Nonlinear Dynamics and Chaotic Phenomena: An Introduction

396 Pages·2014·4.29 MB·English

by  Bhimsen K. Shivamoggi (auth.)

Nonlinear Dynamics and Chaotic Phenomena:
An Introduction
FLUID MECHANICS AND ITS APPLICATIONS
Volume103
SeriesEditor: AndreTHESS
DepartmentonMechanicalEngineering
IlmenauUniversityofTechnology
98684Ilmenau,Germany
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Bhimsen K. Shivamoggi
Nonlinear Dynamics
and Chaotic
Phenomena:
An Introduction
Second Edition
BhimsenK.Shivamoggi
Dept.Mathematics
UniversityofCentralFlorida
Orlando,FL,USA
This is the second, completely revised and updated edition of the original work with the same title
publishedasISBN978-0-7923-4772-9bythesameauthorin1997.
Everyefforthasbeenmadetotracecopyrightholdersandtoobtaintheirpermissionfortheuseofcopy-
rightmaterial.Thepublisherapologizesforanyerrorsoromissionsandwouldbegratefulfornotification
ofanycorrectionsthatshouldbeincorporatedinfuturereprintsoreditionsofthisbook.
ISSN0926-5112 FluidMechanicsandItsApplications
ISBN978-94-007-7093-5 ISBN978-94-007-7094-2(eBook)
DOI10.1007/978-94-007-7094-2
SpringerDordrechtHeidelbergNewYorkLondon
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Salutationstothatguru whoopenedtheeye
oftheoneblindduetothe(cover) darkness
ofignorance withtheneedle(coated)with
ointmentofknowledge
SkandaPura¯na
To Jayashree, Vasudha andRohini
Preface to the Second Edition
FollowingtheformulationofthelawsofmechanicsbyNewton,Lagrangesoughtto
clarifyandemphasizetheirgeometricalcharacter.PoincaréandLiapunovsuccess-
fullydevelopedanalyticalmechanicsfurtheralongtheselines.Inthisapproach,one
representstheevolutionofallpossiblestates(positionsandmomenta)bytheflow
in phase space, or more efficiently, by mappings on manifolds with a symplectic
geometry, and tries to understand qualitative features of this problem, rather than
solvingitexplicitly.
Oneimportantoutcomeofthislineofinquiryisthediscoverythatvastlydifferent
physicalsystemscanactuallybeabstractedtoafewuniversalforms,likeMandel-
brot’sfractalandSmale’shorse-shoemap,eventhoughtheunderlyingprocessesare
notcompletelyunderstood.This,ofcourse,impliesthatmuchoftheobserveddiver-
sityisonlyapparentandarisesfromdifferentwaysoflookingatthesamesystem.
Thus, modern nonlinear dynamics1 is very much akin to classical thermodynam-
icsinthattheideasandresultsappeartobeapplicabletovastlydifferentphysical
systems.
Chaos problem, which occupies a central place in modern nonlinear dynamics,
refers to a deterministic developmentwith chaotic outcome. Computers have con-
tributedconsiderablytoprogressinchaosresearchviaimpressivecomplexgraphics.
However, this approach lacks organization and therefore does not afford complete
insight into the underlying complex dynamical behavior.2 Study of this dynami-
cal behavior mandates concepts and methods from such areas of mathematics and
physicsasnonlineardifferentialequations,bifurcationtheory,Hamiltoniandynam-
ics, number theory, topology, fractals, and ergodic theory. Ergodic theory makes
use of probabilistic methods to consider ensembles of trajectories rather than the
detailedbehaviorofindividualtrajectories.
This book has grown out of my lecture notes for an interdisciplinary graduate
levelcourseonnonlineardynamicsIhavetaughtforthepastseveralyearstoamix
1Inviewoftheubiquityofnonlinearphenomena(alongwiththescarcityoflinearphenomena),it
isratherawkwardtocallthesubjectnonlineardynamics.AsStanislausUlamputit(Gleick1987),
“Callingthesubjectnonlineardynamicsislikecallingzoology‘non-elephantstudies’”.
2Indeed,VictorWeisskopf(1991)remarked,“Weshouldnotbecontentwithcomputerdata.Itis
importanttofindmoredirectinsightsintowhatatheorysays...”.
ix
x PrefacetotheSecondEdition
of students in applied mathematics, physics, and engineering. I have endeavored
to describe the basic concepts, language and results of nonlinear dynamical sys-
tems. This book is accessible, therefore, to first-year graduate students in applied
mathematics,physics,andengineering,andisuseful,ofcourse,toanytheoretically
inclinedresearcherinthephysicalsciencesandengineering.Inordertofacilitatean
interdisciplinaryreadershipandhighlightphysicalimplications,Ihaveadoptedan
