Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing
Abstract
Quasiperiodic perturbations with two frequencies $(1/\varepsilon ,\gamma /\varepsilon )$ of a pendulum are considered, where $\gamma $ is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for $\varepsilon $ small enough. The value of the splitting, that turns out to be ${\rm O} (\exp (-{\rm const} /\sqrt{\varepsilon }) )$, is correctly predicted by the Melnikov function.
Full text
LOWER AND UPPER BOUNDS FOR THE SPLITTING OF SEPARATRICES OF THE PENDULUM UNDER A FAST QUASIPERIODIC FORCING AMADEU DELSHAMS, VASSILI GELFREICH, ANGEL JORBA, AND TERE M. SEARA Revised version, January 1997 Abstract. Quasip erio dic p erturbationswith two frequencies (1 =" =" )ofa p endulum are considered, where is the golden mean number. We study the splitting of the three-dimensionalinvariant manifolds asso ciated to a twodimensional invariant torus in a neighb ourhoo d of the saddle p ointof the p endulum. Provided that some of the Fourier co ecients of the p erturbation (the ones asso ciated to Fib onacci numb ers) are separated from zero, it is proved that the invariant manifolds split for " small enough. The value of the splitting, that turns out to b e O ; exp ; ; const = p " , is correctly predicted by the Melnikov function. 1. Introduction The rapidly (and p erio dically) forced p endulum has b een widely used as a mo del for the motion near a resonance of Hamiltonian systems with two degrees of freedom. As it is well-known in several situations DS92, Gel93], the separatrices of the p erturb ed system do not coincide, giving rise to the so-called splitting of separatrices, which seems to b e the main cause of sto chastic b ehaviour in Hamiltonian systems. In this announcementwe consider a quasip erio dic high-frequency p erturbation of the p endulum (it can b e considered as a mo del near a resonance of a Hamiltonian system with three degrees of freedom), describ ed by the Hamiltonian function ! I " + h ( x y " ) (1) where ! I = ! 1 I 1 + ! 2 I 2 h ( x y " )= y 2 2 +cos x + " p m ( 1 2 )cos x with symplectic form d x ^ d y +d 1 ^ d I 1 +d 2 ^ d I 2 .We assume that " is a small p ositive parameter and that p is a p ositive parameter. Mainly due to a technical limitation imp osed by the Extension Theorem (Theorem 2), we will restrict ourselves to the case p> 3. We also assume that the frequency is of the form !=" for ! =(1 ), where =(1+ p 5) = 2 is the golden mean. The equations of motion related to Hamiltonian (1) are: 1991 Mathematics Subject Classication. Primary 34C37, 58F27, 58F36 Secondary 11J25. Key words and phrases. Splitting of separatrices, quasip erio dic forcing, homo clinic orbits, normal forms. 1
2 A. DELSHAMS, V. GELFREICH, A. JORBA, AND T.M. SEARA _ x = y _ y =(1+ " p m ( 1 2 )) sin x _ 1 = 1 " _ I 1 = ; " p cos x @m @ 1 ( 1 2 ) (2) _ 2 = " _ I 2 = ; " p cos x @m @ 2 ( 1 2 ) : The function m is assumed to b e a 2 -p erio dic function of twovariables 1 and 2 .Thus it can b e represented as a Fourier series: m ( 1 2 )= X k 1 k 2 m k 1 k 2 e i ( k 1 1 + k 2 2 ) : We assume that, for some p ositivenumb ers r 1 and r 2 , sup k 1 k 2 m k 1 k 2 e r 1 j k 1 j + r 2 j k 2 j < 1 (3) and that there are p ositivenumbers a and k 0 ,suchthat j m k 1 k 2 j a e ; r 1 j k 1 j; r 2 j k 2 j (4) for all k 1 , k 2 such that j k 1 j = F n +1 and j k 2 j = F n , where F n and F n +1 are Fib onacci numb ers, which are dened by the following recurrent formula F 0 =1 F 1 =1 F n +1 = F n + F n ; 1 n 1 : (5) We call the corresp onding terms in the p erturbation to b e resonant or Fibonacci terms . For example, the function m ( 1 2 )= cos 1 cos 2 (cosh r 1 ; cos 1 )(cosh r 2 ; cos 2 ) satises these conditions. The upp er b ound (3) implies that the function m is analytic on the strip fj= 1 j < r 1 g fj= 2 j <r 2 g . Equation (4) implies that this function can not b e continued analytically onto a larger strip. Let us select 2 (0 1]. Estimate (3) implies that j m ( 1 2 ) j K" ; 2 (6) on the strip j= 1 j r 1 ; " j= 2 j r 2 ; " : Formula (4) implies that the upp er b ound (6) can not b e improved. It will b e seen that the value of the splitting dep ends essentially on the width of these strips. Moreover, formula (4) will allow us to estimate in the next section the size of the Melnikov function in terms of the separatrix of the unp erturb ed p endulum. (See remark 4.) The function m under consideration has a singularity \of the second order", in the sense that the upp er b ound (6) for the maximum of the mo dulus is quadratic with resp ect to the inverse of the distance to the b oundary of the strip. In a similar way the case of a singularityof any \order" q can b e considered. In this case m k 1 k 2 should b e replaced by m k 1 k 2 = j k j q ; 2 in (3) and (4). The Hamiltonian (1) can b e considered as a singular p erturbation of the p endulum h 0 = y 2 2 +cos x: (7)
SEPARATRIX SPLITTING FOR FAST QUASIPERIODIC FORCING 3 The unp erturb ed system has a saddle p oint(0 0) and a homo clinic tra jectory given by x 0 ( t ) = 4 arctan( e t ) y 0 ( t )= _ x 0 ( t ) : The complete system (2) has a whiskered torus T :(0 0 1 2 ). The whiskers are 3 D -hyp ersurfaces in the 4 D -dimensional extended phase space ( x y 1 2 ). These invariant manifolds are close to the unp erturb ed p endulum separatrix. Our main result (Theorem 3) is that, if condition (4) is veried, then for p> 3 and small "> 0theinvariant manifolds split, and the value of the splitting (i.e., the distance function b etween these invariant manifolds), is correctly predicted by the Melnikov function, whichis O (e ; const = p " ). Remark 1 . Our mo del (1) is based on a previous work byC.Simo Sim94], where Neishtadt's Averaging Theorem Nei84] was generalized to quasip erio dic systems, giving rise to upp er estimates of the splitting which are exp onentially small with resp ect to the parameter of p erturbation " . Related upp er estimates can b e found in CG94, Gal94, BCG95, Ben96 ]. In contrast to these results, Theorem 3 provides both lower and upp er b ounds for our mo del. Remark 2 . As an example from DGJS96b]shows, the splitting can b e of the order of some p ower of " if the function m is not analytic. This makes a rst qualitative dierence b etween p erio dic and quasip erio dic p erturbations. Indeed, in the p erio dic case, only the C 1 dep endence with resp ect to of the p erturb ed Hamiltonian is needed to prove that the splitting is O (e ; c=" ), where c is the width of the analyticity strip of the unp erturb ed separatrix. (In b oth cases, the analyticityof the unp erturb ed system is essential.) Remark 3 . In the case of an entire function m ,we think that the metho d used in the present pap er can b e mo died in order to improve the estimate of the error and to prove that the Melnikov function gives the actual asymptotics at least when the resonant terms decrease not muchfasterthan1 =k !. 2. The Melnikov function As it is well-known, the Melnikov function M ( 1 2 " )= Z 1 ;1 f h 0 h g ( x 0 ( t ) y 0 ( t ) 1 + t=" 2 + t=" ) dt (8) gives a rst order approximation of the dierence b etween the values of the unp erturb ed p endulum energy h 0 on the stable and unstable manifolds. Using the Fourier series of m ( 1 2 ), one can compute the Fourier co ecients of M ( 1 2 )as M k 1 k 2 ( " )= ; 2 i" p ( k 1 + k 2 ) 2 " 2 sinh ( ( k 1 + k 2 ) = (2 " )) m k 1 k 2 : In order to b ound the Melnikov function, it is imp ortantto know for eachxed " which are the indices ( k 1 k 2 ) corresp onding to the biggest Fourier co ecient M k 1 k 2 ( " ). From the expression ab ove, M k 1 k 2 ( " ) is a pro duct of two factors. For " xed and small enough, the rst factor of M k 1 k 2 ( " ) b ehaves as " p ; 2 ( k 1 + k 2 ) 2 e ; ( k 1 + k 2 ) = (2 " ) , and it turns out that it b ecomes bigger for small k 1 + k 2 , i.e., just for the resonant terms where the second factor m k 1 k 2 decreases with resp ect to ( k 1 k 2 ) according to the b ehaviour (3) and (4). Here wehave the main dierence between the quasip erio dic case and the p erio dic one: for a p erio dic p erturbation, the
4 A. DELSHAMS, V. GELFREICH, A. JORBA, AND T.M. SEARA rst Fourier co ecients M 1 ( " ) of the Melnikov function M ( " )give generically the main contribution to the Melnikov function M ( " ), since M k ( " )= O (e ;j k j =" ). However, in the quasip erio dic case the biggest Fourier co ecient dep ends strongly on the exp onent k (1 )= k 1 + k 2 of the Fourier co ecient, i.e., on the rational approximations of . For =(1+ p 5) = 2, it is very well-known that its b est approximation by rational numb ers is given by the quotient of successive Fib onacci numb ers (5). Indeed, it is easy to check that for large values of n one has the following approximation of by Fib onacci numb ers F n ; F n ; 1 =( ; 1) n C F F n ; 1 + O 1 F 3 n ; 1 C F = 1 + ; 1 whereas for the other integer numb ers one has the following result. Lemma 1. If N 2 N is not a Fibonacci number, then for al l integer k j k ; N j > C F N : Using this lemma one can see that the indices ( k 1 k 2 ) corresp onding to leading Fourier co ecients M k 1 k 2 ( " ) dep end on " . In fact, the largest terms corresp ond to ( k 1 k 2 )= ; F n ( " )+1 ; F n ( " ) ,where F n ( " ) is the Fib onacci numb er closest to F ( " )= p 0 =" , where 0 = = (2( + ; 1 )( r 1 + r 2 )). Except for a small neighb ourho o d of " = " ; n , with " given in (10), there is a unique Fib onacci number closest to F ( " ), and then only two corresp onding terms dominate in the Fourier series. Studying the size of this term (which also dep ends on r 1 and r 2 ), one can observe that it is O (e ; c= p " ). With a more detailed analysis, one can see that a b etter estimate is provided taking c not as a constant function but as a b ounded oscillating one: c ( )= C 0 cosh ; 0 2 for 2 0 ; log 0 +log ] (9) where C 0 = s 2 ( r 1 + r 2 ) + ; 1 0 = log " " = ( + ; 1 ) 2 2 ( r 1 + r 2 ) (10) and continued p erio dically onto the whole real axis. In this way, the function c is piecewise-analytic, continuous and 2 log -p erio dic. This is summarized in the following lemma. Lemma 2 (Prop erties of the Melnikov function) . The Melnikov function dened by (8) is a 2 -periodic function of 1 and 2 , such that 1) M ( 1 ; T =" 2 ; T =" " ) is analytic in the product of strips fj= 1 j <r 1 g fj= 2 j <r 2 gfj= T j <= 2 g 2) the maximum of the modulus of the Melnikov function taken on real arguments, max ( 1 2 ) 2 T 2 j M ( 1 2 ) j ,can beboundedfrom above and from below by terms of the form const " p ; 1 exp ; c (log " ) p " (11) with dierent " -independent constants, where the function c in the exponent is denedby (9)
SEPARATRIX SPLITTING FOR FAST QUASIPERIODIC FORCING 5 3) for a xed smal l " only 4 terms (at most) dominate in the Fourier series for the Melnikov function and the rest can be estimatedfrom above by O (e ; C 1 = p " ) , wherethe constant C 1 > max c ( )= C 0 cosh (log p ) . As wehave established that for most small values of " only the terms with ( k 1 k 2 )= ( F n ( " )+1 ; F n ( " ) ) are imp ortant, the Melnikov function is essentially M ( 1 2 " ) 2 M F n ( " )+1 ; F n ( " ) sin ; F n ( " )+1 1 ; F n ( " ) 2 + ' ( " ) : The zeros of the Melnikov function corresp ond to homo clinic tra jectories. The ab ove formula implies that the zeros of the Melnikovfunction formtwo lines on the torus. As already noticed byC.Simo Sim94], the averaged slop es of those lines approachto when " ! 0. Remark 4 . We note that the Melnikov function is not invariant with resp ect to canonical changes of variables. After a change, e.g., after a step of the classical averaging pro cedure, a lot of non-zero harmonics, whichwere not presentinthe original system, can app ear. If in the original system the Fib onacci terms were not big enough, these new harmonics maygive larger contribution to the splitting. This idea was used in Sim94] to detect the splitting for a system with only 4 p erturbing terms. Remark 5 . The hyp othesis that ! in the frequency vector is just (1 )can be relaxed. The generalization of the present result to the case when is a quadratic numb er is straightforward, with a similar expression (11) for the size of the Melnikov function. The case in which ! =( ! 1 ! 2 ), with the ratio ! 1 =! 2 b eing of constant typ e (the continued fraction expansion has b ounded co ecients), but not quadratic, can b e similarly analyzed, but in this case c ( ) is no longer a p erio dic function. In some sense one can say, prop erly sp eaking, that there are no asymptotics. But it seems that there still exist upp er and lower b ounds, with the factor p " in the denominator of the exp onential term. The case of two frequencies whose ratio ! 1 =! 2 is not of constanttyp e, as well as the case of more than two p erturbing frequencies, is more complicated. In the following sections wesketch the metho d used to justify that the prediction given by the Melnikov function is correct. The metho d used here is a generalization to the quasip erio dic case of the metho d used in Laz84, DS92, Gel93]. 3. Normal Form and Local Manifolds The rst step is to give a description of the dynamics near the 2 D -dimensional invariant torus T .So,we will show the existence of a convergent normal form in a neighbourhood of T . As wehave seen during the analysis of the Melnikov function, the size of the splitting dep ends essentially on the widths of the analyticity strip ( r 1 r 2 )ofthe angular variables 1 , 2 ,aswell as on the width of the analyticity strip of the separatrix ( x 0 ( t ) y 0 ( t )). Therefore, to detect the splitting in the quasip erio dic case the loss of domain in the angular variables must b e very small (i.e., O ( " ), where dep ends on the Diophantine prop erties of the frequencies). This makes another dierence with the p erio dic case, where the size of the splitting do es not dep end on the width of the analyticity strip of the angular variable , but only on the width of the analyticity strip of the separatrix ( x 0 ( t ) y 0 ( t )). When dealing with the frequencies (1 ) one needs a reduction of O ( p " ) at most. Hence, during the
6 A. DELSHAMS, V. GELFREICH, A. JORBA, AND T.M. SEARA pro of of the convergence of the normal form one has to b ound carefully the loss of domain (with resp ect to the angular variables) in order to achievesuch a small reduction. Finally,wewant to stress that if the amount of reduction is something bigger, one can only pro duce upp er b ounds for the splitting of separatrices. Theorem 1 (Normal Form Theorem) . Let " 2 (0 " 0 ) . In a neighbourhoodofthe hyperbolic torus T thereisacanonical change of variables ( x y ) ! ( X Y ) , which depends 2 -periodical ly on 1 and 2 , such that the Hamiltonian (1) takes the form H ( XY " )= H 0 ( XY )+ " p ; 1 H 1 ( XY " ) where H 0 is the normal form Hamiltonian for the unperturbedpendulum. Moreover, the change of variables has the form x = x (0) ( X Y )+ " p ; 1 x (1) ( X Y 1 2 " ) y = y (0) ( X Y )+ " p ; 1 y (1) ( X Y 1 2 " ) (12) where ( x (0) y (0) ) are normal form coordinates for the unperturbedpendulum. The functions H 0 , H 1 , x (0) , y (0) , x (1) and y (1) are analytic and uniformly bounded in the complex domain denedby j X j 2 + j Y j 2 <r 2 0 j= 1 j <r 1 ; p " j= 2 j <r 2 ; p " for r 1 and r 2 from (3) and some positive constant r 0 > 0 . The pro of of this theorem can b e found in DGJS96a]. The Normal Form Theorem provides a convenient parametrization for the lo cal invariant manifolds. Let be H 0 (0 " ). Then, x = x s ( T 1 2 ) x (0 e ; T 1 2 ) y = y s ( T 1 2 ) y (0 e ; T 1 2 ) for T T 0 (13) and x = x u ( T 1 2 ) x (e T 0 1 2 ) y = y u ( T 1 2 ) y (e T 0 1 2 ) for T ; T 0 (14) where wehave used the change (12). Theorem 1 also implies that, in the domains ab ove, x ( T 1 2 ) ; x 0 ( T ) C" p ; 1 y ( T 1 2 ) ; y 0 ( T ) C" p ; 1 for = s u: 4. Extension Theorem The Normal Form Theorem provides a lo cal approximation for the unstable manifold in terms of the unp erturb ed separatrix, whichis O ; " p ; 1 . The following theorem extends this lo cal approximation for solutions of system (2) to a global one. Since the unp erturb ed separatrix ( x 0 ( T ) y 0 ( T )) has a singularityon T = = 2, we will restrict ourselves to j= T j = 2 ; p " , i.e., up to a distance to the singularity T = = 2 of the same order as the loss of domain in the angular variables. Besides, the extension time t + T will b e chosen big enough in order that the unp erturb ed separatrix reaches again the domain of convergence of the normal form. This pro cedure follows the same ideas as in the Extension Theorem of DS92], and its complete pro of can b e also found in DGJS96a].
SEPARATRIX SPLITTING FOR FAST QUASIPERIODIC FORCING 7 Theorem 2 (Extension Theorem) . Assume p> 2 . Then, there exists " 0 > 0 such that the fol lowing extension property holds: For any positive constants C and T 0 thereexistsaconstant C 1 , such that for any " 2 (0 " 0 ) , every solution of system (2) that satises the initial conditions j x ( t 0 ) ; x 0 ( t 0 + T ) j C" p ; 1 j y ( t 0 ) ; y 0 ( t 0 + T ) j C" p ; 1 j= 1 ( t 0 ) j r 1 ; p " j= 2 ( t 0 ) j r 2 ; p " for some T 2 C , t 0 2 R with j= T j = 2 ; p " ; T 0 t 0 + < T< 0 can be extendedfor ; T 0 t + < T T 0 , verifying j x ( t ) ; x 0 ( t + T ) j C 1 " p ; 2 j y ( t ) ; y 0 ( t + T ) j C 1 " p ; 2 : In particular, Theorem 2 can b e applied to the lo cal invariant unstable manifold given in (14). As we will see in Lemmas 3 and 4, the approximation ab ove of these invariant manifold in such a complex domain will allowustoderive suitable b ounds of the error on the real axis to detect the splitting. Before closing this section let us note that, as a direct consequence of the Extension Theorem, the dierence of unp erturb ed energies along the invariant manifolds can also b e estimated. Corollary 1. The fol lowing estimate holds h 0 ( x u y u ) ; h 0 ( x s y s )= M ( 1 ; T=" 2 ; T =" )+ O " 2( p ; 2) where h 0 is evaluated on the invariant manifolds corresponding to T 1 2 , < T 2 ( T 0 ; R T 0 ) j= T j = 2 ; p " j= k j r k ; p " k =1 2 (15) for any positive constants T 0 and R , R<T 0 . 5. First return By Theorem 1, the lo cal unstable invariant manifold is " p ; 1 -close to the unp erturb ed separatrix. By Theorem 2, it can b e continued for ; T 0 t + < T T 0 , provided that the parameters ( 1 2 T ) b elong to the complex domain (15), and remains " p ; 2 -close to the unp erturb ed separatrix. Since this unp erturb ed homoclinic orbit comes back to the domain of the normal form, the same happ ens to the unstable manifold, which can b e compared with the lo cal stable manifold. In order to describ e the dierence b etween the global unstable manifoldand the lo cal stable one, it is convenient to take H and T = ; log Y=H 0 ( XY ) as canonical co ordinates near the stable separatrix. The equation of the lo cal stable manifold is then H = 0. In this co ordinate system the unstable manifold is the graph of a function H u : H = H u ( T 1 2 ), which dep ends 2 -p erio dically on 1 and 2 ,and has zero mean, due to the Hamiltoniancharacter of the p erturbation. Using the parametrization provided by the normal form, it turns out that H u is quasip erio dic in T : H u ( T 1 2 )= H u (0 1 ; T=" 2 ; T =" ) :
8 A. DELSHAMS, V. GELFREICH, A. JORBA, AND T.M. SEARA Moreover, by the Extension Theorem and its Corollary 1, H u is given in rst order by the Melnikov function for ( T 1 2 ) in the complex domain (15): H u ( T 1 2 )= M ( 1 ; T =" 2 ; T =" )+ O ; " 2 p ; 4 : (16) It is imp ortant to notice that the term F = O ; " 2 p ; 4 in equation (16) is an analytic function in the complex domain (15) which dep ends 2 -p erio dically on 1 and 2 , has zero mean, and is quasip erio dic in T . The following general Lemma allows us to b ound its Fourier co ecients. Lemma 3. Let F ( 1 + s=" 2 + s=" ) bea 2 -periodic function of the variables 1 , 2 , analytic in the product of strips j= 1 j r 1 , j= 2 j r 2 and j= s j ,and j F j A for these values of the variables. Then for al l k 1 , k 2 2 Z j F k 1 k 2 j A e ;j k 1 j r 1 ;j k 2 j r 2 e ; j k 1 + k 2 j =" : Finally, the next lemma gives the exp onentially small upp er b ound for the function F for real values of the variables. Lemma 4. Consider the (2 log ) -periodic function c r 1 r 2 ( ) dened on the interval log " ; log log " + log ] by c r 1 r 2 ( ) = C 0 cosh ; log " 2 " = ( + ; 1 ) ( r 1 + r 2 ) 2 C 0 =2 s ( r 1 + r 2 ) + ; 1 : and continuedby 2log -periodicity. Let F satisfy the conditions of Lemma 3. If =(1+ p 5) = 2 is the golden mean number and the mean value of the function F is zero, then j F ( 1 2 ) j const A exp ; c r 1 r 2 (log " ) p " (17) on the real values of its arguments. The constant depends continuously on r 1 > 0 and r 2 > 0 . Applying these two lemmas to the error function F = O ; " 2 p ; 4 in equation (16), we obtain the desired exp onentially small estimates. Nowwe can summarize the results ab ove ab out the splitting function H u ( T 1 2 ) in the following