Complex Time-Delay Systems: Theory and Applications

336 Pages·2010·3.333 MB·English

by  Abhijit Sen, Ramana Dodla, George L. Johnston (auth.), Fatihcan M. Atay (eds.)

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Understanding Complex Systems
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Fatihcan M. Atay
Editor
Complex Time-Delay
Systems
Theory and Applications
123
Editor
FatihcanM.Atay
MaxPlanckInstituteforMathematics
intheSciences
Inselstr.22-26
04103Leipzig
Germany
[email protected]
ISBN978-3-642-02328-6 e-ISBN978-3-642-02329-3
DOI10.1007/978-3-642-02329-3
SpringerHeidelbergDordrechtLondonNewYork
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(cid:2)c Springer-VerlagBerlinHeidelberg2010
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Preface
One of the major contemporary challenges in both physical and social sciences is
modeling, analyzing, and understanding the self-organization, evolution, behavior,
and eventual decay of complex dynamical systems ranging from cell assemblies
tothehumanbraintoanimalsocieties.Themulti-facetedproblemsinthisdomain
require a wide range of methods from various scientific disciplines. There is no
question that the inclusion of time delays in complex system models considerably
enriches the challenges presented by the problems. Although this inclusion often
becomesinevitableasreal-worldapplicationsdemandmoreandmorerealisticmod-
els,theroleoftimedelaysinthecontextofcomplexsystemssofarhasnotattracted
theinterestitdeserves.Thepresentvolumeisanattempttowardfillingthisgap.
Thereexistvarioususefultoolsforthestudyofcomplextime-delaysystems.At
theforefrontisthemathematicaltheoryofdelayequations,arelativelymaturefield
inmanyaspects,whichprovidessomepowerfultechniquesforanalyticalinquiries,
alongwithsomeothertoolsfromstatisticalphysics,graphtheory,computerscience,
dynamicalsystemstheory,probabilitytheory,simulationandoptimizationsoftware,
and so on. Nevertheless, the use of these methods requires a certain synergy to
addresscomplexsystemsproblems,especiallyinthepresenceoftimedelays.
Thefollowingseriesofchapterscombineexpertisefrommathematics,physics,
engineering, and biology to address several current issues from the forefront of
research in the field. To unify the various problems and approaches presented, the
language of dynamical systems is heavily used throughout the book. Dynamical
systems, be it in isolation or in interaction with other systems, can display a rich
spectrum of behavior. At one end of the spectrum is the simplest point attractor,
namelyastableequilibrium.Despiteitsdynamicalsimplicity,itcommandsconsid-
erable interest from a control perspective, since it represents the desired behavior
in many applications. However, stable equilibria can also spontaneously arise as a
resultofcoupling,particularlyindelayednetworks.Theso-calledamplitudedeath
referstotheemergenceofsuchstabilityinanotherwiseoscillatoryorevenchaotic
network. In some fields like chaos control, the aim may be to stabilize a periodic
solution instead of a fixed point. Further along the ranks is synchronization phe-
nomenainitsmanyforms,wheretheattractoristypicallyasubsetofthediagonal
of the system’s state space. Nearby such orderly behavior are regimes of cluster
formationandincoherence, aswellastheco-existence ofseveralattractors.Order
v
vi Preface
and disorder can even exist in the same system at different spatial locations, as in
the case of chimera solutions. Though sequences of bifurcations as the parameter
values or inputs are varied, the system can visit the whole plethora of dynamical
regimes,onlyafractionofwhichmaybeknownoramenabletoexistingtechniques
of analysis. This is certainly a challenging field both from theoretical and applied
perspectives.
