Poincaré-Melnikov-Arnold method for twist maps
Abstract
The Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Melnikov function (or Melnikov integral). If this function is not identically zero, the separatrices split. Moreover, the non-degenerate zeros of this function are associated to transversal intersections of the perturbed invariant (stable and unstable) manifolds. There exists a similar theory for planar maps, and in this case the Melnikov function is not a continuous sum anymore, but an infinite and (a priori) analytically uncomputable (discrete) sum. In a previous work, we have given a method to compute explicitly this kind of sums in terms of elliptic functions, under hypotheses of meromorphicity over the functions in the sum. This method allows us to obtain a strong non-integrability criterion and to apply it to perturbations of elliptic billiards and integrable standard-like maps like the McMillan map. Explicit estimates of the splitting angles are also given. Our aim is extend this method to the study of the splitting of doubly asymptotic manifolds (separatrices) associated to hyperbolic fixed points of twist maps in arbitrary dimensions. We work with maps generated globally by a generating function. Using the variational principle satisfied by these maps, we associate the non-degenerated critical points of a scalar function (here called Melnikov potential) to the transversal intersections of the perturbed asymptotic manifolds. We want to stress the difference of this point of view with the usual one in the literature, that is based in the study of non-degenerated zeros of a vectorial function. The symplectic structure and the variational principle play a fundamental role in our construction. As a first example where this theory can be applied, we study standard-like perturbations of a $2d$-dimensional twist map given by~R. McLachlan, for $d\ge 2$. This map is a multidimensional generalization of the McMillan map. We prove, among other results, that any entire perturbation destroys the separatrix of the McLachlan map.
Full text
POINCAR E-MELNIKOV-ARNOLD METHOD FOR TWIST MAPS AMADEU DELSHAMS AND RAFAEL RAMREZ-ROS 1. Intro duction A general theory for p erturbations of an integrable planar map with a separatrix to a hyp erb olic xed p oint has been develop ed in a previous lecture 5]. 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 the case of planar twist maps, M has zero mean and therefore there exists a p erio dic function L (called the Melnikov potential ) suchthat 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). The aim of this lecture is to present a similar theory for more dimensions. The natural frame is to consider twist maps on cotangent bundles. Once the suitable denition of unp erturb ed separatrix has b een intro duced (a non-trivial problem in the high-dimensional case), a scalar function L can be dened on it, in such a way that L veries the same prop erties than in the planar case. The derivation of L is easily related to variational principles, and the prop erty of b eing a scalar function instead of a vectorial function like the classical Melnikov function, makes it more useful for computations and geometrical understanding. Even more, it allows the application of Morse theory to establish the minimal numb er of transverse homo clinic orbits. The results to be presented in this lecture are valid for exact symplectic maps on arbitrary exact symplectic manifolds, that is, the twist character is not essential. Wehave restricted ourselves to twist maps only for simplicity. Full details of the ideas presented here are contained in 4], where another more general situation (i.e., the exact symplectic case) is studied. Related ideas can b e found in 1, 15, 14, 8, 7]. 2. The maps A twist map F is a map from a connected subset U of the cotangent bundle of a manifold M (which can be non-compact) into U , which comes equipp ed with a twist generating function L : MM ! 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 )intheobvious way. The symplectic form ! 0 on T M reads
2 as ! 0 = d x ^ d y 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 b e done globally (i.e., U = T M ) only when M is dieomorphic to ab er of T M ,for example when M is the covering space of T n or a manifold of constantnegative curvature. Finally, let us denote by : T M!M the canonical pro jection. 