Therefore 0 is a pole of yn of exact order 5n  2, but it should only be of order 4n if y C had a CAsE as " ! 0 near x D 0. By the way, it is possible to apply our theory of CAsEs to Eq. (5.53) at the turning point X D 1; Y D 2, but this will not be developed here. Exercise 5.19. Prove that, for  small enough, the solution Y C of Eq. (5.53) has poles in any interval Œ1  ı; 1. Hint: make the change of unknown Y D 1=Z and compute the slow curve of the equation in Z. Draw a picture and check that the

118

5 Composite Expansions and Singularly Perturbed Differential Equations

solution Y C corresponds to a solution which follows the branch asymptotic to X 3 and crosses the real axis for some X close to 1. Comment: Since Eq. (5.53) is a Riccati equation, its solutions are naturally defined on the cylinder X 2 R; Y 2 R [ f1g. As  becomes small, these solutions have poles between X D 0 and X D 1 The number of these poles is of order 1= D "1=5 . Exercise 5.20. The classical Van der Pol equation is given by "xR C .x 2  1/xP C x D 0:

(5.54)

dz D 1  x 2  xz . 1. Considering xP as a function5 z of x, show that z satisfies " dx Then compute the slow curve of the equation satisfied by u D 1z . We focus on the turning point x D 1. 2. Prove that y given by 1z D x1 xCy satisfies an equation of the form (5.36) (with a turning point at x D 1 instead of x D 0) which satisfies the assumptions of Corollary 5.16 with r D 1 and p D 2. X   n 3. Deduce that y and z have CAsEs of the form an .x/ C gn xC1  , resp. 

X   n bn .x/ C hn xC1  , with  D "1=2 . 

n0

n0

Comment: The Van der Pol equation is well-known to have a relaxation cycle. Although they were never used in the literature [15, 35, 45, 46], these CAsEs are very useful to compute the periods of this cycle. The fact that our generalization fits with the Van der Pol equation is a good test in our opinion.

More precisely, consider z D xıx P 1 with the inverse function x 1 of x D x.t /. For convenience, the independent variable is named x now. We hope the fact that x is a function in the original equation, but the independent variable of the new equation is not too confusing for the reader.

5

Chapter 6

Applications

We present three problems in which CAsEs are useful. Our first application concerns canard solutions near a multiple turning point for an equation analytic, not only with respect to the variable x, but also with respect to the small parameter ". On the one hand the problem is to find a necessary and sufficient condition, that can be tested on the coefficients of the equation, for the existence of a solution close to the slow curve on some open interval containing the turning point. Such a solution will be called a local canard. On the other hand we show that, if there is such a local canard, then there is also a global canard. In other words, there is no phenomenon of buffer points for this kind of equation. This absence of buffer points had already been proved in [25] for a simple turning point and for a multiple turning point with a somewhat restrictive condition (see Remark 1 below Theorem 6.1), then by De Maesschalck in [10] for a general multiple turning point. Here we add the necessary and sufficient condition in terms of the coefficients. We present this result first for the quasi-linear equation of Sect. 5.2, then for the generalization of Sect. 5.3. The second application concerns canards called non-smooth or angular, because they follow a non-differentiable slow curve. These canards had already been studied by Diener, Isambert and Gautheron [14, 29, 32], and we give a new proof of some of their results. The theory of CAsEs brings two improvements: it yields uniform approximations of canard solutions and it provides Gevrey estimates of their asymptotic expansions. In this context, we present an example of a convergent CA sE . Finally we solve a problem of resonance in the sense of Ackerberg–O’Malley. Originally, the problem of resonance discussed in [1] is a boundary problem for a linear equation of second order. In this memoir, we do not state the original problem but a similar problem without boundary conditions. Also we do not describe its relationship with overstability for the associated Riccati equation; we refer the reader to [25] and the literature cited there.

A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 6, © Springer-Verlag Berlin Heidelberg 2013

119

120

6 Applications

6.1 Canards in a Multiple Turning Point We consider equation

"y 0 D f .x/y C "P .x; y; "/

(6.1)

where f is analytic in a complex neighborhood U of some real interval Œa; b, a < 0 < b, f real on Œa; b, xf .x/ > 0 if x 2 Œa; b n f0g, and P analytic in some complex neighborhood of Œa; b  f0g  f0g C3 , P .x; y; "/ real if x; y; " are real. Moreover we assume that x D 0 is a multiple turning point, i.e. f .x/ D x p1 1 C  O.x/ if x ! 0 where p is even, p  4 and > 0. For convenience, we reduce to

D p. This can be done by a scaling in x or ". A local canard is a solution of (6.1) bounded on some open interval containing 0. Recall that “bounded” means uniformly with respect to ". A global canard is a solution of (6.1) bounded on all Œa; b. Here the conditions of Proposition 5.14 are satisfied. Therefore, on each of the two mountains containing the real axis. there exists a solution having a Gevrey CA sE . Let M  be the mountain to the west containing Œa;0Œ and M C to the east X   an .x/ C gn x n its containing 0; b; let y  denote a solution on M  and n1

Gevrey CAsE, for  2 S D S.ı; ı; 0 / and x 2 V . ı;  Cı; jaj; jj/, ı; 0 > 0 small enough and some  > 0. Similarly on M C , with similar notation, for  2 S and x 2 V .ı; ı; b; jj/ there exists a solution y C with y C .x; /  1

p

X   an .x/ C gnC x n : n1

Recall that coefficients an and gn˙ are independent of the choice of the solutions. Theorem 6.1. The following statements are equivalent. (a) (b) (c) (d)

There exists a local canard. There exists a global canard. For all n 2 N, gn and gnC can be continued analytically to R and coincide. For all n 2 N, one has gn .0/ D gnC .0/.

Remarks. 1. In [25], we established a link between formal solutions, overstable solutions and so-called C 1 -canards, i.e. whose derivatives are bounded on some interval   ı; ıŒ uniformly with respect to ". Using these formal and overstable solutions, we proved in [25] the equivalence between the existence of a local C 1 -canard and a global C 1 -canard. Concerning canards not necessarily C 1 , the first proof of the equivalence between the existence of a local and a global canard is due to De Maesschalck, cf. [10], using geometric tools, e.g. blow-up, as well as Gevrey results of Dumortier/De Maesschalck[13]. Here we add a necessary and sufficient condition in terms of CAsEs: there is a canard if and only if the functions of the fast part associated with the two mountains containing the real axis coincide.

6.1 Canards in a Multiple Turning Point

121

2. In his thesis [20], Forget considered a variant of equation (6.1) containing an additional parameter1 a "y 0 D px p1 y C "P .x; y; "/ C "˛

(6.2)

on some interval Œa; b; a < 0 < b, and showed the following results. There is a function ˛ D ˛."/ such that (6.2) with ˛ replaced by ˛."/ has a canard solution on Œa; b; this was already known. Furthermore the corresponding canard solution with y.b; "/ D 0 has a uniform expansion of the form y.x; "/ 

X   an .x/ C hn x n ; n1

where p D ", an and hn are independent of " and hn are in a certain algebra of functions. The construction of this algebra has been indicated in Sect. 1.6. 3. For m  1, a C m -canard is a solution of (6.1) whose derivatives up to order m are bounded on some interval   ı; ıŒ (uniformly with respect to "). We can complete the above statement: there is a C m -canard (with m  1) if, and only if, firstly one of the equivalent statements (a)–(d) is satisfied and secondly gnC gn 0 for n D 1; : : : ; m  1. 4. The proof shows the equivalence between the existence of a local canard and a solution bounded for x in some domain containing not only Œa; b but also some sectors around Œa; b (corresponding to the mountains M  and M C and parts of their adjacent valleys) and for  in some sector. 5. In practice, the condition gn gnC can be checked on the formal solutions of the inner equation. We proceed similarly to the remark at the end of Sect. 5.2.1. The inner equation obtained using x D X; y.x/ D Y .X /; " D p has the form dY D pX p1 Y C G.X; Y; / (6.3) dX with G.X; Y; / D X p f1 .X /Y CP .X; Y; p /, f1 given by f .x/ D px p1 C x p f1 .x/. Each of the two solutions Y C andP Y  corresponding to y C and y  has an expansion in powers of  of the form n1 Yn˙ .X /n where the Yn˙ are given recursively by Y0˙ D 0;

YnC .X / D Yn .X / D

1

We have simplified his setting somewhat.

Z Z

X C1 X 1

exp.X p  s p / GnC .s/ds; exp.X p  s p / Gn .s/ds;

122

6 Applications ˙ where Gn˙ is the coefficient (depending Y1˙ ; : : : ; Yn1 ) of the  onP  term of order ˙ n  1 in  obtained by developing G X; 1
(and thus to YnC .0/ D Yn .0/). Since Yn˙ and gn˙ differ by polynomials and these polynomials are the same for cases “C” and “”, this latter condition is equivalent to (d). This already shows the equivalence between (c) and (d). 6. The condition (d) is a sequence of polynomial conditions in the coefficients .ck /kp and .qjkl /.j;k;l/2N3 of the Taylor expansions f .x/ D px p1 C

X

ck x k and P .x; y; "/ D

X

qjkl x j y k "l :

.j;k;l/2N3

kp

To see this, one shows by induction that the functions Yn .X / of the above remark are polynomials in .ck /kp and .qjkl /.j;k;l/2N3 with coefficients in the smallest algebra F  of functions containing 1 and X , stable by the operator T  defined by 

T .H /.X / D e

Xp

Z

X

e T H.T / dT: p

1

We notice that all these functions are of polynomial growth as X ! 1 and therefore the integral always converges. Similarly, the functions YnC .X / are polynomials in .ck /kp , .qjkl /.j;k;l/2N3 whose coefficients are in the set F C analogous to F  for C1 instead of 1. Condition (d), equivalent to the conditions YnC .0/ D Yn .0/, n D 0; 1; : : : of Remark 5, is thus a series of polynomial conditions in .ck /kp , .qjkl /.j;k;l/2N3 whose coefficients are some a priori known numbers, but defined in a “transcendental” manner. In particular if f and P are polynomials in all their arguments, we have a finite number of ck and qjkl . By the Hilbert Nullstellensatz, a finite number of these polynomial conditions suffices. Proof of Theorem 6.1: We show that (a) implies (c) and that (c) implies (b). The equivalence between (c) and (d) has been done in Remark 4 and the implication from (b) to (a) is obvious. Assume (a) and let y denote this local canard. Since it must be exponentially close to y  when x slightly descends the landscape from a, y has the same CAsE as y  for  in an interval I D 0; 0 Œ and for x in a quasi-sector V  ./ D V . ı;  C ı; jaj  ı; jj/ around R , for some ı; 0 ;  > 0. Similarly, y also has the same CA sE as y C for  in I and for x in a quasi-sector V C ./ D V .ı; ı; b  ı; jj/ (with the same ı; 0 ; , reducing them if needed). Hence we have, for all  2 I , all x 2 V  ./ \ V C ./ and all N 2 N

6.1 Canards in a Multiple Turning Point

123

˜b

a˜

F (˜a)

F (˜b)

Fig. 6.1 The domains D ˙ .ı; d / and their images by F .x/ D x 4 . For clarity, the lengths for the images by F have been modified

N 1  X

an .x/ C

nD0

 

gn x

 D n

N 1  X

an .x/ C gnC

 x  

n C O.N /

nD0

from which we deduce that gn .X / D gnC .X / for all n 2 N and all jX j < , hence gn and gnC are analytic continuations to each other. In order to prove the implication from (c) to (b), we use an argument analogous to that in [25]. We consider the solution y  with initial condition y  .e a; / D 0 and the solution y C with initial condition y C .e b; / D 0, where e a < a < b <e b are chosen such that, first f and P are analytic for x inR a complex neighborhood U of Œe a; e b x e and secondly F .e a/ ¤ F .b/, where F .x/ D 0 f . To fix ideas, we assume F .e a/ < a ; 0 whose F .e b/. Given ı > 0 small, let D  .ı; d / denote the domain containing e image by F is the union of the triangle of vertices F .e a/; iF .e a/ tan ı; iF .e a/ tan ı and the disk of center 0 and radius d (Fig. 6.1). Similarly, let D C .ı; d / be the domain containing Œ0; e bŒ whose image by F is the union of the triangle of vertices F .e b/; iF .e b/ tan ı; iF .e b/ tan ı and the disk of center 0 and radius d . If we choose ı sufficiently small, for e ı > ı, e ı arbitrarily close to ı and for 0 small enough, the    e  ı  C solutions y , resp. y , are defined for  in the sector S D S  2p C pı ; 2p e ; 0 p and for x in D  .ı; ıjjp /, resp. D C .ı; ıjjp / and have a Gevrey CAsE except for x close to e a resp. e b. This can be proved in a way analogous to Remark 7 of Sect. 5.2.4, but will be detailed together with the proof of (6.4) below. Consider the function ' W S ! C;  7! y C .0; /  y  .0; /. Since the solutions ˙ y have Gevrey CAsEs with the same coefficients according to (c), their difference at 0 hasGevrey asymptotic expansion zero. As is well-known, this implies './ D  O exp  r.e ı/=jjp for some r D r.e ı/ > 0 depending on e ı. We will show later, e e for ı; 0 > 0 small enough and for all ı 2 0; ıŒ, that   a/ sin ı= jjp './ D y C .0; /  y  .0; / D O  F .e  as  2 S D S 

