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AsymptoticPerturbationMethods AsymptoticPerturbationMethods ForNonlinearDifferentialEquationsinPhysics
AttilioMaccari
Author
Prof.AttilioMaccari ViaAlfredoCasella3
00013Rome Italy
CoverImage: ©CallistaImages/Getty Images
Allbookspublishedby WILEY-VCH arecarefully produced.Nevertheless,authors,editors,and publisherdonotwarranttheinformation containedinthesebooks,includingthisbook, tobefreeoferrors.Readersareadvisedtokeep inmindthatstatements,data,illustrations, proceduraldetailsorotheritemsmay inadvertentlybeinaccurate.
LibraryofCongressCardNo.: appliedfor BritishLibraryCataloguing-in-PublicationData Acataloguerecordforthisbookisavailable fromtheBritishLibrary.
Bibliographicinformationpublishedby theDeutscheNationalbibliothek TheDeutscheNationalbibliotheklists thispublicationintheDeutsche Nationalbibliografie;detailedbibliographic dataareavailableontheInternetat <http://dnb.d-nb.de>
©2023WILEY-VCHGmbH,Boschstraße12, 69469Weinheim,Germany
Allrightsreserved(includingthoseof translationintootherlanguages).Nopartof thisbookmaybereproducedinanyform–by photoprinting,microfilm,oranyother means–nortransmittedortranslatedintoa machinelanguagewithoutwrittenpermission fromthepublishers.Registerednames, trademarks,etc.usedinthisbook,evenwhen notspecificallymarkedassuch,arenottobe consideredunprotectedbylaw.
PrintISBN: 978-3-527-41421-5
ePDFISBN: 978-3-527-84172-1
ePubISBN: 978-3-527-84173-8
oBookISBN: 978-3-527-84174-5
Typesetting Straive,Chennai,India
Contents
AbouttheAuthor xi
Foreword xiii
Introduction xv
1TheAsymptoticPerturbationMethodforNonlinear Oscillators 1
1.1Introduction 1
1.2NonlinearDynamicalSystems 3
1.3TheApproximateSolution 5
1.4ComparisonwiththeResultsoftheNumericalIntegration 10
1.5ExternalExcitationinResonancewiththeOscillator 11
1.6Conclusion 16
2TheAsymptoticPerturbationMethodforRemarkable NonlinearSystems 19
2.1Introduction 19
2.2PeriodicSolutionsandTheirStability 21
2.3GlobalAnalysisoftheModelSystem 27
2.4Infinite-periodSymmetricHomoclinicBifurcation 35
2.5AFewConsiderations 41
2.6APeculiarQuasiperiodicAttractor 42
2.7BuildinganApproximateSolution 44
2.8ResultsfromNumericalSimulation 46
2.9Conclusion 52
3TheAsymptoticPerturbationMethodforVibrationControl withTime-delayStateFeedback 53
3.1Introduction 53
3.2Time-delayStateFeedback 53
3.3ThePerturbationMethod 56
3.4StabilityAnalysisandParametricResonanceControl 59
3.4.1TheFrequency–ResponseCurveIs 62
3.5SuppressionoftheTwo-periodQuasiperiodicMotion 63
3.6VibrationControlforOtherNonlinearSystems 68
4TheAsymptoticPerturbationMethodforVibrationControlby NonlocalDynamics 69
4.1Introduction 69
4.2VibrationControlforthevanderPolEquation 72
4.3StabilityAnalysisandParametricResonanceControl 74
4.4SuppressionoftheTwo-periodQuasiperiodicMotion 79
4.5Conclusion 82
5TheAsymptoticPerturbationMethodforNonlinear ContinuousSystems 83
5.1Introduction 83
5.2TheApproximateSolutionforthePrimaryResonanceofthe nth Mode 86
5.3TheApproximateSolutionfortheSubharmonicResonanceofOrder One-halfofthe nthMode 91
