resonance tongues and instability pockets in the quasi ...broer/pdf/bps.pdf · 3.2 order of...

Resonancetonguesandinstabilitypocketsin thequasi–periodicHill–Schrodingerequation

HenkBroer��

, JoaquimPuig��

andCarlesSimo��

(1) Dept.of MathematicsandComp.Sci.,Univ. of Groningen,POBox 800,9700AV Groningen,TheNetherlands

(2) Dept.deMatematicaAplicadai Analisi, Univ. deBarcelona,GranVia 585,08007Barcelona,Spain

AbstractThis paperconcernsHill’ s equationwith a (parametric)forcing that is real analyticandquasi-periodicwith frequency vector �� anda ‘frequency’ (or ‘energy’) parameter� anda smallparameter�� The 1-dimensionalSchrodingerequationwith quasi-periodicpotentialoccursasaparticularcase.In theparameterplane�� for smallvaluesof � we show thefollowing.Theresonance‘tongues’with rotationnumber �� k �!�#"$��%&(' have )+* -boundarycurves. Ourargumentsarebasedonreducibilityandcertainpropertiesof theSchrodingeroperatorwith quasi-periodicpotential. Analogousto the caseof Hill’ s equationwith periodicforcing (i.e., ,-�/. ),several further resultsareobtainedwith respectto the geometryof the tongues.Oneresult re-gardstransversalityof theboundariesat �0�213� Anotherresultconcernsthegenericoccurrenceofinstability pockets in the tonguesin a reversiblenear-Mathieucase,thatmaydependon severaldeformationparameters.Thesepockets describethe genericopeningandclosingbehaviour ofspectralgapsof theSchrodingeroperatorin dependenceof theparameter�� This resultusesa re-finedaveragingtechnique.Also consequencesaregivenfor thebehaviour of Lyapunov exponentandrotationnumberin dependenceof � for fixed ��Contents

1 Intr oduction, main results 21.1 Backgroundandmotivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Hill’ s equationandthequasi-periodicSchrodingeroperator . . . . . . . 31.1.2 TheperiodicHill equationrevisited . . . . . . . . . . . . . . . . . . . . 3

1.2 Towardthemainresult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Thequasi-periodicHill equation:Rotationnumber, spectralgapsandres-

onancetongues.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Formulationof theMain Theorem . . . . . . . . . . . . . . . . . . . . . 51.2.3 Instabilitypockets,collapsedgapsandstructureof thespectrum . . . . . 61.2.4 Outlineof thepaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Towards a proof of the Main Theorem 1 72.1 Dynamicalproperties.Reducibilityandrotationnumbers. . . . . . . . . . . . . 82.2 Proofof Theorem1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Non-collapsedgap( 465�21 ) . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Collapsedgap( 47�21 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Conclusionof Proofof Theorem1 . . . . . . . . . . . . . . . . . . . . . 13

1

3 Applications and examples 143.1 A criterionfor transversalityat thetonguetip . . . . . . . . . . . . . . . . . . . 143.2 Orderof tangency at thetonguetip andcreationof instabilitypockets . . . . . . 153.3 A reversiblenear-Mathieuexamplewith aninstabilitypocket . . . . . . . . . . . 16

4 Proofs 174.1 Non-collapsedgap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Collapsedgap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Differentiabilityof rotationnumberandLyapunov exponentfor afixedpotential. 24

5 Conclusionsand outlook 245.1 Larger 8 �98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Analyticity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A Lipschitz property of tongueboundariesin the large 25

B Proof of Theorem 3 27

C Structur e of the sets:<;>=@? 31

1 Intr oduction, main results

ConsiderHill’ sequation ACB B�D ;>� D �FE3;HGI?!? A �213� (1)

where �C�� arerealparametersandtherealanalyticfunctions E is quasi-periodicin GF� with afixedfrequency vector �J�K;H� � �L�L�L��!� ?M-� � If the function E is even,Hill’ s equationis reversible,but for the main resultof this paperwe shall considergeneral,including non-reversible,Hill’ sequations.Thequasi-periodicitymeansthat if NO P�Q;H�#RTS9UV'7? is the , -dimensionaltorus,thereexist a real analytic function WYXZN \[ � suchthat, E3;HGI?]�^W_;HG`�#? . The frequency-vector�a� moreover, is assumedto be Diophantinewith constants42bc1 and dQef,hgi.T� i.e., that8 =j�!�#"L8ke\4�8 =l8nmpo , for all =�q' gr13� Thesetof such�\s� is known to have largemeasureas4 is small,e.g.,see[7].

Theobjectsof our interestareresonancetonguesasthey occurin theparameterspace� � ��C�� Ourmainresultstatesthatin thepresentanalyticcase,for small 8 �98t� thetongueboundariesareinfinitely smoothcurves. This resultis usedto studythegeometryof theresonancetongues.Here we restrict to reversiblenear-Mathieu cases,which are a small perturbationof the exactMathieu equationwhere Ek;HGI?h�^u vxw � 4 vzy�{�| ;H� v G!?$� with 4 � �L�L�L��I4 real constants. In the firstremarkof Section1.2.3a geometricreasonis given for restrictingto reversiblesystemswhenlooking for instabilitypockets.An exampleof anear-Mathieucasewith ,6�}S andadeformationparameter~ is givenby EL��;HGI?�� y�{�| ;H� � G!? D y�{�| ;H� � GI? D&� y�{�| ;H� � D � � ?�GF� (2)

It is shown that the occurrenceof instability pockets is genericin the reversiblesettingandaconcisedescriptionof its complexity if given in termsof singularitytheory. We shall draw sev-eral consequencesregardingthe spectralbehaviour of the correspondingSchrodingeroperator,in particularregardingthe effect of instability pockets on the collapsingof gaps. We developexampleswherecollapsedspectralgapsoccurin a way that is persistentfor perturbationof the� -parametrized,reversiblefamily.

Theset-upof thispaperis similar to Broer& Simo [15], wherecertaincasesof Hill’ sequationwith quasi-periodicforcing of two frequencieswerestudiedin a moreexperimentalway. Unlike

2

in theperiodiccase,smoothnessof thetongueboundariesis not easyto obtain.This novel resultusesareducibility resultby Eliasson[21]. Thismakesananalysispossibleasin theperiodiccase.Howeverdueto accumulationof tonguesweneedadelicateaveragingtechnique.

1.1 Background and motivation

Ourmotivationontheonehandrestsontheanalogywith theperiodicHill equation,whereseveralresultsin thesamedirectionwereknown. On theotherhand,thepresentresultsweremotivatedby theinterestthey have for certainspectralpropertiesof theSchrodingeroperator.

1.1.1 Hill’ sequationand the quasi-periodicSchrodinger operator

Hill’ sequationwith quasi-periodicforcing is ageneralizationandextensionof theclassical,peri-odicHill equation.Boththeperiodicandthequasi-periodiccaseoccurasafirst variationequationin thestability analysisof periodicsolutionsandlower dimensionaltori in Hamiltonianwith fewdegreesof freedom.It wasdevisedby GeorgeHill in the19thcenturyto studythemotionof theMoon [31].

For fixed ��-� , Hill’ s equationshows up astheeigenvalueequationof theone-dimensionalquasi-periodicSchrodingeroperator;>�_�� A ?�;HGI?��g A B B ;HGI? D �L��;HG!? A ;HG!?$� (3)

where ��;HGI?q�^g7E3;HGI? , which is an essentiallyself-adjointoperatoron �l�9;H�#? . In this setting,theparameter� is calledtheenergy- or thespectral-parameter. Indeed,theeigenvalueequationhasthe format �_�� A �� A � For generalreferencesee[44, 17, 39]. Quasi-periodicSchrodingeroperatorsoccurin thestudyof electronicpropertiesof solids[5]. Moreover, theseoperatorsareimportantfor solutionsof KdV equationswith quasi-periodicinitial data[29].

Presently, � is not consideredasa constant,but asa parameter. This will give a betterunder-standingof certainspectralphenomenaasthesewereobservedfor fixedvaluesof �� Oneexampleconcernsthe fact that genericallyno collapsedgapsoccur, asshown by Moser& Poschel[33].Including � asa parameter, givesa deeperinsight in thegenericopeningandclosingbehaviourof suchgapsin dependenceof �� Thereforeour maininterestis with thequasi-periodicanalogueof the stability diagramsas theseoccur for the periodic Hill equationin the parameterplane� � �Q�� Thespectralpropertiesof thequasi-periodicSchrodingeroperatorobtainedin thispaperarerelatedto thedynamicalpropertiesof Hill’ sequation.

1.1.2 The periodic Hill equation revisited

We briefly reconsiderHill’ s equationwith periodicforcing (the case,r��. ), compareBroer &Levi [12], Broer& Simo [16], who studyresonancesin thenear-MathieuequationA B B D ;>� D ��Ek;HGI?!? A �213��E3;HG D STU@?#�2Ek;HG!?$� (4)

with E even andwhere � and � are real parameters.As is well-known, in the ;>�C��L? -plane,forall %��a� resonancetonguesemanatefrom thepoints ;>��?��;!;�� ? � �I1�?$� Insidethesetongues,or instability domains,the trivial periodicsolution

A � A B ��1 is unstable.CompareVan derPol & Strutt [46], Stoker [45], Hochstadt[27], Keller & Levy [30], Magnus& Winkler [31] orArnol

Bd [3, 2, 1]. For relatedwork on nonlinearparametricforcing, seeHale [26] andBroer et

al. [9, 10, 11, 6, 14,8]. For nonlineardiscreteversionssee[36, 37].The stability propertiesof the trivial solution of the periodic Hill equationare completely

determinedby theeigenvaluesof thelinearperiodmap,alsocalledstroboscopicor Poincare map

3

�� $� Notethatdueto theconservative characterof Hill’ s equationwe have�O�L� �Z(��;�S��!��?$� the

spaceof S��rS -matriceswith determinant1. In fact, elliptic eigenvaluescorrespondto stabilityandhyperboliceigenvaluesto instability.

Thegeometryof thetongueboundarieswasstudiedin [12] and[16]. It turnsout thatgeneri-cally theboundariesof a giventonguemayexhibit severalcrossingsandtangencies,therebyalsocreatinginstability pockets,seeFigure1. This termwascoinedby Broer-Levi promptedby theterm ‘instability interval’ [30] as it occursfor fixed valuesof �� In [12, 16] normal forms andaveragingtechniquesprovide asettingfor singularitytheory. It turnsout thatin thenear-Mathieucasecloseto the % XCS resonance,onecanhave between0 and %_g¡. instability pockets,with allkindsof intermediatetangencies:thewholescenariohasat leastthecomplexity of thesingularity¢ � � m � � compare[4].

Remark. For adescriptionandanalysisof moreglobalphenomenain theperiodiccase,see[13].

1.2 Toward the main result

In this sectionwe formulateour Main Theoremregardingthe smoothnessof the boundariesofresonancetonguesin Hill’ s equationwith quasi-periodicforcing.

1.2.1 The quasi-periodic Hill equation: Rotation number, spectral gapsand reso-nancetongues.

Preliminaryto formulatingourmainresult,weneedsomedefinitions.Westartrewriting thequasi-periodicHill equation(1) asavectorfield on N �<� � ��£k��; A �!¤p?�� where £P�¥;>£ � �L�L�L��I£ ? areanglescountedmod STUj� Thisyieldsavectorfield ¦q� in systemform givenby£ B � �ACB � ¤ (5)¤ B � g�;>� D �LW_;>£�?!? A �Observe that evennessof W leadsto time-reversibility, which hereis expressedas follows: if§ XjN �-�¥�r� [ N �-�¥�-� is given by

§ ;>£k� A �!¤k?P�¨;�g0£k� A ��g0¤k?$� then§Z© ;�¦�?��ªg«¦<�

Reversibility will notbeassumedfor themainresult.Ourinterestis with theinvariant , -torus N �«�¬;>13�I1�?��6®N �� whichcarriesquasi-periodic

dynamicswith frequency vector �a� wherewe study propertiesof the normal linear behaviour.Eachtrajectoryinsidethe , -torusdenselyfills this torus,from which it maybeclearthat,unlikein theperiodiccase,P�¯.T� for ,�e\S no appropriatetwo-dimensionalPoincare mapis defined.

Neverthelessresonancetonguescanbedefinedby meansof the rotationnumber° {9± ;>��?$� aconceptliving bothin theperiodicandthequasi-periodicsetting.Wefreelyquotefrom [29]. Therotationnumberof equation(1) is definedas° {9± ;>�C��L?l� ²x³x´µ�¶+· * ¸ °I¹M;

A B ;�ºM? D&»�A ;�ºM?!?STU�º �where

Ais any non-trivial solutionof (1). Thisnumberexistsandis independentof theparticular

solution.Themap ;>��L?a<� ��¼[ ° {9± ;>�C��L?is continuousand,for fixed � , is a non-decreasingfunctionof �� Also ° {9± ;>�C��L?7�¯1 if � is suffi-ciently small. Moreover, thespectrumof theSchrodingeroperator�½�� is thesetof points � forwhich thethemap � ¼[ ° {9± ;>��? is not locally constant.

4

PSfragreplacements ¾

¿Collapsedgap

Non collapsedgap

Instability pocket

Tongueboundaries

Figure1: Resonancetonguewith pocket in the ;>�C��L? -planegiving riseto spectralgapson eachhorizontalline with constant�� Notehow collapsesof gapscorrespondto crossingsof thetongue-boundariesat theextremitiesof aninstabilitypocket.

We recall that the complementof the spectrumis called resolvent set. The openintervalswheretherotationnumberis constant,arecalledspectralgaps.In thesegaps,therotationnumbermustbeof theform À � =j�!�#"S �where =ÁÂ'a is a suitablemulti-integer suchthat

=j�!�#"(eÃ1 . This is referredas the GapLabellingTheorem[29]. ThesetÄ · ;H�#?��Å �� =j�!�#"0<�ÇÆÆÆ =r<' and

=j�!�#"ae®1ÉÈis calledthemoduleof positive half-resonancesof � . Whenfor a certainresonance

À Ä · ;H�#?thecorrespondingspectralgapvanishes,the unique � for which ° {9± ;>��L?�� À givesrise to thecollapsedgap ��C��

Now we candefinetheresonancetongue,theobjectof ourmainpresentinterest.

