Table Of ContentSynergetic Phenomena in Active Lattices
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undSynergetik CenterforComplexSystems
derUniversitätStuttgart FloridaAtlanticUniversity
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YoshikiKuramoto ManuelG.Velarde
DepartmentofPhysics UniversidadComplutensedeMadrid
GraduateSchoolofSciences InstitutoPluridisciplinar(USM)
KyotoUniversity PaseoJuanXXIII,No.1
Kyoto606-8592,Japan 28040Madrid,Spain
Vladimir I. Nekorkin
Manuel G. Velarde
Synergetic Phenomena
in Active Lattices
Patterns, Waves, Solitons, Chaos
With207Figures
1 3
ProfessorVladimirI.Nekorkin
RadiophysicalDepartment
NizhnyNovgorodStateUniversity
23GagarinAvenue
NizhnyNovgorod603600,Russia
ProfessorManuelG.Velarde
UniversidadComplutensedeMadrid
InstitutoPluridisciplinar(USM)
PaseoJuanXXIII,No.1
28040Madrid,Spain
DieDeutscheBibliothek-CIP-Einheitsaufnahme
Nekorkin,VladimirI.:
Synergeticphenomenainactivelattices:patterns,waves,solitons,chaos/
VladimirI.Nekorkin;ManuelG.Velarde.-Berlin;Heidelberg;NewYork;
Barcelona;HongKong;London;Milan;Paris;Tokyo:Springer,2002
(Springerseriesinsynergetics)
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ISSN0172-7389
ISBN 978-3-642-62725-5 ISBN 978-3-642-56053-8 (eBook)
DOI 10.1007/978-3-642-56053-8
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Preface
In recent years there has been growing interest in the study of the nonlinear
spatio-temporal dynamics of problems appearing in various fields of science
and engineering. In a wide class of such systems an important place is oc-
cupied by active lattice dynamical systems. Active lattice systems are, e.g.,
networks of identical or almost identical interacting units ordered in space.
Theactivityoflatticesisprovidedbytheactivityofunitsinthemthatpossess
energy or matter sources. In real (1D, 2D or 3D) space, processes develop
by means of various types of connections, the simplest being diffusion. The
uniquenessoflatticesystemsisthattheyrepresentspatiallyextendedsystems
while having a finite-dimensional phase space. Therefore, active lattice sys-
temsareofinterestforthestudyofmultidimensionaldynamicalsystemsand
the theory of nonlinear waves and dissipative structures of extended systems
as well. The theory of nonlinear waves and dissipative structures of spatially
distributedsystemsdemandsusingtheoreticalmethodsandapproachesofthe
qualitative theory of dynamical systems, bifurcation theory, and numerical
methodsorcomputerexperiments.Inotherwords,theinvestigationofspatio-
temporal dynamics in active lattice systems demands a multitool, synergetic
approach, which we shall use in this book.
In this book we deal with basic concepts and models, with methodolo-
gies for studying the existence and stability of motions; for understanding
mechanisms of formation of patterns and waves, their propagation and in-
teractions in active lattice systems; and concerning how much cooperation
or competition between order and chaos is crucial for synergetic behavior
and evolution. The results described in the book have both inter- and trans-
disciplinary features and fundamental character.
When writing this monograph we had in mind graduate students and re-
searchersfromthenonlinear sciencesincluding physics, biophysics, biomath-
ematics, bioengineering, neurodynamics, electrical and electronic engineer-
ing, mathematical economics and computer sciences. We have done our best
to make the material, and hence the methodologies developed, reasonably
self-contained and accessible to the reader. Accordingly, we have provided
numerous illustrations and computer results supporting the theory. We have
also provided, as much as we were able, heuristic arguments and detailed
discussions of time and/or space scales involved in the problems treated.
VIII Preface
Weexpectthatourbookwillprovetobeusefulasatextbookforgraduate
coursesintheabove-mentioneddisciplines.As,toalargeextent,chaptersare
independent units and relatively self-contained, the book is also expected to
beusefulassupplementaryreadingtograduatecoursesindynamicalsystems,
differentialequations,appliedmathematicsandnonlinearscience,ingeneral.
As an honest warning, we ought to alert the potential reader that in
order to really profit from our book he or she must be acquainted with the
basics of dynamical systems theory, (linear and nonlinear) ordinary differen-
tialequations,(Lyapunov)stabilitytheory,andwaveequations(Afraimovich,
Nekorkin, Osipov and Shalfeev, 1994; Shilnikov, 1997; Shilnikov, Shilnikov,
TuraevandChua,1998;ThompsonandStewart,1986;Andronov,Leontovich,
Gordon and Maier, 1973; Andronov, Vitt and Chaikin, 1966; Arecchi, 1996;
Hirsch and Smale, 1974). It is our understanding that such background ma-
terial is currently taught either in senior-level courses (fourth year in the
curriculum before graduation or licence diploma) or in the first year of most
graduate schools.
Following a succinct introduction in Chap. 1, to help the reader place
our book in perspective and to introduce the type of model problems to be
discussed later on, we proceed to six chapters in which a substantial part of
the research effort by the authors in the past decade is covered. Chapter 8
includes a few speculative ideas and suggestions for further work.
