Melnikov potential for exact symplectic maps
Abstract
The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of $n$ degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least $4$ primary homoclinic orbits ($4n$ in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a $2n$-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc.
Full text
Melnikov potential for exact symplectic maps Amadeu Delshams and Rafael Ramrez-Ros Departament de Matematica Aplicada I Universitat Politecnica de Catalunya Diagonal 647, 08028 Barcelona, Spain Revised version, March1997 Abstract The splitting of separatrices of hyp erb olic xed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical p oints of a real-valued function (called the Melnikov p otential) are asso ciated to transverse homo clinic orbits and an asymptotic expression for the symplectic area between homo clinic orbits is given. Moreover, if the unp erturb ed invariant manifolds are completely doubled, it is shown that there exist, in general, at least 4 primary homo clinic orbits (4 n in antisymmetric maps). Both lower b ounds are optimal. Two examples are presented: a2 n -dimensional central standard-like map and the Hamiltonian map asso ciated to a magnetized spherical p endulum. Several topics are studied ab out these examples: existence of splitting, explicit computations of Melnikov p otentials, transverse homo clinic orbits, exp onentially small splitting, etc. AMS Sub ject Classication (1991): 33E05, 34C37, 34E10, 57R19, 57R70, 58F05 1Intro duction In a previous work DR96], the authors were able to develop a general theory for p erturbations of an integrable planar map with a separatrix to a hyp erb olic xed p oint. The splitting of the p erturb ed invariant curves was measured, in rst order with resp ect to the parameter of p erturbation, by means of a p erio dic Melnikov function M dened on the unp erturb ed separatrix. In case of area preserving p erturbations, M has zero mean and therefore there exists a p erio dic function L (called the Melnikov potential ) such that M = L 0 . Consequently,if L is not identically constant (resp ectively,has non-degenerate critical p oints), the separatrix splits (resp ectively, the p erturb ed curves cross transversely). Moreover, under some hyp othesis of meromorphicity,the Melnikov 1
2 A. Delshams and R. Ramrez-Ros potential is elliptic and there exists a Summation Formula (see App endix) to compute it explicitly. The aim of this pap er is to develop a similar theory for more dimensions. The natural frame is to consider exact symplectic p erturbations of a 2 n -dimensional exact map with a n -dimensional separatrix asso ciated to a hyp erb olic xed point. Exact symplectic maps F : P ! P are dened on exact manifolds, i.e., 2 n -dimensional manifolds P endowed with a symplectic form ! which is exact: ! = ; d and they are characterized by the equation F ; = d S for some function S : P ! R , called generating function of F . The typical example of an exact symplectic manifold is provided by a cotangent bundle T M , together with the canonical forms 0 , ! 0 , which in cotangent co ordinates ( x y ) read as 0 = y d x , ! 0 = d x ^ d y . Typical exact symplectic maps are the socalled twist maps, which satisfy F ( y d x ) ; y d x = Y d X ; y d x = d L ( x X ), where ( X Y ) = F ( x y ). The fact that the generating function S can be written in terms of old and new co ordinates: S ( x y )= L ( x X ), is the twist condition that gives the name to these maps. The function L is called twist generating function. As in Eas91 ], we will not restrict ourselves to this typical case, since the results to be presented in this pap er are valid on arbitrary exact symplectic manifolds and the twist condition is not needed. The exact symplectic structure plays a fundamental role in our construction, since it allows us to work neatly with geometric ob jects. For example, it is used to intro duce two homo clinic invariants: the action of a homo clinic orbit and the symplectic area between two homo clinic orbits, called simply homo clinic area. Namely, let p 1 2P be a hyp erb olic xed point of F ,whichlies in the intersection of the n -dimensional invariant manifolds W u s . Given a homoclinic orbit O =( p k ) k 2 Z of F , i.e., O ( W u \W s ) nf p 1 g and F ( p k ) = p k +1 , we dene the homoclinic action of the orbit O as W O ]:= X k 2 Z S ( p k ) where, in order to get an absolutely convergent series, the generating function S has been determined by imp osing S ( p 1 )=0. Given another homo clinic orbit O 0 of F , the homoclinic area between the two homo clinic orbits O , O 0 is dened as the dierence of homo clinic actions W O O 0 ] := W O ] ; W O 0 ]. These two ob jects are symplectic invariants , i.e., they neither dep end on the symplectic co ordinates used, nor on the choice of the one-form .It is worth noting that in the planar case, the homo clinic area is the standard (algebraic) area of the lob es b etween the invariant curves MMP84, Mat86, Eas91] and also measures the ux along the homo clinic tangle, which is related to the study of transp ort MMP84, RW88, Mei92]. The unp erturb ed role will be played by an exact symplectic dieomorphism F 0 : P ! P , dened on a 2 n -dimensional exact manifold P ,which p ossesses a hyp erb olic xed point p 1 and a n -dimensional separatrix W u 0 \W s 0 , where W u s 0 denote the invariant manifolds asso ciated to p 1 .
Melnikov p otential for exact symplectic maps 3 Consider now a family of exact symplectic dieomorphisms f F " g ,as a general p erturbation of the situation ab ove, and let S " = S 0 + "S 1 + O ( " 2 ) b e the generating function of F " . The main analytical results of this pap er are stated and proved in section 2. There, the Melnikov p otential is intro duced as the real-valued smo oth function L : ! R given by L ( p ):= X k 2 Z b S 1 ( p k ) p k = F k 0 ( p ) where b S 1 : P ! R is dened as b S 1 ( p ) = S 1 ( p ) ; ( F 0 ( p )) F 1 ( p )], and F 1 is the rst order variation in " of the family f F " g , that is, F 1 ( p )= @F " ( p ) =@ " ] j " =0 . Obviously, S 1 is determined by imp osing b S 1 ( p 1 )=0, in order to get an absolutely convergent series. In theorem 2.1 it is established that (i) the Melnikov p otential L is F 0 -invariant: L F 0 = L , (ii) if L 6 constant, the p erturb ed invariant manifolds W u s " split for 0 < j " j 1, (iii) the non-degenerate critical points of L are asso ciated to transverse intersections of the p erturb ed invariant manifolds, (iv) the ab ove-mentioned homo clinic invariants are given in rst order by L . As amatter of fact, the p erturb ed homo clinic orbits detected by the Melnikov potential are all of them primary homoclinic orbits O " of F " , i.e., they are smo oth in " for j " j small enough. The Melnikov potential admits several reformulations. For example, if F " is a twist map on a cotangent bundle T M ,withtwist generating function L " = L 0 + " L 1 + O ( " 2 ), b S 1 has the simple form b S 1 ( p )= L 1 ( ( p ) ( F 0 ( p ))), where : T M!M is the natural pro jection. Consequently, the Melnikov potential reads as DRS97] L ( p )= X k 2 Z L 1 ( x k x k +1 ) x k = ( p k ) where L 1 is determined by imp osing L 1 ( x 1 x 1 ) = 0, and x 1 = ( p 1 ). Another interesting situation, that allows us to compare the continuous and discrete frames, is to consider Hamiltonian maps .Let H " : P R ! R be a time-p erio dic Hamiltonian of period T , and F " = T " , where t " ( p )is the solution of the asso ciated Hamiltonian equations with initial condition p at t = 0. If H " = H 0 + "H 1 + O ( " 2 ), then b S 1 ( p )= ; R T 0 H 1 ( t 0 ( p ) t )d t , so the Melnikov p otential takes the form (already known to Poincare) L ( p )= ; Z R H 1 ( t 0 ( p ) t )d t where H 1 is determined by imp osing H 1 ( t 0 ( p 1 ) t ) 0, or simply H 1 ( p 1 t ) 0, if H 0 is autonomous. An essential ingredient for the pro of of theorem 2.1 is the fact that the invariant manifolds W u s " are exact Lagrangian immersed submanifolds of P and therefore can b e
4 A. Delshams and R. Ramrez-Ros expressed in terms of generating functions L u s " . The Lagrangian prop ertyoftheinvariant manifolds was already noticed by Poincare Poi99]for ows, although we learned it for maps from E. Tabacman Tab95], as well as the expression for the invariant manifolds given in prop osition 2.1, in the twist frame. The relationship between L u s 1 and S 1 , the rst order variations in " of the generating functions L u s " and S " ,gives then the formula for the Melnikov p otential. The to ols utilized are very similar to those of D. Treschev Tre94]. However, D. Treschev considers autonomous Hamiltonian ows, and the conservation of energy makes easier the deduction of the continuous version of equation (2.5). In that frame (Hamiltonian-Lagrangian ows), it is worth noting that a variational approach to the Melnikov metho d was carried out by S. Angenent Ang93] for Hamiltonian systems with 1 1 2 degrees of freedom, and that a mechanism for nding homo clinic orbits in p ositively denite symplectic dieomorphisms is due to S. Bolotin Bol94], based on interp olating them by Hamiltonian ows. Section 2 contains also some remarks on the non-symplectic case: a vector-valued Melnikov function M is then dened, whose non-degenerated zeros are asso ciated to transverse homo clinic orbits. The last part of section 2 is devoted to gain information on the number of primary homo clinic orbits after p erturbation. Since the Melnikov potential L is F 0 -invariant, it can be dened on the reduced separatrix := =F 0 ,which is the quotient of the separatrix by the unp erturb ed map. The reduced separatrix is a compact manifold without b oundary, provided that the unp erturb ed invariant manifolds are completely doubled ,i.e., W u 0 = W s 0 and W u s 0 nf p 1 g is a submanifold of P and not only an immersed submanifold of P . This is equivalent to require that the separatrix is = W u s 0 nf p 1 g . Several dynamical consequences of this fact can be pointed out using top ological to ols. In particular, Morse theory gives lower bounds on the number of primary transverse homo clinic orbits, under conditions of generic p osition: in theorem 2.2 it is stated that the number of primary homo clinic orbits is at least 4. Moreover, if the maps F " have a common symmetry I : P!P ( F " I = I F " , and F " ( p 1 )= I ( p 1 )= p 1 )such that the one-form is preserved by I : I = ,then the Melnikov p otential is I -invariant (see lemma 2.6). Consequently, it can b e considered as a function over the quotient manifold I := = f F 0 I g . If, in addition, I is an involution ( I 2 = Id ) such that DI ( p 1 ) = ; Id, the family f F " g will be called antisymmetric . In this case, in theorem 2.2 it is stated that the number of primary homo clinic orbits is at least 4 n and that they app ear coupled in (anti)symmetric pairs: O " is a primary homo clinic orbit if and only if I ( O " ) also is. It is worth mentioning that any family of odd maps F " : R 2 n ! R 2 n (with the standard symplectic structure) is antisymmetric. To prove theorem 2.2, it is enough to check that the sum of the Z 2 -Betti numb ers of and I are 4 and 2 n , resp ectively.This is accomplished by computing the Z 2 - homology of and I . Both lower b ounds are optimal, as it is shown in several p erturbations of maps with a central symmetry,sothattheunperturbedinvariant manifolds are completely doubled.
