Nonlinear Dynamics New Directions: Theoretical Aspects

223 Pages·2015·3.01 MB·English

by  Hernán González-Aguilar

Nonlinear Systems and Complexity
Volume 11
SeriesEditor
AlbertC.J.Luo
SouthernIllinoisUniversity
Edwardsville
Illinois
USA
Nonlinear Systems and Complexity provides a place to systematically summarize
recentdevelopments,applications,andoveralladvanceinallaspectsofnonlinearity,
chaos,andcomplexityaspartoftheestablishedresearchliterature,beyondthenovel
andrecentfindingspublishedinprimaryjournals. Theaimsofthebookseriesare
to publish theories and techniques in nonlinear systems and complexity; stimulate
moreresearchinterestonnonlinearity,synchronization,andcomplexityinnonlinear
science; and fast-scatter the new knowledge to scientists, engineers, and students
in the corresponding fields. Books in this series will focus on the recent develop-
ments, findings and progress on theories, principles, methodology, computational
techniquesinnonlinearsystemsandmathematicswithengineeringapplications.The
Seriesestablisheshighlyrelevantmonographsonwiderangingtopicscoveringfun-
damentaladvancesandnewapplicationsinthefield.Topicalareasinclude,butare
not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos Compu-
tational methods for nonlinear systems Stability, bifurcation, chaos and fractals in
engineeringNonlinearchemicalandbiologicalphenomenaFractionaldynamicsand
applicationsDiscontinuity,synchronizationandcontrol
Moreinformationaboutthisseriesathttp://www.springer.com/series/11433
Hernán González-Aguilar • Edgardo Ugalde
Editors
Nonlinear Dynamics New
Directions
Theoretical Aspects
2123
Editors
HernánGonzález-Aguilar EdgardoUgalde
AutonomousUniversityofSanLuisPotosí AutonomousUniversityofSanLuisPotosí
SanLuisPotosi SanLuisPotosi
Mexico Mexico
ISSN2195-9994 ISSN2196-0003(electronic)
ISBN978-3-319-09866-1 ISBN978-3-319-09867-8(eBook)
DOI10.1007/978-3-319-09867-8
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014951913
© SpringerInternationalPublishingSwitzerland2015
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Preface
Theideaofthisbookaroseduringthemeeting“NonlinearDynamics: NewDirec-
tions”,heldinhonorofProfessorValentinAfraimovichinGuanajuatoMexicoduring
May2010.Themeetingtookplaceinthe“CentrodeInvestigaciónenMatemáticas”,
one of the finest places to do mathematics. During the meeting we had the oppor-
tunity, not only to share our passion for nonlinear dynamics, but also to discuss
itsfoundationsandtheemergingapplicationsmainlyinbiology.Itwasaswellthe
occasion to share life experiences and celebrate this magnificent person, Valentin
Afraimovich.Thebookcoversallscientificaspectsofthemeeting.Foranaccount-
ingofitshumanaspectswehaveLevTsrimling’sphotogallery.Weareawaitinga
convincingtranslationofMichaRavinovich’spoeticwork.
Thistwo-volumebookcoverspartofthevastspectrumofinterestsofProfessor
Afraimovich,whichrangefromfractalanalysistoveryspecificapplicationsofthe
theoryofdynamicalsystemstobiology.Thefirstvolumeofthisbookisdevotedto
fundamentalaspectsandincludesanumberofimportantnewcontributionsaswell
assomereviewarticles,whichemphasizenewdevelopmentprospects.Thesecond
volumecontainsmostlynewapplicationsofthetheoryofdynamicalsystemstoboth
engineeringandbiology.
Theauthorscontributingtothesetwovolumes,allofthemacademicallyrelated
to ProfessorAfraimovich, are among the most prominent specialists in nonlinear
dynamics.Thetopicsaddressedinthesetwovolumesincludearigoroustreatment
offluctuationindynamicalsystems,topicsinfractalanalysis,studiesofthetransient
dynamics in biological networks, synchronization in lasers, and control of chaotic
systems,amongothers.
Weareveryhappytohavefinallycompletedthiscompilationandwethankallthe
contributorsfromthebottomofourhearts.Wealsothankalltheagenciesthatcon-
tributedtofinancingthe2010meeting,inparticularFENOMEC-UNAM,CONACyT,
CIMAT,andUASLP.Ifdespiteoureffort,somemistakesremain,weareaccountable
forit.
SanLuisPotosí HernánGonzález-Aguilar
EdgardoUgalde
v
Contents
ANoteontheLargeDeviationsforPiecewiseExpandingMultidimensional
Maps ............................................................ 1
R.AiminoandS.Vaienti
DirectionalMetricEntropyandLyapunovExponentsforDynamical
SystemsGeneratedbyCellularAutomata ............................ 11
MauriceCourbageandBrunonKamin´ski
OntheComplexityofSomeGeometricalObjects ..................... 29
P.Collet
FluctuationsofObservablesinDynamicalSystems:FromLimitTheorems
toConcentrationInequalities ....................................... 47
Jean-RenéChazottes
