Nonlinear Dynamics New Directions: Models and Applications

243 Pages·2015·11.5 MB·English

by  Hernán González-Aguilar

Nonlinear Systems and Complexity
Volume 12
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
Models and Applications
2123
Editors
HernánGonzález-Aguilar EdgardoUgalde
AutonomousUniversityofSanLuisPotosí AutonomousUniversityofSanLuisPotosí
SanLuisPotosi SanLuisPotosi
Mexico Mexico
ISSN2195-9994 ISSN2196-0003(electronic)
ISBN978-3-319-09863-0 ISBN978-3-319-09864-7(eBook)
DOI10.1007/978-3-319-09864-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014951913
© SpringerInternationalPublishingSwitzerland2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe
materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand
executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this
publicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’s
location,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissions
forusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliableto
prosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication,
neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrorsor
omissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothe
materialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
TheideaofthisbookaroseduringthemeetingheldinhonorofProfessorValentin
Afraimovich in Guanajuato, Mexico, in May 2010. The meeting took place in the
“CentrodeInvestigaciónenMatemáticas,”oneofthefinestplacestodomathematics.
Duringthemeeting,wehadtheopportunitynotonlytoshareourpassionfornonlinear
dynamicsbutalsotodiscussitsfoundationsandtheemergingapplicationsmainly
inbiology.Itwasalsotheoccasiontoshareandcelebratethelifeexperiencesofthe
magnificentperson,ValentinAfraimovich.Thebookcoversallscientificaspectsof
the meeting. For an account of its human aspects, we have LevTsrimling’s photo
gallery.WeareawaitingaconvincingtranslationofMichaRavinovich’spoeticwork.
Thistwo-volumebookcoverspartofthevastspectrumofinterestsofProfessor
Afraimovich,whichrangesfromfractalanalysistoveryspecificapplicationsofthe
theory of dynamical systems to biology. The first volume of this book is devoted
tofundamentalaspectsandincludesanumberofimportantandnewcontributions
aswellassomereviewarticleswhichemphasizenewdevelopmentprospects.The
secondvolumecontainsmostlynewapplicationsofthetheoryofdynamicalsystems
tobothengineeringandbiology.
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.
We are very happy to have finally completed this compilation and we thank all
thecontributorsfromthebottomofourhearts.Wealsothankalltheagencieswhich
contributedtofinancethe2010meeting,inparticular,FENOMEC-UNAM,CONA-
CyT,CIMAT,andUASLP.Ifdespiteofoureffort,somemistakesremained,weare
accountableforit.
SanLuisPotosí EdgardoUgalde
HernánGonzález-Aguilar
v
Contents
PatternsofSynchronyinNeuronalNetworks:TheRoleofSynapticInputs 1
IgorBelykhandMartinHasler
OnTopologicalandHyperbolicPropertiesofSystemswithHomoclinic
Tangencies........................................................ 29
SergeyGonchenko,AlexanderGonchenkoandMing-ChiaLi
HomoclinicΩ-Explosion:HyperbolicityIntervalsandTheirBifurcation
Boundaries ....................................................... 57
SergeyGonchenkoandOlegStenkin
Self-OrganizedRegularityinLong-RangeSystems.................... 79
XavierLeoncini
Reducing the Sequential Dynamics of Excitatory Neural Networks
toSynapticCellularAutomata ...................................... 111
V.I.Nekorkin,A.S.DmitrichevandD.V.Kasatkin
SynchronizationofDelayed-FeedbackSemiconductorLasersandIts
ApplicationinOpticalCommunication .............................. 129
AlexanderN.PisarchikandFlavioR.Ruiz-Oliveras
TransientDynamicsontheEdgeofStability.......................... 157
IrmaTristanandMikhailRabinovich
PhaseControlofChaoticMaps ..................................... 175
SijoK.JosephandMiguelA.F.Sanjuán
VoltageIntervalMappingsforanEllipticBurstingModel.............. 195
JeremyWojcikandAndreyShilnikov
Levenshtein’sDistanceforMeasuringLexicalEvolutionRates ......... 215
FilippoPetroni,MaurizioServaandDimitriVolchenkov
vii
Contributors
IgorBelykh Departmentof MathematicsandStatisticsandNeuroscienceInstitute,
