Table Of ContentSpringer Series in Synergetics
Shanmuganathan Rajasekar
Miguel A.F. Sanjuan
Nonlinear
Resonances
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PeterSchuster,TheoreticalChemistryandStructuralBiology,UniversityofVienna,Vienna,Austria
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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
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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
