Geography of resonances and Arnold diffusion

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Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems
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IOPPUBLISHING NONLINEARITY
Nonlinearity22(2009)1997–2077 doi:10.1088/0951-7715/22/8/013
Geography of resonances and Arnold diffusion in
a priori unstable Hamiltonian systems
AmadeuDelshams1andGemmaHuguet2
1DepartamentdeMatema`ticaAplicadaI,UniversitatPolite`cnicadeCatalunya,Diagonal647,
08028Barcelona,Spain
2CentredeRecercaMatema`tica,Apartat50,08193Bellaterra(Barcelona),Spain
E-mail:[email protected]@upc.edu
Received4December2008,infinalform3June2009
Published20July2009
Onlineatstacks.iop.org/Non/22/1997
RecommendedbyC-QCheng
Abstract
In this paper we consider the case of a general Cr+2 perturbation, for r large
enough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees of
freedom, and we provide explicit conditions on it, which turn out to be C2
genericandareverifiableinconcreteexamples,whichguaranteetheexistence
ofArnolddiffusion.
This is a generalization of the result in Delshams et al (2006 Mem. Am.
Math. Soc.) wherethecaseofaperturbationwithafinitenumberofharmonics
intheangularvariableswasconsidered.
The method of proof is based on a careful analysis of the geography of
resonancescreatedbyagenericperturbationanditcontainsadeepquantitative
description of the invariant objects generated by the resonances therein. The
scattering map is used as an essential tool to construct transition chains of
objectsofdifferenttopology. Thecombinationofquantitativeexpressionsfor
boththegeographyofresonancesandthescatteringmapprovides,inanatural
way,explicitcomputableconditionsforinstability.
MathematicsSubjectClassification: 37J40,37C29,35B34,34C29,37C50
(Somefiguresinthisarticleareincolouronlyintheelectronicversion)
1. Introduction
The goal of this paper is to present a generalization of the geometric mechanism for global
instability (popularly known as Arnold diffusion) in a priori unstable Hamiltonian systems
introduced in [DLS06a]. That paper developed an argument to prove the existence of
0951-7715/09/081997+81$30.00 ©2009IOPPublishingLtdandLondonMathematicalSociety PrintedintheUK 1997
1998 ADelshamsandGHuguet
globalinstabilityinaprioriunstablenearlyintegrableHamiltoniansystems(theunperturbed
Hamiltonian presents hyperbolicity, so that it cannot be expressed globally in action-angle
variables)andappliedittoamodelwhichpresentedtheso-calledlargegapproblem. However,
in that case, the perturbation was assumed to be a trigonometric polynomial in the angular
variables. InthispaperweperformanaccurateprocessoftruncationoftheFourierseriesof
theperturbationandwepresentadeeperstudyofthegeographyofresonances. Usingthis,we
areabletoextendandsimplifysomeoftheresultsin[DLS06a]andapplythemtoanapriori
unstableHamiltoniansystemwithagenericperturbation.
ThephenomenonofglobalinstabilityinHamiltoniansystemshasattractedtheattention
ofbothmathematiciansandphysicistsinthelastyearsduetoitsremarkableimportancefor
theapplications. Itdeals,essentially,withthequestionofwhatistheeffectonthedynamics
whenanautonomousmechanicalsystemissubmittedtoasmallperiodicperturbation. More
precisely,whethertheseperturbationsaccumulateovertimegivingrisetoalongtermeffect
orwhethertheseeffectsaverageout.
The instability problem was formulated first by Arnold in 1964. In his celebrated
paper[Arn64],Arnoldconstructedanexampleforwhichheprovedtheexistenceoftrajectories
that avoided the obstacles of KAM tori and performed long excursions. The mechanism is
basedontheexistenceoftransitionchainsoftransitionwhiskeredtori, thatis, sequencesof
toriwithtransitivedynamicsgivenbyaquasi-periodicflow,suchthattheunstablemanifold
(whisker)ofoneofthesetoriintersectstransversallythestablemanifold(whisker)ofthenext
one. Byanobstructionargument,thereisanorbitthatfollowsthistransitionchain,givingrise
toanunstableorbit.