GeorgiaStateUniversity,Atlanta,GA,USA
A.S.Dmitrichev NonlinearDynamicsDepartment,InstituteofAppliedPhysicsof
theRussianAcademyofSciences,NizhnyNovgorod,Russia
Alexander Gonchenko Department of Calculated Mathematics and Cybernetics,
NizhnyNovgorodStateUniversity,NizhnyNovgorod,Russia
Sergey Gonchenko Research Institute ofApplied Mathematics and Cybernetics,
NizhnyNovgorodStateUniversity,NizhnyNovgorod,Russia
MartinHasler SchoolofComputerandCommunicationSciences,EcolePolytech-
niqueFédéraledeLausanne(EPFL),Lausanne,Switzerland
SijoK.Joseph NonlinearDynamics,ChaosandComplexSystemsGroup,Depar-
tamentodeFísica,UniversidadReyJuanCarlos,Móstoles,Madrid,Spain
D.V. Kasatkin Nonlinear Dynamics Department, Institute ofApplied Physics of
theRussianAcademyofSciences,NizhnyNovgorod,Russia
Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, Mar-
seillecedex,France
Ming-Chia Li Department of Applied Mathematics & Center of Mathematical
Modeling and Scientific Computing, National Chiao Tung University, Hsinchu,
Taiwan
V. I. Nekorkin Nonlinear Dynamics Department, Institute ofApplied Physics of
theRussianAcademyofSciences,NizhnyNovgorod,Russia
AlexanderN.Pisarchik CentrodeInvestigacionesenOptica, Leon, Guanajuato,
Mexico
MikhailRabinovich BioCircuitsInstitute,UniversityofCaliforniaSanDiego,La
Jolla,CA,USA
ix
x Contributors
FlavioR.Ruiz-Oliveras CentrodeInvestigacionesenOptica,Leon,Guanajuato,
Mexico
MiguelA.F.Sanjuán NonlinearDynamics,ChaosandComplexSystemsGroup,
DepartamentodeFísica,UniversidadReyJuanCarlos,Móstoles,Madrid,Spain
Maurizio Serva Dipartimento di Matematica, Università dell’Aquila, L’Aquila,
Italy
Andrey Shilnikov Department of Computational Mathematics and Cybernetics,
LobachevskyStateUniversityofNizhniNovgorod,NizhniNovgorod,Russia
OlegStenkin ResearchInstituteofAppliedMathematicsandCybernetics,Nizhny
NovgorodStateUniversity,NizhnyNovgorod,Russia
Irma Tristan Instituto de Investigacion en Comunicacion Optica, Universidad
AutonomadeSanLuisPotosi,SanLuisPotosi,Mexico
Dimitri Volchenkov Cognitive Interaction Technology-Center of Excellence,
Universit-tBielefeld,Roma,Italy
DipartimentodiScienzeEconomicheedAziendali,UniversitàdiCagliari,Bielefeld,
Germany
JeremyWojcik AppliedTechnologyAssociates,Albuquerque,NM,USA
Patterns of Synchrony in Neuronal Networks:
The Role of Synaptic Inputs
IgorBelykhandMartinHasler
DedicatedtoValentinS.Afraimovichontheoccasionofhis
65thbirthday
Abstract Westudytheroleofnetworkarchitectureandsynapticinputsintheforma-
tionofsynchronousclustersinsynapticallycouplednetworksofburstingneurons.
Through analysis and numerics, we show that the stability of the completely syn-
chronous state, representing the largest cluster, only depends on the number of
synapticinputseachneuronreceives,independentfromallotherdetailsofthenet-
worktopology.Wealsogiveasimplecombinatorialalgorithmthatfindssynchronous
clustersfromthenetworktopology.Wedemonstratethatnetworkswithacertainde-
greeofinternalsymmetriesarelikelytohaveclusterdecompositionswithrelatively
largeclusters,leadingpotentiallytoclustersynchronizationatthemesoscalenetwork
level.Weaddresstheasymptoticstabilityofclustersynchronizationinexcitatorynet-
works of bursting neurons and derive explicit thresholds for the coupling strength
thatguaranteesstableclustersynchronization.
1 Introduction
Brain networks have an hierarchy of different levels, ranging from the microscale
via the mesoscale to the macroscale. The microscale is represented by individ-
ual neurons and their local synaptic connections. The mesoscale level involves a
network of columns and minicolumns, connecting populations of neurons. At the
macroscale, large numbers of neuronal populations are arranged into large-scale
I.Belykh((cid:2))
DepartmentofMathematicsandStatisticsandNeuroscienceInstitute,
GeorgiaStateUniversity,30PryorStreet,Atlanta,GA30303,USA
e-mail:[email protected]
M.Hasler
SchoolofComputerandCommunicationSciences,
EcolePolytechniqueFédéraledeLausanne(EPFL),
Station14,1015Lausanne,Switzerland
e-mail:martin.hasler@epfl.ch
©SpringerInternationalPublishingSwitzerland2015 1
H.González-Aguilar,E.Ugalde(eds.),NonlinearDynamicsNewDirections,
NonlinearSystemsandComplexity12,DOI10.1007/978-3-319-09864-7_1