Splitting and melnikov potentials in hamiltonian systems
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SPLITTING AND MELNIKOV POTENTIALS IN HAMILTONIAN SYSTEMS AMADEU DELSHAMS AND PERE GUTI ERREZ Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona We consider a p erturbation of an integrable Hamiltonian system, p ossessing hyperb olic invariant tori with coincident whiskers. Following an idea due to Eliasson, we intro duce a splitting potential whose gradientgives the splitting distance b etween the perturb ed stable and unstable whiskers. The homo clinic orbits to the perturb ed whiskered tori are the critical p oints of the splitting p otential, and therefore their existence is ensured in b oth the regular (or strongly hyp erb olic, or a-priori unstable) and the singular (or weakly hyp erb olic, or a-priori stable) case. The singular case is a model of a nearly-integrable Hamiltonian near a single resonance. In the regular case, the Melnikov potential is a rst order approximation of the splitting p otential, and the standard Melnikov(vector) function is simply the gradientof the Melnikovpotential. Non-degenerate critical p oints of the Melnikov p otential give rise to transverse homo clinic orbits. Explicit computations are carried out for some examples. 1 Intro duction For more than 2 degrees of freedom, the problem of measuring the splitting of the whiskers of hyp erb olic invariant tori is closely related with the existence of instability in nearly-integrable Hamiltonian systems, i.e. with the Arnold diusion. In this lecture, the splitting is studied in a wide setting, and a general Poincare{Melnikov theory is develop ed. Setup We start with a p erturbation of a hyperbolic integrable Hamiltonian, with n +1 3 degrees of freedom. In canonical variables z =( x y ' I ) 2 D T R T n R n , with the symplectic form d x ^ d y +d ' ^ d I , consider a Hamiltonian of the form H ( x y ' I )= H 0 ( x y I )+ H 1 ( x y ' I ) (1) H 0 ( x y I )= h ! I i + 1 2 h I I i + y 2 2 + V ( x )+ h I i y (2) Manuscript submittedto World Scientic on May 18, 1999 1
where is a p erturbation parameter. The Hamiltonian equations asso ciated to H are: _ x = y + h I i + @ y H 1 ( x y ' I ) _ y = ; V 0 ( x ) ; @ x H 1 ( x y ' I ) _ ' = ! + I + y + @ I H 1 ( x y ' I ) _ I = ; @ ' H 1 ( x y ' I ) : It will shown in section 2 that, under weak assumptions, the unp erturb ed Hamiltonian H 0 has n -dimensional whiskered tori (hyp erb olic invarianttori) with coincident ( n + 1)-dimensional whiskers (invariant manifolds). For a given whiskered torus of H 0 , its (unique) whisker is lled by homo clinic orbits (biasymptotic to the torus). Our aim is to study the splitting of the whiskers, and the p ersistence of some homoclinic orbits , for 6 =0. Main achievements To deal with this problem, the to ols used are Poincare{Melnikovtheory,and a geometric metho d based on Eliasson's approach. Our contributions can b e summarized as follows: A general Poincare{Melnikov theory for Hamiltonian systems is develop ed, dening a scalar function L ( Melnikov potential )whose gradient M ( Melnikov function )gives the splitting distance at rst order in . There exists a scalar function L ( splitting potential )such that, in suitable variables, its gradient M ( splitting function ) gives exactly the splitting distance. Besides, the splitting p otential L is approximated at rst order in by the Melnikov p otential L . The results are signicant for more than 2 degrees of freedom. Motivation The study of the splitting in the Hamiltonian (1{2) is closely related to the problem of Arnold diusion in a general nearly-integrable Hamiltonian system : H ( J )= h ( J )+ "f ( J ) (3) in angle{action variables ( J ) 2 T n +1 R n +1 . Here, the small p erturbation parameter is " . Near single resonances ,itisknown 1 2 3 that one step of (resonant) normal form pro cedure can be performed and leads, under some generic hypotheses Manuscript submittedto World Scientic on May 18, 1999 2
and after a scaling, to a Hamiltonian of the typ e (1{2), taking as H 0 the truncated normal form . Tomake this clearer, consider a selected action J = 0, and assume that its asso ciated frequency vector @ J h (0) 2 R n +1 has a single resonance (this means h k @ J h (0) i = 0 for a certain k 2 Z n +1 nf 0 g and h k @ J h (0) i 6 = 0 for any k 2 Z n +1 not co-linear to k ). It can be assumed that @ J h (0) = (0 ! ), with ! 