Theexampleproposedin[Arn64]turnsouttoberatherartificialbecausetheperturbation
was chosen in such a way that it preserved exactly the complete foliation of invariant tori
existinginth√eunperturbedsystem. However,agenericperturbationofsizeεcreatesgapsat
mostofsize εinthefoliationofthepersistingKAMtori,whereasitmovesthewhiskersonly
byanamountε. TheseKAMtori, whicharejustacontinuationoftheonesthatexistedfor
theintegrablesystem(ε =0)arecommonlyknownasprimarytori. Thegapsinthefoliation
oftheprimarytoriarecentredaroundresonances,thatis,resonanttorithataredestroyedby
the perturbation. This is what is known in the literature as the large gap problem (see, for
instance,[Moe96]foradiscussionaboutthelargegapproblemand,indeed,oftheproblemof
diffusion).
In the last ten years there has been a notable progress in the comprehension of the
mechanisms that give rise to the phenomenon of instability and a variety of methods has
been suggested. As an example of this, we will mention that the large gap problem
has been solved simultaneously by a variety of techniques: different geometrical methods
[DLS00,DLS06a,DLS06b](scatteringmap)and[Tre04,PT07](separatrixmap);topological
methods [GL06b,GL06a] and variational methods [CY04,CY09]. For more information
regardingtheproblemofArnolddiffusionintheabsenceofgapsaswellastimeestimates,the
readerisreferredto[DGLS08].
Ofparticularinterestforthispaperare[DLS00,DLS06a,DLS06b]. Thestrategyinthe
mentionedpapersisbasedonincorporatinginthetransitionchainnewinvariantobjects,created
bytheresonances,likesecondarytori,whichareinvariantKAMtoricontractibletoaperiodic
orbit, aswellasthestableandunstablemanifoldsoflowerdimensionaltori. Thescattering
map,introducedbythesameauthors(see[DLS08]forageometricstudy),istheessentialtool
fortheheteroclinicconnectionsbetweeninvariantobjectsofdifferenttopology.
Inthispaperweextendthegeometricmechanismintroducedinthementionedpaperstoa
widerclassofmodelsystemsforwhichtheperturbationdoesnotneedtohaveafinitenumber
ofharmonicsintheangularvariables. Inparticular,theHamiltonianstudiedinthispaperhas
GeographyofresonancesandArnolddiffusion 1999
thefollowingform:
(cid:1) (cid:2)
H (p,q,I,ϕ,t)=± 1p2+V(q) + 1I2+εh(p,q,I,ϕ,t;ε), (1)
ε 2 2
wherep ∈(−p0,p0)⊂R,I ∈(I−,I+)⊂Rand(q,ϕ,t)∈T3.
The main result of this paper is theorem 2.1, stated in section 2.2 with the concrete
hypothesesforHamiltonian(1),fromwhichwecandeducethefollowingshortversion:
Theorem1.1. ConsidertheHamiltonian(1)andassumethatV andhareCr+2functionswhich
areC2generic,withr >r ,largeenough. Thenthereisε∗ >0suchthatfor0<|ε|<ε∗and
0
foranyinterval[I−∗,I+∗]∈(I−,I+),thereexistsatrajectoryx˜(t)ofthesystemwithHamiltonian
(1)suchthatforsomeT >0
I(x˜(0))(cid:1)I∗; I(x˜(T))(cid:2)I∗.
− +
Remark1.2. Avalueofr whichfollowsfromourargumentisr =242(seeremark2.3).
0 0
Our strategy for the proof follows the geometric mechanism proposed in [DLS06a].
Indeed, in order to organize the different invariant objects that we will use to construct a
transition chain, we will first identify the normally hyperbolic invariant manifold (NHIM)
presentinthesystem. ThisNHIMwillhaveassociatedstableandunstableinvariantmanifolds
that,generically,intersecttransversally. Therefore,wecanassociatewiththisobjecttwotypes
ofdynamics:theinnerandtheouter. Theouterdynamicstakesintoaccounttheasymptotic
motionstotheNHIMandisdescribedbythescatteringmap. Theinnerdynamicsistheone
restrictedtotheNHIMandcontainsCantorfamiliesofprimaryandsecondaryKAMtori. Since
genericallythesefamiliesofKAMtori,invariantfortheinnerdynamics,arenotinvariantfor
theouterdynamics,thecombinationofbothdynamicswillallowustoconstructatransition
chain.
