Homoclinic orbit to a center manifold

Patri k Bernard

january 2000

Introdu tion

A saddle- enter xed point of a Hamiltonian system is a xed point with pre iselyone pair

ofpurely imaginaryeigenvalues, and other eigenvalues all having non-zeroreal part. Su h a

xedpoint is ontained ina two dimensionalinvariant manifold, alled the enter manifold,

asso iated with the pair of imaginary eigenvalues and lled with periodi orbits. Ea h of

these periodi orbits is the transversal interse tion between its energy shell and the enter

manifold,andishyperboli withrespe ttoitsenergyshell. Weareinterestedintheexisten e

oforbits homo lini to these periodi traje tories.

Letus onsideraninitialsystemwithasaddle- enterxedpointandan orbithomo lini

to it. The orbit stru ture near this homo lini orbit of the initial system and of perturbed

systems anbestudiedusingappropriatelo alse tionsandthePoin arereturnmapalongthe

homo lini . ThishasbeeninitiatedbyConleyin[12 ℄,andusedin[21 ℄,[23 ℄ and[26 ℄ toprove

the existen e of homo lini orbits in perturbed systems. Under suitable hypotheses, and if

thephase spa e is fourdimensional,these papersshowthe followingbehavior. Hamiltonian

systems suÆ iently loseto the initialsystem have a saddle- enter xedpointwith a enter

manifold. We ansupposewithoutlossofgeneralitythatthesaddle- enterxedpointalways

haszeroenergyinthesystemsunderinterest. ForanysuÆ ientlysmallxedpositiveenergy,

theperiodi motiononthe entermanifoldat thatenergyhasahomo lini orbitinasystem

suÆ iently loseto the initialsystem. However, inaxed system loseto theinitial system,

the periodi orbits losest to the xed point are not proved to have any homo lini orbit.

The homo lini orbits of smallest energy are rst destroyed by the perturbation. This is

notsurprisingsin ethesaddle- enterxed pointitself doesnothave anyhomo lini orbitin

general. Theseworksprovideamu hmoredetaileddes riptionoftheorbitstru turethanwe

dointhispaper,buttheirrangeislimitedtothestudyofperturbationsofinitialsystemswith

ahomo lini orbittothesaddle- enter, whi hisanex eptional ase, andto fourdimensional

phasespa es.

Variational methodsprovide globalexisten e results on homo lini orbitsto a

hyperbol-i xed point, see [5 ℄, [13 ℄ and many other papers, that an be viewed asnon-perturbative

analogs of the theory of Melnikov that studiesthe persisten e of homo lini s under

pertur-bation. In thesame spirit, we attempt to provide anon-perturbativeanalog of the behavior

des ribed above around saddle- enter xed points. This paper is losely onne ted to [3℄,

where we study the existen e of homo lini orbits to some hyperboli periodi orbits of a

Hamiltoniansystem inC n

. Sin ethe smallestperiodi motioninthe entermanifoldhaving

a homo lini orbit seems to go away from the xed point when thesystem goes away from

in order to ndaglobal(i.e. non-perturbative)result. In [3 ℄,we studythe vi inityof a

pre-s ribedenergyshellsuÆ ientlyfarfromtheorigin,andobtainedhomo lini orbitsinadense

familyof energy shells around thepres ribed one. We abandon all pretension to ndmany

homo lini orbits, but fo us on ndingthe orbit losest to the saddle- enter. It is yet very

unlikely that the orbit we ndis indeedthe losest to the saddle- enter, but it is probably

the losest amongthose whi hsatisfya ertainestimate. Weshall larifythispointlater.

We studyamodelsystemwherethe entermanifoldisaplane withharmoni os illations

on it. We suppose that these periodi motions are hyperboli with respe t to their energy

shells,sothatthe entermanifoldisanormallyhyperboli manifold. Thesettingisthusquite

similartothesettingof[3 ℄,butweassumeherethatthetotalphasespa eistheprodu tofthis

planewiththe otangentbundleofa ompa tmanifoldM,insteadofR 2 n

in[3℄. Thisprodu t

stru tureis a key to our result, sin e we shall obtain homo lini orbits by omparison with

produ t (un oupled) ows. The existen e of homo lini orbits forprodu t ows is redu ed

towell-knownexisten eresultsonhomo lini motionsto hyperboli xed pointson ompa t

Riemannianmanifolds,see [5 ℄. We shall moreover assume that theHamiltonian is berwise

onvex on the total phase spa e T 

(M R), so that a Lagrangian a tion fun tional an be

used. It should be possibleto avoid thisrestri tion sin e theHamiltonian a tion fun tional

an be well studied in this ontext, see [19 ℄ or [10 ℄. The Lagrangian fun tional remains

simpler,and the resultsofthispaperwillbeexpressed intermsof Lagrangian systems.

Let us stress that, although the setting is Lagrangian, our result is very dierent from

lassi al ones on homo lini orbits in Lagrangian systems sin e the periodi orbits of the

entermanifolddo notsatisfythe minimalityhypothesisneeded intheseresults. In fa t,the

entermanifoldasawholesatisesthishypothesis,andwe shalllookfororbitshomo lini to

thismanifold. An orbithomo lini to the entermanifoldishomo lini tooneoftheperiodi

orbits,byenergy onservation. ThediÆ ultyisthatthe entermanifoldisnot ompa t, and

thatwe have to nda wayto lo alizeorbits.

Under suitable hypotheses, we prove the existen e of an orbit homo lini to one of the

os illationsof the entermanifoldandgive an estimateofits a tionand ofits energy. These

estimates arethe mainnovelties ompared with [3 ℄,they allowinteresting new appli ations.

Theenergyis losetozero(theenergyofthexedpoint)whenthesystemis losetoaprodu t

system and the homo lini we nd should be seen as the ontinuation, when a oupling is

orbits, longer and loser, may appear. Both the enter manifold and the homo lini orbit

are preserved by a small perturbation of the system i.e. a perturbed system still has an

invariantmanifolddieomorphi to a plane and foliated byperiodi orbits one of whi hhas

a homo lini .

Amongtheappli ationsletusgivetheexampleofthestielasti spatialpendulum. This

isapendulumwherethebarhasbeenrepla edbyastispringwhi hhasvariablelengthbut

remainsalways straight, see Figure 1. The enter manifoldhere is the set of os illations of

thespringinunstableequilibrium. Weobtainanorbithomo lini to oneofthese os illations

whenthespringisstienough. Thishomo lini ismoreoverpreservedbyasmallperturbation

ofthesystem. Itisaverygeneralpro esstointrodu eanadditionaldegree offreedomhighly

onned to zero in a me hani al system (a previously frozen binding is now granted some

freedomto os illate). Under ertain hypotheses, we see thata hyperboli xed point with a

homo lini orbit of the frozen system is turned to a saddle- enter xed point with a enter

manifoldand an orbithomo lini to the enter manifoldintheextendedsystem.

A major interest of homo lini orbits is their link with haoti behavior. The orbit

stru turenear a transversal homo lini orbitto a hyperboli xed point of a periodi

time-dependentsystemhasbynowbeenwelldes ribed. Thenaturalanalogofthisstru tureexists

inan autonomoussystemaroundatransversalhomo lini orbitto anhyperboli xedpoint.

Itshouldbenotedhoweverthatthebehaviorasso iatedwithhomo lini orbitstohyperboli

xedpointsofautonomoussystemsisnotaswellunderstood,see[15 ℄and[8℄forsomeresults

onthissubje t. Oneoftheinterestsofourworkisthatthehomo lini wend,iftransversal,

lead to the well des ribed ase, i.e. to a Bernoullishift with positive entropy. Consider for

examplea lassi alplane pendulum,ourresultsprovideanewwayto breakintegrabiltyand

introdu e haoti behavior. Insteadof onsideringthatthereissomesmallin uen efromthe

exterior(a timedependent perturbation), one an onsiderthatthe barhas some elasti ity.

Inthis ase, the unstable equilibriumis surroundedbyunstable os illations. We prove that

oneof these os illationshave a homo lini orbit,thishomo lini an bemade transversalby

aperturbation,andthe systemthenhaspositive topologi alentropy.

The questions dis ussed in this paper were asked to me by my advisor, Eri Sere. It is

a pleasureto a knowledge his de isive helpsand en ouragements. I also wish to thank Ivar

Ekeland forhisinteresting omments.

1 Results, omments and appli ations

LetM bea ompa tmanifold,TM 

!M its tangent bundle. We provideM withametri

g,and note kzk= q g (z) (z;z)

thenorm of a tangent ve tor z 2TM. There is an asso iated metri on TM, and we note

d(z;z 0

) thedistan e between two pointsof TM asso iated withthis metri . Letus onsider

thesmoothLagrangianon T(MR) =TMR 2 givenby L(z;q;v)=a v 2 ! 2 q 2
+G(z;q;v) (z;q;v)2TM RR; (1)

where a and ! are positive real numbers and G : TM R 2

! R is a smooth fun tion

0 0 0 0

(q;v) 2R 2

.

HG2 : Thereexists ab>0 su h thatG(z;q;v) >bd(z;z

0 ) 2 : HG3 : G(z;q;v) > 1 2  q G q (z;q;v)+v G v (z;q;v)  +bd(z;z 0 ) 2 :

Moreover, we assume that there exist two smooth berwise onvex fun tions U and W on

TM su hthat

U(z)6G(z;q;v)6W(z) (2)

forall(z;q;v)2TMR 2

,and thatthey bothsatisfy

HU1: U(z 0 )=0 and dU(z 0 )=0, HU2: U(z)>bd(z;z 0 ) 2 .

Finally,weassume thatthe LagrangianL isberwise onvex,

HL : Therestri tionof L to ea hberT

 MT

q

R is onvex.

No more ontrol at innity is ne essary for our results to hold true, but we will use in the

proofssystemssatisfyingtheadditional hypothesis

HG4 : There existsa fun tionG

1

on TM, a number >0, a ompa t set K TM and

a ompa t set B  K R 2 su h that G(z;q;v) = G 1 (z) outside B, and G 1 (z) = kzk 2 outsideK.