Melnikov p otential for exact symplectic maps 5 It is imp ortant to notice that the invariant manifolds of a pro duct of uncoupled planar maps with double lo ops are not completely doubled, see remark 2.3, and hence, the top ological results do not hold in this case. Indeed, the number of primary homo clinic orbits may b e rather dierent under p erturbation for instance, it is p ossible to construct explicitly p erturbations with an innite numb er of primary homo clinic orbits, all of them b eing transverse. The study of this kind of phenomena is currently being researched. In section 3, as a rst example, we consider the family of twist maps on R 2 n : F " ( x y )= y ; x + 2 y 1+ j y j 2 + " r V ( y ) ! > 1 " 2 R with V : R n ! R determined by imp osing V (0) = 0. The map ab ove is ap erturbation of the McLachlan map McL94], which is a multi-dimensional generalization of the McMillan map McM71], which in its turn is a particular case of the standard-like Suris integrable maps Sur89]. The McLachlan map has a central symmetry that makes the dynamics over the separatrix essentially one-dimensional. This is the key fact that allow us to p erform a complete analysis, since the natural parametrizations (3.2) can b e intro duced. If the p otential V is entire and not identically zero, in theorem 3.1 it is proved that the manifolds W u s " of the map F " split, for 0 < j " j 1. This result is obtained simply bychecking that the Melnikov p otential is not constant. Moreover, if V is a p olynomial, the Melnikov potential can b e computed explicitly. In particular, if V is a quadratic form: V ( y )= y > By for some symmetric n n matrix B , in prop osition 3.1 it is stated that under generic conditions on B (det( B ) 6 = 0 and B do es not havemultiple eigenvalues), the p erturb ed invariant manifolds are transverse along exactly 4 n primary homo clinic orbits. If V is linear: V ( y ) = b > y for some vector b 2 R n nf 0 g ,in prop osition 3.2 it is stated that the p erturb ed invariant manifolds are transverse along exactly 4 primary homo clinic orbits. The dierence between b oth kinds of p erturbations is that quadratic potentials V give rise to odd maps, whereas linear ones do not. Moreover, prop ositions 3.1 and 3.2 give the unp erturb ed homo clinic orbits that survive and the rst order (in " ) of the homo clinic areas between the dierent primary homo clinic orbits. The weakly hyp erb olic case 0 <h 1, cosh( h ):= ,is also studied for the case of a quadratic p otential V , and asymptotic expressions for the homo clinic areas are given at the end of section 3. It turns out that, for some distinguished pairs, interlaced in the same way as in the case of 1 degree of freedom, the homo clinic area predicted by the Melnikov p otential is exp onentially small with resp ect to the hyp erb olicity parameter h . Of course, this do es not prove that the splitting size is exp onentially small in singular cases, i.e., when " and h tend simultaneously to zero. The last section is devoted to the study of the Hamiltonian maps arising from timeperiodic p erturbations of an (undamp ed) magnetized spherical p endulum. This mo del was intro duced by J. Gruendler Gru85] as a rst example of application of the Melnikov
6 A. Delshams and R. Ramrez-Ros metho d for high-dimensional (continuous) systems. The Hamiltonians considered have the form Gru85] H " : R 2 n R ! R H " ( x y t )= v 2 = 2+( r 4 ; r 2 ) = 2+ "V ( x t=h ) h> 0 " 2 R where v = j y j , r = j x j ,and V = V ( x ' ) is 1-p erio dic in ' . We determine V by imp osing V (0 ' ) 0. Note that small values of h corresp ond to a quick forcing. General p erturbations, and not only symplectic ones, are considered in Gru85]. As a consequence, the homo clinic orbits are given in the general case by non-degenerate zeros of a vector-valued Melnikov function, instead of non-degenerate critical p oints of the real-valued Melnikov p otential. We have computed the Melnikov p otential for the Hamiltonian p erturbations studied in Gru85], and have veried that his Melnikov function is the gradient of our Melnikov p otential. Most of the results stated ab ove for the McLachlan map also hold for this Hamiltonian map. There is, however, a signicant dierence. One cannot deduce a priori that the Melnikov p otential is not identically constant without computing it. This has to do with the fact that the Melnikov p otential is simply periodic and regular for the p olynomial p erturbations considered, in contrast with the complex p erio d and singularities that the Melnikov p otential has for the entire p erturbations of the McLachlan map. To nish the account of results, let us pointout that a similar Melnikov analysis for p erturb ed ellipsoidal billiards has not b een included for the sake of brevity and will app ear elsewhere. Such billiards are a high-dimensional version of p erturb ed elliptic billiard tables ,which have already b een studied in several pap ers LT93, Tab94, DR96, Lom96a]. After this research was complete, we b ecame aware of some recent pap ers Lom93, Lom96b] of H. Lomel for twist maps on the annulus A n = T T n = T n R n that resemble our metho d. However, they do not contain explicit computations (i.e., in terms of known functions) of the Melnikov p otential, since complex variable metho ds are not used. Besides, in those pap ers it is assumed that the separatrix is globally horizontal, a condition that do es not hold for homo clinics in R 2 n , since the separatrix must fold to go back to the xed point. Another related pap ers are Sun96, BGK95], but their approach is rather dierent, since they deal, like Gru85], with the general case, with no symplectic structure, and therefore a vector-valued Melnikov function is needed. This makes an imp ortant dierence not only from a computational pointof view (there are not explicit (analytic) computations in these works), but also from a theoretical point of view, since Morse theory cannot be applied in the general situation. We also want to mention the work BF96], where p erturbations of n -dimensional maps having homo-hetero clinic connections to compact normally hyp erb olic invariant manifolds are considered. 2 Main results For the sake of simplicity,we will assume that the ob jects here considered are smo oth. For a general background on symplectic geometry we refer to Arn76, GS77,AM78].
Melnikov p otential for exact symplectic maps 7 The basic prop erties of immersed submanifolds can be found in GG73,pages 6{11]. 2.1 Exact ob jects A 2 n -dimensional manifold P together with an exact non-degenerate two-form ! over it, is called an exact symplectic manifold .Then, ! = ; d for some one-form , usually called Liouvil le form , symplectic potential or action form . A map F : P ! P is called exact symplectic (or simply, exact) if H = H F for all closed path P or, equivalently, if F ; = d S for some function S : P ! R , called generating function of F . A n -dimensional submanifold P is called an exact Lagrangian submanifold (or simply, an exact submanifold) if H = 0 for all closed path or, equivalently, if { = d L for some function L : ! R , called generating function of . Here { : , !P stands for the inclusion map. Unfortunately, the invariant manifolds that we will deal with are not submanifolds, but just immersed submanifolds. Thus, the intro duction of some technicalities seems unavoidable in order to give a rigorous exp osition of the sub ject, and more precisely,to intro duce the notion of separatrix, where the distance between the p erturb ed invariant manifolds will b e measured. Given a manifold N , we recall that a map g : N!P is called an immersion when its dierential d g ( z ) has maximal rank at any p oint z 2N .If g is one-to-one onto its image W = g ( N ), there is a natural way to make W a smo oth manifold: the top ology on W is the one which makes g a homeomorphism and the charts on W are the pullbacks via g ; 1 of the charts on N . The manifold W constructed in this way is called an immersed submanifold of P and its dimension is equal to the dimension of N . It is imp ortantto notice that the top ology of the immersed manifold need not be the same as the induced one via the inclusion W P or, in other words, that W need not be a submanifold of P in the usual sense. Figure 1 shows an example of a double lo op W = g ( R ) to p 1 =lim z !1 g ( z )foran immersion g : R ! R 2 . At p 1 , the induced top ology on W via the inclusion W R 2 is not the same as the induced one via g . Both g ( B ), for all op en b ounded interval B R , and Wnf p 1 g are submanifolds, but not W .This situation is a particular case of the following elementary result GG73, page 11]. Lemma 2.1 Let g : N!P be a one-to-one immersion and set W = g ( N ) . (i) Let B be an open subset of N with compact closure. Then, g j B : B ! P is an embedding, that is, a homeomorphism onto its image g ( B ) . Thus, g ( B ) is a submanifold of P , which wil l be cal led an embedded disk in W . (ii) Let W be the set of points where the two topologies on W (the one inducedby the inclusion WP and the one that makes g a homeomorphism) dier. Then, = W n is a submanifold of P . Indeed, W is not a submanifold of P just at the points of .