Definition 1 Let =-q'a . Theresonancetongueof thequasi-periodicHill equation(1) associatedto = is theset Ê ;>=@?l�iËC;>��L?a<� � ÆÆ ° {9± ;>�C��L?��Á�� =j�!�#"CÌ0�Thisstatementmeansthat,for any fixed �FÍ andany resonance�� =j�!�#"a Ä · ;H�#? , thesetof all �for which ;>��FÍ�? belongsto theresonancetongue

Ê ;>=@? is preciselytheclosureof thespectralgapof �_��Î!� (eithercollapsedor non-collapsed)correspondingto this resonanceby theGapLabellingTheorem.SeeFigure1 for illustration.

1.2.2 Formulation of the Main Theorem

As saidbefore,thepresentpaperis concernedwith thegeometryandregularityof theirboundariesfor thequasi-periodicHill equation(1) in theparameterplane ��6�� wherethefunction Eis fixed.

EachtongueÊ ; À Í�?�� Ê ;>=@? hastheformÊ ; À Í�?��iË�;>��?a<� � 8�� m ;��Ï À Í�?�Ð®��Ð®� · ;��Ï À Í�?kÌ (6)

with

À Í� Ä · ;H�#? and � · ;>13Ï À Í�?a�Ñ� m ;>13Ï À Íz?a� À �Í . Indeed,if �M�Ñ1 and �sbÒ1 , thesolutionsof (1), which now is autonomous,arelinearcombinationsof Ó�Ô vxÕ �$Ö . By theabove definitionoftherotationnumberit follows that ° {9± ;>�C�I1�?#�Ø× ��

5

Mostly the valueof

À Í is fixed, in which casewe suppressits occurrencein the boundaryfunctions� Ô � Notethatin (6) onecanask,in general,for notmorethancontinuityof themappings� Ô X�� [ �É� sinceweareimposingthat � m Ð®� · � Comparewith theperiodiccase[12, 16].

Nevertheless,recall that in theperiodiccase,s�K. thereexist realanalyticboundarycurves� � � � �� ;��L? suchthat � m �´_³ÚÙC�� I� � � and � · �´ ¸zÛ �� I� � �� For thepresentcase,heÒSwehave thefollowing result.

Theorem 1 (Smoothnessof tongueboundaries) Assumethat in Hill’ s equation(1)A�B B�D ;>� D �FEk;HG!?!? A �21with ��M<� , thefunction E is realanalyticandquasi-periodicwith Diophantinefrequencyvector�®<� � with ,�e\S�� Then,for someconstant)Ò�Ø)½;>E¬�!�#? andfor any

À ÍZ Ä · ;H�#?$� there exist) * -functions� � �2� � ;��L? and � � �2� � ;��L? , definedfor 8 �T8kÜJ) , satisfying� m �¡´_³ÚÙ�� I� � � and � · �2´ ¸zÛ �� I� � ��Remark. In thesequel,beyonda proof of this theorem,constructive methodsaregivento obtain)MÝ -approximationsof the tongueboundaries.Thesemethodscanbe applied,a fortiori, to theperiodiccase,6�.T�1.2.3 Instability pockets,collapsedgapsand structure of the spectrum

We sketchthe remainingresultsof this paper, regardinginstability pocketsandthe ensuingbe-haviour of spectralgaps.

In thequasi-periodicHill equation,instability pocketscanbedefinedasin theperiodiccase.Thefact thata tonguehasa boundarycrossingat ;>� Í �� Í ? meansthat �� Í � is a collapsedgapfortheSchrodingeroperator(3) with �0�Ø�$Íz� An exampleof thisoccursat thetonguetip �a�213�

Moser& Poschel[33] show that, for small analyticquasi-periodicpotentialswith Diophan-tine frequencies,collapsedgapscanbeopenedby meansof arbitrarily smallperturbations.Thisimpliesthat it is a genericpropertyto have no collapsedgapsfor fixedvaluesof �� In this paperwegoonestepbeyond,studyinghow gapsbehavewhenthesystemis dependingontheparameter� in agenericway.

By Theorem1weknow thatfor analyticforcing(potential),for small 8 �98 andfor aDiophantinefrequency vector �a� the tongueboundariesare infinitely smooth. From this it follows that thecomputationaltechniquesregardingnormalformsandsingularitytheory, for studyingthetongueboundaries,carryover from theperiodicto thequasi-periodicsetting.In particularthis leadsto anaturalconditionfor the tongueboundariesto meettransversallyat the tip ��¥13� implying thattherearenocollapsedgapsfor small 8 �T8�5�}13� As aresultwefind, thatfor reversibleHill equationsof near-Mathieu type, after excluding a subsetof Diophantinefrequency vectors � of measurezero,thesituationis completelysimilar to theperiodiccase.Comparewith thedescriptiongivenbeforein Section1.1.1.

We shall presentexamplesof families of reversiblequasi-periodicHill equationsof near-Mathieutypewith instability pockets. Theseexamplesarepersistentin their (reversible)setting.To our knowledge,sofar theexistenceof collapsedgapsin quasi-periodicSchrodingeroperatorshasonly beendetectedby DeConciniandJohnson[19] in thecaseof algebraic-geometricpoten-tials. Thesepotentialsonly have a finite numberof non-collapsedgaps,while all othergapsarecollapsed.In view of thepresentpaper, this is a quitedegeneratesituation. SeeFigure2 for anactualinstability pocket for which normalform methodsareneededup to secondorder, the re-sultsof whicharecomparedwith directnumericalcomputation.Thetechniquesjustdescribedareusefulwhenstudyinga fixedresonance.We note,however, that for investigating‘all’ resonance

6

tonguesatonce,evenin aconcreteexample,wewill usecertaindirectmethods,whichamounttorefinedaveragingtechniques.Comparewith theperiodiccase[16].

Remarks.

- In the non-reversiblecasegenericallyno instability pockets canbe expected. To explainthis,considertheclassicalperiodiccase,½�¥.T� compare[12]. Recallthatherethestabilitydiagramcanbe describedin termsof Hill’ s map,which assignsto every parameterpoint;>��L? thePoincare matrix

�� Þ-�j�!;�S��!�É?$� which is the3-dimensionalLie groupof S½�hS -matricesof determinant1. The tongueboundariesjust are pull-backsunderHill’ s mapof the unipotentcone,which hasdimension2 (except for singularitiesat ß«à�á ). In the3-dimensionalmatrix spacethe surfacesformed by the coneand the imageunderHill’ smapof the ;>��L? -planegenericallymeetin a transversalway. However, the intersectioncurves(whichcorrespondto thetongueboundaries)genericallydo notmeetaway from thetip ��â13� Boundarycrossingshowever do occurgenericallyunderthe extra conditionofreversibility, which reducesthedimensionof theambientmatrix spaceto 2.

- At this momentwe like to commenton global aspectsof the geometry, asrelatedto thespectrumof the correspondingSchrodingeroperator� � . Unlike in the periodiccasetheunionof resonancetonguesis adensesubsetof ��Ñ��C�� This is dueto thefactthatthemoduleof positive half-resonances,

Ä · ;H�#? is densein � · andthe fact that the rotationnumberis continuous.So if no collapsedgapsoccur, we caneven say that the union ofinteriorsof the tonguesformsandopenanddensesubset.Sincethe lattersetis containedin the complementof the spectrum,it is natural to ask whetherthis spectrumitself, forfixed �� is a Cantorset. Johnsonet al. [28, 22] show that for genericpairsof ;H�a�Vã��?<�� Þ��)Zäz;>Nj T? , with 1_ÐJå�ÜÒ. , thespectrumof � � , see(3), with �½;HGI?�� ã��;H��GI? , indeedis aCantorset.In theanalyticcase,Eliasson[21] usingKAM theory, provesthatfor Diophantinefrequenciesandsmallpotentials,Cantorspectrumhasgenericoccurrence.Notably, heretheCantorsethaspositive measure.In a2-dimensionalstripwhere 8 �98 is sufficiently small,thisgives a Cantorfoliation of curves in betweenthe densecollectionof resonancetongues.For generalbackgroundabout this antagonismbetweentopology and measuretheory inEuclideanspaces,see[38].

1.2.4 Outline of the paper

Let usbriefly outlinetherestof this paper. In Section2 we presentthe ingredientsfor our proofof Theorem1, including the notion of reducibility. Only a sketchof this proof is presented,adetailedproof is postponedto Section4. In fact, mostof the proofsarepostponedto the lattersection.

Section3 containsapplicationsof Theorem1. For thecriterionfor transversalityof thetongueboundariesatthetip seeSection3.1.A morethoroughasymptoticsat thetonguetip ��}1 andtheensuingcreationof instability pocketsis studiedin a classof reversiblenear-Mathieuequationswhich is containedin Section3.2. A proof is given in AppendixB. The zeromeasuresetofDiophantinefrequency vector � to be excludedfor this analysis,is consideredin AppendixC.A concreteexamplewith instability pockets is studiedin Section(3.3) Finally in AppendixA aLipschitzpropertyof thetongueboundariesis givenundervery generalconditions.

2 Towards a proof of the Main Theorem1

Weconsiderparametervalues ;>�¬ÍT��FÍ�? ata tongueboundary, i.e.,atanendpointof aspectralgap,whichmaypossiblybecollapsed.At aboundarypoint ;>�¬ÍT��FÍ�? therotationnumber° {9± ;>�¬Íz��FÍ�?��

7

0

0.1

0.2

0.3

0.4

0.5

0.6

1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0 0.1 0.2 0.3 0.4 0.5

Figure2: Left: Numericalcomputationof the instability pocket of thenear-Mathieuequationwith EL��;HGI?�� y�{�| ;H� � G!? D y�{�| ;H� � GI? D ~ y�{�| ;H� � D � � ?�GF� see(2), in the ;>�C��L? -planewith � � �.T�!� � �¯;�. D × æ ?!RTS�� and ~#�213�nç�� Solid linescorrespondto theapproximationof theboundariesby secondorderaveragingin ;>�C��L?$Ï Dashedlines correspondto direct numericalcomputation.Right: Differencebetweentheaveragingandthedirectnumericalapproximationasafunctionof� . Solid linescorrespondto thetongueboundarythatfor small � turnsto theleft, dashedlinestotheboundaryasit turnsto theright.�� =j�!�#" , i.e. it is ‘rational’. This suggestsa Van der Pol (covering) transformation,leadingtoa systemof co-rotatingcoordinates,e.g., compare[8, 29]. In the co-rotatingcoordinatesthetongueboundarynear ;>�¬Íz��FÍ�? getsasimplerform, thatcanevenbefurthersimplifiedby repeatedtime-averaging,wherethetime-dependenceis pushedto higherorderin thelocalizedparameters;>�C��L?$�2.1 Dynamical properties.Reducibility and rotation numbers

Recallfrom Section1.2.1thesystemform (5)£ B � �A B � ¤¤ B � g�;>� D ��W½;>£¬?!? Aof thequasi-periodicHill equation(1), which is a vectorfield ¦ on Nj ��q��P�¯��£k��; A �!¤p?�� Alsorecallthatevennessof W leadsto time-reversibility.

Sincethis is a linearequationwith quasi-periodiccoefficients,amaintool to studyits dynam-ical behaviour is its possiblereducibility to constantcoefficientsby a suitabletransformationofvariables.We alwaysrequirethat the transformationis quasi-periodicwith the samebasicfre-quenciesastheoriginal equation(or a rationalmultiple of these).Thereducedmatrix, which isnot uniquelydetermined,is calledtheFloquetmatrix. Note that for , �Ç. reductionto Floquetform is alwayspossible[26, 31, 40].

Remark. In the presentsettinggenerically, for Liouville-type rotationnumbers(i.e., which areneitherrational nor Diophantine)the normal behaviour of the invariant torus Nj <�J�¬;>13�I1�?�� isirreducible.In fact,thereexist nearbysolutionsthatareunbounded,wherethegrowth is lessthanlinear [21]. We recall that theLiouville-type rotationnumbersform a residualsubset(denseè ä �secondBairecategory) of thepositive half line. Notably, for largevaluesof 8 �T8 andnot too large� irreducibility holdsfor asetof positive measurein theparameterplane� � �� [23].

In thecaseof Hill’ sequation,whenever (5) is reducible,againdueto theconservative charac-ter of thesystem,theFloquetmatrix canbechosenin é��!;�S��!�É?$� i.e., with tracezero. Even if the

8

Floquetmatrix of a reduciblesystemis not unique,the realpartsof theeigenvaluesare; indeedthey exactly coincidewith the Lyapunov exponents. It canbe shown that for a smoothquasi-periodicpotential, ;>�C��L? is in the interior of a resonancetongueif andonly if it is reducibleto ahyperbolicFloquetmatrixsee[33]. In thecaseof quasi-periodicdifferentialequationson é��!;�S��!�#?(which includesHill’ s equation)this latter propertyis equivalentto exponentialdichotomy;seeSacker & Sell [42] for otherequivalentdefinitions.

Reducibilityoutsideof andat theboundaryof resonancetonguesin generaldoesnot have tohold,seetheabove remark.Thefollowing resultby Eliasson[21] howeverprovesreducibility forsmall valuesof � andsuitableconditionson the forcing. Also comparewith previous resultsbyDinaburg & Sinai[20] andMoser& Poschel[33].

Theorem 2 ([21]) Considerthequasi-periodicHill equation(1), or, equivalentlythesystemform(5). Assumethat thefollowingconditionshold

- Thefrequencyvector � is Diophantinewith constants4Zb\1 and d<e®,Pg®.T� i.e., 8 =j�!�#"L8ke4�8 =l8 mpo , for all =rq'a �g��1��- Thefunction W/X�N <[ � is analytic on a strip aroundthe , -torus N �Q� R3;�STUV' ?6êë �R3;�STUV'a �? , givenby 8 à�´®£�8¬Ü®ì .

Thenthere existsa constant)Ò�Ò)_;>d3�Iì@?�b®1 such that if8 �98 |�íkîï ðòñPóLï ôCõ 8öW_;>£�?L8�ÜJ)«� (7)

while the rotation numberof (1) is either rational or Diophantine, thenthere existsan analyticmap £_<N M¼[ø÷ ;>£�?ÉsèM�Þ;�S��!�a?$�anda constantmatrix ù�;>��L?Þ]é��!;�S��!�#?$� such that a fundamentalsolutionof (5) canbewrittenas ú ;HG!?l� ÷ ;H��G!RTS�?�û Û î ;>ù<;>��?�GI?��Moreover, if condition(7) is satisfiedthen

1. ù<;>��L? is nilpotentandnon-zero if andonly if � is an endpointof a non-collapsedspectralgap;

2. ù<;>��L? is zero if andonly if ��C� is a collapsedgap.