Theresultsin[DLS06a]canbeappliedstraightforwardlyforthepersistenceoftheNHIM
andthetransversalityofitsassociatedstableandunstablemanifolds. Theargumentspresented
inthispaperfocusontheinnerdynamicsandthestudyoftheinvariantobjectspresentinthe
NHIM.
ForHamiltonian(1),resonancescorrespondtotheplaceswherethefrequencyI =−l/k
for (k,l) ∈ Z2 is rational and the associated Fourier coefficient h of the perturbation h is
k,l
non-zero. Ontheseresonances,thefoliationofKAMtoriintheNHIMisdestroyedandgaps
betweentheCantorfamilyofinvarianttoriintheNHIMofsizeO(ε1/2|h |1/2)arecreated,
k0,l0
for(k ,l )suchthatl/k = l /k andgcd(k ,l ) = 1(seeequation(88)). Foraperturbation
0 0 0 0 0 0
h which is a Cr+2 function and C2 generic, when we restrict it to the NHIM and we write
it in adequate coordinates we are left with a Cr perturbation (see the subsection ‘restriction
to NHIM’ in section 2.3.3), so that |h | ∼ |(k,l)|−r, and therefore the above gaps are of
k,l
size O(ε1/2|(k ,l )|−r/2). Moreover, other invariant objects, like secondary tori and lower
0 0
dimensionaltori,arecreatedinsidethegap. Theycorrespondtoinvariantobjectsofdifferent
topologythatwerenotpresentintheunperturbedsystembutaregeneratedbytheresonances.
In order to study their existence and give an approximate expression for them we will
combinemstepsofaveragingplusaKAMtheorem. Notethatinourcase,sincetheperturbation
is generic, we will have an infinite number of resonances. Our approach for this study will
be to consider an adequate truncation up to some order M, depending on ε, of the Fourier
seriesoftheperturbationhinsuchawaythatwedealonlywithafinitenumberofharmonics
|(k,l)|(cid:1)M andthereforeofresonances.
Anotherremarkabledifferencewithrespecttotheresultsobtainedin[DLS06a]isthatin
that case the size of the gaps created in the foliation of invariant tori was uniform, whereas
in our case, since the size is O(ε1/2|(k ,l )|−r/2), we have a heterogeneous sea of gaps of
0 0
2000 ADelshamsandGHuguet
different sizes. Among them, we will distinguish between small gaps and big gaps, which
are strongly related to the mentioned large gap problem. Indeed, big gaps are those of size
biggerthanorequaltoεandthereforetheyaregeneratedbyresonances−l /k oforderone,
0 0
suchthat|(k ,l )|<ε−1/r or,equivalently,|(k ,l )|−r/2 (cid:2)ε1/2 (seesection3.3.3forprecise
0 0 0 0
results).
Fromamoretechnicalpointofview(seesection3.2fordetails),wewouldliketoremark
that the main difficulties arise from the fact that in order to perform a resonant averaging
procedure, we need to isolate resonances corresponding to |(k,l)| (cid:1) M, for M depending
onε. Consequently,thewidthLoftheresonantdomaincannotbechosenindependentlyofε,
aswasthecasein[DLS06a]. Moreover,alongtheaveragingprocedureweneedtokeeptrack
of the C(cid:3) norms of the averaged terms and the remainders, and these blow up as a negative
powerofL. Hence,wewillseethatagoodchoiceforLaroundaresonanceI = −l/k will
be L = L ∼ ε1/n/|k| (see hypotheses of theorem 3.11), where n is the required regularity
k
toapplyKAMtheoremaftertheaveragingprocedure. NotethatLisnotuniformalongthe
resonancesbutdependsonthevalue|k|oftheresonance.