2 R n nonresonant. Near J , the unp erturb ed Hamiltonian h in (3) can b e written as: h ( J )= h @ J h (0) J i + 1 2 @ 2 J h (0) JJ + O 3 ( J ) : We write =( x ' ) 2 T T n and J =( y I ) 2 R R n ,and @ 2 J h (0) = 2 > where wehave put 2 > 0 in order to x ideas, 2 R n , and is an ( n n )- matrix. With some scaling, we can assume =1, and our Hamiltonian written in the form H ( x y ' I )= h ( y I )+ "f ( x y ' I ) h ( y I )= h ! I i + 1 2 h I I i + y 2 2 + h I i y + O 3 ( y I ) : Performing one step of resonant normal form pro cedure, we can construct a symplectic map such that H = H 0 + H 1 , with H 0 ( x y I " )= h ! I i + 1 2 h I I i + y 2 2 + "V ( x )+ h I i y H 1 ( x y ' I " )= "R ( x y ' I )+ O 3 ( y I )+ O ; " 2 and V ( x ) is the p erio dic function obtained byaveraging with resp ect to the angles ' : V ( x )= f ( x 0 0) = 1 (2 ) n Z T n f ( x 0 ' 0)d ' x 2 T : In the normalized expression for H , note that H 0 (the truncated normal form) is an integrable Hamiltonian, and then H 1 can b e considered as a p erturbation of some size " where can b e determined in terms of " . In this sense, the expression obtained generalizes the Lo chak's example 4 (which, in its turn, generalizes the famous Arnold's example, 5 designed to describ e the diusion). Under generic hyp otheses, it can be shown that the Hamiltonian H 0 has whiskered tori with coincident whiskers associated to this hyp erb olic point (see Manuscript submittedto World Scientic on May 18, 1999 3
section 2). Therefore, although there is no hyperb olicityin h , the p erturbation f provides some weak hyperbolicity , which app ears in the truncated normal form H 0 . This hyp erb olicity disapp ears for " ! 0, b ecause the Lyapunov exp onents of the whiskered tori of H 0 are of the form p " . Tohavexed exp onents, we replace y , I by p "y , p "I (a non-canonical linear change), and divide the Hamiltonian by " . Then the new system is still Hamiltonian, and we obtain obtain for H = H 0 + H 1 an expression of the form (1{2), with ! = ! p " = O " 1 = 2 : It has to b e p ointed out that, after this pro cedure, in general the truncated normal form H 0 is a coupled Hamiltonian: 6 =0in (2). So the motivation for the coupling term h I i y is that this term app ears in a natural waywhen one studies a nearly-integrable Hamiltonian, in a region close to a single resonance. As a particular case, note that if = 0 in (2), then the unp erturb ed Hamiltonian H 0 is somewhat simpler b ecause it is formed byapendulumand n rotors: we then saythat H 0 is uncoupled . We will show in section 4 that the formulation of Poincare{Melnikov theory is simpler in this sp ecial case. Although the (homoclinic) splitting between the whiskers of hyp erb olic tori in single resonances is very imp ortantinthe detection of Arnold diusion (through the construction of transition chains), we p oint out that there are other important diculties related with this problem. These diculties are the study of the transition prop erties of the tori, the detection of heteroclinic intersections b etween whiskers of dierent tori, and jumping the gaps asso ciated to double resonances. Regular and singular cases According to the motivation ab ove, it is convenient in (1{2) to allow ! to dep end on an additional parameter " , considering fast frequencies ! = ! = p " . The parameters " and can b e whether indep endentorlinked by a relation of the typ e = " p with some p> 0 these two cases will b e called, resp ectively, regular and singular . Wehaveshown that, in the study of a general nearlyintegrable Hamiltonian, the actually relevant case is the singular one (with p =1 = 2), and that this feature is directly related to the weak hyperb olicity of the truncated normal form. Concerning the regular situation, we recall that the strategy of keeping " > 0 xed and letting ! 0(having in this way a regular system) was intro duced by Arnold 5 in order to avoid dealing with a singular p erturbation problem. In this case, Poincare{Melnikov theory can be applied directly to the Manuscript submittedto World Scientic on May 18, 1999 4
detection of the splitting, but only if the parameter is taken exponential ly smal l with resp ect to " . This is due to that the Melnikovintegrals involved are exp onentially small in " , as in the second example shown in section 5 (for the rst example shown, the integrals are not exp onentially small, b ecause the p erturbation is not analytic in this case). In the singular case, one assumes that the parameters " and satisfy a power-like relation of the typ e = " p (the smaller p the b etter), and one lets " ! 0. In this case, the problem of detecting the splitting from the Melnikov integrals is much more intricate, b ecause of the exp onentially small character of the integrals involved. However, some recentworks 6 7 8 suggest that, under some weak conditions, the Melnikovintegrals givethe right predictions for the splitting. Nevertheless, the existence of homo clinic orbits has b een established in several works. 1 9 10 This result is valid for regular and singular systems, and we recall it in section 6. 2 The unp erturb ed Hamiltonian Assumptions In this section, we take = 0 and study the unp erturb ed Hamiltonian H 0 dened in (2). Note that the given ingredients of H 0 are the vectors ! 2 R n , the symmetric ( n n )-matrix , and the function V ( x )of x 2 T . We require the following assumptions: The function V ( x ) has a unique and nondegenerate global maximum. To x ideas, we require V (0) = 0 V 0 (0) = 0 V 00 (0) < 0 V ( x ) < 0 8 x 6 =0 (mo d 2 ) : (4) The following nondegeneracy condition holds: det ; ; > =det 1 > 6 =0 : (5) The vector ! is assumed to satisfy a Diophantine condition : for some n ; 1and > 0, jh k ! ij j k j ; 8 k 2 Z n nf 0 g : (6) Manuscript submittedto World Scientic on May 18, 1999 5