We will outline here the noticeable results and methodological develop-
ments to be found in this book. In Chap. 2, a detailed study is provided of
the (1D) Boussinesq–Korteweg–de Vries (BKdV) equation augmented with
dissipationandaninput–outputenergybalance.Thelatteriscapableofexcit-
ing and supporting solitons due to the (local) balance between nonlinearity
and dispersion. The BKdV equation (Boussinesq, 1877; Korteweg and de
Vries, 1895; Drazin and Johnson, 1989; Christov, Maugin and Velarde, 1996;
Remoissenet, 1996; Scott, Chu and McLaughlin, 1973; Bishop, Krumhansl
andTrullinger,1980;Scott,1999)istheparadigmaticmodelequationforthe
study of solitons and nonlinear (conservative) integrable systems. We show
that contrary to common belief dissipation does not spoil solitons. Rather
if energy input, leading to instability, helps past threshold to dynamically
balance(andhenceovercome)dissipation,solitonscansurviveverymuchlike
steady convective structures (Nicolis and Prigogine, 1977; Haken 1975, 1983
a, b, 2000; Colinet, Legros and Velarde, 2001; Nepomnyashchy, Velarde and
Colinet, 2002; Velarde, Nepomnyashchy and Hennenberg, 2000; Scott, 1999;
Koschmieder, 1993; Nicolis, 1995; Velarde and Normand, 1980; Gaponov-
Gekhov and Rabinovich, 1992; Rabinovich, Ezersky and Weidmann, 2000).
Indeedsolitarywaves,periodicwavetrains,solitonsandsolitonboundstates,
if traveling with constant velocity, are steady (localized) structures in the
corresponding moving frame. The input–output energy balance first selects
the velocity and then helps maintain it while the energy supply lasts.
Preface IX
The results about the dissipation-modified BKdV equation are obtained
by using a methodology developed for (3D) dynamical systems, which is
a methodology used once and again throughout the book. For this reason
we provide details of the methodology allowing the construction of stable
and unstable manifolds and solutions (homoclinic and heteroclinic orbits).
Single-ormultiloop/humps waves,soliton-bound states,chaoticwavetrains,
etc., are constructed first by identifying their domain of existence and then
by tracking them in what can be considered as numerical experiments.
In order to provide further methodological details and to help train the
reader in its use, Chap. 3 is devoted to a succinct discussion of another
paradigmatic equation in nonlinear science, the (1D) perturbed sine-Gordon
(PSG) equation for long Josephsonjunctions (see,e.g.,Ustinov, 1998) which
appear widely in various branches of science and engineering.
Another building block of nonlinear science and engineering is the Chua
circuit (Madan, 1993; Mira, 1997), whose rich panoply of dynamic behav-
iors (bistability, chaotic oscillations, etc.) make it particularly attractive for
applied oriented uses of cellular neural networks or, with greater generality,
cellularnonlinearnetworks(Chua,1998;Haken,1996;Manganaro,Arenaand
Fortuna,1999).Accordingly,Chua’scircuitistakenasamodelunit,neuron-
like element, and Chap. 4 is devoted to a thorough, albeit nonexhaustive,
studyofitsdynamicsanditsrolein(linearandcircular)arraysand(2D)lat-
tices,wheremanysuchunits appearcoupledtogetherbyresistors(diffusion)
or by inductions. The emergence of (steady) patterns and waves of various
types (pulses, solitons, and spiral, periodic and chaotic waves) is analysed
in great detail using the methodologies presented in Chaps. 2 and 3 about
homoclinic and heteroclinic orbits of the underlying dynamical system. In
Chap. 4 these methodologies are augmented with further subtle and sophis-
ticated details to help the reader master the subject. Analytical results are
once more checked with numerical experiments.
Abistableunit,eventuallyexcitableandhenceanactiveelement,appears
as one of the most universal and versatile building blocks in a complex,
synergetic system, be it a portion of the brain or of a man-made machine
(Changeux, 1983; Hoppensteadt, 1986; Izhikevich, 2000; Llin´as, 1988, 2001;
Milton,1996;Scott,1995;Shepherd,1991,1998;Tass,1999).Forthisreason,
we have paid a great deal of attention to it in Chaps. 5 and 6. Particular
care is taken with regard to coupled nonisochronous bistable oscillators and
their role in lattices where (amplitude, frequency or phase) synchronization
and clustering may occur. Thus, again the emergence of patterns and waves
is the subject of our study, using once more the methodologies given in
Chaps. 2 and 3. For mathematical simplicity we have made ample use of
the FitzHugh–Nagumo–Schlo¨gl (FNS) model functional law which accounts
forthesalientfeatures,e.g.,oftheHodgkin–Huxley axonalneurondynamics
(see, e.g., Cronin, 1987).
X Preface
Linear or circular arrays and (2D and 3D) lattices with bistable units
and diffusive bonds/couplings mimick well reaction–diffusion systems (Fife,
1979).Chapter5providesawealthofinformationaboutsynergeticreaction–
diffusion features in various lattice geometries with bistable units.