Finally, after m steps of averaging, we will show that the remainder tail, that is, the
Fourier coefficients h such that |(k,l)| > M can be neglected. This will be ensured by a
k,l
fastenoughdecreasingrateofthecoefficientsandthereforealargeenoughregularityr ofthe
perturbation. Thus,therequiredregularityr willbedeterminedaccordingtothenumbermof
stepsofaveragingperformed,aswellastheneededregularityntoapplyKAMtheoremafter
theaveragingprocedure.
WeareusingaversionoftheKAMtheoremthatrequirestohavetheHamiltoniansystem
writteninactionanglevariables. Sinceneartheresonancesweapproximatethesystembythe
onewhichisclosetoapendulum,theactionvariablebecomessingularontheseparatrix. This
fact, together with the requirement to have the invariant objects close enough (at a distance
smallerthanε),impliesthattheperturbationoftheaveragedHamiltonianhastobeextremely
small in the resonant regions. The immediate consequence of this fact is that, in the case
wearestudying, onehastoperformatleast m = 10 stepsofaveraging(seetheorem 3.28).
TheneededregularityntoapplyKAMtheoremaftermaveragingstepsisn = 2m+6(see
proposition3.24). Sincetheregularityr requiredtoensurethattheremaindertailissmaller
thantheaveragingremainderturnsouttober > (n−2)m+2,seeremark3.20,onehasto
imposer >r =242.
0
We do not claim that this is an optimal result. Actually, another version of the KAM
theoremthatallowedustoavoidthechangeintoaction-anglevariableslike[LGJV05,FLS09]
couldimprovetheresultsintermsoftheneededregularity(seealso[HLS09]foranumerical
implementation). However,itisworthmentioningthatwemanagedtodecreasetherequired
stepsofaveragingintheresonantdomainswithrespecttotheresultsin[DLS06a]. Sincein
the resonances the behaviour of KAM tori is different depending on how close they are to
the separatrix (tori are flatter as they are further from the separatrix), we consider different
regions where we perform different scalings. This strategy, which was already introduced
in [DLS06a], has been improved in this paper introducing a new sequence of domains in
theorem3.30. Whenappliedtothecasewithafinitenumberofresonancesasin[DLS06a],
m=9stepsofaveragingandr (cid:2)26areenough(seeremark3.32). Thisclearlyimprovesthe
neededregularityr whichwasr (cid:2)60in[DLS06a]becausemwaschosenasm=26.
Sections 3.3.3, 3.3.4 and 3.3.5 contain a quantitative description of the geography of
resonances and a detailed study of the invariant objects generated by the resonances. The
effectoftheresonancesinasystemconstitutesafundamentalproblemnotonlyfordiffusion
butalsoformanyotherphysicalapplicationsandithasbeenanimportantobjectofstudyin
the physical literature, see for instance [Chi79,Ten82]. The study performed in this paper
GeographyofresonancesandArnolddiffusion 2001
contributes to a better understanding of the different types of resonances and the geometric
objectsthatonecanfindthereinunderanarbitrarilysmallvalueoftheperturbationparameter
ε. Therefore,thisstudycanbeveryhelpfulinmanyphysicalproblems,althoughinconcrete
problems,thesizeofεisnotnecessarilyverysmall,andotherdiffusionmechanismscantake
place,likeChirikov’soverlappingofresonances[Chi79].
Moreover, we think that this study can be extended to a class of models that presents
multipleresonances,see[DLS07].
We would like to emphasize that in our case, and this is different from the results
in [DLS06a], only the resonances of order 1, that is the ones that appear at the first step
ofaveraging,createbiggaps;whereasin[DLS06a],bothresonancesoforders1and2could
generatebiggaps. Thisisbecausewearedealingwithaperturbationthatgenericallywillhave
alltheharmonicsdifferentfromzero. Thismeansthattheeffectoftheresonancesassociated
with the biggest Fourier coefficients (low frequencies) will be detected at the first step of
averaging. Sincethesizeofthegapdependsonboththeorderoftheresonanceandthesizeof
theFouriercoefficientassociatedwiththatresonance,theonesthatappearatthesecondstep
ofaveragingalreadycorrespondtosmallFouriercoefficientsandthesizeoftheirgapwillbe
smallerthanε. Theimmediateconsequenceofthisfactisthatintheforthcomingtheorem2.1
wecangiveallconditionsexplicitlyintermsoftheoriginalperturbationh.