Asa onsequen eof [HL℄, thetraje toriesof L on MR are theproje tions of theintegral

urves ofave tor-eldY

L

on TM,thatis onjugatedtothe Hamiltonianve toreld X

H on

T 

M,whereH istheberwisedual ofL

H(;q;p)= sup z2 1 (  ());v2R h;zi+pv L(z;q;v) (;q;p)2T  M R 2 :

SeeSe tion2 formoredetails. The owofY

L

hasaninvariantmanifold,the entermanifold,

ofequationz=z

0

. The entermanifoldislledwithperiodi orbits,whi haretheliftingsof

O

r

(t)=(

0

;r os (!t));

and an be des ribedalso by

O r =f(z 0 ;q;v)2TMR 2 =v 2 +! 2 q 2 =! 2 r 2 g:

We are lookingfor orbits homo lini to O

r

, i.e. traje tories x =(;q) :R ! M R su h

that6 0 and lim t!1  (t); _ (t);q(t)_ 2 +! 2 q(t) 2  =  0 ;0;! 2 r 2
:

WewillseeinSe tion5 thattoanyfun tionU on TM satisfying[HU1,2℄ we an asso iate a

numberI(U)su hthat

U 6W =)I(U)6I(W) and     1 I(W) I(U)     6sup z jW(z) U(z)j bd 2 (z;z 0 ) (3)

for all U and W satisfying[HU1-2℄. Re all that b is the onstant of [HU2℄. The value I(U)

an be thought of asthea tionofan orbithomo lini to z

0

fortheLagrangian systemU on

TM, although we an only prove that there is a homo lini of a tion below I(U). We are

and

U(z)6G(z;q;v)6W(z)

with U and W satisfying [HU1,2℄. There isa radius

r6 r

I(W) I(U)

2a!

(4)

su h that the periodi orbit O

r

has a homo lini orbit X

1 = (

1 ;q

1

). This orbit moreover

satises Z R G(X 1 ) 1 2 q 1 G q (X 1 ) 1 2 _ q 1 G v (X 1 )6I(W): (5)

In the expression above, X is the lifting of X, see Se tion 2. This paper is organized as

follows. First we omment thetheorem, and give some appli ationsin thenext subse tions.

InSe tion2,we re allsomegeneralfa ts aboutHamiltonianandLagrangiansystems. These

fa tswillbeusedthroughoutthepaper. Thedetailedanalysisofthelo albehaviorofthe ow

inSe tion6maybeofindependentinterest, whilese tion5providesa on isea ount about

the existen e of homo lini orbits in Lagrangian Systems of the kind U on TM, and gives

the pre ise denition of the number I(U). The proof of Theorem 1 is explained in Se tion

3, and detailed in the last se tions of the paper. We show in Se tion 4 how to hange the

Lagrangianfun tionat innityinorder to be redu edto aLagrangian satisfying[HG4℄.

Remarks:

1. A very similar result is obtained in [3℄. Beyond the fa t that the setting is dierent,

the maininterestof thisresult is that we obtainan expli itestimate ofthe maximum

radius (4), whi h, ombined with (3), allows in ertain instan es to prove that the

homo lini wendisa tually losetothesaddle- enter. Thisenablesnewappli ations.

The estimate(5)isalsonew,we haveto relaxitin[3 ℄ tolo alizethehomo lini orbits.

Ourbeliefisthatthehomo lini weobtainisthe losesttothexedpointamongthose

whi h satisfy(5). The pri efor these estimates is that we obtainonly one homo lini

orbit, whileinnitelymanyare found in [3℄. It shouldbe possible, although noteasy,

to arry over theresultsofthispaperto thesetting of[3℄, andtheresultsof [3 ℄ to this

setting.

2. Asa onsequen eof thehypotheses[HR1,2℄,theorbitO

r

ishyperboli withrespe tto

its energy shelland thexed point (

0

;0) is of saddle enter type, with 2nhyperboli

dimensionsand 2 ellipti dimensionsinphase spa e. Thisis proved inSe tion6.

3. The hypothesis [HG3℄ an alsobe written

L(z;q;v)> 1 2  q L q (z;q;v)+v L v (z;q;v)  + d(z;z 0 ) 2 ;

or intheHamiltonianform

H(;q;p)+ d 2 (H  ;z 0 )6 1 2  q H q +p H p  +h;H  i; where H 

2TM is thederivative ofH withrespe tto theberto T 

E(O r )=a! 2 r 2 ;

and theenergy ofthehomo lini obtainedfrom Theorem1 satises

06E(X 1 )6E 0 = ! 2 I(W) I(U)
: 5. Theintegral R R

L(X)isnotdenedforahomo lini orbitbe auseithasanos illating

tail. Thisis linked to thefa tthat thea tion

Z t 0 L(O r (s))ds

is notidenti allyzero. We an neverthelessintegratebypartstheexpression

Z _ q 2 +! 2 q 2 =[qq℄_ Z q(q ! 2 q);

and usingtheEuler-Lagrangeequation

2a(q+! 2 q)= G q d dt  G v  ;

and a se ond integration bypartswe obtain

L(X 1 )=[q 1 _ q 1 ℄+ Z R G(X 1 ) 1 2 q 1 G q (X 1 ) 1 2 _ q 1 G v (X 1 );

thusthe integral an bethought asthea tionof thehomo lini orbit.

6. Although thesettingis Lagrangian,thetheoryof Bolotin[6 ℄ an notbeappliedto our

problem. Here the whole enter manifoldenjoys a minimizingpropertyas used in[6℄,

but the orbits O

r

themselves do not. This is onne ted with the fa t that the enter

manifoldishyperboli withrespe tto thefullphase spa e,whiletheperiodi orbitO

r

is not. For thatreason,we shallrathersear horbits homo lini to the enter manifold

as a whole, and this is whywe do not know pre isely whi h of the periodi orbits O

r

have a homo lini .

7. It should be possible to extend Theorem 1 to more general Hamiltonian systems by

usingtheanalysis of [19 ℄ orpseudo-holomorphi urvesasin[9 ℄ and [10℄,[11℄.

1.1 Normalizationofthe entermanifoldandpersisten eofthe hypotheses

ThehypothesesofTheorem1mayappeartobeveryrigid. Theyimplyforexamplethatthere

isaninvariantplanewithellipti linearmotiononit. Weseeinthisse tionageneralmethod

for normalizing entermanifolds i.e. bringingthem to a linearellipti plane. Thisrequires a

hangeof oordinatesand areparametrisation. Theseoperationspreservehomo lini orbits.

This method an be applied to prove that the homo lini obtained by Theorem 1 is not

destroyed by aC 3

E 0 > ! 2 I(W) I(U)

be a xed energy and let K be a ompa t set of TMR 2

ontaining fE 6E

0

+1g. There

is a >0 su h that any perturbed Lagrangian L

 satisfying kL  Lk C 3 (K) 6 has a

saddle- enter xed point p() and a enter manifold C() interse ting ea h energy shell fE = eg,

E



(p())<e6E

0

transversally along a losed integral urve of the asso iated ve tor eld Y

 .

Ea h of these periodi orbits

C()\fE=eg; E



(p())6e6E

0

ismoreover hyperboli withrespe ttoitsenergyshell,andoneofthemhasa homo lini orbit.

Let us start with some general omments before we prove Theorem 2. We all a

non-degenerate xed point p of a Hamiltonian ve tor eld on a 2n+2-dimensional symple ti

manifolda saddle- enter if the linearized ve tor eld at p has one pair of purely imaginary

eigenvalues i! and if the 2n other eigenvalues have nonzero real part. By a theorem of

Lyapunov, thereexistsa uniquelo al enter manifold,whi his aninvariant twodimensional

symple ti manifold. There are symple ti oordinates (x

i ;y

i )

06i6n

around p su h that the

lo al enter manifoldis aneighborhoodof the origininthe plane (x

0 ;y

0

). Theindu ed ow

on thistwo-dimensionalplane is integrable, and we an hoose the oordinates(x

0 ;y

0 ) su h

that the indu edHamiltonian is H(x

0 ;y 0 ;0;::: ;0) = f(x 2 0 +y 2 0

), with a smoothfun tion

f su h that f 0

(0) = ! > 0. There is an in reasing fun tion g : ( 1;h

0 ℄ ! R su h that g= f on an interval [0;h 0 ℄, with some h 0

> 0. The Hamiltonianfun tion ~ H = g 1 (H=2) is dened on fH 6 2f(h 0

)g. The point p is a saddle- enter xed point of ~

H, its enter

manifold is the plane (x

0 ;y

0

) in lo al harts, and ~ H(x 0 ;y 0 ;0;:::;0) = x 2 0 +y 2 0
=2 when x 2 0 +y 2 0 6h 0 . Wesaythat ~

H hasanormalized entermanifold. The importantpoint isthat

there is a homo lini for H if there is a homo lini for ~

H. Su h ahomo lini may befound

underadditional hypotheses byapplyingTheorem 1 to a Hamiltoniansystem extending ~

H.

This an bedoneforexamplewhenH isaperturbationofasystemsatisfyingthehypotheses

ofTheorem1. Let usnow fo us ourattention on thissituation.

The entermanifoldis globallypreserved by aperturbation. To make thispre ise, let us

onsideraLagrangian given by(1), satisfying[HL℄and [HG1,2℄, anda oneparameter family

ofLagrangiansL  satisfying kL  Lk C 3 6

andsu hthatL



L=0 outsidesome xed ompa tsubsetK of TMR 2 . Theasso iated Hamiltonian fun tion H  satises kH  Hk C 3 6 o  (1) and H  H = 0 outside K. The

periodi orbitslling(

0

;0)R 2

fortheunperturbedsystemLarehyperboli ,thisisproved

inSe tion 6. The persisten e of theinvariant manifold an be seen as a parti ularlysimple

ase of the theory of normally hyperboli manifolds in the sense of [18 ℄ or [17 ℄, or proved

dire tlysin ethepersisten e of agiven periodi orbit an beredu edto the persisten e of a

hyperboli xedpointaftertakingase tionandrestri tingtotheenergyshell. Theperturbed

manifoldis smooth, and an be redressed by a global symple tomorphism. This is arried

outindetailsin[4℄,whereweprovethefollowing: Thereis afamilyof ompa tly supported

symple ti dieomorphisms   with k  idk C 2 = o  (1) and a family f  of fun tions with kf  idk C 2 =o 

(1)su h thatthemanifold

 ( 0 ;0)R 2
is invariantforH  ,and f  ÆH  Æ  ( 0 ;0;q;p)=H 0 ( 0 ;0;q;p):

We dene the normalized HamiltonianH  =f  ÆH  Æ 

,the asso iatedLagrangian L

 an

bewrittenas(1)witha fun tion ~

G



satisfying[HG1℄,wehavegloballynormalizedthe enter

manifold. Letusnow ompute

d 2 ~ H  (x)
(u;u)=f 0  H  Æ  (x)
d 2 H  (  (x)) d  (x)
u;d  (x)
u
+f 0  H  Æ  (x)
dH  (  (x))Æd 2   (x)
(u;u) +f 00  H  Æ  (x)
dH  (  (x))Æd  (x)
u
2 : We obtainthat ~ H  H C 2 ! !0 0;

whi himpliesthat

~ L  L C 2 ! !0 0; (6)

and wealso easilysee that

~

L



L=0 outsideK : (7)

We usethisnormalized formto proveTheorem2.