3. The theory Assume now that we are given a smo oth twist dieomorphism F 0 on the cotangent bundle T M . Let L 0 be its twist generating function. We assume that there exists a hyperbolic xedpoint z 1 0 of F 0 ,suchthatits n -dimensional (unstable and stable) invariant manifolds W u s 0 are doubled , that is, they coincide: W := W u 0 = W s 0 . In the planar case, the separatrix consists of the intersection of the invariant curves, except for the hyp erb olic xed p oint, which is the only p oint where suchinvariant manifolds are not submanifolds of the cotangent bundle. In the high-dimensional case, the situation is more complicated. We can consider three top ologies on the set W : the one induced by the inclusion W T M , and the two ones induced by the inclusions W W u s 0 . We rst dene the bifurcation set of this problem as the subset of W formed by the p oints such that the three top ologies do not coincide. Then, the separatrix is dened as its complementary in W , i.e., := Wn : With this denition, it turns out that is a doubly asymptotic exact submanifold of T M ,invariantby F 0 . Next, consider a p erturb ed twist map F " , and let L " = L 0 + " L 1 + O ( " 2 ) b e the twist generating function of F " .For 0 < j " j 1, there exists a hyp erb olic xed p oint z 1 " of F " , close to z 1 0 , and it is not restrictive to normalize the twist generating function by imp osing L " ( x 1 " x 1 " )=0, where x 1 " = ( z 1 " ). In particular, L 1 ( x 1 0 x 1 0 ) = 0, where x 1 0 = ( z 1 0 ). Wenow dene the Melnikov potential in the same way as for planar twist maps 5]: L : ;! R L ( z )= X k 2 Z L 1 ( x k x k +1 ) x k = ( z k ) z k = F k 0 ( z ) z 2 : (1) The Melnikov theory is based on the following prop erties of the Melnikov p otential 4]: ; L : ! R is well-dened, smo oth and invariant under the action of the unp erturb ed map: L F 0 = L . Consequently, L canbedenedonthe reducedseparatrix = =F 0 . ; The dierential of the Melnikov potential M = d L (called the Melnikov function ), measures, in rst order in " , the distance b etween the p erturb ed invariant manifolds, and is also dened on the reduced separatrix . ; If L 6 constant, then the p erturb ed invariant manifolds W u s " split for 0 < j " j 1, i.e., they do not coincide. ; If L has a critical p ointat z = z 0 then, for 0 < j " j 1, W u s " intersect transversally on a homo clinic p oint near z 0 . ; If the unp erturb ed invariant manifolds are completely doubled (i.e., = f z 1 0 g ), the reduced separatrix is a compact n -dimensional manifold without b oundary. Actually, if denotes the sign of the pro duct of the eigenvalues with mo dulus greater that one of DF 0 ( z 1 0 ), and S n ; 1 stands for the unit sphere of R n ,then is homeomorphic to
3 S S n ; 1 for = +, and it is somewhat more complicated for = ; (for more details, see 4]). In this situation, Morse theory applied to the Melnikovpotential, thought as a function over ,gives the minimal numb er of transverse homo clinic orbits, under conditions of generic p osition. ; There exists a variational principle ,in an analogous way to the one of the planar case 9, 6], which establishes that the homo clinic orbits of a twist map with twist generating function L are the extremals of the homoclinic action W O ]:= X k 2 Z L ( x k x k +1 ) O =( x k ) k 2 Z and a homoclinic area can b e dened for every pair of homo clinic orbits O =( x k ) k 2 Z , O 0 = ( x 0 k ) k 2 Z ,and is given by the dierence of homo clinic actions W O O 0 ] = W O 0 ] ; W O ]. In terms of the Melnikov p otential, there is also a nice expression for the homo clinic area: W O O 0 ]= " ; L ( z 0 0 ) ; L ( z 0 ) + O ( " 2 ) : We nish this survey of results with two remarks ab out dierent, but related, settings. 1. Regarding Hamiltonian ows, let H " : T M R ! R b e a time-p erio dic Hamiltonian of p erio d T ,and F " = T " under the conditions of this section, where t " ( z ) is the solution of the asso ciated Hamiltonian equations with initial condition z at t =0. If H = H 0 + "H 1 + O ( " 2 ), one can see 4] that the Melnikov potential takes the form (already known to Poincare) L ( z )= ; Z R H 1 ( t 0 ( z ) t )d t where H 1 is determined by imp osing H 1 ( t 0 ( z 1 0 ) t ) 0, or simply H 1 ( z 1 0 t ) 0, if H 0 is autonomous. 2. In relation with a non-symplectic setting, let us assume now that M = R n , that is, T M = R 2 n . In that case, using a dierent pointof view 13, 2], one can assume that the unp erturb ed map F 0 : R 2 n ! R 2 n p ossesses n indep endent rst integrals H 1 :::H n on the separatrix (not necessarily in involution, since this concept requires a symplectic structure), and consider a p erturbation F = F 0 + "F 1 + O ( " 2 ) (not necessarily symplectic). Then, the Melnikov vectorial function M : ! R n can b e written as (compare with the planar case in 5]): M =( M 1 ::: M n ) > M j ( z )= X k 2 Z hr H j ( z k +1 ) F 1 ( z k ) i z k = F k 0 ( z ) z 2 : 4. The example Let us consider central standard-like maps on R 2 n = T R n , that is, F 0 ( x y )=( y ; x + r V 0 ( y )) or L 0 ( x X )= ;h x X i + V 0 ( X ) (2) where V 0 ( x )= V c ( k x k 2 ) for some function V c :0 1 ) ! R . Then, the \angular momenta" A ij ( x y )= x i y j ; x j y i are rst integrals and the ( n + 1)-dimensional manifold in R 2 n of zero angular momenta is A n +1 0 := f ( x y ): A ij ( x y )=0 g = f ( qa pa ): a 2 S n ; 1 ( q p ) 2 R 2 g .