The paper is organized in the following way. In section 2 we state theorem 2.1,
which establishes the existence of diffusing orbits for the model considered under precise
conditions. SincetherequiredhypothesesarecheckedtobeC2 generic,theorem1.1follows
straightforwardly. The proof of theorem 2.1 is given in section 2, except for two technical
results,theorem3.1andproposition4.1,whicharepostponedtothefollowingsections.
Thus, in section 3 we prove theorem 3.1, which provides a quantitative existence of
invariantobjectsfortheinnerdynamicsintheNHIMfollowingthestepsindicatedinsection2.
In section 4, we use the scattering map to prove proposition 4.1 about the existence of
heteroclinicconnectionsbetweentheinvariantobjectsobtainedintheprevioussection.
We would like to remark that, in contrast to [DLS06a], and thanks to the new results
about the scattering map obtained in [DLS08], we use the Hamiltonian function generating
thedeformationofthescatteringmapinsteadofthescatteringmapitself,inordertocompute
theimagesoftheleavesofacertainfoliationunderthescatteringmap.
Finally,insection5wehaveincludedforillustrationaconcreteexample,forwhichwe
sketch how the hypotheses of theorem 2.1 can be checked. We plan to come back to this
exampleinafuturepaperforamoredetaileddescriptionofthemechanism. Intheappendix,
wehavebroughtsometechnicalresultsusedinthepaper.
2. Statementofresults
Beforestatingthemainresultinthispaperweneedtointroducesomenotation.
2.1. Notationandpreliminaries
Let r be a positive integer and D ⊂ Rn a compact set with non-empty interior D˚. We will
denotethesetofCr functionsfromD˚ toRmandcontinuousonDbyCr(D,Rm). Whenm=1,
wesimplywriteCr(D)insteadofCr(D,Rm). Givenf ∈Cr(D,Rm),weconsiderthestandard
Cr norm,
(cid:3)m (cid:3)r (cid:3) |Dαf (x)|
|f|Cr(D) = sup i , (2)
i=1 (cid:3)=0|α|=(cid:3)x∈D α!
2002 ADelshamsandGHuguet
wheref denotestheithcomponentofthefunctionf,fori =1,...,m. Weomitthedomain
i
inthenotationwhenitdoesnotleadtoconfusion.
We use the standard multi-index notation: if α = (α ,...,α ) ∈ Nn and x =
1 n
(x ,...,x )∈Rnonesets
1 n
|α|=α +···+α ,
1 n
α!=α !α !···α !,
1 2 n
∂|α|
Dα = .
∂xα1...∂xαn
1 n
Inthecasethatthefunctionf dependsonlyonafewofthevariables,wewilldenoteitin
thesameway,thatis|f|Cr =|f|Cr(D),andconsideritasafunctionofmorevariablesdefined
intheappropriatedomain.
Notethatwedenote|f|C0 =supx∈D|f(x)|,whichisthestandardsupremumnorm,sothe
|·|Cr(D)normcanbeexpressed,equivalently,as
|f|Cr(D) :=(cid:3)m (cid:3)r (cid:3) |Dαfi|C0(D).
α!
i=1 (cid:3)=0|α|=(cid:3)
ThespaceofCr(D)functionsendowedwiththeCrnormisaBanachalgebra(see[Con90]),
thatisitisaBanachspacewiththepropertythatgivenanytwofunctionsf,ginCr(D),they
satisfy
|fg| (cid:1)|f| |g| .