Remarks.

- At first sightit mayseemthatthis theoremimpliesthatfor smallvaluesof 8 �98 thesituationismoreor lesssimilar to theperiodiccase,_�¯.T� Thatthis is not truefollows from theaboveremarkon irreducibility for Liouville-type rotationnumbers.

- Theorem2 only gives analytic dependenceof the reducingtransformationon the angles£ , but not on the parameters;>�C��L? . However, aswe shall arguebelow, smoothnessof thetongueboundariescanbederived from it anyway. We shall useTheorem2 only to arriveat a suitableperturbative settingaroundany point ;>� Í �� Í ? at a tongueboundary. We shallconstructa formal power seriesfor the tongueboundary, which is shown to be the actualTaylor expansionat ;>�¬ÍT��FÍ�?$� Herewe make a direct useof thedefinition of thederivativeasa differentialquotient.Limits aretakenby constructinga seriesof shrinkingwedgelikeneighbourhoodsof thecurve,with increasingorderof tangency at ;>�¬ÍT��FÍ�?$� Theconstructionof thewedgesusesdynamicalpropertiesof Hill’ sequation,e.g.,concerningthevariationoftherotationnumberoutsidethetongueandits constancy in theinterior. Notethat this willdetermineregionsof exponentialdichotomyin theinterior of thetongue.

9

Usingthefundamentalsolutionprovidedby Theorem2, weturnto co-rotatingcoordinatesas-sociatedto parametervaluesat thetongueboundaries.Let ;>��ÍT��FÍ�?É<� � beata tongueboundary.ThenTheorem2 providesuswith a fundamentalmatrixof theformú ;!;>��ÍT��$Í�?!?ü;HG!?��/ý¡þ �!� ;HG!? þ � � ;HGI?þ � � ;HG!? þ �!� ;HGI?6ÿ ý . 4FG1 .}ÿ � (8)

where 4 is a constantandwhere þ v�� ;HGI?�� ÷ v�� ;H��GIRTS�?$� Observe that 4«�Ò1 if andonly if ;>�¬ÍT��FÍ�? isanextremepoint of aninstabilitypocket. Also observe that þ v�� ;HGI?$�3.ZÐ » ��Ð\S�� is quasi-periodicwith frequency vector �� a�To constructtheco-rotatingcoordinates,againconsiderHill’ s equation(1)A B B D ;>� D �FEk;HG!?!? A �21andperformthelinear, G -dependentchangeof variablesý AA B ÿ � ýJþ �!� ;HGI? þ � � ;HGI?þ � � ;HGI? þ �!� ;HGI?6ÿ � � (9)

where� &� � � Observe that thechangeof variablesin G is quasi-periodicwith rotationnumber° {9± ;>�¬ÍT��FÍ�?�� =j�!�#"$� Thedifferentialequationfor

�reads� B � ý0ý 1 41 1 ÿ D å��E ýJþ �!� þ � � þ �� g þ ��!� g þ �!� þ � � ÿ7ÿ � � (10)

where�h�¯;>��g½��ÍT�� g��FÍ�? is thenew localmultiparameterandwhereå��¬E+�¯;>��g½��Í�? D ;�� g��FÍ�?`E .Proposition1 In theabovecircumstances:

1. Thefunctionsþ �!� þ � � , þ ��!� and þ �� are quasi-periodicin G with frequencyvector � .

2. In thecaseof even E wehaveú ;�gaGFÏ�;>�C��L?!? � ú m � ;HGFÏ�;>��?!? and,thus,þ � � ;�gaGI? � g þ � � ;HGI?þ �!� ;�gaGI? � þ �!� ;HGI?$�3. Let å�° {9± ;>��L? betherotationnumberof (10). Thenå�° {9± ;>�C��L?l�2° {9± ;>�C��L?jg¯�� =j�!�#"$�

Moreover, thetongueboundariescoincidewith theboundariesof thesetwhere å�° {9± ;>��L?��13�4. Wecanchoose÷ h�j�!;�S��!�É?$�

Weomit aproof,sincethis is elementary. Thethird itemagainuses[21].

2.2 Proof of Theorem 1

By repeatedaveragingwe recursively pushthe time dependenceof the equation(10) to higherorderin thelocal parameter�-�¥;>�6gr��ÍT��g(�FÍ�?$� See[16, 8, 43] for details.As before,by £ wedenotetheangularvariablesin Nj «�¯;>�lRk;�S�UV'7?!? �

FromTheorem2 recallthatthefunction WÒ�ØW½;>£¬? is complex analyticonastrip 8 à`´ £�8¬Ü®ì �At eachaveragingstep,for someì � Ü®ì � a linearchangeof variables �¯;� D § ;>£p��?!? � �is found,which is complex analyticin £ on thestrip 8 à�´K£�8@Ü¯ì � andin the local parameters�in a neighbourhoodof 13� Furthermore,if the systemis reversible,thenthe changeof variablespreservesthis reversibility. To bemoreprecisewe have

10

Proposition2 In theabovesituation,after stepsof averaging system(10) takestheform� B � �� Ý�� ;��? 4 D �� Ý�� ;��?g«� � Ý�� ;��? gÞ� � Ý�� ;��?�� D�� Ý�� ;H��GF��? � � � (11)

where � � Ý��v ;��?�� Ý � � Ý�� v ;��?for» �Á.T��S��ç andwhere the functions� � Ý�� v � for

» �Á.T��S��ç and .�Ðié<Ð! �� havethe followingproperties:

1. � � Ý�� v ;��? are homogeneouspolynomialsof degree é in ��Ï2. � � Ö �� v � � � � �� v for é�Ð G�Ð� 3Ï3. � � � �� v �¯;>�Þg<�¬ÍL?#" þ �� v�$ D ;��gq�$Í�?#" E � þ �� v�$ , where "�% $ denotesthetimeaverage, for

» �.T��S , and� � � �� ¯;>�Pgr��Í�?#" þ �!� þ � � $ D ;��Ég��FÍL?#" E � þ �!� þ � � $ ;4. Theremainder

� � Ý��$;>£k��? is complex analytic in both £ and �� (when 8 à�´K£�8 and 8 �#8 aresufficientlysmall)while it is of order D . , that is, thefunction;>£k��? ¼[ ÆÆÆ � � Ý��v�� ;>£k��?zÆÆÆ / 8 �#8 Ý · � �.�Ð » ��Ð\S�� is boundedon a neighbourhoodof �h�213�

In thecaseof even E wehave �� Ý�� 21 .In theapplicationof this result,key ideais thattheequation� � Ý�� ;��?'&�4 D � � Ý�� ;��?)(<g&� � Ý�� ;��? � �}13� (12)

which is thedeterminantof theaveragedpartof equation(11), determinesthederivativesof thetongueboundariesup to order z� In theanalysisof (12) we distinguishbetweenthecases4s5�¥1(non-collapsedgap)and 47�21 (collapsedgap).Thiswill bedonenext.

2.2.1 Non-collapsedgap ( *,+-/. )Wefirst treatthecase4�5�21 of anon-collapsedgap.Wewill assumethat 4Mb®1 , whichmeansthat;>�¬Íz��FÍ�? is at theright boundaryof aresonancetongue.Thecaseof 4MÜ®1 canbetreatedsimilarly.Let è � Ý�� ;��?��Ò� � Ý�� ;��?'&�4 D � � Ý�� ;��?)(<g(� � Ý�� ;��? � �We solve theequationè � Ý�� ;��?É�Ò1 by theImplicit FunctionTheorem,which providesa polyno-mial � � Ý�� ;��L?��2��Í D ��C�0� Ý21 � ;��g��FÍL? � �Thecoefficients 1 � , .½ÐÒ%�Ð3 z� areuniquelydeterminedby thefunctions � � Ý�� v �C._ÐÑé½Ð4 �� » �.T��S��ç and è � Ý��$;!;>� � Ý��;��L?3g��¬ÍT�� g��FÍL?!?l�65 Ý · � ;�� g��$Í�?$� Hereandin whatfollows, 7 ;�8�?#�65:9P;�8�?$�meansthat ÆÆÆÆ 7�;�8�?8 8C8 9 ÆÆÆÆis boundedaround8��21 .

11

In orderto applytheImplicit FunctionTheorem,we compute;; � è � Ý�� ;��? ï � w Í �24<" þ ��!� $ b®13�Thisyieldsauniquepolynomial � � Ý�� 2� � Ý�� ;��L? with thepropertiesstatedabove.

Ournext purposeis to show that,if � ¼[ � ;��L? is a tongueboundarywith ��;��FÍ�?��2��Í , then²x³x´� ¶ � Î 8 � ;��L?jgr� � Ý��;��L?L88 �#g(�$Í�8 Ý �213�Morepreciselywe have

Proposition3 Considerequation(11) with 4+bJ1 . There exist positiveconstants= and > , suchthat if �@?BA and �@?DC aredefinedby�@?BE�;��L?��2� � Ý�� ;��L?üß�=�8 �Ég��FÍ�8 Ý · � �thefollowing holds.For 1_Ü�8 �Ég�� Í 8�ÜF>

1. therotationnumber° {9± ;>�@? A ;��L?$��? is different from ° {9± ;>��ÍT��FÍ�?$�2. thesystem(11) (or equivalently(5)) for �h�i;>� ?DC�;��L?�g��ÍT��pg6�FÍ�? haszero rotationnumber.

Theproof is postponedto Section4. As adirectconsequencewe have

Corollary 1 Let ;>�¬ÍT��FÍ�? be at the tongueboundaryas above and assumethat ��¬Íz� is not acollapsedgap. Then, there exists a function � ¼[ � ;��L? definedin a small neighbourhoodof��Ø�$Í , such that in this neighbourhood,

1. ;>� ;��L?$��L? is at thetongueboundaryof thesametongueas ;>��ÍT��$Í�?$�2. Themap � ¼[ � ;��L? at � Í is -timesdifferentiableat � Í andcanbewrittenas��;��L?��2��Í D ��C�G� Ý 1 � ;��g��FÍ�? � D 5 Ý · � ;��Ég��FÍ�?$�

Proof: Fromnow on,assumethat 1_Ü�8 �Ég�� Í 8�ÜF>�� Then,by Proposition3, thesetË ;>�@?DC�;��?$��L?aX�1_Ü�8 �Ég��FÍ�8�ÜF> Ìis a subsetof the tongue’s interior. Again by Proposition3, for each1]ÜÇ8 �Þg��FÍ�8OÜH> , thesetËC;>�@? A ;��L?$��L?�X�1_Ü�8 �Ég��FÍ�8�ÜF>hÌ is a subsetof the complementof the tongue. Now, for eachfixed �� themap � ¼[ å�° {9± ;>��L? is monotonous,while, moreover, ° {9± ;>�C��L? is continuousin � and�� Therefore,for each 1(ÜÁ8 �«g®�FÍ�8jÜI> thereexists a unique � ;��L? suchthat ;>��;��L?$��L? is at thetongueboundary.

Putting ��;��FÍ�?-�^��Íz� the map � ¼[ ��;��L? is continuouslyextendedto � �^�FÍT� The aboveargumentalsoimpliesthatfor 16Ü�8 ��g�� Í 83ÜF><�� ?DC ;��?ÉÐ\��;��L?�Ð®� ?BA ;��L? (13)

andas � ?DCj;��$Í�?l�2��;��FÍ�?��2�@?BA�;��FÍ�?��2��Í , this inequalitydirectlyextendsto �0�}�FÍ . Thus,duetotheform of both � ?JA and � ?KC , we have thatfor 8 �#g(�$Í�8�ÜL> ,ÆÆÆ ��;��L?jg-� � Ý�� ;��? ÆÆÆ ÐM= 8 �#g(�$Í�8 Ý · � �from which thecorollaryfollows. NRemark. ThecasewheretheFloquetmatrixhasnon-zeroelementbelow thediagonalrunssimi-larly.

12

2.2.2 Collapsedgap ( * -I. )In thecase47�21 of acollapsedgap,system(11) reads� B � �� Ý�� ;��? � � Ý�� ;��?g«� � Ý�� ;��?YgÞ� � Ý�� ;��? � D�� Ý�� ;H��GF��? � � � (14)

Thus,theanalogueof (12)now isè � Ý�� ;��?��Ø�� Ý�� ;��?!�� Ý�� ;��?jg(�� Ý�� ;��? � �213�Wewill seein section4 thatthereexist two polynomialsof order , � � Ý�� ;��L? and � � Ý�� ;��L? suchthatè � Ý�� & �2� Ý��v ;��L?jgr��ÍT��#g(�$Í ( �65 Ý · � ;��g��FÍ�?and,usingthesametoolsasin thecaseof anon-collapsedgapthefollowing resultis true,whoseproof is postponedto section4,

Proposition4 Undertheaboveassumptions,thereexist positiveconstants= and > , such that if8 �#g(�$Í�83ÐL> , then 8 � · ;��?�g-´ ¸zÛvòw � � � ��2� Ý��v ;��L?��8ÇÐ = 8 ��gr�FÍ�8 Ý · � and8 � m ;��?�g¡´½³xÙvxw � � � �� Ý��v ;��L?��8ÇÐ = 8 ��gr� Í 8 Ý · � �As adirectconsequencewe now have

²x³Ú´� ¶ ��Î 8 � · ;��L?jgr´ ¸zÛ vxw � � � ��2� Ý��v ;��L?��88 ��g��FÍ�8 Ý �21and ²x³x´� ¶ ��Î 8 � m ;��L?jg-´_³xÙ vxw � � � ��2� Ý��v ;��L?��88 ��g��FÍ�8 Ý �21andwe canchoose� · and � m in sucha way (skippingtherestriction � m ÐÑ� · ) thatbothmapsarecontinuousand timesdifferentiableat � Í � Moreover their Taylorexpansionsup to order at�$Í aregivenby � � Ý�� and � � Ý�� . CompareCorollary1 of Proposition3.