Proofof Theorem2: Therststepisto repla etheLagrangianL

 byanewLagrangian, stillnoted L  ,whi h satises kL  Lk C 3 6C K 

and su h that L



=L outside K. This an be donewith a onstant C

K

dependingonly on

K. Whenissmallenough, theasso iatedve tor eldhasaglobal enter manifold. Wenow

onsiderthenormalized Lagrangian

~ L  =a v 2 ! 2 q 2
+ ~ G  (z;q;v) (z;q;v)2TMR R;

as dened above. Let us stress that L



has an orbit homo lini to a periodi traje tory

C()\fE =egforsomee2[E

 (p());E 0 ℄if ~ L 

hasanorbithomo lini toaperiodi traje tory

O

r

for some r satisfying a!r 2

6E

0

. We apply Theorem 1 to nd su h a homo lini orbit.

Thereremainsto he kthatthehypothesesofTheorem1aresatisedby ~ L  . TheLagrangian ~ L 

hasbeen onstru ted toobtain[HG1℄. The hypothesis[HL℄is adire t onsequen e of(6)

and (7) when  is small enough. It is notharder to see that [HG2℄ holds with the onstant

b=2 instead of b for suÆ iently small . We also obtain from (6) and (7) the existen e of a

fun tion ()>0with lim

!0 ()=0 su h that   L  L   6 ()d 2 (z;z 0 ) and       q L q (z;q;v)+v L v (z;q;v)  q  ~ L  q (z;q;v)+v  ~ L  v (z;q;v) !     6 ()d 2 (z;z 0 ):

The hypothesis [HG3℄ is thus satised with the onstant b=2 when  is small enough. We

moreoverhave theinequality

U  =U ()d 2 (z;z 0 )6 ~ G  6W + ()d 2 (z;z 0 )6W  :

  I(W  ) I(U  )   ! !0   I(W) I(U)   :

It is possibleto applyTheorem 1 to ~

L



when  is smallenough, and geta homo lini orbit

to O r with a!r 2 6 ! 2 I(W  ) I(U  )
6E 0 : 

1.2 Perturbation from produ t systems

Letusrst onsiderthe asewhere Gdoesnotdependon (q;v). We an setU =G=W in

thenotations ofTheorem1, and

Q(q;v)=a(v 2 ! 2 q 2 ); (q;v)2R 2 :

The system L is the un oupled produ t between the linear os illating Lagrangian system

Q on R and the Lagrangian system U on M. It is well known that if [HU1,2℄ hold the

LagrangiansystemU hasanorbith(t)homo lini to

0

,seeSe tion5. This anbere overed

from Theorem1. The hypothesis [HG3℄ always holds inthis ase, and Theorem 1 givesthe

existen eof an orbithomo lini to (

0

;0) forL,whi hof ourse impliesthe existen eof the

homo lini ofU. Alltheorbits O

r

have a homo lini forL inthis ase, given by

h

r

(t)=(h(t);r os (!t)):

Letusnow ome ba ktothegeneral aseofafun tionG(z;q;v). Thetheorem1 anbeseen

as a perturbation result when a oupling is introdu ed in a produ t as above. Elementary

dimension onsiderationsshowthatthesaddle- enterxedpointdo nothave anyhomo lini

orbitinageneri oupledsystem. Thetheorem1yetgivestheexisten eofanorbithomo lini

tosomeperiodi orbitO

r

if[HG3℄holds. Inviewoftheestimate(3)thequantityI(W) I(U)

isameasureofthe oupling,andweobtainthattheradiusr tendstozerowhen the oupling

tends to 0. The orbit obtained by Theorem 1 an be onsidered asthe ontinuation of the

orbit homo lini to the xed point that existed in the un oupled system. Moreover, the

hypothesis[HG3℄ issatisedwhen the oupling issmallsin eit an be written

1 2  q C q +v C v  C6U d 2 (z;z 0 )

ifwe separate themain part U and the oupling perturbation C of R : G(z;q;v) =U(z)+

C(z;q;v),and it issatisedforexamplewhen

jCj+     q C q     +     v C v     6d 2 (z;z 0 )

with a suÆ iently small . Shortly, The homo lini orbit to the xed point that existed in

theun oupledsystemisturnedto anorbithomo lini toO

r

whenthe ouplingisintrodu ed,

with r as smallas the oupling is small, and thishomo lini exists as long as[HG3℄ holds.

 su h that L 0 (z;q;v)=a v 2 ! 2 q 2
+U(z); (z;q;v)2TMRR

satises[HL℄and[HU1,2℄. Thereisan

0

>0andafun tione()>0satisfyinglim

!0 e()=

0su hthat for6

0

the systemL



hasa saddle- enterxedpoint p()anda entermanifold

C()interse tingtransversallythe energylevelE 1

 e()

along ahyperboli periodi traje tory

whi h has a homo lini orbit. 

1.3 Singular perturbation

The ase ! !1 isof physi alinterest. Letus onsidera system

L ! (z;q;v)=a(v 2 ! 2 q 2 )+G(z;q); (z;q;v)2TMR R; andset G 0 (z)=G(z;0); 2M:

When! is large,theterma! 2

q 2

inL an be seenasa potential onningthesystem onthe

subspa eMf0gofthetotal ongurationspa eMR. Taking! !1approximatesthe

aseof a holonomi binding,see[1 ℄, hapter 4. At thelimit,the ongurationof thesystem

is for ed to stay in M =M f0g, and its evolution is des ribed by the Lagrangian ow of

G

0

on TM. Let ussupposethat there is a riti al point z

0 =( 0 ;0) 2TM of G 0 su h that G 0 (z 0 )=0and

HG2 : Thereis a b>0 su h that G

0 (z)>bd 2 (z;z 0 ): The point z 0 = ( 0

;0) is then a hyperboli rest point of the limit ow (the ow of G

0 ) and

thereisan orbitofG

0

homo lini to thisxed point,seeSe tion5. It isinteresting tostudy

thelimitpro essand des ribewhatremainsofthishomo lini orbitinthetotal owforlarge

butnite!. We willfurthermore assumethat Gsatises

HG1 lo : Thereis isan >0 su h thatG(z

0

;q)=0and dG(z

0

;q)=0 whenjqj6.

Example : Let us onsidera pendulum,in theplane orinspa e, wherethe baris repla ed

by a stispring whi h has variable lengthbutremains always straight, see gure 1, page2.

TheLagrangian of thissystem an be written

L(; _ ;q;q)_ =q_ 2 ! 2 q 2 +(l 0 +q) 2 _  2 +(l 0 +q)( os'() 1) where  2 S 2

is the dire tion of the spring, '() is the angle between the spring and the

verti al axis pointing up, and l

0

+q is the length of the spring, l

0

being its length in the

unstable equilibrium position. Let us all 

0

the verti al dire tion pointing up, that is the

dire tionoftheunstableequilibrium. Itisnothardto he kthatbothhypothesesabovehold

forthatsystem. Thereisanunstableinvariantmanifold(; _

)=(

0

;0)lledwithos illations

of the spring. In view of the appli ation below, one of these os illations have a homo lini

orbitifthespringisstienough. Thewholestru ture, entermanifoldandhomo lini orbit,

ispreservedbyasmallperturbation. Thehomo lini ofthestielasti pendulum anbeseen

as the ontinuation of the homo lini that exists in the rigid pendulum,whi h is the limit

systemwhen thestiness tendstoinnity. Notethattheenergyofthehomo lini orbitdoes

nottendto zero ingeneral when the stinesstends to innity(orat leastwe an notprove

thatitdoes) althoughthelengthof thespringis onverging to l

0

. The homo lini has small

Appli ation 2 The point (z

0

;0;0) is a saddle- enter xed point of L . It has a enter

manifold z=z

0

, whi h is lled with the periodi orbits O ! r =fv 2 +! 2 q 2 =!g. There isan energy E 1 >0 su h that

 When ! islargeenoughthereisan orbit h ! = z ! ;q ! ;q_ !
of L ! homo lini toO ! r with r 6 1 ! r E 1 a ;  The orbits h !

onverge to M in onguration spa e:

kq ! k 1 ! !!1 0;  The fun tion! ! R d 2 (z ! ;z 0 ) isbounded;

 For any sequen e!

n

!1, there isa subsequen e p

n

, a nite number m of orbits Z i of L 0 homo lini toz 0 and m sequen es t i p

su h that lim

p!1 t i+1 p t i p
=1 and z ! p (t t i p ) C 1 lo ! p!1 Z i (t):

 If ! is large enough and xed, there is an  > 0 su h that any Lagrangian system ~ L satisfying k ~ L L ! k C

3 6 also has a saddle- enter xed point with a enter manifold

and an orbit ~

h homo lini to this entermanifold and su h that ~ E( ~ h)6E 1 . Remarks :

1. The limit onguration spa e M = M f0g is not invariant for L !

hen e the xed

point (z

0

;0;0) does not have any homo lini orbit in general (its stable and unstable

manifoldhave dimensionn ina2n+1-dimensionalenergyshell).

2. The energy E !

(h !

) is bounded,but doesnot onverge to zero, or at least we an not

prove that it does. It shouldbe interesting to understand whether this is only a side

ee t duetoour approa h, orwhether ithasa physi almeaning.

