Stability and bifurcation theory for non-autonomous differential equations : Cetraro, Italy 2011

314 Pages·2013·1.971 MB·English

by  Anna Capietto, Peter E Kloeden, J Mawhin, Sylvia Novo, Rafael Ortega, All authors

Lecture Notes in Mathematics 2065
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
FondazioneC.I.M.E.,Firenze
C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International
MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,
Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year
foryear,tothisday.
Many mathematicians from all over the world have been involved in a way oranother in
C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre
maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes
from pure and applied mathematics are offered byapplication to mathematicians from all
countries. ASessionis generally basedonthree orfourmaincourses given byspecialists
ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural
locationinItaly.
TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins,
development, and perspectives of some very active branch of mathematical research. The
topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere
ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational
collaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
Universita`diFirenze Universita`diFirenze
ViaS.Marta,3 vialeG.B.Morgagni67/A
50139Florence 50134Florence
Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
-INdAM(IstitutoNazionalediAltaMatematica)
-MIUR(Ministerodell’Universita’edellaRicerca)
Anna Capietto Peter Kloeden Jean Mawhin
(cid:2) (cid:2)
Sylvia Novo Rafael Ortega
(cid:2)
Stability and Bifurcation
Theory for Non-Autonomous
Differential Equations
Cetraro, Italy 2011
Editors:
Russell Johnson
Maria Patrizia Pera
123
AnnaCapietto SylviaNovo
Universita`diTorino E.deIngenier´ıasIndustriales
DipartimentodiMatematica DepartamentodeMatema´ticaAplicada
Torino,Italy UniversidaddeValladolid
Valladolid,Spain
PeterKloeden
Universita¨tFrankfurt RafaelOrtega
Institutfu¨rMathematik UniversidaddeGranada
Frankfurt,Germany DepartamentodeMatema´ticaAplicada
Granada,Spain
JeanMawhin
Universite´CatholiquedeLouvain
InstitutdeRecherche
enMathe´matiqueetPhysique
Louvain-la-Neuve,Belgium
ISBN978-3-642-32905-0 ISBN978-3-642-32906-7(eBook)
DOI10.1007/978-3-642-32906-7
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012951625
MathematicsSubjectClassification(2010):34B15,37B55,34C25,37E40,37G35,34K12
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Preface
TheCIMEsession“StabilityandBifurcationTheoryforNon-AutonomousDiffer-
entialEquations”washeldatCetraroItaly,from19to25June2011.Thisvolume
containsthenotesofthefivelecturecourseswhichwereheldonthatoccasion.
One of ourgoalsin organizingthe session was to fostera comparisonbetween
the“topological”and“dynamical”approachestothestudyofnonautonomousdif-
ferentialequations.Anothergoalwastofacilitatetheinteractionbetweenspecialists
versed in the one approachor the other. We are amply convincedthat those goals
werefulfilled.
In these course notes, the reader will find a systematic introduction to many
of the themes and methodswhich make up the modern theory of nonautonomous
differential and dynamical systems. Topics pertaining to differential equations in
finite and infinite dimensions receive sustained attention. Also, discrete equations
and systems have an important place in the notes. This is natural both because a
differentialequation is often studied via an appropriatediscretization process and
because nonautonomous discrete systems are of fundamental importance in their
ownright.
Here is a partial list of the various themes which were taken up in the course
ofthelectures:boundedorbitsandstabilityinnon-periodicmonotonetwistmaps;
properties of the minimal subsets of nonautonomousmonotone differential-delay
systems;resonancephenomenain nonautonomousordinarydifferentialequations;
existenceandpropertiesofpullbackattractorsinskew-productdynamicalsystems;
andtheuseoftheMaslovindexinbifurcationproblemsregardingnonautonomous
Hamiltoniansystems.Ofcourseanimpressiverangeofothertopicswasconsidered
aswell,andanamplequantityofspecificproblemswasdiscussed.
Themethodsintroducedbythespeakersinthetheoreticaldevelopmentsandin
the treatment of specific problems may be divided into two classes. First, the use
of “classical” techniques drawn from the topological degree theory, the calculus
of variations, the search for upper/lower solutions, and others of a similar vein.
Second, the use of “dynamical” constructs such as processes and skew-product
flows, minimal sets and omega-limit sets, pullback attractors, invariant measures,
andthelike.
v
vi Preface
Hereisabriefdescriptionofeachofthecourseswhichmadeupthesession.