5.4Conclusion 93
6TheAsymptoticPerturbationMethodforDispersiveNonlinear PartialDifferentialEquations 95
6.1Introduction 95
6.2ModelNonlinearPDESObtainedfromtheKadomtsev–Petviashvili Equation 97
6.3TheLaxPairfortheModelNonlinearPDE 98
6.4AFewRemarks 100
6.5AGeneralizedHirotaEquationin2 + 1Dimensions 100
6.6ModelNonlinearPDEsObtainedfromtheKPEquation 101
6.7TheLaxPairfortheHirota–MaccariEquation 103
6.8Conclusion 105
7TheAsymptoticPerturbationMethodforPhysics Problems 107
7.1Introduction 107
7.2DerivationoftheModelSystem 108
7.3IntegrabilityoftheModelSystemofEquations 111
7.4ExactSolutionsfortheC-integrableModelEquation 112
7.4.1NonlinearWave 112
7.4.2Solitons 112
7.4.3Dromions 113
7.4.4Lumps 116
7.4.5RingSolitons 116
7.4.6Instantons 117
7.4.7MovingBreather-LikeStructures 117
7.5Conclusion 120
8TheAsymptoticPerturbationModelforElementaryParticle Physics 121
8.1Introduction 121
8.2DerivationoftheModelSystem 122
8.3IntegrabilityoftheModelSystemofEquations 124
8.4ExactSolutionsforthe C-integrableModelEquation 125
8.4.1NonlinearWave 125
8.4.2Solitons 126
8.4.3Dromions 126
8.4.4Lumps 127
8.4.5RingSolitons 127
8.4.6Instantons 129
8.4.7MovingBreather-likeStructures 129
8.5AFewConsiderations 130
8.6HiddenSymmetryModels 130
8.7DerivationoftheModelSystem 133
8.8CoherentSolutions 138
8.8.1NonlinearWave 138
8.8.2Solitons 138
8.8.3Dromions 139
8.8.4Lumps 139
8.8.5RingSolitons 140
8.8.6Instantons 141
8.8.7MovingBreather-likeStructures 142
8.9ChaoticandFractalSolutions 143
8.9.1Chaotic–ChaoticandChaotic–PeriodicPatterns 143
8.9.2ChaoticLineSolitonSolutions 145
8.9.3ChaoticDromionandLumpPatterns 145
8.9.4NonlocalFractalSolutions 147
8.9.5FractalDromionandLumpSolutions 147
8.9.6StochasticFractalDromionandLumpExcitations 148
8.10Conclusion 150
9TheAsymptoticPerturbationMethodforRogueWaves 151
9.1Introduction 151
9.2TheMathematicalFramework 153
9.3TheMaccariSystem 154
9.4RogueWavePhysicalExplanationAccordingtotheMaccariSystemand BlowingSolutions 156
9.5Conclusion 158
10TheAsymptoticPerturbationMethodforFractalandChaotic Solutions 159
10.1Introduction 159
10.2ANewIntegrableSystemfromtheDispersiveLong-waveEquation 161
10.3NonlinearCoherentSolutions 165
10.3.1NonlinearWave 165
10.3.2Solitons 165
10.3.3Dromions 166
10.3.4Lumps 166
10.3.5RingSolitons 167
10.3.6Instantons 167
10.3.7MovingBreather-LikeStructures 168
10.4ChaoticandFractalSolutions 168
10.4.1Chaotic–ChaoticandChaotic–PeriodicPatterns 168
10.4.2ChaoticLineSolitonSolutions 168
10.4.3ChaoticDromionandLumpPatterns 169
10.4.4NonlocalFractalSolutions 169
10.4.5FractalDromionandLumpSolutions 169
10.4.6StochasticFractalExcitations 170
10.4.7StochasticFractalDromionandLumpExcitations 170 10.5Conclusion 171
11TheAsymptoticPerturbationMethodforNonlinear RelativisticandQuantumPhysics 173
11.1Introduction 173
11.2TheNLSEquationfor a1 > 0 174
11.3TheNLSEquationfor a1 < 0 176
11.4APossibleExtension 178
11.5TheNonrelativisticCase 180
11.6TheRelativisticCase 183
11.7Conclusion 185
12Cosmology 187
12.1Introduction 187
12.2ANewFieldEquation 188