(6.4)

 ı  ı  e Ce ; e p ;0 /. For pj arg j D 2  ı, the estimate (6.4) 2p p 2p  a/ with q D sin ı arbitrarily close to 1. By the implies './ D O exp  q F.e p sine ı

124

6 Applications

Phragmen–Lindel¨of theorem, this remains valid for arg  D 0. The Gronwall lemma then shows that the solution y C is defined and bounded for values of x arbitrarily close to e a, thus on all Œa; b. For the proof of (6.4),since we assumed that f vanishes only in 0, there is g real analytic such that F g.t/ D t p . By the change of variable x D g.t/ (and keeping the notation x instead of t) we reduce the setting to the case F .x/ D x p and therefore f .x/ D px p1 . Now we use the existence of  > 0 and, for all   > 0, of an integer L > 4p and numbers 0 ; % > 0 small enough, such that there .x; /; ` D L; : : : ; L of (6.1) defined, holomorphic and bounded are solutions z˙ `    as  2 S` D S.` 2pL  ; ` 2pL C ; 0 / and x 2 V`˙ D ˙V . 3 C ` 2pL C 2; 3 C 2p 2p  1 ` 2pL  2; %;  jj/ which have CAsEs of Gevrey order p z˙ ` .x; /  1

N 1  X

an .x/ C gn˙

p

 x  

n :

(6.5)

nD0

This is a consequence of Theorem 5.12, applied after rotations  D e i e , x D  ˙e i e x with D ` 2pL ; the fact that the functions an and gn˙ are independent of ` is due to the uniqueness of the formal solution of (6.1), i.e. to Theorem 5.4. Now, according to assumption (c) and Proposition 3.7, for all ` the functions zC .0; / and z ` ` .0; / have the same asymptotic expansion. As it is a Gevrey expansion of order p1 in , they are exponentially: there exists s > 0 such that p

 s=jj zC / ` .0; /  z` .0; / D O.e

(6.6)

for ` 2 fL; : : : ; Lg and  2 S` . Reducing % if necessary, we may assume that %p < s. Now consider e ı > 0 such that F .e ˛ / sine ı < %p and ı 2 0; e ıŒ. Let ` 2 fL; : : : ; Lg and  2 S` \ S , arg  D . Then y C .x; / and zC ` .x; / are holomorphic bounded on a neighborhood of the segment x 2 Œ0; T ei  where p 0 p1 T p D F .e ˛ / sin ı. As their difference D D y C  zC C ` satisfies  D D px  C p C p  2 P .x; z` .x; /; y .x; /;  / D; we deduce that    i p p y C .0; /  zC ` .0; / D O exp  .T e / = D O e T

p =jjp

as  2 S` \ S:

(6.7)

  T p =jjp as  2 S` \ S . Similarly, we show that y  .0; /  z .0; / D O e ` With (6.7) and (6.6), this implies (6.4) as  2 S` \ S . As the union of the S` contains S , the proof of (6.4) is complete. t u Theorem 6.1 can be generalized to Eq. (5.48) of Corollary 5.16, i.e. to "y 0 D f .x/y C "h.x; "/ C y P .x; y; "/

6.1 Canards in a Multiple Turning Point

125

where f is analytic in D1 , f .x/ D px p1 C O.x p / as x ! 0, p even and where h and P are analytic bounded in D1  ˙, resp. D1  D2  ˙ and having each one a uniform asymptotic expansion of Gevrey order 1 as ˙ 3 " ! 0, where D1 is a neighborhood of the interval Œa; b, D2 D D.0; r2 / and ˙ D S.ı; ı; "0 /, and where a; b; r2 ; "0 ; ı > 0, ı small enough. It is further assumed that the values of the functions f; h; P are real when their arguments are real and that there is r 2 f1; : : : ; p  1g such that, on the one hand h.x; 0/ D O.x r1 /; x ! 0 and on the other hand X P .x; y; 0/ D pkl x k y l : k0;l1; kCrlp1

By Corollary 5.16, the equation has solutions y ˙ .x; / holomorphic in all .x; / with jarg j < ı, jj < 0 , x 2 V C ./ D V .ˇ; ˇ; b; jj/, respectively x 2 V  ./ D V .  ˇ;  C ˇ; jaj; jj/ with some ı; 0 ; ˇ > 0 and some  < 0 if ı is small enough. These solutions have CAsEs y ˙ .x; /  1

p

p1 X nDr

gn˙

1  x  n X   an .x/ C gn˙ x n :  C 

(6.8)

nDp

Contrary to Eq. (6.1), the domains of the two solutions have in general empty intersection. This requires changes to the statement of the theorem and its proof. Theorem 6.2. Under the previous conditions and notations, the following statements are equivalent. (a) There exists a local canard y with y.x; / D O.r / uniformly in a real neighborhood of x D 0. (b) There exists a global canard y with y.x; / D O.r / uniformly on Œa; b. (c) The functions gn˙ .X / can be continued analytically in a neighborhood of R and these extensions coincide, i.e. gnC gn for all n 2 N. Proof. The proof that (c) implies (a) is almost identical to that of Theorem 6.1. The only observation to add is as follows: since grC D gr are analytic on R, we can apply Theorem 5.17 and we obtain that y C .x; / and its CAsE extend in some quasi-sector V .e ˇ; e ˇ; b; e jj/ with e ˇ; e  > 0; for y  and the z˙ ` of the proof of Theorem 6.1, the analogous statement is true. The rest of the proof is unchanged. The proof that (c) is a consequence of (a) requires the use of the inner equation. The theorem on the CAsEs of (5.48) only says that the local canard y.x; / has a CAsE on Œa C ı; L, L D jj, (and even in V C ./ \ D.0; jaj  ı/) and another CAsE on ŒL; b  ı. For the solution Y .X; / D r y.X; / of the inner equation, this implies that its values U ˙ ./ D Y .˙L; / have asymptotic expansions of Gevrey order p1 as  ! 0. By assumption, the function Y is bounded on ŒL; L0; . Since the values of Y 0 .X; / are expressed by the inner equation e .X; Y; / Y 0 D pX p1 Y C P

126

with

6 Applications

e .X; Y; / D 1r h.X; / C 1p YP .X; r Y; p /; P

and thus by a regularly perturbed differential equation, these values are also bounded. The Arzela–Ascoli Theorem shows that any sequence .k /k2N of positive numbers   tending to 0 has a sub-sequence .kl /l2N such that the sequence Y .X; kl / l2N converges uniformly on ŒL; L; the limit is necessarily the solution G of the reduced inner equation Y 0 D pX p1 Y C cX r1 C Y Q.X; Y /; Q.X; Y / D

X

pkl X k Y l ; (6.9)

k0;l1

kCrlDp1

with initial condition G.L/ D lim!0 U C ./. So here we obtain that the first terms gr˙ .X / of the CAsEs can be continued to R and coincide. As above, we conclude that the CAsEs can be continued on quasi-sectors with some e  > 0. The rest of the proof is like that of Theorem 6.1. t u Remark. The conditions are no longer equivalent to a series of polynomial conditions as in Remark 6 after Theorem 6.1; they are transcendental in c and the pkl of (6.9).

6.2 Non-smooth Canards 6.2.1 Equations of “Union Jack” Type We consider a differential equation of the form "y 0 D y.y  x/.y C x/ C P .x; y; "/ C "c;

(6.10)

where P is analytic on D1  D.0; r/  D.0; "1 /, D1 a neighborhood of some interval Œa; b, a < 0 < b, r > max.jaj ; b/ and c 2 C an additional parameter. In the original problem [14, 29, 32], P has real values for real arguments x; y; ", but nothing changes if this is not the case. It is assumed that P .0; 0; "/ D O."2 / and that the homogeneous valuation of P .x; y; 0/ is at least 4, i.e. there are pkl 2 C such that X P .x; y; 0/ D pkl x k y l ; for jxj ; jyj small enough. (6.11) kCl4

The slow set of (6.10), of equation y.yx/.yCx/CP .x; y; 0/ D 0, can be desingularized by a blow-up y D xz. We obtain the equation z.z1/.zC1/CxQ.x; z/ D 0 where Q.x; z/ D x 4 P .x; xz; 0/ is analytic in D1  D.0; 1 C ı/; ı > 0, to which we can apply the implicit function theorem at .0; 0/ and .0; ˙1/. It follows that the slow set locally consists of three analytic slow curves y0 .x/ D O.x 2 / and

6.2 Non-smooth Canards

127

Fig. 6.2 The branches y , yC and y0 of the slow set y.y  x/.y C x/ C P .x; y; 0/ D 0

y+

y0

y−

Fig. 6.3 Three phase portraits of Eq. (6.12), for c D 0, c D 0:3621759411 and c D 1

y˙ .x/ D ˙x C O.x 2 / (Fig. 6.2). Diener gave the name Union Jack to Eq. (6.10) because the slow set of the “model” equation "y 0 D y.y  x/.y C x/ C "c consists of the three straight lines y D 0 and y D ˙x, and therefore resembles the flag of the United Kingdom. Thus the three slow curves y0 , yC and y of the general case form a “modified Union Jack”. Furthermore, the lines y D 0 and y D ˙x are also particular solutions of the model equation for certain values of c: the solution y 0 for c D 0, y x for c D 1 and y x for c D 1. It is the same for the reduced inner equation of (6.10), obtained by putting x D X , y D Y , " D 3 and by letting  ! 0, i.e. Y 0 D Y .Y  X /.Y C X / C c:

(6.12)

Whatever the value of c, Eq. (6.12) has a unique solution Y` .X; c/ such that Y` .X; c/ ! 0 as X ! 1; indeed, this equation is of the form Y 0 D X 2 Y C O.1 C jX j/ when Y is bounded. For a similar reason, there exist, for any value of c, two unique solutions Yr˙ .X; c/ such that Yr˙ .X; c/  ˙X as X ! C1. These three solutions are the repelling rivers that can be seen on each of the phase portraits of Fig. 6.3. We can check that Y` .X; c/ D O.X 2 / as X ! 1 and Yr˙ .X; c/ D ˙X C O.X 2 / as X ! 1; this is the case uniformly for c in compacts. It was shown in [14] that there is a unique value c D c0 2 0; 1Œ such that the solutions Y` and YrC of (6.12) coincide, i.e. Y` .X; c0 / YrC .X; c0 / W Y0 .X /.

128

6 Applications

This value of c is a long canard value for the corresponding equation "y 0 D y.y  x  x/.y C x/ C "c because the solution y.x; "/ D Y0  is attracting as x  ı < 0 and repelling as x  ı > 0; it is called a non-smooth canard because the uniform limit z.x/ of y.x; "/ as " ! 0 is z.x/ D 0 as x  0 and z.x/ D x as x > 0: the associated slow curve is non-differentiable. By the way, the value c D c0 is also a non-smooth canard value: by symmetry the solutions Y` and Yr coincide for this value of c. The following result answers the natural question whether this phenomenon persists for the complete equation (6.10) and if we can describe the canard values and the corresponding canard solutions. This question has been solved by Diener [14] and Isambert [32]. Our theory of CAsEs is particularly well suited here and can provide additional information: a uniform approximation of the canard solutions and the Gevrey character of the asymptotic expansions. Theorem 6.3. With the above assumptions and notation,  we assume that y0 is x; y attracting on Œa; 0Œ, i.e. 3y0 .x/2  x 2 C @P .x/; 0 < 0 for all x 2 Œa; 0Œ, 0 @y while yC is repelling on 0; b. Then equation (6.10) has a non-smooth canard value c D c./ and a corresponding canard solution y.x; / such that y.x; /  z.x/ D O./, where z.x/ D y0 .x/ as x  0 and z.x/ D yC .x/ as x > 0. Moreover, the function c D c./ has an asymptotic expansion of Gevrey order 13 of the form 1 X c./  1 cn n 3

nD0

the first term of which is the value c0 introduced above. Similarly, the function y has 1 CA sE s of Gevrey order 3 y.x; /  1 y0 .x/ C 3

1  X   agn .x/ C bgn x n

(6.13)

nD1

as  ! 0 and x 2 Œa; L for L > 0 arbitrary and y.x; /  1 yC .x/ C 3

1  X   adn .x/ C bdn x n

(6.14)

nD1

as  ! 0 and x 2 ŒL; b, where the agn are analytic on a (complex) neighborhood of Œa; ı, the adn on a neighborhood of Œı; b and where the bgn and the bdn are analytic on a neighborhood of R. The functions bgn have consistent expansions of Gevrey order 13 in the sense of (3.1) as X ! 1, the bdn as X ! C1. Finally, the analogous statement is true for y .x/ instead of yC .x/.