Cr Cr Cr
Since we will also deal with Cr functions defined on a compact domain D = I ×Tn,
whereI ⊂ Rn isacompactsetwithnon-emptyinterior,wecanalsoconsiderthefollowing
seminorm,thattakesintoaccountthedifferentregularitiesandtheestimatesforthederivatives
ineachtypeofvariable:
(cid:4) (cid:4)
|f|(cid:3)1,(cid:3)2 :=m(cid:3)1(cid:3)=10m(cid:3)2(cid:3)=20|α1|=α1m,(cid:3)α12,|∈α2N|n=m2 α1!1α2!(Is,ϕu)p∈D(cid:4)(cid:4)(cid:4)∂m∂1+Imα21f∂ϕ(Iα,2ϕ)(cid:4)(cid:4)(cid:4), (3)
for0(cid:1)(cid:3)1+(cid:3)2 (cid:1)r. (cid:5)
Notethat|f|C(cid:3) = (cid:3)m=0|f|m,(cid:3)−m,for0(cid:1)(cid:3)(cid:1)r.
Wewillusethefollowingnotation,whichisratherusual. Givenα =α(ε)andβ =β(ε),
wewillwriteα (cid:6) β andalsoα = O(β)ifthereexistε andaconstantC independentofε,
0
such that |α(ε)| (cid:1) C|β(ε)|, for |ε| (cid:1) ε . When we have α (cid:6) β and β (cid:6) α we will write
0
α ∼β. However,insomeinformaldiscussionswewillabusenotationandwewillsaythatα
isoforderεp ifandonlyifα ∼εp.
Wewillsaythatafunctionf =OCr(D)(β)when
|f| (cid:6)β.
Cr(D)
2.2. Setupandmainresult
We consider a 2π-periodic in time perturbation of a pendulum and a rotor described by the
non-autonomousHamiltonian(1),
H (p,q,I,ϕ,t)=H (p,q,I)+εh(p,q,I,ϕ,t;ε)
ε 0
=P±(p,q)+ 1I2+εh(p,q,I,ϕ,t;ε), (4)
2
GeographyofresonancesandArnolddiffusion 2003
where
(cid:1) (cid:2)
P±(p,q)=± 1p2+V(q) (5)
2
andV(q)isa2π-periodicfunction. WewillrefertoP±(p,q)asthependulum.
Theterm 1I2describesarotorandthefinaltermεhistheperturbationtermanddepends
2
periodicallyontime,sothathcanbeexpressedviaitsFourierseriesinthevariables(ϕ,t):
(cid:3)
h(p,q,I,ϕ,t;ε)= h (p,q,I;ε)ei(kϕ+lt). (6)
k,l
(k,l)∈Z2
Itwillbeconvenienttoconsidertheautonomoussystembyintroducingtheextravariables
(A,s):
H˜ (p,q,I,ϕ,A,s)=A+H (p,q,I)+εh(p,q,I,ϕ,s;ε)
ε 0
=A+P±(p,q)+ 1I2+εh(p,q,I,ϕ,s;ε) (7)
2
where the pairs (p,q) ∈ R×T, (I,ϕ) ∈ R×T and (A,s) ∈ R×T are symplectically
conjugate.
The extra variable A does not play any dynamical role and it simply makes the system
autonomous. So,weareonlyinterestedinstudyingthedynamicsofvariables(p,q,I,ϕ,s)
givenbythefollowingsystemofequations:
∂h
p˙ =∓V(cid:8)(p)−ε (p,q,I,ϕ,s;ε),
∂q
∂h
q˙ =±p+ε (p,q,I,ϕ,s;ε),
∂p
∂h
I˙=−ε (p,q,I,ϕ,s;ε), (8)
∂ϕ
∂h
ϕ˙ =I +ε (p,q,I,ϕ,s;ε),
∂I
s˙ =1.
Thedomainofdefinitionweconsiderisacompactsetoftype
D :=S×[I−,I+]×T2×[−ε0,ε0],
whereS ⊂R×Tisaneighbourhoodoftheseparatrix(P−1(0))ofthependulum.
±
Then,themaintheoremofthispaperis
Theorem2.1. Consider a Hamiltonian of the form (1) where V and h are Cr+2 in D, with
r >r ,sufficientlylarge. Assumealsothat
0
H1 The potential V : T → R has a unique global maximum, say at q = 0, which is
non-degenerate(i.e.V(cid:8)(cid:8)(0) < 0). Wedenoteby(p (t),q (t))anorbitofthependulum
0 0
P±(p,q)in(1),homoclinicto(0,0).