The unperturbed torus and its homoclinic whisker The integrable Hamiltonian H 0 can easily b e studied. Let us introduce P ( x y )= y 2 2 + V ( x ) b P ( x y I )= P ( x y + h I i ) (7) then H 0 can b e rewritten as H 0 = h ! I i + 1 2 ; ; > I I + b P ( x y I ) : We see that, on every plane I = const, the Hamiltonian H 0 reduces to a 1degree-of-freedom Hamiltonian: a generalizedpendulum (the standard p endulum b eing given by V ( x ) = cos x ; 1). This p endulum has ( x y )=(0 ;h I i ) as a hyp erb olic equilibrium point, with (homo clinic) separatrices given by y + h I i = p ; 2 V ( x ). The Lyapunov exp onents of the hyp erb olic p oint are , where we dene = p ; V 00 (0). Therefore, the Hamiltonian H 0 p ossesses an n -parameter family of n - dimensional whiskered tori given by the equations I = const, y = ;h I i , x = 0, with ( n + 1)-dimensional whiskers. The stable and unstable whiskers of each torus coincide, and hence all orbits on this (unique) whisker are homoclinic , i.e. biasymptotic to the torus. We will fo cus our attention on a concrete hyp erb olic torus, that we assume lo cated at the origin: I =0, x = y = 0. Note that the vector ! , assumed Diophantine, consists of the frequencies of this torus: _ ' = ! . In view of the nondegeneracy condition (5), the neighbor tori have dierent frequencies. Parameterizations for the unperturbed Hamiltonian We denote T 0 the whiskered torus of H 0 having frequency vector ! .This torus can obviously b e parameterized by T 0 : z 0 ( ' )=(0 0 ' 0) ' 2 T n : As mentioned ab ove, the stable and unstable whiskers of the torus T 0 coincide this homo clinic whisker is given by the equations I = 0, P ( x y )=0. We denote W 0 the p ositive part ( y > 0) of the homo clinic whisker (it is often called separatrix ). To give a suitable parameterization for W 0 , we consider the 1-degree-of-freedom Hamiltonian P ( x y ), and denote ( x 0 ( s ) y 0 ( s )) the asso ciated homo clinic tra jectory, with x 0 (0) = , y 0 (0) > 0. Note that x 0 ( s ) go es from 0 to 2 when s go es from ;1 to 1 . It is clear that we can give the whisker W 0 the parameterization W 0 : z 0 ( s ' )=( x 0 ( s ) y 0 ( s ) ' +( x 0 ( s ) ; ) 0) s 2 R ' 2 T n Manuscript submittedto World Scientic on May 18, 1999 6
where the term ( x 0 ( s ) ; ) expresses the phase drift undergone byanytrajectory when traveling along W 0 . This drift is asso ciated to the coupling term. Note that, with our denition, the dynamics on W 0 is given by the equations _ s =1, _ ' = ! . One has lim t !1 z 0 ( s + t ' + !t ) ; z 0 ( ' + !t )] = 0 and this implies that that every tra jectory on W 0 is biasymptotic to two dierent tra jectories on the invarianttorus T 0 . If is an integer (a very sp ecial case) then these two tra jectories on T 0 coincide. 3 Preservation of the whiskered torus and its whiskers The local normal form Before studying the splitting, wehave to establish the surviving under p erturbations of our Diophantine whiskered torus, as well as its lo cal whiskers. Then wehave to extend them to global whiskers in order to compare the stable and the unstable ones. The surviving of the torus and its lo cal whiskers under a small p erturbation can b e ensured by means of the hyperbolic KAM theorem ,aversion of the KAM theorem adapted to this problem. Roughly sp eaking, the hyperb olic KAM theorem provides a symplectic transformation taking our Hamiltonian into a local normal form e H = H (in some domain), having a simpler expression in which the p erturb ed torus b ecomes transparent, as well as its whiskers. This kind of result follows from aconvergent KAM-like iterativescheme. We are interested in a normal form dened in a whole neighb orho o d of our concrete torus, 1 11 according to the \Kolmogorov's approach" to KAM theory. This approach allows us to control a neighb orho o d of the lo cal stable whisker, which can b e ensured in this wayto contain also a piece of the global stable whisker (this feature is used in section 6). On the contrary,in the \Arnold's approach" (used in other pap ers) the normal form only holds on a Cantor set, although a large family of surviving tori is obtained. Some more comments and references to pap ers following b oth approaches are given in a recent pap er of the authors. 10 In most pap ers (like for instance 11 ), the hyp erb olic KAM theorem is dealt in terms of some lo cal variables in a neighborho o d of the torus, in suchaway that the whiskers b ecome coordinate planes. A signicantly new approach was intro duced by Eliasson, who rewrote the hyp erb olic KAM theorem and expressed it directly in the \original variables". 1 This is more suitable to our Manuscript submittedto World Scientic on May 18, 1999 7