6.2 Non-smooth Canards

129

Remarks. 1. The asymptotic seriesb c./ of c./ and the composite formal series of the theorem are computed as before from the inner and outer expansions. Here we must begin with the inner expansion, however, in order to determine b c./; this has been done by Isambert in [32]. Just as in Remark 5 after Theorem 6.1, the calculation procedure of slow and fast parts (cf. Comment P 5.2.4.2) shows n that both inner expansions should coincide, i.e. y.X; /  1 nD1 un .X / DW b.X; / with a formal solution of the inner equation Y b.X; / with c D b Y c./. Since, by Proposition 2.16, this inner expansion is made of the fast part of the CA sE and a polynomial part, the coefficients un .X / have polynomial growth as X ! C1 and X ! 1. We show (as in [32]) that this uniquely determines the values of cn and the functions un .X /. For instance, one has the relationship bg1 .X / D u1 .X / D Y0 .X / D X C bd1 .X /. The slow parts of CAsEs (6.13) and (6.14) are then determined as usual as the non-polar parts of the outer formal solutions of (6.10) with c D b c./ for the slow curves y0 .x/, respectively yC .x/. This implies e.g. agn D 0 and adn D 0 for n D 1 and 2. 2. As usual for canard problems, there is neither a unique canard values c./ nor a unique canard solution associated to any canard value: of c./ or  a change p the initial condition by an exponentially small term, O e K=jj with K > 0 sufficiently large, does not change the conclusion of the theorem. Conversely, two canard values are exponentially close. This can be shown by variation of constants, cf. e.g. [3, 5, 9], or [26] and the references therein. Proof of Theorem 6.3: In a neighborhood of 0, say jxj <  , we make the change of variable y D y0 .x/ C z. The resulting equation is first written as "z0 D Q.x; z; "/ C "c     WD z C y0 .x/ z C y0 .x/  x z C y0 .x/ C x C   P x; z C y0 .x/; "  "y00 .x/ C "c: e W .x; z; 0/ 7! Q.x; z; 0/  By construction, we have Q.x; 0; 0/ 0; the function P z.z  x/.z C x/ satisfies a property analogous to (6.11). Therefore the previous equation can be rewritten   "z0 D r.x/z C " c C s.x; "/ C zR.x; z; "/;

(6.15)

where we decompose Q.x; z; "/ D r.x/z C "s.x; "/ C zR.x; z; "/. Here the 1 2 3 coefficient of z is r.x/ D @Q @z .x; 0; 0/ D x C O.x /, s.x; "/ D " Q.x; 0; "/ satisfies s.0; 0/ D 0 and R satisfies R.x; 0; 0/ D 0. From our assumption on P and e , the function R can be developed for " D 0 the previous observation on P R.x; z; 0/ D z2 C

X k0;l1;kCl3

as jxj <  , jzj small.

Rkl x k zl ;

130

6 Applications

p 3 After the rescaling x !  3x, Eq. (6.15) falls within the framework of Corollary 5.16 with p D 3 and r D 1. Moreover, its reduced inner equation is Z 0 D Z.Z  X /.Z C X / C c. The solution of this last equation with asymptotic behavior of the form const. X 2 as X tends to 1 is Y` . When c D c0 , this solution coincides with YrC ; in particular, it can be analytically continued on R. Given M > 0, there is a neighborhood jc  c0 j < % such that Y` can be continued on Œ1; M . Now we apply Theorem 5.17. We obtain that (6.15) has a solution z D z.x; c; / D O./ with CAsE as  ! 0, x 2 V .  ˇ;  C ˇ; ; L/ and jc  c0 j < % with some L > 0. Now, it is well-known that our assumption of attractiveness of y0 .x/ on Œa; 0Œ implies that the solution of (6.15) with initial condition 0 at a point just before a has an asymptotic expansion in " of Gevrey order 1 without constant term in the interval Œa; =2, say. Hence it is exponentially close to the solution z D z.x; c; / constructed above, uniformly for x 2 Œ; =2 and jc  c0 j < %. Thus we obtain the existence of a solution y` .x; c; / of (6.10) analytic for  2 S.˛; ˛; 1 /, x 2 V .  ˇ;  C ˇ; jaj ; L/ and jc  c0 j < %, for some L; 1 ; ˛; ˇ > 0, with a CAsE of Gevrey order 13 y` .x; c; /  1 y0 .x/ C

1  X

3

agn .x; c/ C bgn

 n ; c  ; 

x

(6.16)

nD1

where bg1 .X; c/ D Y` .X; c/ and ag1 D 0. As the equation obtained from (6.10) by y D yC .x/ C z also falls within the framework of Theorem 5.17, using now the repulsiveness of yC on 0; b we obtain the existence of a solution yr .x; c; / of (6.10) analytic for  2 S.˛; ˛; 1 /, x 2 V .ˇ; ˇ; b; L/ and jc  c0 j < %, having a CAsE of Gevrey order 13 yr .x; c; /  1 yC .x/ C 3

1  X   adn .x; c/ C bdn x ; c n ;

(6.17)

nD1

where bd1 .X; c/ C X D YrC .X; c/ and ad1 D 0. Non-smooth canards values of the theorem are obtained by solving the equation y` .0; c; / D yr .0; c; /:

(6.18)

Therefore we must show that the solution c D c./ exists and has the required  properties, as well as the functions y`;r x; c./;  . We apply the implicit function theorem to (6.18), slightly modified,   f .c; / D 0; where f .c; / D 1 y` .0; c; /  yr .0; c; / : (6.19) We first have the equality lim!0 1 y` .0; c0 ; / D Y0 .0/ D lim!0 1 yr .0; c0 ; / and thus lim!0 f .c0 ; / D 0. It will be shown below that lim 1 !0 

@y` @yr .0; c0 ; / ¤ lim 1 .0; c0 ; / !0 @c @c

(6.20)

6.2 Non-smooth Canards

131

hence lim!0 @f @c .c0 ; / ¤ 0. The conditions of the implicit function theorem are satisfied and we obtain the existence of a solution c D c./ of (6.18) with c.0/ D c0 . A priori, this theorem only says that c is a continuous function, but formula Z c./ D

1 2 i

jxc0 jD%=2

x @f .x; / @c dx f .x; /

together with the compatibility of Gevrey expansions with the elementary operations shows that it has an expansion of Gevrey order 13 as  ! 0 in a sector S.˛; ˛; 2 / with some 2 > 0.   Using Theorem 4.7 (a), we obtain that the compositions .x; / 7! y` x; c./;  and .x; / 7! yr .x; c./; / have CAsEs of Gevrey 1 order  3 . As solutions of the same  equation (6.10) with the same initial condition y` 0; c./;  D yr 0; c./;  , they coincide. This proves the statements of the theorem, in particular (6.13) and (6.14) whose coefficients can be obtained by    developing agjd;n x;b c./ D agjd;n .x; c0 / C : : : respectively bgjd;n X;b c./ by the Taylor formula in (6.16) and (6.17). For the proof of (6.20), we use that the CAsEs of ygjd are uniform with respect to c in a complex neighborhood of c0 . Thus we can obtain the partial derivatives with respect to c using the Cauchy formula; as a consequence these partial derivatives also have CAsEs and these CAsEs are obtained by differentiating term by term those of y`;r . A comparison with the outer expansion (see Sect. 2.4) shows that @Y` 2 ` ag1 D 0 and ad1 D 0, hence @y @c .0; c0 ; / D Z` .0; c0 / C O. / with Z` D @c C

r and @y .0; c0 ; / D Zr .0; c0 / C O.2 / with Zr D @Y@cr . Now the functions @c X 7! Z`jr .X; c0 / are solutions of Eq. (6.12) differentiated with respect to c, taken at c D c0 and for Y D Y` .X; c0 / D Y0 .X /, resp. Y D YrC .X; c0 / D Y0 .X /. In other words, these are solutions of

  Z 0 D 3Y0 .X /2  X 2 Z C 1: More precisely, Z` .X; c0 / is the solution tending to 0 as X ! 1 and Zr .X; c0 / that tending to 0 as X ! C1. Let I0 .X / be the antiderivative of 3Y0 .X /2 vanishing at X D 0 and J0 .X / that of 3Y0 .X /2  3X 2 vanishing at X D 0. Since 3Y0 .X /2 D O.X 1 / as X ! 1 and 3Y0 .X /2  3X 2 D O.X 1 / as X ! C1, these antiderivatives have at most logarithmic growth at 1, resp. C1. The variation of constants then gives Z Z` .0; c0 / D

1

Z

and

0

1

Zr .0; c0 / D 

  exp X 3 =3  I0 .X / dX > 0

  exp  2X 3 =3  J0 .X / dX < 0:

0

This proves (6.20) and the proof of the theorem is now complete.

t u

132

6 Applications

Of course, the theory of CAsEs is not essential to prove the existence of canard values for (6.10) or similar equations, nor the fact that the right hand side of the differential equation is analytic. Indeed, we could use the fixed point theorem as in the proof of Theorem 5.15 in order to show the existence of y` .x; c; / and yr .x; c; / for x 2 Œa; L, resp. x 2 ŒL; b, and then extend them to Œa; 0 resp. Œ0; b using the inner differential equation. A possible canard value c D c./ is then the solution of equation y` .0; c; / D yr .0; c; / whose existence is ensured by the implicit function theorem. The crucial fact that some partial derivative does not vanish is then shown—as above—by reducing it to a certain linear differential equation for the parameter  D 0. From this point of view, the problem of the existence of canard values treated in this part is simpler than, for example, that of necessary and sufficient conditions for the existence of canards for ordinary differential equations without parameters treated in Sect. 6.1 or that of Ackerberg– O’Malley resonance treated in Sect. 6.3. The theory of CAsEs provides two new properties on non-smooth canards values: it gives a uniform approximation of canard solutions, precisely our CAsE, and the asymptotic expansions of the function c D c./ and the canard solutions turn out to be Gevrey. This can serve e.g. to prove that a summation “to the smallest term” in the spirit of [24] provides a canard value.

6.2.2 Non-smooth Equations The canards of Theorem 6.3 are called non-smooth because the associated slow curve has an angle at x D 0 but of course, for each ", the canard solutions of the analytic equation (6.10) are analytic in x. In [29, 32], Isambert and Gautheron also studied canard solutions of non-smooth ordinary differential equations. In what follows, we indicate how the theory of CAsEs also applies to this situation. These are equations of angular canard type    "y 0 D y  f .x/ y  g.x/ C "c;

(6.21)

where f .x/ D AxCO.x 2 /, g.x/ D BxCO.x 2 /, A ¤ B and where the restrictions of f and g on intervals Œ0; b, resp. Œa; 0 are real analytic functions, but where f and g are no longer assumed C 1 on Œa; b. We assume that the slow  curve y D f .x/ is attracting for x 2 Œa; 0Œ and repelling for x 2 0; b, i.e. x f .x/  g.x/  0 on Œa; b; in particular one has A > B (Fig. 6.4). The classical example of [32] corresponds to f .x/ D jxj3 =3 and g.x/ D x C jxj3 =3. In another example of [29, 32], we have f .x/ D x.1 C jxj/ and g D f . First let us discuss the solutions on the interval Œ0; b. We write the differential equation on a complex neighborhood of Œ0; b with the analytic continuation fC ; gC of the restrictions f jŒ0;b and g jŒ0;b "y 0 D .y  fC .x// .y  gC .x// C "c:

(6.22)

6.2 Non-smooth Canards

133

Fig. 6.4 The slows curves of (6.21) and the orientation of the field

y = f (x)

y = g(x)

The change of variable y D fC .x/ C z then leads to equation     "z0 D fC .x/  gC .x/ z C z2 C " c  fC0 .x/ which falls, after a rescaling of the variable x, in the framework of Corollary 5.16 with p D 2, r D 1. For a neighborhood jcj < %, % small enough, we can again use Theorem 5.17; as in the above proof the repulsiveness yields a solution with CAsE up to b. We obtain that there is a holomorphic solution y D yr .x; c; / of (6.22) for  2 S.˛; ˛; 1 /, x 2 V .ˇ; ˇ; b; L/ and jcj < % with some ˛; 1 ; ˇ; L; % > 0 small enough and that it has a CAsE of Gevrey order 12 yr .x; c; /  1 b y r .x; c; / WD fC .x/ C 2

