Complex and Adaptive Dynamical Systems: A Primer

433 Pages·2015·11.9 MB·English

by  Claudius Gros (auth.)

3
COMPLEXITY
Claudius Gros
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A Primer
Fourth Edition
123
Springer Complexity
Springer Complexity is an interdisciplinary program publishing the best research and
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Complex Systems are systems that comprise many interacting parts with the ability to
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the spontaneous formation of distinctive temporal, spatial or functional structures. Models
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EditorialandProgrammeAdvisoryBoard
HenryAbarbanel,InstituteforNonlinearScience,UniversityofCalifornia,SanDiego,USA
DanBraha,NewEnglandComplexSystemsInstituteandUniversityofMassachusettsDartmouth,USA
Péter Érdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of
Sciences,Budapest,Hungary
KarlFriston,InstituteofCognitiveNeuroscience,UniversityCollegeLondon,London,UK
HermannHaken,CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany
ViktorJirsa,CentreNationaldelaRechercheScientifique(CNRS),UniversitédelaMéditerranée,Marseille,
France
JanuszKacprzyk,SystemResearch,PolishAcademyofSciences,Warsaw,Poland
KunihikoKaneko,ResearchCenterforComplexSystemsBiology,TheUniversityofTokyo,Tokyo,Japan
ScottKelso,CenterforComplexSystemsandBrainSciences,FloridaAtlanticUniversity,BocaRaton,USA
Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick,
Coventry,UK
JürgenKurths,NonlinearDynamicsGroup,UniversityofPotsdam,Potsdam,Germany
AndrzejNowak,DepartmentofPsychology,WarsawUniversity,Poland
HassanQudrat-Ullah,SchoolofAdministrativeStudies,YorkUniversity,Canada
LindaReichl,CenterforComplexQuantumSystems,UniversityofTexas,Austin,USA
PeterSchuster,TheoreticalChemistryandStructuralBiology,UniversityofVienna,Vienna,Austria
FrankSchweitzer,SystemDesign,ETHZurich,Zurich,Switzerland
DidierSornette,EntrepreneurialRisk,ETHZurich,Zurich,Switzerland
StefanThurner,SectionforScienceofComplexSystems,MedicalUniversityofVienna,Vienna,Austria