purp ose of carrying out a global control of the whiskers in order to study their splitting (see section 6). Another key fact is the use of exact symplectic transformations to normal form in the hyp erb olic KAM theorem. To recall what an exact symplectic transformation is, consider the 1-form = ; ( y d x + I d ' ), whose dierential is the standard symplectic 2-form: d =d x ^ d y +d ' ^ d I . Then a transformation is symplectic if the 1-form ; is closed, and it is exact symplectic if this 1-form is exact (= d S , globally, for some scalar primitive S ). Eliasson 1 used the exactness of the normalizing transformation as a crucial to ol in order to detect homo clinic intersections b etween the whiskers, in both regular and singular systems (although he did not compute the splitting). A similar result was also obtained by Bolotin. 9 In a further step, in the present lecture the exactness allows us to put the splitting function as the gradient of a splitting potential (see section 6). Another pap er that has inuenced our version of Eliasson's theorem is a recent one by Niederman. 11 This pap er deals with a similar framework (using the Kolmogorov's approach but not working in the original variables), and obtains more accurate estimates for the normal form. Let us intro duce rst some notations. Concerning the domain, we dene for r> 0 the complex set B r = f ( x y ' I ): j x j j y j j I j j Im ' j r g : For a function f ( x y ' I ) analytic on some domain D (and continuous on its closure), we denote j f j D its supremum norm. Theorem 1 (Eliasson's theorem) Let H = H 0 + H 1 as described in (1{ 2) and in the assumptions (4{6), with > n ; 1 . Assume H analytic on B r ( r r 0 ). Then for j j smal l enough, thereexists an exact symplectic transformation =( ): B r ;! B r (analytic with respect to ( x y ' I ) and ), 0 < < 1 , and there exist a = a ( ) , b = b ( ) (analytic in ), such that e H = H takes the form e H = h ! I ; a i + b b P ( x y I )+ O 2 b P ( x y I ) I ; a : (8) Besides, one has = id + O ( ) , a = O ( ) , b =1+ O ( ) . The most imp ortant p oint ab out this result is that, thanks to the use of the original variables x , y , the lo cal normal form e H can be put in terms of the generalized p endulum b P ( x y I ). By using this feature, a \global" control of the whiskers, very useful in order to compare them and study the splitting, can b e carried out. 1 10 In is not hard 10 to establish the validityof theorem 1 in the singular case, with = " p and ! = ! = p " ,for j " j small enough. Manuscript submittedto World Scientic on May 18, 1999 8
Parameterization of the perturbed torus It is clear that the normal form e H given in (8) has a whiskered torus of frequency vector ! . We denote this torus as e T , and its asso ciated lo cal whiskers as f W + lo c (stable) and f W ; lo c (unstable). The torus e T has the following obvious parameterization: e T : ~ z ( ' )=(0 ;h a i 'a ) ' 2 T n : This torus can be translated to a whiskered torus T of the original p erturb ed Hamiltonian H : T : z ( ' )=(~ z ( ' )) ' 2 T n : In section 4, it will be useful to give a rst order approximation in for the shift suered by the p erturb ed torus T with resp ect to the unp erturb ed torus T 0 , along the I -direction. We will denote I ( ' )the I -comp onent of z ( ' ). To describ e this approximation, we consider the (zero average) scalar function ( ' ) solving the following smal l divisors equation : h ! @ ' i + H 1 (0 0 0) = H 1 (0 0 0) : (9) where the notation f denotes the ' -average of a function f . The existence of is ensured by the Diophantine condition (6). The function ,intro duced byTreschev, 12 provides a rst order approximation 10 for the perturb ed torus: I ( ' )= ( ; @ ' ( ' )) + O ; 2 ' 2 T n (10) where we dene = ; ; ; > ; 1 ; @ I H 1 ; @ y H 1 (0 0 0). Parameterizations of the perturbed whiskers As in section 2, we can also take parameters on the p erturb ed local whiskers of the normal form e H : f W lo c : ~ z ( s ' )=( x 0 ( bs ) y 0 ( bs ) ;h a i ' +( x 0 ( bs ) ; ) a ) for s s 0 , ' 2 T n , with some s 0 = s 0 ( r ). For the original Hamiltonian H , the lo cal whiskers can b e parameterized as follows: W lo c : z lo c ( s ' )=(~ z ( s ' )) s s 0 ' 2 T n : In the parameters s , ' , the dynamics of H on W lo c is given by _ s =1, _ ' = ! . We need to extend these lo cal whiskers to global whiskers ,in order to measure the splitting b etween them. The parameterizations of the whiskers Manuscript submittedto World Scientic on May 18, 1999 9