H2 Consider the Poincare´ function, also called Melnikov potential, associated with h (and
withthehomoclinicorbit(p ,q )):
0 0
(cid:6)
+∞
L(I,ϕ,s)= − (h(p (σ),q (σ),I,ϕ+Iσ,s+σ;0)
0 0
−∞
−h(0,0,I,ϕ+Iσ,s+σ;0))dσ (9)
2004 ADelshamsandGHuguet
H2(cid:8) GivenrealnumbersI− <I+,assumethat,foranyvalueofI ∈(I−,I+),thereexists
anopensetJ ∈T2,withthepropertythatwhen(I,ϕ,s)∈H,where
I (cid:7)
H = {I}×JI ⊂(I−,I+)×T2, (10)
I∈(I−,I+)
themap
τ ∈R(cid:11)→L(I,ϕ−Iτ,s−τ)
hasanon-degeneratecriticalpointτ whichislocallygivenbytheimplicitfunction
theoremintheformτ =τ∗(I,ϕ,s),withτ∗asmoothfunction.
H2(cid:8)(cid:8) IntroducethereducedPoincare´ functionL∗definedby
L∗(I,ϕ):=L(I,ϕ−Iτ∗(I,ϕ,0),−τ∗(I,ϕ,0)), (11)
on (cid:7)
H∗ ={(I,θ˜):θ˜ =ϕ−Is,(I,ϕ,s)∈H}= {I}×J∗, (12)
I
I∈(I−,I+)
and assume that there exists an open set J∗+ ⊆ J∗ (respectively J∗− ⊆ J∗), such
I I I I
thatforθ˜ =ϕ−Is ∈J∗+,thefunction
I
θ˜ (cid:11)→ ∂L∗(I,θ˜) (13)
˜
∂θ
isnon-constantandpositive(respectivelynegative). Wedenote
(cid:7)
H∗ = {I}×J∗+ (14)
+ I
(cid:8)I∈(I−,I+)
(respectively,H∗ = {I}×J∗−).
− I∈(I−,I+) I
H3 Fix 1/(r/6 − 1) < ν (cid:1) 1/16, for any 0 < ε < 1 and for any (k ,l ) ∈ Z2
0 0
with gcd(k ,l ) = 1 and |(k ,l )| < M , where |(k ,l )| = max(|k |,|l |) and
0 0 0 0 BG 0 0 0 0
M =ε−(1+ν)/r,introducethe2π-periodicfunction
BG
(cid:3)
Uk0,l0(θ)= htk0,tl0(0,0,−l0/k0;0)eitθ,
t∈Z−{0},
|t||(k0,l0)|<M
whereθ =k ϕ+l s andM =ε−1/(26+δ),for0<δ <1/10,forwhichweassume
0 0
H3(cid:8) ThefunctionUk0,l0 hasanon-degenerateglobalmaximum.
H3(cid:8)(cid:8) For |(k ,l )| < C ε−1/r, where C is given explicitly in (153), we assume that the
0 0 1 1
2πk -periodicfunctionf givenby
0 (cid:9) (cid:10) (cid:9) (cid:10)
f(θ)= k0U(cid:8)k0,l0(θ)∂∂Lθ˜∗ −k0l0,kθ0 (cid:9)+2Uk0(cid:10),l0(θ)∂∂2θL˜2∗ −k0l0,kθ0 (15)
2∂2L∗ −l0, θ
∂θ˜2 k0 k0
isnon-constantforθ/k ∈J∗ .
0 −l0/k0
H3(cid:8)(cid:8)(cid:8) ForC ε−1/r (cid:1) |(k ,l )| (cid:1) C ε−1/r,whereC andC aregivenexplicitlyin(153)
1 0 0 2 1 2
and(155)respectively,weassumethenon-degeneracyconditionstatedexplicitlyin
equation(156).