8 A. Delshams and R. Ramrez-Ros R W R 2 g p 1 = g (0) Figure 1: g =( g 1 g 2 ): R ! R 2 , where g 1 ( z )= 3 2 z= (1 + z 2 ), g 2 ( z )= g 1 (2 z ). For the sake of clearness, submanifolds and immersed submanifolds will b e denoted by dierent letters, namely and W , resp ectively. For immersed submanifolds W , the map { W : W!P stands for the inclusion map, as b efore. It should b e noted that { W is smo oth, even when W is not a submanifold of P , b ecause of the dierential structure given to W . Moreover, if P is a (closed) path, we will say that is a (closed) path in the immersed submanifold W if and only if is contained in W and it is continuous in the topology of W . For example, if is one lo op of gure 1, it is a closed path in R 2 but not in W . With these notations and denitions, we are naturally led to dene exact immersed submanifolds in the same way as exact submanifolds. A n -dimensional immersed submanifold WP is called exact if H =0 for all closed path in W or, equivalently, if { W = d L for some function L : W ! R , called generating function of W . The symplectic p otential is determined except for the addition of a closed zeroform, and the generating functions of maps or (immersed) submanifolds are determined except for an additive constant. Henceforth, the symbol W R q p ,with p q 2 W ,will denote the integral of along an arbitrary path from p to q in W .It only makes sense for an exact immersed submanifold W ,since then the integral do es not dep end on the path. The dierence of values of L can be expressed as an integral of this kind: L ( q ) ; L ( p )= Z q p d L = W Z q p 8 p q 2W : (2.1) Lemma 2.2 Let W be a connected exact immersed submanifold of P , invariant under an exact map F . Let L and S be their respective generating functions. Then, S ( p ) + constant= L ( F ( p )) ; L ( p ) 8 p 2W : (2.2)
Melnikov p otential for exact symplectic maps 9 W u 'W s p p 0 D D 0 p 1 W u ' R p 1 p p 0 W s ' R p p 0 p 1 Figure 2: The invariant manifolds W u and W s are dierent as smo oth manifolds, and are not submanifolds of R 2 . There exist no paths u s in W u s from p to p 0 such that u = s . Moreover, if p 1 2W is a xed point of F , the constant is ; S ( p 1 ) . Pro of. From d S = F ; and d L = { W we get d S jW = { W d S = F jW d L ; d L = d L F jW ; L where S jW = S { W and F jW =( { W ) ; 1 F { W are the restrictions of S and F to W . Thus, S ; L F + L is constant over W by connectedness and (2.2) is proved. To end the pro of we only need to evaluate equation (2.2) at p = p 1 . 2 Let p 1 2 P be a hyp erb olic xed point of F . The point p 1 lies in the intersection of the n -dimensional unstable and stable invariant manifolds of the map F asso ciated to p 1 : W u := p 2P : lim k !;1 F k ( p )= p 1 W s := p 2P : lim k ! + 1 F k ( p )= p 1 : The manifolds W u s need not be submanifolds of P , but just connected immersed submanifolds, see gure 2. In fact, W u s = g u s ( R n ) for some one-to-one immersions g u s : R n !P ,such that g u s (0) = p 1 and dg u s (0) R n ]is the tangent space to W u s at p 1 PM82, II x 6]. Since F is exact, they are exact immersed submanifolds: if is a closed path in W u ( W s ), then H = H F k ;! H p 1 =0, when k !;1 ( k ! + 1 ). It should be noted that if P is closed and contained in W u (resp. W s ), but it is not a path in W u (resp. W s ), the ab ove argument fails. (For instance, if is one lo op of gure 2.) We denote by L u s the generating functions of W u s and we determine the generating functions S , L u s by imp osing S ( p 1 )= L u s ( p 1 )=0. The next prop osition gives a nice interpretation of the generating functions of the stable and unstable invariant manifolds in terms of the generating function of the map.
16 A. Delshams and R. Ramrez-Ros (iii) This result follows directly from the geometric interpretation of the Melnikov potential and the Implicit Function Theorem. (iv) Let O " = ( b p k ( " )) k 2 Z , p k = b p k (0) = F k 0 ( p ), and v k =(d b p k = d " )(0). From equation (2.3), d S 0 = F 0 ; ,and d F 0 ( p k ) v k ]= v k +1 ; F 1 ( p k ), we obtain: W O " ] = X k 2 Z S " ( b p k ( " )) = X k 2 Z f S 0 ( p k )+ " ( S 1 ( p k )+ d S 0 ( p k ) v k ]) + O ( " 2 ) g = X k 2 Z S 0 ( p k )+ " X k 2 Z f S 1 ( p k )+ ( p k +1 )d F 0 ( p k ) v k ]] ; ( p k ) v k ] g + O ( " 2 ) = W O 0 ]+ " X k 2 Z n b S 1 ( p k )+ ( p k +1 ) v k +1 ] ; ( p k ) v k ] o + O ( " 2 ) = W O 0 ]+ "L ( p )+ O ( " 2 ) : Finally, the asymptotic formula for the homo clinic area follows from its denition, using (iii) of lemma 2.5. 2 Remark 2.5 The actions of homo clinic orbits arising from dierent connected comp onents of the separatrix need not b e equal at " = 0, see remark 2.1, whereas the splitting size is of order O ( " ). Thus, it seems inappropriate to measure the splitting comparing the action of homo clinic orbits arising from dierent comp onents of . For instance, in the planar case with a double lo op, the geometric sense of the area between primary homo clinic orbits arising from dierent lo ops is very unclear. Remark 2.6 If L has some non-degenerate critical p oint, the p erturb ed invariant manifolds of F " have a transverse intersection and, in particular, a top ological crossing. Thus, using some recent results contained in BW95], the p erturb ed maps have p ositive top ological entropy, for 0 < j " j 1. Let us see now that the Melnikov p otential is invariant under additional dieomorphisms, if the family f F " g has suitable symmetries. We recall that given a dieomorphism I : P ! P the family f F " g is called I -symmetric if F " I = I F " and F " ( p 1 )= I ( p 1 )= p 1 , for all " . Lemma 2.6 Assume that the family f F " g is I -symmetric, and that the symplectic potential is preserved by the symmetry: I = . Then, the Melnikov potential L is I -invariant: L I = L . Pro of. Let p 2W = W u s 0 and q = I ( p ). Using that F k 0 I = I F k 0 for all k 2 Z ,we get lim k !1 F k 0 ( q )= lim k !1 I ( F k 0 ( p )) = I lim k !1 F k 0 ( p ) = I ( p 1 )= p 1 : This proves that W is I -invariant. Thus, the separatrix also is, by the same argument as in (ii) of lemma 2.5, and the expression L I makes sense on .