We recall some basic facts concerning these numb ers, that will b e used b elow. Dening C F = 1 $+$ ; 1 = 1 p 5 wehave F n = C F $ n +1 ; ( ; 1) n +1 $ ; ( n +1) n 0 : The b est rational approximations of $ are given by the convergents F n =F n ; 1 . In other words, the indexes k ( n ) =( F n ; F n ; 1 )(and also ( ; F n F n ; 1 )) are the ones that givethedominant b ehavior among the small divisors h k ! i . More precisely, one has: D k ( n ) ! E = F n ; F n ; 1 $= ( ; 1) n $ n = ( ; 1) n C F F n ; 1 + O 1 F 3 n ; 1 n 1 and also the following inequality: 6 for any k =( k 1 ; k 2 )such that k 2 > 0is not a Fib onacci numb er, jh k ! ij = j k 1 ; k 2 $ j > $ C F k 2 : (20) Note that the frequency vector (19) satises the Diophantine condition (6) with =1. This frequency vector is considered in the two examples that we next study. Lower bounds: An example with nite-order splitting Nowwe consider a concrete example in the dierentiable case (16), analogous to the one of Delshams et al. , 18 and obtain lower b ounds for the maximum of the Melnikov p otential L , and for the determinantof @ 2 ' L at a critical p oint. For the p erturbation, we consider the following function: f ( ' )= X n 1 1 $ nr cos D k ( n ) ' E : In this function, the only nonvanishing Fourier co ecients are the ones asso ciated to the Fib onacci indexes k ( n ) . Since k ( n ) = F n +1 $ n +2 ,the co ecients decrease as in (16). Applying (17), the Melnikov p otential L ( ' )isgiven by the series L ( ' )= X n 1 S n cos D k ( n ) ' E Manuscript submittedto World Scientic on May 18, 1999 16
with S n = 2 p " 1 $ n ( r +1) sinh 2 n p " : Note that all the co ecients are positive. The main part of this expression is given by S 0 n = 4 p " e ; b 0 n b 0 n = b 0 n ( " )= n ( r +1)log$ + 2$ n p " : To determine the dominantbehavior, we lo ok for the minimum exp onent b 0 n , for n 1. This is reached for N 0 = N 0 ( " ), with $ N 0 = 2( r +1) p " and this gives the largest co ecient: S N 0 S 0 N 0 " r= 2 . This co ecient itself constitutes a lower b ound for the maximum of the Melnikov p otential, b ecause all the co ecients are p ositive and max ' 2 T 2 j L ( ' ) j = L (0) = X n 1 S n : (21) We can also get an upp er b ound, which coincides with the one of (18). Let us break the series (21) in two parts. For n N 0 , note that S 0 n S 0 n ; 1 = 1 $ r +1 exp 2$ n +1 p " 1 $ r +1 exp 2$ N 0 +1 p " = e 1 = $ r +1 > 1 : Using also S n 3 2 S 0 n (from the fact that sinh x e x = 3for x 1), the sum P n N 0 S n has an upp er b ound of the same order as S N 0 . On the other hand, for n>N 0 wehave the inequality S n 4$ ; nr (simply using that sinh x x ), and we can b ound P n>N 0 S n as a geometric series. In this way,we obtain for the maximum value of the Melnikovpotential, an upp er b ound and a lower b ound, b oth of the same order: max ' 2 T 2 j L ( ' ) j " r= 2 : The Melnikov p otential L ( ' ) has ' = 0 as a critical point. Now, we wantto show that this critical point is nondegenerate, estimating also the Manuscript submittedto World Scientic on May 18, 1999 17
eigenvalues of @ 2 ' L (0) as a measure for the transversality. Wehave @ 2 ' L (0) = ; X n 1 S n k ( n ) k ( n ) > and then det @ 2 ' L (0) = 0 @ X n 1 F 2 n S n 1 A 0 @ X n 1 F 2 n ; 1 S n 1 A ; 0 @ X n 1 F n F n ; 1 S n 1 A 2 = X nm 1 F n F m ; 1 ( F n F m ; 1 ; F n ; 1 F m ) S n S m = X 1 m<n ( F n F m ; 1 ; F n ; 1 F m ) 2 S n S m = X 1 m<n F 2 n ; m ; 1 S n S m where wehave used the formula F n F m ; 1 ; F n ; 1 F m =( ; 1) m +1 F n ; m ; 1 .Note that all terms in this series are also positive. To estimate the determinant, note that F 2 n ; m ; 1 S n S m $ 2 n S n $ ; 2 m S m and hence the indexes n and m can b e separated. This allows us to nd the indexes N 1 ( " ) and M 1 ( " )that give the dominantterm in the series of the determinant, in the same way as b efore. In this way, we easily obtain an upp er b ound and a lower b ound for the determinant: det @ 2 ' L (0) " r : In fact, we should estimate the minimum eigenvalue of @ 2 ' L (0). This eigenvalue can b e put in terms of =tr @ 2 ' L (0) and =det @ 2 ' L (0). Again, note that " r= 2 (applying the same metho d). The minimum eigenvalue is given by ; p 2 ; 4 2 and this has clearly a lower b ound of order " r= 2 . Then it is a consequence of theorem 2 that, for = o ; " r= 2 ,the critical p oint ' =0of L ( ' )gives rise to a transverse homoclinic orbit. This result makes a dierence with the second example, next considered, and is due to the non-analyticity of the p erturbation. Manuscript submittedto World Scientic on May 18, 1999 18