3. It shouldbepossible,whenM isnotsimply onne ted,to prove thatthez !

isa tually

onverging to asingle homo lini of L

0 .

4. The hypothesisHG1 lo is anunpleasantrestri tion,assumedinorder thatTheorem1

an bereadilyapplied. Itisnothardto seehoweverthatevenwithoutthisassumption

a saddle- enter existsinL !

forlarge!,and itmay be possibleusingthe te hniquesof

Se tion 1.1to prove that thephenomenondes ribed intheappli ation stillo urs.

5. Addingmore thanone degree of freedommakesthingsmu h harder. Even intheideal

ase where a enter manifold foliated by quasi-periodi tori would exist, there would

remain the problemthat the interse tion between the enter manifoldand an energy

shell would ontain familiesof su h quasi-periodi tori, in ontrast with our situation

where ea h periodi orbit is the interse tion between its energy shell and the enter

manifold. Moreover,thisideal aseisnotasrigidasour ase,sin esomeoftheinvariant

G 0 (; _ )=k _ k 2 +(1 os); 2S 1 :

It is wellknown thatintegrability an be destroyed and haoti behaviorturnedon by

a time-dependentsmallperturbation. Letus onsiderasystem

L ! (; _ ;q;q)_ =q_ 2 ! 2 q 2 +G(; _ ;q); (;q)2S 1 R withG(; _ ;0) =G 0 (; _

),satisfyingthehypothesesoftheappli ation. Thehomo lini

orbitobtainedbytheappli ation anbemadetransversalbyasmallperturbationofG.

Thisisanewway,alsophysi allyrelevant,tointrodu e haoti behaviorinthe lassi al

pendulum.

Proof : We areinterested in traje tories lo ated around q =0, and it is rst ne essaryto

hangetheLagrangianfun tionoutsideaneighborhoodofq=0. We needasmoothfun tion

':[0;1℄ ! [0;1℄ su h that 'j [0;1℄ = 1 and 'j [2;1) = 0 and 0 >' 0 > 2. Let us xÆ >0 anddene G Æ (;q)='(q=Æ)G(;q)+ 1 '(q=Æ)
G 0 (): It is lear that G Æ

satises HG1 when Æ is smallenough. To he k the other hypotheses of

Theorem1let usrst noti ethatthere is a onstant D>0 su h that

jG G 0 j6DÆd 2 (z;z 0 ) and     G q     6Dd 2 (z;z 0 )

when jqj62Æ. Itfollows fromtherst estimateabovethat G

Æ (;q)>bd 2 (z;z 0 )=2 when Æ is

smallenough. Inview of the al ulation

G Æ 1 2 q G Æ q ='(q=Æ)G+(1 '(q=Æ))G 0 + 1 2 q  '(q=Æ) G q +(G G 0 )' 0 (q=Æ)=Æ  >G 0 '(q=Æ) ' 0 (q=Æ)
jG G 0 j Æ'(q=Æ)     G q     >G 0 4ÆDd 2 (z;z 0 );

thehypothesesHG2andHG3arebothsatisedwiththe onstantb=3whenÆissmallenough.

We also obtainthat

U Æ =G 0 (z) DÆd 2 (z;z 0 )6G Æ 6G 0 (z)+DÆd 2 (z;z 0 )=W Æ ; and (3) yields I(W Æ ) I(U Æ )63DÆ=b:

We are now in a position to apply Theorem 1 to L ! Æ = a v 2 ! 2 q 2
+G Æ (z;q;v), and obtainan orbith ! Æ =( z ! Æ ;q ! Æ ;q_ ! Æ )homo lini to O r with r6B p Æ=!,where B is a onstant

that depends neither on ! nor on Æ. The energy fun tion E ! Æ asso iated with L Æ is E ! Æ = a v 2 +! 2 q 2
+E 0 (z);whereE 0

,theenergyasso iatedtoG

0

,isboundedfrombelow. Writing

energy onservation along h ! Æ yields ! 2 j q ! Æ j 2 6B 2 Æ!+C :

The homo lini h Æ is thusan orbitofL if ! 2 Æ 2 >B 2 Æ!+C :

Letusnow hooseÆ==!,where isasuÆ ientlylargexednumber,theinequalityabove

is satised and the homo lini h ! = h ! =! = (z ! ;q ! ;q_ ! ) is a traje tory of L ! , of bounded energy E ! (h ! )6E 1 =aB 2 ,and satisfying Z R G =! (h ! ) 1 2 q ! G =! q (h ! )6I(W =! ):

In view ofHG2, thisestimateyields

Z d 2 (z ! ;z 0 )6C : The fun tionG q

(z)=G(z;q) is a berwise onvex Lagrangian on TM for all q. Let us all

Y

q

the asso iatedve toreld. The urves z !

satisfytheEuler-Lagrangeequation

_ z ! (t)=Y q ! (t) (z ! (t)) hen e z ! is boundedin C 1 (R;TM). Moreover, sin e kq !

k ! 0, any limit urve z 1

of z !

satisestheEuler-Lagrangeequation

_ z 1 (t)=Y 0 (z 1 (t)); whereY 0

istheEuler-Lagrangeve toreldofG

0

. Itisnothardtoseethatthereisa onstant

C >0independentof!su hthatallorbitZ ofG

0 homo lini toz 0 ,satises R d 2 (Z ;z 0 )>C ;

and all orbit X = (Z ;Q; _

Q ) of L !

homo lini to the enter manifold z = z

0 and lying in E ! 6 E 1 satises kd(Z ;z 0 )k 1

> C : One now applies the on entration ompa tness

prin iple, see [27 ℄, 4.3, to the fun tion d 2

(z !

(t);z

0

) in order to prove the last point of the

theorem, see [13 ℄ for the use of on entration ompa tness with homo lini orbits. In our

situation, vanishing is impossible sin e kd 2 (z ! ;z 0 )k 1 > C 2

; while only a nite number of

bumps an appear sin eea h bump satises R d 2 (Z i ;z 0

) >C : To nish,the persisten e is a

dire t onsequen eof Theorem2. 

2 Lagrangian and Hamiltonian systems

We re all in thisse tion some standard fa ts aboundLagrangian and Hamiltoniansystems.

This is an opportunity to introdu e some notations and to state a simple estimate of the

energyfun tionthatwillbe usedthroughoutthe paper.

Let N be a manifold, TN 

! N the tangent bundle and T 

N 



! N the otangent

bundle. The liftingx of a urve x:R !N isthe urve

x:R !TN

t7 !dx(t;1):

We onsidera smoothLagrangian fun tionL :TN !R, that is uniformly onvex on ea h

ber. TheberderivativeL

v

of Lis welldened,and theappli ation

:TN !T  N v7 !L v (v)

E:TN !R

v7 !h(v);vi L(v)

and a Hamiltonianfun tion

H:T  N !R p7 !EÆ 1 (p)=hv; 1 (v)i L( 1 (v)):

There isa anoni alsymple ti stru tureon T 

M,and we asso iateto H itsHamiltonian

ve toreldXdenedbytheequationi

X

= dH. Letusdenethea tionL(x)ofasmooth

urve x:[t 0 ;t 1 ℄ !N L(x)= Z t 1 t0 L(x(t))dt;

we say that x is a traje tory of L if it is a riti al point of the a tionwith respe t to xed

endpoints variations. A urve x :R !N is a traje tory of L ifand only ifits restri tions

to nite time intervals are traje tories of L. We will pay spe ial attention to the periodi

traje toriesof L. LetT >0 beaxed period,a T-periodi urve x:R !N is atraje tory

ofL ifand onlyif theloop x

j[0;T℄ isa riti alpoint of L T (x)= Z T 0 L(x) on C 1 T = fx 2 C 1

([0;T℄;M)=x(0) = x(T)g: There is a one to one orresponden e between

traje tories xof L andintegral urvesz of X,given by

x !z=(x); z !x= 

(z):

Asa onsequen e, thereisave tor-eldY onTM su hthatxisatraje toryofLifand only

ifx isan integral urveof Y,and we have

Y(z)=(d

z )

1

(X((z)):

Inany anoni al hart(q;v)ofTM,thetraje toriesofLsatisfytheEuler-Lagrangeequations

d dt L v (q(t);q(t))_ = L q (q(t);q(t)):_

The Hamiltonian fun tion H is invariant along integral urves of X, and the energy E is

invariant along integral urvesof Y hen e E(x) is onstant if x is a traje tory of L. This

onstru tion anbereversed. LetH:T 

N !R beaHamiltonianfun tion. Ifthemapping

:TN !TN

z7 !H

v (z)

isa dieomorphism,whi hhappenswhen H is berwise onvexand proper,we dene

L(z)=(z; 1

(z)) H( 1

(z));

theasso iated mapping isthe dieomorphism= 1

;and the orresponden e des ribed

abovebetweenorbitsofLand integral urvesof Hholds. Letusnow ome ba kto ourmain

subje tofinterestandestimatetheenergyEasso iatedto(1). Weassume[HG1-4℄and[HL℄.

onstant C>0 su h that   E(z;q;v) a v 2 +! 2 q 2
   6Cd 2 (z;z 0 ) (8) for all (z;q;v) 2TMR R.

Proof : Let us onsider the energy E

1

on TM asso iated with the Lagrangian G

1 as

denedin[HG4℄. It an be omputed inlo al oordinatesthat

E(z;q;v)=a(v 2 +! 2 q 2 )+E 1 (z)

whenz62B,and that

E

1

(z)=kzk 2

outsideK. Re allthat K and B aredenedin[HG4℄. Itfollows that thefun tion

jE 1 (z)j=d 2 (z;z 0 )

isboundedat innity,and it an be he ked from [HG1℄ (usinglo al expression(10) below)

thatitis bounded aroundz

0

,and thusbounded. It followsthat thefun tion

jE(z;q;v) a v 2 +! 2 q 2
j=d 2 (z;z 0 )

is bounded outside B. It also follows from [HG1℄ and the lo al expression (10) below that

thisfun tionis boundedin a neighborhood of B\fz =z

0

g, and thus bounded everywhere.