1  X   adn .x; c/ C bdn x ; c n

(6.23)

nD1

with ad1 D 0 and bd1 .X; c/ D Ur .X; c/, where Ur is the solution of the reduced inner equation U 0 D .A  B/X U C U 2 C c (6.24) tending to 0 as X ! C1. We verify, as at the end of the proof of Theorem 6.3, that @Ur .0; 0/ < 0. @c For the interval Œa; 0, Eq. (6.21) also reduces to an analytic equation    (6.25) "y 0 D y  f .x/ y  g .x/ C "c with the analytic continuations f and g of the restrictions f jŒa;0 and g jŒa;0 on a complex neighborhood of Œa; 0. Here we obtain the existence of a holomorphic solution y D y` .x; c; / of (6.25) for  2 S.˛; ˛; 1 /, x 2 V .  ˇ;  C ˇ; jaj ; L/ and jcj < % with some ˛; 1 ; ˇ; L; % > 0 small and having a CAsE of Gevrey order 12 y` .x; c; /  1 b y ` .x; c; / WD f .x/ C

1  X

2

nD1

agn .x/ C bgn

 x  n  

(6.26)

134

6 Applications

with ag1 D 0 and bg1 .X; c/ D U` .X; c/, where U` is the solution of the same reduced ` inner equation (6.24) tending to 0 as X ! 1. We check that @U @c .0; 0/ > 0. By applying the implicit function theorem to equation 1 y` .0; c; / D 1 yr .0; c; / in a neighborhood of c D 0 as in the proof of Theorem 6.3, we deduce again the existence of non-smooth canard values c D c./ with asymptotic expansionb c./ of Gevrey order 12 and the existence of Gevrey CAsEs for the canard solution defined     by y.x; / D y` x; c./;  resp. y.x; / D yr x; c./;  on the intervals Œa; 0 resp. Œ0; b. P1 n A priori, b c./ D nD1 cn  is a formal series in powers of ; the theory of CA sE s already tells us that this series is Gevrey. To complete this study, we reprove and improve below a result of [32]. This is an opportunity to present a convergent CA sE , the only non-trivial one in this memoir besides those of Exercise 2.15. This CA sE is actually obtained as the Taylor expansion of some special function. Proposition 6.4. In the case of the classical angular canard, i.e. (6.21) with f .x/ D jxj3 =3 and g.x/ D x C jxj3 =3, the formal series associated to the canard values 1 X c./ D c4m 4m (6.27) mD1 4

only contains powers of  and it converges. Remarks. 1. Isambert [32] has shown the form (6.27) of the formal series associated to the canard values. We show, moreover, its convergence. 2. For this model equation, we can also express the solution corresponding to the sum of the series (6.27) in a simpler way and we obtain very special CAsEs as we shall see in the proof and in the remark at the end of this section. 3. Again, canard values and solutions are not unique here, but the convergence allows to distinguish one of them. Proof of Proposition 6.4: In the sequel, we only use p", since most of the functions involved are functions of this variable, and not  D ". It has therefore to be shown that the canard value c D c."/ is holomorphic in a neighborhood of 0 and is even. Consider first the equation on Œ0; C1Œ:    3 3 y C x  x3 C "c: (6.28) "y 0 D y  x3 The change of variables y D

x3 3

C z reduces to the simple equation

"z0 D xz C z2 C ".c  x 2 /; we can further simplify. Let d D d."/ be the solution of d C d 2 D " holomorphic in a neighborhood of " D 0 and such that d.0/ D 0: Then the change of variables z D d."/x C u leads to equation     "u0 D 1 C 2d."/ xu C u2 C " c  d."/ :

6.2 Non-smooth Canards

135

The rescaling x D t=."/, u D ."/v with the holomorphic function ."/ satisfying .0/ D 1 and ."/2 D 1 C 2d."/ leads to the equation "

c  d."/ dv D tv C v2 C "C."/; where C."/ D : dt ."/2

For this equation, we can eliminate the small p parameter p ", except in the argument of C . Indeed, the change of variables t D " T , v D " V leads to dV D TV C V 2 C D dT

(6.29)

with D D C."/, which is the reduced inner equation, except that D is not independent of ": it is an analytic function D D C."/, with C.0/ D c. Now, for D arbitrary in C, let Vr .T; D/ denote the solution of (6.29) tending to 0 as T tends to C1. It can be expressed by solutions of the Weber equation, but this is not useful here. Simply note that it is an entire function of D, meromorphic in T , that it has an 1 asymptotic expansion of Gevrey with respect to D in any compact, P1 order 2 uniform 2m1 of the form Vr .T; D/  1 W .D/T with W0 .D/ D D and finally mD0 m 2

r that Vr .T; 0/ D 0 and @V @D .0; 0/ < 0. In summary, if one takes into account all the changes of variables, for jcj 3 small (6.28) has a unique solution yr such that yr .x; c; "/  x3  d."/x tends to 0 as x tends to C1. It is the function   p 3 yr .x; c; "/ D x3 C d."/x C "."/Vr ."/ px" ; cd."/ : 2 ."/

p This solution is holomorphic for " 2 S.˛; ˛; "1 /, x 2 V .ˇ; ˇ; 1; L "/ and jcj < % with ˛; ˇ; "1 ; L; % > 0 maybe small. On   1; 0, the classical angular canard equation is  "y 0 D y C

x3 3



yCxC

x3 3



C "c:

(6.30)

Similarly, and using that (6.29) does not change under the transformation T ! T; V ! V , we obtain that the function   p 3 y` .x; c; "/ D  x3 C d."/x  "."/Vr ."/ px" ; cd."/ 2 ."/ 3

x tends is its unique solution such that y` .x; c; "/ C x3  d."/x tends to 0 as p to 1. It is holomorphic for " 2 S.˛; ˛; "1 /, x 2 V .  ˇ;  C ˇ; 1; L "/ and jcj < % with some ˛; ˇ; "1 ; L; % > 0. The equation determining the non-smooth canard value c D c."/ is thus     cd."/ D ."/V 0; : ."/Vr 0; cd."/ r 2 2 ."/ ."/

(6.31)

136

6 Applications

To this equation, we can apply the implicit functions theorem for holomorphic functions. We first obtain that c D c."/ is a holomorphic function of " in a neighborhood of 0 and then that it is even because of the symmetry of (6.31). This proves the statement. t u Remark. Not only the formal canard value c D c."/, but also the formal canard solutions y` and yr , are convergent. For x  0, one canard solution is given by y.x; "/ D yr .x; c."/; "/ D

x3 3

C d."/x C

  p "."/Vr ."/ px" ; c."/d."/ 2 ."/

with the  functions yr and Vr of the  proof. Now  the function .X; "/ 7! ."/Vr ."/X; C."/ with C."/ D c."/  d."/ =."/2 is holomorphic and bounded in V .˛; ˛; 1; L/  D.0; "1 /, if ˛; L; "1 > 0 are small enough. As c.0/ D d.0/ D 0, we deduce that the Taylor series 1   X ."/Vr ."/X; C."/ D Wn .X /"n nD1

converges uniformly on V .˛; ˛; 1; L/ and on p any compact of D.0; "1 /. Therefore, the CAsE of y is a convergent series in  D ". This CAsE consists of a slow part containing only even powers of , with leading term x 3 =3 and other terms that are scalar multiples of x, and a fast part containing only odd powers of  and whose coefficients are functions Wn .X / with asymptotic expansions of Gevrey order 12 as X tends to C1. As for the function Vr , these last expansions contain only odd powers of X 1 . The properties of the canard solution for x  0 are similar.

6.3 Ackerberg–O’Malley Resonance We consider the linear ODE of order 2 "z00  f .x; "/z0 C g.x; "/z D 0

(6.32)

where f and g are analytic in a complex neighborhood U of .0; 0/ and have real values for real x; ". This assumption is natural as the problem comes from a boundary value problem (cf. [25]), but it is not essential for the sequel. In [25], we had considered, among other topics, local C 1 -resonant solutions of (6.32). These are solutions z D z.x; "/ defined for 0 < "  "0 and ı  x  ı with some "0 ; ı > 0, which tend to a non-trivial solution of the reduced differential equation  f .x; 0/z0 C g.x; 0/z D 0

(6.33)

as " ! 0, uniformly on  ı; ıŒ, and such that all derivatives z.m/ are bounded on   ı; ıŒ (uniformly as " tends to 0). We also studied local resonant solutions, i.e.

6.3 Ackerberg–O’Malley Resonance

137

solutions that tend to a non-trivial solution of (6.33) as " ! 0 uniformly on  ı; ıŒ, but the derivatives of which are not necessarily bounded uniformly with respect to ". In this section, we assume f .x; 0/ D ˛x p1 C O.x p / and g.x; 0/ D ˇx p2 C O.x p1 /

(6.34)

with p a positive even integer, ˛; ˇ 2 R and ˛ > 0. We also assume that f .x; 0/ only vanishes at x D 0 in U . The result that we present (Theorem 6.5) gives necessary and sufficient conditions, in terms of the formal solutions of (6.32) and of the associated inner equation, for the existence of local resonant and C 1 -resonant solutions. It is therefore appropriate to introduce this inner equation; it is obtained by the change of variables x D X , Z.X / D z.X / and " D p d 2Z e dZ Ce g.X; /Z D 0;  f .X; / dX 2 dX

(6.35)

e.X; / D 1p f .X; p / and e where f g.X; / D 2p g.X; p /. The assumption (6.34) implies that e.X; / D f

1 X nD0

pn .X /n ; e g.X; / D

1 X

qn .X /n

nD0

are series converging uniformly for X in any compact subset of R and for jj small enough, where pn ; qn are polynomials, the first are p0 .X / D ˛X p1 and q0 .X / D ˇX p2 . Before stating the result, we describe the outer and inner formal solutions of (6.32) resp. (6.35). The outer formal solutions of (6.32) are of the formb z.x; "/ D P 1 n z .x/" ; their coefficients satisfy the recursion formula n nD0 f .x; 0/z0n  g.x; 0/zn D hn .x/; where h0 0 and where hn .x/ is the coefficient of "n in the Taylor expansion of   0   "b z 00 .x; "/  f .x; "/  f .x; 0/ b z .x; "/ C g.x; "/  g.x; 0/ b z.x; "/: The functions hn depend on f; g, on z0 ; : : : ; zn1 and on their derivatives. It is easily seen that these are unique up to a constant factor.2 More P formal solutions n precisely, if b z0 D n0 zn .x/" is a formal solution of (6.32) with z0 ¤ 0, then the formal solutions of (6.32) are the formal series of the form b c."/b z0 where P n b c."/ D 1 c " . Note that the coefficients z may have singularities at x D 0, n n nD0 but can be continued analytically along any path in the neighborhood U avoiding 0.

2

Here and hereafter, the word “constant” means constant with respect to x.

138

6 Applications

b The inner formal solutions of (6.35) are of the form Z.X; "/ D their coefficients satisfy the recursion

P1 nD0

Un .X /n ;

d Un d 2 Un C ˇX p2 Un D Hn .X / ;  ˛X p1 2 dX dX

(6.36)

where H0 0 and Hn .x/ is the coefficient of n in the Taylor expansion of    b  e.X; /  ˛X p1 d Z .X; /  e b g.X; /  ˇX p2 / Z.X; f /: dX e;e g , on U0 ; : : : ; Un1 and on their derivatives. These The functions Hn depend on f inner formal solutions are not unique, even up to a constant factor b c./, but depend on two parametersb c 1 ./ andb c 2 ./. The functions Un are entire since the coefficients e;e of f g are polynomials. If one asks furthermore that the coefficients Ui are functions of at most polynomial growth as X tends to C1 (resp. 1), one obtains inner formal solutions P ˙ n b ˙ .X; "/ D 1 denoted by Z nD0 Un .X / which are unique up to a constant factor and will play an important role in the sequel. We point out that both types of formal solutions may contain logarithmic terms: the outer formal solutions at the singularity x D 0 and the inner formal solutions in the asymptotic behavior of their coefficients as X ! ˙1. This will somewhat complicate the proof. Now we are ready to state the result. Theorem 6.5. (a) Under assumption (6.34), Eq. (6.32) has local resonant solutions if and only if the two following conditions are satisfied: • the quotient D D ˇ=˛ is a non-negative integer congruent to 0 or 1 modulo p and P n b • there is a non-trivial formal solution Z.X; "/ D Z0 .X / C 1 nD1 Un .X / of (6.35) with coefficients of polynomial growth, both as X ! C1 and as X ! 1. (b) Under assumption (6.34), Eq. (6.32) has local C 1 -resonant solutions if and only if: • D is a non-negative integer congruent to 0 or 1P modulo p and n • there is a non-trivial formal solutionb z.x; "/ D 1 nD0 zn .x/" of (6.32) with coefficients analytic in a neighborhood of 0. P n z.X; p / D 1 This is the case if and only if Db nD0 Zn .X / , is a non-trivial formal solution of (6.35) where Zn are polynomials of degree at most n C D. In both items of the statement, if we assume that the leading term Z0 is non-zero, then it is a polynomial of exact degree D; it is the same in both statements up to a constant factor.