Lower bounds: A singular example For the sake of completeness, we also include an example 10 in the analytic case (16). For the p erturbation, we consider a \full" Fourier series, with the co ecients j f k j = e ;j k j 8 k 2 Z 2 nf 0 g : Note that a non-even function f ( ' )isallowed, so we are not assuming that the p erturbation H 1 ( x ' )isreversible, unlike other pap ers. 19 7 8 Following the metho d by Delshams et al. 6 (though the context is somewhat dierent), it is shown 10 that the dominant harmonics in the Fourier series of the Melnikov p otential L ( ' ) are the ones asso ciated to the Fib onacci indexes k ( n ) . Denoting S n = L k ( n ) , from (17) one directly obtains j S n j = 2 $ n p " e ; F n +1 sinh 2 n p " n 1 : The main part of this expression is given by S 0 n = 4 $ n p " e ; b 0 n b 0 n = b 0 n ( " )= C F $ n +2 + 2$ n p " : (22) For a xed "> 0, to nd the dominant harmonic among the Fib onacci ones, one has to lo ok for the minimum exp onent b 0 n , n 1. Let us dene " n = D 0 $ n +1 4 = " 0 $ 4 n D 0 = r 2 C F : The minimum exp onent among the b 0 n is reached for an only integer N 0 = N 0 ( " ), such that log " N 0 is the closest to log " ,among the log " n .Then the co ecient S 0 N 0 is the dominant one among the S 0 n , and it is not hard to check that the \whole" co ecient S N 0 is also dominant among the S n . One can also check from (20) that the non-Fib onacci co ecients L k , with k 6 = k ( n ) , do not dominate. In terms of " , the value of the minimum exp onent dep ends on " in the following way: b 0 N 0 = c (log " ) " 1 = 4 where c ( ) is a continuous function, dened as the (4 log $)-perio dic extension of c ( )= C 0 cosh ; 0 4 j ; 0 j 2log$ Manuscript submittedto World Scientic on May 18, 1999 19
with C 0 =$ p 2 C F 0 = log " 0 : The extreme values of this function are given by C 0 c ( ) $ 3 = 2 C 0 2 =(1 : 029085 ::: ) C 0 : In this way, the maximum value of the Melnikov p otential can be approximated by its dominant Fibonacci harmonic, and one obtains the following upp er and lower b ound: max ' 2 T 2 j L ( ' ) j 1 " 1 = 4 exp ; c (log " ) " 1 = 4 : It is also shown 10 that the Melnikovpotential L ( ' ) has nondegenerate critical points. In order to detect these points, one has to consider an approximation given by at least the 2 dominant harmonics, b ecause with only 1 harmonic the approximation to the matrix @ 2 ' L would be degenerate. In the discussion ab ove, it can also be considered the integer N 1 ( " )reaching the \second" minimum among the b 0 n this integer satises j N 1 ; N 0 j =1. Calling N = N ( " ) = min ( N 0 N 1 ), it turns out that " N +1 <"<" N , and the Fib onacci co ecients with indexes N and N +1 givethe 2 dominant harmonics in the Fourier expansion of the Melnikov p otential. The two dominant harmonics give the main part of the Melnikov p otential L ( ' ). In the trigonometric form, this main part can b e written as L ( ' ) 2 j S N j cos D k ( N ) ' E + N +2 j S N +1 j cos D k ( N +1) ' E + N +1 where N , N +1 are some phases. The numb er of critical points is given by the determinant % N = det k ( N ) k ( N +1) = F N ; 1 F N +1 ; F 2 N =( ; 1) N +1 which implies that, for " small enough, L ( ' ) has exactly 4 critical p oints. At every critical p oint ' , one has det @ 2 ' L ( ' ) 4 j S N S N +1 j 6 =0 : Then L ( ' ) is a Morse function, b ecause all its critical points are nondegenerate. Note also that 4 = 2 2 is the minimum number of critical p oints for a Morse function on T 2 . To estimate the size of the determinant, we use (22) again: 4 S 0 N S 0 N +1 = 64 2 $ 2 N +1 " e ; ( b 0 N + b 0 N +1 ) b 0 N + b 0 N +1 = c 1 (log " ) " 1 = 4 Manuscript submittedto World Scientic on May 18, 1999 20
where c 1 ( ) is another (4 log $)-p erio dic function, dened from c 1 ( )=$ 3 = 2 C 0 cosh ; 0 0 4 j ; 0 0 j 2log$ with 0 0 =log p " 0 " 1 . The extreme values of this function are given by $ 3 = 2 C 0 c 1 ( ) $ 3 C 0 2 : Thus, one obtains an upper b ound and a lower b ound for the determinantat the 4 critical p oints: det @ 2 ' L ( ' ) 1 " 1 = 2 exp ; c 1 (log " ) " 1 = 4 : Pro ceeding as in the previous example, one also nds an estimate for the minimum eigenvalue. Since, in this case, , the minimum eigenvalue can b e approximated by = . This leads to an estimate of the type 1 " 1 = 4 exp ; c 2 (log " ) " 1 = 4 where c 2 ( )= c 1 ( ) ; c ( ). This is also a p ositive p erio dic function, with $ 3 = 2 C 0 2 c 2 ( ) $ ; 1 2 C 0 : Then it is a direct consequence of theorem 2 that there exist 4 transverse homo clinic intersections, for = o ; exp ; c 2 (log " ) " ; 1 = 4 .The estimate obtained gives a measure for the transversality of the splitting. However, it has to be recalled again that this is actually a regular situation, and a justication in the singular case = " p ,for some p > 0, do es not follow directly from theorem 2. 6 Flow-b oxvariables and splitting p otential The aim of this section is to sketchthe pro of of the result 10 that, using suitable variables, the \whole" splitting distance (and not only its rst order approximation) is the gradient of some function, in order to establish the existence of homo clinic orbits even in the singular case. Manuscript submittedto World Scientic on May 18, 1999 21