Then, there exists ε∗ > 0 such that for 0 < |ε| < ε∗ and for any interval [I∗,I∗] ∈
− +
(I−,I+),thereexistsatrajectoryx˜(t)ofthesystem(1)suchthatforsomeT >0
I(x˜(0))(cid:1)I∗; I(x˜(T))(cid:2)I∗.
− +
(respectively,
I(x˜(0))(cid:2)I∗; I(x˜(T))(cid:1)I∗).
+ −
GeographyofresonancesandArnolddiffusion 2005
Remark2.2. Asamatteroffact,theonlyrestrictionneededfordiffusioninH2(cid:8)(cid:8) isthatthe
function θ˜ (cid:11)→ L∗(I,θ˜) has to be non-constant. Typically, when it is non-constant there are
somedomainsH+ andH− whereitsderivativeispositiveandnegative,respectively. These
∗ ∗
arethesetschosentohaveanincreasinganddecreasingdiffusioninI,respectively.
Remark2.3. r depends on the number m of some averaging steps performed in the proof:
0
r = 2(m+1)2 and m (cid:2) 10 (see hypotheses of theorem 3.1 in section 3). If we take just
0
m=10thenr =242isenough.
0
Remark2.4. ThetruncationorderM inhypothesesH3dependsontheregularitynrequired
fortheapplicationoftheKAMtheoremalongtheproof:M = ε−1/(n+δ),forn = 2m+6and
0 < δ < 1/m,wheremisthenumberofaveragingstepsperformedintheproofandissuch
thatm (cid:2) 10(seehypothesesoftheorems3.11and3.1andremark3.20). Hence,wechoose
m=10andthereforeM =ε−1/(26+δ)inhypothesesH3.
Remark2.5. NotethatforeveryfixedεwehaveoneconditionH3forevery(k ,l )suchthat
0 0
|(k ,l )| < M , that depends explicitly on (k ,l ). Hence, the number of non-degeneracy
0 0 BG 0 0
conditionsH3isfinitebutgrowswithε.
Remark2.6. Notethatbythedefinitionofτ∗(I,ϕ,s),thefunction
f(I,ϕ,s)=L(I,ϕ−Iτ∗(I,ϕ,s),s−τ∗(I,ϕ,s))
satisfiestheequation
I∂ f(I,ϕ,s)+∂ f(I,ϕ,s)=0.
ϕ s
Thereforeitisoftheformf(I,ϕ,s)=L∗(I,ϕ−Is),sowecanalternativelydefine
L∗(I,ϕ−Is)=L(I,ϕ−Iτ∗(I,ϕ,s),s−τ∗(I,ϕ,s)).
Remark2.7. Themainfeatureoftheorem2.1,asalreadysaidintheintroduction,isthathis
notrequiredtobeatrigonometricpolynomialinthevariables(ϕ,s),whichisanon-generic
assumption,aswasthecasein[DLS06a].
Beforeprovingtheorem2.1letusseethattheorem1.1statedintheintroductionisjusta
consequenceoftheorem2.1. Indeed, foreveryfixedε, conditionsH1andH2areopenand
dense,thatistheyholdforanopenanddensesetofHamiltoniansintheC2 topology.
For every fixed ε, the number of non-degeneracy conditions H3 is finite but grows
with ε (the number of conditions depends on (k ,l ) ∈ Z2 such that gcd(k ,l ) = 1 and
0 0 0 0
|(k ,l )| (cid:6) ε−1/r). Whenε tendsto0wehaveacountablenumberofconditionsintermsof
0 0
thefunctions
(cid:3)
U∞k0,l0(θ)= htk0,tl0(0,0,−l0/k0;0)eitθ,
t∈Z−{0}
which are the same as those in hypotheses H3 but without any truncation. This countable
number of conditions involve only derivatives up to order 2 of the Hamiltonian. Hence the
set of Hamiltonians satisfying them is a residual set in the C2 topology, that is a countable
intersectionofopenanddensesetsintheC2 topology.
Therefore the hypotheses of the theorem are C2 generic in the set of Cr+2 Hamiltonians
oftheform(1). So,theshortversionoftheorem2.1statedintheorem1.1intheintroduction
followsstraightforwardly.