12.3ExactSolutionintheRobertson–WalkerMetrics 191
12.4EntropyProduction 195
12.5Conclusion 197
13ConfinementandAsymptoticFreedominaPurelyGeometric Framework 199
13.1Introduction 199
13.2TheUncertaintyPrinciple 201
13.3ConfinementandAsymptoticFreedomfortheStrongInteraction 203
13.4TheMotionofaLightRayIntoaHadron 207
13.5Conclusion 208
14TheAsymptoticPerturbationMethodforaReverse Infinite-PeriodBifurcationintheNonlinearSchrodinger Equation 209
14.1Introduction 209
14.2BuildinganApproximateSolution 210
14.3AReverseInfinite-PeriodBifurcation 212
14.4Conclusion 215
Conclusion 217 References 219 Index 235
AbouttheAuthor AttilioMaccariisafree-lanceresearcherinnonlinearphysics.HereceivedhisPhD fromLaSapienzaRomeUniversityin1990.Hehaspublishedaboutonehundred papers,mainlyoncoupledoscillatorsandnonlinearpartialdifferentialequationsin physicsandengineering.
Amonghismostimportantaccomplishments,werecalltheMaccarisystemfor roguewavesaswellastheHirota–Maccariequationfornonlinearsystems.Both equationsareintegrableandwithremarkablenonlinearwaves.Hisrecentwork hasbeendevotedtovibrationcontrol,andhefoundtwonewmethodsinorderto performthisveryimportanttask,time-delaystatefeedbackcontrolandnonlocal feedbackcontrol.
Foreword Thistextbookisdevotedtononlinearphysics.
Theasymptoticperturbationmethodisusedasamathematicaltoolandis explainedinsomedetail,andthetheoryisdevelopedsystematically,startingwith nonlinearoscillators,limitcyclesandtheirbifurcations,followedbyiteratednonlinearmaps,continuoussystems,nonlinearpartialdifferentialequations(NPDEs), andculminatingwithinfinite-periodbifurcationinthenonlinearSchrodinger equationandfractalandchaoticsolutionsinNPDEs.
Aremarkablefeatureofthebookistheemphasisonapplications.Thereareseveralexamples,andthescientificbackgroundisexplainedatanelementaryleveland closelyintegratedwiththemathematicaltheory.
Thisbookisidealforanintroductorycourseattheseniororfirst-yeargraduate level.Itisalsoadvisableforascientistwhohasnotadeepknowledgeaboutnonlinearphysicsbutnowwantstobeginacompletestudy.
Theprerequisitesaremultivariablecalculusandintroductoryphysics.
Introduction Nonlinearsystemsareparamountinengineeringandscience.Manyperturbation methodscanbeusedtostudythesesystemsinordertopredictremarkablebifurcations(aqualitativechangeintheirbehavior).Inthisbook,wewillusetheasymptoticperturbation(AP)methodbothfornonlinearordinarydifferentialequations (NODEs)andnonlinearpartialdifferentialequations(NPDEs).
InChapters1–4,wewillstudyNODEsandwillderiveasuitablemodelsystem tofindthemostimportantnonlinearsystemcharacteristics.Themainfindingis thatanonlinearmodelsystemofequationsdescribestheirbehavior.Inparticular,in Chapter2,wewilldescribeaninfinite-periodbifurcationforaparametricallyexcited Liènardsystemandfindapeculiarattractorforaweaklynonlinearoscillatorwitha two-periodquasiperiodicforcing.
InChapter3,weconsidervibrationcontrolwithtime-delaystatefeedback andperformasuccessfulcontrolstrategy.InChapter4,weillustrateanother vibration-controlmethodbasedonnonlocaldynamics.Numericalsimulation confirmsourmethod’svalidity.
