%**start of header \newcount\mgnf\newcount\tipi\newcount\tipoformule \newcount\aux\newcount\piepagina\newcount\xdata % \mgnf=0 \aux=1 %1 produce aux \tipoformule=0 %0 usa aux; 1 no (usa i simboli dati) \piepagina=2 %0 =data e #par.#pag; 1=data e #pag; 2=#pag \xdata=0 %0 data del giorno, 1 data fissa da \Di: \def\Di{1 settembre 1997} \ifnum\mgnf=1 \aux=0 \tipoformule =1 \piepagina=1 \xdata=1\fi \newcount\bibl %\bibl= ? % 0= rif [XXX], 1= rif. numerici \ifnum\mgnf=0\bibl=0\else\bibl=1\fi \bibl=0 % Per poter cambiare a piacimento il formato dei riferimenti % bibliografici in .tex: % % 1: citare nella forma esemplificata da \ref{B}{2}{20}} % ove XXX e' un simbolo per le iniziali e 2 distingue i lavori con % le stesse iniziali, 7 e' il numero SIMBOLICO del riferimento per XXX2. % Il numero 7 puo' essere ARBITRARIO e viene automaticamente % riaggiustato al momento della compilazione (vedi punto 4) % % 2: Se si sceglie \bibl=0 si cita nella forma [XXX2]; se si sceglie % \bibl=1 si cita nella forma [numero di ordine di prima citazione]. % % 3: La bibliografia va scritta nella forma \def{\qqq}{} % in ordine alfabetico per autore attribuendo un simbolo % qualsiasi al testo che (usando ref.b) produce fin.tex e .tex % con i riferimenti giusti % in \bibl=1 e la si ricompila e stampa. 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\def\tst{\textstyle}\def\st{\scriptscriptstyle} \def\*{\vskip0.3truecm} \def\lis#1{{\overline #1}}\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }} \def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }} \def\ie{\hbox{\it i.e.\ }} \def\fiat{{}} \def\\{\hfill\break} \def\={{ \; \equiv \; }} \def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}} \def\sign{{\rm sign\,}}\def\atan{{\,\rm arctg\,}} \def\annota#1{\footnote{${}^#1$}} \ifnum\aux=1\BOZZA\else\relax\fi \ifnum\tipoformule=1\let\Eq=\eqno\def\eq{}\let\Eqa=\eqno\def\eqa{} \def\equ{{}}\fi \def\defi{\,{\buildrel def \over =}\,} \def\pallino{{\0$\bullet$}}\def\1{\ifnum\mgnf=0\pagina\else\relax\fi} \def\W#1{#1_{\kern-3pt\lower6.6truept\hbox to 1.1truemm {$\widetilde{}$\hfill}}\kern2pt\,} \def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,}\def\DD{{\cal D}} \def\FINE{ \* \0{\it Internet: Authors' preprints downloadable (latest version) at: \centerline{\tt http://chimera.roma1.infn.it} \centerline{\tt http://www.math.rutgers.edu/$\sim$giovanni} \0Mathematical Physics Preprints (mirror) pages.\\ \sl e-mail: giovanni@ipparco.roma1.infn.it, gentileg@ipparco.roma1.infn.it,\\ vieri@@ipparco.roma1.infn.it }} \def\GG{{\cal G}}\def\JJ{{\cal J}}\def\xx{{\V x}} \def\ff{{\V f}}\def\TJ{{\tilde J}}\def\II{{\cal I}} \def\NN{{\cal N}}\def\CC{{\cal C}}\def\EE{{\cal E}} \def\KK{{\cal K}}\def\FF{{\cal F}} \def\yy{{\V y}}\def\zz{{\V z}} \def\ff{{\V f}}\def\TJ{{\tilde J}}\def\II{{\cal I}} \def\NN{{\cal N}}\def\CC{{\cal C}}\def\EE{{\cal E}} \def\KK{{\cal K}}\def\FF{{\cal F}}\def\UU{{\cal U}} \def\MM{{\cal M}}\def\mm{{\V\m}} \def\AA{{\V A}} \def\IG{{\cal I}}\def\OO{{\cal O}}\def\IGd{{\lis\IG^{\,2}}} \def\FFF{{\cal F}} \def\mm{{\V\m}}\def\eln#1{{e^{i\l\nn\cdot\oo\,#1}}} \def\erm#1{{e^{i\r\V\m\cdot\oo\,#1}}} \def\Eln{{E^\l_\n}}\def\Erm{{E^\r_\m}} \def\el#1#2{{e^{i\l_#1\nn_#1\cdot\oo\,#2}}} \def\El#1{{E^{\l_#1}_{\n_#1}}} \def\KJ{{\bf K}}\def\dn{{\,{\rm dn}\,}}\def\sn{{\,{\rm sn}\,}} \def\cn{{\,{\rm cn}\,}}\def\am{{\,{\rm am}\,}}\def\atan{{\,{\rm arctg}\,}} \def\aa{{\V \a}}\def\nn{{\V\n}} \def\pps{{\V \ps}}\def\giu{{\downarrow}}\def\su{{\uparrow}} \def\oo{{\V \o}}\def\nn{{\V\n}} %**end of header \fiat %\headline{\nota\hss Draft \#13} \centerline{\titolo Pendulum: separatrix splitting.} \*\* \centerline{\bf G. Gallavotti, G. Gentile, V. Mastropietro} \* \centerline{Universit\`a di Roma 1,2,3 } \centerline{\Di} \vskip1.truecm {\bf Abstract:} {\it An exact expression for the determinant of the splitting matrix is derived: it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in {\rm Ann. Inst. H. Poincar\'e, B-60, 1, 1994}. The asymtotic validity of Melnikov's formulae is proved for the class of models considered, which include polynomial perturbations.} \*\* \0{\bf\S1. Introduction.} \numsec=1\numfor=1\* Recently V. Gelfreich noted that a ``theorem'' in [CG] contains an error. The theorem gave a lower bound on the splitting angles in a three degrees of freedom system and it was needed to show the existence of heteroclinic chains in a class of hamiltonian systems with the aim of an application to a celestial mechanics problem. We correct it here by providing the correct lower bound and, at the same time, exposing again and in a more meditated form some ideas of [CG]. In the present paper we do not discuss the existence of Arnold's diffusion, [A2], in our systems. We do not discuss the celestial mechanics application of [CG] either, as two parts of it (see below) relied on the erroneous statement, and more work is needed. The present paper is, therefore, {\it not} a correction of the implications of the error in [CG] but only of the error itself. In order to derive the same implications further work is necessary as the erroneous result was used several times in the last three sections of [CG]. Each use has therefore to be treated separately. Our analysis deals mainly with systems with {\it three} different time scales with the ratio between the smallest to the largest being $\ll1$: \eg the largest is $\h^{-1/2}$, the intermediate is $1$ and the smallest is $\h^a$, $a\ge0$ (the $a=0$ case being a limiting two scales problem). The result will be called the {\it large angles theorem}, (theorem 2 in \S5, proved in \S6,\S7 for the systems defined in \equ(2.1),\equ(7.1) below), and describes a property of a pendulum subject to both a slow periodic force and to a rapid periodic force (incommensurate to the former) that we imagine generated by a pair of rotators, whose positions are given by two angles $\a,\l$ while the pendulum position is given by an angle $\f$. The angles $\a,\l$ will be respectively called {\it slow} and {\it fast}. When the system is perturbed some of the quasi periodic motions performed by rotators (or clocks) while the pendulum is in its unstable equilibrium persist. They become motions in which the pendulum is slightly moving quasi periodically around its unstable position (``without ever falling down''): one has therefore an invariant $2$--dimensional torus in phase space. For each position $(\a,\l)$ of the rotators there is well defined value of the pendulum variables $(I,\f)$ as well as of the rotators actions $(A,B)$. And $(\a,\l)$ evolve quasi--periodically (\ie $(\a,\l)$ can be written as $\a=\ps_1+ \D_1(\ps_1,\ps_2),\l=\ps_2+\D_2(\ps_1,\ps_2)$ and the time evolution is simply linear $\pps\to\pps+\oo t$ for a suitable $\oo$). Obviously the above quasi periodic motion is unstable and perturbations will make the pendulum ``fall''. The $2$--dimensional invariant tori have therefore stable and unstable manifolds: which coincide in absence of perturbations (because they correpond to the separatrix motions and the pendulum separatrix is degenerate). However under perturbations they ``split'' (\ie become distinct $3$ dimensional surfaces). Often, simply by symmetry, there is one trajectory that lies on both manifolds. Physically it corresponds to a motion