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Applied Mathematical Sciences Volume 59 Editors S.S.Antman J.E.Marsden L.Sirovich Advisors J.K.HaleP. HolmesJ.Keener J.Keller B.J.Matkowsky A.Mielke C.S.Peskin K.R.Sreenivasan Applied Mathematical Sciences 1. John:PartialDifferentialEquations,4thed. 33. Grenander:RegularStructures:Lecturesin 2. Sirovich:TechniquesofAsymptoticAnalysis. PatternTheory,Vol.III. 3. Hale:TheoryofFunctionalDifferential 34. Kevorkian/Cole:PerturbationMethodsin Equations,2nded. AppliedMathematics. 4. Percus:CombinatorialMethods. 35. Carr:ApplicationsofCentreManifold 5. vonMises/Friedrichs:FluidDynamics. Theory. 6. Freiberger/Grenander:AShortCoursein 36. Bengtsson/Ghil/Ka¨lleń:Dynamic ComputationalProbabilityandStatistics. Meteorology:DataAssimilationMethods. 7. Pipkin:LecturesonViscoelasticityTheory. 37. Saperstone:SemidynamicalSystemsin 8. Giacaglia:PerturbationMethodsinNon- InfiniteDimensionalSpaces. linearSystems. 38. Lichtenberg/Lieberman:RegularandChaotic 9. Friedrichs:SpectralTheoryofOperatorsin Dynamics,2nded. HilbertSpace. 39. Piccini/Stampacchia/Vidossich:Ordinary 10. Stroud:NumericalQuadratureandSolutionof DifferentialEquationsinRn. OrdinaryDifferentialEquations. 40. Naylor/Sell:LinearOperatorTheoryin 11. Wolovich:LinearMultivariableSystems. EngineeringandScience. 12. Berkovitz:OptimalControlTheory. 41. Sparrow:TheLorenzEquations:Bifurcations, 13. Bluman/Cole:SimilarityMethodsfor Chaos,andStrangeAttractors. DifferentialEquations. 42. Guckenheimer/Holmes:Nonlinear 14. Yoshizawa:StabilityTheoryandtheExistence Oscillations,DynamicalSystems,and ofPeriodicSolutionandAlmostPeriodic BifurcationsofVectorFields. Solutions. 43. Ockendon/Taylor:InviscidFluidFlows. 15. Braun:DifferentialEquationsandTheir 44. Pazy:SemigroupsofLinearOperators Applications,FourthEdition. andApplicationstoPartialDifferential 16. Lefschetz:ApplicationsofAlgebraic Equations. Topology. 45. Glashoff/Gustafson:LinearOperationsand 17. Collatz/Wetterling:OptimizationProblems. Approximation:AnIntroductiontothe 18. Grenander:PatternSynthesis:Lecturesin TheoreticalAnalysisandNumerical PatternTheory,Vol.I. TreatmentofSemi-InfinitePrograms. 19. Marsden/McCracken:HopfBifurcationand 46. Wilcox:ScatteringTheoryforDiffraction ItsApplications. Gratings. 20. Driver:OrdinaryandDelayDifferential 47. Haleetal.:DynamicsinInfiniteDimensions. Equations. 48. Murray:AsymptoticAnalysis. 21. Courant/Friedrichs:SupersonicFlowand 49. Ladyzhenskaya:TheBoundary-Value ShockWaves. ProblemsofMathematicalPhysics. 22. Rouche/Habets/Laloy:StabilityTheoryby 50. Wilcox:SoundPropagationinStratified Liapunov’sDirectMethod. Fluids. 23. Lamperti:StochasticProcesses:ASurveyof 51. GolubitskylSchaeffer:BifurcationandGroups theMathematicalTheory. inBifurcationTheory,Vol.I. 24. Grenander:PatternAnalysis:Lecturesin 52. Chipot:VariationalInequalitiesandFlowin PatternTheory,Vol.II. PorousMedia. 25. Davies:IntegralTransformsandTheir 53. Majda:CompressibleFluidFlowandSystem Applications,ThirdEdition. ofConservationLawsinSeveralSpace 26. Kushner/Clark:StochasticApproximation Variables. MethodsforConstrainedandUnconstrained 54. Wasow:LinearTurningPointTheory. Systems. 55. Yosida:OperationalCalculus:ATheoryof 27. deBoor:APracticalGuidetoSplines, Hyperfunctions. RevisedEdition. 56. Chang/Howes:NonlinearSingular 28. Keilson:MarkovChainModels-Rarityand PerturbationPhenomena:Theoryand Exponentiality. Applications. 29. deVeubeke:ACourseinElasticity. 57. Reinhardt:AnalysisofApproximation 30. Sniatycki:GeometricQuantizationand MethodsforDifferentialandIntegral QuantumMechanics. Equations. 31. Reid:SturmianTheoryforOrdinary 58. Dwoyer/Hussaini/Voigt(eds):Theoretical DifferentialEquations. ApproachestoTurbulence. 