2.2.3 Conclusionof Proof of Theorem 1

Summarizingwe concludethatTheorem1 follows from theprevioussubsections,sincewe haveshown thatthetongueboundariesareinfinitely smooth.Indeed,by Eliasson’s Theorem2 a posi-tiveconstant) existsonly dependingon � and W (see(5)), suchthatfor any 6<� thefollowingholds.For any 8 �FÍ�8�ÜJ)«� polynomials�2� Ý�� and �O� Ý�� exist of degree in ;��Ég��FÍ�?$� suchthat

²x³Ú´� ¶ ��Î 8 � · ;��L?jgr´ ¸zÛ vxw � � � �� Ý��v ;��L?��88 ��g�� Í 8 Ý �21and ²x³x´� ¶ ��Î 8 � m ;��L?jgr´½³xÙ vòw � � � �� Ý��v ;��?��88 �#g(� Í 8 Ý �213�This indeedproves Theorem1. Moreover thesesubsectionsprovide a methodto computetheTaylor expansionsof the tongueboundaries,provided thata certainnumberof harmonicsof thereducingmatrix ÷ is known, compareTheorem2.

13

3 Applications and examples

In this sectionthemethodsandresultsof Section2 areappliedto studythegeometricstructureof resonancetonguesin Hill’ s equationswith quasi-periodicforcing (1). In theprevioussectionwe saw that the tongueboundariesaresmootharound ��Á1 . Also we found that their Taylorexpansionsaroundacertainpointcanbeobtainedby anaveragingprocedurefor whichoneneedsto know thereducingmatrixat thatpoint. In generalthis is notknown unless�a�21 , i.e.,whenthesystemhasconstantcoefficients. In this sectionthis factwill beusedto obtaingeneralizationsofresultsasthesehold for Hill’ s equationswith periodiccoefficients,compare[12, 16].

3.1 A criterion for transversality at the tonguetip

Thefirst applicationwill beacriterionfor thetransversalityof thetongueboundariesattheorigin,i.e.,atthetonguetip. In theperiodiccaseit is known [2, 12, 16] thatthetwoboundariesof acertainresonancetonguearetransversalat �Þ��1 if, andonly if, thecorrespondingharmonic(or Fouriercoefficient) of E doesnot vanish.In thequasi-periodiccase,thesituationis thesame.

Proposition5 In thequasi-periodicHill equationA B B D ;�� D ��Ek;HGI?!? A �}13�where Ek;HGI?P��W½;H��G!? and W XjN s[ � , assumethat W is real analytic and that the frequencyvector �®<�� is Diophantine. Thenthetongueboundariesof the = th resonance,

À Ía�Á�� =j�!�#"�Ä · ;H�#? , À Í 5�}1 , meettransversallyat �a�21 if andonly if the = th harmonicof W doesnotvanish.

Proof: Let �¬Í+� À �Í and

À ÍZ�ª�� =j�!�#"Z Ä · ;H�#?$� Thena fundamentalsolutionfor ��Í and��}1 is givenby ú ;HG!?��ý y�{�| ; À Í�GI? �P Î | ³ÚÙ ; À Í�GI?g À Í | ³xÙü; À Í�GI? y�{�| ; À Í�GI?6ÿ � (15)

Following thenotationof theprevioussection,let

þ �!� ;HG!?�� y�{�| ; À Í�GI? and þ � � ;HGI?�� .À Í | ³xÙV; À Í�G!?$�Then

þ ��!� ;HGI?^� y�{�| � ; À Í�GI?�� .S D .S y�{�| ;�S À Í�G!?$�þ �� ;HGI?^� .��Í | ³xÙ � ; À Í�G!?�� .ST�¬Í g .ST��Í y�{�| ;�S À Í�GI?$�þ �!� ;HGI? þ � � ;HGI?^� .S À Í | ³xÙü;�S À Í�GI?Denotingthetongueboundariesby � v �2� v ;��L?$� for

» �.T��S , theirderivativesat �a�21 areobtainedaveragingonceandconsideringtheequation� � � �� ;>�Pg��¬Íz��g��FÍ�?!� � � �� ;>��g-�¬ÍT��gr�FÍ�?Og(� � � �� ;>��gr��ÍT��Ég��FÍ�? � �213�UsingProposition2, it is seenby meansof a computationthat� B � ;>1�?^� . D 8öWRQ 8t�� B � ;>1�?^� .agJ8öWRQ 8t�from which follows that � B � ;>1�?Z5�2� B� ;>1�? if, andonly if, WSQs5�213� This concludesourproof. N

14

3.2 Order of tangencyat the tongue tip and creation of instabilitypockets

We now focuson a specialclassof quasi-periodicHill equationsof the reversiblenear-Mathieutype: ACB B�DUTV � D � TV ��Fw � 4 �@y�{�| ;H� � G!? D � y�{�| ; = © �!�#"�G!?XWYZWY A �213� (16)

comparewith Section1. Here�

is asmalldeformationparameterand�&�;H� � �L�L�L��!� ? µ is aDio-phantinefrequency vector. Also we take 4 � 5�21 for all �P�.T�L�L�L��I, andfix = © �¯;�% ©� �L�L�L��% © ? µ anon-zerovectorin ' . Weoftenabbreviate

=@"�� =j�!�#"$�Theorem 3 Considerthe reversible near-Mathieu equationwith quasi-periodicforcing (16) asabove. Then

(i) If� ��1 , theorder of tangencyat ��1 of the = © th resonancetongueis greateror equal

than 8 = © 8 andit is exactly 8 = © 8 if, andonly if, � doesnotbelongto :<;>= © ? , where :�;>= © ? is asubsetof theDiophantinefrequencyvectors of measure zero.

(ii) If� 5�21 , �Ø5<:<;>= © ? is Diophantineand 8 � 8 is smallenough,there existsat leastonepocket

at the = © th resonancetonguewith ends��1 and �P��T; � ?½5��1 . Here�

needsto haveasuitablesign.

Remarks.

1. Notethattheabove resultonly appliesto quasi-periodicnear-Mathieuequationsof thetype(16). For moregeneralquasi-periodicforcings,theproblemof theorderof tangency of thetongueboundariesat �0�}1 is at leastascomplicatedasin theperiodiccase,see[16].

2. Thesets:�;>= © ? arenot emptyin general.For examplesandsomepropertiesof thesesets,seeAppendixC.

3. Insteadof fixing�

onecanalsofix 8 �FÍ�8 sufficiently smallandshow that,for asuitablevalueof� � � ;��FÍ�?$� onecancreatean instability pocket in equation(16) with endsat ��i1 and��Ç� Í � A suitablechoiceof the componentsof 4 alsoallows several pockets (associated

to different = © ) with endsat ��¥1 and ��¥�FÍ andfor thesamevalueof�. In generalthe

samecomplexity holdshereasin theperiodiccase,comparewith thegeneraldiscussioninSection1.1.2.

A proof of Theorem3 is givenin AppendixB. Oneconsequenceof theTheoremis

Corollary 2 Assumethat in Hill’ s equationACB B�D ;�� D �FE3;HGI?!? A �21the forcing E is an evenfunction,real analytic, quasi-periodicand with Diophantinefrequencyvector �a� Supposethat, for some = © 5�\[ , the = © th harmonicof E doesnot vanishand that��R<:�;>= © ? . Then,thefollowingequationACB B�DUTV � D � TV ��Fw � 4 �@y�{�| ;H� � GI? D Ek;HG!?)WY WY A �21 (17)

hasa pocket at the = © th resonancetongueprovidedthat the 8 4 � 8 aresufficiently large.

15

Proof: Let� b21 bea smallparameteranddefine ã4 � �Ò4 � R � , for �½��.T�L�L�L��I, . Writing ã�Þ�Ñ�LR � ,

theequation(17) readsA B B D TV � D ã� TV ��$w � ã4 � y�{�| ;H� � GI? D�� Ek;HGI? WY WY A �213�Sincethe = © th harmonicE]Q<^ of E doesnot vanish,thisevenfunctioncanbesplit intoE3;HGI?��2E0Q_^ y�{�| ; = © �!�#"�G!? D ãE�;HGI?where ãE is anevenfunctionwhose= © th harmonicvanishes.Let ã� � � E]Q_^ . In thesenew parameterstheequation(17)getstheformA B B D TV � D ã� TV ��Fw � ã4 �üy�{�| ;H� � GI? D ã� y�{�| ; = © �!�#"�G!? D ã�E]Q<^ ãE3;HGI? WY,WY A �}13�The only differenceof the latter equationwith (16) is the term ãE�� But sinceits = © th harmonicvanishes,theconclusionsof Theorem3 concerningtheexistenceof pocketshold here,provided�Ç5®:<;>= © ? is Diophantine,ã4 � do not vanishand

�is sufficiently small. The latter conditionis

equivalentto the 4 � beingsufficiently large. N3.3 A reversiblenear-Mathieu examplewith an instability pocket

In this sectionthefollowing concreteexampleof a reversiblenear-Mathieuequationwith quasi-periodicforcing is investigated:A B B D ;�� D �T; y�{�| G D y�{�|a` G D ~ y�{�| ;�. D ` ?�GI?!? A �}13� (18)

Here ` is a Diophantinenumberand�

a deformationparameter. We considerthe resonanceÀ Í � �� ;�. D ` ?$� whichmeansthat ;>�C��~$? will benear&Ob �� ;�. D ` ?Xc � �I13�I1 ( �Since is stronglyincommensurablewith . andtheforcing is entireanalytic,thereexistsa con-stant)Ò�Ø)½; ` ? suchthatif 8 �T8kÜ\) and 8 ~�83ÜØ. , thenthereis reducibilityatthetongueboundaries[21]. CompareSection2. After a twofold averagingandothersuitablelinear transformationswhichdonotaffect theresonancedomains,thesystemis transformedinto� B �/ý7ý 1 ú ; 1 ? D�d ; 1 ?g ú ; 1 ? Ded ; 1 ? 1 ÿ D�� ; 1 ? ÿ � �where1 �¯;>��gr�¬Íz��~$? andwhere

úand

daregivenby thefollowing expansionsin 1ú ; 1 ?��g .. D ` ý �Pgr� Í. D ` g ;>�Pgr� Í ?`�;�. D ` ? � g �F�f ;�. D ` ? � )½;�~$? ÿ

and d ; 1 ?��¯g �. D ` ý g7~Sk;�. D ` ? D ~�;>�Pgr� Í ?;�. D ` ? � D �Sk;�. D ` ? � ;�. D .` ? ÿ �Here )_;�~F?�� ~ �Sk;�. D ` ? D .S D ` D .. D S ` D . D .` �

16

Hence,in thenotationof Section2 wehave � � �� ; 1 ?�� ú ; 1 ?9g d ; 1 ? and � � �� ; 1 ?�� ú ; 1 ? Dgd ; 1 ? .Thus �� ; 1 ?É�Ò1 (resp. �� ; 1 ?É�Ò1 ) if andonly ifú ; 1 ?�� d ; 1 ? (resp.

ú ; 1 ?��g d ; 1 ? ). TheTaylorexpansionsof thetongueboundariesup to secondorderin ;��~$? aregivenby

� � �� ;��~$?��ý . D `S ÿ � g ��~S D �F�f ;�. D ` ? ýCç D ç` D .S D ` D .. D S ` ÿand � � �� ;��~$?��ý . D `S ÿ � D ��~S D �F�f ;�. D ` ? ýüg+.ag .` D .S D ` D .. D S ` ÿ �ThereforethesecondorderTaylorexpansionsof thetongueboundarieshaveatransversalcrossingboth at ;��~F?r� ;>13�I1�? and at the point ;��~$?�� ; ` ~L��~F? if ~Ø5� 1 . By Theorem1 we know

that theboundaryfunctions �2� �� and �2� �� areof class )�* in �� With little moreeffort, onealsoestablishesthis samedegreeof smoothnessin the parameter~L� Following the argumentof theprevioussubsection,onehas

Corollary 3 For the reversible near-Mathieu equation(18) there exists a positiveconstant )such that, if 8 �98 Ü�) and 8 ~�8�Ü¥. , thenthetongueboundariesof theresonancecorrespondingtoÀ Í0�¯;�. D ` ?!RTS are )+* functionsof ;��~F?$� while

1. For ~+5�21 thetongueboundarieshavetwotransversal crossings,oneat ;>��L?��¯;>��ÍT�I1�? and

theotherat ;>� � �� ; ` ~L��~F? D 5 � ;�~$?$� ` ~ D 5 � ;�~F?!? ,2. For ~É�21 thetongueboundariesat �0�}1 havea secondorder tangency.

Remark. NotethatCorollary3 exactly describesthe¢ � -scenario,compare[4]. For theperiodic

analoguesee[12], whereHill’ smaphasaWhitney cuspsingularity. Comparewith Section1.1.2.

4 Proofs

Main aim of this sectionis to prove thePropositions3 and4 of Section2. We recall thesettingof Section2. Around a point ;>�¬Íz��FÍ�? Ñ�� with 8 �FÍ�8 sufficiently small, at the boundaryof aresonancezoneby Theorem2 and Proposition1 a fundamentalmatrix exists of the form (8)wherethe ÷ matrix is symplectic.Let �h�¯;>�7g<��ÍT��gq�$Í�?��XC; À ��hO? bethenew localparametersandhenceå��¬E�� À D h@E�� Thechangeof variables(9) reducestheequationfor

� � ; � � � � � ? totheform (10).