4 We nowintro duce the reduced map in A n +1 0 of F , as the planar standard-like map f : R 2 ! R 2 dened by f ( q p )=( p ; q +2 V 0 c ( p 2 ) p ). We note that f ( q p )=( Q P ) () F ( qa pa )=( Qa P a ) 8 ( q p ) 2 R 2 a 2 S n ; 1 (3) so that the non-trivial dynamics on the separatrix is induced by the reduced map. Totakeadvantage of the results for planar twist maps in the lecture 5], weintro duce nowthe McLachlan map 10] as the central standard-like map with potential V 0 ( y ) = ln(1 + k y k 2 ), > 1. It has the expression F 0 ( x y )= y ; x + 2 y 1+ k y k 2 ! > 1 : (4) Its reduced map is nothing else but the McMillan map 11] whose separatrix ; = ; + has the following natural parameterization 3]: ;= f z 0 ( t )=( q 0 ( t ) p 0 ( t )) g q 0 ( t )= p 0 ( t ; h ) p 0 ( t ) = sinh h sech t where h> 0 is determined by the equation cosh h = : Now, it is easy to checkthat 1. The origin is a hyp erb olic xed p ointof F 0 , and Sp ec DF 0 (0)] = f e h g . 2. The invariant manifolds of F 0 are completely doubled, and the separatrix is given by = f ( qa pa ):( q p ) 2 ; a 2 S n ; 1 g : 3. The function z 0 : R S n ; 1 ;! given by z 0 ( t a )=( p 0 ( t ; h ) a p 0 ( t ) a ) p 0 ( t )=sinh h sech t (5) is a natural parameterization of the separatrix, i.e., z 0 is a dieomorphism that satises F 0 ( z 0 ( t a )) = z 0 ( t + h a ), for t 2 R and a 2 S n ; 1 . As exp ected, wenow consider a general p erturbation of (4) that preserves the standard character, i.e., F " ( x y )= y ; x + 2 y 1+ k y k 2 + " r V ( y ) ! > 1 " 2 R (6) with V : R n ! R determined by imp osing V (0) = 0. The generating function of F " that vanishes at the origin is L " = L 0 + " L 1 , where L 0 ( x X )= ;h x X i + ln(1 + k X k 2 ) and L 1 ( x X )= V ( X ). The Melnikovpotential is simply L : R S n ; 1 ! R L ( t a )= X k 2 Z V ( p 0 ( t + hk ) a ) p 0 ( t )= sinh h cosh t : (7) Since L is h -p erio dic in t (this is the invariance of the Melnikov potential under the action of the unp erturb ed map), wecan consider t dened mo dulo h , i.e., L dened over the reduced separatrix S 1 S n ; 1 . Rep eating the arguments for the case of the planar twist maps 5], we see that if V is a non-constant real entire function, then V ( p 0 ( t ) a ) has the same isolated singularities in the complex variable t as p 0 ( t ), and it is not di cult to check that they remain as singularities for the Melnikov p otential, whichmust b e non-constant. In this waywe have established the following result.
5 Theorem 1 If V is a non-constant real entire function, then the perturbed invariant manifolds of the standard-like map (6) split, for 0 < j " j 1 . References 1. S.V. Bolotin. Homo clinic orbits to invariant tori of Hamiltonian systems. Preprint, 1994. To app ear in Adv. Sov. Math. 2. T.C. Bountis, A. Goriely, and M. Kollmann. A Melnikovvector for n -dimensional mappings. Phys. Lett. A , 206:38{48, 1995. 3. A. Delshams and R. Ramrez-Ros. Poincare-Melnikov-Arnold metho d for analytic planar maps. Nonlinearity , 9(1):1{26, 1996. 4. A. Delshams and R. Ramrez-Ros. Melnikov potential for exact symplectic maps. Preprint, May 1996. To app ear in Comm. Math. Phys. 5. A. Delshams and R. Ramrez-Ros and T.M. Seara. Splitting of separatrices in Hamiltonian systems and symplectic maps. In Simo 12]. 6. R.W. Easton. Transport through chaos. Nonlinearity , 4:583{590, 1991. 7. H.E. Lomel. Saddle connections and hetero clinic orbits for standard maps. Nonlinearity , 9:649{668, 1996. 8. 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 appear in Ergo dic Theory Dynamical Systems. 9. R.S. MacKay, J.D. Meiss, and I.C. Percival. Transp ort in Hamiltonian systems. Phys. D , 13:55{81, 1984. 10. R.I. McLachlan. Integrable four{dimensional maps of standard typ e. Phys. Lett. A , 177:211{214, 1994. 11. E.M. McMillan. A problem in the stability of p eriodic 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. 12. C. Simo, editor. Hamiltonian Systems with Three or MoreDegrees 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. 13. J.H. Sun. Melnikovvector function for high-dimensional maps. Phys. Lett. A , 216:47{52, 1996. 14. E. Tabacman. Variational computation of homo clinic orbits for twist maps. Phys. D , 85:548{562, 1995. 15. D.V. Treschev. Hyp erbolic tori and asymptotic surfaces in Hamiltonian systems. Russ. J. Math. Phys. , 2(1):93{110, 1994. Internet access: All the authors' quoted and related preprints are available at http://www-ma1.upc.es in the Preprints pages, or at ftp://ftp-ma1.upc.es ,in the pub/preprints directory.