32. Meis/Markowitz:NumericalSolutionof PartialDifferentialEquations. (continuedafterindex) J.A. Sanders F. Verhulst J. Murdock Averaging Methods in Nonlinear Dynamical Systems Second Edition JanA.Sanders FerdinandVerhulst JamesMurdock FaculteitExacte MathematischInstituut Dept.Mathematics WetenschappenDivisie StateUniversityofUtrecht IowaStateUniversity WiskundeenInformatica TheNetherlands Iowa,USA FreeUniversityofAmsterdam 6Budapestlaan 478CarverHall TheNetherlands Utrecht3584CD Ames50011 1081DeBoelelaan [email protected] [email protected] Amsterdam1081HV Editors: S.S.Antman J.E.Marsden L.Sirovich DepartmentofMathematics ControlandDynamical LaboratoryofApplied and Systems,107-81 Mathematics InstituteforPhysicalScience CaliforniaInstituteof Departmentof andTechnology Technology BiomathematicalSciences UniversityofMaryland Pasadena,CA91125 MountSinaiSchool CollegePark,MD20742-4015 USA ofMedicine USA [email protected] NewYork,NY10029-6574 [email protected] USA [email protected] MathematicsSubjectClassification(2000):34C29,58F30 LibraryofCongressControlNumber:2007926028 ISBN-10:0-387-48916-9 ISBN-13:978-0-387-48916-2 eISBN-13:978-0-387-48918-6 (cid:1)C 2007SpringerScience+BusinessMedia,LLC Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermis- sionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY10013, USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwith anyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilaror dissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. 9 8 7 6 5 4 3 2 1 springer.com Preface Preface to the Revised 2nd Edition Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that we are presenting a rather drastic revision of the first edition of the averaging book. Chapters 1 – 5, 7 – 10 and the Appendices A, B and D can be found, more or less, in the first edition. There are, however, many changes, corrections and updates. Part of the changes arose from discussions between the two authors of the first edition and Jim Murdock. It was a natural step to enlist his help and to include him as an author. Onenoticeablechangeisinthenotation.Vectorsarenowinboldface,with componentsindicatedbylightfacewithsubscripts.Whenseveralvectorshave the same letter name, they are distinguished by superscripts. Two types of superscripts appear, plain integers and integers in square brackets. A plain superscript indicates the degree (in x), order (in ε), or more generally the “grade”ofthevector(thatis,wherethevectorbelongsinsomegradedvector space). A superscript in square brackets indicates that the vector is a sum of termsbeginningwiththeindicatedgrade(andgoingup).Aprecisedefinition is given first (for the case when the grade is order in ε) in Notation 1.5.2, and then generalized later as needed. We hope that the superscripts are not intimidating;theequationslookclutteredatfirst,butsoonthenotationbegins to feel familiar. Proofs are ended by (cid:164), examples by ♦, remarks by ♥. Chapters6and11–13arenewandrepresentnewinsightsinaveraging,in particularitsrelationwithdynamicalsystemsandthetheoryofnormalforms. AlsonewaresurveysoninvariantmanifoldsinAppendixCandaveragingfor PDEs in Appendix E. We note that the physics literature abounds with averaging applications and methods. This literature is often useful as a source of interesting mathematical ideas and problems. We have chosen not to refer to these results as all of them appear to be formal, proofs of asymptotic validity are generally vi Preface not included. Our goal is to establish the foundations and limitations of the methodsinarigorousmanner.