Melnikov p otential for exact symplectic maps 17 From F " ; = d S " , I = ,and F " I = I F " we have d( S " I )= I (d S " )= I F " ; I = F " I ; = F " ; = d S " : Hence, S " I ; S " is a constant function that evaluated at p 1 vanishes, so S " (and in particular S 1 ) are I -invariant. The rst order terms of F " I = I F " give F 1 I = DI ( F 0 ) F 1 ]. Using this equality, we see that the function ( F 0 ) F 1 ] is also I -invariant: ( F 0 I ) F 1 I ]= ( I F 0 ) DI ( F 0 ) F 1 ]] = I ( F 0 ) F 1 ]= ( F 0 ) F 1 ] : Thus, the dierence b S 1 = S 1 ; ( F 0 ) F 1 ]is I -invariant, to o. Finally, L I = P k 2 Z ( b S 1 F k 0 I )= P k 2 Z ( b S 1 I F k 0 )= P k 2 Z ( b S 1 F k 0 )= L . 2 As wehave seen, the dierential of L measures the distance b etween invariant manifolds and thus M = d L is called the Melnikov function of the problem. It can be also constructed in the non-symplectic case, although it is not longer the dierential of a function. We recall now this construction, but we will not go further in this direction, since the non-symplectic framework is out of the spirit of this pap er. For the sake of simplicity,we only consider P = R 2 n . Assume that a dieomorphism F 0 : R 2 n ! R 2 n has a separatrix and n rst integrals H 1 :::H n , indep endentover the separatrix (but not necessarily in involution, since this concept requires a symplectic structure), and let F " = F 0 + "F 1 + O ( " 2 )be ageneral p erturbation of F 0 . Given p 2 , let % p be the n -dimensional linear variety spanned by the p oint p and the vectors r H j ( p ) (1 j n ). Since % p is transverse to at p , there exist p u s ( " ) 2W u s " \ % p , dep ending in a smo oth way on " , such that p u s (0) = p . A natural measure of the distance between the invariant manifolds is given by the dierence of rst integrals (\energies") ( p " )= H ( p u ( " )) ; H ( p s ( " )) = "M ( p )+ O ( " 2 ) H =( H 1 :::H n ) > where M : ! R n is the vector-valued Melnikov function of the problem. It is easy to generalize (actually, rewrite) the pro of given in DR96] for the planar case to see that M ( p )= X k 2 Z DH ( p k +1 ) F 1 ( p k )] p k = F k 0 ( p ) : (2.8) Remark 2.7 Some similar results can be found in BGK95], although with a less geometrical (and more functional) setting. They only can prove that a necessary condition for the existence of primary homo clinic orbits is the existence of zeros for M . Our geometrical construction shows that the existence of non-degenerate zeros for M is a sucient condition for the existence of transverse primary homo clinic orbits, even in the non-symplectic case. However, it should be noted that BGK95] deals with a broader range of maps for example, the existence of rst integrals is not needed.
18 A. Delshams and R. Ramrez-Ros 2.4 Twist maps Now, we present another formulation of the metho d that is useful for the physical problems that verify the twist condition, since the formula for the Melnikov p otential is simpler. For more details on twist maps, the reader is referred to Gol94a, Gol94b, BG96]. We follow closely the notations and denitions of the later reference. An exact symplectic twist map (or simply,twist map) F is a map from a connected subset U of the cotangent bundle of a manifold M (which can b e non-compact) into U , which comes equipp ed with a twist generating function L : M M ! R that satises F ( y d x ) ; y d x = Y d X ; y d x = d L ( x X ) ( X Y )= F ( x y ) where ( x y ) are any cotangent co ordinates on T M ,that is, x are co ordinates on M , extended to co ordinates ( x y ) in the obvious way. The canonical form 0 on T M reads as 0 = y d x in cotangent co ordinates. This can also be written in a co ordinate free manner. Given L , one can retrieve the map (at least implicitly) from y = ; @ 1 L ( x X ), and Y = @ 2 L ( x X ). This can be done globally (i.e., U = T M )only when M is dieomorphic to a b er of T M , for example when M is the covering space of T n or a manifold of constant negative curvature. The form F 0 ; 0 is exact, so F is exact. Let S : U ! R b e the generating function of F , in the geometric sense of the previous denitions. Then, S ( x y )= L ( x X ). The fact that S can be written in terms of old and new co ordinates: ( x X ), is the twist condition. In a co ordinate free formulation it reads as S ( p )= L ( ( p ) ( F ( p ))) 8 p 2 U (2.9) where : T M!M is the canonical pro jection. Now, we carry out the generalization of (2.9) for families of twist maps, dep ending (in a smo oth way) on a small parameter " .That is, we search for the relationship between the rst order variations in " of the twist and geometric generating functions. Lemma 2.7 Let f F " g be a smooth family of twist maps. Let L " (resp. S " ) be the twist (resp. geometric) generating function of F " . Set L " = L 0 + " L 1 + O ( " 2 ) and S " = S 0 + "S 1 + O ( " 2 ) . Then, b S 1 ( p )= L 1 ( ( p ) ( F 0 ( p ))) 8 p 2 U (2.10) where b S 1 is the function given in (2.6). Pro of. Fix p 2 U and let ( x y ) b e cotangent co ordinates in a neighb ourho o d of p . If we denote ( X " Y " )= F " ( x y )=( X 0 Y 0 )+ " ( X 1 Y 1 )+ O ( " 2 ), the O ( " ) terms of the equality S " ( x y )= L " ( x X " )give S 1 ( x y )= L 1 ( x X 0 )+ @ 2 L 0 ( x X 0 ) X 1 = L 1 ( x X 0 )+ Y 0 X 1 : Thus, from the denition of b S 1 and using 0 = y d x we get b S 1 ( x y )= L 1 ( x X 0 ). 2
Melnikov p otential for exact symplectic maps 19 Assume now that F 0 has a hyp erb olic xed point p 1 with a separatrix U and that F " : U ! U are exact dieomorphisms. The choice b S 1 ( p 1 ) = 0 reads as L 1 ( x 1 x 1 ) = 0 in the twist frame, where x 1 = ( p 1 ). From equation (2.10), it follows directly that the Melnikov p otential (2.7) can be written as L ( p )= X k 2 Z L 1 ( x k x k +1 ) x k = ( p k ) p k = F k 0 ( p ) : (2.11) This formula is simpler than (2.7), since only the rst order term of the twist generating function L " app ears in it. 2.5 Hamiltonian maps One of the main ideas in dynamical systems is to study maps in order to understand ows. For example, the description of Hamiltonian systems can be carried out considering the timeT maps of their ows, which are exact maps. Thus, it is interesting to present the previous results from the Hamiltonian p ointofview. Besides, this allows us to compare the discrete and continuous frameworks. Recall that a non-autonomous Hamiltonian system over an exact symplectic manifold ( P ! = ; d )isgiven by a real-valued function (called the Hamiltonian ) H : P R ! R . Then, the equations of motion have the form _ p = X H ( p t ), p 2 P , t 2 R , where for every xed t , X H ( t ) is the Hamiltonian eld generated by H ( t ): d H ( p t ) = ! ( p )( X H ( p t ) ), 8 p 2 P . In symplectic co ordinates ( x y )on P , we have = y d x , ! = d x ^ d y and the Hamiltonian equations take the canonical form _ x = @H @y ( x y t ) _ y = ; @H @x ( x y t ) : It is clear that X H do es not change if a function dep ending only on time is added to the Hamiltonian H . We will restrict ourselves to Hamiltonians suchthat generate a Hamiltonian ow, i.e., all the tra jectories of X H are dened for all time. A Hamiltonian map F is the timeT map of some Hamiltonian H and for some T > 0, i.e., F = T : P ! P , where t ( p ) stands for the solution of the Hamiltonian equations of H , with initial condition p at t = 0. Obviously, Hamiltonians maps are dieomorphisms isotopic to the identity. Besides, they are exact over exact manifolds if i ( X ) ! stands for the inner pro duct of a form ! byaeld X ,and': P R !P R is given by '( p t )=( t ( p ) t ), we get F ; = ( T ) ; ( 0 ) = Z T 0 d d t ( t ) ]d t = Z T 0 ' f i ( X H )d +d( i ( X H ) ) g d t = d " Z T 0 ' ( i ( X H ) ; H )d t # : Thus, the generating function S of F is given by S ( p )= Z ( F ( p ) T ) ( p 0) = ; H d t (2.12)