Flow-box variables In order to provide a clearer formulation for the problem of measuring the splitting, it is convenient to intro duce new symplectic variables in which the Hamiltonian equations are very simple. The ow-box variables W = ( S E J ) are constructed 10 with the help of the ow asso ciated to the normal form e H given in (8), from a suitable Poincare section containing the set S = ~ z ( s 1 T n ) f W + lo c ,with some xed s 1 >s 0 . The new variables are then given by an exact symplectic transformation ( x y ' I )=&( S E J ), dened on a real neighborhood of b S =& ; 1 ( S )=(0 0 T n a ). Thanks to the use of the Kolmogorov's approachtothe hyp erb olic KAM theory, the neighborho o d where the ow-boxvariables are dened contains a piece of b oth whiskers. In the construction of the variables, one can makethe lo cal stable whisker b ecome a co ordinate plane (see (24)), and then the global unstable whisker can b e seen as a graphic over the local stable one. In this way, the splitting distance and the homo clinic intersections b etween the two whiskers appear much more transparently. Our Hamiltonian takes, in the ow-b oxvariables, a very simple form: b H = e H &= E + h ! J ; a i and hence the asso ciated Hamiltonian equations are _ S =1 _ E =0 _ = ! _ J =0 : (23) We recall that analogous ow-b oxvariables have already b een used 20 6 in some case where the symplectic change can b e dened explicitly from the expression of the normal form, whichisintegrable. In our case, the normal form e H is, in general, not integrable, and the construction of the ow-b ox variables is more involved (it uses implicit functions). Parameterizations in the ow-box variables Let us describ e more precisely how the whiskers can b e parameterized in the ow-b oxvariables. Let us denote c W + lo c =& ; 1 f W + lo c the lo cal stable whisker (or more precisely a piece of it). This whisker becomes a co ordinate plane, given by E =0, J = a , and can b e parameterized as follows: c W + lo c : W + lo c ( s ' )=& ; 1 (~ z ( s 1 + s ' ))=( s 0 'a ) s 2I ' 2 T n (24) where I is some interval containing s =0 (wehave replaced s ; s 1 by s for a clearer notation: in this waywehave s =0 on b S ). Manuscript submittedto World Scientic on May 18, 1999 22
Now, we dene c W ; =& ; 1 ; 1 ( W ; )as an invariant manifold of b H , which is the equivalentinthe ow-boxvariables for (a piece of ) the global unstable whisker. Let us parameterize: c W ; : W ; ( s ' )=& ; 1 ; 1 ; z ; ( s ' ) s 2I ' 2 T n : In comp onents, we write W ; ( s ' )= ; S ; ( s ' ) E ; ( s ' ) ; ( s ' ) J ; ( s ' ) : There is splitting of the whiskers when J ; ( s ' ) 6 = a or E ; ( s ' ) 6 =0. Nevertheless, it suces to control the J -comp onent b ecause the whisker is contained in the zero energy level: E ; + h ! J ; ; a i =0. The approximation given in theorem 2, expressed there in the original variables, remains true after changing to the ow-boxvariables. So the Melnikov function M = @ ' L also provides a rst order approximation in for the splitting distance J ; ( s ' ) ; a , at least in the regular case. But, as a new feature, in the ow-b oxvariables this splitting distance becomes a quasiperiodic function, only dep ending on ' ; !s . Indeed, from the simple form (23) of the Hamiltonian equations, one sees that J ; ( s ' )= J ; (0 ' ; !s ) : This prop ertyisvery important in the singular case, b ecause it can give rise to exp onentially small estimates for the splitting. The key point 6 20 is to use that a function of ' ; !s , with ! = ! = p " ,having a p olynomial b ound on a complex domain, b ecomes exp onentially small in " for real values of s , ' . Splitting potential and existenceof homoclinic orbits Using the sp ecial formulation of theorem 1 (in the original variables) it is p ossible 1 to intro duce in some neighborhood an exact symplectic map ' that takes the lo cal stable whisker onto the global unstable one. In fact, one can go a bit farther, 10 and express the map ' in the ow-boxvariables. The exactness of the symplectic map ' and the fact that ' = id + O ( ), imply that for j j small enough there exists a generating function ; S ( E ( J , dened in some neighb orho o d of b S and global in the angles 2 T n , such that the map ' : ; ( S ( E ( ( J 7! ( S E J )isgiven by S = ( S ; @ E ( E = E ; @ S = ( ; @ J ( J = J ; @ : (25) In order to compare the whiskers c W + lo c and c W ; , it will b e useful to express the splitting distance J ; ( s ' ) ; a as a gradient. This cannot be deduced Manuscript submittedto World Scientic on May 18, 1999 23
