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Springer Series in Synergetics Shanmuganathan Rajasekar Miguel A.F. Sanjuan Nonlinear Resonances Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics,medicine,neuroscience,socialandcomputerscience. Complex Systems are systems that comprise many interacting parts with the ability to generate anew qualityof macroscopic collectivebehavior themanifestations of whichare the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like theclimate,thecoherentemissionoflightfromlasers,chemicalreaction-diffusionsystems, biologicalcellularnetworks, thedynamicsofstockmarketsandoftheinternet,earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in socialsystems,tonamejustsomeofthepopularapplications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence,dynamicalsystems,catastrophes,instabilities,stochasticprocesses,chaos,graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. ThethreemajorbookpublicationplatformsoftheSpringerComplexityprogramarethe monograph series“Understanding ComplexSystems”focusing on thevariousapplications of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative theoreticalandmethodological foundations,andthe“SpringerBriefsinComplexity”which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporatesindividualtitlesrangingfromtextbookstomajorreferenceworks. EditorialandProgrammeAdvisoryBoard HenryAbarbanel,InstituteforNonlinearScience,UniversityofCalifornia,SanDiego,USA DanBraha,NewEnglandComplexSystemsInstituteandUniversityofMassachusettsDartmouth,USA PéterÉrdi,CenterforComplexSystemsStudies,KalamazooCollege,USAandHungarianAcademyofSciences, 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,YorkUniversity,Toronto,Ontario,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 Springer Series in Synergetics FoundingEditor:H.Haken TheSpringerSeriesinSynergeticswasfoundedbyHermanHakenin1977.Since then,theserieshasevolvedintoasubstantialreferencelibraryforthequantitative, theoreticalandmethodologicalfoundationsofthescienceofcomplexsystems. Throughmanyenduringclassictexts,suchasHaken’sSynergeticsandInforma- tion andSelf-Organization,Gardiner’sHandbookof StochasticMethods,Risken’s The Fokker Planck-Equationor Haake’s QuantumSignaturesof Chaos, the series hasmade,andcontinuestomake,importantcontributionstoshapingthefoundations ofthefield. Theseriespublishesmonographsandgraduate-leveltextbooksofbroadandgen- eralinterest,withapronouncedemphasisonthephysico-mathematicalapproach. Moreinformationaboutthisseriesathttp://www.springer.com/series/712 Shanmuganathan Rajasekar (cid:129) Miguel A.F. Sanjuan Nonlinear Resonances 123 ShanmuganathanRajasekar MiguelA.F.Sanjuan SchoolofPhysics DepartmentofPhysics BharathidasanUniversity UniversidadReyJuanCarlos Tiruchirappalli,TamilNadu Móstoles,Madrid,Spain India ISSN0172-7389 ISSN2198-333X (electronic) SpringerSeriesinSynergetics ISBN978-3-319-24884-4 ISBN978-3-319-24886-8 (eBook) DOI10.1007/978-3-319-24886-8 LibraryofCongressControlNumber:2015956614 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) To ProfessorM.Lakshmanan To myparentsIsidroandFlorentina Preface Resonance is one of the fundamental phenomena exhibited by nonlinear systems andis importantinphysics,engineeringandbiology.Itrefersto a realizationof a maximum response of a dynamicalsystem. In an oscillating system, the response is primarily due to the ability of the system to store and transfer energy received from an external forcing source into an internal oscillating mode. Resonance can bedeterministicorstochasticandcanberealizedinmicroscopicandmacroscopic systems. Both single and coupled systems can display resonance behaviour. It is beneficialinmanyapplicationsandalsoleadstoinstabilityanddisastersincertain systems. Resonance can be induced in dynamical systems by means of external forces. Different kinds of external forces give rise to distinct types of resonances. An enhancedmaximumresponseofasystematafrequencyduetoanexternalperiodic force is termed a forced resonance or simply a resonance. Some of the other fascinating types of resonances are (1) stochastic resonance—induced by a weak noise at the frequency of the applied periodic force, (2) vibrational resonance— induced by a high-frequency force at the low-frequency of the external force, (3) coherence resonance—induced by an external noise in the absence of any external periodic force, (4) ghost resonance—inducedby a multi-frequencyforce at a frequency absent in the external force, (5) parametric resonance—induced by a periodic variation of a parameter of the system, (6) autoresonance—induced by an external force with time-dependent frequency and (7) chaotic resonance due to a perturbation of chaotic nature. Among these types, the resonance due to an additive force with a single frequency and parametric resonance occur both in linear and nonlinear systems. Other resonances take place only in nonlinear systems. In addition to the above resonances, in certain systems the amplitude of the responseis foundto be eitherzero orminimumat oneor morefrequenciesor atcertainvaluesofacontrolparameter.Thisphenomenonistermedantiresonance. Salient features of the above-mentioned resonances and antiresonance have been investigatedinmathematicalmodelequationsofphysicallyinterestingsystemsand in real experimental systems. Theoretical procedures and statistical measures are developedtoidentifyandexplorethevariousfeaturesofresonances. vii viii Preface In the near future, the features of resonances are expected to lead to advanced technologicalapplications. For example, many macro, micro and nanoscale oscil- latorsanddevicesworkingin