6.3 Ackerberg–O’Malley Resonance

139

Remarks. 1. For the link with the original problem of resonance of Ackerberg– O’Malley [1] and with overstability, one may consult [25]. 2. In the case p D 2, the first proof of the sufficiency of the condition in (b) for resonance was given by Sibuya [53]. In [25], we conjectured item (b) and proved it in two cases: firstly in the case p D 2 and secondly in the case p > 2 and ˇ D 0, i.e. g.x; 0/ D O.x p1 /. In [11], De Maesschalck presents similar results. We describe them after Theorem 6.8. 3. In the case ˇ D 0, the reduced equation f .x; 0/z0 D g.x; 0/z has no singularity at x D 0, thus Z0 1. As a consequence, the passage to the Riccati equation at the beginning of the proof below does not introduce poles if ˇ D 0. If ˇ ¤ 0, however, the solutions y ˙ of this Riccati equation may have poles in ŒL; L (where L is introduced at the beginning of the proof); this is one reason why we return to solutions of the corresponding linear equation of order two. 4. If we fix ˛; ˇ such that D D ˇ=˛ is a non-negative integer congruent to 0 or 1 mod p, then the condition of Theorem 6.5 (a) becomes a sequence of polynomial conditions in the coefficients anm and bnm of the series f .x; "/ D ˛x p1 C

X

a0m x m C

mp

XX

anm x m "n

n1 m0

respectively g.x; "/ D ˇx p2 C

X

b0m x m C

mp1

XX

bnm x m "n :

n1 m0

This is shown analogously to Remark 6 after Theorem 6.1. Proof of Theorem 6.5: The first step, following an idea of Jean-Louis Callot, is to go to the corresponding Riccati equation. This is done by setting y D "z0 =z and yields "y 0 D f .x; "/y  "g.x; "/  y 2 :

(6.37)

From our assumption, Corollary 5.16 can be applied with r D p  1. We obtain that there are 0 ; x0 ; L > 0 such that (6.37) has two solutions y ˙ .x; /, defined for 1=p  D "1=p 2 0; "0  and ˙x 2 ŒL; x0 , and that these solutions have Gevrey CAsEs ˙ y ˙ .x; /  1 gp1 p

1   x  p1 X   n ˙ x a   C .x/ C g n n  

(6.38)

nDp

as  tends to 0, uniformly on ˙ŒL; x0 . Here and in the sequel, we combine both statements for x positive resp. x negative using the symbol ˙; the functions an are real analytic on Œx0 ; x0  and the functions gn˙ are real analytic on ˙ŒL; C1Œ. The definition of Gevrey CAsE is the usual Definition 3.6, with sectors and quasi-sectors replaced by intervals. Notice that, from (6.37), the derivatives of y ˙ also have CAsEs of Gevrey order p1 , thanks to the compatibility of Gevrey CAsEs with the elementary

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6 Applications

operations; these CAsEs of the derivatives are therefore obtained by differentiating term by term, due to the remark after Lemma 2.7. Furthermore, from Remark 3 after Proposition 2.16, the functions an in (6.38) are the same for y C and for y  . ˙ Plugging these CAsEs in (6.37), we obtain that gp1 is the unique solution of the reduced Riccati equation dY D ˛X p1 Y  ˇX p2  Y 2 dX

(6.39)

˙ which tends to 0 as X tends to ˙1. Hence we have gp1 .X / D

D CO.X 2 /; X ! X 0 Z0˙ .X /=Z0˙ .X / where Z0˙

˙ ˙1 with D D ˇ=˛. We will also use that gp1 .X / D is the only solution of the reduced linear equation associated to (6.35)

dZ d 2Z  ˛X p1 C ˇX p2 Z D 0 dX 2 dX such that

Z0˙ .X /

1

p

(6.40)

  1 X mp as X tends to ˙1. This will .˙X / 1 C dm X D

mD1

facilitate the passage from the CAsE (6.38)  the linear equation.  R to This passage is done by z D exp 1" y and thus we have to integrate a CAsE. P1 m and the fact that gnm is zero Using the expansions gn˙ .X /  1 mD1 gnm X p when n C m 6 0 mod p, we obtain by Proposition 2.9 and Exercises 3.11 or 4.8 1 "

Z

x

b˙ .x; /  Y b˙ .˙r; / y ˙ .; / d   1 Y p

˙r

with    b˙ .x; / D log D Z0˙ x C A0 .x/ C R."/ b log.x p C "/ C Y 

(6.41)

1  X   An .x/ C Gn˙ x n nD1

Z

x

where An .x/ D

b D anCp ./ d , R."/

0

Gn˙ .X /

1 X

glp1;1 "l1 and

lD2

Z D

1 p

X ˙1

  ˙ gnCp1 .T /  gnCp1;1 T p1 .T p C 1/1 dT: ˙

b .˙r; / is identified with the formal series in As in the proof of Proposition 2.9, Y p powers of  obtained by developing   log.r C"/ and using the asymptotic expansions  D ˙ ˙r ˙ ˙r of log  Z0 .  / and Gn  as  ! 0.

6.3 Ackerberg–O’Malley Resonance

141

Using the compatibility of the CAsEs with composition (here with exponentiation) and multiplying by a function of  only having the asymptotic  ˙  b .˙r; / , we deduce the existence of two solutions having the D exp Y generalized CAsE described in (6.42) below. We describe these solutions in a separate statement since it might be useful for the study of second order linear equations, independently of the problem of resonance. Proposition 6.6. Under assumption (6.34), there are x0 ; L > 0, a formal series b D R."/ b R of Gevrey order 1 without constant term, real analytic functions Bn on Œx0 ; x0  with B0 .0/ D 1 and Hn˙ on ˙ŒL; 1Œ and two solutions z˙ of (6.32), such that    R."/ log.x p C"/ B0 .x/ C z˙ .x; /  1 Z0˙ x eb p

 1  X   n ˙ x Bn .x/ C Hn  

(6.42)

nD1

as  ! 0, uniformly on ŒL; x0 , resp. Œx0 ; L, where Z0˙ .X / are the unique  solutions of (6.40) satisfying Z0˙ .X /  .˙X /D 1 C O.X 1 / as X ! ˙1. More precisely, the relation (6.42) means that, for any function R W 0; "0  ! R b having R."/ as asymptotic expansion of Gevrey order 1 (with respect to "), the function e R."/ log.x

p C"/

z˙ .x; /=Z0˙

x  

 1 B0 .x/ C

1  X

p

Bn .x/ C Hn˙

 x  

n

nD1

has a CAsE of Gevrey order p1 . ˙ Remarks. 1. In terms of the functions  An and Gn above, the functions Bn and ˙ Hn are defined by B0 .x/ D exp A0 .x/ and by

B0 .x/ C

1  X   Bn .x/ C Hn˙ x n

 X 1    n ˙ x An .x/ C Gn   : D B0 .x/ exp nD1

(6.43)

nD1

2. The uniqueness of the CAsEs of the solutions of the Riccati equation for x positive resp. negative implies that the corresponding expansions (6.42) of the formal solutions of (6.32) are unique up to a multiplicative constant b c./. 3. As this is the case for the functions an in (6.38) and the functions An in (6.41), as well as the asymptotic expansions of the functions Hn˙ as X ! ˙1, the C  functions Bn .x/ are the  same for z and z . 4. For fixed x, D Z0˙ x and the two other factors of the expansions (6.42) of z˙ have an asymptotic expansion in powers of p D " as  ! 0. This is the reason why we chose `.x; / D log.x p C p / when we applied Proposition 2.9.

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6 Applications

5. If Re .D/ > 0, then D z˙ .x; / tends to x D B0 .x/ uniformly on ˙ŒL; x0  as  ! 0. Indeed, as a product of two functions having CAsEs,  D ˙ p e R."/ log.x C"/ x z .x; / admits, as CAsE of Gevrey order p1 , the formal series obtained by developing  x D 

  1  X     n   ˙ x Bn .x/ C Hn   D B0 .x/CG0˙ x CO./ B0 .x/C

Z0˙ x

nD1

with some function G0˙ .X / having an asymptotic expansion without constant   term as X ! ˙1. Multiplying by x D and using x D G0˙ . x / D O min.1;Re D/ , we obtain the wanted result. 6. A comparison with the expansions presented in the sequel shows that the products Z0˙ .X /Hn˙ .X / are analytic in a neighborhood of the real axis; since Z0˙ might have zeroes, this does not prevent Hn˙ from having singularities, even on the real axis, but these singularities are necessarily poles. The products Z0˙ .X /Hn˙ .X /, however, can have singularities of another type outside the real axis. This comes from the fact that one factor of the asymptotics is a series containing log.X p C 1/ which has branch points at the zeroes of X p C 1. 7. As above, one can also differentiate these expansions term by term. Continuation of the proof of Theorem 6.5: On intervals of the form ˙ŒL; M  with some M > 0, we have therefore Z ˙ .X; / WD z˙ .X; / D Z0˙ .X / C O./. Moreover Z ˙ is a solution of the regularly perturbed equation (6.35) which reduces to (6.40) as  D 0. Therefore the analytic continuation of z˙ on ŒM ; M  satisfies  ˙ ˙ x z .x; / D Z0  C O./ by the theorem of analytic dependence of ordinary differential equations with respect to parameters. The existence of a local resonant solution of (6.32) thus needs first that Z0C is proportional to Z0 . We will later show the following lemma. Lemma 6.7. The function Z0C is proportional to Z0 if and only if D D ˇ=˛ is a non-negative integer congruent to 0 or 1 modulo p. Moreover, in this case, Z0 WD .˙1/D Z0˙ is a polynomial of degree D. The condition on D is assumed from now on. Then, using Proposition 3.7, relation (6.42) implies the following inner expansions for ˙X 2 ŒL; M   R."/ log.X p C1/ R."/ log " Z0 .X /C  1 eb z˙ .X; /e b p

1 X 

Zn .X / C

Kn˙ .X /





(6.44)

n



nD1

with polynomials Zn of degree at most n C D and functions Kn˙ 2 GL˙ , where GL˙ is the set of all real analytic functions on ŒL; C1Œ having an asymptotic expansion without constant term as X tends to C1.

6.3 Ackerberg–O’Malley Resonance

143

˙ More precisely, let Pn denote the polynomial  1 part of the product Z0 Hn ; its ˙ as X ! ˙1. Hence we have degree is at most D  1 since Hn .X / D O X Kn˙ D Z0 Hn˙  Pn . The polynomial Zn is obtained by developing the functions X 7! Bk .X /k ; k  n using the Taylor formula:

Zn .X / D Pn .X / C Z0 .X /

n .k/ X B

nk .0/

kŠ

Xk:

kD0

Developing the exponential on the right hand side of (6.44), we thus obtain functions Un˙ 2 GL˙ ŒX; log.X p C1/ (i.e. polynomials in X and log.X p C1/ with coefficients in GL˙ ) such that R."/ log " z˙ .X; /e b  1 Z0 .X / C

1 X

p

Un˙ .X /n

(6.45)

nD1

as  ! 0 uniformly for ˙X 2 ŒL; M . Furthermore the formal series in (6.45) is a formal solution of (6.35), hence its coefficients can be continued analytically to entire functions. By construction, the growth of U ˙ .X / as X tends to ˙1 is polynomial. Since (6.35) is regularly perturbed (and linear), we can apply the theorem of analytic dependence and Proposition 3.8. We obtain that (6.45) remains valid for X 2 ŒM; M , as well as (6.44). As before, (6.44) and (6.45) may be differentiated term by term with respect to X . First suppose that (6.32) has a local resonant solution, denoted by z.x; /, defined and bounded as ı  x  ı and 0 <   0 . Then, reducing ı if needed, the function y W .x; / 7! "z0 .x; /=z.x; / is a solution of (6.37) with bounded initial conditions at ˙ı. Therefore it has CAsEs (6.38) as 0 <   0 and ˙x 2 ŒL; ı with˚ some R x L > 0 and reduced 0 ; ı. Up to a constant factor, the functions exp 1" ˙ı y./ d  hence have the generalized CAsEs (6.42) and also the inner expansions (6.45). By construction, these functions are proportional to the given solution z, hence each one is proportional to the other. Thus the quotient of both solutions must have an asymptotic expansion P in powers of , denoted by b c./. 1 C n As a consequence the formal series Z .X / C U .X / must be equal to 0 nD1 n   P  n b c./. This implies that the coefficients of the two .1/D Z0 .X / C 1 U .X / nD1 n series have polynomial growth in both directions C1 and 1, which proves the necessity for the first item of the theorem. To prove that it is a sufficient condition, we use the existence of two solutions satisfying (6.42) and thus also (6.44) and (6.45) for X 2 ŒM; M . The existence of a formal solution of (6.35) with coefficients of polynomial as X tends P1 growth C n to C1 and 1 implies that the formal series Z .X / C U .X / must be 0 nD1 n P1 D  n proportional to Z0 .X / C nD1 .1/ Un .X / . Since the slow coefficients Bn .x/ are the same in (6.42), the quotient has to be 1. At this point, we use that our CAsEs are Gevrey; in particular (6.45). Moreover, these are the same expansions for zC and z and in particular these two functions and their derivatives have the same Gevrey