(Anotherpointisthatthesephysicsresultsare usually missing out on the subtle aspects of resonance phenomena at higher orderapproximationsandnormalizationthatplayanessentialpartinmodern nonlinear analysis.) When preparing the first and the revised edition, there were a number of privatecommunications;thesearenotincludedinthereferences.Wemention results and remarks by Ellison, Lebovitz, Noordzij and van Schagen. WeowespecialthankstoTheoTuwankottawhomadenearlyallthefigures and to André Vanderbauwhede who was the perfect host for our meeting in Gent. Ames James Murdock Amsterdam Jan Sanders Utrecht Ferdinand Verhulst Preface to the First Edition Inthisbookwehavedevelopedtheasymptoticanalysisofnonlineardynamical systems. We have collected a large number of results, scattered throughout the literature and presented them in a way to illustrate both the underlying commontheme,aswellasthediversityofproblemsandsolutions.Whilemost of the results are known in the literature, we added new material which we hope will also be of interest to the specialists in this field. The basic theory is discussed in chapters 2 and 3. Improved results are obtained in chapter 4 in the case of stable limit sets. In chapter 5 we treat averagingoverseveralangles;herethetheoryislessstandardized,andevenin our simplified approach we encounter many open problems. Chapter 6 deals with the definition of normal form. After making the somewhat philosophical point as to what the right definition should look like, we derive the second ordernormalformintheHamiltoniancase,usingtheclassicalmethodofgen- eratingfunctions.Inchapter7wetreatHamiltoniansystems.Theresonances in two degrees of freedom are almost completely analyzed, while we give a survey of results obtained for three degrees of freedom systems. Theappendicescontainamixofelementaryresults,expansionsonthetheory and research problems. In order to keep the text accessible to the reader we havenotformulatedthetheoremsandproofsintheirmostgeneralform,since itisourownexperiencethatitisusuallyeasiertogeneralizeasimpletheorem, thantoapplyageneralone.Theexceptiontothisruleisthegeneralaveraging theory in chapter 3. SincetheclassicbookonnonlinearoscillationsbyBogoliubovandMitropol- sky appeared in the early sixties, no modern survey on averaging has been Preface vii published. We hope that this book will remedy this situation and also will connect the asymptotic theory with the geometric ideas which have been so important in modern dynamics. We hope to be able to extend the scope of this book in later versions; one might e.g. think of codimension two bifurcations of vectorfields, the theory of which seems to be nearly complete now, or resonances of vectorfields, a difficult subject that one has only very recently started to research in a systematic manner. In its original design the text would have covered both the qualitative and the quantitative theory of dynamical systems. While we were writing this text,however,severalbooksappearedwhichexplainedthequalitativeaspects better than we could ever hope to do. To have a good understanding of the geometrybehindthekindofsystemsweareinterestedin,thereaderisreferred to the monographs of V. Arnol0d [8], R. Abraham and J.E. Marsden [1], J. Guckenheimer and Ph. Holmes [116]. A more classical part of qualitative theory,existenceofperiodicsolutionsasitistiedinwithasymptoticanalysis, hasalsobeenomittedasitiscoveredextensivelyintheexistingliterature(see e.g. [121]). A number of people have kindly suggested references, alterations and corrections. In particular we are indebted to R. Cushman, J.J. Duistermaat, W. Eckhaus, M.A. Fekken, J. Schuur (MSU), L. van den Broek, E. van der Aa, A.H.P.vanderBurgh,andS.A.vanGils.Manystudentsprovideduswithlists of mathematical or typographical errors, when we used preliminary versions of the book for courses at the ‘University of Utrecht’, the ‘Free University, Amsterdam’ and at ‘Michigan State University’. We also gratefully acknowledge the generous way in which we could use the facilities of the Department of Mathematics and Computer Science of the Free University in Amsterdam, the Department of Mathematics of the UniversityofUtrecht,andtheCenterforMathematicsandComputerScience in Amsterdam. Amsterdam, Utrecht, Jan Sanders Summer 1985 Ferdinand Verhulst List of Figures 0.1 The map of the book ............... .................... xxiii 2.1 Phase orbits of the Van der Pol equation x¨+x=ε(1−x2)x˙ ... 23 2.2 Solution x(t) of x¨+x= 2 x˙2cos(t), x(0)=0, x˙(0)=1. ....... 