informalstyle,keptthemathematicalformalismtoaminimum,andomittedmuch
detail.Thus,Ihaveabandonedtheabstractlanguageofbundlesandconnectionsand
havestatedafewtheoremsfor whichtheproofs areonlysketchedorarereplaced
byillustrativeexampleswhichmakethegeneralprinciplesplausible.
The book starts with a discussion of nonlinear ordinary differential equations,
bifurcationtheoryandHamiltoniandynamics.Itthenembarksonasystematicdis-
cussionofthetraditionaltopicsofmodernnonlineardynamics—integrablesystems,
Poincaré maps, chaos, fractals and strange attractors. The Baker’s transformation,
thelogisticmapandLorenzsystemarediscussedindetailinviewoftheircentral
place in the subject. There is a detailed discussion of solitons centered around the
Korteweg-deVriesequationinviewofitscentralplaceinintegrablesystems.Then,
there is a discussion of the Painlevé property of nonlinear differential equations
which seems to provide a test of integrability. Finally, there is a detailed discus-
sionoftheapplicationoffractalsandmulti-fractalstofully-developedturbulence—
aproblemwhoseunderstandinghasbeenconsiderablyenrichedbytheapplication
oftheconceptsandmethodsofmodernnonlineardynamics.Ontheapplicationside,
onemaydiscernaspecialemphasisonsomeaspectsoffluiddynamicsandplasma
physicsreflectingmyinvolvementintheseareasofphysics.Ihaveprovidedafew
exercisesthatrangefromsimpleapplicationstooccasionalconsiderableextension
of the theory. Finally, the list of references given at the end of the book contains
primarilybooksandpapersIhaveusedindevelopingmylecturematerial(andthis
book).3Thislistisnottobeconstruedascompleteandsomeimportantpapersmay
bemissing.
TheSecondEdition,inadditiontoconstitutinganextensiverewriteofthetext,
incorporatesthefollowingfeatures,
• refinementandenhancementoftheclarityandprecisionofthetext;
• extensiveupdatingandamplificationofseveralitemsinthebook;
• addition of new material like theory of nonlinear differential equations, La-
grangianchaosinfluids,solitonsandcriticalphenomenaperspectivesonthefluid
turbulenceproblem;
• additionofnewexercises.
Orlando,FL,USA BhimsenK.Shivamoggi
2013
3One is reminded in this context of the observation made by Sheldon Glashow (Zee 2003)—
“Tapestriesaremadebymanyartisansworkingtogether.Thecontributionsofseparateworkers
cannotbediscernedinthecompletedwork,andthelooseandfalsethreadshavebeencoveredover.”
Acknowledgements
As I mentioned in the First Edition, my deep gratitude is to Professor John Can-
non, our former Chairman, Professor Peter Rautenstrauch, our former Assistant
ChairmanandProfessorLarryAndrews,ourformerGraduateCoordinatorfortheir
encouragement and support in getting our graduate course sequence on Nonlinear
Dynamicsoff the ground and then sustaining it as this course sequencedeveloped
further.IamgreatlyindebtedtoProfessorPiotrMikusinski,ourDepartmentChair-
man, Professor Joseph Brennan, our Associate Chairman and Graduate Coordina-
tor,ProfessorXinLi,ourformerGraduateCoordinatorfortheirencouragementand
supportasthiscoursesequenceevolvedfurtherandtheworkonthesecondedition
of this book got under way. I wish to thank my students and colleagues who ren-
deredvaluableassistanceinthiseffort.Theresponsibilityforanyremainingdefects
is however mine alone. As with the First Edition, in preparing the second edition
material, I have enormously drawn from the ideas and formulations developed by
manyeminentauthors.IhaveacknowledgedthevarioussourcesintheReferences
given at the end of this book and I wish to apologize to any sources I may have
inadvertentlymissed.
MysincerethanksareduetoLindaPerez-Rodriguezforthetremendouspatience
anddedicationwithwhichshedidthewholetypingjobandputupwithmycount-
lesschanges.MythanksareduetoSiglindeQuirkforherexcellentjobindoingthe
figures.IamverythankfultoMrs.NathalieJocobs,SpringerVerlagPublishers,for
theirtremendouscooperationwiththiswholeproject.Thesecondeditionworkwas
afiveyearprojectwhichdemandedIsitawayengagedonitweekendafterweekend,
holidayafterholidaytotrytodojusticetothegreatintellectualstridesinthisarea
madebyseveraleminentpioneersofthe20thcentury.Myimmensethanksaredue
to my wife Jayashree and my daughters, Vasudha and Rohini for their enormous
understandingandcooperationwiththisendeavor.
BhimsenK.Shivamoggi
xi