InChapter5,weenlargeourperspectiveandstudynonlinearcontinuoussystems, inparticularthevibrationsofanEuler–Bernoullibeamrestingonanonlinearelasticfoundationandwithanexternalperiodicexcitation.Frequency–responseand externalforce–responsecurvescanbeeasilyfoundandcomparedwithnumerical simulation.
InChapter6,theAPmethodisusedtostudyNPDEs,andweareableto findtwonewandintegrablenonlinearequations,theMaccarisystemandthe Hirota–Maccariequation.
Atthesametime,theAPmethodcanbeusedinordertofindapproximatesolutionstorelevantphysicsproblems.InChapter7,westudytheZakharov–Kusnetsov equationandshowtheexistenceofinteractinglocalizedsolutionsbecausethe ZKequationcanbedescribedthroughaC-integrable(solvableviaanappropriate changeofvariables)systemofnonlinearevolutionequations.Dromions,lumps, ringsolitons,andbreathersexistforthisremarkablenonlinearequation.
InChapter8,westudytheconnectionbetweentheAPmethodandelementary particlephysics.
Introduction
InChapter9,wetrytoexplaintheroguewavesappearanceinnonlinear systems.
InChapter10,wearriveatoneofthemostimportantfindingsinthisbook, fractalandchaoticsolutionsarepossiblefornonlinearsystemsandperhapsata veryfundamentallevelwemustlettheparticleconcept(i.e.acoherentsolution) downbecausewecanstatethatingeneralsolutionshavefractalandchaotic properties.
InChapter11,weusetheAPmethodinordertoarriveatnonlinearquantum mechanicsandachievetheEinstein–deBroglesoliton-particleconceptbystudyingtheweaklynonlinearKlein–Gordonequationforaparticleconfinedinabox. InChapter12,weillustratehowtomodifytheEinsteinequationsoastoexplain theacceleratingandirreversibleevolutionoftheuniverse.AccordingtoPrigogine’s ideas,theentropyincreaseisconnectedwithmatterproduction.
InChapter13,thisnewtheoryisusedtofindhowconfinementandasymptotic freedomcanbeexplainedinaframeworkwhereparticlesarelikesmallblackholes. Finally,Chapter14isdevotedtoareverseinfinite-periodbifurcationforthenonlinearSchrodingerequationin2 + 1dimensions.
Manyteachingyearsallowedmewritingthisbook,andIwouldliketothank mystudentsatFolignoinPerugiaUniversity,Italy,fortheirhelpfulandvaluable suggestions.
TheAsymptoticPerturbationMethodforNonlinear Oscillators 1.1Introduction
Oscillationsareafundamentaltopicinphysics.Whenasystemisnearitsequilibriumpoint,itbeginstooscillate,butifthedisplacementincreases,thenthenonlineartermsarenotnegligible.Thestartingpointisthedifferentialequationforthe harmonicoscillator
d2 X dt + �� 2 X (t)= 0(1.1)
where X (t)isthedisplacementand ω thecircularfrequency.Themostgeneralsolutionis
where �� and �� arefixedbytheinitialconditions(theCauchyproblem) if X (0) = X 0 forthedisplacement and ̇ X (0)= ̇ X 0 fortheinitialvelocity thenweeasilyget
Now,wecanconsideraweaklynonlinearpartinthedifferentialEq.(1.1)or,onthe contrary,astronglynonlinearpartbutwithsmallsolutions.Thefirstconsequenceis thattheamplitudeandthephaseareslowlyvaryingwithtime,sowecanintroduce anotherslowtime ��
where �� isabookkeepingdeviceand q isarationalnumberthatwillbechosenafterwards.Ifwewanttostudytheasymptoticsolutionbehavior(t → ∞)and �� → 0,then
AsymptoticPerturbationMethods:ForNonlinearDifferentialEquationsinPhysics, FirstEdition.AttilioMaccari. ©2023WILEY-VCHGmbH.Published2023byWILEY-VCHGmbH.