144

6 Applications 0

asymptotic expansion at x D 0. The differences zC .0; /  z .0; / and zC .0; /  z0 .0; / are therefore exponentially small (of order p in , i.e. of order 1 in "). Similarly to the proof of Theorem 6.1, we deduce by the Gronwall Lemma that zC and z are exponentially close on an interval Œı; ı with ı > 0 independent of . The continuations of the solutions D z˙ .x; /, on an interval independent of  containing 0 in its interior, thus tend to x D B0 .x/ uniformly in  by Remark 5 after Proposition 6.6. Hence these are local resonant solutions. Now suppose that (6.32) has a local C 1 -resonant solution. Then the functions zC and z must have bounded derivatives (uniformly with respect to ") in a neighborhood of 0 independent of ". Then the functions Z ˙ W .X; / 7! z˙ .X; / .m/ must satisfy Z ˙ .X; / D O.m / uniformly on ˙ŒL; M  for all m 2 N. If one of b or one of the functions Kn˙ were non-zero, and therefore if one the coefficients of R ˙ of the Un were not a polynomial in X , then differentiating (6.45) term by term, we .m/ would obtain that Z ˙ .X; / could not be O.m / for m large enough. Therefore b D 0 and Kn˙ D 0 for all n. With this information, (6.42) implies we must have R."/ an expansion for z˙ of the form D ˙

 z .x; / 

1 X nD0

Cn .x/ WD  Z0 n

D

1 x X  nD0

Bn .x/ C

1 X

D Pn

x 

n

(6.46)

nD1

    here D Z0 x and the D Pn x are polynomials in x and  and C0 .x/ D x D B0 .x/ 6 0. In the case where there is a local C 1 -resonant solution, one has D z˙ .x; /  P1 n nD0 Cn .x/ and, in particular, the right hand side is a non-trivial formal solution of (6.32) with coefficients analytic at x D 0. Multiplying this solution by a series b c./ if necessary, we can impose for instance C0 .0/ D 1; C1 .0/ D    D Cp1 .0/ D 0, and we then check that Cn D 0 if n 6 0 mod p; thus it is a formal solution in powers of " D p . To see that it is once again a sufficient condition, we use once more the existence of the two solutions satisfying (6.42). The existence of a non-trivial formal solution of (6.32) without singularity at x D 0 implies, replacing x D X , that (6.35) has a formal solution with coefficients in X . Since the formal solutions P polynomial ˙ n of (6.35) Z0 .X / C 1 U .X / , with Un˙ .X / of polynomial growth as X tends nD1 n to infinity, are unique up to a constant factor, the functions Un˙ are polynomials. A b D 0 and that Kn˙ 0 for all n. Using their comparison with (6.44) shows that R."/ definition (below (6.44)), we thus obtain that the functions z˙ satisfy (6.46). This asymptotic expansion can be considered as a CAsE without fast part, and this for x 2 ŒL; r resp. x 2 Œr; L with some L > 0. We extend the validity to ˙Œ0; r as above using the fact that X 7! z˙ .X; / satisfies a regularly perturbed differential equation on ˙Œ0; L. Finally, we use once again that (6.42), and therefore (6.46), are Gevrey. Since it is the same expansion for zC and z , we deduce as above by the Gronwall lemma that the functions z˙ are local resonant solutions. Differentiating term by term the CAsE (6.46), we can do the same reasoning for all their derivatives; therefore they are local C 1 -resonant solutions. t u

6.3 Ackerberg–O’Malley Resonance

145

Remark. A proof of the sufficiency for the second item as in [9] is possible, see also [25]. We first prove the Gevrey character of the formal solution b z (up to a factor b c."/); then we construct a quasi-solution of (6.32) and finally we show that the solution with the same initial condition at 0 is a local C 1 -resonant solution. We end this section with the proof of Lemma 6.7: Suppose that Z0C D c Z0 for some c 2 R. As p is even, the change of variable X 7! X leaves (6.40) unchanged. From their definition by their asymptotics, we therefore have Z0C .X / D Z0 .X /. As a consequence, one also has Z0 D c Z0C , hence c D ˙1. In the case c D 1, the function Z0 WD Z0 D Z0C is even, therefore Z00 .0/ D 0. In the case c D 1, the function Z0 WD Z0 D Z0C is odd, hence Z0 .0/ D 0. By the general theory of irregular singular  points of second order linear equations, we have Z0 .X / D X D 1 C O.X 1 / in the sector jarg.X /j < 3 2p (corresponding to a mountain and two adjacent valleys for the associated Riccati equation). Equation (6.40) has other symmetries. Set % D exp.2 i=p/; the change of e 0 defined by Z e 0 .X / D variable X 7! %X leaves (6.40) unchanged. The function Z   D 1 e 1 C O.X Z .%X / thus satisfies (6.40) and Z .X / D exp.2D i=p/X / as 0 ˇ ˇ 0 ˇ 3 ˇ ˇarg X C 2 p ˇ < 2p . e 0 D Z0 and thus the In the case c D 1, the initial values at X D 0 imply that Z  uniqueness of the asymptotic behavior in the intersection fX I arg X 2 Œ 3 2p ;  2p g of the two sectors mentioned above implies that exp.2D i=p/ D 1 and thus D has k to be integer and a multiple  of p. Using  Z0 .% X / D Z0 .X /, k D 0; 1; : : : ; p  1, D 1 we obtain Z0 .X / D X 1 C O.X / as jX j ! C1 for any arg X . This implies that D is non-negative and that Z0 .X / is a polynomial of degree D. e 0 D %Z0 and thus exp.2D i=p/ D % D In the case c D 1, we obtain Z exp.2 i=p/. This implies that D  1 is a multiple of p; as before we deduce that D is non-negative and that Z0 .X / is a polynomial of degree D. This proves the necessity of the condition of the lemma. It is easily shown that (6.40) has a polynomial solution in both cases, which is necessarily proportional to Z0˙ ; thus the condition is sufficient. t u The previous results are of local nature. As in Sect. 6.1, the conditions of Theorem 6.5 also give global results. Theorem 6.8. Let a < 0 < b and f; g be analytic in a complex neighborhood U of Œa; b. We make the assumption (6.34) and assume moreover that f .x; 0/ is real on the real axis and that xf .x; 0/ > 0 when x 2 Œa; b n f0g. Then the conditions of Theorem 6.5 (a) are equivalent to the existence of a global resonant solution of (6.32), i.e. a solution which tends to a non-trivial solution of the reduced equation (6.33) uniformly on Œa; b as " ! 0. Similarly, the conditions of Theorem 6.5 (b) are equivalent to the existence of a global C 1 -resonant solution of (6.32), i.e. a solution which tends to a non-trivial solution of the reduced equation (6.33) uniformly on Œa; b and whose all derivatives are bounded uniformly on Œa; b as " ! 0.

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6 Applications

Remark. In [11], De Maesschalck proves that the condition of Theorem 6.5 (b) is sufficient for the existence of a global C 1 -resonant solution, cf. Theorem 2. He also shows, cf. Theorem 5 of the same article, that the existence of a local resonant solution implies that of a global resonant solution. Nevertheless, the necessity of the condition in Theorem 6.5 (b) and the necessity and sufficiency of the condition in Theorem 6.5 (a) for the existence of a local or global resonant solution are new. Proof. The necessity of the conditions of Theorem 6.5 (a) for the existence of a global resonant solution is evident, since they are already necessary for the existence of a local resonant solution. The proof of the sufficiency is analogous to that of the implication   (c))(b) of Theorem 6.1. By a change of variable x D h.t/ such that F h.t/ D t p for 0 Rx p1 F0 .x/ D 0 f .t; 0/ dt, we may suppose that f .x; 0/ D px . The assumption on f ensures that this change of variable can be done globally on some neighborhood   of Œa; b. By a change of variable z D e G.x;"/e z with G 0 .x; "/ D 1" f .x; "/  px p1 , we may suppose that f .x; "/ D px p1 . Without loss of generality, we can suppose the existence of e a; e b such that Œe a; e b U , e a
p ="

.z1 z02  z01 z2 /.x; "/:

By the Liouville theorem, it is a function of " only. The wronskian is a bilinear and antisymmetric function. One easily checks, for several solutions zi of (6.32), the formulas Œz1 ; z2 z3 C Œz2 ; z3 z1 C Œz3 ; z1 z2 D 0;

(6.47)

Œz1 ; z2 Œz3 ; z4   Œz1 ; z3 Œz2 ; z4  C Œz1 ; z4 Œz2 ; z3  D 0:

(6.48)

Now, with the assumptions of the theorem, consider the solutions za ; zb of (6.32) determined by a; "/ D 1; z0a .e a; "/ D 0; zb .e b; "/ D 1; z0b .e b; "/ D 0: za .e It is well-known that these solutions have outer asymptotic expansions on Œe aC d; d , respectively Œd; e b d , where d > 0 is arbitrarily small and that they extend on all U , but the expansions of the continuations may be different or even may not exist. We will prove that Œza ; zb ."/ is exponentially small. More precisely we prove below that there is C > ap such that   Œza ; zb ."/ D O e C =" as " ! 0:

(6.49)

Then, we will prove that this property implies that zb is a global resonant solution.

6.3 Ackerberg–O’Malley Resonance

147

For that purpose, we introduce other solutions and we proceed analogously to the proof of (6.4). First we use the existence of  2 R, and for any sufficiently small   > 0 of a integer L > 4p and 0 ; % > 0 small enough, such that there are solutions    .x; /; ` D L; : : : ; L of (6.32) holomorphic as  2 S` D S ` 2pL  ; ` 2pL C z˙ `    3  3  ˙ ; 0 and x 2 V` ./ D ˙V  2p C ` 2pL C 2; 2p C ` 2pL  2; %;  jj having generalized CAsEs ˙ z˙ ` .x; /  1 Z0 p

  1  X x b   n R."/ log.x p C"/ ˙ x B B (6.50) e  .x/ C .x/ C H 0 n n   nD1

with the functions Z0˙ , Bn and Hn˙ of Proposition 6.6; the Bn are holomorphic 3 bounded in D.0; %/, the Hn˙ in ˙V  3 2p C2; 2p 2; 1;  and have asymptotic expansions without constant term as X ! ˙1. This is proven similarly to Proposition 6.6 using the Riccati equation (6.37) and rotations  D e i e , x D  ˙e i e x with D ` 2pL ; the fact that the functions an and gn˙ are independent of ` is due to the uniqueness of the formal solution of (6.37), i.e. to Theorem 5.4, and to the construction of the solutions of Proposition 6.6 from the solutions of (6.37). Let us notice that (6.50) implies, for  ! 0  D  D z˙ ` .x; / D   x B0 .x/ C O./ for x 2 ˙V` ./; jxj > d; ˙ x z˙ ` .x; / D Z0  C O./ for jxj  K jj :

(6.51)

for all K > 0. As in the proof of Theorem 6.5, the last relation is a consequence of (6.50) and of the fact that the inner equation is regularly perturbed. As we have seen in the proof of Theorem 6.5, the condition (a) is equivalent to the conditions Z0C D Z0 DW Z0 and HnC D Hn DW Hn for all n. Since the asymptotic expansion is of Gevrey order p1 in , this implies that the wronskians  ŒzC ` ; z`  are exponentially small. Hence there exists s > 0 such that  s=jjp   ŒzC ` ; z` ./ D O e

(6.52)

as ` 2 fL; : : : ; Lg and  2 S` . Reducing % if needed, we can suppose %p < s. We also need solutions with an asymptotic behavior complementary to the x p =" functions z˙ w which ` . To construct them, we make the change of variable z D e transforms (6.32) in "w00 Cpx p1 w0 Ce g .x; "/w D 0 where e g .x; "/ D g.x; "/Cp.p1/x p2 : (6.53) This equation is very similar to (6.32), but the sign of the coefficient of the derivative is changed. We treat it analogously to (6.32)—the only difference is that the mountains and the valleys of the Riccati equation g .x; "/  u2 "u0 D px p1 u  e

(6.54)

