Composite Asymptotic Expansions

168 Pages·2013·1.786 MB·English

by  Augustin Fruchard, Reinhard Schäfke (auth.)

Lecture Notes in Mathematics 2066
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
•
Augustin Fruchard Reinhard Scha¨fke
(cid:2)
Composite Asymptotic
Expansions
123
AugustinFruchard ReinhardScha¨fke
LaboratoiredeMathe´matiques, InstitutdeRechercheMathe´matique
InformatiqueetApplications Avance´e
Universite´deHauteAlsace Universite´deStrasbourg
Mulhouse Strasbourg
France France
ISBN978-3-642-34034-5 ISBN978-3-642-34035-2(eBook)
DOI10.1007/978-3-642-34035-2
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012953999
MathematicsSubjectClassification(2010):41A60,34E,34M30,34M60
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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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
functionsofthequotientofthesetwovariables.Thesecompositeasymptoticexpan-
sions(CAsEs for short) are particularly well suited to the description of solutions
of singularly perturbed ordinary differential equations. Their use is classical for
boundarylayers,butlessfamiliarnearturningpoints.
Letusdescribethecontextinafewwords.Consideranequationoftheform
dy
" D˚.x;y;"/ (1)
dx
where˚ isinfinitelydifferentiable,xandyarerealorcomplexvariables,and"isa
smallparameter,realpositiveorinasectorofthecomplexplane.Suchanequation
mightbeobtainedfromanautonomousslow–fastsystem
du
D"f.u;v;"/
dt
(2)
dv
Dg.u;v;"/
dt
byeliminationofthevariablet,i.e.byconsideringthefunctiony Dvıu(cid:3)1.
Theslowsetof(1)isthesubsetL ofC2determinedby˚.x;y;0/D0.Apoint
.x;y/ of L is called regularif @˚.x;y;0/ ¤ 0; otherwise it is called a turning
@y
point.
Ina neighborhoodofa regPularpoint.ex;ey/ 2 L,oneeasilyseesthat(1)hasa
unique formal solutionby D y .x/"n and it is known that certain solutions
n(cid:4)0 n
of(1)haveby asasymptoticexpansioninappropriatedomains.Theclassicaltheory
ofcompositeexpansions(seee.g.Vasi’leva/Butuzov[59]) isusefultodescribethe
boundarylayer (also called inner layer) of a solution which takes atex an initial
valuecloseenoughtoey.Forexample,intherealframework,ifthe point.ex;ey/is
v
vi Preface
attracting,i.e. @˚.ex;ey;0/ < 0,thenanapproximationofthesolutiony D y.x;"/
@y P (cid:2)
can be given, uniformlyon some intervalŒex;ex Cı(cid:2), of the form y .x/C
(cid:3) (cid:4)(cid:5) n(cid:4)0 n
z x(cid:3)ex "n,containingtheformalsolutionby aswellasfunctionsz exponentially
n " n
decreasingatinfinity.SeealsoBenoˆıt/ElHamidi/Fruchard[4] fordetails.
Ataturningpoint.x(cid:5);y(cid:5)/,thecoefficientsoftheformalsolutionmayhavepoles
orramifiedsingularitiesandtheclassicaltheoryofcompositeexpansionsnolonger
applies. The most commonmethodto obtain an approximationof the solutionsis
thematchingoftheso-calledinnerandouterexpansions;inourmemoirwewould
liketoproposeanalternative.
WepresentanextensivestudyofinfiniteCAsEsoftheform
X(cid:2) (cid:3) (cid:4)(cid:5)
a .x/Cg x(cid:3)x(cid:2) (cid:3)n; (3)
n n (cid:3)
n(cid:4)0
(cid:3) a certain rootof ", thatare uniformlyvalid for x in some sector with vertexx(cid:5)
satisfying Kj(cid:3)j (cid:3) jx(cid:4)x(cid:5)j (cid:3) L with some K;L > 0, and sometimes even in
a full neighborhood(cid:2)of x(cid:5)(cid:5)of size O.j(cid:3)j/. Such CAsEs provide outer expansions
by re-expanding g x(cid:3)x(cid:2) with respect to (cid:3) and inner expansions by replacing
n (cid:3)
x D (cid:3)X andre-expandinga .(cid:3)X/.Oneadvantageofcompositeexpansionsisthat
n
theyprovideapproximationsalsoin“intermediate”rangesKj(cid:3)j˛ (cid:3) jxj (cid:3) Lj(cid:3)jˇ,
where0<ˇ (cid:3)˛ <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,orSkinner[54],p.4.We proposetoworkdirectlywith
composite expansions using the many properties and theorems we present and
comparewithinnerandouterexpansionsinasecondstep.Moreremarksconcerning
therelationswithpreviousworkwillbegiveninChap.7.
There is a vast literature on singular perturbation methods for ordinary differ-
ential 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
followingfeatures.
(cid:129) Weworkwith infiniteasymptoticexpansions,discusstheiralgebraicandanalytic
properties,andusethemtoobtainnewresults.
(cid:129) The expansions are mainly presented in the complex domain. Their regions of
validityaremodifiedsectorswithvertexataturningpoint.
(cid:129) 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/Scha¨fke/Sibuya[9]) whereasherewe have(quasi-)
sectorswithvertexataturningpoint.
(cid:129) We exhibitthreeapplications,in two ofwhichnewresultsare obtained,thanks
totheGevreytheoryofCAsEs.
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
asymptoticexpansionsarenatural,evenunavoidable,nearturningpoints.
ThesecondchapterpresentsthedefinitionofCAsEsandtheircompatibilitywith
algebraic and analytic operations and discusses the relation with the method of
matched asymptotic expansions.The third chapter containsthe Gevreyversion 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;(cid:3)/ 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
differencesareexponentiallysmall.Weemphasizethathere,unlikeintheclassical
Gevrey theory(cid:2), two kinds(cid:5)of exponential smallness appear: the differences in the
x-planeareO e(cid:3)cjxjp=j(cid:3)jp andhenceonlysmallawayfromtheoriginwhereasthe
(cid:2) (cid:5)
differencesinthe(cid:3)-planeareO e(cid:3)c=j(cid:3)jp asusual.
Chapter5appliesthetheoryofCAsEstosingularlyperturbednonlinearordinary