20 A. Delshams and R. Ramrez-Ros where the one-form is the so-called Poincare-Cartan invariant integral ,denedonthe (extended) phase space P R , and the path of integration is the tra jectory '( p t ), 0 t T ,of the (extended) ow. Now, we carry out the generalization of equation (2.12) for families of Hamiltonian maps, dep ending (in a smo oth way) on a small parameter " . That is, we lo ok for the relationship between the rst order variations in " of the Hamiltonians and the generating functions of their Hamiltonian maps. Lemma 2.8 Let H " be a smooth family of non-autonomous Hamiltonians, and t " ( p ) the solution of its Hamiltonian equations with 0 " ( p )= p . Let F " and S " be the Hamiltonian map T " and its generating function, respectively. Set H " = H 0 + "H 1 + O ( " 2 ) and S " = S 0 + "S 1 + O ( " 2 ) . Then b S 1 ( p )= ; Z T 0 H 1 ( t 0 ( p ) t )d t 8 p 2P (2.13) where b S 1 is the function given in (2.6). Pro of. Let ( p " )be the path in the (extended) phase space ( t " ( p ) t ), 0 t T . Set A " ( p t )= ( t " ( p )) _ t " ( p )] ; H 0 ( t " ( p ) t ), where the dot means the derivative with resp ect to the time t . We will use through the pro of the following notations for the rst variation of the considered ob jects: F 1 ( p )= @F " @" ( p ) " =0 t 1 ( p )= @ t " @" ( p ) " =0 A 1 ( p t )= @A " @" ( p t ) " =0 : Besides, we will prove b elow that A 1 ( p t )= _ B 1 ( p t ) B 1 ( p t )= ( t 0 ( p )) t 1 ( p )] : (2.14) From S " ( p )= R ( p" ) ; H " d t ], A 1 = _ B 1 , T 1 = F 1 and 0 1 0, we get S " ( p ) = Z ( p" ) ; H 0 d t ] ; " Z ( p" ) H 1 d t + O ( " 2 ) = Z T 0 A " ( p t )d t ; " Z T 0 H 1 ( t " ( p ) t )d t + O ( " 2 ) = S 0 ( p )+ " Z T 0 _ B 1 ( p t )d t ; " Z T 0 H 1 ( t 0 ( p ) t )d t + O ( " 2 ) = S 0 ( p )+ " ( F 0 ( p )) F 1 ( p )] ; " Z T 0 H 1 ( t 0 ( p ) t )d t + O ( " 2 ) and the terms O ( " ) in this equation give (2.13). To end the pro of, it only remains to checkthat (2.14) holds. For simplicity, we prove it using symplectic co ordinates. Given p 2 P and t 2 R ,let ( x y )be symplectic co ordinates in a neighb ourho o d of t 0 ( p ). We denote the co ordinates of t " ( p ) by ( x " y " )=( x 0 y 0 )+ " ( x 1 y 1 )+ O ( " 2 ). Thus, A " ( p t ) = y " _ x " ; H 0 ( x " y " t ) = A 0 ( p t )+ " y 0 _ x 1 + y 1 _ x 0 ; @ x H 0 ( x 0 y 0 t ) x 1 ; @ y H 0 ( x 0 y 0 t ) y 1 ]+ O ( " 2 ) = A 0 ( p t )+ " d y 0 x 1 ] = d t + O ( " 2 )
Melnikov p otential for exact symplectic maps 21 where we have used the canonical form of Hamiltonian equations in symplectic co ordinates. Finally, since in this co ordinates B 1 = y 0 x 1 ,equation (2.14) follows. 2 Henceforth, we restrict ourselves to time-p erio dic Hamiltonians H " ,being T their period. Assume now that F 0 has a hyp erb olic xed p oint p 1 with a separatrix . In the Hamiltonian frame, the choice b S 1 ( p 1 ) = 0 b ecomes R T 0 H 1 ( t 0 ( p 1 ) t )d t =0. Indeed, it is p ossible (and more usual) to determine the Hamiltonian in suchawaythatitveries the stronger condition H 1 ( t 0 ( p 1 ) t ) 0. From equation (2.13), it follows easily that the Melnikov potential (2.7) can be written as L ( p )= ; Z R H 1 ( t 0 ( p ) t )d t (2.15) since t 0 ( F k 0 ( p )) = t + kT 0 ( p ), for all integer k and real t ,and H 1 is T -p erio dic in t .(This is the reason to consider only p erio dic Hamiltonians.) Wewant to emphasize that the Hamiltonian version of the Melnikov p otential can b e deduced directly in the continuous frame, without app ealing to discrete to ols. However, taking into account the theory already develop ed in this pap er, it has been easier to work directly on Hamiltonian maps. Remark 2.8 Usually, the unp erturb ed Hamiltonian H 0 is time indep endent. In fact, in most of the applications it is Liouville integrable. Remark 2.9 Using the Lagrangian formalism instead of the Hamiltonian one, a similar formula to (2.15) can b e obtained for Lagrangian maps (i.e., timeT maps of some EulerLagrangian ow), but with ; H 1 replaced by the rst order in " of the Lagrangian. 2.6 Lower Bounds Along this subsection, we will assume without explicit mention that: (a) the invariant manifolds are doubled, that is, W u 0 = W s 0 ,and (b) the bifurcation set is minimal, i.e., = f p 1 g . (Remember that the hyp erb olic xed point p 1 is always contained in the bifurcation set , see (i) of lemma 2.5.) These hyp othesis are equivalent to require that the separatrix is = W u s 0 nf p 1 g . Wewillsay that the invariant manifolds are completely doubled in this case. Besides, we also assume n > 1, to avoid trivial degenerate cases. (In particular, the separatrix is connected.) To avoid a tedious exp osition, several standard computations ab out Betti numb ers are omitted. The exp ert reader in dierential and algebraic top ology will b e able to ll in the gaps without di$culty, and we prefer to give the appropriate references for the novice one, instead of writing here a treatise. Thus, for ageneral discussion of Morse theory we refer to Hir76], and for thorough discussions of homology the reader is urged to consult Swi75, GH81]. The quotient manifold := =F 0 , consisting of unp erturb ed homo clinic orbits of , will b e called the reducedseparatrix (of the unp erturb ed map). It is shown b elowthat
22 A. Delshams and R. Ramrez-Ros is a compact manifold without b oundary. Since the Melnikov p otential L is invariant under F 0 , we can consider it dened over the reduced separatrix. (The new function is called L ,to o.) We search for lower bounds of the number of homo clinic orbits and the main idea is to apply the Morse's inequalities to the map L : ! R . The presence of symmetries and/or reversions usually leads to b etter results concerning the existence of homo clinic orbits. Let us intro duce the (anti)symmetries that allow us to improve the lower b ounds. Wewillsay that the family f F " g is antisymmetric if f F " g is I -symmetric, for some involution I preserving the symplectic p otential such that DI ( p 1 ) = ; Id. As it is well-known, involutions are lo cally conjugate to their linear parts at xed points. Thus, there exist co ordinates z = ( z 1 :::z 2 n ) in some neighbourhood of p 1 such that I ( z )= ; z , that is, the maps F " are odd in some co ordinates dened close to p 1 . The denition ab oveofantisymmetric maps is intended to translate the main features of odd maps on ( R 2 n d x ^ d y ) to maps on general exact manifolds. Under these hyp otheses, lemma 2.6 claims that the Melnikov p otential is I -invariant. Thus, we can consider L dened over the quotientmanifold I := = f F 0 I g ,whichhas aricher top ological structure than , in the sense that Morse theory gives b etter lower bounds of the number of homo clinic orbits. We recall that a real-valued smo oth function over a compact manifold without boundary is called a Morse function when all its critical p oints are non-degenerate. It is very well-known that the set of Morse functions is op en and dense in the set of real-valued smo oth functions Hir76, page 147]. Thus, to be a Morse function is a condition of generic p osition. Now we can state a result ab out the number of primary homo clinic orbits that p ersist under a general p erturbation. In section 3, we will verify the optimalityof this result for sp ecic examples. Theorem 2.2 Assume that L : ! R is a Morse function. Then the number of primary homoclinic orbits is at least 4 . If the family f F " g is antisymmetric, there exist at least 2 n antisymmetric pairs of primary homoclinic orbits, and so at least 4 n primary homoclinic orbits. Pro of. From the celebrated Morse's inequalities, a Morse function over a n -dimensional compact manifold without b oundary X has at least SB ( X R ):= P n q =0 q ( X R ) critical points, where q ( X R ) are the R - Betti numbers of X and R is any eld. Let us recall that q ( X R ) is the dimension of the q -th singular homology R -vector space of X , noted H q ( X R ). In the antisymmetric case, I 2 ( p ) = p 6 = I ( p ), for all p 2 . Thus ( %) is a covering space of I of two sheets, where % : ! I is the canonical pro jection onto the quotient of by the antisymmetry I .In particular, L : I ! R is a Morse function if and only if the same happ ens to L : ! R ,and each critical point Q of L : I ! R corresp onds to an antisymmetric pair of critical p oints % ; 1 ( Q )= fO I ( O ) g of L : ! R , for some unp erturb ed homo clinic orbit O2 . Nowthe theorem follows from the formulae SB ( Z 2 ) = 4 and SB ( I Z 2 )=2 n . The rest of the pro of is devoted to checkthat these formulae hold.