Thepresentvolumestartswithachapteronthecollectivedynamicsofcoupled
oscillators, authored by Sen, Dodla, Johnston, and Sethia, which already exhibits
the range of dynamics from equilibrium to chimera solutions. Chapter 2 by Atay
focuses on the suppression of oscillations by time delays infeedback systems and
complex networks, in particular making the connection between stability and net-
worktopology.Chapter3byNiculescu,Michiels,Gu,andAbdallahexaminessta-
bilityof equilibria by delayed output feedback from a control-theoretical point. In
Chap.4,authoredbySchöll,Hövel,Flunkert,andDahlem,theemphasisisshifted
tothestabilizationofperiodicsolutions,studyingarangeofapplicationsfromlasers
tocoupledneurons.TheinvestigationofneuralsystemsiscontinuedinChap.5by
Hutt,thistimewithadifferentnetworkmodel,namelyacontinuumfielddescription
ofcollectiveneuralactivity.Chapter6byLongtinaddressesstochasticdynamicsof
neurons, after a discussionof stochastic delay differential equations. Chapter 7 by
Lu and Chen gives a comprehensive coverage of the stability of neural networks.
Chapter 8 by Crauste looks at stability in systems with distributed delays, with
an application to oscillations in stem cell populations. Finally, Chap. 9 by Sipahi
and Niculescu gives a survey and latest results on a novel application to complex
systems,namelytime-delayedtrafficflow.
This book is aimed at researchers and students from all disciplines who are
interested in time-delay systems. The chapters contain the state-of-the-art in their
respectivefields,inadditiontothecurrentresearchofthecontributors.However,the
emphasishasbeentomakethebookaself-containedvolumebyprovidingsufficient
introductory material in every chapter, as well as ample references to the relevant
literature.Inthisway,thereaderwillbeexposedtotherecentresultsandatthesame
timebeprovidedwithdirectionsforfurtherresearch.
Leipzig, FatihcanM.Atay
January2010
Contents
1 AmplitudeDeath,Synchrony,andChimeraStatesinDelayCoupled
LimitCycleOscillators .......................................... 1
AbhijitSen,RamanaDodla,GeorgeL.Johnston,andGautamC.Sethia
1.1 Introduction ................................................ 1
1.2 AMinimalCollectiveModel .................................. 3
1.2.1 Timedelayeffects..................................... 6
1.3 N-OscillatorModels ......................................... 15
1.3.1 GlobalCoupling ...................................... 16
1.3.2 NearestNeighborCoupling............................. 18
1.3.3 Non-LocalCoupling................................... 26
1.4 SummaryandPerspectives .................................... 40
References ...................................................... 41
2 Delay-InducedStability:FromOscillatorstoNetworks.............. 45
FatihcanM.Atay
2.1 Introduction ................................................ 45
2.2 ABriefSynopsisofAveragingTheory .......................... 46
2.3 StabilitybyDelayedFeedback................................. 48
2.4 AmplitudeDeathinNetworksofOscillators ..................... 52
2.5 DiffusivelyCoupledNetworks................................. 55
2.6 Discrete-TimeSystems ....................................... 58
2.7 ConcludingRemarks......................................... 60
References ...................................................... 61
3 DelayEffectsonOutputFeedbackControlofDynamical
Systems ........................................................ 63
Silviu-IulianNiculescu,WimMichiels,KeqinGu,
andChaoukiT.Abdallah
3.1 Introduction ................................................ 63
3.1.1 ExistingMethodologies ................................ 64
3.1.2 ProblemFormulationandRelatedRemarks ............... 66
3.1.3 MethodologyandApproach ............................ 67
vii
viii Contents
3.2 MainResults................................................ 67
3.2.1 Notation............................................. 67
3.2.2 StabilizabilityintheDelayParameter .................... 70
3.2.3 ControllerDesign ..................................... 75
3.3 IllustrativeExamples......................................... 76
3.3.1 Second-OrderSystem.................................. 77
3.3.2 StabilizingaChainofOscillators ........................ 78
3.3.3 MultipleCrossingFrequenciesToward(in)Stability ........ 80
3.4 ConcludingRemarks......................................... 82
References ...................................................... 83