directly from (25), but this obstruction is easily overcome, 20 intro ducing new parameters that substitute the initial ones s , ' on the whiskers: S = S ; ( s ' ) = ; ( s ' ) : In terms of the new parameters S , , the unstable whisker c W ; app ears nicely as a graphic over the stable whisker c W + lo c , through the parameterization: c W ; : ~ W ; ( S )= S ~ E ; ( S ) ~ J ; ( S ) S 2I 2 T n (26) (the interval I can have undergone a reduction). It is then natural to intro duce the splitting potential as the following scalar function, p erio dic in : L ( S )= ( S 0 a ) S 2I 2 T n : (27) This function also dep ends on , and is determined up to an additive constant. The (vector) splitting function can then be dened as the gradientof L with resp ect to the angles: M ( S )= @ L ( S ) : The next theorem is easily deduced from the equations (25). Theorem 3 The functions L and M only depend on ; !S : L ( S )= L (0 ; !S ) M ( S )= M (0 ; !S ) : Besides, these functions arerelated with (24) and (26) in the fol lowing way: ~ E ; ( S )= @ S L ( S ) ~ J ; ( S ) ; a = M ( S ) : According to this theorem, the function M gives the splitting distance (expressed in the parameters S , ). It is imp ortantto stress that the fact that the splitting distance can b e put as the gradient of some p otential is a reection of the Lagrangian properties of the whiskers. As a corollary of theorem 3, one can recover a result due to Eliasson: 1 there exist at least n +1 homoclinic orbits (not necessarily transverse), biasymptotic to the whiskered torus T . This result, valid for b oth the regular case and the singular case, comes from the fact that a function on T n has at least n + 1 critical p oints (not necessarily nondegenerate), according to the Lyusternik{Schnirelman theory. 21 Then for a xed S , the splitting p otential L ( S ) has at least n + 1 critical p oints, whichgive rise to respective homoclinic intersections b etween the whiskers W , and hence to homo clinic orbits, contained in b oth whiskers. Manuscript submittedto World Scientic on May 18, 1999 24
First order approximation for the splitting potential Finally, using Poincare{Melnikovtheory, we can obtain rst order approximations for the splitting p otential L ,intro duced in (27), and for the splitting function M . At rst order in , these approximations are given, resp ectively, by the Melnikov p otential L and the Melnikov function M dened in section 4, but they are go o d enough only for the regular case. Theorem 4 For S 2I and 2 T n ,onehas L ( S )= L ( ; !S )+ O ; 2 M ( S )= M ( ; !S )+ O ; 2 : We nish with some remarks about the additional diculties of the singular case. Note that theorem 4 provides an O ; 2 error term that is not small enough in the singular case = " p with p > 0, due to the fact that the functions L and M are exp onentially small with resp ect to " . (This is illustrated in the second example of section 5). Nevertheless, one can exp ect that, under some weak hyp otheses on the p erturbation, the predictions of the splitting given by the Melnikov p otential L are also valid in the singular case, for some p> 0. To get b etter bounds of the O ; 2 -term for real values of the variables S , , one should bound this term on a complex strip of these variables. This requires some improvements of the results presented here. First, one needs a more precise version of theorem 1, carrying out a careful control on the loss of complex domain in the angular variables. Such an improvementof the normal form theorem has already been p erformed by the authors, 10 and in fact analogous results had previously b een obtained 2 6 for somewhat dierent contexts. On the other hand, one needs an extension theorem and the ow-b oxvariables extended to a suitable complex domain, whichwould lead to a signicant renement of theorem 4, of the type L ( S )= L ; ; S! = p " + O ; 2 " ; p for S , on a complex strip j Im S j = 2 ; " 1 = 4 , j Im j ; " 1 = 4 .Then one could obtain, for real values of S , ,exp onentially small upper bounds for the error term, whichwould be dominated by the rst order approximation provided byPoincare{Melnikovtheory, under some general hyp otheses on the p erturbation. If this is true, then the Poincare{Melnikov theory gives the right predictions for the splitting even in the singular case. The problem of giving asymptotics for the exp onentially small splitting of separatrices is now b eing researched by the authors. In fact, the strategy describ ed ab ove has b een followed 20 6 in simpler situations in which the normal Manuscript submittedto World Scientic on May 18, 1999 25