that consists of just one swing of the pendulum from nearby the unstable position back to it. If we observe the swinging trajectory at the moment it passes through the stable equilibrium position, $\f=\p$, we see just a point that can be identified by the value $\aa_0$, at that moment, of the $\aa$ coordinates (where the two manifolds meet). If we move away from that point the $\AA$ coordinates on the two manifolds become different, at the same $\aa$, and their difference is a ``splitting'' vector $\V Q(\aa)$. The $2\times2$ matrix $D=\dpr_\aa\V Q(\aa_0)$ is called the {\it intersection matrix}. The homoclinic angles can be defined as the angles whose tangents are the eigenvalues of the intersection matrix. The {\it splitting} is usually defined as $\det D$. When the perturbing frequencies are held fixed and the perturbation is sent to $0$ there is a well known asymptotic expression for the splitting, called {\it Melnikov's formula} and coinciding with the ``first order perturbation theory result''. The splitting problem is to find under which conditions {\it Melnikov's formula} holds when {\it both} the perturbation and the shortest forcing period are sent to $0$. Informally our result is (see theorem 2 in \S5 for a formal statement): \* \0{\it Suppose that the forcing frequency is very large, say $\h^{-1/2}$ times the pendulum frequency, with $\h$ very small. Suppose also that the second characteristic frequency is relatively very small, say $\h^a$ times that of the pendulum with $a\ge0$. Then there are perturbations of size $\e=O(\h^c)$ (for some $c>0$) such that the separatrix splitting has size given asymptotically by {\it Melnikov's formula} provided the frequencies of the motion verify a suitable diophantine condition. In fact this property holds generically, under the same conditions, for perturbations of size $\e=O(\h^c)$ which are trigonometric polynomials.} \* There are many examples of systems for which the above property does not hold, see [S], [DJGS], [RW]. But the forcing frequencies relations are different. The correct lower bound estimate on the splitting makes an analysis {\it to all orders} necessary: the analysis, performed here, becomes ``marginal'' but the final result on the existence of homoclinic splitting (\ie a lower bound on it) remains valid. Therefore the present paper corrects the error in \S10 of [CG] as far as its implications on the size of the splitting are concerned. The techniques we use here {\it to bypass perturbation theory} were started in the Appendix A13 of [CG]: they were not pushed too far because the error made any developments unnecessary for the purposes of [CG]. The techniques were subsequently developed in [G3] (which {\it does not} repeat the computational error). The methods of [G3] were not developed to treat three time scales problems: the aim there being to prove the smallness of the splitting in {\it two time scales} problems (\ie in systems not considered here with all rotators with comparably large velocity, $a=-\fra12$). But they can be {\it easily} extended to three time scales problems and even lead to a remarkable {\it nonperturbative and exact} computation of the leading order (exponentially small) of the intersection matrix, see \equ(6.12),\equ(7.19): V. Gelfreich stressed necessity of a nonperturbative analysis by a simple cogent argument. The reason why we study three time scales systems is simply that they arise in a celestial mechanics problem of interest to us, see \S12 of [CG]. But most of our results hold also if the slow time scale is the same as that of the pendulum, \ie the parameter $a$ above (see also \equ(2.1)) is $a=0$, see also comments in \S9: this is also an interesting case and it has been considered, to some extent, in [RW]. The recent work of [RW] considerably overlaps with ours: even when $a=0$ (a two time scales problem) it cannot be used to achieve all our results because our assumptions can violate eq. (15) of [RW] (\ie the bound on the constant called $b$ in [RW], see \equ(2.2), below, for the isochronous case and \equ(7.3) for the anisochrounus). We also require, as an essential assumption, that the perturbations be trigonometric polynomials while the examples of [RW] require, as an essential assumption, that the perturbation has infinitely many non vanishing modes (see eq. (19) of [RW]). Another essential difference with respect to [G3] and [RW] is that we {\it only} study the splitting {\it at the homoclinic point} while they study it everywhere: hence our work is much more limited in scope. On the other hand we recover some of the results of [RW] because our proofs also apply nontrivially to two time scale cases with $3$ degrees of freedom (the ones in [RW]). Nevertheless we feel that the main difference between our work and [RW] lies in the techniques: here we show that the techniques of [G3] do apply immediately to the problem. See concluding remarks for a more technical comparison. \* \0{\bf\S2. Isochronous clock--pendulum system.} \numsec=2\numfor=1\* Our main result concerns anisochronous systems, \equ(7.1). Isochronous models will be considered only to illustrate the simplest cases: the cancellations that we find in the anisochronous case would look miraculous otherwise. Calling $(I,\f),(A,\a),(B,\l)$ pairs of canonical coordinates suppose them action--angle variables: $(\f,\a,\l)$ is a triple of angles (varying on the $3$--dimensional torus $T^3=[0,2\p]^3$) and $(I,A,B)\in R^3$. The hamiltonian will be: $$H= \h^a A+\fra {I^2}{2}+\h^{-1/2}B+\,g^2\, (\cos \f -1) + \m\, \h^c\,f(\f,\a,\l),\ \e\defi\m\h^c\Eq(2.1)$$ % where $\h,\m>0$ and $f$ is an {\it even trigonometric polynomials} in the angles $(\f,\a,\l)$, \ie for instance one could take $f(\f,\a,\l)=\big(\cos (\a+\f)+\cos(\l+\f)\big)$. The parameter $\h$ sets the ratios between time scales and is a free parameter which we take, eventually, to be close to zero in order to study asymptotic properties as $\h\to0$. If $\m=0$ the $2$--dimensional torus: $A=B=I=0,\,\f=0$ and $\aa=(\a,\l)$ arbitrary, is invariant and run quasi-periodically with rotation velocity $\oo=(\h^a,\h^{-1/2})$: it will be called the {\it unperturbed torus}. In the above case, {\it \ie in the isochronous case only}, we shall suppose (not ``for simplicity'': but as an essential hypothesis) that the vector $\V\o=(\h^a,\h^{-1/2})$ is a {\it diophantine vector}: $$|\V \o\cdot\nn|> C^{-1}\h^d{|\nn|^{-\t}}\Eq(2.2)$$ % for some {\it diophantine constant} $C,\t$ and some $d>0$. This restricts the values of $\h$ that we can consider. For $a\in [0,\fra12]$ it still allows sequences $\h_k$, such that \equ(2.2) holds with prefixed $\t>2, d=a$ and $C$ large enough, with $\h=\h_k=\h_1 k^{-1}$, for some $\h_1\in [\fra12,1]$ and all integers $k$ large enough. \* \0{\it Remarks:} (1) A related $H$ with $a=\fra12$ arises in a celestial mechanics problem near a {\it double resonance}, responsible for the time scales differences (see (12.39) in [CG], after scaling away the factors $\o_T$ to put $H$ in dimensionless form and with several factors replaced here by constants, for simplicity). A first simplification of \equ(2.1) compared to the ``realistic'' model in \S12 of [CG] is the absence of an additive isochrony breaking term $\h\fra{A^2}2$: taking it into account does not essentially change the analysis of the splitting results (even in its quantitative aspects on the asymptotics as $\h\to0$, see \S7). A second simplification is the absence in \equ(2.1) of a further perturbation $\b f_0$ which is not small (\ie $f_0,\b$ are $\h,\m$--independent) but depends only on the ``fast'' angle $\l$ and on $\f$: $f_0=f_0(\f,\l)$. Taking it into account is a problem not discussed here. However it does not change the qualitative aspects but only the quantitative ones {\it as long as the system is isochronous}, see the discussion at the end of \S8. (2) $g^2$ in the above hamiltonian is {\it fixed} (\ie it is $\h,\m$ independent): eventually we take $g\=1$ for simplicity. The parameters $\m,\h$ are free and we shall be interested in them having a ``small value'': note that if $\h\to0$ the rotation vector $\oo$ of the unperturbed torus ($I=\f=0$, $B=A=0$) has a size tending to $\io$. The even parity assumptions on the ``interactions'' $f$ simplifies, possibly in an essential way, the analysis. In the precession problem of [CG] $\h$ is the deviation from spherical shape of a planet precessing around its baricenter, which moves on an ellipse of (fixed) eccentricity $\e=\m\h^c$. (3) A general physical interpretation of \equ(2.1) is that of a system consisting in (i) a forcing clock with angular velocity $\h^a$ (\ie a point moving on a circle with angular velocity $\h^a$, position $\a$ and action variable $A$); (ii) a pendulum (\ie a point moving on a vertically placed circle with angular momentum $I$ and position $\f$ counted by taking $\f=0$ as the unstable equilibrium position of the pendulum); (iii) a second forcing clock with angular velocity $\h^{-1/2}$ and action variable $B$. Equivalently one can delete the $A\h^a,B\h^{-1/2}$ terms and replace $\a,\l$ by $\oo t=(\h^a t,\h^{-1/2} t)$ thus regarding the system as a time dependent one, consisting of a pendulum subject to a quasi periodic force with periods $2\p\h^{-a}, 2\p\h^{1/2}$. The ``characteristic time'' of the pendulum system is $T_0=g^{-1}$. (4) Finally the ``coupling constant'' in \equ(2.1) is written as $\m \h^c$ and not just $\m$, because the convergence radius of the expansions around the unperturbed torus is expected to be of the order of some power of $\h$: it would be nice to know the best value of $c$ (thus replacing the constant $c$ by its optimal value) but it seems not known, see [HMS], [ACKR] (it seems that $c>\fra12$ might be the right condition, but below we make no attempt at getting even close to such a small value). \* \0{\bf\S3. Separatrices and non degeneracy.} \numsec=3\numfor=1\* Supposing $\m=0$ in \equ(2.1) we look at the unstable quasi periodic motion with $A=0, B=0, I=0,\f=0 $ where $\a=\a_0+{\h}^{a}t$ and $\l=\l_0+\h^{-1/2} t$. This is a family of motions whose initial data are parameterized by $\V\a\defi(\a_0,\l_0)$ and therefore form an invariant torus: we shall call it a {\it hyperbolic torus}. The energy of such motions is $H=0$. If we fix $A,B$ at other values we find a continuum of invariant hypebolic tori, with rotation vector $\oo\defi (\h^{a},\h^{-1/2})$. The energy of such motions is $ \h^{-1/2} B+\h^a A$. The torus is unstable and its unstable manifold $W^-$ is $3$--dimensional and contains the set parameterized by $(\f,\a,\l)$ via the equations: $$A=0, \qquad I=I(\f), \qquad (\a,\l)\in [0,2\p]^2,\, \f\in(0,2\p)\Eq(3.1)$$ % where $I(\f)=I^0(\f)\defi g\sqrt{2(1-\cos\f)} $ is the pendulum {\it separatrix}. But also the set with $I=-I^0(\f)$ is part of the torus unstable manifold because of the meaning of $\f$ as an angle. The stable manifold $W^+$ has equations $I(\f)=\pm I^0(\f)$: this means that $W^+\=W^-$: a well known degeneracy of the pendulum motion. A nice way of seeing the separatrix motions is by representing them in the (canonical) {\it Jacobi's coordinates}: these are coordinates $(p,q)$ in terms of which the neighborhood of the (two) lines $I=\pm I^0(\f)$ in \equ(3.1) is represented, near $I=\f=0$, as: $$I= R_0(p,q),\qquad \f= S_0(p,q)\Eq(3.2)$$ % where the functions $R_0,S_0$ are defined near the origin, where they vanish, and the pendulum motion becomes, in such coordinates, simply $p(t)=p_0 e^{-gt}, q(t)=q_0 e^{gt}$. Therefore the $(p,q)$ coordinates describe globally, as $t\to+\io$, the evolution of initial data $(p_0,0)$ and globally, as $t\to-\io$, those with data $(0,q_0)$. In such coordinates the equation of the stable manifold is simply $p=0$ while that of the unstable manifold is $q=0$. The functions $S_0,R_0$ have well known holomorphy properties: the latter imply that the singularities of $R_0(pe^{gt},0), R_0(0,qe^{-gt})$, at fixed $p$ or $q$, occur at $t=\pm i\fra\p2 g^{-1}$, and the same holds for $S_0$. This can be seen from the explicit solution of the pendulum equation in terms of elliptic integrals: an elementary analysis is in Appendix A9 of [CG], or in Appendix A1 of [G5]. The first result that we use to set up a general picture but strictly speaking not even necessary as shown in [Ge1], is the (well known, see [Gr]) stability of such unperturbed hyperbolic torus and of its stable and unstable manifolds. This will be stated as: \* \0{\bf Theorem 1}: {\it Suppose $c$ in \equ(2.1) large enough, and suppose that $\h$ is such that the rotation vector $\V\o\defi(\h^a,\h^{-1/2})$ verifies the diophantine property \equ(2.2) and $\m_0>0$ is small enough. One can define functions $\G,\X,\L,\Th$ divisible by $\m$ and analytic in the variables $(\psi_1,\psi_2, p,q,\m)$, varying respectively on the torus $[0,2\p]^2$, on a neighborhood of $p=q=0$, and in $|\m|\le\m_0$ with $\m_0$ small enough, such that setting, as $\V\ps$ varies on $T^2=[0,2\p]^2$: % $$\eqalign{ A=&A_0+ \X(\V\ps,p,q),\quad B=B_0+\G(\V\ps,p,q),\quad \a=\psi_1,\quad \l=\ps_2\cr I=& R_0(p,q) +\L(\V\ps,p,q),\quad\f=S_0(p,q)+ \Th(\V\ps,p,q)\cr}\Eq(3.3)$$ % one defines, for all $A_0,B_0$, an invariant set on which the motion described by \equ(2.1) takes place following $t\to (\V\psi+\V\o t, p e^{-\lis g t}, q e^{+\lis g t})$ (as long as the $(p,q)$ stay in the domain of definition of $\X,\G,\L,\Th$), with $\lis g=(1+\g(\m))g$ and $\g$ analytic in $\m$, near $\m=0$ and divisible by $\m$. The functions $\G,\X$ evaluated at $p=q=0$ have zero average with respect to $\V\ps$; also the time average of $H$ vanishes on the above motions and (therefore) $H=0$ for all of them. The radius of convergence $\m_0$ is uniform in $\h<1$, as long as $\V \o$ verifies the diophantine condition.} \* {\it Remarks:} (1) The theorem implies that if $p=q=0$ then the parametric equations $\f=\Th(\V\ps,0,0)$, $\a=\psi_1$, $\l=\psi_2$, $I=\L(\V\ps,0,0)$, $A=\X(\V \ps,0,0)$ and $B=\G(\V \ps,0,0)$ describe, as $\V\ps$ varies on the $2$--dimensional torus $[0,2\p]^2$, an invariant torus. The quasi periodic rotation $\V\ps\to\V\ps+\oo t$ with $\oo=(\h^a,\h^{-1/2})$ gives, for all $\V\ps$, a solution to the equations of motion. This means that the invariant torus of dimension $2$ that we considered in \S2 (\ie $A=B=I=0,\,\f=0$) {\it survives the onset of perturbation} and persists, {\it slightly deformed}, with the same rotation vector $\oo$ and a slightly varied pair of Lyapunov exponents (\ie $\pm\lis g=\pm(1+\g)\,g$ rather than $\pm g$). The zero average property for $\G,\X$ means that the perturbed torus is in the average (over $\V\ps$) located at the same position as the unperturbed one from which it ``emanates'': this is useful as it allows us to parameterize the invariant tori by their average position in action space. (2) Setting $p=0, q\ne0$ one finds a surface of dimension $3$ which is {\it a part} of the {\it unstable} manifold $W^-$ of the torus, while setting $q=0, p\ne0$ one gets a part of the stable manifod $W^+$. Such manifolds are colorfully called local {\it whiskers}. They are in fact ``local'' parts of larger ``global'' manifolds (see item (4) below) and can be called, for this reason, {\it local stable and local unstable manifolds}. (3) The motion of $A,B$ is computed from that of the other coordinates by quadrature (\eg $\dot B=-\e \,\dpr_\l f(\f,\a,\l)$, while $B$ itself does not occurr in the equations of the other coordinates, and similarly for $A$). The symmetry of $f$ does not imply that $\L,\Th$ have zero average over $\V \ps$, if $\f=\p$, [CG], and also the Lyapunov exponent $\lis g$ in general changes by a quantity $\g$ of order $O(\m)$ with respect to the unperturbed value $g(pq)$. The motions energy also changes, in general, by an amount analytic in $\m$ and divisible by $\m$ (\ie of $O(\m)$). (4) Once the {\it local} parts of the torus whiskers have been defined as above we can extend them to {\it global} objects by simply applying time evolution to their points. (5) The case $\m=0$ (one of the most classical results in Mechanics) is a non trivial exercise: it is developed in Appendix A9 of [CG] or Appendix A1 of [G5]. (6) The stable and unstable manifolds do not coincide, in general, for $\m\ne0$. (7) in the anisochronous cases of \S7 a very similar result holds with the difference that the relation between $\aa$ and $\pps$ is also non trivial and described by a function $\V \D(\pps,p,q)=(\D(\pps,p,q),0)$ with zero average over $\pps$ and divisible by $\m$ and with the same domain of definition of the other functions in \equ(3.3). In this case all the functions in \equ(3.3), and $\V\D$ as well, depend also on $A_0$ which must be restricted so that $\oo=(\h^a+\h A_0,\h^{-1/2})$ verifies a diophantine property with suitable diophantine constants $C(\h),\t$: the size of the radius of convergence and the value of $c$ will depend on the selected $C(\h)$ (like \equ(2.2) or \equ(7.3)). (8) The above theorem is well known; it is explicitly proved (in a much more general case) in the above form in \S5 of [CG], see p. 38. Its proof is a rather straightforward adaptation of Arnold's method of proof of the KAM theorem, [A1], as exposed for instance in [G1]: the only element of ``novelty'' is perhaps the {\it normal form} of the motion in the coordinates $\V\ps,p,q$, as long as the latter two remain in their domain of definition. A more modern proof, based on Eliasson's method, [E], for proving KAM (as exposed in [G2],[GG]) and extending it to the problem of tori of one dimension less than the maximal, can be derived from [Ge1] (where only the cases $p=0$ or $q=0$ are studied). Also unusual is the absence of the twist condition (present, and necessary, in the more general proof in [CG]): it can be eliminated because of the special structure of the hamiltonian \equ(2.1). But it is a long and uninteresting proof for us here. We shall not need, however, the normal form and, as it will be clear, we only need the classical results in the weaker form discussed in [Gr],[Ge1]. \* \0{\bf\S4. Splitting angles. A recursive determination.} \numsec=4\numfor=1\* The symmetry of $f$ implies an intersection at $\f=\p$ and $\V\a=\V0$ (see below or, for instance, p. 363 of [G3]) between stable and unstable manifolds of the torus into which the unperturbed torus (\ie $A=B=0,I=0,f=0$) is deformed by the perturbation. Therefore we set up an algorithm to study such intersection. For this purpose it is convenient to work in the original canonical coordinates and write the stable and unstable whiskers $W^\pm_\m$ as: $$W^\pm_\m=\{(\f,\aa,I,\AA)=\big(\f,\aa, I^\pm_\m(\f,\aa),\AA^\pm_\m(\f,\aa)\big)\}\Eq(4.1)$$ % with $ \aa\in T^2,\,\e< |\f|<2\p-\e$ where $\e>0$ can be fixed \ap as small as we please provided we diminish the value of the analyticity radius $\m_0$ in theorem 1. In other words the whiskers deformation is of $O(\m)$ in every closed subinterval of $(-2\p,2\p)$: therefore they remain parametrizable by $\f,\aa$ for $\f$ in any closed subinterval if $\m$ is small enough (just note that for $\m=0$ they {\it are} parametrizable). We say that, in such region of $(\f,\aa)$, they are {\it graphs} over the angle variables. We only need that $\f=\p$ is allowed (hence $\e=\fra\p2$ will do). We define the {\it splitting vector} $\V Q(\aa)$, the {\it splitting } or {\it intersection matrix} and the {\it splitting} between $W^+_\m$ and $W^-_\m$ at $\f=\p$ and $\V\a=\V0$, respectively, as: $$\V Q(\aa)\defi \AA^+_\m(\p,\aa)-\AA^-_\m(\p,\aa),\ D\defi \dpr_\aa \V Q(\aa)\big|_{\aa=\V0},\ {\rm and}\ \det D\Eq(4.2)$$ % A relevant remark (Lochak and Sauzin, private communication) is that the whiskers are {\it lagrangian manifolds} so that, for a suitable generating function $S^\pm(\f,\aa)$, the whiskers have equation: $\AA^\pm=\dpr_\aa S(\f,\aa), I^\pm=\dpr_\f S^\pm(\f,\aa)$ around every point $(\f_0,\aa_0)$ where they can be locally regarded as graphs over the angles $(\f,\aa)$, \eg at $\f=\p$ and for $\m$ small. This implies that $D$ is {\it symmetric}, as we indeed find in \equ(6.10), \equ(7.10). We now derive recursive formulae for $I^\pm_\m$, $\AA^\pm_\m$ in \equ(4.1) and their time evolution, keeping in mind that for $\m=0$ it is $I^+_0=I^-_0=I^0(\f)=g \sqrt{2(1-\cos\f)}, \AA^\pm_0=\V0$. The unperturbed motion is simply: $X^0(t)\=(\f^0(t), \aa+\oo t,I^0(\f^0(t),\V 0)$, where $(\f^0(t),I^0(\f^0(t)))$ is the free (\ie with $\m=0$) separatrix motion, generated by the pendulum in \equ(2.1) setting the origin of time when the pendulum swings through $\f=\p$. The following \S4,\S5 are presented here only for completeness: although selfcontained {\it they are not} meant as a substitute of the work done in [G3] but serve the purpose of guiding the reader to dig out of that paper what he may want to see in more detail. Let $X_\m^\s(t;\aa)$, $\s=\pm$, be the evolution of the point on $W^\s_\m$ whose initial coordinates are given by $(\p,\aa,I^\s_\m(\p,\aa),\AA^\s_\m(\p,\aa))$; from now on we shall fix initial data with $\f=\p$ (which amounts to studying the whiskers at the ``Poincar\'e's section" $\{\f=\p\}$). The analyticity in $\m$ implied by the above theorem 1 and the analyticity properties of the Jacobi functions $R_0,S_0$ allow us to consider the {\it convergent} (if $c$ in \equ(2.1) theorem 1 is large) Taylor series expansions, in $\m$, of the whiskers equations. Let: % $$X^\s_\m(t)\=X^\s_\m(t;\aa)\defi \sum_{k\ge 0} X^{k\s}(t;\aa) \e^k\= \sum_{k\ge 0} X^{k\s}(t) \e^k,\qquad \s=\pm\Eq(4.3)$$ % be the power series in $\e=\m \h^c$ of $X^\s_\m$, (convergent for $\m$ small); note that $X^{0\s}\=X^0$ is the degenerate unperturbed whisker. We shall often omit writing explicitly the $\aa$ variable among the arguments of various $\aa$ dependent functions, to simplify the notations. Theorem 1, \S3, tells us that the $t$--dependence of $X^\s_\m(t)$ has the form: % $$X^\s_\m(t)=X^\s_\m(\oo t,t;\aa)\Eq(4.4)$$ % where $X^\s_\m(\V\ps,t;\aa)$ is a real analytic function, of {\it all} its arguments ($\m$ included), which is periodic in $\V\ps$ and $\aa$. And in fact in our {\it isochronous} case the dependence on $\V\ps$ and $\aa$ is via $\aa+\V\ps$ (``no phase shift'' in the sense of [CG]). The two functions $X^{k\s}(t)$ will be regarded as forming a single function $X^k(t)$: % $$X^k(t)=\cases{X^{k+}(t)& if $\s={\rm sign}\,t=+$\cr X^{k-}(t)& if $\s={\rm sign}\,t =-$\cr}\Eq(4.5)$$ % {\it We label the $6$ components of $X$ with an index $j$, $j=0,1,\ldots,5$, and write them ({\rm notation used in [G3]}), with the convention: % $$X_0\defi X_-,\quad (X_j)_{j=1,2}\defi \V X_\giu,\quad X_3\defi X_+,\quad (X_j)_{j=4,5}\defi \V X_\su\Eq(4.6)$$ % and we write first the angle variables ($(\f,\a,\l)=(X_-,X_\giu)$), then the action variables ($(I,A,B)=(X_+,\V X_\su)$); first the pendulum, then the rotator and then the clock variables.} Therefore at order $0$ in $\m$: % $$\eqalign{ X_0(t)=&\f^0(t),\quad X_1(t)=\a_0+\h^a t, \quad X_2(t)=\l_0+ \h^{-1/2}t\cr X_3(t)=& I^0(\f^0(t)),\quad X_4=0=X_5,\cr }\Eq(4.7)$$ % where $\f^0(t)$ is the free separatrix motion (\ie $\f^0(t)=4\atan e^{-g t}$, see Appendix A1). Inserting \equ(4.3) in Hamilton's equations for \equ(2.1) and comparing the various orders in $\m$, the coefficients $X^{k\s}(t)\=$\- $X^{k\s}(\oo t,t;\aa)$ are seen to satisfy the hierarchy of equations: % $${d\over dt} X^{k\s}\= \dot {X}^{k\s}=L X^{k\s}+F^{k\s}\Eq(4.8)$$ % where $L$ is a $6\times 6$ matrix with only two non vanishing elements $L_{03}=1$ and $L_{30}=g^2\cos\f^0(t)$. Expanding $X^\s$ in powers of $\e$ and imposing that the equations of motion are verified, we find recursively the expressions for $F$. For instance: % $$\eqalignno{ F^1_+=&-\dpr_\f f,\qquad \V F^1_\su=-\dpr_\aa f\cr F^2_+=&-{g^2\over2}\sin\f\, (X^1_-)^2-\dpr_{\f^2} f X^1_- -\dpr_{\f\aa} f \V X^1_\giu\cr \V F^2_\su=&-\dpr_{\aa\f}f X^1_--\dpr_{\aa^2}f \V X^1_\giu&\eq(4.9)\cr F^3_+=&-{g^2\over2}\,2X_-^1X_-^2\sin\f-{g^2\over 3!}\cos\f(X^1_-)^3-\dpr_{\f^2}f\,X_-^2+\cr &-\dpr_{\aa\f} f \V X^2_\giu-{1\over2}\dpr_{\f\aa^2}f \V X^1_\giu \V X^1_\giu-\dpr_{\f^2\aa}f \V X^1_\giu X^1_--{1\over2}\dpr_{\f^3}f\,(X^1_-)^2\cr \V F^3_\su=&-\dpr_{\aa\f}fX^2_--\dpr_{\aa^2}f \V X_\giu^2-{1\over2}\dpr_{\aa^3}f \V X^1_\giu \V X^1_\giu-\dpr_{\f\aa^2}f\,\V X^1_\giu X^1_-- {1\over2}\dpr_{\aa\f^2}f X^1_-X^1_-\cr} $$ % where the functions are evaluated at $\f(t)=\f^0(t),\aa(t)\defi \aa+\oo t$. More generally, $F^{k}$ depend upon $X^0,...,X^{k-1}$ {\it but not on $X^{k}$}. It is $F^k_-=\V F^k_\giu\=0$ for all $k\ge1$. Expressing the solution of a linear inhomogeneous equation like the one in \equ(4.8) can be conveniently done in terms of the {\it wronskian matrix}. We recall therefore the notion of {\it wronskian matrix} $W(t)$ of a solution $t\to x(t)$ of a differential equation $\dot x= f(x)$ in $R^n$. It is a $n\times n$ matrix whose columns are formed by $n$ linearly independent solutions of the linear differential equation obtained by linearizing $f$ around the solution $x$ and assuming $W(0)=$ identity. In other words $W(t)$ is, in our case, the solution of the differential equation $\dot W(t)=L(t)\,W(t),\, W(0)=1$. The solubility by elementary quadrature of the free pendulum equations leads, on the separatrix, to the following formulae that have importance because the wronskian of the free separatrix motion can be expressed in terms of them. If $\f^0(t)=4 \arctan e^{-gt}$: % $$\eqalign{ \cos \f^0=&1- {2\over (\cosh gt)^2}= 1- 8 {x^2\over (1+x^2)^2}\cr \sin \f^0=& 2 {\sinh gt\over (\cosh gt)^2} = 4\s x {1-x^2\over (1+x^2)^2}\cr}\Eq(4.10)$$ % for $x=e^{gt}$ {\it or, as well, for} $x=e^{-gt}$. Eq. \equ(4.10) leads, see \equ(A1.1) in Appendix A1, to the following expression for the wronskian $W(t)$ of the {\it separatrix motion} for the pendulum appearing in \equ(2.1), with initial data at $t=0$ given by $\f=\p,I=2g$: % $$W(t)=\pmatrix{ {1\over\cosh gt}&{{\lis w}\over4}\cr -{\sinh gt\over\cosh^2 gt}& (1-{{\lis w}\over4}{\sinh gt\over\cosh^2gt})\cosh gt\cr}, \qquad{\lis w}\={2gt+\sinh 2gt\over\cosh gt}\Eq(4.11)$$ % Notationally we follow here [G3] (in [CG] $I,\f$ are exchanged). The evolution of the $X_\pm$ (\ie $I,\f$) components can be determined by using the above wronskian: % $$\pmatrix{X^{k\s}_-\cr X^{k\s}_+\cr}=W(t) \pmatrix{0\cr X^{k\s}_+(0)\cr} + W(t)\ig_0^t{W\,}^{-1}(\t)\pmatrix{0\cr F^{k\s}_+(\t)\cr}\ d\t \Eq(4.12)$$ % Thus, denoting by $w_{ij}$ ($i,j=0,3$) the entries of $W$, we see immediately that: % $$\eqalign{\textstyle X^{k\s}_-(t)=&\textstyle w_{03}(t)X^{k\s}_+(0)+w_{03}(t) \ig^t_0w_{00}(\t) F^{k\s}_+(\t) d\t %\cr& \textstyle-w_{00}(t)\ig^t_0w_{03}(\t) F^{k\s}_+(\t)\,d\t\cr \textstyle X^{k\s}_+(t)=&\textstyle w_{33}(t)X^{k\s}_+(0)+w_{33}(t) \ig^t_0 w_{00}(\t) F^{k\s}_+(\t) d\t %\cr& \textstyle-w_{30}(t)\ig^t_0 w_{03}(\t) F^{k\s}_+(\t)\,d\t\cr}\Eq(4.13)$$ % Integrating on the separatrix \equ(4.8) for the $\su,\giu$ components is ``easier'': one can find it directly or, more systematically, by writing the full $6\times6$--wronskian matrix of the equation \equ(4.8), which is trivially related to $W(t)$ above; we shall not write it here: see eq. (4.21) in [G3] for an explicit expression. The result is: % $$\V X_\giu^{k\s}(t)= \V 0,\qquad \V X_\su^{k\s}(t)=\V X_\su^{k\s}(0)+\ig_0^t\V F^{k\s}_\su(\t)d\t\Eq(4.14)$$ % having used that the $\V X^{k\s}_\giu(0)\=\V 0$, because the initial datum is fixed and $\m$--independent. If \equ(2.1) is modified by adding an isochrony breaking term $\fra\h2 A^2$ the first component of the first of \equ(4.14) becomes: $$X^{k\s}_1(t)=\h \,\big(t X^{k\s}_1(0)+\ig_0^t (t-\t) F_1^{k\s}(\t)\,d\t\big)\Eq(4.15)$$ % where $X_1^{k\s}$ is the first compnent of $\V X_\su^{k\s}$, see eq. (2.18) in [G3]. Equations \equ(4.13), \equ(4.14) can be used to find a reasonably simple algorithm to represent whiskers to all orders $k\ge1$ of the perturbation expansion. It is very important to keep in mind that the initial data in \equ(4.13), \equ(4.14) {\it are not constants}: according to the convention following \equ(4.3) they can depend on the $\aa$ variables of the initial data. This means that the functions $X$ depend separately on $\aa$ and $\oo t$. {\it Except} when, as in \equ(2.1), the hamiltonian is linear in the $A,B$ variables. In the latter case the dependence on $\aa$ and $\oo t$ of the r.h.s. of \equ(4.4) (where the notation is complete and all variables are indicated explicitly) {\it must be} through $\aa+\oo t$, since the $\aa$ angles vary as $\dot{\aa}=\oo$. Note that the case \equ(2.1) is {\it non trivial and, in fact, very interesting: it is equivalent to a problem on a non linear quasi periodic Schr\"odinger equation}, see [G2], [BGGM]. The extension to anisochronus cases (\ie with a quadratic term in $A$ added to \equ(2.1)) is worked out in [CG] up to third order: a general analysis can be found in [G3]. The initial data (still unknown) in \equ(4.13),\equ(4.14) are determined by imposing the correct behavior at $\pm\io$, and the correct dependence on $\aa$ and $\oo t$ (\ie a dependence on these two vectors through