resonantmodesasfilters,nonlinearmixers,sensors, atomicscaleimagingandamplifiersarefoundtogiverisetohigherefficiencyand greatlyimprovedperformance. This book presents basic aspects and salient features of the above-mentioned variousnonlinearresonancesandantiresonance.Particularly,foreachresonance,it coverstheoreticalconcepts,illustration,casestudies,mechanism,characterization, numerical simulation, experimental realization, quantum analogue, applications and significant progresses made over the years. It is self-contained, mathematical derivations show all the main steps, and the techniques involved in numerical simulation are clearly described so that a reader is able to reproduce the results presented. This is written in a simple language and prototype and paradigmatic modelequationsare used to illustrate the mechanismsof resonancesand describe thetheoreticalprocedures.Thisisprimarilydevelopedasatextatthepostgraduate levelandalsoasa referencebookforresearchersworkingand/orinterestedinthe dynamicsofresonances. This book is structured into 14 chapters and 3 appendices. The book begins with a detailed introduction to the phenomenonof a forced resonance in Chap.1. Througha theoreticaltreatment, the occurrenceof resonancein certain physically interesting nonlinear systems driven by a single deterministic periodic external forceisdiscussed.Somenotableapplicationsofforcedresonancesarementioned. Chapter2presentstheory,characterizationandapplicationsofstochasticresonance, andaquantumanalogueofstochasticresonanceispointedout.Chapter3isdevoted to the theoretical analysis of the biharmonic force induced vibrational resonance in monostable nonlinear systems. Resonance with nonsinusoidal periodic forces, the effect of noise and the role of asymmetry of a potential on resonance are discussed.Thefeaturesofbothstochasticandvibrationalresonancesinmultistable andexcitablesystemsarepresentedinChap.4.Occurrencesofbothstochasticand vibrational resonances in spatially periodic potential (pendulum) system and in a modifiedChua’scircuitmodelequationwithperiodiccharacteristiccurveofChua’s diode are analysed in Chap.5. The role of the number of equilibrium points on resonances is explored. Furthermore, the characteristic differences between these twotypesofresonancesareenumerated. Chapter 6 mainly deals with vibrational resonances in the Duffing oscillator withtime-delayedfeedback.Notableexamplesoftime-delayindifferentbranches of science are enumerated. The role of constant single time-delay, multiple time- delay,integrativetime-delay,distributedtime-delayandstate-dependenttime-delay andtheinfluenceofstrengthoftime-delayedfeedbackareanalysed.Next,Chap.7 reports on the resonance enhanced signal propagation in unidirectionally coupled nonlinear systems. Particularly, the influence of forced, vibrationaland stochastic resonances on signal propagation is explored. Chapter 8 covers experimental observationofvibrationalresonancein singleChua’scircuit,overdampedbistable system,verticalcavitysurfaceemittinglasersystem,anexcitableelectroniccircuit and unidirectionallycoupled Chua’s circuits. In the next chapter, ghost-stochastic Preface ix resonanceinducedbynoiseandghost-vibrationalresonanceduetoahigh-frequency forcearecovered.Bothsingleandcoupledsystemsaretakenforanalysis.Thephe- nomenonofparametricresonanceisdealtinChap.10.Particularly,itsoccurrencein certainlinearandnonlinearsystems,analogcircuitsimulationresultsanditscertain signaturesinquantumsystemsarepresented. Chapter 11 is devoted to autoresonance.The role of limiting phase trajectories onautoresonanceisexamined.Thefeaturesofautoresonanceinopticalwaveguides and four-wavemixing are elucidated. Its quantumanalogueis shown to be ladder climbing of energy with time. Coherence resonance and chaotic resonance are treated in Chap.12. Theory of coherence resonance is presented with reference to a two-state noise driven system. Resonance-like behaviour induced by certain kindsofchaoticperturbationsandGaussianwhitenoiseisanalysed.Theresponse of linear and nonlinear systems to time-varying frequency of the driving external force and monotonic increase of a parameter with time are discussed in the next chapter.Suchperturbationsareshowntogiverisetoaslowpassageoftrajectories through resonance and resonance tongues. The role of initial conditions on the maximumresponseisdiscussed.Thefinalchapterisprimarilyconcernedwiththe identificationandanalysis ofantiresonancein bothsingle and coupledoscillators. Analog circuit simulation results are also presented. Finally, equilibrium point analysis,analyticalexpressionforthe rootsof a cubicequationanddescriptionof analogcircuitconstructionforanordinarydifferentialequationarepresentedinthe appendices. We are indebted to J. Kurths, K. Thamilmaran, V. Chinnathambi, V.M. Gand- himathi, S. Jeyakumari, V. Ravichandran, S. Arathi, C. Jeevarathinam, R. Joth- imurugan, J.M. Casado, M. Coccolo, A. Daza, J. Seoane, G. Litak, J. Used, A. Wagemakers, J.H. Yang, S. Zambrano, K. Abirami and S. Rajamani for many yearsof collaborationwithoutwhichitwouldnothavebeenpossibleto write this book.One of us (S.R) thanksP. Philominathan,T. Arivudainambi,A. Venkatesan and K. Murali for their suggestions and encouragement. All the analog circuit simulation studies presented in this book are prepared in collaboration with K.ThamilmaranandR.Jothimurugan.Itisagreatpleasuretothankthem.Finally, weextendourthankstoourfamilymembersfortheirgreatsupport,cooperationand encouragementduringthecourseofpreparationofthiswork. Tiruchirapalli,TamilNadu,India S.Rajasekar Móstoles,Madrid,Spain MiguelA.F.Sanjuan July2015
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