Table Of ContentNonlinear Dynamics and Chaos:
Where do we go from here?
Edited by
John Hogan, Alan Champneys,
Bernd Krauskopf, Mario di Bernardo,
Eddie Wilson, Hinke Osinga
and Martin Homer
Department of Engineering Mathematics,
University of Bristol, UK
Institute of Physics Publishing
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Contents
Preface ix
1 Bifurcationanddegeneratedecompositioninmultipletimescale
dynamicalsystems 1
JohnGuckenheimer
1.1 Definitionsandbackground 4
1.2 Slow–fastdecompositions 8
1.3 Degeneratedecompositionandbifurcation 10
1.4 TheforcedVanderPolequation 13
Acknowledgment 18
References 19
2 Many-bodyquantummechanics 21
RobertSMacKay
2.1 Signsofnonlinearityinquantummechanics 24
2.2 Discretebreathers 27
2.3 Experimentalevidenceforquantumdiscretebreathers 33
2.4 Towardsamathematicaltheoryofquantumdiscretebreathers 35
2.5 Obstructionstothetheory 39
2.6 Aproposedsolution 41
2.7 Atentativeapplicationto4-methylpyridine 47
2.8 Conclusion 51
Acknowledgments 52
References 52
3 Unfoldingcomplexity:hereditarydynamicalsystems
—newbifurcationschemesandhighdimensionalchaos 55
UweanderHeiden
3.1 Hereditarysystems 55
3.2 Differenceequationswithcontinuousargument:idealized
turbulence 60
3.3 First-orderdifference-differentialequations:asingular
perturbationproblemwithbifurcationgaps 61
3.4 Primenumberdynamicsofaretardeddifferenceequation 64
vi Contents
3.5 Second-ordernon-smoothdifference-differentialequations 67
3.6 Outlook 70
Acknowledgment 70
References 70
4 Creatingstabilityoutofinstability 73
ChristopherKRTJones
4.1 Nonlinearopticalfibres 74
4.2 Cantwounstablewavesmakeastablepulse? 84
4.3 Somepredictions,suggestionsandquestions 88
References 89
5 Signalornoise? Anonlineardynamicsapproachto
spatio-temporalcommunication 91
GregoryDVanWiggeren,JordiGarcia-OjalvoandRajarshiRoy
5.1 Communicationwithdynamicallyfluctuatingstatesoflight
polarization 94
5.2 Spatio-temporalcommunicationwithsynchronizedopticalchaos 105
5.3 Conclusions 112
Acknowledgments 113
References 113
6 Outstandingproblemsinthetheoryofpatternformation 117
EdgarKnobloch
6.1 Patternselectiononlattices 128
6.2 Imperfectionsensitivity 132
6.3 CoupledGinzburg–Landauequations 138
6.4 Thenearly-inviscidFaradaysystem 141
6.5 Nonlinearwavesinextendedsystemswithbrokenreflection
symmetry 149
6.6 Summaryandconclusions 158
Acknowledgments 159
References 159
7 Ischaosrelevanttofluidmechanics? 167
TomMullin
7.1 Taylor–Couetteflow 167
7.2 Preliminaryobservations 170
7.3 Symmetryconsiderations 173
7.4 Codimension-twobifurcations 174
7.5 Imperfectgluingbifurcations 180
7.6 Conclusion 182
Acknowledgments 183
References 184
Contents vii
8 Time-reversedacousticsandchaos 187
MathiasFink
8.1 Time-reversalmirrors 188
8.2 Time-reversalexperiments 192
8.3 Timereversalinchaoticcavities 198
8.4 Conclusion 206
Acknowledgments 206
References 206
9 Reductionmethodsappliedtonon-locallycoupled
oscillatorsystems 209
YoshikiKuramoto
9.1 Whencouplingnon-localitybecomescrucial 211
9.2 Rotatingspiralwaveswithoutphasesingularity 213
9.3 Simplercase: coexistenceofcoherentandincoherentdomains 218
9.4 Concludingremarks 224
References 225
10 Aprimenumberofprimequestionsaboutvortexdynamics
innonlinearmedia 229
ArtWinfree
10.1 Stableorganizingcentres 230
10.2 Persistentorganizingcentres 248
10.3 Ignitingresistancetocurvature-inducedcontraction 251
10.4 DynamicC:synchronization 258
10.5 Ball-parkestimationofpertinentquantities 260
10.6 Passinginsilencefrom1994to2002 262
Acknowledgments 264
Addendumwhileinpress 264
References 264
11 Spontaneouspatternformationinprimaryvisualcortex 269
PaulCBressloffandJackDCowan
11.1 TheTuringmechanismanditsroleincooperativecortical
dynamics 274
11.2 AcontinuummodelofV1anditsintrinsiccircuitry 281
11.3 OrientationtuningandO(2)symmetry 285
11.4 Amplitudeequationforinteractinghypercolumns 288
11.5 CorticalpatternformationandE(2)symmetry 292
11.6 SpatialfrequencytuningandSO(3)symmetry 307
11.7 Futuredirections 316
References 317
viii Contents
12 Modelsforpatternformationindevelopment 321
BardErmentroutandRemusOsan
12.1 Competition 326
12.2 Spatialscales 330
12.3 Orientationmapsandfeaturemaps 336
12.4 Kohonenmapsandabstractfeaturemodels 341
12.5 Conclusions 345
References 346
13 Spatio-temporalnonlineardynamics: anewbeginning 349
WilliamLDitto
13.1 Fibrillationoftheheart 350
13.2 Neurocomputing 353
13.3 Wheredowegofromhere? 355
Acknowledgments 355
References 355
Authorindex 357
Preface
This book was born out of the lingering suspicion that the theory and practice
of dynamical systems had reached a plateau of development and that someone
had to ask the question ‘Where do we go from here?’ The idea is not to look
back warmly at ‘chaos theory’, give the main protagonists a slap on the back,
and suggesthow the mathematicalstate ofthe artcan be tinkeredwith. Instead
each invited contributor was asked to reflect on what has been learnt from the
phenomenal rise in nonlinear dynamics and, more importantly, to address the
question of what still needs to be learnt. Thanks to the efforts of our authors,
we now have a plethora of suggestions, but the over-ridingmessage is clear: if
dynamicalsystems theoryis to make a significantlong-termimpact, it needsto
getsmart,becausemostsystemsareill-definedthrougheitherstochasticity,delay,
spatialextentandinhomogeneityorthefinite-timenatureofreal-worlddata.