The correspondingHamiltonian,written in autonomousform by introducingnew momentai q�� , readsj ; � � � � � �I£p� i ?l� i �!�#" D .S 4 � �� D å��3W ý .S þ ��!� � � � D þ �!� þ � � � � � � D .S þ �� ÿ �Thefirst two termsof the right handsideform theunperturbedHamiltonian

j Í , the lastoneis

j� �

The rotation numberof this Hamiltonianis å�° {9± ;>�C��L?s�c° {9± ;>��L?«gÇ�� =j�!�#"$� The tongueboundariesarethe boundariesof the set å�° {9± ;>��?��Ç13� After stepsof averaging,the systemtakestheform of (11),� B � �� Ý�� ;��? 4 D � � Ý�� ;��?g«� � Ý�� ;��? gÞ� � Ý�� ;��? � D�� Ý�� ;H��GF��? � � �

17

seeProposition2. In what follows, theexpressionof theprevious equationin polarcoordinateswill beused.Writing k-� ¸ °I¹«; � � D&» � � ? , thedifferentialequationfor k becomesk B �¯;�� Ý�� D�� Ý�� ? | ³xÙ � k D Sk;�� Ý�� DM� � Ý�� ? | ³xÙlk y�{�| k D ;>4 D � � Ý�� D�� Ý�� ? y�{�| � kl� (19)

which is aquadraticform with matrix� �� Ý�� Ý�� Ý�� 4 D �� Ý�� D � � � Ý�� Ý�� Ý�� Ý�� (20)

We recall that� � Ý�� m5 Ý · � ;I8 �#8ö? uniformly in £ in a complex neighbourhoodof N � It is now

importantto distinguishbetweenthecasesof a non-collapsedgap( 4_5�}1 ) andof a collapsedgap( 47�¡1 ).4.1 Non-collapsedgap

Supposewe are in the caseof a non-collapsedgap, i.e., with 4}5�Ã13� Presentaim is to proveProposition3 which dealswith thecase4Zb\1 . Thecase4ZÜ\1 is treatedsimilarly. Recallthat,inSection2.2.1for any 6eØ. we obtaineda polynomialof order in �Ég��FÍ , � � Ý�� ;��L? suchthat,ifè � Ý�� ;��?��Ò�� Ý�� ;��? & 4 D �� Ý�� ;��? ( g(�� Ý�� ;��? � �then è � Ý�� & � � Ý�� ;��L?jg-� Í ��gr� Í ( �65 Ý · � ;��Ég�� Í ?$�In orderto prove Proposition3 we will show that thereexist constants=fb21 , sufficiently large,and >Kb®1 , sufficiently small,suchthatif 1_Ü�8 �Ég��FÍ�8�ÜF>

1. Equation(11) for ;>�C��L?l�i;>� � Ý�� ;��L? D =�8 ��g]�FÍ�8 Ý · � ��L? hasrotationnumberstrictly differentfrom zero.

2. Equation(11) for ;>��L?l�¯;>� � Ý��;��L?jgn=�8 �Ég��FÍ�8 Ý · � ��? haszerorotationnumber.

In what follows we write again ; À ��hO?M�Ç;>�_g��ÍT��0g �FÍ�? and

À � Ý�� ;�hO?«�i� � Ý�� ;��?$� Let, for some= b®1 , § Ô� ;�hO?#�Ø� � Ý�� ; À � Ý�� ;�hO?üß�=�8 hÉ8 Ý · � ��hO?$� ��.T��ç��§ Ô� ;�hO?��24 D � � Ý�� ; À � Ý�� ;�hj?üße= 8 hÉ8 Ý · � ��hO?and

� Ô ;>£p��hO?�� Ý�� b £k��; À � Ý�� ;�hO?@ße=�8 h�8 Ý · � ��hj? c . With thesedefinitions,matrix (20)becomes

ý § Ô � § Ô�§ Ô� 4 D § Ô� ÿ D ý� Ô� � Ô�� Ô� � Ô� ÿ � (21)

Let§ Ô bethefirst termof thepreviousexpression.First of all, notethat,since; è � Ý��; À 8 � w Í7�24<" þ ��!� $ �

thenápû ± § Ô ;�hO?�� § Ô � ;�hO?Kb 4 D § Ô� ;�hj? c g4b § Ô� ;�hO? c � �Øß6b�4#=o" þ ��!� $ Dep c 8 hÉ8 Ý · � D 5 Ý · � ;�hO?$�beingthe time-dependentterm

puniformly boundedfor all £s&Nj . This meansthat = and hCÍ

canbechosensothat 8`ákû ± § Ô ;�hO?�8ke 4#=S " þ ��!� $ 8 hÉ8 Ý · � �18

provided 8 hÉ8�Ü/hCÍ , andthe sign of ápû ± § Ô is ß . The elementsof the time-dependingpart, the� Ô� ;>£k��hO? , canbeuniformly boundedby ? q " þ ��!� $ if = and h�Í aresuitablymodified.Themodulusof theeigenvaluesof

§ Ô canbeboundedfrombelow by =½RTçp8 hÉ8 Ý · � and ST4�RTç . Now wedistinguishbetweenthecasesof

§ ·and

§ m .In thecaseof

§ ·, thesymmetricmatrix (21) is definitepositive andfor all £6qN , k B in (19)

is boundedfrom below by =½R�.�Sp8 hÉ8 Ý · � , sincetheminimumof k B , ignoringthecontribution of thetime-dependentpart, is =½RTçp8 hÉ8 Ý · � . This implies that therotationnumberis differentfrom zero,if 1_ÜÑ8 hÉ83Ü�hCÍ .

In thecaseof§ m , matrix (21) is indefiniteandthereexist k � and k � for which thederivative

of k at thesepointsis boundedfrom below by .�RTç andfrom aboveby gr=½R�.�Sp8 hÉ8 Ý · � respectively,for all £6qNj . In particular, therotationnumbermustbezero. NRemark. The Normal Form for 425�¨1 canbe obtainedwithout termsin

� � � � and withoutchangingthe term in

� �� Indeed,at eachstepof normalizationthe homologicalequationis ofthe form " èP�I�_Í $ � � � where

�containsknown termsof the form

� �ts� � �)u� û Û î ; i =j�I£¬"!? with� � D � � � S�� Let us seethe systemto solve for a fixed = . Let º � � � � D º � � � � � D º � � �� thetermshaving û Û î ; i =j�I£¬"!? asa factor in the expressionof

�and

p � � � � D3p � � � � � Dvp � � �� thecorrespondingtermsto befoundin è . In matrix form we haveTV i

=j�!�#" 1 1ST4 i =j�!�#" 11 4 i

=j�!�#" WY TV p �p �p � WY � TVØº �º �º � WY �If =J5�}1 thematrix invertible. If =h�}1 onecannot cancelº � , which mustbekeptin theNormalForm, but the terms º � i º � canbe cancelledby suitablechoicesof

p � � p � � Thevalueofp � is

arbitrary.

4.2 Collapsedgap

Presentaimis to proveProposition4, i.e.,assumingthat 40�213� Herewefollow ideassimilarto theabovecase465�213� Weshallseethatthetongueboundariescanbedividedoversectors,determinedby whetherthemodulusof themodifiedrotationnumberis greaterthansomeconstantor whethertherotationnumberis zeroandthereis exponentialdichotomy, comparewith Figure3. Fromthisweobtainthetangency of therequiredorder.

Recallthatthefirst stepof averaginggives i �!�#" D ý .S � � � �� D � � � �� D .S � � � �� ÿ D 5 � ;�� I£¬?$�where 5 � denotestermswhichare 5_;I8 �#8 � ? (andquadraticin

�anddependingon timethrough£ )

and� � � �� À " þ ��!� $ D h�" W þ ��!� $ � � � � �� À " þ �� $ D h�" W þ �� $ � � � � �� À " þ �!� þ � � $ D h�" W þ �!� þ � � $ �seeProposition2. Hencethe coefficients of

Àin �� and �� are positive and " þ �!� þ � � $ � Ü" þ ��!� $ �M" þ �� $ � a key fact in what follows. To order the coefficient � � � �� is replacedby � � Ý�� for�P�.T��S��ç�� of theform describedbefore,and 5 � by 5 Ý · � �

After stepsof normalizationthematrixof thesystemis� �� Ý�� Ý��g«� � Ý�� gÞ� � Ý�� D ý � � � �g � � g � � ÿ �19

PSfragreplacements

wx w<yx y

Tongueboundariesrot z w@{ x}|�~ rot z w y { x y |andexponentialdichotomy

rot z wa{ x�|�� rot z w y { x y |rot z w@{ x}|�� rot z w y { x y |

Figure3: Areasof exponentialdichotomyinsidea resonancetongueasguaranteedby Lemma2 andareaswith rotationnumberdifferentfrom rot �� _�t��0� outsidethetongueasguaranteedbyLemma2. Solid linesdenotetongueboundaries.

wherethe �� termsdependof � analyticallyon thesamedomainas � andareof order ��3� in� ��First we analyzethepartcomingfrom theNormalForm. As it is well-known, theboundaries

of theresonancezonecorrespondto � -valuessuchthatthedeterminantof thesystem� � � ��D�� ¡��¢¤£ ��¡��¥ ¢is equalto zero.As thetermsof degree1 in � in the �a� give riseto a positive definitepart in the

Hamiltonian,thereexistsa canonicalchangeof variables(a rotationandscalings)suchthat � �� and ��¡��¢ startas ¦ � (for some¦¨§M© ) and ��¥ containsno lineartermin � � By scaling

�wecan

assume¦ª�!� in thepreviousexpressions.Hence,we areleft with�� e« ��D�¬� �B � � ��D�}�� ¢ � � �e« ¢ ��D�¬� �B ¢ � � ��D�}��¥ � « ¥ ��D�¬� �B ¥ � � ��D�}�where « � arepolynomialsin � of maximaldegree � andstarting,in principle,with linear termsand � arepolynomialsin � �� of maximaldegree� £ �_� If « �¯®° © let ±0� theminimaldegreein « �_�for ²³�!�_�t´ �tµ � Otherwisewe set ± � �6¶v� UsingNewton’s polygonarguments(see,e.g.,[24]) tolook to therelevanttermsof thezerosetof

� � onecanneglectthe � terms.Assumefirst ±M�¸·¯¹»º½¼G± �t± ¢ �t± ¥¿¾ÁÀ �¿� Introducingthe changeof variables� �mÂ��JÃ the

function�

canbewrittenas� ¢ Ã�Ä Â ¢ �3��Å �Æ Ã �oÅ ¢�Æ Ã ��Âg�oÅ �Æ Ã Å ¢�Æ Ã £ Å ¢¥�Æ Ã �MÇ¯��K�XÈ��where Å � Æ Ã denotesthecoefficient of degree ± in « � �É²Z�Ê�_�t´ �tµ (someof themcanbezero,butnotall). Factoringout � ¢ Ã andneglectingthe Çg��D� termthezeros,Â and Â ¢ � of theequationforÂ aresimple,unlessÅ �Æ Ã £ Å ¢�Æ Ã and Å ¥�Æ Ã �Ë©@� Hence,theImplicit FunctionTheoremimpliesthattherearetwo differentanalyticfunctionsÌ � ��D��Í� Ã �ÎÂ � �MÇ¯��K��}�¤²³�!Ï_�tÐ

20

in thezerosetof ÑÓÒ whichdiffer atorder ÔZÕ�Ö¿×If ØÚÙ�Û Ü�ÝÞØàß�Û Ü and Øàá�Û Ü½â3ã let usintroduce äå â å�æ ØÚÙ�Û Ü0ç Ü andrewrite Ñ in termsof äå Ò�ç�×

Werename äå againas å × Thenthenew equationfor å Ò�ç is asbeforewhere Ô is at leastreplacedby Ô æéè andwherethemaximaldegreeof the ê ë and ì¿ë polynomialsalsocanincrease.If theequationfor thenew í hastwo differentrootsoneobtainstwo curves î]ëðï�çKñ in thezerosetof Ñ ,asbefore.Otherwisetheprocedureis iteratedandendswhentwo differentcurvesareobtainedorwhenavalue ÔZò�Ö is reached.

If Ô,âHó theprocedureis stoppedimmediately. In this case,or whenwe reachÔ�òvÖ in theiterativeprocess,afterachangeof variables äå â å Ýàôõï�çDñ}Ò whereô is apolynomialof degreeÖ ,theproblemis equivalentto theinitial one.Herethe ��ö¡÷�øë polynomialsarereplacedby ��ùë , where� ùÙ â äå,æ ê ùÙ ï�çDñ æ äå ì ù Ù ï_äå Ò�çKñ}Ò� ùß â äå,æ ê ùß ï�çDñ æ äå ì ùß ï_äå Ò�çKñ}Ò��ùá â êBùá ï�çDñ æ äå ì@ùá ï_äå Ò�çKñ}Òandwheretheminimal degreeof the ê ùë is at leastÔ æÍè × Hence,afterafinite numberof stepsweobtain

Lemma 1 Considerthe Normal Form after Ö stepsof normalizationin the case úFâûã@× LetÑüï å Ò�çKñgâýã be the definingequationof a boundaryof the resonancezone. Thenthere existsçOþRòMã such that, for ÿ ç�ÿ��çOþ¿Ò oneof thefollowingstatementsholds:

a) Thezero setof Ñ consistsof twoanalyticcurveså âLî]ëðï�çKñ}Ò��Óâ è Ò�� Ò with î ß ï�çDñÉÝÓî Ù ï�çDñ�â� ç Ü ï è�æ�� ï�çKñ�ñ}ÒtÔàÕ�Ö¿Ò � òMã@× FurthermoreÑüï å Ò�çDñ�âHï å ÝÚî Ù ï�çKñ�ñ#ï å ÝÚî ß ï�çDñ�ñgï å Ò�çDñwhere is an analyticfunctionwith gï�ã@Ò�ãðñ:òMã@×

b) There existsa curve å â ôõï�çDñ}Ò with ô a polynomialof degree Ö¿Ò and a constant�Êò/ãsuch that thezero setof Ñ is containedin thedomainboundedby ôõï�çDñ ��ÿ ç�ÿ ÷�� Ù ×

Proof: To completethe proof of the first item it is only necessaryto remarkthat, from thepreviousdiscussion,only two branchesof Ñ�âvã canemergefrom ï�ã@Ò�ãðñ . HenceÑüï å Ò�çKñï å ÝÚî Ù ï�çDñ�ñ#ï å ÝÚî ß ï�çDñ�ñis aninvertiblefunction.Thefactthat gï�ã@Ò�ãðñlòLã follows from thepositive definitecharacterofthelineartermsin å ×

Concerningthe seconditem, using the variable äå â å ÝFôõï�çDñ onecanwork with the ��ùëfunctions. Let us denoteas Ñ ù ï_äå Ò�çKñ the expression� ùÙ � ùß Ý\� ùá ß Ò that is, the valueof Ñ in thenew variables.Replacing äå by ��½ÿ ç�ÿ ÷�� Ù thefunction ÑRù becomespositive if � is largeenough.It remainsto show that thezeroset is not empty, but this will be an immediateconsequenceofLemma2. �

Next we considerthevariationsof therotationnumberin differentdomainsof theparameterplane. That is, we want to estimate��¿ï��OÒ�� ñ which in the currentparameterswill be denotedsimply by ��Gï å Ò�çDñ}× The differential equationfor � â�� ï"! ß æ i ! Ù ñ , i.e., equation(19) forú â3ã , reads�$#OâHï�� ö¡÷�øÙ æ&% Ù ñ('�)+* ß � æ �aï�� ö¡÷�øá æ&% á ñ('�)+*,�.-/�0'1� æ ï�� ö�÷�øß æ�% ß ñ(-/�0' ß ��×Lemma 2 Considerthe rotation numberì32�â4��Gï å Ò�çDñ of thedifferential equation(19) in co-rotatingcoordinates.Thenthere exist constants5ªÒ�ç þ òMã such that, for ã6�4ÿ ç�ÿ��ç þ ,

21

a) In casea) of Lemma1 letî�7'ï�çDñ�â986)+*;: î Ù ï�çDñ}ÒXî ß ï�çDñ�< Ò î � ï�çDñ'â986�>=1: î Ù ï�çDñ}ÒXî ß ï�çDñ�< ×Thenonehasì?�Mã if å �eî(7'ï�çDñ�Ý@5�ÿ ç�ÿ ÷�� ÙBA ìÞòMã if å òeî � ï�çDñ æ 5�ÿ ç�ÿ ÷�� Ù/Aì³â3ã if î(7'ï�çDñ æ 5�ÿ ç�ÿ ÷�� Ù � å �eî � ï�çKñKÝ@5�ÿ ç�ÿ ÷�� Ù ×

b) In caseb) of Lemma1 onehasìC�Mã if å �Fôgï�çDñ�ÝD5�ÿ ç�ÿ ÷�� Ù A ìZòMã if å �Fôgï�çDñ æ 5�ÿ ç�ÿ ÷�� Ù ×Proof: Let usconsiderthequadraticform in theexpressionof � # , i.e., equation(20) for ú³â/ã ,in casea) obtainedby skippingthe % ë termsandwhere å is takenequalî0E�ï�çKñ}× For definiteness

let F�aëõââ��ö�÷�øë ïÎî0E ï�çDñ}Ò�çDñ}× This quadraticform is degenerate.As, in general,thediscriminantofthequadraticform is ÝlÑÓÒ theform is indefinitefor å in ïÎî�7�ÒXî � ñ anddefiniteoutside G î(7�ÒXî � H × Ifå â4î(7 theform is negative definiteeverywhereexceptat onedirection. Similarly, if å âéî � itis positive definiteeverywhereexceptat onedirection.