3 Sket h of proof of Theorem 1

We obtain the homo lini orbit as limit set of a sequen e of periodi orbits obtained by a

variational method. We rst have to solve a te hni aldiÆ ulty. The statement of Theorem

1involvesno growth onditionswhilesu h onditionsareneeded to deneappropriate

fun -tionals. These onditions anbearti iallyobtainedby hangingtheHamiltonianatinnity,

sin ethe behavior we are des ribingis lo alized ina ompa tzone E 6E

0

,where E is the

(proper) energyfun tionand

E 0 = ! 2 I(W) I(U)
:

Proposition 1 If the on lusions of Theorem 1 hold for any Lagrangian fun tionsatisfying

all the hypotheses of Theorem 1 and the additional hypothesis [HG4℄, then Theorem 1 holds.

This proposition is proved in Se tion 4 by hanging the Lagrangian fun tion at innity.

It is thus suÆ ient to prove Theorem 1 for Lagrangian fun tions satisfying the additional

Hypothesis [HG4℄. We willuseperiodi orbits of

L l (z;q;v)=a v 2 l 2 ! 2 q 2
+G(z;q;v); (z;q;v)2TMRR

obtainedas riti al pointsoftheLagrange a tionfun tional

L l (X)= Z T 0 L l (X)

l

of L

l

. We will see in Se tion 8, that it is possible to nd a riti al value

T

(l) of L

l for all

T =2=!, 2N, and all l2(1;1+1=):This riti alvalue satises

I T (U)6 T (l)6I T (W); the numbers I T

(U) are dened in Se tion 5 together with the numbers I(U). Sin e the

fun tionl !

T

(l)is non-in reasing,thereisanl(T)2(1;1+1=)su hthat 0 T (l(T))exists andj 0 T (l(T))j6(I T (W) I T

(U)):Weusethistonda riti alpointX

T =( T ;q T )ofL l (T) atlevel T (l(T))su h that 2l(T)a! 2 kq T k 2 2 =     L l (X T ) l     l (T) 61+j 0 T (l(T))j6(I T (W) I T (U))+1:

The periodi orbit we obtain is not triviali.e. 

T 6

0

, be ause it has nonzero a tion. All

thisis detailedinSe tion8,wherewe prove

Proposition 2 For all  2N and T = 2=!, there exist a parameter l(T) in the interval

(0;1=) and a traje tory X T =( T ;q T ) of L l (T) su h that 1 T kq T k 2 2 6 1 4a!  I T (W) I T (U)+ 1   ; L l (T) (X T ) = T (l(T));  T 6  0 :

We now use the periodi orbits given by Proposition2 to build the homo lini orbit. Sin e

I(U)=liminf

T!1 I

T

(U), seeSe tion5, we an extra t asubsequen e T

n of T su h that 1 T n kq T n k 2 2 !r 2 =2 with aradius r6 r I(W) I(U) 2a!
We obtainasequen e X n ofT n -periodi orbits of L l (T n ) whi h satises  l(T n ) !0  T n !1,  L l (T n ) (X n )6I(W),  kq n k 2 =T n !r 2 =2,   n 6 0 ,

and we prove inSe tion 7that a homo lini orbit an be foundasan a umulation point of

All the systems onsidered in thispaper are autonomous, and preserve an energy fun tion.

It isthuspossibleto hangetheLagrangianfun tionatinnitywithout hangingthe owon

pres ribed ompa tenergyshells. Thisobservationisthebasisof theproofofProposition1.

The detailsare notsimple sin e berwise onvexity has to be preserved duringthe pro ess.

We now prove Proposition1.

Letus onsidera Lagrangianfun tionL=a(v 2

! 2

q 2

)+Gsatisfyingallthehypotheses

of Theorem1. We builda new Lagrangian fun tion that is equalto theold one on E 6E

0

andthat satisesall thehypothesis ofTheorem1 and [HG4℄. Re allthat

E 0 = ! 2 I(W) I(U)
:

First step : LetK

0

bea ompa t subsetof TM,we rst builda fun tionG

1 on TMR 2 su hthat G 1 (z;q;v)=G(z;q;v) when z2K 0 ; G 1 (z;q;v)=kzk 2 when z62K

forsome >0and some ompa t setKTM,andG

1 +av

2

isberwise onvex. Letusset

d=sup

z2K

0

kzk. Thereexistsa d

1

>d su hthatforall >0 thereexistsa onvexfun tion

f  :[0;1) ! [0;1) satisfying f  (x) =0 when x 6d and f  (x) =x 2 when x> d 1 . We

now onsiderafun tion':R +

![0;1℄su hthat'(x)=1whenx6d

1 and'(x)=0when x>d 2 forsome d 2 >d 1 and we set G 1 (z;q;v)='(kzk)G(z;q;v)+f  (kzk 2 ):

Itiseasy to seethatthefun tionG

1 +av

2

isberwise onvex ex eptmaybe onthe ompa t

setd 1 6kzk6d 2 :Toprove thatG 1 +av 2

isalso berwise onvex on d

1

6kzk6d

2

we will

usea lo al anoni al hart z=(;) of TM and prove that thefun tionG

1 +av 2 is onvex in( ;v) on d 1 6kzk6d 2 :We rst notethat d 2 v (G 1 +av 2 )='(kzk)d 2 v G+2a is positive sin e d 2 v

G+2a is positive. On the other hand, it is not hard to see that given

 >0 one an take  large enough so that d 2  G 1 > Id on d 1 6 kzk 6 d 2

: Sin e the ross

derivativesd  d v (G 1 +av 2

) donotdependon ,one an he kthat theHessian

 d 2  (G 1 +av 2 ) d v d  (G 1 +av 2 ) d  d v (G 1 +av 2 ) d 2 v (G 1 +av 2 ) 

ispositive denite when  is largeenough. The fun tion G

1 +av

2

is thus berwise onvex,

aswell asthe fun tionL

1 =a(v 2 ! 2 q 2 )+G 1 . The fun tion G 1

satises [HG1-3℄ withthe

same onstantb. Let usdenethefun tions

U 1 (z)='(kzk)U(z)+f  (kzk 2 ); (9) andW 1

inthe same way,sothat

U 1 6G 1 6W 1 :

1 1 U 1 (z)=U(z) and W 1 (z)=W(z) whenz2K 0 and U 1 (z)=W 1 (z)=kzk 2 when z62K : IfK 0

issuÆ ientlylarge,thedenitionofI isI(U)=I(U

1

)andI(W)=I(W

1

)(seeDenition

2 of Se tion5). So themaximalenergy E

0

hasnotbeen hanged.

se ondstep: Wenowwantto ontrolthebehaviorforlargev. LetB

0

bea ompa tsubset

of TM R 2

, we will dene a fun tion ~ G su h that ~ G = G 1 on B 0 and ~ G = U 1 outside a ompa t subsetB B 0

. To do thiswe rst observe that U

1 G

1

is bounded,whi h easily

impliesthatd v G 1 isboundedsin ed 2 v G 1

> 2a. Wenowtakea ompa tlysupportedfun tion

:R 2

! [0;1℄ su h that (q;v) = 1 when there existsa z2 TM with(z;q;v) 2B

0 , and su hthat kd k C 1 6and qd q +vd v 60. Thefun tion ~ G(z;q;v)= (q;v)G 1 (z;q;v)+(1 (q;v))U 1 (z); satises[HG4℄ withG 1 =U 1

. Withthe notationz=(;),wederive

d 2  ~ G= d 2  G 1 +(1 )d 2  U 1 d v d  ~ G= d v d  G 1 +d v (d  G 1 d  U 1 ) d 2 v ~ G= d 2 v G 1 +d 2 v (G 1 U 1 )+2d v d v G 1 ; and get d 2 (v;) ( ~ G+av 2 ) d 2 (v;) (G 1 +av 2 ) (1 )d 2 (v;) (U 1 +av 2 ) 1 ! !0 0;

whi h impliesthat ~

G is berwise onvex when is small enough. The fun tion ~

G moreover

satises [HG1-2℄withthe same onstant band [HG3℄ follows from:

~ G 1 2 qd q ~ G+vd v ~ G
=  G 1 1 2 qd q G 1 +vd v G 1
 +(1 )U 1 + 1 2 (U 1 G 1 ) qd q +vd v
> bd 2 (z;z 0 )+(1 )bd 2 (z;z 0 )=bd 2 (z;z 0 ): The fun tion ~ L=a(v 2 ! 2 q 2 )+ ~

G satises all thehypotheses of Theorem1 with the same

onstantsa, band!,and withU and W repla edbyU

1

andW

1

. LetusassumethatK

0 and

B

0

have beentaken largeenough sothat

fE6E 0 gB 0 \ K 0 R 2
:

If the on lusionsof Theorem1 holdfor ~

L,they givethe existen eof ahomo lini of energy

below E

0

, this orbit is also an orbit of L sin e the fun tion have not been hanged in this

We now denethe numberI(U) fora berwise onvex Lagrangian U :TM !R satisfying

[HU1-2℄. Thesehypothesesimplythatz

0

isahyperboli xedpointofthesystem,seeSe tion

6,andthatithasahomo lini orbit. Homo lini sforthiskindofLagrangianwererststudied

variationallybyBolotin[5 ℄,andthenbyseveralauthors(seee.g. [2℄,[25 ℄). Theseworkshave

also been extendedto more general Hamiltoniansystems in[16 ℄ and [10 ℄, [11 ℄. We just give

here a presentation of these resultsthat willbe usefulin the proof of Theorem1. We shall

rststudyLagrangians U withtheadditional hypothesis

HU3: Thereexistsan >0su h thatU(z)=kzk 2

outside a ompa t setof TM.

Althoughwe areinterestedmainly inthehomo lini orbit, we shalluse avariational setting

forT-periodi orbitsof U: Let 

T be themanifoldofH 1 -loops :S T =R= TZ!M;

thea tionfun tional

U T : T !R 7 ! Z T 0 U( (t))dt

is smooth and satises thePalais-Smale ondition. We note H the rational ohomology. A

pointed set (S;s) is a set S with a distinguishedelement s 2 S. We will use the notation

H(S;s)forthe relative ohomologyH(S;fsg). Forany losedsubsetS of

T

ontainingthe

onstant loop

0

we onsiderthe morphism

i  S :H( T ; 0 ) !H(S; 0 )

asso iatedwith thein lusion.