26 15 2.3 Exact and approximate solutions of x¨+x=εx. .............. 27 2.4 ‘Crude averaging’ of x¨+4εcos2(t)x˙ +x=0. ................. 28 2.5 Phase plane for x¨+4εcos2(t)x˙ +x=0. ..................... 29 2.6 Phase plane of the equation x¨+x−εx2 =ε2(1−x2)x˙. ....... 42 (cid:80) 4.1 F(t)= ∞ sin(t/2n) as a function of time.................. 85 n=1 4.2 The quantity δ /(εM) as a function of ε. .................... 86 1 5.1 Phase plane for the system without interaction of the species... 94 5.2 Phase plane for the system with interaction of the species...... 95 5.3 Response curves for the harmonically forced Duffing equation. . 98 5.4 Solution x starts in x(0) and attracts towards 0. .............101 5.5 Linear attraction for the equation x¨+x+εx˙3+3ε2x˙ =0......110 6.1 Connection diagram for two coupled Duffing equations. .......116 6.2 Separation of nearby solutions by a hyperbolic rest point. .....121 6.3 A box neighborhood ......................................124 6.4 A connecting orbit. .......................................130 6.5 A connecting orbit. .......................................131 6.6 A dumbbell neighborhood .................................132 6.7 A saddle connection in the plane............................140 7.1 Oscillator attached to a flywheel ...........................154 8.1 Phase flow of φ¨+εβ(0)sin(φ)=εα(0) ......................173 8.2 Solutions x =x (t) based on equation (8.5.2) ................179 2 10.1 One normal mode passes through the center of the second one..219 x List of Figures 10.2 The normal modes are linked...............................219 10.3 Poincaré-map in the linear case.............................219 10.4 Bifurcation diagram for the 1:2-resonance...................226 10.5 Poincaré section for the exact 1:2-resonance. ................226 10.6 Projections for the resonances 4:1, 4:3 and 9:2.............235 10.7 Poincaré map for the 1:6-resonance of the elastic pendulum. ..236 10.8 Action simplex...........................................242 10.9 Action simplex for the the 1:2:1-resonance. ................251 10.10Action simplex for the discrete symmetric 1:2:1-resonance. ..252 10.11Action simplex for the 1:2:2-resonance normalized to H1.....253 10.12Action simplex for the 1:2:2-resonance normalized to H2.....254 10.13Action simplex for the 1:2:3-resonance.....................256 10.14The invariant manifold embedded in the energy manifold ......256 10.15Action simplex for the 1:2:4-resonance for ∆>0............258 10.16Action simplices for the 1:2:5-resonance. ..................261 List of Tables 10.1 Various dimensions........................................209 10.2 Prominent higher-order resonances of the elastic pendulum ....237 10.3 The four genuine first-order resonances. .....................240 10.4 The genuine second-order resonances. .......................240 10.5 The 1:1:1-resonance.....................................246 10.6 Stanley decomposition of the 1:2:2-resonance ..............246 10.7 Stanley decomposition of the 1:3:3-resonance...............247 10.8 Stanley decomposition of the 1:1:2-resonance...............247 10.9 Stanley decomposition of the 1:2:4-resonance...............247 10.10Stanley decomposition of the 1:3:6-resonance...............247 10.11Stanley decomposition of the 1:1:3-resonance...............247 10.12Stanley decomposition of the 1:2:6-resonance...............248 10.13Stanley decomposition of the 1:3:9-resonance...............248 10.14Stanley decomposition of the 1:2:3-resonance...............248 10.15Stanley decomposition of the 2:4:3-resonance...............248 10.16Stanley decomposition of the 1:2:5-resonance...............248 10.17Stanley decomposition of the 1:3:4-resonance...............249 10.18Stanley decomposition of the 1:3:5-resonance...............249 10.19Stanley decomposition of the 1:3:7-resonance...............249 10.20Integrability of the normal forms of first-order resonances. .....259