Melnikov p otential for exact symplectic maps 23 Since Betti numb ers are top ological invariants, we lo ok for top ological spaces homeomorphic to and I whose homologies can be easily computed. To accomplish it, let us consider the restriction f u s of F 0 to W u s 0 ,and denote B u s = Df u s ( p 1 ). Since F 0 is symplectic, det( B u ) det( B s ) = 1, so det( B u )and det( B s )have the same sign. When these signs are p ositive (resp. negative) the map F 0 preserves (resp. reverses) the orientation of , and we denote by = + (resp. = ; ) the so-called index of orientation .In the following lemma it is shown that the top ological classication of f u only dep ends on .This will allowus to classify and I just in terms of . Lemma 2.9 Let A : R n ! R n be the linear isomorphisms given by: A ( x )=2 x x =( x 1 :::x n ) x =( x 1 x 2 :::x n ) : Then, there exists a global topological conjugation between f u and A ,that is, a homeomorphism g : R n ! W u 0 such that f u g = g A . In the antisymmetric case, the conjugation g can be chosen in such a way that g ( ; x )= I ( g ( x )) . Pro of. We note that p 1 is ahyp erb olic xed pointof f u ,andall the eigenvalues of B u have mo dulus greater than one. From PM82 , Th. 5.5, II x 5], we get that f u is lo cally conjugated at p 1 to A + (resp. A ; )in the orientation-preserving (resp. orientationreversing) case. This lo cal conjugation can be extended to a global one, using that f u and A are global repulsors. The existence of an antisymmetric conjugation (certainly, avery intuitive fact) follows the same lines. We omit the details. 2 Thanks to Lemma 2.9, we now easily intro duce time-energy coordinates ( t a ) on . First, wegive some notations. We denote by S n , T n ,and P n , the n -dimensional sphere, the n -dimensional torus, and the n -dimensional pro jective space, resp ectively.Besides, we intro duce the n -dimensional manifold X n := R S n ; 1 and the homeomorphism : X n ! R n n f 0 g , ( t a ) = 2 t a , whose inverse is given by ; 1 ( x ) = ( b t ( x ) b a ( x )) = (log 2 j x j x= j x j ). Then, b t ( A x ) = b t (2 x ) = b t ( x )+1 and b a ( A x )= b a (2 x )=( b a ( x )) ,so A = , where the map : X n ! X n is ( t a )=( t +1 a ) a =( a 1 :::a n ) a =( a 1 a 2 :::a n ) : Thus, F 0 : ! and : X n ! X n are top ologically conjugated by g ,where g is the conjugation given in lemma 2.9. This proves that = =F 0 and X n := X n = are homeomorphic. Hence, SB ( Z 2 )= SB ( X n Z 2 ). Concerning the antisymmetric case, we note that | = ; ,where | : X n ! X n | ( t a )=( t ; a ) : Thus, the pairs of maps F 0 I : ! and | : X n ! X n are simultaneously top ologically conjugated by g . This proves that I = = f F 0 I g and Y n := X n = f | g are homeomorphic. Hence, SB ( I Z 2 )= SB ( Y n Z 2 ).
24 A. Delshams and R. Ramrez-Ros Consequently, it only remains to prove that SB ( X n Z 2 )=4 and SB ( Y n Z 2 )=2 n . First, we consider the case =+. In this case, X n + = S 1 S n ; 1 and Y n + = S 1 P n ; 1 , since S 1 = R = f t = t +1 g and P n ; 1 = S n ; 1 = f a = ; a g . Therefore, from the well-known Z 2 -homologies H q ( S m Z 2 ) = ( Z 2 if q =0 m 0 otherwise H q ( P m Z 2 ) = ( Z 2 if 0 q m 0 otherwise and the K)unneth's Formula H q ( X Y Z 2 ) = L q p =0 H p ( X Z 2 ) H q ; p ( Y Z 2 ), we get H q ( X 2 Z 2 ) = 8 > < > : Z 2 if q =0 2 Z 2 Z 2 if q =1 0 otherwise H q ( X n Z 2 ) = ( Z 2 if q =0 1 n ; 1 n 0 otherwise for all n> 2, and H q ( Y n Z 2 ) = 8 > < > : Z 2 if q =0 n Z 2 Z 2 if q =1 :::n ; 1 0 otherwise for all n> 1. Adding dimensions, we get SB ( X n + Z 2 )=4 and SB ( Y n + Z 2 )=2 n . Finally, a standard Mayer-Vietoris sequence argumentshows that the Z 2 -homologies of X n and Y n do not dep end on ,so SB ( X n ; Z 2 )=4 and SB ( Y n ; Z 2 )=2 n . 2 Remark 2.10 Since the case = ; is more intricate, one could b elieve that it is b etter to replace the maps with their squares to get =+. However, it should be noted that the lower b ounds obtained in this way are worse since a single homo clinic orbit consist of two dierent ones for the square map: one gets2and2 n , instead of 4 and 4 n , as the number of homo clinic orbits. Thus, the case = ; deserves its own separate study. We also remark that this case cannot app ear in the continuous frame, since the maps generated by aow are isotopic to the identity. 3 Standard-like maps As a rst example we deal with standard-like maps over the symplectic manifold ( P ! )= ( R 2 n d x ^ d y ), n > 1, which are ones of the most celebrated examples of twist maps. Among them, we consider p erturbations of maps with central symmetry , since then the dynamics over the unp erturb ed separatrix is essentially one-dimensional and gives rise to explicit computations, as already announced in DR97c]. In the sequel, given x y 2 R n , x > y and j x j stand for the scalar pro duct P n i =0 x i y i and the Euclidean norm p x > x .
Melnikov p otential for exact symplectic maps 25 3.1 Central standard-like maps Let V : R n ! R be a function. The map F : R 2 n ! R 2 n with equations F ( x y ) = ( y ; x + r V ( y )) is called the standard-like map with potential V .It is immediate to check that L ( x X )= ; x > X + V ( X )isa twist generating function of F ,so F is a twist map. When V is even, F is odd. It is worth mentioning that standard-like maps are also expressed in the literature as F ( x 0 y 0 ) = ( x 0 + y 0 + r U ( x 0 ) y 0 + r U ( x 0 )), for some function U .The symplectic linear change of variables ( x 0 y 0 )=( y y ; x ) is the bridge b etween these two equivalent formulations, and the relation between the p otentials is given by V ( y ) = j y j 2 + U ( y ). Thus, it makes no dierence which formulation is used, since we deal with symplectic invariants. A central standard-like map is a standard-like map with a central p otential, i.e., V ( y )= V c ( j y j 2 ) for some function V c :0 1 ) ! R . Central standard-like maps are odd and have the \angular momenta" A ij ( x y )= x i y j ; x j y i as rst integrals. We denote by A n +1 0 = f ( x y ): A ij ( x y )=0 g the ( n + 1)-dimensional manifold in R 2 n of zero angular momenta. Clearly, A n +1 0 = f ( qa pa ): a 2 S n ; 1 ( q p ) 2 R 2 g . Let F be a central standard-like map with p otential V , and f : R 2 ! R 2 the standard-like area preserving map dened by f ( q p )=( p ; q +2 V 0 c ( p 2 ) p ). We will call f the reduced map (in A n +1 0 )of F . This denition b ecomes clear when it is noted that f ( q p )=( Q P ) () F ( qa pa )=( Qa P a ) 8 ( q p ) 2 R 2 a 2 S n ; 1 : (3.1) Our interest in central standard-like maps is motivated by the following lemma, which follows easily from (3.1). Lemma 3.1 Let F be a central standard-like map and f its reduced map. Assume that Sp ec Df (0)] = f e h g , for some h > 0 , and hence that the origin is ahyperbolic xed point of f . Then: (i) The origin is ahyperbolic xed point of F . Moreover, Sp ec DF (0)] = f e h g . (ii) Suppose now that f has a separatrix ; . Then, the invariant manifolds of F are completely doubled, giving rise to the separatrix = f ( qa pa ):( q p ) 2 ; a 2 S n ; 1 g : (iii) Let = ( q p ) : R ! ; be a natural parametrization of the separatrix ; , i.e., is a dieomorphism that satises f ( ( t )) = ( t + h ) ,for al l t 2 R . Then, the dieomorphism : R S n ; 1 ! dened by ( t a ):=( q ( t ) a p ( t ) a ) satises F ( ( t a )) = ( t + h a ) 8 t 2 R a 2 S n ; 1 : (3.2) Wenotethat f is o dd, so when it has a separatrix, it has in fact a double (symmetric) lo op.
32 A. Delshams and R. Ramrez-Ros (i) The origin is ahyperbolic xed point of F 0 .Moreover, Sp ec DF 0 (0)] = f e h g . (ii) The invariant manifolds of F 0 arecompletely doubled, giving rise to the separatrix = f ( ra _ ra ): _ r 2 = r 2 ; r 4 r 6 =0 a 2 S n ; 1 g : (iii) The dieomorphism : R S n ; 1 ! denedby ( t a )=( r ( t ) a _ r ( t ) a ) r ( t ) = sech t (4.1) veries s 0 ( ( t a )) = ( t + s a ) 8 t s 2 R a 2 S n ; 1 : (4.2) 4.2 Perturb ed problem Let us consider a p erturbation that preserves the natural character, i.e., the p erturb ed Hamiltonians are H " ( x y t )= T ( y )+( j x j 4 ;j x j 2 ) = 2+ "V ( x t=h ) h> 0 " 2 R where V = V ( x ' ) is 1-p erio dic in ' . We determine V by imp osing V (0 ' ) 0. Small values of h corresp ond to a rapidly forced p endulum of angular frequency (radians per second) ! = 2 =h . We denote by F " the Hamiltonian map h " ,where t " ( p )is the solution of the Hamiltonian equations of H " , with initial condition p .(The dep endence on the parameter h is omitted to simplify the notation.) Using equations (2.15), (4.2) and (4.1), the Melnikov p otential L : R S n ; 1 ! R of the problem turns out to b e L ( t a )= ; Z R V ( r ( t + s ) a s=h )d s = ; Z R V ( r ( s ) a ( s ; t ) =h )d s r ( s ) = sech s: (4.3) Now, we consider p olynomial p erturbations, that is, we assume that the TaylorFourier expansion of the p otential V has a nite number of terms. We write V ( x ' )= X ( k` ) 2K C k` ( x )cos(2 k' )+ S k` ( x ) sin(2 k' )] (4.4) where K is a nite subset of f ( k ` ) 2 Z 2 : k 0 ` 1 g and C k` , S k` are homogeneous p olynomials of degree ` . In this case, the Melnikov p otential can b e explicitly computed. The result is summarized in the following lemma, whose pro of is straightforward. Lemma 4.2 Let P ` ( ! ) ( ` 0 ) be the polynomials generated by the recurrences P 0 ( ! )=1 P 1 ( ! )= ! P ` +1 ( ! )= ! 2 + ` 2 ` ( ` +1) P ` ; 1 ( ! ) : (4.5) Then, the Melnikov potential (4.3) with V given in (4.4) is L ( t a )= X ( k` ) 2K f sech( k!= 2) P ` ; 1 ( k! ) C k` ( a ) cos( k!t ) ; S k` ( a ) sin( k!t )] g (4.6) where ! =2 =h is the frequency of the perturbation.