Denition1 We all

T

the familyof all ompa t subsets of

T

ontaining 

0

andhaving

indu ed ohomology, i.e. su h that i 

 6=0.

Thedistinguishedlevel

I T (U)= inf 2 T sup  U T satises:

Lemma2 There exists a onstant M >0 independent of T su h that 0<I

T

(U)6M:

Proof: ToprovethatI

T

(U)>0,wetakeasmalldiskD2M enteredat

0 ,andlet T (D) betheset ofH 1

loopsinD. Itisnothardtoseethat 

T (D) is ontra tible,thusi   T (D) =0 and i   = 0 forall   T (D) ontaining  0

. From this follows that all  2  must ontain

a urve leavingD. Su h a urve has its a tion bounded away from 0. To prove the se ond

inequality,letusintrodu etheset ofloopsstartingat 

0  0 T =f(t)2 T =(0)= 0 g: We needthe

Lemma3 There exists a ompa t subset K 0 1 su h that i  K 6=0.

simply onne ted, there is a non ontra tible urve 2  0 1 . We take K =f ; 0 g, and see that i  K (H 0 ( 1 ; 0

))6=0. Things are mu h harderwhen M is simply onne ted. Let us set

C =C 0 (S 1 ;M)andC 0 =f 2C 0 (S 1 ;M)= (0)= 0 ),thein lusioni  0 :( 0 ; 0 ) !(; 0 )

is homotopy equivalentto the in lusioni :(C 0

;

0

) ! (C ;

0

). Atheorem of Sullivangives

the existen e of innitely many nonzero rational Betti numbers of the spa e C 0 (S 1 ;M) if  1

(M)=0,see [28 ℄,page46. Then,we onsidertheSerrebration

C ! M 7 ! (0) of ber C 0 to prove that i  is nonzero, hen e i   0

is nonzero. We now use broken geodesi s

approximation,see [7℄,to nda ompa t K representing this ohomology. 

Proof of Lemma 2: For anyT >1,we an extendloopsin 0

1

to [0;T℄byxingthem in



0

outside[0;1℄, thisdenes theinje tion

j T :( 0 1 ; 0 ) ! ( 0 T ; 0 ) (t) 7 ! j T ((t))=(min(1;t))

whi his homotopi to thedieomorphism

( 0 1 ; 0 ) ! ( 0 T ; 0 ) (t) 7 ! S T ((t))=(t=T): Itfollows thatj T (K)2 T ,thus I T (U)6 sup j T (K) U T =sup K U 1

be ause the traje tory t 7 ! 

0

has zero a tion. This ends the proof of the lemma sin e

sup

K U

1

isa nitenumber. 

There is a T-periodi traje tory

T su h that U T ( T ) =I T

(U): We shall not prove it sin e

itis very lassi al, and involves arguments simplerthan those of Se tion8. Here non-trivial

meansthat

T 6

0

. We an denethe number

I(U)=liminf

T !1 I

T (U):

Theremustbe anontrivialhomo lini orbitto z

0 su hthat Z 1 1 U( _(t))dt 6I(U);

weobtainitasana umulationpointofthesequen e

T

of periodi orbits, ompareSe tion

7. The followingpropositionis usefulforappli ations

Proposition 3 Thefun tion U 7 !I(U) is in reasing and ontinuous:

U 6W =)I(U)6I(W)     1 I(W) I(U)     6sup z jW(z) U(z)j bd 2 (z;z 0 )

=sup z jW(z) U(z)j bd 2 ((z); 0 ) ;

we obtainusing[HU2℄ that

(1 )U 6W 6(1+)U: This yields (1 )I T (U)6I T (W)6(1+)I T (U):

We thushaveforanyT

    1 I T (W) I T (U)     6;

and weobtainthepropositionbytaking thelimit. 

Let us ome ba k to Lagrangian systems U on TM satisfying only [HU1,2℄ butnot [HU3℄.

The energy fun tionE

U

isproperand the sets E e

U =fE

U

6egare ompa t. Let E

U

bethe

set of all Lagrangians U

1

satisfying [HU1-3℄ and su h that U

1 = U on E e U for some e > 0. Elementsof E U

an be onstru tedbythemethodsof Se tion4.

Denition 2 For all Lagrangian fun tionU satisfying [HU1,2℄, we set

I(U)=I(U 1 ) for any U 1 2E U

. Theproposition 3 holds for U and W satisfying [HU1,2℄ withthis extended

denition of I.

Proof : One has to prove that the number I(U

1

) does not depend on the hoi e of the

LagrangianU

1 2E

U

. LetustaketwoLagrangiansU

1 andU 0 inE U ,deneU t =tU 1 +(1 t)U 0 , t2[0;1℄, andletE t

betheenergyfun tionasso iatedwithU

t

. Thereisanenergye>0su h

thatU

t

(z) =U(z) for all z2E e

U

and all t2[0;1℄. The LagrangiansU

t

satisfy[HU1,2℄ with

the same onstant b, and U

t =  t kzk 2 at innity hen e E t =  t kzk 2 at innity. Sin e  t ,

t2[0;1℄isboundedthereisa onstantC >0independentoftsu hthatE

t 6Cd 2 (z;z 0 ),see

Lemma1. Forall T >0 thereisaT-periodi traje tory t T ofU t su hthatU t T ( t T )=I T (U t ).

One anbuildbythemethodsofSe tion4aLagrangianU

2 2E U su hthatU 2 >max(U 0 ;U 1 ) andthusU 2 >U t

forallt2[0;1℄. It followsthat U t T ( t T )=I T (U t )6I T (U 2 )isbounded,and E t ( t T )6 C T Z d 2 ( t T ;z 0 )6 C Tb Z U t ( t T )6 C 0 T

As a onsequen e, there exists a T

0

>0 su h that all the periodi orbits  t T with T >T 0 are ontained in fE t

6 eg, whi h is nothing but E e U . The urves t T with T > T 0 are

thus all traje tories of U and of U

0

, and the value I

T (U

t

) is riti al forthe Lagrangea tion

U 0

T

asso iated with U

0

. The set of riti al values of U 0

T

has measure zero. This is a

non-trivialappli ation of Sard's Theorem, see for example [11 ℄, Lemma 3.1, for a result of this

kind. On the other hand, we see from Proposition 3 that the fun tion t 7 ! I

T (U

t ) is

ontinuous, hen e onstant sin e it takes values in a set of measure zero. We have proved

that I T (U 1 ) = I T (U 0

) when T is large enough, hen e I(U

1

) = I(U

0

), and the denition is

meaningful. We now prove that Proposition 3 holds with this extended denition. Let us

Se tion 4to builddistinguishedelementsofE U andE W . WetakeK 0 ontainingE U andE W

inits interior,and dene U

1 2E U and W 1 2E W

bythesame expression(9). It is learthat

U

1 6W

1

ifU 6W,and thatjI(U

1

) I(W

1

)j6jI(U) I(W)j. Proposition3 forU

1

and W

1

thus impliesProposition3 forU and W. 

Letusnow ome ba kto thefullsystem.

6 Lo al stru ture.

In this se tion, we fo us our attention o the vi inityof the enter manifold z = z

0

. Let us

denethe ballsD

Æ =B( 0 ;Æ) 2M and B Æ =B(z 0

;Æ)2TM. We willwork ina lo al hart

of M around  0 , that is we identify D Æ with a neighborhood of 0 in R n

. The lo al form of

theLagrangianfun tion is

L(; ;q;v)=a v 2 ! 2 q 2
+G(; ;q;v); (; ;q;v)2D Æ R n RR;

andwe an ompute theasso iatedenergyfun tion, seeSe tion2

E(; ;q;v)=a v 2 +! 2 q 2
+   ; G   +  v; G v  G: (10)

Thehypothesis [HG2℄ implies

[HG2 lo ℄ :  2 G (;) 2 (0;0;q;v) >b:

Wewillonlyusethislo alminimizingpropertyinthisse tion. Thefollowinglemmawillnot

be usedinthesequel, butLemma5 belowis thekeyto nontriviality.

Lemma4 (Des ription of the lo al orbit stru ture) Ifthehypotheses [HR1-2℄are

sat-ised, the ow has a saddle- enter xed point (0;0) 2 B

Æ R

2

with a 2-dimensional ellipti

spa e and a 2n-dimensional hyperboli spa e. The entermanifold of this saddle- enter xed

point is the invariant plane f0gR 2

B

Æ R

2

: The ow on the enter manifold is linear

ellipti , andthe enter manifoldis foliated by the traje tories

O

r

(t)=(0;r os (!t); !rsin(!t)):

Ea h of these periodi orbits is the interse tion between its energy shell and the enter

man-ifold, and is hyperboli with respe t to its energy shell (but not with respe t to the full phase

spa e). Proof : Let  : TM R 2 ! T  M R 2

be the dieomorphism dened in Se tion 2, we

have (f0gR 2

)=f0gR 2

and theHamiltonianH=EÆ 1 an be written H(;;q;p)= 1 4a p 2 +a! 2 q 2 + 1 2 (g 1  ;)+ ~ R (;;q;p) where ~ R=O(kk 2 +kk 2

). Itfollowsthattheplanef0gR 2

isinvariantfortheHamiltonian

ow, and foliated bytheperiodi orbits

~

O

r

weapply to obtaintheexpressionof theasso iatedorbitsofY. We nowprove

hyperbol-i ity. The hypersurfa e

=f(z;q;v)2TMR 2

=q >0 andv =0g=TMR +



istransversaltothe owaroundf0gR +



,andwedenetheasso iatedPoin are returnmap

. Letus xa r >0, we want to study theeigenvaluesof modulus1 of the linearizedmap

d(0;r). Note that  jf0gR +  = Id, thus d(0;r) jf0gR

= Id. It follows that for all  > 0

thereis a fullyresonant approximation of d(0;r) su hthat

k (q;z) d(0;r)(q;z)k6kzk:

Byfullyresonant,we meanthat alltheeigenvaluesofmodulus1 of arerootsoftheunity.