their sum). These conditions arise from the fact that the motion must be asymptotic to the quasi periodic motion on the invariant torus whose whiskers are described by $X^+$ or $X^-$. The scheme to do so is the following, see [G3]. Note that $w_{03},w_{33}$ in \equ(4.11) behave, as $t\to\s\io$ with $\s=\pm1$, asymptotically as $\s e^{gt\s}/4$ and $-e^{gt\s}/4$, while the other two matrix elements become exponentially small. Hence we see that the tems in \equ(4.13) proportional to $w_{33}(t)$ or $w_{30}(t)$ diverge, in general, as $t\to\s\io$ exponentially fast (supposing the integrals convergent, \ie supposing $F^{k\s}_+(\t)$ not growing faster than a polynomial in $t$, as it will turn out to be). But they are multiplied by: $\big(X^{k\s}_+(0)+\ig_0^t w_{00}(\t) F_+^{k\s}(\t)\,d\t\big)$ so that the condition to determine $X^{k\s}_+(0)$ is: % $$X^{k\s}_+(0)+\ig_0^{\s\io} w_{00}(\t) F_+^{k\s}(\t)\,d\t=0\Eq(4.16)$$ % Likewise, in the isochronous case, from the second of \equ(4.14) we determine $\V X^{k\s}_\su(0)$ by imposing that $\V X_\su^{k\s}(t)$, \ie the momentum component corresponding to the isochronous angles $\a,\l$, depends asymptotically on $t$ only via $\aa+\oo t$: this will determine $\V X^{k\s}_\su(0)$ up to a constant. And the average over $\aa$ of $\V F^{k\s}_\su(t)$ must tend to $0$ as $t\to\s\io$ (otherwise the second of \equ(4.14) could not possibly be bounded as $t\to\s\io$: but it has to be such because $\V X_\su^{k\s}(t)$ has to be bounded, by theorem 1). This means that the constant is not a function of $\aa$ and can be fixed arbitrarily: however we want that the averages of the $\Xi,\G$ in \equ(3.3) are $0$ so that $X^{k\s}(0)$ is completely determined. One finds, for {\it both components}: $$\V X^{k\s}_\su(0)+\ig_0^{\s\io} \V F^{k\s}_{\su}(\t) d\t=\V0\Eq(4.17)$$ % where the integral is usually improper: see \equ(5.1) below for a proper definition (derived by simply looking at the meaning of the conditions imposed to determine $\V X^{k\s}_\su(0)$). Eq. \equ(4.17) {\it remains the same} even in the anisochronous cases (see eq. (4.5) in [G3]). The key to concrete calculations is that, $f$ in \equ(2.1) being a trigonometric polynomial, the function $F^1$ (see \equ(4.9)) belongs to the class $\MM$ of linear combinations of terms like: % $$M(t)={(g\s t)^h\over h!} x^k \s^\th e^{i\r\oo\cdot\nn t}\Eq(4.18)$$ % with $h\ge 0,k$ integers, $\th=0,1$, $\r=\pm1$ and $gk \pm i\oo\cdot \nn\ne 0$ and we set $x=e^{-gt\s}$ with $\s=\sign\,t=\pm1$. In fact $F^1$ is a {\it finite linear combination} of harmonics $\nn$. By \equ(4.10) we see that \equ(4.11) is an analytic function of the variable $x=e^{gt}$ and of $t$, or of $x=e^{-gt}$ and of $t$: the explicit $t$--dependence arises because of the term $\lis w$ in \equ(4.11). Expressing $F^1$ as a sum of monomials like \equ(4.18) requires a (convergent) infinite sum over $k,h$. In the case of $F^1$, in fact, there are {\it no monomials} with a power of $t$ higher than $0$ (\ie with $h>0$ in \equ(4.18)). By induction {\it all} the $F^k$ have the property of being expressible as sums of monomials like \equ(4.18). However, starting with the second order, one sees that powers of $t$ do appear (this reflects, see theorem 1, that the function $\g$, describing the change of the Lyapunov exponents to $\pm (1+\g) g$ with $\g$ analytic in $\m$, is not identically zero; see, however, [Ge2]). A full description of the induction can be found in [G3]. For the analyticity properties of the series introduced above we refer to \S2 of [G3] and we proceed to a quick discussion of the determination of the the initial constants. \* \0{\bf\S5. Improper integrals and the operators $\II,\OO,\OO_0$.} \numsec=5\numfor=1 \* The integrations in \equ(4.13),\equ(4.14) can be expressed in terms of an operator $\II$ acting linearly on finite linear combinations of monomials like \equ(4.18) with $k^2+(\oo\cdot\nn)^2>0$: % $$\eqalign{ &\II M(t)\defi \ig_{\s\io}^t M(\t) d\t, \qquad {\rm with:}\cr &\II M(t)=- \s^{\th +1} x^k e^{i\r\oo\cdot\nn t} \sum_{p=0}^h{ (\s t)^{h-p} \over(h-p)!} {1 \over(k- i \r \s \oo\cdot\nn)^{p+1}}\cr} \Eq(5.1)$$ % where the first row is a formal definition whose mathematical meaning is given by the second row (note that if $k\le0$ the first line is an improper integral), and we set $g=1$. Note that the $\II$ is {\it not defined} on the polynomials of $t,\s$, \ie if $k=0$ and $\oo\cdot\nn=0$ (so that no exponentials are present in the monomial defining $M$). It can be naturally extended, for $j\ge0$, to the polynomials by setting $\II t^j\s^\th=\fra{\s^\th}{j+1} t^{j+1}$, see (3.7) in [G3]. The $\II$ is an integration with respect to $t$ with special initial data: in fact {\it at fixed $\s$}: % $$\dpr_t\II M\=M\Eq(5.2)$$ % If $M$ is such that $M(t)\=M(\oo t,\s)$ for some $M(\pps,\s)$ defined on the torus, then: % $$\II M(t)=(\oo\cdot\V\dpr_\pps)^{-1}M(\oo t,\s)\Eq(5.3)$$ % The integrals in \equ(4.13),\equ(4.14) can be expressed in terms of the operators: % $$\eqalign{ \OO F(t)= &w_{03}(t)\bigl(\IG (\,w_{00} F))\bigr)(t)-w_{00}(t)\bigl(\IG ( w_{03}F)\bigr)(\t)|^t_{0^\s}\cr \OO_+ F(t)=& w_{33}(t)\bigl(\IG (\,w_{00} F)\bigr)(t)- w_{30}(t)\bigl(\IG w_{03})F \bigr)(\t)|^t_{0^\s}\cr \lis\II^2 F(t)=&\II^2 F(t)-\II^2 F(0^\s)\cr }\Eq(5.4)$$ % where $\s=\sign t $. Then one finds, in the general anisochronous case: % $$X^h_-(t)=\OO F^h_+(t),\quad X^h_+(t)=\OO_+ F^h_+,\quad \V X^h_\giu(t)=\h E\lis\II^2 F^h_\su(t), \quad \V X^h_\su(t)=\IG \V F^h_\su(t)\Eq(5.5)$$ % where $E$ is the projection over the first component, see \equ(4.15), and $F^{h}$ have to be expressed in terms of the $X^{h'}$ with $h'0$): % $$ (k^i_j)_{\V m,p}\=(k^0_1,\ldots,k^0_{m_0},k^1_1,\ldots,k^1_{m_1}, \ldots,k^{3}_1,\ldots,k^{3}_{m_{3}})\qquad {\rm s.t.\ }\sum k^i_j=p\Eq(5.6)$$ % referring to \S2 of [G3] for more details (if needed). If $f_1\defi f, f_0\defi g^2\cos\f$, we find: % $$\eqalignno{ & F_-^{k\s}\=0\ ,\ \V F_\giu^{k\s}\=\V 0\ ,\ \V F_\su^{k\s} = - \sum_{|\V m|\ge0,\d=0,1}\fra1{\V m!} (\dpr^{\V m}\dpr_\aa f_\d) \sum_{(k^i_j)_{\V m,k-1}} \prod_{i=0}^{l-1}\prod_{j=1}^{m_i} X^{k^i_j\s}_i\cr & F_+^{k\s} \=- \sum_{|\V m|\ge 2,\d=0,1}^* \fra1{\V m!} (\dpr^{\V m}\dpr_\f f_\d) \sum_{|\V m|\ge0} (\dpr_0 f_\d)_{\V m} \sum_{(k^i_j)_{\V m,k-1}} \prod_{i=0}^{l-1}\prod_{j=1}^{m_i} X^{k^i_j\s}_i&\eq(5.7)\cr}$$ % where $(k^i_j)_{\V m,k},(k^i_j)_{\V m,k-1}$ are defined in \equ(5.6). The $*$ means that if $\d=0$ only vectors $\V m$ with $|\V m|\ge2$ have to be considered in the sums. Note that if $\d=0$ the sum in the expression for $F^h_+$ can only involve vectors $\V m$ with $m_j=0$ if $j\ge1$, because the function $f_0= g^2\cos\f$ depends only on $\f$ and not on $\aa$, (hence also $k^i_j=0$ if $i>0$). The functions are evaluated at $\f(t)=\f^0(t),\aa(t)\defi \aa+\oo t$. The indices in \equ(5.7) are mutually contracted with a natural rule that we leave to the reader to work out. The relations $\V F^h_\giu=\V0$ and $F_-^h\=0$ are general (as the equations for $\dot {{\V X}}_\giu, \dot X_-$ are linear) and in the {\it isochronous cases}: % $$\eqalign{ \V X_\su^1(t)=&-\II(\dpr_\aa f),\quad \V X^2_\su(t)=-\II\Big(\dpr_{\aa\f} f\,\OO(-\dpr_\f f)\Big) \cr \V X^3_\su(t)=&-\II\Big(\dpr_{\aa\f} f\OO\big( -{1\over2}\sin\f\OO(\dpr_\f f)\OO(\dpr_\f