Melnikov p otential for exact symplectic maps 33 A typical dierence between the continuous and discrete frames is revealed here: the Melnikov potential (4.6) is an entire p erio dic function in the complex variable t , whereas the Melnikov p otential (3.6) is adoubly periodic one with singularities. Another dierence is that a theorem like 3.1 do es not hold for the p endulum, since there exist p erturbative p otentials V ( x ' ) such that the Melnikov p otential (4.6) vanishes identically. We also notice that sech( k!= 2) = sech( k 2 =h ) e ; 2 =h , when h ! 0. Thus, a discussion on a priori exp onentially small splittings for this rapidly forced magnetized p endulum, along the lines of the previous section, can be given for any p olynomial p erturbation. As in the previous section, the exp onentially small asymptotic expressions predicted by the Melnikov metho d are far of being proved for n > 1. However, it is well-known that for some p erturbations of the rapidly forced planar p endulum DS92], the Melnikov metho d gives the rightanswer. Finally,we consider the p erturbative p otential V ( x 1 x 2 ' )= 2 ! 2 +1 x 2 ( x 2 1 + x 2 2 ) cos(2 ' ) whichwas already studied in Gru85]. In that pap er, the general (non-Hamiltonian) case is considered, and consequently the symplectic structure is not taken into account, even in the examples where it was p ossible, likethe one ab ove. Using the formula (4.6), we get the Melnikov p otential L ( t a )= sech ! 2 sin # cos !t , where a = (cos # sin # ) 2 S 1 . Its gradientisjustthe vector-valued Melnikov function used in Gru85] to measure the splitting. Obviously, it is easier to compute a real-valued function than a vector-valued one. For higher dimensional cases, the saving of work is still bigger. App endix: Elliptic functions A function that plays an imp ortant role in the computation of the innite sums that app ear in Melnikov p otentials, is a complex function satisfying the following prop erties, where T h > 0 are given parameters: (C1) is meromorphic on C . (C2) is T i -p erio dic and its derivativeis h -p erio dic. (C3) The set of p oles of is h Z + T i Z ,and all of them are simple and of residue 1. Remark A.1 Conditions (C1){(C3) determine a function except for an additive constant: if 1 satises also (C1)-(C3), ( ; 1 ) 0 is an entire doubly p erio dic function, and it must b e a constant thus, ( z ) ; 1 ( z )= az + b ,but a =0duetothe T i -p erio dicity. The function can b e expressed in terms of Jacobian elliptic functions, Theta functions, or Weierstrassian functions. The Jacobian elliptic functions are well adapted
34 A. Delshams and R. Ramrez-Ros to p encil-and-pap er computations, whereas the Theta functions are the b est from the numerical p oint of view, and the Weierstrassian functions are the natural choice for theoretical work on account of their symmetry in the p erio ds. Here, we deal with p encil-and-pap er computations, so our choice are the Jacobian elliptic functions. For a general background on elliptic functions of anykind, we refer to AS72, WW27]. We followthe notation of the rst reference. Given the parameter m 2 0 1], werecall that K = K ( m ):= Z = 2 0 (1 ; m sin # ) ; 1 = 2 d # E = E ( m ):= Z = 2 0 (1 ; m sin # ) 1 = 2 d # are the complete el liptical integrals of the rst and second kind and that E ( u )= E ( u j m ):= Z u 0 dn 2 ( v j m )d v is the incomplete el liptic integral of the second kind , where dn is one of the well-known Jacobian el liptic functions . Moreover, intro ducing K 0 = K 0 ( m ) := K (1 ; m ), E 0 = E 0 ( m ):= E (1 ; m ), we also recall that the nome q , j q j < 1, is dened by q = q ( m ):= e ; K 0 =K .If any of the numb ers m , q , K , K 0 , E , E 0 or K 0 =K is given, all the rest are determined. From a numerical point of view, it is b etter to x rst the nome q , and after compute the rest of parameters and elliptic functions, since the q -series are rapidly convergent. It is not di$cult to check (see DR96]) that T ( z )=(2 K T =h ) 2 ( E 0 T =K 0 T ; 1) z +(2 K T =h ) E (2 K T z=h + K 0 T i j m T ) veries (C1)-(C3), where the nome is determined by q = q T =e ; T =h ,and m T , K T , K 0 T , E T , E 0 T are the asso ciated parameters. (The dep endence on h is not explicitly written.) Thus, K 0 T =K T = ; 1 log(1 =q T )= T=h: (A.1) Given an isolated singularity z 0 2 C of a function f , let us denote a ; j ( f z 0 ) the co e$cientof( z ; z 0 ) ; j in the Laurent expansion of f around z 0 .Obviously, a ; j ( f z 0 )=0 if z 0 is a p ole of f and j is greater than its order. Prop osition A.1 (Summation Formula) Let f be a function verifying: (P1) f is analytic in R and has only isolated singularities on C . (P2) f is T i -periodic for some T > 0 . (P3) j f ( t ) j A e ; c j< t j when j< t j!1 ,for some constants A c 0 . Then, ( t ):= P k 2 Z f ( t + hk ) is analytic in R , has only isolated singularities in C , and is doubly periodic with periods h and T i . Moreover, ( t ) can be expressed by the fol lowing sum ( t )= ; X z 2 Sing T ( f ) res( T ( ; t ) f ( ) z )= ; X z 2 Sing T ( f ) X j 0 a ; ( j +1) ( f z ) j ! ( j ) T ( z ; t ) (A.2) where Sing T ( f ) is the set of singularities of f in I T = f z 2 C :0 < = z<T g .
Melnikov p otential for exact symplectic maps 35 Pro of. See DR96, Prop. 3.1]. 2 If f is meromorphic in C ,the same happ ens to , and then is elliptic. From a computational p oint of view, this is the interesting case, since then (A.2) is a nite sum and can b e explicitly computed, as the following lemma, used in section 3, shows. Lemma A.1 Let ` ( t )= P k 2 Z f ` ( t + kh ) ,where f = sech . Then: 1 ( t ) = 2 K 2 h p m 2 cn 4 K 2 t h m 2 +dn 4 K 2 t h m 2 2 ( t ) = 2 K h 2 " E 0 K 0 ; 1+dn 2 2 K t h m # : Pro of. Clearly, f = sech satises prop erties (P1)-(P3) with T = 2 . Moreover, the singularities of f in I 2 = f z 2 C : 0 < = z < 2 g are simple p oles: i = 2and 3 i = 2, with a ; 1 ( f i = 2) = ; a ; 1 ( f 3 i = 2) = ; i. Thus, from (A.2) we get 1 ( t )= i 2 ( i = 2 ; t ) ; 2 (3 i = 2 ; t )] : From equation (A.1) with T = 2 ,and using that E ( u +2 K 0 i) ; E ( u ) is a constant, and that E ( ; u )= ; E ( u ), we have 1 ( t )= i(2 K 2 =h ) i ( K 0 2 ; E 0 2 ) ; E ( v= 2+ K 0 2 i j m 2 )+ E ( v= 2 j m 2 )] where v = u ; K 0 2 i and u =4 K 2 t=h . In WW27, pages 520 and 508] we nd the following formulae E ( v + K 0 i ) ; E ( v )= i( K 0 ; E 0 ) + cn( v ) ds( v ) cn( v= 2) ds( v= 2) = dn( v )+cn( v ) sn( v ) =ds( v ) + cs ( v ) : Therefore, we arrive at the following expression for 1 1 ( t )= ; i(2 K 2 =h )ds( u ; K 0 2 i j m 2 ) + cs( u ; K 0 2 i j m 2 )] and the formula for 1 follows from ds( u ; K 0 i) = i p m cn( u ) and cs ( u ; K 0 i) = idn( u ). The formula for 2 is easier, since f 2 = sech 2 also veries the prop erties (P1)-(P3), but with T = instead of T =2 .It has only one singularity in I : i = 2. Moreover, i = 2 is a double p ole with a ; 1 ( f 2 i = 2) = 0 and a ; 2 ( f 2 i = 2) = ; 1. Thus, by (A.2) we get 2 ( t )= 0 ( i = 2 ; t ). But E 0 ( u )=dn 2 ( u ) is an even 2 K 0 i -p erio dic function, so the formula for 2 follows from (A.1) for T = . 2
36 A. Delshams and R. Ramrez-Ros Acknowledgements This work has been partially supp orted by the EC grant ERBCHRXCT-940460 and the NATO grant CRG950273. Research by Amadeu Delshams is also supp orted by the Spanish grant DGICYT PB94-0215 and the Catalan grant CIRIT 1996SGR{00105. Research by Rafael Ramrez-Ros is also supp orted by the U.P.C. grant PR9409. Both authors wish to express our appreciation to J. Amoros, M.A. Barja and P.Pascual for their help on the topics related to algebraic top ology. It is also a pleasure to thank V. Gelfreich, A. Haro, R. de la Llave, J. Ortega, C. Simo, E. Tabacman and D. Treschev for very stimulating discussions and fruitful remarks. References AA89] V.I. Arnold and A. Avez. Ergodic Problems of Classical Mechanics . Advanced Bo ok Classics. Addison-Wesley,1989. AM78] R.H. Abraham and J.E. Marsden. Foundations of Mechanics . Benjamin/Cummings, Reading, Mass., 1978. Ang93] S. Angenent. Avariational interpretation of Melnikov's function and exp onentially small separatrix splitting. In D. Salamon, editor, Symplectic geometry , volume 192 of London Math. Soc. Lecture Note Ser. , pages 5{35. Cambridge Univ. Press, 1993. Arn76] V.I. Arnold. Les methodes mathematiques de la mecanique classique .Mir, Moscou, 1976. AS72] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions .Dover, 1972. BF96] M. Baldoma and E. Fontich. Poincare-Melnikov theory for n -dimensional dieomorphisms. Preprint, 1996. To app ear in Applicationes Mathematicae. BG96] P. Boyland and C. Gole. Lagrangian systems on hyp erb olic manifolds. Preprint 1996/1, Institute for Mathematical Sciences, Stony Bro ok, January 1996. BGK95] T.C. Bountis, A. Goriely, and M. Kollmann. A Melnikov vector for n -dimensional mappings. Phys. Lett. A , 206:38{48, 1995. Bol94] S.V. Bolotin. Homo clinic orbits to invariant tori of Hamiltonian systems. Preprint, 1994. ToappearinAdv. Sov. Math. BW95] K. Burns and H. Weiss. A geometric criterion for positive top ological entropy. Comm. Math. Phys. , 172:95{118, 1995. DR96] A. Delshams and R. Ramrez-Ros. Poincare-Melnikov-Arnold metho d for analytic planar maps. Nonlinearity , 9(1):1{26, 1996. DR97a] A. Delshams and R. Ramrez-Ros. Exp onentially small splitting of separatrices for standard-like maps. Preprint, 1997.