We wantto seetheeffect of addingthe % ë termsandthechangein thevalueof å . Fromthe

expressionof the ��ö¡÷�øë ï å Ò�çKñ onehas� ö�÷�øÙ ïÎî 7 ï�çDñKÝ@5�ÿ ç�ÿ ÷�� Ù Ò�çDñ æ�% Ù ïÎî 7 ï�çDñKÝ@5�ÿ ç�ÿ ÷�� Ù Ò�ç�Ò�IÉñJ� F� Ù Ý 5 � ÿ ç�ÿ ÷�� Ù Ò��ö�÷�øß ïÎî(7�ï�çDñKÝ@5�ÿ ç�ÿ ÷�� Ù Ò�çDñ æ�% ß ïÎî(7'ï�çDñKÝ@5�ÿ ç�ÿ ÷�� Ù Ò�ç�Ò�IÉñJ� F� ß Ý 5 � ÿ ç�ÿ ÷�� Ù Òÿö��ö¡÷�øá ïÎî(7�ï�çKñ�ÝD5�ÿ ç�ÿ ÷�� Ù Ò�çDñ æ&% á ïÎî�7�ï�çKñKÝ@5�ÿ ç�ÿ ÷�� Ù Ò�ç�Ò�IÉñDÝKF� á ÿL� 5 M ÿ ç�ÿ ÷�� Ù Ò

uniformly in I , if 5 is largeenoughand ã6��ÿ ç�ÿN��çOþ for someçOþ . Thecurrentquadraticform isboundedfrom above byF� Ù '�)+* ß � æ � F� á '�)+*O�.-/�0'P� æ F� ß -/�0' ß ��Ý 5 � ÿ ç�ÿ ÷�� Ù ï�'�)Q* ß � æ '�)+*,�.-/�0'P� æ -/�0' ß �Kñ}×Hence� # �/Ý 5 M ÿ ç�ÿ ÷�� Ù , proving thefirst of theassertionsin a). Thesecondassertionis proved

in thesameway.To prove thethird statementin a) it is betterto shift å by î(7'ï�çDñ}× Let äå â å Ýnî�7�ï�çDñ}× Then

the � ë functionsareof theform ä� Ù â äå,æ äê Ù ï�çDñ æ äå äì Ù ï<äå Ò�çDñ}Òä� ß â äå,æ äê ß ï�çDñ æ äå äì ß ï<äå Ò�çDñ}Òä� á â äê á ï�çDñ æ äå äì á ï<äå Ò�çDñ}×It is clearthatwhen äå â3ã wehave Ñ4âvã by construction,andtheotherroot is î � ï�çDñ Ý³î(7�ï�çDñ�â� ÿ ç�ÿ Ü ï è:æR� ï�çDñ�ñ}Ò � ò/ã . Therefore, äê Ù ï�çDñ æ äê ß ï�çDñ�â¸Ý � ÿ ç�ÿ Ü ï è:æR� ï�çDñ�ñ and äê Ù ï�çDñ<äê ß ï�çDñSâï]äê á ï�çDñ�ñ ß × Furthermorethe äê ë functionshave Ô asminimaldegreefor ��â è Ò�� Ò�S × For definitenesslet äê ëðï�çDñ�âUTðë'ï è�æ�� ï�çDñ�ñ Ò with TðëCVâvã .

We setnow äå âW5�ÿ ç�ÿ ÷�� Ù andaddthe % ë termsto the ä�aë functions.Thenew determinantisof theform X ï�5 æ&Y ñ ÿ ç�ÿ ÷�� Ù æ äê Ù�Z X ï�5 æ�[ ñ ÿ ç�ÿ ÷�� Ù æ äê ß�Z Ý X]\ ÿ ç�ÿ ÷�� Ù æ äê á�Z ß

22

where ÿ Y ÿ�ÒGÿ [ ÿ�ÒGÿ \ ÿ areuniformly boundedfor all I by quantitieswhichare � Ù ï�çDñ}× Thereforethedeterminantis uniformly boundedfrom aboveby Ý � 5�ÿ ç�ÿ Ü � ÷^� Ù if 5 is largeenough.Thisshowsthatthequadraticform is indefinitefor all I .

Furthermore,when äå â_5�ÿ ç�ÿ ÷�� Ù the ä�aë functionsare � ï�ÿ ç�ÿ Ü ñ}× This, combinedwith theboundon thediscriminantandthedifferenttermscontributing to the ä�aë shows that theslopesofthedirectionsin the ï"! Ù Ò`! ß ñ -planefor which � # âvã areof theformú Ù ï�çDñ �6ÿ ç�ÿba]c�dfe0gh ï�ú ß æ ú á ï�ç�Ò�I ñ�ñ}Òwhereú Ù and ú á areanalyticfunctionsof their argumentsandú Ù ï�ãðñ'â!Ý T áT Ù Vâ3ã@ÒOú ß â i � 5ÿ T Ù ÿand ÿ ú á ï�ç�Ò�IÉñ ÿj�Êú ß�k � Ò uniformly in Ia× The time dependenceappearsonly in the ú á term. Oneof the directionsis attractingfor the dynamicsof � in l Ù andthe other is repelling. We recallthat thesedirectionsdependon m�× However theslopesof bothdirectionsareboundedaway fromú Ù ï�çDñ uniformly in I and thereforein m}× Let �Dù÷ ïbm�ñ the argumentof a repellingdirection. Anyvalueof theform � ù÷ ïbm�ñ æ Øon is alsorepelling. Considertwo consecutive repellingcurves. Forany fixed ç with ÿ ç�ÿ ��ç þ smallenough,they arecontainedin a strip of the form ï�� ï�ú Ù ï�çDñ�Ý�_ú ß ñ}Ò�� ï�ú Ù ï�çDñ æ �_ú ß æ nJñ}× Any initial condition ï"! Ù Ò`! ß ñ betweentheserepellingcurvesremainsin thestrip for all m�× This shows that ì³âvã , asdesired.

To prove theassertionfor å âéî � ï�çDñ'Ýp5�ÿ ç�ÿ ÷�� Ù oneproceedsin a symmetricway. Thenitfollows for thefull interval asin thestatement,by monotonicityof ì with respectto �O×

Finally, weproceedto caseb). By introducing äå â å ÝZôgï�çDñ oneobtains� functionslike the� ùë definedabove,with ê ùë ï�çDñ�â � ÷�� Ù ï�çDñ}× Then

��ùÙ ïXÝO5�ÿ ç�ÿ ÷�� Ù Ò�çDñ æ&% Ù ï å Ò�ç�Ò�I ñq� Ý 5 � ÿ ç�ÿ ÷�� Ù Ò��ùß ïXÝO5�ÿ ç�ÿ ÷�� Ù Ò�çDñ æ&% ß ï å Ò�ç�Ò�I ñq� Ý 5 � ÿ ç�ÿ ÷�� Ù Òÿö� ùá ïXÝO5�ÿ ç�ÿ ÷�� Ù Ò�çDñ æ&% á ï å Ò�ç�Ò�I ñ ÿr� 5 M ÿ ç�ÿ ÷�� Ù Ò

uniformly in I , if 5 is large enoughand ÿ ç�ÿj�/çOþ for some ç�þ . The currentquadraticform is

boundedfrom above asin thea) caseby Ý 5 M ÿ ç�ÿ ÷^� Ù . This provesthefirst assertionin b) andthe

secondoneis proved in a similar way. Furthermore,aswasannouncedin Lemma1, thezerosetof Ñ is containedbetweenthesetwo curvesbecausetherotationnumberpassesfrom �6ã to theleft to òMã to theright.

Thisfinishestheproofof Lemma2 andthelastpartof Lemma1, caseb). �Proposition4 is now immediate.Indeed,let �2ö¡÷�øÙ and �2ö¡÷�øß theTaylor expansionsup to order Ö

in �rÝ��þ of � þ æ î Ù ïf�:Ý��þ0ñ and �Éþ æ î ß ïf�rÝ&��þ]ñ respectively. Then,letting s âËçOþ , thereis aconstant5 , givenby theprevious lemma,suchthat if � � ïf� ñ and �t7'ïf� ñ denotetheright andleftboundaryof thetongue,thenÿ � � ïf�#ñ�ÝD86�>=uwv Ù�Û ß :x�2ö¡÷�øu ïf� ñ�< ÿ Õ 5�ÿ ��Ý@��þðÿ ÷^� Ù andÿ �y7�ïf�#ñ�Ýz8{)+*u+v Ù�Û ß :x� ö¡÷�øu ïf� ñ�< ÿ Õ 5�ÿ ��Ý@��þðÿ ÷^� Ù Òfor ÿ ��Ý|� þ ÿ��}s , aswe wantedto show.

23

4.3 Differ entiability of rotation number and Lyapunov exponentfora fixed potential

In thissectionwefix theparameter��þ in asufficiently smallneighbourhoodof theorigin, toensurereducibilityaccordingto [21], seeSection2. In this casewe studyrotationnumberìgâÍì ï��añ and(maximal)Lyapunov exponent~ â�~¬ï��@ñ of thequasi-periodicHill equation(1), or equivalently(5), in dependenceof theparameter��× Theresultsin this settingarecompletelyanalogousto theperiodiccase,andproofscanbeobtainedfrom thoseof theprevioussection.

Corollary 4 In theabovesituation,let � þ bean endpointof a spectral gap.Then

1. If � þ is in theleft (resp.right) endpointof a non-collapsedspectral gap,thenthefunctionså3� ïXÝ è Ò è ñ�� ì ï�� þ Ý å ß ñ and åp� ïXÝ è Ò è ñ��~Jï�� þ æ�å ß ñ(respåp� ïXÝ è Ò è ñj�� ì ï�� þ æªå ß ñ and åp� ïXÝ è Ò è ñj��~¬ï��Éþ¬Ý å ß ñ ) aredifferentiableat zero.

2. If :x� þ�< is a collapsedspectral gap,thenthefunctions�6�� ì ï��@ñ and �6��~Jï��añare differentiableat �Éþ .

In particular, in anynon-collapsedspectral gap G �y7�Ò�� H the function ��Óï��añ�2�â¸Ý�~¬ï��@ñ ß isanalyticin ï��y7�Ò�� ñ andhaslateral derivativesat �üâ9�y7�Ò�� ×The sameresult was obtainedin [34, 35] in more generalcontexts (e.g., for the Schrodingerequationwith almostperiodicor ergodicpotential).Ourmethodof proof is very similar to [34].

Remark. With a little moreeffort, onecanrecover the fact that for fixed, small potentialthefunction ��³ï��@ñ in a gap G �t7�Ò�� H is of class

\�� ï�ï��y7�Ò�� ñ�ñ�� \�� ï^G �t7�Ò�� H ñ}Ò seeMoserandPoschel[33].

5 Conclusionsand outlook

Summarizing,this paperstudiesthe geometricstructureof resonancetonguesin a classof Hillequationswith quasi-periodicforcing �B�¯ïb��m�ñ}× Severalresultswereobtained,analogousto thepe-riodic case,regardingsmoothnessof tongueboundariesandtheoccurrenceof instabilitypockets.Herewe usedreducibilityof theequationsat thetongueboundariesfor small ÿ �_ÿ .5.1 Lar ger ��Accordingto numerical([15]) andanalyticalwork ([23]), it seemsthatfor eachtongueboundarythereexistsacritical value � crit Ò suchthatfor ÿ �<ÿ�� crit reducibilityholds,while for ÿ �<ÿ@ò}� crit notevencontinuousreduction(to Floquetform) seemspossible.

Resonancetongues,however, canbedefinedby therotationnumber. Sincethis definition isindependentof reducibility, we canstill speakof tongueboundariesfor larger ÿ �_ÿ�× Thereforetheproblemremains,whetherandto whatextent thepossiblenon-reducibilityof thesystemaffectsthe regularity of the tongueboundaries. In Appendix A we show that tongueboundariesarealwaysglobally Lipschitz, but is this the bestpossibleresult in general?Is it possiblethat thetongueboundariesbecontinuedwith somedegreeof regularity above thecritical value?

24

5.2 Analyticity?

It shouldbenotedthatour Main Theorem1 on theregularity of the tongueboundariesfor smallanalyticpotentialsjustexpressesinfinite differentiability, whereasin theperiodiccaseanalyticityholds.

In the presentquasi-periodiccase,analyticity of the tongueboundarieswould follow if theaveragingprocessof Section2.2 wereconvergent. However genericallythis is not thecase.In-deed,suchconvergenceof theaveragingprocessin apointata tongueboundarywould imply theexistenceof a spectralinterval (that is, a whole interval in thespectrum)besidesthepoint in thetongueboundary, wheregenericallywe have Cantorspectrum,see[32, 28, 21].

