Marzena Ciszak ANTICIPATED SYNCHRONIZATION

194 Pages·2006·10.18 MB·English

Marzena Ciszak
ANTICIPATED SYNCHRONIZATION:
NUMERICAL AND THEORETICAL STUDY
Ph.D. Thesis

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ANTICIPATED SYNCHRONIZATION:
NUMERICAL AND THEORETICAL STUDY
——————————————————————
PhD Thesis of Marzena Ciszak
Supervisors: Claudio Mirasso and Rau´l Toral
University of Balearic Islands

ToJaninaJaworska

Contents
Acknowledgements ix
Resumen 1
Abstract 3
1 Introduction 5
2 Background 9
2.1 Modellingexcitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Chaoticsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Lyapunovexponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Reconstructionofanattractor . . . . . . . . . . . . . . . . . . . . . . 26
2.2.3 Controlofchaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Reaction-diffusionsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Delayeddifferentialequations . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Artificialneuralnetworksasadynamicalsystems . . . . . . . . . . . . . . 39
2.6 Coupleddynamicalsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.1 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.2 Anticipatedsynchronization . . . . . . . . . . . . . . . . . . . . . . 49
3 Anticipatedsynchronization: characterization 51
3.1 Anticipatedsynchronizationinexcitablesystemsdrivenbynoise . . . . . 51
3.1.1 Numericalcharacterizationofstability . . . . . . . . . . . . . . . . 51
3.1.2 Linearstabilityanalysis . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.3 Coupledneuronsinachain . . . . . . . . . . . . . . . . . . . . . . . 66
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3.1.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Anticipatedsynchronizationinspatiotemporalsystems . . . . . . . . . . . 74
3.2.1 CoupledcomplexGinzburg-Landauequations . . . . . . . . . . . . 74
3.2.2 Modellingpredictionofchemicalreactions . . . . . . . . . . . . . . 82
3.2.3 Zero-lagsynchronizationinexcitablesystems . . . . . . . . . . . . 92
3.2.4 Cellularautomata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Dynamicalmechanismsofanticipatedsynchronization 103
4.1 Mechanismofanticipatedsynchronizationinexcitablesystems . . . . . . 103
4.1.1 Characterizationoftheresponsetime . . . . . . . . . . . . . . . . . 103
4.1.2 Effectofthedelayedcouplingtermontheexcitablesystem . . . . 106
4.1.3 Analysisofthedelayedcouplingterm . . . . . . . . . . . . . . . . . 111
4.2 Modified-systemapproachtochaoticsystems . . . . . . . . . . . . . . . . . 116
4.2.1 Diffusivedelayedcouplingscheme . . . . . . . . . . . . . . . . . . 116
4.2.2 Completereplacementscheme . . . . . . . . . . . . . . . . . . . . . 123
4.3 Anticipatedsynchronizationasadelayedfeedbackcontrol . . . . . . . . . 127
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Practicalapproachtopredictabilitywithanticipatedsynchronization 133
5.1 Predict-preventcontrolmethodforexcitablesystems . . . . . . . . . . . . 133
5.1.1 Predictionofthemasterresponse . . . . . . . . . . . . . . . . . . . . 134
5.1.2 Suppressionofthemasterresponse . . . . . . . . . . . . . . . . . . 134
5.1.3 Synchronizationofsystemswithparametermismatches . . . . . . 139
5.2 Neuralnetworksandanticipatedsynchronization: acomparison . . . . . 140
5.2.1 Theoreticallimitsforanticipationtimesinchaoticsystems . . . . . 141
5.2.2 Coupledsystemsinachain . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.3 Anticipatedsynchronizationwithneuralnetworkreplicas . . . . . 146
5.2.4 Standardpredictionwithneuralnetworks . . . . . . . . . . . . . . 149
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Summaryandopenproblems 155
Appendices 163
A Stabilityanalysisforlineardifferentialequationswithdelay 163
B Numericalintegration 171
Bibliography 175
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Curriculumvitae 183
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