Howdid wegethere?
Since the pioneering works of Lorenz (1963)1 and May (1976)2, enormous
mathematical strides have been taken3, and almost every scientific field has
been swamped by computations of nonlinear differential equations and maps.
Most of the existing theory and its relation to applications lies in systems of
low dimension. Thisis significantbecausethe theoreticalunderpinningisoften
geometric in nature as many systems lend themselvesto representationin three
dimensionsorless.
However, most systems of practical importance are of much higher
dimension.Chemicalsystemscaninvolvehundredsofcouplednonlinearordinary
differential equations, realistic neural systems are several orders of magnitude
larger, and the partial differential equations involved in fluid mechanics are
1 LorenzEN1963Deterministicnon-periodicflowJ.Atmos.Sci.20130–41
2 MayRM1976SimplemathematicalmodelswithverycomplicateddynamicsNature261459–67
3 Formoreintroductory reading see, forexample; Gleick J 1987Chaos—Making aNew Science
(NewYork:VikingPenguin),MullinT1993TheNatureofChaos(Oxford:OxfordUniversityPress),
Cvitanovic P(ed) 1989Universality in Chaos (Bristol: Institute ofPhysics Publishing), Ruelle D
1991Chance&Chaos(Princeton,NJ:PrincetonUniversityPress)andStewartI1990DoesGodPlay
Dice?TheNewMathematicsofChaos(Oxford:Blackwell).
ix
x Preface
infinite-dimensional; even the numerical representation of fluid mechanics by
finite element or finite difference codes has dimensions running into many
thousands. In these areas, theory advances understanding either by system
reduction methods leading to low-dimensional systems4 or by approaches
involvinglengthynumericalcomputations.
However, most large systems cannot be reduced simply, and numerical
computationwithmanyfreeparametersseldomleadstoanunderstandingofthe
systemunderinvestigation. Itisthisgeneralsituationwefeltmustbeaddressed.
However, it is not clear what the future problemsshould be, still less how they
shouldbesolved.
Hence we brought together theoreticians, experimentalists and the next
generation of nonlinear scientists, so that they could together define an agenda
forfutureresearchdirections.
Thus our unique meeting ‘Nonlinear dynamics and chaos: where do we
go from here?’ was held in Bristol in June 2001. Over three gloriously
sunnyand rain-freedays, 70 scientists fromover a dozencountriesenjoyed the
Victorian grandeur of Burwalls, a University of Bristol dedicated conference
centre. Whether playing croquet, eating in the oak-panelled dining room, or
looking out of the window at Brunel’s majestic suspension bridge next door,
the participants enjoyed a stimulating atmosphere that genuinely addressed the
conferencetitle’s question. In addition to 14 invited speakers, the meeting was
shaped by the contributions of a large number of graduate students and young
researchers.
Where do wego from here?
Thisquestion is centralto all13 chaptersofthis book. Eachchapterrepresents
a write-up of an invited lecture, which has been edited (and in some cases
transcribed)byus.
Werecognizedattheoutsetthatthefuturedirectionsandchallengesforthe
theory of nonlinear dynamicsare far too many and varied to be coveredin one
book.Thereforethreebroadinter-relatedthemeshavebeenselected:
(i) neuralandbiologicalsystems,
(ii) spatiallyextendedsystemsand
(iii) experimentationinthephysicalsciences.
It was a pleasant surprise to us that many of the resulting chaptersof this book
address two or more of these themes. As you will see, other synergies also
emerge. Each author has offered suggestions as to what they see as the way
forward,andeachindividualchaptergivesanoverviewofitsparticularfieldand
raisesmanyspecificandmoregeneralquestionsforfutureresearch.
4 See, forexample, Carr J1981 Applications ofCentre Manifold Theory (Applied Math. Sci. 35)
(NewYork:Springer).