148

6 Applications

satisfied by u D "w0 =w have been switched with respect to that of (6.37). We thus obtain, with ,  , L, 0 and % as before, solutions v˙ ` .x; /; ` D L; : : : ; L    of (6.32) defined, holomorphic and bounded as  2 S` D S ` 2pL  ; ` 2pL C ; 0   e˙ ./ D ˙V   C `  C 2; 5 C `   2; %;  jj having and x 2 V ` 2p 2pL 2p 2pL generalized CAsEs v˙ ` .x; /

e˙ 1 Z 0 p

x 

e

b R."/ log.x p C"/

  1  X   n ˙ x e e e B n .x/ C H n   B 0 .x/C (6.55) nD1

 e ˙ are the unique solutions of (6.40) with Z e ˙ .X /  e X p X DpC1 1 C where Z 0 0  en are analytic on D.0; %/, B e0 does not O.X 1 / as arg.˙X / D p , jX j ! 1, the B   e0 .0/ D 1 and finally the H e n are holomorphic in ˙V  vanish, B C 2; 5  2p 2p  2; 1;  and have asymptotic expansions without constant term as X ! ˙1. Let us also notice that (6.55) implies, as for the z˙ ` ,   x DpC1 e x p =p e˙ ./; jxj  d; . B .x; / D e / .x/ C O./ for x 2 V v˙ 0 ` `  (6.56)   e ˙ x C O./ for jxj  K jj v˙ .x; / D Z `

0



 x p =p  as  ! 0 and thus that v˙ as  > 0 and ˙x 2 Œ0; %Œ; the fact ` .x; / D O e that the negative real axis descends in a valley of (6.54) implies that this remains true for v ` and the whole interval Œa; 0. Using the outer expansions of zajb , the estimates (6.51) and (6.56), and the definition of the wronskians, we obtain D1 D1 ˙ 1 Œza ; v ; Œzb ; vC ; Œz˙ ` ./   ` ; v` ./   ` ./  

(6.57)

 1 as well as ŒvC ` ; v` ./ D O. /, where the symbol f ./  g./ means here that the quotient f ./=g./ tends to a non-zero limit as  ! 0. The proof of (6.49) uses the Phragmen–Lindel¨of theorem and follows the lines of the proof of (6.4). Given ı; d > 0 small, let D  .ı; d / denote the domain containing e a ; d  whose image by F .x/ D x p is the triangle having the vertices F .e a/; iF .e a/ tan ı; iF .e a/ tan ı without the disk of center 0 and radius d . Similarly, let D C .ı; d / denote the domain containing Œd; e bŒ whose image by F is the triangle of vertices F .e b/; iF .e b/ tan ı; iF .e b/ tan ı without the disk of center 0 and radius d . If we choose ı small enough, for e ı > ı, e ı arbitrarily close to ı and for 0 ; d small enough, the solutions za .x; /, resp. zb .x; /, are holomorphic and bounded in the    ı  ı sector S D S  2p Ce ; e ; 0 and for x in D  .ı; d /, resp. D C .ı; d /. This p 2p p is classical (Fig. 6.5). Now consider e ı > 0 such that F .e a/ sin e ı < %p and ı 2 0; e ıŒ arbitrary. Let ` 2 fL; : : : ; Lg and  2 S` \ S , arg  D . Then zb .x; / and zC ` .x; / are holomorphic and bounded in a neighborhood of the point x D T e i where T p D F .e a/ sin ı. We deduce with (6.51) that

6.3 Ackerberg–O’Malley Resonance

149

˜b

a˜

F (˜a)

F (˜b)

Fig. 6.5 The domains D  .ı; d /, D C .ı; d / and their images by F .x/ D x 4 . Here once again the scaling has not been respected between the two drawings for a better visibility

     p p D Œzb ; zC exp  .T e i /p =p D O jjD e T =jj ` ./ D O jj

(6.58)

  D T p =jjp as  2 as  2 S` \ S . Similarly, we show that Œza ; z ./ D O e jj ` S` \ S: C  Formula (6.48) applied to za ; z ` ; z` and v` is written C  C    C  Œza ; z ` Œz` ; v`   Œza ; z` Œz` ; v`  C Œza ; v` Œz` ; z`  D 0 :

With (6.52), the estimate of Œza ; z ` above and (6.57),  this implies that the wronskian

D T C Œza ; zC e `  satisfies Œza ; z` ./ D O jj

p =jjp

C . Applying (6.48) to za ; z ` ; v`

C D1 and v ; similarly, we obtain ` , we obtain Œza ; v` ./   D1 : Œzb ; v ` ./  

(6.59)

C C Applying (6.48) time  to za ; z` ; zb and v` and using (6.58), we obtain  this p p Œza ; zb ./ D O e T =jj as  2 S` \ S . Since the union of the S` covers S , this gives  p p Œza ; zb ./ D O e T =jj

for  2 S . For pj arg j D

 2

   a/  with e ı, this estimate implies Œza ; zb ./ D O exp q F.e p

ı arbitrarily close to 1. By the Phragmen–Lindel¨of theorem, this remains q D sine sin ı valid for arg  D 0. This finally proves (6.49). C Formula (6.47) applied to zb ; zC 0 and v0 gives C C C C C Œzb ; zC 0 v0 C Œz0 ; v0 zb C Œv0 ; zb z0 D 0:

 x p =p  for x 2 Œ0; d , With the estimates (6.58) and (6.57), and with vC 0 .x; / D O e  > 0, this shows that zb is exponentially close to a solution proportional to D zC 0 on Œ0; d  and thus tends to a limit also on this interval, if d < T . The proportionality coefficient,

Œzb ;vC 0 

C ŒzC 0 ;v0 

D , depends on  but tends to a non-zero limit as  tends to 0.

150

6 Applications

 Using (6.59), we treat similarly zb , z 0 and v0 on Œd; 0. Finally (6.47) applied to  zb ; za and v0 gives   Œzb ; za v 0 C Œza ; v0 zb C Œv0 ; zb za D 0:

 x p =p  on Œa; 0, The estimate (6.49) then implies, with (6.57) and v 0 .x; / D O e that zb remains exponentially close to a solution proportional to za on Œa; d , too; therefore it is a global resonant solution. If the condition of Theorem 6.5 (b) is satisfied, then so is that of Theorem 6.5 (a). The solution zb is therefore a global resonant solution and it remains to show that all its derivatives are bounded. By the proof of Theorem 6.5 (b), the derivatives of D z˙ ` .x; / are bounded on ˙Œ0; d ; since zb is exponentially close to them, this remains true for zb . Finally because of the differential equation, all the derivatives of za are bounded on Œa; d ; since zb is exponentially close to it, it follows that zb is a global C 1 -resonant solution on all Œa; b. t u

Chapter 7

Historical Remarks

The literature on matching, i.e. the method of matched asymptotic expansions is abundant. Already in the fifties and sixties this method was common, see e.g. the work of Kaplun/Lagerstrom [40], Erd´elyi [18]. The matching is the main subject of Eckhaus’ book [17] and the Chaps. VII and VIII of Wasow’s book [62]. In the latter book, the method is presented for linear systems of singularly perturbed differential equations. To describe the boundary layers, CAsEs in the regular case (as in Sect. 5.1, but without studying their Gevrey character) were studied in several works. For a systematic treatment, we mentioned the works of Vasil’eva/Butuzov [59] and Benoˆıt/El Hamidi/Fruchard [4] only in the preface, but these CAsEs are widely used. Actually they are already present in the early literature on matching and sometimes cited as the Lagerstrom–Kaplun principle. This principle consists of the following. If there are an inner and an outer expansion and if the regions of validity of these expansions overlap, then a single composite expansion can be formed by adding both expansions and removing their common terms. Besides the works of Eckhaus [17] and Wasow [61, 62] cited above, we would like to mention the works of Fraenkel [23] and Skinner [54–57] and describe them in a few words. In [23], Fraenkel defines operators Ep and Hp which, in our context, essentially take the first p terms of the outer, resp. inner, expansion of a function f D f .x; "/. He then shows that the function .Ep C Hp  Ep Hp /f is an approximation up to order p of f in ", uniformly for x 2 Œ0; r. Erd´elyi presents the same method in [18], using analogous notation. As an example, in [18] p.113, a composite expansion is given by y c D y o C y i  .y o /i D y o C y i  .y i /o . In his book [54], Skinner refines the method and judiciously spreads the terms of Ep Hp f to regroup some with Ep f and others with Hp f . Then result the first terms of a CAsE for f , see pp. 4–5 of [54]. The first chapter of Skinner’s book presents the general theory of matching in a very concise and elegant way. Then the following chapters implement the theory for equations of increasing difficulty, of order one or two. A large part of the book is devoted to the explicit calculation of these CAsEs, using computer algebra software. A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2 7, © Springer-Verlag Berlin Heidelberg 2013

151

152

7 Historical Remarks

In our opinion, this book is one of the best and accessible presentations of the matching method for singularly perturbed ODEs. We would like to emphasize that all authors we mentioned above prove that composite expansions give uniform approximations by proving beforehand that the functions have an inner and an outer approximation and that the regions of validity intersect (“overlapping assumption”). There seems to be no systematic study of the compatibility of composite expansion with product, differentiation, integration or composition to the left and right by functions of one variable. Moreover, mostly only finite sums are taken into account. Wasow’s book [61] also presents an alternative to the matching and CAsEs for 2 by 2 systems: the uniform simplification to an equation of special functions. This method is also the subject of Sibuya’s memoir [52]. For example, the equation  0 1 "y 0 D A.x; "/y in C2 with A.x; 0/ D can (under certain conditions) x 2 =4 0 ! 0 1 be reduced to "z0 D x 2 z by a transformation y D T .x; "/z, where 4 C "c."/ 0 T .x; "/ has an asymptotic expansion as " ! 0 uniformly a neighborhood of  in  x2 2 00 x D 0. The reduced equation is equivalent to " u D 4 C "c."/ u and can  2  2 x therefore be reduced to dd XU2 D X4 C c."/ U by U.X; "1=2 / D u. "1=2 ; "/. We could see this uniform simplification as separation of the asymptotics in a slow part T .x; "/ and a fast part coming from the reduced equation. In this regard, we mention work in progress by Charlotte Hulek. She proves the main result of Sibuya’s memoir using our CAsEs and accurate tracking of their Gevrey type. We only found one work on composite expansions not using matching: the thesis of Forget [20] and the corresponding articles [21,22]. In his thesis, the author uses a variant of CAsEs to obtain uniform approximations of canard solutions (and values) of equations similar to (5.11); this has been mentioned in Sects. 1.4 and 1.6, and in Remark 2 after Theorem 6.1. The modern theory of geometric singular perturbation was initiated by Fenichel [19] and widely developed by several authors, among them one can cite e.g. Jones [33] and Kopell [36, 37]. This theory may be seen as a systematic and advanced treatment of the matching using a variant of the theory of invariant manifolds. The theory is complete in the case of normally hyperbolic points. Unfortunately in applications one often has to tackle degenerate (i.e. non-hyperbolic) points where the Fenichel theory no longer applies directly. To overcome this difficulty, Dumortier/Roussarie developed in [16] a technique of desingularization (“blowup”). Several authors use this technique. One can cite Panazzolo [47], Krupa, Szm´olyan, Wechselberger [38, 39, 58], De Maesschalck [10–13], Van Gils [30], Hek [31]. Almost all these works in geometric singular perturbation deal with real problems. Asymptotic expansions are scarce, a fortiori Gevrey asymptotics, with the notable exception of the work of De Maesschalck et al. [7, 10–13]. Nevertheless, in these latter articles the Gevrey aspect is generally well separated from the geometric tools, and there are no CAsEs.

7 Historical Remarks

153

Finally, for the convenience of the reader, we list the connections with previous work discussed at various occasions in our memoir. The relation with the work of Matzinger [42–44] has been detailed in Sect. 5.3. The relation between CAsEs on an annulus in X D x= and the monomial asymptotics of [8] is detailed in the first remark after Definition 2.9 and for the Gevrey theory after Sect. 3.2. The works of Diener, Isambert and Gautheron [14,29,32] are the basis of our Sect. 6.2 and referred to throughout that section. The relation to the results of De Maesschalck on canard solutions and resonance [10, 11] was mentioned in Remark 1 after Theorem 6.1 and in the remark after Theorem 6.8.