f)\big)\Big)+\cr &-\II\Big(\dpr_{\aa\f} f\OO\big(\dpr_{\f^2} f\OO(\dpr_\f f)\big)\Big) -{1\over2}\II\Big(\dpr_{\aa\f^2} f\OO(\dpr_\f f)\OO(\dpr_\f f)\Big)\cr}\Eq(5.8)$$ % (see the examples in \equ(4.9)). We fix our attention on the models with $f$ a trigonometric polynomial (``trigonometric perturbation'') of degree $N$: % $$\eqalign{ &f(\aa,\f)=f_S(\f,\a,\l)+f_F(\f,\a,\l)\cr & f_j(\f,\a,\l)= \sum_{n,\nn}^N f_{j,(n,\nn)}\,\cos(\n_1\a+\n_2\l+ n\f),\quad j=S,F\cr}\Eq(5.9)$$ % with $|n|,|\nn|\le N$ and $f_{S,(n,\nn)}=0$ unless $\n_2=0$, \ie $\nn$ is a {\it slow} mode, while $f_{F,(n,\nn)}=0$ unless $\nn_2\ne0$, \ie $\nn$ is a {\it fast} mode. We also say that $f_S$ depends only on slowly rotating angles and $f_F$ depends on fastly rotating angles. A nontrivial example can be: % $$f(\a,\l)=\big(\cos(\a+\f)+\cos(\l+\f)\big)\Eq(5.10)$$ % The intersection matrix to order $h$, $D^h_{ij}$, can be expressed, to order $h=1,2,3$ as: $$\eqalignno{ D^1_{ij}=&\ig_{-\io}^\io dt \,\dpr_{ij}f,\quad D^2_{ij}=- \ig_{-\io}^\io dt\,\big[\dpr_{ij0}f\,\OO(\dpr_0 f)+ \dpr_{j0} f \OO(\dpr_{i0} f)\big]\cr D^3_{22}=&\ig_{-\io}^\io dt\,\Big[ w_{30}\,\OO(\dpr_{220}f)\,\OO(\dpr_0 f)^2+2w_{30}\,\OO(\dpr_{20} f)^2 \,\OO(\dpr_0 f)+&\eq(5.11)\cr &+ \dpr_{200}f\, \OO(\dpr_{20}f)\,\OO(\dpr_0f)+\dpr_{00}f\,\OO(\dpr_{220}f) \,\OO(\dpr_0 f)+\cr & +\dpr_{00}f\,\OO(\dpr_{20}f)\,\OO(\dpr_{20} f)+\fra12 \dpr_{2200} f\,\OO(\dpr_0 f)^2+ \dpr_{200}f \OO(\dpr_{20}f)\,\OO(\dpr_{0}f)\Big]\cr}$$ % where the derivatives of the $f$'s are evaluated at $\f(t),\aa+\oo t$, with $\f(t)\=\f^0(t)$ the free separatrix motion, see \equ(4.10). We set $\dpr_0\defi \dpr_\f$ and $\dpr_i\defi \dpr_{\a_i}$; the $\aa$'s have to be set equal to $\00$ after evaluating derivatives. The expressions contain improperly convergent integrals (in general) and must be understood by thinking $\ig_{-\io}^{+\io}$ as $\ig_{-\io}^{0}+\ig_{0}^{+\io}$ and by using the definition \equ(5.1), see [G3]. It is convenient to split the operation $\OO$, see (6.5) in [G3], as: $$\eqalign{ %\OO_+(F)=& \OO^+_0(F)+\s w_{33}(t)\, G^{(0)}(F)+\, w_{30}(t)\,G^{(1)}(F)\cr \OO(F)=& \OO_0(F)+|w_{03}(t)|\, G^{(0)}(F)+\, w_{00}(t)\, G^{(1)}(F)\cr \OO_0(F)=&\fra12\Big(\ig_{-\io}^t d\t (w_{03}(t)\,w_{00}(\t) -w_{00}(t)\,w_{03}(\t))\,F(\t)+ \ig_{+\io}^t d\t\,(same)\Big)\cr %\OO^+_0(F)=&\fra12\Big(\ig_{-\io}^t d\t (w_{33}(t)w_{00}(\t) %-w_{30}(t)w_{03}(\t))\,F(\t)+\ig_{+\io}^t d\t\,(same)\Big)\cr G^{(0)}(F)=&\fra12\ig_{+\io}^{-\io} d\t\,w_{00}(\t)\,F(\t), \qquad G^{(1)}(F)= \fra12\,\ig_{+\io}^{-\io} d\t\,|w_{03}(\t)|\,F(\t)\cr}\Eq(5.12)$$ % The identity, see (6.12) and Appendix A2 of [G3]: % $$\ig_{-\io}^{+\io} dt\, F(t)\OO(H)(t)=\ig_{-\io}^{+\io} dt\, H(t)\OO(F)(t)\Eq(5.13)$$ % implies symmetry of the above matrices $D_{ij}$, at least for the first three orders (see \equ(5.11)): symmetry follows to all orders as said after \equ(4.2), or as it will be seen in \S6. In the anisochronous case, \ie in \S7, we shall also use the splitting: $$\eqalign{ &\lis\II^2(F)=\II^2_0(F)+|t| G^{(2)}(F)+G^{(3)}(F)\cr &\II^2_0(F)= \fra12\Big(\ig_{-\io}^t d\t\, (t-\t)\,F(\t)+\ig_{+\io}^t d\t\,(same)\Big) \cr &G^{(2)}(F)=\fra12\ig_{+\io}^{-\io}d\t\,F(\t),\qquad G^{(3)}(F)= \fra12\ig_{+\io}^{-\io}d\t\,|\t|\,F(\t)\cr}\Eq(5.14)$$ % see eq. (6.3),(6.6) in [G3]. The key remark, to understand the asymptotic behaviour of \equ(5.11) as $\h\to0$, is that whenever the integrand is analytic it becomes possible to shift the integrations, over $t$ and the $\t$'s, to an axis close to $\Im t$, $\Im\t=\pm(\fra\p2-\h^{1/2})$: see \S8 in [G3] (choosing the free parameter $\d$ appearing in [G3] as $\h^{1/2}$). Using that $G^{(1)}(F)=G^{(0)}(F)=0$ if $F$ is odd (as the odd derivatives of $f$ are, when evaluated at $\aa=\V0$) and using also that $\OO_0$ leaves parity unchanged, in general, we shall find that the {\it non analytic } terms (\ie those containing integrals of a non analytic function, \eg $|w_{03}(\t)|$) {\it cancel each other} in their contribution to the determinant of $D_{ij}$ {\it to all orders} $k\ge1$. The result will be the proof of the following theorem: \* \noindent{\it {\bf Theorem 2} ({\sl``Large angles theorem''}): Consider a system described by the hamiltonian \equ(2.1) or \equ(7.1) with $f$ an even trigonometric polynomial. Let $\m$ be small ($|\m|<\m_0$) and $c$ large, enough. Consider an invariant torus with diophantine rotation $\oo$ among those described in theorem 1 above. At the homoclinic point with $\f=\p,\aa=\V0$ the intersection matrix determinant is exponentially small as $\h\to0$ and it is generically asymptotic to its first order value (``Melnikov' formula''). The choice given by \equ(5.10) is a concrete example of this result, generically holding for \equ(5.9).} \* \noindent{\it Remark}: The result in the case $f$ is given by \equ(5.10) is $\det D=24 \p \h^{-1/2} \e^2 e^{-\fra\p2 \h^{-1/2}}$ in both cases (isochronous, \equ(2.1), and anisochronous, \equ(7.1)) with $a>0$. This theorem has an analogue with and $a=-\fra12$, \ie only two time scales: an early review is in [G3]. In the latter case it has been extended to analytic perturbations, \ie beyond the trigonometric case, and to cover the exact asymptotic value of the splitting, \ie far beyond [G3], see [DGJS], [RW]. The above three scales case, \equ(2.1), is quite different from the two scales case (discussed in [G3]): but arises naturally in celestial mechanics problems near a double resonance, as in the case of the precession problem in [CG] to which we hope to apply, eventually, the results of this paper. The case in \equ(2.1) is that of a pair of ``clocks'' and a pendulum. The case of a ``clock'', a ``rotator'' and a pendulum is exemplified by the hamiltonian obtained by adding $\fra12 \h A^2$ to \equ(2.1), see \equ(7.1) below. Both cases are treated in full detail in \S6,\S7. \* \0{\bf\S6. Nonperturbative splitting analysis in presence of fast and slow rotations.} \numsec=6\numfor=1\* It would be easy to show that the determinant of the intersection matrix is exponentially small to order $\e^4$: this requires evaluating the intersection matrix only to third order. But it cannot be done without due care, as the error in [CG] was precisely due to the belief that it was not necessary to evaluate the matrix element $D_{22}$ because it was exponentially small. In fact it is not exponentially small and it has the right value to make, instead, the whole determinant exponentially small. The real problem is to compute the determinant to all orders and to show that {\it to all orders} it is exponentially small: \ie to all orders the determinant is a sum of ``large'' terms (not exponentially small) which ``cancel each other'' with a result that is exponentially small. So small that the first order calculation dominates in the limit as $\h\to0$. {\it It is remarkable that in fact one can give an exact expression for the leading corrections to the first order of perturbation expansion.