Melnikov p otential for exact symplectic maps 37 DR97b] A. Delshams and R. Ramrez-Ros. Melnikov metho ds and exp onentially small splitting of separatrices for planar maps: An exp erimental study. Preprint in progress, 1997. DR97c] A. Delshams and R. Ramrez-Ros. Poincare-Melnikov-Arnold metho d for symplectic maps. In Simo Sim97]. DRS97] A. Delshams, R. Ramrez-Ros, and T.M. Seara. Splitting of separatrices in Hamiltonian systems and symplectic maps. In SimoSim97]. DS92] A. Delshams and T.M. Seara. An asymptotic expression for the splitting of separatrices of the rapidly forced p endulum. Comm. Math. Phys. , 150:433{463, 1992. Eas91] R.W. Easton. Transp ort through chaos. Nonlinearity , 4:583{590, 1991. FS90] E. Fontich and C. Simo. The splitting of separatrices for analytic dieomorphisms. Ergodic Theory Dynamical Systems , 10:295{318, 1990. GG73] M. Golubitsky and V. Guillemin. Stable Mappings and Their Singularities ,volume 14 of Graduate Texts in Math. Springer, New York, 1973. GH81] M.J. Greenb erg and J.R. Harp er. Algebraic Topology| A First Course , volume 58 of Math. Lecture Note Ser. Addison-Wesley, Reading, Massachussets, 1981. Gol94a] C. Gole. Optical Hamiltonians and symplectic twist maps. Phys. D ,71:185{195, 1994. Gol94b] C. Gole. Perio dic orbits for Hamiltonian in cotangent bundles. Trans. Amer. Math. Soc. , 343(1):327{347, 1994. GPB89] M.L. Glasser, V.G. Papageorgiou, and T.C. Bountis. Melnikov's function for twodimensional mappings. SIAM J. Appl. Math. , 49(3):692{703, 1989. Gru85] J. Gruendler. The existence of homo clinic orbits and the metho d of Melnikov for systems in R n . SIAM J. Math. Anal. , 16(5):907{931, 1985. GS77] V. Guillemin and S. Sternb erg. Geometric Asymptotics . Amer. Math. So c., Providence, R.J., 1977. Hir76] M.W. Hirsch. Dierential Topology ,volume 33 of Graduate Texts in Math. SpringerVerlag, New York, Heidelb erg, Berlin, 1976. Lom93] H.E. Lomel. Applications of the Melnikov metho d to twist maps in higher dimensions using the variational approach. Preprint, Univ. of Minnesota, 1993. To app ear in Ergo dic Theory Dynamical Systems. Lom96a] H.E. Lomel. Perturbations of elliptic billiards. Phys. D , 99:59{80, 1996. Lom96b] H.E. Lomel. Saddle connections and hetero clinic orbits for standard maps. Nonlinearity , 9:649{668, 1996.
38 A. Delshams and R. Ramrez-Ros LT93] P. Levallois and M.B. Tabanov. Separation des separatrices du billard elliptique p our une p erturbation algebrique et symetrique de l'ellipse. C. R. Acad. Sci. Paris Ser. I Math. , 316:589{592, 1993. Mat86] J.N. Mather. A criterion for non-existence of invariant circles. Publ. Math. I.H.E.S. , 63:153{204, 1986. McL94] R.I. McLachlan. Integrable four{dimensional maps of standard typ e. Phys. Lett. A , 177:211{214, 1994. McM71] E.M. McMillan. A problem in the stability of p erio dic systems. In E. Brittin and H. Odabasi, editors, Topics in modern physics, a tribute to E.V. Condon , pages 219{244. Colorado Asso c. Univ. Press, Boulder, CO, 1971. Mei92] J.D. Meiss. Symplectic maps, variational principles, and transp ort. Rev. Modern Phys. , 64(3):795{848, 1992. MMP84] R.S. MacKay, J.D. Meiss, and I.C. Percival. Transp ort in Hamiltonian systems. Phys. D , 13:55{81, 1984. PM82] J. Palis, Jr. and W. de Melo. Geometric Theory of Dynamical Systems .SpringerVerlag, New York, 1982. Poi99] H. Poincare. Les methodes nouvel les de la mecanique celeste , volume 1, 2, 3. Gauthier-Villars, Paris, 1892{1899. Rom95] V. Rom-Kedar. Secondary homo clinic bifurcations theorems. Chaos , 5(2):385{401, 1995. RW88] V. Rom-Kedar and S. Wiggins. Transp ort in two-dimensional maps. Arch. Rational Mech. Anal. , 109:239{298, 1988. Sim97] C. Simo, editor. Hamiltonian Systems with Three or More Degrees of Freedom . NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Held in S'Agaro, Spain, 19{30 June 1995. Kluwer Acad. Publ., Dordrecht, Holland, to app ear in 1997. Sun96] J.H. Sun. Melnikov vector function for high-dimensional maps. Phys. Lett. A , 216:47{52, 1996. Sur89] Yu B. Suris. Integrable mappings of the standard typ e. Functional Anal. Appl. , 23:74{76, 1989. Swi75] R.M. Switzer. Algebraic Topology| Homotopy and Homology , volume 212 of Grund lehren Math. Wiss. Springer-Verlag, New York, Heidelb erg, Berlin, 1975. Tab94] M.B. Tabanov. Separatrices splitting for Birkho 's billiard in a symmetric convex domain, close to an ellipse. Chaos , 4(4):595{606, 1994. Tab95] E. Tabacman. Variational computation of homo clinic orbits for twist maps. Phys. D , 85:548{562, 1995.
Melnikov p otential for exact symplectic maps 39 Tre94] D.V. Treschev. Hyp erb olic tori and asymptotic surfaces in Hamiltonian systems. Russ. J. Math. Phys. , 2(1):93{110, 1994. Wei73] A. Weinstein. Lagrangian submanifolds and Hamiltonian systems. Ann. of Math. (2) , 98:377{410, 1973. Wig91] S. Wiggins. Introduction to applied nonlinear dynamical systems and chaos ,volume 3 of Texts Appl. Math. Springer, New York, 1991. WW27] E.T. Whittaker and G.N. Watson. A course of modern analysis .Cambridge Univ. Press, 1927. Internet access: All the authors' quoted preprints are available at http://www-ma1.upc.es in the preprints page, or at ftp://www-ma1.upc.es ,in the pub/preprints directory. E-mail addresses of the authors: [email protected] , [email protected] .