Of coursetheseconsiderationsdo not forbid analyticity of the tongueboundariesand it isinterestingto know whetheranalyticity is the caseor not. Work in this direction, in a moregeneralcontext, is in progress(see[41]).

Appendix

A Lipschitz property of tongueboundariesin the large

In thepaperwe approachedtheregularity of the tongueboundariesusingreducibility. However,thereexistsnumerical[15] andanalyticalevidence[23], thatin casesfarfrom constantcoefficientsthis approachcannotbe used. Presentlywe reconsiderthe quasi-periodicHill equation(1), orequivalently (5), wherewe only assumethecomponentsof � to be rationally independent(i.e.,notnecessarilyDiophantine)andwherethefunction ��2��¡ is just continuous.

Proposition6 In theabovesituation,let\ â¢'�£N¤¥§¦�¨�© ÿª�¯ï�I ñ ÿ�×and � � 9�� ïf� ñ � bea (left or right) tongueboundary. Thenfor all �GÒ�� # � wehave«« � ïf� ñKÝ@� ïf� # ñ «« Õ \ «« ��Ý|� # «« ×Our proof is basedon Sturm-like argumentsfor the oscillationof the zeroesof a secondorderlineardifferentialequation.Firstwereview someusefulpropertiesof therotationnumber[29, 35].

For any non-trivial solution ¬Dïbm�ñ of equation(1) we definea function 5oï® A ¬ ñ}Ò which is thenumberof zeroesof ¬Dïbm�ñ in theinterval G ã@Ò¯ H . Fromtheform that(1) takesin polarcoordinates(by theso-calledPrufer transformation),it follows thatall thezeroesof ¬Kïbm�ñ aresimpleandthatthelimit ° )+8±1²�� n³5eï® A ¬ ñexists for all initial conditions,agreeingwith the rotation numberof (1), or equivalently, thesystem(5).

Our ideato prove Proposition6 is to usea suitableSturmOscillationTheoremto control thezeroesof a variation � æ �� of the original potential �³Ò with thepropertythat ��¯ï�IÉñ is eitherpositive or negative for all I � � � ×Lemma 1 Assumethat the maps �³Ò��´2µ��o�� are continuousand that ��¯ï�IÉñ¯òIã for allI � � � × Let ì Ù betherotationnumberof¬ # # æ �gï�IÉñ]¬Zâvã@Ò3I # â9�

25

and ì ß therotationnumberof¶ # # æ ï"�¯ï�IÉñ æ �>�gï�IÉñ�ñ ¶ âÍã@Ò3I # âz� ×Thenì Ù Õ�ì ß ×Lemma1 is a directconsequenceof theSturmComparisonTheorem,see,e.g.,[18]. Indeed,bythis result,thenumberof zeroes5eï® A ¬ ñ in theinterval G ã@Ò¯ H is lessthanor equalto thenumberof zeroes5eï®:Ò ¶ ñ of ¶ in the sameinterval, assumingthat we have the sameinitial conditions¬Dï�ãðñ�â ¶ ï�ãðñ}Òy¬ # ï�ãðñ'â ¶ # ï�ãðñ . Therefore,by theabove considerationsì Ù â ° )+8±P²�� n³5oï® A ¬ ñ Õ ° )+8±P²�� n³5oï® A ¶ ñ âvì ß Òaswasto beshown.

We proceedshowing how Lemma1 canbe usedto checkthe Lipschitz conditionstatedinProposition6. First, notethat, if the condition �� ò ã is replacedby �� ã , thenwe haveì ÙO· ì ß ×

In thesettingof Proposition6, condition�x�³Ý \ ÿ ��<ÿ@òFã (22)

impliesthat ��¯ï�I ñ�âR�� æ ��B�¯ï�I ñ�òMã for all I � � � andthus,by Lemma1, that��0ï��Ò�� ñ�Õ&��Gï�� æ ��OÒ�� æ \ �>� ñ}×Now, if ï��Ò�� ñ is at theboundaryof a certaintongue(for simplicity assume� is theright endpointof the correspondingspectralgap), this meansthat for arbitrarily small perturbationsin the �direction,therotationnumberis strictly larger thanthatof theoriginal equation.That is, for any� # �ÞòMã , ��0ï��Ò�� ñ¸��¿ï�� æ �x#+��Ò��#ñ}×TheLemmathenyieldsthat,if ïf��OÒ��>� ñ satisfies(22),also��¿ï��OÒ�� ñ¹�&��Gï�� æ ��#w� æ �x��Ò�� æ \ �� ñ}×As � # � maybearbitrarily small theperturbationsïf��OÒ��>� ñ in thesectordefinedby condition(22)do not containany point in theboundaryof thesametongueas ï��OÒ�� ñ}× Therefore,in our proof ofProposition6 we have � ïf� Ù ñ�Ý@� ïf� ß ñ�Õ \ ÿ � Ù Ý|� ß ÿ#×In orderto prove theremaininginequality, observe thatperturbationsin thesector�x� æ \ ÿ ��<ÿ��Fã (23)

containnopointsin theleft boundaryof thetongueof ï��Ò��#ñ}× By contradictionassumethatsuchapoint in theleft boundaryexistsandlet ïf��OÒ��>� ñ satisfying(23)bethecorrespondingperturbation.Then,dueto the opennessof the above condition,thereexists a positive constant� # � suchthatïf�x� æ � # ��Ò�� ñ still satisfies(23). Moreover, aswe areassumingï�� æ �x��Ò�� æ �� ñ to be in theendpointof theleft spectralgapand � # �ÞòMã��0ï��Ò�� ñ'âz��0ï�� æ �x��Ò�� æ �>�#ñ¸�&��Gï�� æ � # � æ ��Ò�� æ �� ñ}×Ontheotherhand,theperturbationïXÝ,�x�³Ý|� # �OÒ]Ý�� ñ satisfiescondition(22)andtherefore��Gï�� æ � # � æ �x��Ò��>�#ñ:Õ&��¿ï��OÒ�� ñ}Ò

26

which implies ��Gï��Ò��#ñ��º��0ï��OÒ�� ñ}× This is thedesiredcontradiction,wherebyProposition6 isproved.

Remark. TheLipschitzpropertyin Proposition6 regardingtongueboundariesalsoholdsin theperiodiccase,wheretheproofrunsexactlythesame,andwherethisis referredto asthedirectionalconvexity of stability andinstability domains,see[48]. The propertyalsoprovidesa boundonthederivativesof thetongueboundarieswhenever they exist. This boundcoincideswith theoneobtainedin theaveragingprocessof Section2.2.

B Proof of Theorem3

Our proof follows from ananalysisof thenormalform to order ÿ »Bù<ÿ . Thereareseveralnormal-izationtechniquesandany suchmethodfor arbitrary ÿ » ù ÿ canbecumbersome.Thereforeweonlyusetheformatof thenormalform of order ÿ » ù ÿ to find outwhichtermsarerelevant.Subsequently,thecoefficientsof thosetermsareobtainedby analternative, recurrentandsimplermethod.

Let usset å þ:â½¼�» ù�¾ k � and �üâ å ßþ æ�å . Next, a scaling¬Zâ ¿i å þ Ò ¶ âzÀ i å þandpassageto complex coordinates

¿ âÂÁ æpÃwÄi � Ò@Àgâ Ã Á æ�Äi � Ògive thefollowing form for thetime-dependentHamiltonianÅ ï Á Ò Ä Ò^m�ñ'â å þ Ã Á Äüæ Á ß Ý Ä ß æ � Ã Á Ä� ÆÇ å� å þ æ �M å þ ÆÇ �Èë v Ù úXë$-/�0'0ïb� ë�m�ñ æ&É -/�0'Kï^¼�» ù ¾ m�ñÊËÌÊË!× (24)

Now let Í becanonicallyconjugateto thetime m , andlet Î�ëlâ9Ï/=(¤¬ï Ã signï Ô ùë ñ]� ë/m�ñ for �Óâ è Ò × × ×0Ò � .Then(24)canbewrittenasthesumof anintegrablepart

Å þ andaperturbationÅ ÙÅ þ âÐÍ æüå þ Ã Á Ä Ò Å Ù â ä� X Á ß Ý Ä ß æ � Ã Á Ä Z ÆÇ äåZæ �Èë v Ù ú ë�Ñ Î ë æ Î 7 ÙëKÒ æ&É Ñ Î0Ó ^ æ Î 7 Ó ^ Ò ÊË Ò (25)

where äå âR� å k �GÒ ä��âU� k ïfÔ å þGñactasperturbationparameters.

To carry out the normalization(averaging)onecanuseany Lie seriesmethod,for instancethe Giorgilli-Galgani algorithm [25] as it is was donein [16]. Startingwith

Å þ Û þnâ Å þ andÅ Ù�Û þrâ Å Ù , theterms Å ë Û Ülâ ÜÈ Õ v Ù �Ô G�Ñ Õ Ò Å ë Û Ü 7 Õ H Ò �Óâvã@Ò è Ò�ÔZòMã@Òwhere G�Ö ÒBÖ H denotesthePoissonbracket,arecomputedrecurrently. A termas

Å ë Û Ü containsä� ë � Ü asa factor. Thefunctions Ñ�× aredeterminedfor cancellingthetime dependenceasfar aspossible,i.e., if no resonancesoccur. To beprecise,assumethat Ñ Ù Ò × × ×�Ñ ×�7 Ù arealreadycomputed.Then

27

all termsinÅ Ù�Û ×�7 Ù æ Å þ Û × areknown excepttheonescomingfrom G�Ñ�×�Ò Å þ Û þ H × Let Ø6× contain

theknown termsat order Ù . Then Ñ�× is determinedby requiring Ø6× æ G�Ñ�× Ò Å þ Û þ H not to containtermsin the Î�ë variables.ThetransformedHamiltonianthenis 5¸â95Óþ æ 5 Ù æ 5 ß æ ÖBÖBÖ , where5�þrâ Å þ Û þ and 5Ú×üâ Å Ù�Û ×�7 Ù æ Å þ Û ×2× In particular, 5.× is of order Ù with respectto ä�G×

It directly follows thatÛ Å þ Û þ¿Ò Á ÷ Ä ß 7 ÷ Î Ó�Ü â Á ÷ Ä ß 7 ÷ Î Ó Ã ï å þ ïf�RÝ|�¿ÖðñDÝ}¼�» ¾ ñ Ò ÖRâ3ã@Ò è Ò�� Òandthis shows thatall termswith å þ<ïf�ÓÝp�¿Öðñ'Ýz¼�» ¾ differentfrom zerocanbecancelledto anyfinite order. Proceedingby inductiononeobservesthat,if � æ Ô¯âÍØ then

Å ë Û Ü hastheformÅ ë Û Ü½â Á ß � Ù Ý Ä ß � ß æpÃ Á Ä ï � á æ ��Ý ñ}Òwith thecorrespondingÑ�Þ of theformÑ Þ â Ã ï Á ß � Ù æßÄ ß � ß ñ æ Á Ä ï � á Ý ��Ý ñ}×Here

� Ù containsthetermswith realcoefficientsof theform äå Þ 7;àþ Î Ó with ÿ »'ÿ2âéÖ¿Ò where á andÖ have thesameparity. Thetermsin� ß canbeobtainedfrom

� Ù by a replacementof Î Ó by Î 7 Ó .Similarly, theexpressionof

� á is identicalto thatof� Ý

but for replacingÎ Ó by Î 7 Ó .Summingup, by a canonicalchangeof variables,the Hamiltonian

Å â Å þ æ Å Ù , up to aremainderof higherorderin ä� , canbereducedto thenormalform5ß!âÐÍ æ �Éþ Ã Á Ä¯æ coefÙ Ã Á Ä³æ coefß Ñ Á ß Î 7 Ó ^ Ý Ä ß Î Ó ^ Ò Ò (26)

where

- coefÙ â äå�æ Ö Ù , where Ö Ù a (real) functiondependingon ï ä�GÒ2äå Ò É Ò�ú]ñ}Ò andcontainingsomepower of ä� asa factor;

- coefß â É ä�Bâ ß ï ä�GÒ�äå Ò É Ò�ú]ñ æ ä�>ã Ó ^ ã��ÓÖ ß , whereÖ ß a(real)functiondependingon ï ä�¿ÒOäå Ò É Ò�ú ñ andwhere â ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñ.Vâvã doesnotdependon ú A

- Theorderof theremainderin ä� is largerthan ÿ »Bù ÿ�×Truncatingaway theremainderandpassingto co-rotatingcoordinatesïbäJÒ^åañ definedbyäàâ Á Ï/=(¤JïXÝ Ã ¼�» ù ¾ m�ñ}Ò å³â Ä Ï/=(¤Dï Ã ¼�» ù ¾ m�ñyieldsthesystem ä # â Ã coefÙ äõÝ|� coefß åOÒ å # â!Ý,� coefß äZÝ Ã coefÙ å�×Therefore,up to the ÿ » ù ÿ th orderthetongueboundariesaregivenby theequation

coefÙ âU�� coefß ×Soif Ö ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñ�Vâvã for É âvã thereis a ÿ » ù ÿ th ordertangency at �râvã@Ò while for É Vâvã thereisaninstabilitypocket. Hence,ourproofof Theorem3 is concludedby checkingwhen Ö ß ï�ã@Ò�ã@Ò�ã@Ò�ú ñvanishes.

To find outwhetherÖ ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñ vanishesor not,it is only necessaryto considerequation(16)for É âvã at theexactresonance

¬ # # æ ÆÇ ¼�»Bù ¾M æ �� ÆÇ �Èë v Ù úXë�ÑxÎ�ë æ Î 7 ÙëæÒ ÊË ÊË9¬Zâ3ã@× (27)

28

Notethat Ö ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñ�â9ç¯ïb� Ò�» ù ñ)ú Ó ^wherenow ç doesnotdependon ú . Thereforeonemayassumethat úXë½â è for �³â è Ò × × ×GÒ � .

According to the normal form (26), any non-trivial solution ¬Kïbm�ñ of (27) canbe expandedin powersof � thefirst ØUÝ è coefficientsof which arequasi-periodicfunctionsandwheretheØ th coefficient is also quasi-periodicif and only if ç¯ïb� Ò�»Bù ñ vanishes. We arenow going tocomputethis expansiondirectly from thedifferentialequations,insteadof usingtheHamiltonianformulation.