Claudius Gros
Complex and Adaptive
Dynamical Systems
A Primer
Fourth Edition
123
ClaudiusGros
InstituteforTheoreticalPhysics
TheGoetheUniversityFrankfurt
Frankfurt
Germany
ISBN978-3-319-16264-5 ISBN978-3-319-16265-2 (eBook)
DOI10.1007/978-3-319-16265-2
LibraryofCongressControlNumber:2015936717
SpringerChamHeidelbergNewYorkDordrechtLondon
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Contents
1 GraphTheoryandSmall-WorldNetworks.............................. 1
1.1 GraphTheoryandReal-WorldNetworks.......................... 1
1.1.1 TheSmall-WorldEffect................................... 1
1.1.2 BasicGraph-TheoreticalConcepts....................... 3
1.1.3 GraphSpectraandDegreeDistributions................. 9
1.1.4 GraphLaplacian........................................... 13
1.2 PercolationinGeneralizedRandomGraphs....................... 15
1.2.1 GraphswithArbitraryDegreeDistributions............. 16
1.2.2 ProbabilityGeneratingFunctionFormalism............. 21
1.2.3 DistributionofComponentSizes......................... 24
1.3 RobustnessofRandomNetworks.................................. 27
1.4 Small-WorldModels................................................ 32
1.5 Scale-FreeGraphs .................................................. 34
Exercises ..................................................................... 39
FurtherReading.............................................................. 41
References.................................................................... 41
2 BifurcationsandChaosinDynamicalSystems ......................... 43
2.1 BasicConceptsofDynamicalSystemsTheory ................... 43
2.2 Fixpoints,BifurcationsandStability............................... 49
2.2.1 FixpointsClassificationandJacobian.................... 51
2.2.2 BifurcationsandNormalForms.......................... 53
2.2.3 HopfBifurcationsandLimitCycles ..................... 56
2.3 GlobalBifurcations................................................. 58
2.3.1 CatastropheTheory ....................................... 61
2.4 TheLogisticMapandDeterministicChaos....................... 66
2.5 DynamicalSystemswithTimeDelays ............................ 72
Exercises ..................................................................... 75
FurtherReading.............................................................. 77
References.................................................................... 77
vii
viii Contents
3 Dissipation,NoiseandAdaptiveSystems................................ 79
3.1 DissipationandAdaption........................................... 79
3.1.1 DissipativeSystemsandPhaseSpaceContraction...... 79
3.1.2 StrangeAttractorsandDissipativeChaos................ 83
3.1.3 AdaptiveSystems ......................................... 85
3.1.4 ConservingAdaptiveSystems............................ 89
3.2 DiffusionandTransport ............................................ 92
3.2.1 RandomWalks,DiffusionandLévyFlights............. 93
3.2.2 MarkovChains............................................ 95
3.2.3 TheLangevinEquationandDiffusion ................... 98
3.3 Noise-ControlledDynamics........................................ 100
3.3.1 StochasticEscape ......................................... 101
3.3.2 StochasticResonance ..................................... 104
Exercises ..................................................................... 108
FurtherReading.............................................................. 109
References.................................................................... 109
4 SelfOrganizationandPatternFormation ............................... 111
4.1 InterplayBetweenDiffusionandReaction........................ 111
4.1.1 TravellingWavefrontsintheFisherEquation ........... 113
4.1.2 SumRulefortheShapeoftheWavefront................ 116
4.1.3 Self-StabilizationofTravellingWavefronts.............. 117
4.2 InterplayBetweenActivationandInhibition...................... 119
4.2.1 TuringInstability.......................................... 120
4.2.2 PatternFormation ......................................... 121
4.2.3 TheGray-ScottReactionDiffusionSystem ............. 123
4.3 CollectivePhenomenaandSwarmIntelligence................... 126
4.3.1 PhaseTransitionsinSocialSystems ..................... 128
4.3.2 CollectiveDecisionMakingandStigmergy ............. 129
4.3.3 CollectiveBehaviorandSwarms......................... 132
4.3.4 OpinionDynamics ........................................ 134
4.4 CarFollowingModels.............................................. 136
4.4.1 LinearFlowandCarryingCapacity...................... 136
4.4.2 Self-OrganizedTrafficCongestions...................... 137
Exercises ..................................................................... 140
FurtherReading.............................................................. 141
References.................................................................... 141
5 ComplexityandInformationTheory..................................... 143
5.1 ProbabilityDistributionFunctions................................. 143
5.1.1 BayesianStatistics ........................................ 146
5.1.2 TheLawofLargeNumbers .............................. 149
5.1.3 TimeSeriesCharacterization............................. 150
Contents ix
5.2 EntropyandInformation ........................................... 154
5.2.1 InformationContentofaReal-WorldTimeSeries ...... 161
5.2.2 MutualInformation ....................................... 163
5.2.3 Kullback-LeiblerDivergence............................. 168
5.3 ComplexityMeasures............................................... 170
5.3.1 ComplexityandPredictability............................ 172
5.3.2 AlgorithmicandGenerativeComplexity................. 175
Exercises ..................................................................... 176
FurtherReading.............................................................. 179
References.................................................................... 179
6 CellularAutomataandSelf-OrganizedCriticality ..................... 181
6.1 TheLandauTheoryofPhaseTransitions.......................... 181
6.2 CriticalityinDynamicalSystems.................................. 187
6.2.1 1/fNoise................................................... 190
6.3 CellularAutomata .................................................. 191
6.3.1 Conway’sGameofLife................................... 192
6.3.2 TheForestFireModel .................................... 194
6.4 TheSandpileModelandSelf-OrganizedCriticality .............. 195
6.4.1 AbsorbingPhaseTransitions ............................. 198
6.5 RandomBranchingTheory......................................... 199
6.5.1 BranchingTheoryofSelf-OrganizedCriticality......... 200
6.5.2 Galton-WatsonProcesses................................. 205
6.6 ApplicationtoLong-TermEvolution .............................. 206
Exercises ..................................................................... 213
FurtherReading.............................................................. 215
References.................................................................... 215
7 RandomBooleanNetworks................................................ 217
7.1 Introduction......................................................... 217
7.2 RandomVariablesandNetworks .................................. 219
7.2.1 BooleanVariablesandGraphTopologies................ 219
7.2.2 CouplingFunctions ....................................... 221
7.2.3 Dynamics.................................................. 223
7.3 TheDynamicsofBooleanNetworks .............................. 225
7.3.1 TheFlowofInformationThroughtheNetwork......... 225
7.3.2 TheMean-FieldPhaseDiagram.......................... 227
7.3.3 TheBifurcationPhaseDiagram.......................... 229
7.3.4 Scale-FreeBooleanNetworks............................ 233
7.4 CyclesandAttractors............................................... 235
7.4.1 QuenchedBooleanDynamics............................ 235
7.4.2 TheKD1KauffmanNetwork ........................... 238
7.4.3 TheKD2KauffmanNetwork ........................... 240
7.4.4 TheKDNKauffmanNetwork........................... 241