OnFlowBarriersinDiscontinuousDynamicalSystems................ 87
AlbertC.J.Luo
NonstandardAnalysisoftheBehaviorofErgodicMeansofDynamical
SystemsonVeryBigFiniteProbabilitySpaces ........................ 115
E.I.Gordon,L.Yu.GlebskyandC.W.Henson
OnMeasuresResistingMultifractalAnalysis ......................... 153
JörgSchmelingandStéphaneSeuret
AnOverviewofComplexKleinianGroups ........................... 167
AngelCanoandJoséSeade
Semigroups of Mappings and Correspondences: Characters and
RepresentationsinHolomorphicDynamicalSystems .................. 195
CarlosCabrera,PeterMakienkoandPeterPlaumann
vii
Contributors
R.Aimino UMR-6207CentredePhysiqueThéorique, CNRS,Universitésd’Aix-
Marseille,UniversitéduSud,Toulon-VarandFRUMAM,FédéderationdeRecherche
desUnitésdeMathématiquesdeMarseilleCPT,MarseilleCedex9,France
CarlosCabrera InstitutodeMatemáticas,UNAM,Cuernavaca,MOR,Mexico
AngelCano InstitutodeMatemáticas,UnidadCuernavaca,UniversidadNacional
AutónomadeMéxico,Cuernavaca,México
Jean-RenéChazottes CentredePhysiqueThéorique,CNRS-Écolepolytechnique,
PalaiseauCedex,France
P. Collet Centre de Physique Théorique, Ecole Polytechnique, Palaiseau Cedex,
France
Maurice Courbage Laboratoire Matière et Système Complexes and Université
Paris7-DenisDiderot,ParisCedex13,France
L.Yu.Glebsky IICO-UASLP,SanLuisPotosi,SLP,Mexico
E.I.Gordon EasternIllinoisUniversity,Charleston,IL,USA
C.W.Henson UniversityofIllinoisatUrbana-Champaign,Urbana,USA
Brunon Kamin´ski Faculty of Mathematics and Computer Science, Nicolaus
CopernicusUniversity,Torun´,Poland
AlbertC.J.Luo DepartmentofMechanicalandIndustrialEngineering,Schoolof
Engineering,SouthernIllinoisUniversityEdwardsville,Edwardsville,IL,USA
PeterMakienko InstitutodeMatemáticas,UNAM,Cuernavaca,MOR,Mexico
Peter Plaumann Mathematisches Institut, Friedrich-Alexander-Universität
Erlangen-Nürnberg,Erlangen,Germany
JörgSchmeling MathematicsCentreforMathematicalSciences,LundInstituteof
Technology,LundUniversity,Lund,Sweden
ix
x Contributors
José Seade Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional
AutónomadeMéxico,Cuernavaca,México
StéphaneSeuret LAMA,CNRSUMR8050,UniversitéParis-EstCréteil,Créteil
Cedex,France
S.Vaienti UMR-7332 Centre de Physique Théorique, CNRS, Universités d’Aix-
Marseille, Université du Sud, Toulon-Var andFRUMAM, Fédéderation de
Recherche des Unités de Mathématiques de Marseille CPT, Marseille Cedex 9,
France
A Note on the Large Deviations for Piecewise
Expanding Multidimensional Maps
R.AiminoandS.Vaienti
Abstract We present here the large deviation principle for some systems admit-
ting a spectral gap, by using the functional approach of Hennion and Hervé, with
slight modification. Our main application concerns multidimensional expanding
mapsintroducedbySaussol.
1 Introduction
Therearedifferentwaystoestablishlargedeviationprinciples(LDP)fordynamical
systems.Oneofthemistheso-calledLaplaceMethod,whichreliesonthespectral
propertiesofthePerron–Frobenius,ortransfer,operator.Thisstrategyhasbeende-
velopedinaverygeneralandabstractsettingbyHennionandHervéin[17].They
assume that the transfer operator acts on a Banach space of measurable functions,
anditisquasicompact—namelyithasaspectralgap.Theexistenceofaninvariant
probabilitymeasurefollowsimmediately,andbyusingperturbationtheoryforlinear
operators,theyderiveafewothersstatisticalproperties.Thisapproachcoversalot
ofsystems,forinstance,expandingmapsoftheinterval[17,XII.1],Gibbsmeasures
forsubshiftsoffinitetype[17,XII.2]andexpandingYoungtowers[33].Neverthe-
less, this theory seems inappropriate for expanding discontinuous maps in higher
dimension, like those treated by several authors [1, 4, 5, 7, 9–11, 24, 28–31]. In-
deed,HennionandHervéassumethatthetransferoperatoractsonBanachspacesof
measurablefunctionsdefinedeverywhere,whileforthehigherdimensionalsystems
quotedabove,thefunctionalspacescurrentlyused(boundedvariation,quasi-Hölder
R.Aimino((cid:2))
UMR-6207CentredePhysiqueThéorique,CNRS,Universitésd’Aix-Marseille,Universitédu
Sud,Toulon-VarandFRUMAM,FédéderationdeRecherchedesUnitésdeMathématiquesde
MarseilleCPT,LuminyCase907,13288MarseilleCedex9,France
e-mail:[email protected]
S.Vaienti
UMR-7332CentredePhysiqueThéorique,CNRS,Universitésd’Aix-Marseille,UniversitéduSud,
Toulon-VarandFRUMAM,FédéderationdeRecherchedesUnitésdeMathématiquesdeMarseille
CPT,LuminyCase907,13288MarseilleCedex9,France
e-mail:[email protected]
©SpringerInternationalPublishingSwitzerland2015 1
H.Gonzalez-Aguilar,E.Ugalde(eds.),NonlinearDynamicsNewDirections,
NonlinearSystemsandComplexity11,DOI10.1007/978-3-319-09867-8_1