– Anna Capietto (Torino) considered a broad class of boundary value problems
posedfornonautonomousnonlinearHamiltoniansystems.Shestatedandproved
bifurcationresultsof“Rabinowitz”typefortheseproblems.Sheshowedhowthe
Maslovindexcanbeusedasaneffectivetoolinderivingsuchresults.
– PeterKloeden(Frankfurt)gaveanintroductiontothelanguageandconceptsof
nonautonomousdiscretedynamicalsystems.Hediscussedthetheoryofpullback
attractorsandwentontomentionsomeresultsfromnonautonomousbifurcation
theory.Healsotookupsomequestionsintheareaofrandomdynamicalsystems.
– Jean Mawhin (Louvain-la-Neuve) discussed a number of illustrative nonlinear
nonautonomous resonance problems. He effectively used a mix of methods
drawn from the Leray–Schauder theory, the calculus of variations, and the
technique of upper and lower solutions. He presented many results regarding
existenceandmultiplicityofperiodicsolutionsofcertainparadigmaticperiodi-
callyforcedODEs.
– Sylvia Novo (Valladolid) considered a significant class of nonautonomous
functionaldifferentialequationshavingmonotonicityproperties.Shestudiedthe
existenceandthestabilitypropertiesofminimalsets,togetherwiththeexistence
andstructuralpropertiesofglobalattractors.Shegaveseveralapplications,e.g.,
tothetheoryofneuralnetworksandtothatofcompartmentalsystems.
– Rafael Ortega (Granada) first discussed the existence of bounded orbits and
invariantcurvesforexactsymplectictwistmapsonthecylinderandespeciallyon
theplane.Theresultsoninvariantcurveshavestabilitystatementsascorollaries.
HemadeuseofavariationalprincipleofMathertype.Hethenanalyzedcertain
impactproblems,especiallytheso-calledping-pongmodel.
Thesessionwasattendedbyabout50scientistsof“topological”and“dynamical”
extractions. Their good-natured and active participation in the courses and their
individual discussions helped to create a positive atmosphere which certainly
facilitatedtheexchangeofscientificideas.Webelievethattheinteractionbetween
specialists in the topological and in the dynamical approachesto nonautonomous
differentialequationswasgreatlyenrichedbythisCIMEsession.
Firenze,Italy RussellJohnson
19Dec2011 MariaPatriziaPera
Contents
The Maslov Index and GlobalBifurcationfor Nonlinear
BoundaryValueProblems ...................................................... 1
AlbertoBoscaggin,AnnaCapietto,andWalterDambrosio
1 IntroductionandClassicalResults ........................................... 1
2 TheMaslovIndex............................................................. 7
3 TheNumberofMomentsofVerticality...................................... 12
4 ThePhase-AnglesandtheNumberofMomentsofVerticality ............. 16
5 SomeRelatedNotions ........................................................ 21
6 NonlinearFirstOrderSystemsinR2N ....................................... 24
7 NonlinearDirac-TypeSystemsintheHalf-Line............................. 27
References......................................................................... 32
Discrete-TimeNonautonomousDynamicalSystems......................... 35
P.E.Kloeden,C.Po¨tzsche,andM.Rasmussen
1 Introduction.................................................................... 36
2 AutonomousDifferenceEquations........................................... 37
2.1 AutonomousSemidynamicalSystems................................. 39
2.2 LyapunovFunctionsforAutonomousAttractors...................... 40
3 NonautonomousDifferenceEquations....................................... 43
3.1 Processes ................................................................ 44
3.2 Skew-ProductSystems.................................................. 45
4 AttractorsofProcesses........................................................ 50
4.1 NonautonomousInvariantSets......................................... 51
4.2 ForwardsandPullbackConvergence................................... 52
4.3 ForwardsandPullbackAttractors...................................... 53
4.4 ExistenceofPullbackAttractors....................................... 54
4.5 LimitationsofPullbackAttractors..................................... 59
5 AttractorsofSkew-ProductSystems......................................... 61
5.1 ExistenceofPullbackAttractors....................................... 61
5.2 ComparisonofNonautonomousAttractors............................ 64
vii
viii Contents
5.3 LimitationsofPullbackAttractorsRevisited.......................... 66
5.4 LocalPullbackAttractors............................................... 68
6 LyapunovFunctions........................................................... 69
6.1 ExistenceofaPullbackAbsorbingNeighbourhoodSystem.......... 70
6.2 NecessaryandSufficientConditions................................... 72
7 Bifurcations.................................................................... 79
7.1 HyperbolicityandSimpleExamples................................... 80
7.2 AttractorBifurcation.................................................... 86
7.3 SolutionBifurcation .................................................... 88
8 RandomDynamicalSystems................................................. 92