1TheAsymptoticPerturbationMethodforNonlinearOscillators
�� mustassumefinitevalues.So,weassumethatanapproximatesolutionisgivenby
orbetter
where c.c.standsforcomplexconjugateand h.o.t.forhigherorderterms. Followingthispath,wearemixingthemostimportantfeaturesoftwowell-known perturbationmethods,theharmonicbalanceandthemultiplescalemethods(for moredetailsaboutthesetwoperturbationmethods,seeRefs.[202,203,249]).
Ifweconsideraweaklynonlineardifferentialequation
where NL standsforthenonlinearpart,forinstance,
wecaninsertthesolution(1.7)inthenonlinearEq.(1.8)andwithsomealgebra manipulation,wegetfor n = 0
andfor n = 1
then, q = 2forthepropernonlineartermbalanceandwithsomealgebra manipulation
Weobservethatthevariablechange(1.5)impliesthat
whenthetemporaldifferentialoperatoractsonthefunction
FromEq.(1.10),wecanseethattheapproximatesolutionisalwaysperiodic,the amplitudeisconstant,buttheperiodchangesandbecomes
However,if
theperioddoesnotchangeandisequaltothelinearcaseperiod.
Inthischapter,wewanttoextendthismethodandstudyageneralizedVander Pol–Duffingoscillatorinresonancewithaperiodicexcitation
Weusetheasymptoticperturbation(AP)methodbasedonFourierexpansionand timerescaling(seeabove)anddemonstratethroughasecond-orderperturbation analysistheexistenceofoneortwolimitcycles.Moreover,weidentifyasufficient conditiontoobtainadoublyperiodicmotionwhenasecondlowfrequencyappears, inadditiontotheforcingfrequency.Thecomparisonwiththesolutionobtainedby thenumericalintegrationconfirmsthevalidityofouranalysis.
1.2NonlinearDynamicalSystems Thestudyofnonlineardynamicalsystemshasinterestedmanyresearchers,andvariousmethodshavebeenused.Historically,theAPmethodwasfirstappliedinorder tostudythemostimportantcharacteristicsofanonlocaloscillator[112,113,118].
Wenowdevoteourattentiontothefollowingtypeofnonlinearequation
wherethedotdenotesdifferentiationwithrespecttothetimeandthefunctions f (x ) and g(x , y)aresupposedtobeanalytic.
ThelimitcyclesofthemodifiedVanderPolequation
havebeenstudiedinRef.[23]bymeansofatimetransformationmethod.
Phaseportraitsanddynamicalpropertiesoftheequation
havebeeninvestigatedwiththemethodsofdifferentiabledynamics[74]andthe equation
withthemethodofaveraging,theKBMmethod,themethodofmultiplescales,and thePoincaré–Lindstedtmethod[202,203].
NotethatEqs.(1.22)–(1.24)belongtothegeneralclass(1.21)andarecharacterized bythefactthat f (x )isanoddfunctionof x .
1TheAsymptoticPerturbationMethodforNonlinearOscillators
WerestrictourstudytothefollowingparticularcaseofEq.(1.21)
Eq.(1.5)canbeconsideredageneralizedVanderPol–Duffingequationbecauseit includesasparticularcasestheVanderPoloscillator(f 2 , f 3 , g1 = 0and g0 =− g2 ≠ 0) andtheDuffingequation(f 2 = g1 = g2 = 0and g0 = f 3 ≠ 0).ManyauthorshavestudiedtheproblemofapproximatingthelimitcycleoftheVanderPolequation.Stokes [249]usedthenonlinearGalerkinmethodanddevelopedaseriesrepresentation; DepritandSchmidt[47]utilizedthePoincaré–Lindstedtmethodtofindtheamplitudeandfrequencyofthelimitcycle;andGarcia-MagalloandBejarano[57]consideredageneralizedVanderPolequationbymeansoftheharmonicbalancemethod. Thesteady-statebehavioroftheVanderPoloscillatorhasalsobeenstudiedbyintegralmanifoldmethodsandsymbolicmanipulationpackagesbyGilsinn[59,61]. MehriandGhorashi[195]consideredtheperiodicallyforcedDuffingequationin ordertoestablishsufficientconditionstohaveaperiodicsolution,andQaisi[233] studiedasimilarproblemusingananalyticalapproachbasedonthepowerseries method.Inaseriesofpapers[69–71],Hassanusedthehigherordermethodofmultiplescaleswithreconstitutionandtheharmonicbalancemethodtodeterminethe periodicstateresponseoftheDuffingoscillator.