Sincewe areinterestedin the » ù th power in Î2Ò we first considerthe equation(27) only forpositive powersof the Î�ë�2 ¬y# # æ ÆÇ ¼�»Bù ¾ ßM æ �� Èë v Ù Î�ë ÊË ¬Zâvã@× (28)

Scalingtimeby m�âR��è turns(28) intoé¬ æ ÆÇ ¼�» ù ¾ ß æpê �Èë v ÙNë ßë ÊË ¬àâvã@× (29)

wherethe dot denotesderivation with respectto è andwhere ê â´�� is the new perturbationparameter. Also notethat after this changewe have Î�ëZâ ë ßë Ò where ë ëÞâÂÏ/=(¤¬ï Ã signï Ô ùë ñ]� ëxèOñ .Any solutionof equation(29)canbeexpandedin powersof ê asfollows¬ ö Ù ø â9¬�þ æpê ¬ Ù æ�ê ß ¬ ß æ ÖBÖBÖ æpê$ì ¬ ì�æ�� ï ê ì � Ù ñwhereØ âËÿ »Bù ÿ . Substitutionof thisexpansioninto (29)leadsto thefollowing recursiverelations:é¬ ÷ æ ¼�» ù ¾ ß ¬ ÷ â!Ý ÆÇ �Èë v Ùtë ßë ÊËz¬ ÷ 7 Ùfor Ö â è Ò × × ×GÒ�Ø and é¬�þ æ ¼�» ù ¾ ß ¬ þ:âvãfor Öõâ/ã . Oneof thetwo fundamentalsolutionsof the latterequationis ¬�þÓâ ë 7 Ó ^ sothat theequationfor ¬ Ù becomes é¬ Ù æ ¼�» ù ¾ ß ¬ Ù â!Ý �Èë d v Ù�ë 7 Ó ^ � ß¯íî d Òwhereï_ë is the � th elementof thecanonicalbasisof � × A solutionof thelatterequationis givenby ¬ Ù â!Ý �Èë d v Ù ë 7 Ó ^ � ß¯íî d¼�» ù ¾ ß Ý}¼�» ù Ý3��ï¿ë d ¾ ß ×This recursive processcanbecontinuedup to any order. By inductionit directly follows thatatthe Ö th step ¬ ÷ âHïXÝ è ñ ÷ �Èë d Ûªðªðªð Û ë a v Ù ë ï 7 Ó ^ � ß ï íî d � ðªðªð � íî a ñ�ññ ÷ Õ v Ù ï^¼�» ù ¾ ß Ý}¼�» ù Ý|��ò Õ ¾ ß ñwhere ò Õ â�ï¿ë d æ ÖBÖBÖ æ ï_ë]ó for ��â è Ò × × ×GÒ�Ö¿× Note thatwhen ÿ ÖOÿ$�Êÿ » ù ÿ�Ò thedenominatorof theabove expressionnever vanishes.At theorder Ø âËÿ » ù ÿ theequationfor ¬ ì readsé¬ ì�æ ¼�» ù ¾ ß ¬ ìeæ ïXÝ è ñ ì 7 Ù �Èë d Ûªðªðªð Û ë¯ô v Ù ë ï 7 Ó ^ � ß ï í¯î d � ðªðªð � í¯î ô ñ�ññ ì 7 ÙÕ v Ù ï^¼�» ù ¾ ß Ý}¼�» ù Ý|��ò Õ ¾ ß ñ âvã@×

29

In thesummationweareonly interestedin termswith ë Ó ^ × Indeed,all othertermscanberemovedby aproceduresimilar to theoneusedfor theprevious ¬ Ù Ò × × ×0Ò^¬ ì 7 Ù ×

Theremainingtermscanbeindexedby thepathsof length Ø joining õ and » ù in thelatticeö � × Thesetof thesepathswill bedenotedby ÷�ï�»Bù ñ}Ò andfor every path í � ÷�ï�»Bù]ñ we considertheintermediatepositionvectorsò ÷ ïÎíBñ}× In thisway theequationfor ¬ ì becomesé¬ ìoæ ¼�» ù ¾ ß ¬ ìeæ ë Ó ^ gïb� Ò�» ù ñ�âvãwheregïb� Ò�» ù ñ�â Èø ¦>ù ö Ó ^ ø ïXÝ è ñ ì 7 Ùñ ì 7 ÙÕ v Ù ï^¼�» ù ¾ ß Ý�¼�» ù ÝÚ��ò Õ ïÎíBñ ¾ ß ñ â èM Èø ¦>ù ö Ó ^ ø ïXÝ è ñ ì 7 Ùñ ì 7 ÙÕ v Ù ï^¼�» ù Ý�ò Õ ïÎí ñ ¾ Ý�¼�» ù ¾ ñ Òwhich hasnon-vanishingdenominatorsfor all irrationalfrequency vectors� × Thelatterequationhasasanon-trivial solution ¬ ì ï�èOñ�â!Ý,� Ã è ¼�» ù ¾ gïb� Ò�» ù ñ ë Ó ^ Òprovidedthat gïb� Ò�»BùGñ�Vâvã , andthissolutionis clearlynotquasi-periodic.

Next we proceedwith theotherfundamentalsolution ¬ ö ß ø of theequation(29), startingwiththezeroorderterm ë Ó ^ × However it is betterto studythatsolutionvia theconjugateequationé¬ æ ÆÇ ¼�» ù ¾ ß æ�ê �Èë v Ù ë 7 ßë ÊË ¬Zâ3ã@× (30)

This leadsto a recursive processasbeforefor obtainingthecoefficientsof theexpansionof ¬ öw� øin termsof ê . Now, taking ¬ ö E ø â Ùß X ¬ ö Ù ø �p¬ ö ß ø Z asfundamentalsolutions,wegetthefollowingequation é¬ æ ÆÇ ¼�» ¾ ß æ�ê �Èë v Ù Ñ ë ßë æ ë 7 ßë�Ò ÊËz¬àâvã (31)

which,undoingthechangesin è and ê Ò canbetransformedinto (27).In this way we have foundtwo linearly independentsolutions¬ � and ¬ 7 of this system,the

expansionof which in powersof � have quasi-periodiccoefficientsin timeup to order Ø Ý è andwherethe Ø th ordercoefficient is a functionof theform m ë Ó ^ times gïb� Ò�»Bù]ñ}× By comparisonofcoefficientsit follows that gïb� Ò�» ù ñ and ç¯ïb� Ò�» ù ñ areidenticalexceptfor anon-zerofactor.

Note that gïb� Ò�»Bù]ñ is a rationalfunction. We denoteits numeratorby 5oïb� Ò�»Bù0ñ andits de-nominatorby úÚïb� Ò�» ù ñ}× Define ûZï�» ù ñ asthesetof � ’s for which 5oïb� Ò�» ù ñ is non-zero.Weclaimthat û,ï�» ù ñ hasmeasurezero,which follows from thefactthat 5oïÖ Ò�» ù ñ is not identicallyzero.Tocheckthis first notethat if �Mâ ï è Ò × × ×GÒ è ñ ± Ò then úÚïb� Ò�»Bù]ñ doesnot vanish.Secondwe resorttotheperiodiccase[16], noting that theequationnow canbetransformedto theclassicalMathieuequation.It therebyfollows that 5eïb� Ò�»Bù]ñ is non-zerofor this valueand,hencethat 5oïÖ Ò�»BùGñ isnot identically zerofor any » ù × Thereforethe set û,ï�» ù ñ , given by the zeroesof 5oïb� Ò�» ù ñ , is azeromeasuresetandthetheoremfollows. �

Summarizing,thetongueboundariesat the »Bù th resonance,up to order ÿ »Bù ÿ , aregivenby theequation

coefÙ âU�� coefßwhich, in termsof äå , ä� , É and ú becomesäå,æ Ö Ù ï<äå Ò ä�GÒ É Ò�ú]ñ'âU��.Ñ É ä�/â ß ï_äå Ò ä�¿Ò É Ò�ú]ñ æ ä� ã Ó ^ ã Ö ß ï<äå Ò ä�¿Ò É Ò�ú ñ Ò ×

30

Thetongueboundarycrossingsup to order ÿ » ù ÿ correspondto

coefß âÍãanda furtheranalysisrequiresto distinguishbetweenthecasesof evenandodd Ø ×

When Ø is even,thenfor any ãü��ÿ É ÿ�ý è thereis apocket endingat � âvã andat

� tip âÿþ Ý É â ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñú Ó ^ ç ß ïb� Ò�» ù ñ�� dô e(d æ ÖBÖBÖ¬Òwherethedotsdenotehigherordertermsin É × If Ø is oddthenthesignof É mustbeselectedsuchthat Ý É â ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñú Ó ^ ç ß ïb� Ò�» ù ñis positive. If this is thecase,thentherearetwo instability pocketswith endsat �râvã andat

� Etip âU� þ Ý É â ß ï�ã@Ò�ã@Ò�ã@Ò�ú]ñú Ó ^ ç ß ïb� Ò�» ù ñ�� dô e(d æ ÖBÖBÖJ×C Structur eof the sets��An interestingquestionrelatedto Theorem3 is whetherthesetof Diophantinefrequency vectorsin û,ï�»Jñ , for afixedresonance» , is emptyor not.

When� âU� , we canalwaysassumethat �eâ/ï è ÒXíBñ , whereí is a realnumber. Notethatany

real irrational í for which 5eïb� Ò�»Jñ:âËã for some» , is Diophantine,sinceit is algebraic.Directcomputations,performedon gïb� Ò�»Jñ}Ò yield thatif theorderof theresonanceis lessthan Ò all therootsof 5eï�ï è ÒXí ñ}Ò�»Jñ areeitherrationalor complex (i.e.,nonreal).However, for »ªâHïfS Ò�� ñ ,5 ï�ï è ÒXí ñ}Ò0ïfS Ò�� ñ�ñ âU� M ævè� �Gí æ M M í ß æ _ã�Gí á æ ��S��Gí Ý æ M� í�� æ M í��which hasreal irrational zeroes. Direct computationalsoshows that the samehappensfor allresonances�àÕIÿ »'ÿ Õ

with Ô Ù Vâ è Ò Ô ß Vâ è . For� · S thesituationis evensimpler, sincefor»Bùrâ!ï è Ò è Ò è ñ thepolynomial 5eïb� Ò�»Bù]ñ hasrealDiophantinezeroes.

Thereis onecasewhen ÿ » ù ÿ th ordertangency at the » ù th resonancealwayscanbegranted:

Proposition7 In theMathieuequation¬y# # æ ï�� æ �Kïf-/�0'0ïbm�ñ æ -/�0'¿ïÎí;m�ñ�ñ�ñ�¬Zâvã@Ò (32)

where �GíÐVâIã is not a negativeinteger, theorder of tangencyat �³â/ã of the resonancetongueboundariescorrespondingto » ù âËï�Ø Ò è ñ , for any Ø , exactlyis ÿ » ù ÿ<â9Ø ævè .Proof: In this casethe numberof pathsof minimal length in

ö ß joining ï�ã@Ò�ãðñ and » ù exactlyis Ø æ!è andany of thesecanbe labelledby an integer betweenã and Øª× Thesepathswill bedenotedby ê�þGÒ × × ×GÒ�ê ì . To show that theorderof tangency is exactly Ø æéè we mustcomputegï�ï è ÒXí ñ}Ò0ï�ØªÒ è ñ�ñ â�2�â�ïÎí¬Ò�Ø¨ñ , whichamountsto

â�ïÎí¬Ò�Ø¨ñ�â ïXÝ è ñ ìM ìÈë v þ èñ ìÕ v Ù ï^¼�» ù Ý@ò Õ ï�êÉë0ñ ¾ Ý}¼�» ù ¾ ñ (33)

andshow that for í�V� ö , this doesnot vanish.For eachof thepathsê ë , �õâéã@Ò × × ×0Ò�Ø , let å ë bethecontribution to thesumin (33). Thenå ë â èï�Ø¸Ý è�æ í ñ è èï�ØýÝ|� æ í ñ¯� ÖBÖBÖ èï�ØýÝß� æ íBñ�� èï�ØýÝ � ñ#ïw� æ í ñ ÖBÖBÖ èè ï�ØýÝ è�æ í ñ Ò

31

wherethetotal numberof factorsis Øª× UsingtheGamma-functionit follows thatìÈë v þ å ëlâ ìÈë v þ ÷�ï�Ø¸Ý � æ í ñ÷�ïw� æ íBñ��»ï�ØýÝ � ñ�� ÷�ï�Ø æ íBñ÷�ï�Ø æ íBñ ×Since � ïÎí¬Ò�Ø¨ñ�â9úÚï�ï è ÒXí ñ}Ò0ï�ØªÒ è ñ�ñ âHï�ØýÝ è�æ í ñ è Ö × × ×�ÖðïÎí ævè ñ#ï�ØýÝ è ñ ÖðïÎí ñ#ï�Ø¨ñit is clearthat � ïÎí¬Ò�Ø¨ñ'â ÷�ï�Ø æ í ñØ��÷�ïÎíBñimplying that� ïÎí¬Ò�Ø¨ñ ìÈë v þ å ë½â ìÈë v þ þ Ø � � ÷�ï�Ø¸Ý � æ í ñ÷�ïw� æ í ñ÷�ï�Ø æ í ñ÷�ïÎí ñ â è÷�ï�Ø æ í ñ÷�ïÎí ñ ÷�ï�Ø æ �Gí ñ÷�ïÎíBñ ß÷�ïf�Gí ñ Òwherethelast identity is anapplicationof Pochhammer’s formula,see[47]. Therefore,therele-vantcoefficient is

â�ïÎíDÒ�Ø ñ�â ïXÝ è ñ ìM ìÈë v þ å ëlâ èú ÷�ïÎíBñ ß÷�ï�Ø æ í ñ÷�ïÎí ñ÷�ïf�Gí ñ ÷�ï�Ø æ �Gí ñwhich, if �Gí is notanegative integer, is differentfrom zero. �Acknowledgements

TheauthorsthankHakanEliasson,HansJauslin,RussellJohnsonandYingfei Yi for stimulatingdiscussionsin thepreparationof thispaper.

The authorsare indebtedto eachother’s institutionsfor hospitality. Last two authorshavebeensupportedby grantsDGICYT BFM2000-805(Spain)andCIRIT 2000SGR-27,2001SGR-70 (Catalonia). Partial supportof grant INTAS 2000-221is also acknowledged. The secondauthoracknowledgesthePh.D.grant2000FI71UBPG.

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