8.1 RandomDifferenceEquations ......................................... 93
8.2 RandomAttractors...................................................... 94
8.3 RandomMarkovChains................................................ 96
8.4 ApproximatingInvariantMeasures .................................... 98
References......................................................................... 100
ResonanceProblemsforSomeNon-autonomousOrdinary
DifferentialEquations........................................................... 103
JeanMawhin
1 Introduction,NotationsandPreliminaryResults ............................ 103
1.1 Introduction ............................................................. 103
1.2 Notations ................................................................ 105
1.3 ClassesofHomeomorphisms........................................... 107
1.4 ANonlinearProjector................................................... 109
2 TopologicalApproach......................................................... 112
2.1 Introduction ............................................................. 112
2.2 DirichletProblem....................................................... 112
2.3 PeriodicProblem........................................................ 114
2.4 NeumannProblemforSingular(cid:2)...................................... 145
3 LagrangianVariationalApproach ............................................ 147
3.1 Introduction ............................................................. 147
3.2 TheFunctionalFramework............................................. 147
3.3 GroundStateSolutions ................................................. 153
3.4 SaddlePointSolutionsforBoundedNonlinearities................... 159
3.5 MultipleSolutionsNearResonance.................................... 162
3.6 BV Periodic Solutions of the Forced Pendulum
withCurvatureOperator................................................ 169
4 HamiltonianVariationalApproach........................................... 172
4.1 Introduction ............................................................. 172
4.2 AnEquivalentHamiltonianSystemandItsAction ................... 174
4.3 MultiplicityofPeriodicSolutions...................................... 177
References......................................................................... 179
Contents ix
Non-autonomousFunctionalDifferentialEquations
andApplications ................................................................. 185
SylviaNovoandRafaelObaya
1 Introduction.................................................................... 185
2 BasicNotionsandResults .................................................... 189
2.1 FlowsOverCompactMetricSpaces................................... 189
2.2 AlmostPeriodicandAlmostAutomorphicDynamics................ 192
2.3 SomeImportantODEsExamples...................................... 195
2.4 OrderedBanachSpaces:MonotoneSkew-ProductSemiflows ....... 203
3 Non-autonomousFDEswithFiniteDelay................................... 206
3.1 CooperativeandIrreducibleSystemsofFiniteDelayEquations..... 210
3.2 SemicontinuousEquilibriaandAlmostAutomorphicExtensions.... 216
3.3 MonotoneandConcaveorSublinearCases ........................... 220
3.4 ANon-autonomousCyclicFeedbackSystem......................... 225
3.5 CellularNeuralNetworks............................................... 229
4 Non-autonomousCyclicFDEswithInfiniteDelay.......................... 237
4.1 StabilityandExtensibilityResultsforOmega-LimitSets ............ 237
4.2 FDEswithInfiniteDelay ............................................... 240
4.3 MonotoneFDEswithInfiniteDelay................................... 249
4.4 CompartmentalSystems................................................ 255
References......................................................................... 259
TwistMappingswithNon-PeriodicAngles.................................... 265
MarkusKunzeandRafaelOrtega
1 Introduction.................................................................... 265
2 SymplecticMapsinthePlaneandintheCylinder .......................... 270
3 TheTwistConditionandtheGeneratingFunction.......................... 274
4 TheVariationalPrinciple...................................................... 277
4.1 TheFrenkel–KontorowaModel........................................ 277
4.2 AGeneralFramework .................................................. 278
5 ExistenceofCompleteOrbits................................................. 281
6 TheActionFunctionalofaNewtonianEquation............................ 286
7 ImpactProblemsandGeneratingFunctions................................. 290
7.1 TheDirichletProblem.................................................. 291
7.2 TheConditionofElasticBouncing .................................... 293
7.3 ABouncingBall ........................................................ 294
8 CommentsandBibliographicalRemarks.................................... 296
References......................................................................... 299
ListofParticipants............................................................... 301