InourtreatmentofEq.(1.25),noconditionsareimposedonthecoefficients f 2 , f 3 , g1 ,and g2 ,whichcanbeoforder1.Onlythedissipativecoefficient g0 issupposedto beoforder e2 .Eq.(1.25)transformsinto
Inthesecondsection,wecalculatetheapproximatesolutiongoodtotheorder of ��4 andconstructaccurateexpressionsforthelimitcycleofEq.(1.26).Moreover, wedemonstratethat,inthefirstapproximation,thebehaviorofthesolutioncanbe describedbymeansofamodelsystemofdifferentialequations,whichrepresents thecharacteristicsofEq.(1.26)bymeansofareducedsetofparameters.
Usually,perturbationanalysisiscarriedoutonlytothefirstorderbecause,inmany cases,asecondorder-calculationdoesnotchangethequalitativebehaviorofthe solution.However,inSection1.2,wedemonstratethatiftheparametersareappropriatelychosen,wecanfindtwolimitcyclesandcancalculatetheirpositionsonly byasecond-orderperturbationanalysis.
InSection1.3,acomparisonwiththeresultsofthenumericalintegrationpermits discussionofthevalidityoftheAPmethod.
InSection1.4,wetreatanextensionofEq.(1.26)thatisanonlinearoscillator forcedbyasmallperiodicexcitation,oforder e2 ,inresonancewiththenaturalfrequencyoftheoscillator
Wedemonstratethat,underappropriateconditions,astablelimitcycleappears andcalculatetherelativeapproximatesolution.Moreover,wederivesufficient conditionsfortheexistenceofadoublyperiodicmotionwhenthefundamental
1.3TheApproximateSolution 5
oscillationissubjectedtoaslightmodulation,withanamplitudeproportionalto themagnitudeoftheperiodicexcitation.
Finally,inthelastsection,webrieflyrecapitulatethemostimportantresultsand indicatesomepossiblegeneralizationsofthepresentstudy.
1.3TheApproximateSolution TheAPmethodweusetocalculatetheapproximatesolutionwasfirstdevelopedin Refs.[1,2],andtheninthissection,wesketchthemainstepsofthisperturbation technique.
Firstofall,wenowintroducearationalnumber
q = rationalnumber(1.28)
thetemporalrescaling t = eq t (1.29)
wheretherationalnumber q willbefixedafterwardsbecauseitestablishestowhat extentwecanpushthetemporalasymptoticlimitinsuchawaythatthenonlinear effectsbecomeconsistentandnotnegligible.If t → ∞,then �� → 0,when �� assumes afinitevalue.
Ifwetake �� = 0inEqs.(1.26)andneglectnonlinearterms,weseethatitadmits simpleharmonicsolutions X (t) = A exp( it) + c c.,where A isaconstantdependingoninitialconditionsand c. c.standsforcomplexconjugate.Nonlineareffects induceamodulationoftheamplitude A andtheappearanceofhigherharmonics. Themodulationisbestdescribedintermsoftherescaledvariable t thataccountsfor theneedtolookonlargertimescales,toobtainanonnegligiblecontributionfrom thenonlinearterm.