We an moreover take  small enough so that and d(0;r) have the same number of

eigenvalues of modulus 1. Sin e

jf0gR +



= Id there exists a neighborhood of (0;r) 2 

where

j(z;q) d(0;r)(z;q r) (0;r)j6kzk 2

:

As a onsequen e, there exists a fun tion G

1

satisfying [HG1℄ and [HG2℄ with a smaller

onstant b

1

and su h thatPoin are map 

1

of the owasso iatedto

L 1 (; ;q;v)=a v 2 ! 2 q 2
+G 1 (; ;q;v) satises 1

(z;q)=(0;r)+ (z;q r)inaneighborhoodof(0;r). Letus onsideraneigenspa e

of asso iated with a pair of eigenvalues of modulus one, whi h are therefore root of the

unity. This eigenspa e is lled with periodi points, moreover given Æ > 0 there exists a

neighborhoodof 0 intheeigenspa esu h thatall thepointsin thisneighborhood have their

-orbit ontainedinthezonewhere

1

(z;q) =(0;r)+ (z;q r),andsu hthattheperiodi

orbits of L

1

asso iated with these 

1

-orbits are ontained in B

Æ R

2

. We now apply the

lemma 5 below to L

1

and obtain that the periodi orbits we just onstru ted must be the

trivialones, orrespondingto thexed spa ef0gR of . Asa onsequen e, thelinearized

Poin are map d(0;r) an have no eigenvalue of modulus1 ex ept the one asso iated with

thisxed spa e. 

Lemma5 Let L bethe Lagrangian fun tion(1) witha fun tion Gsatisfying [HG1-2℄, there

isa two-parameters family of periodi orbits of L

O(t)=(

0

;r os(!t+));

andthereexists a Æ >0 su h that they arethe onlyperiodi orbits satisfying x2B

Æ R

2

.

Proof : We work inlo al oordinatesasdes ribed above. The traje tories lying inD

Æ R

satisfythestandardEuler-Lagrangeequation

d dt G  = G 

As a onsequen e of[HG2 lo ℄there isa Æ>0su hthat

  ; G   > b 2 kk 2 and  ; G   > b 2 kk 2

in B

Æ

R . Let us now onsider a losed traje tory ((t);q(t)) su h that (; _ ) 2 B Æ , the equation Z  ; G   = Z  ; d dt G   = Z  _ ; G   yields b 2 Z kk 2 6 Z  ; d dt G   6 b 2 Z k _ k 2 : It follows thatkkk _ k0. 

7 Convergen e of sequen es of periodi orbits.

In this se tion,we prove the onvergen e of good sequen es of periodi orbits to homo lini

orbits. Werst state the strongminimizingpropertyof thesubspa ez=z

0 .

Lemma 6 AnyT-periodi traje tory X =(;q) of L satises

L T (X)>b Z T 0 d(;z 0 ) 2 :

Proof : If X=(;q) isa traje tory, q mustsatisfy therstEuler-Lagrange equation

2a q+! 2 q
= G q d dt  G v  :

IfX is losed, we an integratebypartsto writeitsa tion

L(X)= Z aq q+! 2 q
+G(;q;q)_ = Z  G 1 2 q G q  + Z 1 2 q d dt  G v  ;

andintegrating bypart againthe lastterm,

L(X)= Z G 1 2 q G q 1 2 _ q R v >b Z d(;z 0 ) 2 : 

Thislemmaroughlyimpliesthatifthereexistsasequen eofperiodi orbitsofLof

unbound-ed period and bounded a tion, there must be an orbit homo lini to the enter manifold.

Unfortunately, there is no onnement in the q dire tion, and we must have some estimate

oftheq part ofthe periodi orbits inorder tobe ableto prove onvergen e. Asexplainedin

thesket h of proof, we must allowthe parameter! to vary. Consider nowa sequen e !

n of

pulsations, with a limit !, and the asso iated Lagrangian and a tion L

n and L

n

. We have

thefollowing onvergen eproperty:

Proposition 4 If there exist a onstant M, a radius r and a sequen e X

n = ( n ;q n ) of T n -periodi orbitsof L n su h that

n  L n (X n )6M,  kq n k 2 =T n !r 2 =2,   n 6 0 ,

then there exists an orbit X

1 =( 1 ;q 1 ) homo lini toO r

and su h that

Z R G(X 1 ) 1 2 q 1 G q (X 1 ) 1 2 _ q 1 G v (X 1 )6M: Proof : Sin e  n 6 0

,Lemma 5impliesthat 

n

doesnotstay inB

Æ

. We an onsider

n

as aperiodi urve denedon R, andby hangingtimeorigin,we an requirethat

d(

n (0);z

0 )>Æ:

Sin ethesequen e L

n (X

n

) is bounded,we obtainfrom Lemma6 thatthesequen e

Z T n =2 Tn=2 d 2 ( n ;z 0 )

is bounded. Asso iatedwithLemma 1thisyields

E n (X n ) a T n Z _ q 2 n +! 2 n q 2 !0:

On the other side, we obtain using the Euler-Lagrange equations and two integrations by

partsthat a T n Z _ q 2 n ! 2 n q 2 = 1 2T n Z q n G q (X n )+q_ n G v (X n ) !0

be ause [HG1℄ and [HG4℄ imply

q G q (z;q;v)+v G v (z;q;v)6Cd 2 (z;z 0 ):

Combiningthese equationsand thethirdhypothesisyields

E 1 =limE n (X n )=2lim a T n kq_ n k 2 2 =2lim a! 2 n T n kq n k 2 2 =a! 2 r 2 : Sin e E n (X n

) is a bounded sequen e and sin eX

n is an integral urve of Y n the sequen e X n is C 1

-bounded,and byAs oli'sTheoremithasa subsequen e onverginguniformlyon

ompa tsetstoalimit ~

X

1

thatisanintegral urveofY

n

andthustheliftingofaL-traje tory

X 1 of energyE 1 . Re all that Z Tn=2 Tn=2 d 2 ( n ;z 0 )

is bounded. It follows that

Z 1 1 d 2 ( 1 ;z 0 )

1 lim t!1 (t)=z 0 :

Usingon emore thelemma 1weget that

a(q_ 2 1 +! 2 q 2 1 ) ! t!1 E 1 =a! 2 r 2 :

Thisisthe denitionwehave taken forahomo lini orbit. The lastinequalityfollowsfrom

Z Tn=2 T n =2 G(X n ) 1 2 q n G q (X n ) 1 2 _ q n G v (X n )=L(X n )6M

sin etheintegrandis non-negative. 

8 Existen e of periodi orbits.

Let usxa period T =2=!, 2N: Foranyl2R, the fun tional

Q l :C 1 T (R) !R x(t)7 !a Z T 0 _ x(t) 2 l 2 ! 2 x(t) 2

an be omputedusingFourier expansion:

Q l X k q k e ik!t= ! =aT X k  k 2 ! 2  2 l 2 ! 2  jq k j 2 : Itfollows thatQ l an beextended to E T =H 1 (S T =R= TZ;R)

as a ontinuous quadrati form. It has a two dimensional kernel when l 2 Z=, and is

non-degenerateforother valuesofl. Let usset

E + = fq su hthat q k =0when jkj6g E = fq su hthat q k =0when jkj>g;

thereis anorthogonal splitting

E T =E + E ; su hthat Q l j E 

ispositive deniteforall l2(1;1+1=). Noti e thatE isnite

dimen-sional,whi his theusualfeatureof Lagrangianformulations. Let usdenethefun tionals

G:A T = T E T !R x(t) =((t);q(t))7 ! Z T 0 G((t);q(t);q(t))_ dt:

L l :A T = T E T !R x(t) =((t);q(t))7 !L l (x)=Q l (q)+G(x):

We also denetheproje tionP

 : T E T ! T .

Lemma 7 For any lin the interval(1;1=), the fun tional L

l isC

1

and satises the

Palais-Smale ondition. The riti al points of L

l

are the T-periodi smooth traje tories of the

La-grangian L l (z;q;v)=a v 2 l 2 ! 2 q 2
+G(z;q;v); (z;q;v)2TMRR:

Proof: Wewilloftenomitthesubs riptlinthefollowingproof. Re allthat

T

isasmooth

manifold, and thatthemappings

exp :H 1 (O ) !H 1 (S T ;M) (t)7 !exp((t))

are hartsof thismanifold,where 2C 1

(S

T

;M),O

isa suÆ ientlysmallneighborhoodof

the zerose tioninthebundle 

TM oftangentsve tors ofM along ,and exp:TM !M

is theexponentialmap asso iated withsomesprayon M,see [20 ℄. Let

T

:R !S

T

bethe

natural proje tion, the indu ed ve tor bundle  



TM is trivial sin e it is a ve tor bundle

over R, wehave the ommutative diagram

RR n  !    TM ~  !  TM ? ? y ? ? y ? ? y R id ! R  ! S T ;

where isa ve tor bundleisomorphismandwe denethe overing

r =~Æ:RR n !  TM: A H 1 se tion :S T ! 