4 Time-DelayedFeedbackControl:FromSimpleModelstoLasers
andNeuralSystems ............................................. 85
Eckehard Schöll, Philipp Hövel, Valentin Flunkert,
andMarkusA.Dahlem
4.1 Introduction ................................................ 85
4.2 Time-DelayedFeedbackControlofGenericSystems.............. 87
4.2.1 StabilizationofUnstableSteadyStates ................... 87
4.2.2 AsymptoticProperties .................................100
4.2.3 Beyond the Odd Number Limitation of Unstable
PeriodicOrbits....................................... 107
4.2.4 StabilizingPeriodicOrbitsNearaFoldBifurcation .........116
4.3 Time-DelayedControlofOpticalSystems .......................122
4.3.1 Stabilizing Continuous-Wave Laser Emission
byPhase-DependentCoupling.......................... 123
4.3.2 NoiseSuppressionbyTime-DelayedFeedback ............125
4.4 Time-DelayedControlofNeuronalDynamics ...................130
4.4.1 ModelofTwoCoupledNeurons.........................131
4.4.2 ControlofStochasticSynchronization....................133
4.4.3 DynamicsofDelay-CoupledNeurons ....................136
4.4.4 DelayedSelf-FeedbackandDelayedCoupling.............140
References ......................................................144
5 FinitePropagationSpeedsinSpatiallyExtendedSystems............151
AxelHutt
5.1 Introduction ................................................151
5.2 DynamicsintheAbsenceofNoise .............................152
5.2.1 ANeuralFieldModel .................................152
5.2.2 TheGenericModel....................................158
5.3 DynamicsInThePresenceofNoise ............................168
5.3.1 GeneralStabilityStudy ................................168
5.3.2 ApplicationtoaSpecificModel .........................171
References ......................................................175
Contents ix
6 StochasticDelay-DifferentialEquations............................177
AndréLongtin
6.1 Introduction ................................................177
6.2 TheFundamentalIssue .......................................178
6.3 LinearSDDEs ..............................................180
6.4 SmallDelayExpansion.......................................181
6.5 ReductionTechniques ........................................184
6.5.1 ReducingtheDimensionality ...........................184
6.5.2 CrossingTimeProblems ...............................187
6.6 StochasticDelayedNeurodynamics.............................188
6.6.1 NeuralNoiseandDelays ...............................188
6.6.2 NeuralControl .......................................189
6.6.3 NeuralPopulationDynamics............................189
6.6.4 SimplifiedStochasticSpikingModelwithDelay ...........191
6.7 Conclusion .................................................192
References ......................................................193
7 GlobalConvergentDynamicsofDelayedNeuralNetworks...........197
WenlianLuandTianpingChen
7.1 Introduction ................................................197
7.2 StabilityofDelayedNeuralNetworks...........................201
7.2.1 Preliminaries .........................................201
7.2.2 DelayedHopfieldNeuralNetworks ......................204
7.2.3 DelayedCohen–GrossbergCompetitiveandCooperative
Networks ........................................... 209
7.3 PeriodicityandAlmostPeriodicityofDelayedNeuralNetworks ....217
7.3.1 DelayedPeriodicHopfieldNeuralNetworks...............218
7.3.2 Delayed Periodic Cohen–Grossberg Competitive
andCooperativeNeuralNetworks....................... 221
7.3.3 DelayedAlmostPeriodicHopfieldNeuralNetworks........226
7.4 DelayedNeuralNetworkwithDiscontinuousActivations ..........232
7.4.1 Preliminaries .........................................234
7.4.2 StabilityofEquilibrium ................................241
7.4.3 ConvergenceofPeriodicandAlmostPeriodicOrbits........245
7.5 ReviewandComparisonofLiterature...........................252
References ......................................................258
8 StabilityandHopfBifurcationforaFirst-OrderDelayDifferential
EquationwithDistributedDelay ..................................263
FabienCrauste
8.1 Introduction ................................................263
8.2 DefinitionsandHopfBifurcationTheorem.......................265
8.3 StateoftheArtandObjectives.................................269
8.3.1 TheClassicalLinearDiscreteDelayDifferentialEquation ...269