Handbook of differential equations. Ordinary differential equations. Vol.4

719 Pages·2008·3.11 MB·English

by  Flaviano Battelli

HANDBOOK
OF DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS
VOLUME IV
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H
ANDBOOK
D E
OF IFFERENTIAL QUATIONS
O D
RDINARY IFFERENTIAL
E
QUATIONS
V IV
OLUME
Editedby
FLAVIANO BATTELLI
DipartimentodiScienzematematiche
UniversitàPolitecnicadelleMarche
Ancona,Italy
ˇ
MICHAL FECKAN
DepartmentofMathematicalAnalysis
AndNumericalMathematics
ComeniusUniversity
Slovakia
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Preface
ThisbookisthefourthvolumeinaseriesoftheHandbookofOrdinaryDifferentialEqua-
tions. This volume contains six contributions which are written by excellent mathemati-
cians.Wethankthemforacceptingourinvitationtocontributetothisvolumeandalsofor
theireffortandhardworkontheirpapers.Thescopeofthisvolumeislarge.Wehopethat
itwillbeinterestingandusefulforresearch,learningandteaching.
A brief survey of the volume follows. First, the contributions are presented in alpha-
betical authors’ names. The paper by Balanov and Krawcewicz is devoted to the Hopf
bifurcationoccurringindynamicalsystemsadmittingacertaingroupofsymmetries.They
useaso-calledtwistedequivariantdegreemethod.GlobalsymmetricHopfbifurcationre-
sultsarepresented.Applicationsaregiventoseveralconcreteproblems.Thecontribution
ofFabbri,JohnsonandZampogniliesinlinear,nonautonomous,two-dimensionaldiffer-
entialequation.Forinstance,theystudytheminimalsubsetsoftheprojectiveflowdefined
by these equations. They also discuss some recent developments in the spectral theory
andinversespectraltheoryoftheclassicalSturm–Liouvilleoperator.Thequestionofthe
genericityoftheexponentialdichotomypropertyisconsidered,aswell,forcocyclesgen-
erated by quasi-periodic, two-dimensional linear systems. The paper by Lailne is mainly
devoted to considering growth and value distribution of meromorphic solutions of com-
plexdifferentialequationsinthecomplexplane,aswellasintheunitdisc.Bothlinearand
nonlinear equations are studied including algebraic differential equations in general and
theirrelationstodifferentialfields.Ashortpresentationofalgebroidsolutionsofcomplex
differentialequationsisalsogiven.ThepaperbyPalmerdealswiththeexistenceofchaotic
behaviourintheneighbourhoodofatransversalperiodic-to-periodichomoclinicorbitfor
autonomousordinarydifferentialequations.Theconceptoftrichotomyisessentialinthis
study.Also,aperturbationproblemisconsideredwhenanunperturbedsystemhasanon-
transversal homoclinic orbit. Then it is shown that a perturbed system has a transversal
orbitnearbyprovidedthatacertainMelnikovfunctionhasasimplezero.Thecontribution
byA.RontóandM.Miklósinvestigatesthesolvabilityandtheapproximateconstruction
ofsolutionsofcertaintypesofregularnonlinearboundaryvalueproblemsforsystemsof
ordinarydifferential equationson a compactinterval. Severaltypes of problemsare con-
sideredincludingperiodicandmulti-pointproblems.Parametrizedandsymmetricsystems
areconsideredaswell.Mostoftheoreticalresultsareillustratedbyexamples.Somehistor-
icalremarksconcerningthedevelopmentandapplicationofthemethodarepresented.Fi-
nally,thepaperbyZ˙oła¸dekisdevotedtothelocaltheoryofanalyticdifferentialequations.
Classification of linear meromorphic systems near regular and irregular singular point is
described.Also,alocaltheoryofnonlinearholomorphicequationsispresented.Next,for-
v
vi Preface
malclassificationofnilpotentsingularitiesisgivenandanalyticityoftheTakensprenormal
formisproved.
We thank the Editors of Elsevier for their collaboration during the preparation of this
volume.
List of Contributors
Balanov,Z.,NetanyaAcademicCollege,Netanya,Israel(Ch.1)
Fabbri,R.,UniversitàdiFirenze,Firenze,Italy(Ch.2)
Johnson,R.,UniversitàdiFirenze,Firenze,Italy(Ch.2)
Krawcewicz,W.,UniversityofAlberta,Edmonton,Canada(Ch.1)
Laine,I.,UniversityofJoensuu,Joensuu,Finland(Ch.3)
Palmer,K.J.,NationalTaiwanUniversity,Taipei,Taiwan(Ch.4)
Rontó,A.,InstituteofMathematicsoftheASCR,Brno,CzechRepublic(Ch.5)
Rontó,M.,UniversityofMiskolc,Miskolc-Egyetemváros,Hungary(Ch.5)
Zampogni,L.,UniversitàdiPerugia,Perugia,Italy(Ch.2)
Z˙oła˛dek,H.,WarsawUniversity,Warsaw,Poland(Ch.6)
vii
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Contents
Preface v
ListofContributors vii
ContentsofVolume1 xi
ContentsofVolume2 xiii
ContentsofVolume3 xv
1. SymmetricHopfbifurcation:Twisteddegreeapproach 1
Z.BalanovandW.Krawcewicz
2. Nonautonomousdifferentialsystemsintwodimensions 133
R.Fabbri,R.JohnsonandL.Zampogni
3. Complexdifferentialequations 269
I.Laine
4. Transversalperiodic-to-periodichomoclinicorbits 365
K.J.Palmer
5. Successiveapproximationtechniquesinnon-linearboundaryvalueproblemsfor
ordinarydifferentialequations 441
A.RontóandM.Rontó
6. Analyticordinarydifferentialequationsandtheirlocalclassification 593
H.Z˙oła˛dek
Authorindex 689
Subjectindex 697
ix