TM has auniquelifting ~

 :R !R n

su h thatthediagram

RR n r !  TM (id; ~ ) x ? ? x ? ?  R  ! S T

ommutes. Let us take a ompa t neighborhoodU

of the originin R n su h that R U  r 1 (O

),and supposewithoutlossofgeneralitythatO

=r (RU ). Themapping :H 1 (  TM) !H 1 ([0;T℄;R n )  7 ! ~  j[0;T℄

isalinearisomorphismonto itsimageT ~

H H ([0;T℄;R ). Wewillalso note themapping

(;id E T ),and we all ~ H theset ~ H=T ~ H\H 1 ([0;T℄;U

). Let usdenethesmoothmap

~ L :R U R n RR !R (t; ~ ; ;q;v)7 !L expÆr (t; ~ ); d 1 (expÆr )(t; ~ ):1+d 2 (expÆr )(t; ~ ):;q;v
;

andthefun tional

~ L:H 1 ([0;T℄;U )E T !R ( ~ (t);q(t))7 ! Z T 0 ~ L (t; ~ (t); ~  0 (t);q(t);q_(t))dt; we have L = ~

LÆ: One an he k from [HG4℄ and the expression of ~

L

above that the

estimates j ~ L (t; ~ ; ;q;v)j6C(1+q 2 +jj 2 +v 2 )       ~ L  ~  ;  ~ L q ! (t; ~ ; ;q;v)      6C(1+q+jj 2 +v 2 )       ~ L  ;  ~ L v ! (t; ~ ; ;q;v)      6C(1+q 2 +jj+v) holdon RU R n

RR. Thesegrowth onditionsimplybywell-knownresults(see[22 ℄)

that ~

L, and thusL,are ontinuously dierentiable. We also have the lo alexpressionof the

dierential: d ~ L( ~ ;q)= Z T 0  ~ L  ~  d ~ +  ~ L  (d ~ ) 0 +  ~ L q dq+  ~ L v (dq) 0 anddL(;q)=d ~ L( ~

;q)Æ. Letusnowprove thatthePalais-Smale onditionissatised. We

take a Palais-Smale sequen e (

n ;q n ). The sequen e L l ( n ;q n )=Q l (q n )+G( n ;q n ) isbounded. Sin eQ l

isanon-degeneratequadrati form,thereexistsanoperatorA

l :E T ! E T su h that dQ l (q):A l q=jQ l (q)j>Ckqk 2 H 1 :

Let usnowwriteusing[HG4℄and that dL( n ;q n ) = n !0  n kq n k H 1 >dL l ( n ;q n )(0;A l q n )=dQ l (q n ):A l q n +dG( n ;q n ):(0;A l q n ) >Ckq n k 2 H 1 + Z G q
A l q n + G v
A l _ q n >Ckq n k 2 H 1 C 0 kq n k W 1;1 >Ckq n k 2 H 1 C 00 kq n k H 1:

Itfollows thatthe sequen ekq

n k

H 1

is bounded. Pluggingthisinto thea tion

C>L l ( n ;q n )> Z G+Q l (q n )>b Z d 2 ( n ;z 0 ) Ckq n k 2 H 1

yieldsthat k

n k

2

isalsobounded. By astandardappli ationofthetheoremofAs oli,see

[20 ℄,Lemma1.4.4,we anndaC 0

- onvergentsubsequen eof

n

,andbyextra tinganother

subsequen e we an obtainthat q

n

also has a uniform limit. From now on, we willsuppose

that ( n ;q n ) C 0 !(;q):

It remainsto provethatthelimitholdsinE

T

,thatisinH 1

-norms. Sin ethe ontinuous

limit an be approximatedbya smooth urve ,all the urves

n

lieina single hart exp

of for nlargeenough. We all 

n

thelo alrepresentatives of 

n

,and we an usethe lo al

expressionsgivenabove. It isusefulto dene themapping

:R U R n RR !RU R n R R (t; ~ ; ;q;v)7 ! t; ~ ;  ~ L  ;q;  ~ L v ! :

It is straightforwardfromtheexpli itexpressionof ~

L

andfrom [HL℄that

isa

dieomor-phism,and theestimate

1

C

jXj C6

(t;X)6C(jXj+1)

isa onsequen eof [HG4℄. Atheorem of Krasnoselskiiimpliesthat themapping

 :L 2 ([0;T℄;R 2n+2 ) !L 2 ([0;T℄;R 2 n+2 ) X(t)7 ! (t;X(t))

isa homeomorphism. It isnothardto see thatthesequen e

 ~ L  ~  ( ~  n ; ~  0 n ;q n ;q_ n );  ~ L q~ ( ~  n ; ~  0 n ;q n ;q_ n ) ! isboundedinL 1 ([0;T℄;R n+1

),thusitszero averaged primitiveP

n 2W 1;1 ([0;T℄;R n+1 ) hasa

subsequen ethatis onvergent inL 2 ([0;T℄;R n+1 ). We supposethat P n L 2 !P: Sin ekdL( n ;q n )k !0,we have kd ~ L ( ~  n ;q n ) jT ~ H

k !0,and theinequality

     Z h _ P n ;(d ~ ;dq)i+  ~ L  d ~  0 +  ~ L v dq_      6 n k(d ~ ;dq)k H 1

holdsforall variations(d ~ ;dq)2H 1 0 ([0;T℄;R n )T ~ H. The sequen e m n = 1 T Z T 0  ~ L  ( ~  n (t); ~  0 n (t);q n (t);q_ n (t));  ~ L v ( ~  n (t); ~  0 n (t);q n (t);q_ n (t)) ! dt

is bounded,and we an supposetaking a subsequen e that it has a limitm. Integrating by

partsin theinequalityabove yields

*  ~ L  ;  ~ L v ! P n m n ;(d ~  0 ;dq)_ + L 2 6 n k(d ~  0 ;dq)k_ L 2

 ~ L  ;  ~ L v ! P n m n L 2 6 n : We thushave  ~ L  ~  ( ~  n ; ~  0 n ;q n ;q_ n );  ~ L q~ ( ~  n ; ~  0 n ;q n ;q_ n ) ! L 2 !P m;

and thesequen e

 ~  n ; ~  0 n ;q n ;q_ n  = 1 ~  n ;  ~ L  ~  ( ~  n ; ~  0 n ;q n ;q_ n );q n ;  ~ L q~ ( ~  n ; ~  0 n ;q n ;q_ n ) ! hasalimitinL 2 . Thesequen e( ~  n ;q n

)thushasalimitinH 1

([0;T℄;R n+1

),andthesequen e

( n ;q n )= 1 ( ~  n ;q n ) hasa limitinH 1 (O )E T . 

We nowhave to studythetopologyof thefun tional. Let usdene a group of admissible

deformationsof A T : Denition3 A homeomorphism h : A T ! A T

belongs to if and only if there exist a

parameter l2(1;1+1=) anda ontinuous isotopyk :[0;1℄A

T !A T su h that k 0 =Id, k 1

=h, and for all t2[0;1℄ k

t :A T ! A T is a homeomorphism satisfying k t (;q) =(;q) whenQ l (q)+ R U()60.

Forany ompa tsubset 

T

and any h2 we denethe ompa t subset

h:=P  h(E )\E +
:

Lemma8 (Interse tion property ) Let bethe family of ompa t subsets of 

T

dened

in Se tion 5, Denition 1, we have

2and h2 =)h: 2:

Proof : Compare[19 ℄, Proposition1. Letus onsiderthemapping

T s :E !E (z;q)7 !T s (z;q)=q P Æk s (z;q); whereP : T E T

!E is theproje tionasso iatedwiththe splittingE

T =E + E , andk s

isthe homotopybetweenk

0 =Id and k 1 =h. Letusset F s =f(z;q)2E =T s (z;q)=qg and I s =k s (E )\E + =k s (F s ): BothI s and F s ontain ( 0 ;0). Sin e Q l

is negative deniteon E and R

U is bounded on

,there isa >0su h thatQ

l (q)+

R

U()<0forall 2 andq 2E satisfyingkqk> .

Asa onsequen e, themapping T

s

satises

 T

0 =0,

s

 T

s (

0

;q)=0 forall q and alls,

and we an apply Dold's xed point transfer, see [14℄ and [19 ℄, page 433, that asserts the

inje tivityof the morphism P   :H  (; 0 ) ! H  (F s ;( 0

;0)): We now take s=1 and have

the ommutativediagram

H  (I 1 ;( 0 ;0)) h  ! H  (F 1 ;( 0 ;0)) P   x ? ? x ? ? P   H  (h:; 0 ) H  (; 0 ) i  h: x ? ? x ? ? i   H  (; 0 ) h   ! H  (; 0 ); whereh  

is theisomorphismthatmakesthefollowing diagram ommute

H  (E;( 0 ;0)) h  ! H  (E;( 0 ;0)) P   x ? ? x ? ? P   H  (; 0 ) h   ! H  (; 0 ):

Comingba k to the rst diagram, we see that i 

h:

an not be zero be ause P   Æi   Æh   is nonzero. 

Forall Gsatisfying[HR1-4℄and all l2(1;1+1=) we dene

G T (l)= inf 2 inf h2 supL l   h(E )

We havethe estimate:

Lemma9 If G satises (2) then the inequality

I T (U)6 G T (l)6I T (W) holds. Proof : Sin e G ! G T

(l) is an in reasing fun tion this is an easy onsequen e of the

followinglemma.

Lemma10 For all U satisfying [HU1-3℄, we have

U T (l)=I T (U):

Proof : Re all that

U T (l)=inf 2 inf h2 sup (z;x)2h(E ) Q l (x)+U(z) I T (U)=inf 2 sup z2 U(z)

6 inf 2 sup (z;q )2E Q l (q )+U(z)= inf 2 sup z2 U(z)=I:

To obtaintheotherinequality,we applyLemma8 andget

sup (z;x)2h(E ) Q l (x)+U(z)> sup z2h: U(z)>I: 

We arenowina positionto prove Proposition2.

Proof of Proposition 2: First, noti e that the third on lusion is a onsequen e of

thetwo otherones sin etheonlyT periodi solutionofL

l satisfying T  0 isthe onstant urve( 0

;0),andhaszeroa tion,whi hisforbiddenbythese ond on lusionsin e

T

(l(T))>

I

T

(U) >0. Letus now hoose l(T). The fun tion l !

T

(l) is non-in reasing thus almost

everywhere dierentiable. Moreover, theinequality

Z 1+1= 1 0 T (l)dl>I T (U) I T (W)

holdsand we an hoosean l(T) intheinterval (1;1=) su hthat

0 =j 0 T (l(T))j6(I T (W) I T (U)): Letusset = T

(l(T))and re all that

L l (;q)=L(;q) al 2 ! 2 kqk 2 2 :

We shallprove thatthere exists a riti al pointX

T =( T ;q T ) of L l (T) su hthat 2a! 2 kq T k 2 2 61+ : 0

Arguingby ontradi tionweassumethatthereisno riti alpointofL

l (T) atlevel satisfying 2a! 2 kq T k 2 2 6 1+ 0

. We an then ndusing a standard deformation argument an  in the

interval(0; =2) anda homeomorphismh

0 2 satisfying L l (T) (h 0 (X))6L l (T) (X) forallX 2A T

,and su hthat

L l (T) (h 0 (X))6  forallX =(;q)2A T satisfying L l (T) (X)6 + and 2a! 2 kqk 2 2 6 0 +1=2: Letl n

be an in reasing sequen e onverging to l(T), and let

n = T (l n ) and L n =L ln . We an hoose n 2and h n 2 su h that supL n   hn(nE ) 6 n +(l(T) l n )=10: