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\let\bf\elevenbf \renewcommand\thm#1#2{\vglue 0.3cm\noindent {\elevenbf #1}{\sl #2\hfil}\vglue 0.3cm} \renewcommand\section#1#2{\vglue 0.6cm \line{\elevenbf #1. #2\hfil}\vglue 0.4cm \baselineskip=14pt} \renewcommand\subsection#1#2{\line{\elevenit {#1.} {#2}\hfil} \smallskip\baselineskip=14pt} \renewcommand\nota#1#2{\footnote{$^{#1}$}{ \ninerm\baselineskip=11pt \negthinspace\negthinspace{\negthinspace#2}}} \def\W{{W^\pm_{\m,{\rm loc}}(\e,\o)}} \def\Wp{{W^+_{\m,{\rm loc}}(\e,\o)}} \def\Wm{{W^-_{\m,{\rm loc}}(\e,\o)}} \def\T{{\cT_\m(\e,\o)}} \def\wH{{\widetilde H}} \def\const{{\; \rm const\;}} \def\mod{{\; \rm mod \;}} \def\meas{{\; \rm meas\;}} \def\tF{{\widetilde F}} \def\medsk{\vglue 0.4cm} \def\newline{\hfill\vglue0.1truecm} \def\bI{{\bar I}} \def\baf{{\bar \f}} \def\sqe{{\sqrt{\e}}} \def\kp{{k\pm}} \def\cJ{{\cal J}} \def\cH{{\cal H}} \def\wcF{{\widetilde\cH}} %%%%%%%%%%%%%%%%%%%%%%%% %\baselineskip=10pt %\centerline{\tenbf WORLD SCIENTIFIC PUBLISHING CO. PTE. LTD.} %\vglue 12pt \centerline{\tenbf ARNOLD INSTABILITY FOR NEARLY--INTEGRABLE} \vglue 0.2cm \centerline{\tenbf ANALYTIC HAMILTONIAN SYSTEMS} \vglue 1.0cm \centerline{\tenrm LUIGI CHIERCHIA} \baselineskip=13pt \centerline{\tenit Department of Mathematics, Terza Universit\`a di Roma} \baselineskip=12pt \centerline{\tenit via C. Segre 2, 00146 Roma, Italy} \baselineskip=12pt \centerline{\tenit Internet: luigi@matrm3.mat.uniroma3.it} \vglue 0.8cm \centerline{\tenrm ABSTRACT} \vglue 0.3cm {\rightskip=3pc \leftskip=3pc \tenrm\baselineskip=12pt%\parindent=1pc \noindent A notion of instability, which is proposed as a definition for the so--called ``Arnold diffusion", for one--parameter families of nearly--integrable analytic Hamiltonian systems, is introduced. An example of unstable system is given. \vglue 0.8cm } \line{\elevenbf 1. Introduction \hfil} \bigskip \baselineskip=14pt \elevenrm One of the central questions in the theory of nearly--integrable Hamiltonian systems concerns the ``stability" of the ``action" variables, or, equivalently, of the perturbed integrals of motions. In the late fifties KAM theory\nota*{ A.N. Kolmogorov, V.I. Arnold, J.K. Moser; see, e.g., Ref.\ref{1} and references therein.} gave a partial answer to that question, ensuring, under general conditions, ``stability" for ``most" initial data. In 1964, Arnold\ref{2} pointed out a mechanism for ``instability", illustrating it on an example containing two parameters. The name ``Arnold diffusion" for related phenomena was introduced in the physics literature by B.V. Chirikov. Here, we introduce, in the context of one--parameter families of real--analytic Hamiltonian systems, the definition of ``Arnold instability", which captures the main features of the mechanism discovered by Arnold. After a brief review of some known results, we discuss an explicit example (containing only one parameter), of an Arnold--unstable system. The {\sl content} of the rest of the paper is: \medskip {\tenrm\baselineskip=12pt%\parindent=1pc \item{\S 2} Nearly--Integrable Analytic Hamiltonian System \item{\S 3} Definition of ``Arnold Instability" \item{\S 4} A Brief Review of Some Known Result \item{\S 5} An Arnold--Unstable System (Result and Scheme of Proof) \itemitem{\S 5.1} Main Result \itemitem{\S 5.2} Outline of Proof \item{\S 6} On Two Technical Points \itemitem{\S 6.1} Remarks on the KAM Construction \itemitem{\S 6.2} Generalized Poincar\`e Integrals \item{\S 7} References} % \vfill\eject \vglue 0.6cm \line{\elevenbf 2. Nearly--Integrable Analytic Hamiltonian Systems\hfil} \vglue 0.4cm % Let $\cM$ denote the Cartesian product of some open connected set $G\subset \rN$ times the standard $N$--torus $\tN\=\rN/(2\p\zN)$. On the ``phase space" $\cM$, endowed with the standard symplectic form $\sum_{i=1}^N dI_i\wedge d\f_i$, consider one--parameter families of Hamiltonian functions $H(I,\f;\e)$. {\sl Throughout this paper we shall assume the following real--analyticity and near--integrability assumptions}: for some $\bar\e>0$, (i) $H$ is real-analytic on $\cM\times (-\bar\e,\bar\e)$; (ii) $H(I,\f;0)\=H_0(I)$. The study of the trajectories (or ``orbits") generated by such $H(I,\f;\e)$, namely, the solutions of the Hamilton equations $$ \dot I_i=-{\dpr H\over \dpr \f_i}\ ,\qquad\dot \f_i= {\dpr H\over \dpr I_i}\ , \quad\big[i=1,...,N;\ \dot{()}\={d\over dt}()\big]\ , \eqno(1) $$ was considered by H. Poincar\'e ``{\sl le probl\`eme g\'en\'eral de la Dynamique}" (see Ref.\ref{3}, Tome I, chap. I, No. 13). For $\e=0$, $I_1$,...,$I_N$ are $N$ independent integrals in involution and all trajectories are quasi--periodic\nota\dag{A function on $\cM$ is an integral for $H$ if it is constant on the trajectories generated by $H$; ``independent" means ``functionally independent"; two (smooth) functions $f$ and $g$ are in involutions if their Poisson bracket $\{f,g\}=\sum_{i=1}^N { \dpr f\over \dpr I_i} { \dpr g \over \dpr \f_i}- { \dpr g\over \dpr I_i} { \dpr f\over \dpr \f_i}$ vanishes; a trajectory is quasi--periodic if it is conjugated to a linear flow on a torus.}: this is equivalent to say that the system is {\sl completely integrable}. The basic {\sl stability problem} is here whether, for $\e\neq 0$, the evolution of the ``action" variables, $I_1(t)$,...,$I_N(t)$, stays close, for all $t$, to their initial values $I_1(0)$,...,$I_N(0)$. % \vglue 0.6cm \line{\elevenbf 3. Definition of ``Arnold Instability"\hfil} \vglue 0.4cm % Let us denote phase points by $z=(I,\f)$ and by $\phi_H^t$ the Hamiltonian flow generated by $H$ \ie $z(t)\=\phi_H^t(z)\=(I(t),\f(t))$ solves Eq. (1) with initial data $z(0)\=z\=(I,\f)$. A set $\cT\subset \cM$ will be called {\sl invariant for $H$} (or $H$--invariant) if it is closed and $\phi_H^t(\cT)\subset \cT$ for all $t\in\real$. We now give the following definitions (compare with Ref.\ref{2}). \giu {\elevenbf Definition 1.} {\sl An $n$--ple of trajectories $z^\ppu(t)$,...,$z^\ppn(t)$ is called a ``chain of heteroclinic trajectories" if there are $(n+1)$ invariant sets $\cT^\ppu$,...,$\cT^{(n+1)}$ such that, for $1\le i \le n$, $$ \lim_{t\to-\io} \dist\big(z^\ppi(t),\cT^\ppi\big)=0\ ,\qquad \lim_{t\to+\io} \dist\big(z^\ppi(t),\cT^{(i+1)}\big)=0\ . $$ The sets $\cT^\ppu$,...,$\cT^{(n+1)}$ are said to be connected by the chain $z^\ppu(t)$,...,$z^\ppn(t)$. } \giu {\elevenbf Definition 2.} {\sl A chain $z^\ppu(t)$,...,$z^\ppn(t)$ is called a ``transition chain" if, for any $\r>0$, there exists a trajectory $z(t)$ and a time $T>0$ such that: $z(0)$ is in a $\r$--neighbourhood of $\cT^\ppu$; $z(T)$ is in a $\r$--neighbourhood of $\cT^{(n+1)}$; for all $0\le t\le T$, $z(t)$ is in a $\r$--neighbourhood of $Z\=\bigcup_{i=1}^n \{ z=z^\ppi(t):\ t\in\real\}$. } \giu {\elevenbf Definition 3.} {\sl A Hamiltonian system $(\cM,H)$ with $\cM$ and $H$ as in \S 2 is called ``Arnold unstable" if, for some $0<\e_0\le \bar\e$, the following two conditions are fulfilled: \noindent {\rm (i)} for all $\e\in (- \e_0,\e_0)$ there exist invariant sets $\cT(\e)$ and $\cT'(\e)$, which are continuous in the Hausdorff metric\nota{**}{ If $A$ and $B$ are closed subsets of $\cM$, the Hausdorff metric $\r(\cdot,\cdot)$ is defined by $$ \r(A,B)= \max \{ \sup_{x\in A} \dist(x,B)\ ,\ \sup_{x\in B} \dist(x,A)\}\ . $$}, and $p_1(\cT(0))\neq p_1(\cT'(0))$, where $p_1(I,\f)=I$ denotes the projection onto the ``action" variables; \noindent {\rm (ii)} for all $\e\neq 0$ in $(- \e_0,\e_0)$ there exists a transition chain connecting $\cT(\e)$ with $\cT'(\e)$. } \giu In other words, an Arnold--unstable system posses, for all $\e\neq 0$ small enough, trajectories along which (some of ) the ``action" variables undergo a variation of order 1 and, furthermore, such trajectories ``shadow" long chains of heteroclinic connections. Notice that the above instability mechanism is completely different from the ``linear drift" occurring along so--called ``superconductivity channels". The classical example (due to N. N. Nekhoroshev\nota*{See, e.g., \S 3.3, chapter 5 of Ref.\ref{1}.}) is the region $I_1=-I_2$ for the Hamiltonian ${1\over 2} (I_1^2-I_2^2)+\e\sin(\f_1-\f_2)$, which admits the trajectory $I_2(t)=\e t=-I_1(t)$, $\f_1(t)=\f_2(t)=-{1\over 2}\e t^2$. % \vglue 0.6cm \line{\elevenbf 4. A Brief Review of Some Known Results\hfil} \vglue 0.4cm % Even though the above definition is strongly inspired by Arnold's paper\ref{2}, it is not known whether the Hamiltonian system considered there is (in a suitable sense) Arnold unstable or not. Let us be more specific on this point. In Ref.\ref{2}, Arnold considered on $\cM\=\real^3\times \torus^3$ the {\sl two--parameter} family of Hamiltonians given by $$ H=I_1+ {1\over 2}\big( I_2^2+ I_3^2\big) + \e (\cos \f_3-1) + \e\m (\cos \f_3-1) (\cos \f_1 + \sin \f_2)\ . \eqno(2) $$ Since $\dot\f_1=1$, it is $\f_1=t_0+t$ so that the above Hamiltonian describes, effectively, a system with two ``degrees of freedom" depending explicitly (and periodically) on time\nota\dag{ We prefer the ``autonomous" formulation which allows a unified treatment; one should, however, bear in mind that the ``action" $I_1$ is an auxiliary variable introduced so as to make the system autonomous and is not really relevant from a dynamical point of view. In Ref.\ref{2} $\f_1$, $\f_2$, $t$ correspond here to $\f_3$, $\f_2$, $\f_1$ and $I_1$, $I_2$ to $I_3$, $I_2$.}. % The ``resonant" 2--torus $\{I_1=$const, $I_2=$const, $I_3=0=\f_3$, $(\f_1,\f_2)\in\torus^2\}$ is invariant for $H$ {\sl for all $\e$ and $\m$}; such a torus is run by the linear flow $(\f_1,\f_2)\to(\f_1+t,\f_2+ \o t)$ where $\o$ is the constant value of $I_2$: using the frequency $(1,\o)$ to parametrize the above invariant torus we shall denote it by $\cT(\o)$. Of course the fact the $\cT(\o)$ is invariant for all values of the parameters $\e$ and $\m$ is a very special feature of the particular perturbation chosen by Arnold. Using the language introduced in \S 3, the main result in Ref.\ref{2} concerning the system with Hamiltonian as in Eq. (2), can be formulated as follows \thm{Theorem 1 (Arnold\ref{2})}{ For every $\e,r>0$ there exists a $\m_0>0$ such that for all $0<\m<\m_0$ there are invariant tori $\cT(\o)$ and $\cT(\o')$, with $|\o-\o'|>r$, which are connected by a transition chain.} % Note that, for all $\e,\m,\o$, the projection onto the $I_2$--value of $\cT(\o)$ is just $\o$. The proof of this result is perturbative in $\m$ keeping $\e>0$ fixed. For $\m=0$, although one can write down explicitly all trajectories, the system is not ``completely integrable" in the sense described in \S 2. In fact (for $\m=0$) the trajectories with initial values of $(I_3,\f_3)$ on the ``separatrix" ${1\over 2} I_3^2+\e (\cos \f_3 -1)=0$ (and $I_3\neq 0$) are not quasi--periodic. The arguments used in Ref.\ref{2} indicate that $\m_0$ may be taken to be of order $$\exp\big(-{1\over \e^a}\big)\ , \eqno(3)$$ for some constant $a\ge {1\over 2}$, leaving, thus, open the following \thm{Question}{ Does there exist a positive integer $n$ such that the system with Hamiltonian as in Eq. (2) with $\m=\e^n$ is Arnold unstable\nota{**}{ The choice $\m=\e^n$ in Arnold's model is somehow ``natural", not only because it makes $H$ belong to the ``Poincar\`e category" described in \S 2, but also in view of the fact that near a ``simple resonance" the dynamics generated by a ``general" Hamiltonian $h(I)+\e f(I,\f)$ is equivalent to that of a Hamiltonian $\tilde h (I;\e)+\e g(I,\f_N;\e)+ \e^n\tilde f(I,\f;\e)$ (everything being real--analytic in a neighbourhood of, say, $I_0\times \tN$ and of $\e=0$); for more informations see Ref.\ref{1}. Furthermore, in case of models with several perturbation parameters, it seems natural (at least from the applicative point of view) to require that such parameters be related by a power law.}?} % A result analogous to that of Arnold has been recently obtained by Z. Xia\ref{4} in the elliptic restricted three--body problem of Celestial Mechanics. Such a system contains many parameters and by ``similar" we mean the following. First, transition chains are detected in parameter regions where, to a positive value of one of the perturbative parameter, say $\e$, corresponds some ``reference" hyperbolic trajectory (analogous to the separatrix of the pendulum). Then, {\sl keeping fixed $\e$}, perturbation theory in a second parameter (corresponding to $\m$ in Arnold's example) is performed. Again, the argument indicates that the result holds for $0<\m<\m_0(\e)$ with $\m_0(\e)$ as in Eq. (3) and the question whether the elliptic restricted three--body problem is Arnold unstable in parameter regions where the various parameters are related by, say, power laws is open. As of now the only explicit system, which has been proven to be Arnold unstable is the ``eccentric D'Alembert model" of Celestial Mechanics considered in Ref.\ref{5}, which may be described as follows. Consider a planet modelled by a ``homogeneous" rotational ellipsoid whose center of mass revolves on a Keplerian ellipse of eccentricity $\m$. The planet is subject to the gravitational attraction of a fixed star occupying one of the foci of the ellipse. If $(\th_1,\th_2,\th_3)$ denote the Euler angle of the planet, the Lagrangian\nota{*}{See, e.g., Ref.\ref{1} for generalities.} of the system is given by $$ {J_3\over 2} (\dot \th_2\cos \th_1+ \dot \th_3)^2+{J_1\over 2} (\dot \th_1^2 +\dot \th_2^2 \sen^2 \th_1)+ c \int_{E(t)}{ dx\over |x_0(t)+x|} $$ where $J_3$ and $J_2\=J_1$ are the inertia moments of the planet, $c$ is a physical constant\nota\dag{$c=\k m_0 m/V$ where $\k$ is Newton's gravitational constant, $m_0,m$ are the masses of the star and of the planet, and $V$ is the volume of the planet.}, $E(t)$ is the region occupied by the planet at time $t$ and $x_0(t)$ is the position of the center of mass of the planet at time $t$. Using the Andoyer variables (see Ref.\ref{1}) , one can find symplectic coordinates on $\cM\=G\times \torus^3$ ($G$ being some open set of $\real^3$) so that the dynamics generated by the above Lagrangian is equivalent to the dynamics generated by the Hamiltonian $$ H=\o I_1+ \e {I_2^2\over 2} + {I_3^2\over 2} +\e \m f(I,\f;\m) \eqno(4)$$ for a suitable function $f$ real--analytic (on $\cM\times (-\m_0,\m_0)$), whose explicit expression can be found in Appendix 14 of Ref.\ref{5}; here $\e$ is essentially the ratio between the polar and equatorial radii of the planet. In Ref.\ref{5} it is proven the following \thm{Theorem 2}{ There exists $n_0\ge 1$ such that for all integers $n\ge n_0$ the ``eccentric D'Alembert problem"\nota{**}{ That is, the Hamiltonian system on $\cM$ with $H$ as in Eq. (4).} with $\m=\e^n$ is Arnold unstable.} % We remark that in Ref.\ref{1,2,4} there is no mathematical definition of ``Arnold diffusion" (to be precise in Ref.\ref{2} such a terminology does not even appear while in Ref.