Theassumedsolution X (t)of(1.26)canbeexpressedbymeansofapowerseries intheexpansionparameter ��,weformallywrite
with �� n = |n|for n ≠ 0,and �� 0 = r isapositivenumber,whichwillbefixedlateron; inconsequenceoftherealityof(1.30a)
Theassumedsolution(1.30a)canbeconsideredacombinationofthedifferentharmonics,solutionsofthelinearequation,i.e.oftheequationobtainedafterneglecting allthenonlinearterms,andthecoefficientsofthiscombinationdependon �� and ��.
Eq.(1.30a)canbewrittenmoreexplicitly
1TheAsymptoticPerturbationMethodforNonlinearOscillators
Thefunctions �� n (t, ��)dependontheparameter ��,andwesupposethat �� n ’slimit for �� → 0existsandisfiniteand,moreover,theycanbeexpandedinpowerseriesof ��,i.e.
Inthefollowing,forsimplicity,weusetheabbreviations �� (0) n = ��n for n ≠ 1and �� (0) 1 = �� for n = 1.
Notethatthevariablechange(1.29)impliesthat (����n exp(−int))
AfterinsertingthisexpansionintoEq.(1.26),weobtainequationsforeveryharmonicandforafixedorderofapproximation,whicharerightforthepurposeof determiningthecoefficients.
For n = 0,weobtain
Acorrectbalanceoftermsshows r = 2,andthenwederivethefollowingrelation
For n = 2,takingintoaccountEq.(1.32),wehave
andthen
For n = 1,Eq.(1.26)yieldsfortheright-handside
andfortheleft-handside
If q = 2,thefirsttermhasthesamemagnitudeorderofnonlinearterms. TakingintoaccountEqs.(1.33b)and(1.34b),wecanderiveadifferentialequation, whichinvolvesonly �� ,
Substitutingthepolarform
intoEq.(1.36),andseparatingrealandimaginaryparts,wearriveatthefollowing modelsystem:
AswecanseefromEqs.(1.30c),(1.31),and(1.40),theapproximatesolutionof Eq.(1.26)canbewrittenasasumofacontributionoforder �� andacontributionof order ��2
ByinspectionofEq.(1.41),whichcanbeeasilyintegrated,weconcludethata stablesteady-stateresponseispossibleif �� 1 > 0and �� 1 < 0.Inthiscase,weobtaina stableequilibriumpoint,whichcorrespondstoastablelimitcycleforEq.(1.26),and itsapproximateexpressionisgivenby(1.43),with
Thenaturalfrequencyoftheoscillatorissubjecttoaslightmodificationand becomes
Ifwewanttoimprovethevalidityoftheapproximatesolution,wemustinclude higherorderterms.However,wecaneasilyconcludethat
0 (fortheirdefinition,seeEq.(1.31)).Indeed,weconsiderEq.(1.26)for n = 0and Eqs.(1.33b)and(1.34a)for n = 0and n = 2insuchawaytoobtain
Afterinserting(1.26b)into(1.26a),weseethattheresultingequationissatisfied if �� (1) 1 = 0.Recallthatwecanalwaysassumethattheinitialconditionis �� (1) 1 (0)= 0, becausetheinitialconditionsassociatedwithequation(1.25), X (0) = X 0 and X (0)= X 0 ,canbeusedtodetermine �� (0)= ��(0) exp(i�� (0))
Avalidhigherorderapproximationcanbederivedonlyifwetakeintoaccount �� (2) 1 ,�� (2) 2 ,�� (2) 0 For n = 0,wederivethefollowingrelation
8 1TheAsymptoticPerturbationMethodforNonlinearOscillators where h.o.t = higherordertermsand
Theobviousconclusionis
Inasimilarway,for n = 2,weobtain
Ifweneglectonlytermsoforder ��6 orhigher,Eq.(1.33a–c)transformsinto
Theterm d
inEq.(1.50)canbeeliminatedtakingintoaccountthatifwedifferentiateEq.(1.36),wehave
Moreover,from(1.50),weseethatitisnecessarytoconsiderEq.(1.26)for n = 3
Ifweusetheabbreviation