Our greatest achievements are presented in Chap. 6. We refer to the
questionofreplicationofform(patternsorwaves)andimageandinformation
transfer with a controllable degree of fidelity. This was to us a fascinating
subject with potential applicability in the understanding of disparate phe-
nomenarangingfromreplicationprocessesinprebioticevolutiontodesigning
computersinthe(near)future(FeistelandEbeling,1989).Accordingly,great
attention is paid in Chap. 6 to the emergence and replication of patterns
and waves in 3D (multilayered) architectures. Many pictures obtained in
computer experiments, carried out by V.B. Kazantsev, illustrate the use and
results found with the methodologies based on the material presented in
Chaps. 2, 3 and 4. Computer experiments have also permitted us to ob-
tain new results that qualitative study and analytical theory cannot permit
due to the computational (apparent though not real) “complexity” of the
synchronization dynamics involved in the replication process and dynamic
competition between patterns or waves in lattices with a high number of
coupled active units.
In Chaps. 4 to 6 we deal with problems where space is discrete (lattices)
buttimeproceedsasacontinuousvariable.Chapter7introducestimealsoas
adiscretequantity,andhenceitisdevotedtoageneralizationtocoupledmap
lattices(CMLs)ofresultsfoundearlierinthebook.CMLsappeartobeuseful
in modeling dynamical eventsobtained from discretetime series,whichafter
allisalwaysthecaseinexperiments(Chat´eandCourbage,1997;Kanekoand
Tsuda, 2001; Kaneko, 1993). Once more, use is made of the methodologies
based on the qualitative study of dynamical systems. In particular, we make
useofthediscreteversionoftheFHNSmodel.Inaconstructiveanddetailed
processweprovidemethodsforfinding steadyanddynamicalhenceevolving
structures in CMLs of infinite dimension.
Finally, in Chap. 8 we provide a few concluding and speculative remarks,
not really a conclusion, about avenues for future research and the potential
applicability of methodologies and results earlier presented in the book.
Last but not least, we wish to express our appreciation first to Prof.
Dr. Hermann Haken for his illuminating influence over decades. He was in-
strumentalinthescientificsurvivalofoneofus(M.G.V.),indifficulttimesin
Spain, thanks to a cooperative Grant he helped obtaining from the Stiftung
Volkswagenwerk. Furthermore, the attendance of one of us (M.G.V.) at sev-
eral of the Synergetics Workshops held at the Schloss Elmau (near Munich)
decided much of his scientific interests and even largely shaped his approach
to science.
Preface XI
The development of the research leading to this book was greatly in-
fluenced by Prof. Valentin S. Afraimovich, Prof. Gregoire Nicolis and Prof.
John Ross.
Prof.LeonO.Chuawasasourceofinspirationanddrivetobothauthors.
We also gratefully thank Prof. Rodolfo R. Llinas and Prof. Antonio Fer-
nandezdeMolinafortakingtimeandpatienceinhelpingusunderstandbrain
physiology and neurodynamics, a realm of science which we have recently
entered using the methodology developed in this book.
Our progress in the study of the problems treated in this book was
possible thanks to the collaboration with Dr. Mederic Argentina, Dr. Jean
Bragard, Prof. Christo I. Christov, Dr. Pierre Colinet, Dr. Ezequiel del Rio,
Prof. Victor Fairen, Dr. Antonino Giaquinta, Dr. Andrej G. Maksimov,
Prof.AlexA.Nepomnyashchy,Dr.AlexYe.Rednikov,Dr.NikolaiF.Rulkov
and, particularly, Dr. Viktor B. Kazantsev and Dr. Valeri A. Makarov.
The collaboration between the two coauthors was the result of a sug-
gestion made by Prof. Mikhail I. Rabinovich back in 1993. He was right in
foreseeing how muchwecould achieveworking together,and hencehow easy
itwouldbe,andhowfruitfulthiscollaborationcouldbe.Forthisandalsofor
hisenlighteninginspirationinourresearchonnonlinearproblems,weexpress
our gratitude to him.
Inthepreparationofthecompuscript(textandfigures)webenefitedfrom
thetechnicalskillsofViktorKazantsev,ValeriMakarovandIouliiaMakarova.
Theformerperformedforusmanyhoursofpainfultypingandretyping.Our
sincere thanks for their help and friendship.
This book and, indeed, the research behind it was possible thanks to
various institutional grants from the Ministries of Education and Culture
andofScienceandTechnology(Spain) underGrantPB96-599,fromINTAS,
and from the BCH Foundation (Spain). Visiting positions at the Instituto
Pluridisciplinar (Madrid) were held through the years by one of us (V.I.N.)
with support from the BBV Foundation (Spain), from the Ministry of Ed-
ucation and Culture (Spain) and from Universidad Complutense de Madrid
(Spain). To all these institutions we express our gratitude.
July, 2001
Nizhny-Novgorod and Madrid
V.I. NEKORKIN and M.G. VELARDE