} See \equ(6.12),\equ(7.19). This section relies, and in fact it follows almost immediately, from the general theory of the intersection matrix in [G3]. We cannot repeat here the general theory and therefore refer the reader to [G3] for details on the main definitions: we try nevertheless to make what follows readable at least from a formal viewpoint and as a guide to [G3]. The point of [G3] is that the intersection matrix can be quite explicitly calculated to all orders by using a graphical formalism very similar to that used in quantum field theory when the Schwinger functions are expressed via Feynman's graphs. In the present case the graphs will be, topologically, trees: very unusual graphs from the viewpoint of field theory, where loops are often the main source of interest and non triviality. On the other hand the graphs have nodes with arbitrarily large coordination number: also unusual in quantum field theories (with polynomial interactions). Let $\th$ be a tree built with oriented lines all pointing towards a ``highest'' node $r$ that we call the {\it root} and that we suppose to have only one ``incoming'' line, see figure below. The graph will bear a label $\d_v=0,1$ on each node: if $\d_v=1$ it represents $f\defi f_1$ while if $\d_v=0$ it represents $f_0(\f)=g^2\cos\f$. And the labels can be given arbitrarily with the restrictions that all endnodes bear a label $\d=1$ and that all nodes bearing a label $\d=0$ have at least two incoming lines. Each node will also bear a ``time'' label $\t_v$. We define the {\it value} of a graph by building the following symbol. We first lay down a set of parentheses $()$ ordered hierarchically and reproducing the tree structure: in fact any tree partially ordered towards the root can be represented as a set of matching parentheses corresponding to the tree nodes. Matching parentheses corresponding to a node $v$ will be made easy to see by appending to them a label $v$. The root will not be associated with a parenthesis. Inside the parenthesis $(_v$ and next to it we write $-\dpr_0^{p_v+1} f_{\d_v}$ for all nodes $v$ lower than the node $v_0$ preceding the root (``first node''). For $v=v_0$ we write $-\dpr_j\dpr_0^{p_v} f_{\d_{v_0}}$ where $\dpr_0\defi \dpr_\f$ and $\dpr_j\defi \dpr_{\a_j}$, $j=1,2$ (this implies that $\d_{v_0}=1$): the functions have to be evaluated at $(\f^0(t),\aa+\oo t)$. \pagina \* \figini{bggmfig0} \8 \8 \8<0.83333 0.83333 scale 0 90 punto > \8<70 90 punto > \8<120 60 punto > \8<160 130 punto > \8<200 110 punto > \8<240 170 punto > \8<240 130 punto > \8<240 90 punto > \8<240 0 punto > \8<240 30 punto > \8<210 70 punto > \8<240 70 punto > \8<240 50 punto > \8<0 90 moveto 70 90 lineto> \8<70 90 moveto 120 60 lineto> \8<70 90 moveto 160 130 lineto> \8<160 130 moveto 200 110 lineto> \8<160 130 moveto 240 170 lineto> \8<200 110 moveto 240 130 lineto> \8<200 110 moveto 240 90 lineto> \8<120 60 moveto 240 0 lineto> \8<120 60 moveto 240 30 lineto> \8<120 60 moveto 210 70 lineto> \8<210 70 moveto 240 70 lineto> \8<210 70 moveto 240 50 lineto> \8 \8 \figfin \eqfig{199.99919pt}{141.666092pt}{ \ins{-29.16655pt}{74.999695pt}{\it root} \ins{0.00000pt}{91.666298pt}{$j$} \ins{49.99979pt}{70.833046pt}{$v_0$} \ins{45.83314pt}{91.666298pt}{$\d_{v_0}$} \ins{126.66615pt}{99.999596pt}{$v_1$} \ins{120.83284pt}{124.999496pt}{$\d_{v_1}$} \ins{91.66629pt}{41.666500pt}{$v_2$} \ins{158.33270pt}{83.333000pt}{$v_3$} \ins{191.66589pt}{133.332794pt}{$v_5$} \ins{191.66589pt}{99.999596pt}{$v_6$} \ins{191.66589pt}{70.833046pt}{$v_7$} \ins{191.66589pt}{-8.333300pt}{$v_{11}$} \ins{191.66589pt}{16.666599pt}{$v_{10}$} \ins{166.66600pt}{54.166447pt}{$v_4$} \ins{191.66589pt}{54.166447pt}{$v_8$} \ins{191.66589pt}{37.499847pt}{$v_9$} }{bggmfig0}{\hskip.6truecm\eq(6.1)} \kern0.9cm \didascalia{A graph $\th$ with $p_{v_0}=2,p_{v_1}=2,p_{v_2}=3,p_{v_3}=2,p_{v_4}=2$ and $k=12$, $\prod p_v!=2^4\cdot6$, and some decorations. The line numbers, distinguishing the lines, and their orientation pointing at the root, are not shown. The lines length should be the same but it is drawn of arbitrary size. The nodes labels $\d_v$ are indicated only for two nodes.} Outside the parenthesis $(_v$ we write $\OO$ for all the $v< v_0$ and we add to the right of the matching parenthesis the symbol $(\t_v)$; for the first node we simply integrate over $\t_{v_0}$ from $+\io$ to $-\io$. The symbol thus defined has the meaning of a linear combination of products of multiple integrals if one uses the definitions of the symbols $\OO$, see \equ(5.12). We multiply it by $n!^{-1}$ if $n$ is the number of lines in the graph and we shall regard all the lines different (\ie labeled); {\it however} two graphs that can be superposed, {\it labels included}, by successively rotating rigidly around the nodes subtrees that are attached to them have to regarded as identical. This defines the {\it value} of a graph (it is a function of $\aa$). The reader can see that the above is a rather natural construction by working out patiently the definition in the case of trees with one, two or three nodes. The sum over all graphs of ``order'' $k=\sum_v \d_v$ of the graph values gives the coefficient $Q^{k}_j$ of order $k$ of the splitting vector $\V Q(\aa)$: see [G3] where the above construction is performed in Fourier transform to obtain directly an expression for $\V Q^{k}_\nn$. It is convenient to make this more explicit by using the decomposition of $\OO$ in the first line of \equ(5.12). This can be easily done by simply attaching to each node lower than the first a label $\b=O,D, R$ signifying that we select the first of the three terms in the decomposition of $\OO$ (see first line in \equ(5.12)) or the second or the third. We can alternatively imagine drawing a circle around each node $v$ enclosing only the subtree with that node $v$ as first node and then to attach the label to the circle. By \equ(5.12) we see that a circle with a $D$ or $R$ label encircling a node $w$ linked to the higher node $v$ (external to the circle) represents just a function $|w_{03}(\t_v)|$ or $w_{00}(\t_v)$ multiplied by a number that in order to be evaluated requires essentially the same operations required to evaluate the value of a graph $\th$. This allows us to give a {\it nonperturbative} expression for the splitting vector $\V Q(\aa)$. We simply consider the sum of all the values of the graphs bearing a label $O$ on all the nodes except perhaps the endnodes that can bear also $D,R$ labels. We evaluate the graphs values and in the end we replace the number associated with the $R,D$--labeled endnodes by the {\it full perturbation series} that is obtained by imagining that inside the circles with $R$ and $D$ labels there is the most general graph with $O,R,D$ labels in all possible ways. The sum of such perturbation series will be denoted $G^{(0)}(\aa)$ for $D$--labeled circles, and $G^{(1)}(\aa)$ for $R$--labeled circles. The new representation of $\V Q(\aa)$ is therefore a representation in terms of trees with a few ``{\it fruits}'' around some of the endnodes (possibly none or all) that can be $D$--labeled or $R$--labeled ({\it dry} or {\it ripe}, to follow the names of [G3]). Furthermore $G^{(1)}$ and $G^{(0)}$ verify a simple recursion relation that can be found by a more explicit representation of the quantities defined in the same way as $G^{(0)},G^{(1)},\V Q$ {\it but evaluated by considering only trees deprived of fruits}, see \equ(6.4) below. Let $\th$ be a graph whose nodes $v$ carry indices $\d_v=0,1$ and $\t_v$; let $v_0$ be the first node of $\th$ and $v'>v$ be (if $v