References

1. Ackerberg, R.C., O’Malley, R.E.: Boundary layer problems exhibiting resonance. Stud. Appl. Math. 49, 277–295 (1970) 2. Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer, New York (2000) ´ Callot, J.-L., Diener, F., Diener, M.: Chasse au canard. Collect. Math. 31, 37–119 3. Benoˆıt, E., (1981) ´ El Hamidi, A., Fruchard, A.: On combined asymptotic expansions in singular 4. Benoˆıt, E., perturbations. Electron. J. Diff. Equat. 51, 1–27 (2002) ´ Fruchard, A., Sch¨afke, R., Wallet, G.: Solutions surstables des e´ quations 5. Benoˆıt, E., diff´erentielles complexes lentes-rapides a` point tournant. Ann. Fac. Sci. Toulouse Math. VII 4, 627–658 (1998) ´ Fruchard, A., Sch¨afke, R., Wallet, G.: Overstability: Toward a global study. C. R. 6. Benoˆıt, E., Acad. Sci. Paris I 326, 873–878 (1998) 7. Bonckaert, P., De Maesschalck, P., Gevrey normal forms of vector fields with one zero eigenvalue. J. Math. Anal. Appl. 344, 301–321 (2008) 8. Canalis-Durand, M., Mozo-Fernandez, J., Sch¨afke, R.: Monomial summability and doubly singular differential equations. J. Differ. Equat. 233, 485–511 (2007) 9. Canalis-Durand, M., Ramis, J.-P., Sch¨afke, R., Sibuya, Y.: Gevrey solutions of singularly perturbed differential equations. J. Reine Angew. Math. 518, 95–129 (2000) 10. De Maesschalck, P.: On maximum bifurcation delay in real planar singularly perturbed vector fields. Nonlinear Anal. 68, 547–576 (2008) 11. De Maesschalck, P.: Ackerberg-O’Malley resonance in boundary value problems with a turning point of any order. Commun. Pure Appl. Anal. 6, 311–333 (2007) 12. De Maesschalck, P.: Gevrey properties of real planar singularly perturbed systems. J. Differ. Equat. 238, 338–365 (2007) 13. De Maesschalck, P., Dumortier, F.: Canard solutions at non-generic turning points. Trans. Am. Math. Soc. 358, 2291–2334 (2006) 14. Diener, M.: Regularizing microscopes and rivers. SIAM J. Math. Anal. 25, 148–173 (1994) 15. Dorodnitsyn, A.A.: Asymptotic solution of the Van der Pol equation. Priklad. Mat. Mekh. 11, 313–328 (1947) (in russian) 16. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Memoir. Am. Math. Soc. 577, 1996 17. Eckhaus, W.: Asymptotic analysis of singular perturbations. Studies in Mathematics and its Applications, vol. 9. North-Holland, Amsterdam (1979) 18. Erd´elyi, A.: Singular perturbations of boundary value problems involving ordinary differential equations. J. Soc. Indust. Appl. Math. 11, 105–116 (1963)

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45. Mischenko, E.F., Rozov, N.Ch.: Differential Equations with Small Parameters and Relaxation Oscillations. Plenum Press, New York and London (1980) 46. O’Malley, R.E.: Singular perturbation methods for ordinary differential equations. Applied Mathematical Sciences, vol. 89. Springer, New York (1991) 47. Panazzolo, D.: On the existence of canard solutions. Publ. Mat. 44, 503–592 (2000) 48. Ramis, J.-P.: D´evissage Gevrey. Ast´erisque 59–60, 173–204 (1978) 49. Ramis, J.-P.: Les s´eries k-sommables et leurs applications. In: Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lect. Notes Physics, vol. 126, pp. 178–199. Springer, New York (1980) 50. Sibuya, Y.: Gevrey property of formal solutions in a parameter. In: Asymptotic and computational analysis (Winnipeg, MB, 1989). Lecture Notes in Pure and Appl. Math., vol. 124, pp. 393–401. Dekker, New York (1990) 51. Sibuya, Y.: Linear differential equations in the complex domain. Problems of Analytic Continuation. Am. Math. Soc., Providence (RI) (1990) 52. Sibuya, Y.: Uniform simplification in a full neighborhood of a transition point. Memoi. Am. Math. Soc. 149 (1974) 53. Sibuya, Y.: A theorem concerning uniform simplification at a transition point and a problem of resonance. SIAM J. Math. Anal. 12(5), 653–668 (1981) 54. Skinner, L.A.: Singular Perturbation Theory. Springer, New York (2011) 55. Skinner, L.A.: Uniform solution of boundary layer problems exhibiting resonance. SIAM J. Appl. Math. 47, 225–231 (1987) 56. Skinner, L.A.: Matched expansion solutions of the first-order turning point problem. SIAM J. Math. Anal. 25, 1402–1411 (1994) 57. Skinner, L.A.: A class of singularly perturbed singular Volterra integral equations. Asymptot. Anal. 22, 113–127 (2000) 58. Szmolyan, P., Wechselberger, M.: Canards in R3. J. Differ. Equat. 177, 419–453 (2001) 59. Vasil’eva, A.B., Butuzov, V.F.: Asymptotic Expansions of the Solutions of Singularly Perturbed Equations. Izdat. “Nauka”, Moscow (1973) (in Russian) 60. Wallet, G.: Surstabilit´e pour une e´ quation diff´erentielle analytique en dimension un. Ann. Inst. Fourier 40, 557–595 (1990) 61. Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Interscience, New York (1965) 62. Wasow, W.: Linear Turning Point Theory. Springer, New York (1985)

Index

Symbols A.; %/, annulus 59 A.r; 1/, infinite annulus 18 D.0; r/, the disk of center 0 and radius r 17 Rd , landscape function 94 S.˛; ˇ; r/, the sector of vertex 0, radius r and angles ˛; ˇ 17 Sl;lC1 , intersection of consecutive sectors 67 U C , special function 5 U  , special function 4 Uj , special function 12 V .˛; ˇ; r; /, quasi-sector 18 V j , quasi-sector for X 59 j;j C1 e Vl ./, intersection of consecutive quasi-sectors in x 65 j Vl ./, quasi-sector for x 60 A .0 ; r0 ; /, domain covered by quasi-sectors 60

, differential operator 25

2 , difference operator 78, 96, 106 G .V /, the space of bounded holomorphic functions in V 18 e , space of (unbounded) holomorphic H functions 90 H .r0 /, the space of bounded holomorphic functions in D.0; r0 / 19 S, shift operator 4, 19 T, shift operator 18 D, usual differential operator 4 I, canonical inclusion 20 C , the nonzero complex numbers 17 f , the Riemann surface of the logarithm 17 C N, the set of natural numbers, including 0 17 e C .V /, space of (unbounded) composite formal series 90

b C .r0 ; V /, space of composite formal series w.r.t. V and D.0; r0 / 20 `, kind of logarithm 27 b R, series of residues 27, 140 , asymptotic expansion 18  1 , Gevrey asymptotic expansion 47 p

ylext , function of x and  on a good covering 66 j int yl , function of x and  on a good covering 66 j gn , function of X 71 j yl , function of x and  on a good covering 66 A

Ackerberg-O’Malley resonance 136 local C 1 -resonant solution 136 Ackerberg, R. C. 119, 139 Asymptotic expansion at infinity 18 inner expansion vi, 29, 30, 33 matched expansion vi, 29, 30, 33 outer expansion vi, 29, 30, 33 in the sense of Poincar´e 3, 18 B

Balser, Werner 55 ´ Benoˆıt, Eric vi, 21, 82 Borel-Ritt theorem for CA sEs 28 classical 28

A. Fruchard and R. Sch¨afke, Composite Asymptotic Expansions, Lecture Notes in Mathematics 2066, DOI 10.1007/978-3-642-34035-2, © Springer-Verlag Berlin Heidelberg 2013

159

160 Borel-Ritt-Gevrey theorem for CA sEs 55 classical 55 Borel transform 56, 85, 100, 110 Boundary layer v Bounded 2, 9 Butuzov, V. F. v, 21, 82

C

Callot, Jean-Louis 139 Canalis-Durand, Mireille vii, 23, 43, 48, 82, 145 Canard solution or duck C m canard 121 C 1 canard 120 angular canard 132 formal canard 136 global canard 105, 120 local canard 120 non-smooth canard 128, 132 Cauchy-Heine formula 65, 66 Cole, J. D. vi, 118 Composite asymptotic expansion, CA sE v, 5, 22 classical composite formal series 21 composite formal series 19 composition of CA sEs 23 convergent CA sE 29, 134 differentiation of CA sEs 26 fast part of a CA sE 19, 23 generalized CA sE 28 integration of CA sEs 27 slow part of a CA sE 19, 23

D

De Maesschalck, Peter 119, 120, 139, 146 ı-descending 94 Diener, Marc 119, 127, 128 Dorodnitsyn, A. A. 118 Dumortier, Freddy 120, 152

Index Erd´elyi, A. 151 Exponential decay 21 Exponentially small vii F Fast variable 4 Fenichel, Niel 152 Flat 21, 53 in the strong sense 53 in the weak sense 53 Forget, Thomas 12, 15, 121 Fraenkel, L. E. 151 Fruchard, Augustin vi, 17, 21, 82, 84, 119, 120, 123, 132, 139, 145 G Gautheron, V´eronique 31, 119, 132 Geometric singular perturbation 152 Gevrey CA sE 43, 50 composition of CA sEs 76 consistent 43 differentiation of CA sEs 48 integration of CA sEs 48, 52 order 43 simultaneous integration of CA sEs 78 type 43, 44, 105 Good covering vii consistent 60, 63 resolution 61 I Isambert, Emmanuel 134

31, 119, 128, 129, 132,

K Kaplun, S. 151 Kevorkian, J. vi, 118

E

L

Eckhaus, Wiktor 33, 151 El Hamidi, Abdallah vi, 21, 82 Elimination of the time v, 118

Lagerstrom, P. A. 151 Landscape 94, 112 Lobry, Claude 9

Index

161

M ´ Matzinger, Eric 116 Monomial expansion 23, 153 Gevrey expansion 48, 153 summability 48 Mountain 10, 13, 99, 100, 103, 112 Mozo, Jorge 23, 48

Sector 17 annulus 18, 23, 28, 29, 48, 59, 60, 79, 153 infinite quasi-sector 18 quasi-sector 18 Series of residues 27 Sibuya, Yasutaka vii, 43, 68, 82, 145, 152 Skinner, Lindsay A. vi, 29, 151 Slow function 81 Slow set v, 81 Slow variable 4

O T O’Malley, Robert E. 118, 119, 139 Overstability 1, 3, 10, 13 Q Quasi-linear equation

89, 94, 105

Truncated Laplace transform 110 Turning point v, 1, 81, 106

U

R

Union Jack equation

Ramis, Jean-Pierre vii, 43, 68, 82, 145 Ramis-Sibuya theorem for CA sEs 63 classical 68 Regular point v, 81 attracting vi Roussarie, Robert 152

V

127

Valley 10, 99, 100, 103, 112 Van der Pol equation 118 Vasil’eva, Adelaida Borisovna

W S Sch¨afke, Reinhard vii, 17, 23, 43, 48, 82, 84, 119, 120, 123, 132, 139, 145

56, 85, 100,

Wallet, Guy 1, 3, 10, 82 Wasow, Wolfgang 30, 151 Watson lemma 54

v, 21, 82

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Edited by J.-M. Morel, B. Teissier; P.K. Maini Editorial Policy (for the publication of monographs) 1. Lecture Notes aim to report new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. Monograph manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialised lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this “lecture notes” character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series, though habilitation theses may be appropriate. 2. Manuscripts should be submitted either online at www.editorialmanager.com/lnm to Springer’s mathematics editorial in Heidelberg, or to one of the series editors. In general, manuscripts will be sent out to 2 external referees for evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further referees may be contacted: The author will be informed of this. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Authors should be aware that incomplete or insufficiently close to final manuscripts almost always result in longer refereeing times and nevertheless unclear referees’ recommendations, making further refereeing of a final draft necessary. Authors should also be aware that parallel submission of their manuscript to another publisher while under consideration for LNM will in general lead to immediate rejection. 3. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include – a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader. For evaluation purposes, manuscripts may be submitted in print or electronic form (print form is still preferred by most referees), in the latter case preferably as pdf- or zipped psfiles. Lecture Notes volumes are, as a rule, printed digitally from the authors’ files. To ensure best results, authors are asked to use the LaTeX2e style files available from Springer’s web-server at: ftp://ftp.springer.de/pub/tex/latex/svmonot1/ (for monographs) and ftp://ftp.springer.de/pub/tex/latex/svmultt1/ (for summer schools/tutorials).

Additional technical instructions, if necessary, are available on request from [email protected]. 4. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files and also the corresponding dvi-, pdf- or zipped ps-file. The LaTeX source files are essential for producing the full-text online version of the book (see http://www.springerlink.com/ openurl.asp?genre=journal&issn=0075-8434 for the existing online volumes of LNM). The actual production of a Lecture Notes volume takes approximately 12 weeks. 5. Authors receive a total of 50 free copies of their volume, but no royalties. They are entitled to a discount of 33.3 % on the price of Springer books purchased for their personal use, if ordering directly from Springer. 6. Commitment to publish is made by letter of intent rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: Professor J.-M. Morel, CMLA, ´ Ecole Normale Sup´erieure de Cachan, 61 Avenue du Pr´esident Wilson, 94235 Cachan Cedex, France E-mail: [email protected] Professor B. Teissier, Institut Math´ematique de Jussieu, ´ UMR 7586 du CNRS, Equipe “G´eom´etrie et Dynamique”, 175 rue du Chevaleret 75013 Paris, France E-mail: [email protected] For the “Mathematical Biosciences Subseries” of LNM: Professor P. K. Maini, Center for Mathematical Biology, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LP, UK E-mail : [email protected] Springer, Mathematics Editorial, Tiergartenstr. 17, 69121 Heidelberg, Germany, Tel.: +49 (6221) 4876-8259 Fax: +49 (6221) 4876-8259 E-mail: [email protected]