differential equations of first-order near turning points. We present two methods
for obtaining CAsEs. The first is direct, followingthe classical lines: study formal
solutionsoftheform(3),thenprovetheexistenceofananalyticsolutionhavingthis
formalsolutionasa CAsE. Thesecondmethodismoreindirectandusesthe main
result of Chap. 4, embedding a single solution in a family of solutions on a good
coveringofaneighborhoodoftheturningpointandshowingthattheirdifferences
satisfy exponential estimates. A reader familiar with differential equations in the
realdomainmightfindthismethodtoo“complex,”butitis,toourknowledge,the
simplestoneinthecontextofsingularperturbation.
Wefirsttreatatypeofequationsallowinganalyticcontinuationofthesolutionsto
asmallneighborhoodoftheturningpointandthenconsidermoregeneralequations.
ThemostpowerfulresultinthissectionisTheorem5.17.
Inthesixthchapter,weapplythepreviousresultsandusethegeneralproperties
of CAsEs for severalwell-known problemsof singular perturbation:canards solu-
tions near multiple turning points, non-smoothcanards, and Ackerberg–O’Malley
resonance. The linear differential equation of second order appearing in the third
problemisreducedtoafirst-orderRiccatiequationtowhichourresultsareapplied.
Forthefirstandthirdproblems,weobtainnewresults,andforthesecondproblem,
wecanaddGevreypropertiesandGevreyCAsEstotheknownresults.
Thememoirfocusesonordinarydifferentialequationsandonlyconsidersfirst-
ordernonlinearandsecond-orderlinearequations,butweareconvincedthatCAsEs
viii Preface
can be very useful for equationsof higher order and for other types of functional
equations,especiallydifferenceequations.
Wehaveincludedafewexercisesforthebenefitofthereader—inmostofthem,
theresultsofthecorrespondingsectionareappliedtoexamples.
Acknowledgements TheworkofthesecondauthorwassupportedinpartbygrantsoftheFrench
NationalResearchAgency(ref.ANR-10-BLAN0102andANR-11-BS01-0009).
Contents
1 FourIntroductoryExamples ............................................... 1
1.1 FirstExample ........................................................... 1
1.2 Extensions............................................................... 6
1.3 SecondExample ........................................................ 9
1.4 ThirdExample .......................................................... 11
1.5 FourthExample ......................................................... 13
1.6 CompositeExpansions.................................................. 14
2 CompositeAsymptoticExpansions:GeneralStudy ..................... 17
2.1 Notation ................................................................. 17
2.2 CompositeFormalSeries............................................... 19
2.3 CompositeExpansions:DefinitionandBasicProperties ............. 22
2.4 CompositeExpansionsandMatching.................................. 29
2.5 ContinuationofCompositeExpansions ............................... 34
2.6 QuotientsofCAsEs...................................................... 37
2.7 MultipleCAsEs.......................................................... 40
3 CompositeAsymptoticExpansions:GevreyTheory..................... 43
3.1 CompositeGevreyFormalSeries ...................................... 43
3.2 CompositeGevreyAsymptoticExpansions ........................... 47
3.3 FlatGevreyFunctions .................................................. 53
3.4 TheoremsofBorel–RittType .......................................... 55
3.5 ConsistentGoodCoverings............................................. 59
4 ATheoremofRamis–SibuyaType......................................... 63
4.1 StatementoftheTheoremandBeginningoftheProof ............... 63
4.2 TheSlowPart ........................................................... 67
4.3 TheFastPart ............................................................ 70
4.4 StudyoftheFunctionsgj .............................................. 71
n
4.5 TheCase(cid:4)Negative.................................................... 74
4.6 CompositionofaGevreyCAsEandanAnalyticFunction............ 76
ix
x Contents
5 Composite Expansions and Singularly Perturbed
DifferentialEquations....................................................... 81
5.1 ClassicalCAsEsataRegularPoint..................................... 82
5.2 CAsEsataTurningPoint:TheQuasi-linearCase ..................... 89
5.2.1 CompositeFormalSolutions................................... 90
5.2.2 AnalyticSolutionsandCAsEs.................................. 93
5.2.3 TheGevreyCharacterofCAsEs................................ 99
5.2.4 RemarksandExtensions ....................................... 103
5.3 CAsEataTurningPoint:AGeneralization............................ 106
6 Applications .................................................................. 119
6.1 CanardsinaMultipleTurningPoint................................... 120
6.2 Non-smoothCanards.................................................... 126
6.2.1 Equationsof“UnionJack”Type............................... 126
6.2.2 Non-smoothEquations......................................... 132
6.3 Ackerberg–O’MalleyResonance....................................... 136
7 HistoricalRemarks .......................................................... 151
References......................................................................... 155
Index............................................................................... 159