theorem, which is the main result of this pap er. Theorem 3 (Main Theorem) . There exist positive constants T 0 and R , R< T 0 , such that in the coordinate system ( H T 1 2 ) the unstable manifold can berepresentedasthegraph of the function H = H u ( T 1 2 " ) ,where the function H u depends 2 -periodical ly on 1 and 2 . In the domain < T 2 ( T 0 ; R T 0 ) j= T j 2 ; p " j= 1 j <r 1 ; p " j= 2 j <r 2 ; p " this function is analytic and close to the Melnikov function: H u ( T 1 2 )= M ( 1 ; T =" 2 ; T =" )+ O ; " 2 p ; 4 : Moreover, H u ( T 1 2 )= H u (0 1 ; T=" 2 ; T =" )
SEPARATRIX SPLITTING FOR FAST QUASIPERIODIC FORCING 9 and its mean value is zero: Z T 2 H u (0 1 2 ) d 1 d 2 =0 : (18) Furthermore, for p> 3 and real T , 1 and 2 , j H u ( T 1 2 ) ; M ( 1 ; T =" 2 ; T =" ) j const " 2 p ; 4 exp ; c (log " ) p " where c ( ) is dened in (9). If condition (4) is full led, then there exists " 0 > 0 such that, for 0 <"<" 0 , the maximum of the modulus of the Melnikov function is larger than the right hand side of the last upper bound. Acknowledgments. We are indebted to C. Simo for relevant discussions and remarks. Three of the authors (A. D., A. J. and T.M. S.) have b een partially supp orted by the Spanish grant DGICYT PB94{0215, the EC grant ERBCHRXCT940460, and the Catalan grant CIRIT 1996SGR-00105. One of the authors (V. G.) was supp orted by a CICYT grant. References BCG95] G. Benettin, A. Carati, and G. Gallavotti, A rigorous implementation of the JeansLandau-Tel ler approximation for adiabatic invariants ,Preprint, August 1995. Ben96] G. Benettin, On the Landau{Tel ler approximation for adiabatic invariants ,InSimo Sim97]. CG94] L. Chierchia and G. Gallavotti, Drift and diusion in phase space , Ann. Inst. H. PoincarePhys. Theor. 60 (1994), no. 1, 1{144. DGJS96a] A. Delshams, V.G. Gelfreich, A. Jorba, and T.M. Seara, Exponential ly smal l splitting of separatrices under fast quasiperiodic forcing , Math. Preprints Series 199, Univ. Barcelona, Barcelona, 1996. DGJS96b] A. Delshams, V.G. Gelfreich, A. Jorba, and T.M. Seara, Splitting of separatrices for (fast) quasiperiodic forcing ,InSimo Sim97]. DS92] A. Delshams and T.M. Seara, An asymptotic expression for the splitting of separatrices of the rapid ly forcedpendulum , Comm. Math. Phys. 150 (1992), 433{463. Gal94] G. Gallavotti, Twistless KAM tori, quasi at homoclinic intersections, and other cancel lations in the perturbation series of certain completely integrable Hamiltonian systems. a review , Rev. Math. Phys. 6 (1994), no. 3, 343{411. Gel93] V.G. Gelfreich, Separatrices splitting for the rapid ly forcedpendulum , Pro ceedings of the Dynamical Systems Semester (Basel-Boston-Stuttgart) (S. Kuksin, V.F. Lazutkin, and J. Poschel, eds.), Held in St. Petersburg, Russia, 17{30 November, 1991. Birkhauser, Basel-Boston-Stuttgart, 1993, pp. 47{67. Laz84] V.F. Lazutkin, Splitting of separatrices for the Chirikov's standardmap , Preprint VINITI No. 6372{84 (in Russian), 1984. Nei84] A.I. Neishtadt, The separation of motions in systems with rapid ly rotating phase ,J. Appl. Math. Mech. 48 (1984), no. 2, 133{139. Sim94] C. Simo, Averaging under fast quasiperiodic forcing , Hamiltonian Mechanics: Integrability and Chaotic Behaviour (New York) (J. Seimenis, ed.), NATO Adv. Sci. Inst. Ser. B Phys., vol. 331, Held in Toru n, Polland, 28 June{2 July 1993. Plenum, New York, 1994, pp. 13{34. Sim97] C. Simo (ed.), Hamiltonian systems with threeormoredegrees 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.