\ref{1} appears only in a note by the translator). In Ref.\ref{5}, a genearal and detailed terminology (including notions such as ``diffusion paths") is introduced in the context of perturbations of ``a priori unstable systems", which, roughly speaking, are ``integrable systems possessing separatrices". Examples of such systems include the above Arnold's model with $\e$ kept fixed and positive and with $\m$ as perturbation parameter, and the three--body problem in the parameter region considered by Xia. Finally, we point out that Theorem~2 is differently formulated in Ref.\ref{5} (as the notions given in \S 3 are introduced in this paper). \section{5}{An Arnold--Unstable System (Result and Scheme of Proof)} % \subsection{5.1}{Main Result} Let $\cM\=\real^3\times \torus^3$ and consider $$ H(I,\f;\e)\=\e\big({I_1^2\over 2}+ {I_2^2\over 2}\big) + {I_3^2\over 2} + \e (\cos \f_3 -1) + \e^n \big(\cos (\f_1+\f_3)+\cos(\f_2+\f_3)\big)\ , \eqno(5)$$ where $n$ is an integer greater than 1. We shall prove \thm{Theorem 3}{ There exists $n_0\ge 1$ such that for all integers $n\ge n_0$ the above Hamiltonian system $(\cM,H)$ is Arnold unstable.} \noindent {\bf Remark} (i) In some sense $(\cM,H)$ is the ``simplest" Arnold--unstable system with a perturbation\nota*{ In Eq. (5) the perturbation is $H_1=\e\big\{{I_1^2\over 2}+ {I_2^2\over 2} +(\cos \f_3 -1) + \e^{n-1} \big(\cos (\f_1+\f_3)+\cos(\f_2+\f_3)\big)\big\}$, the ``integrable part" is $H_0={I_3^2\over 2}$: $H=H_0+ H_1$. As already mentioned, having a ``separation" (in terms of order in $\e$) may be achieved near a ``simple resonance" by means of symplectic transformations. } exhibiting the main feature of a ``general perturbation". In particular the invariant sets $\cT(\e)$, which will be shown to be connected, for $\e\neq 0$, by heteroclinic trajectories, will be non--trivial continuations of invariant sets for $\e=0$ (i.e., $\cT(0)$ is {\rm not} invariant for $\phi_H^t$ as it happens in Arnold's example). (ii) What is really ``not general" in the Hamiltonian in Eq. (5) is the integrable part $H_0\={I_3^2\over 2}$, which is highly degenerate (depending only on one of the three action variables). Note that also the ``eccentric D'Alembert problem" features a similar (although less dramatic) degeneracy, which is actually quite common in problems arising in Celestial Mechanics. (iii) {\sl No example is known of an Arnold--unstable system on $G\times \real^N$} ($G$ open set in $\real^N$) {\sl with ``non--degenerate" integrable part} (of the form, say, $H_0\={1\over 2} \sum_{i=1}^N I_i^2$). \medsk \subsection{5.2}{Outline of Proof} The system with Hamiltonian $$ h(I,\f_3;\e)\= \e\big( {I_1^2\over 2}+ {I_2^2\over 2}\big) +{I_3^2\over 2} +\e (\cos \f_3-1) $$ may be explicitly solved as it decouples into a bidimensional completely integrable system times a pendulum. The instability region will be sought in a neighbourhood of the ($\e$--dependent) separatrix of the pendulum. Since for $\e$ small and positive\nota\dag{During the proof we shall always assume $\e>0$. The case $\e<0$ can be easily recovered by a simple translation in the $\f_3$ variable.}, $\e^n\ll \e$, it is natural to treat the full Hamiltonian system as a perturbation of the system $(\cM,h)$. To make this remark precise, we replace, in a first moment, $\e^n$ with an {\sl auxiliary parameter} $\m$, and establish perturbative results in $\m$. Such result will be analytic in $\m$ and we shall prove that for a suitable $n$, $\e^n$ is inside the $\m$--analyticity domain allowing to recover the original system (by setting $\m=\e^n$). Therefore, consider the {\sl modified system} on $\cM$ with Hamiltonian $$ \widetilde H (I,\f;\e,\m)\= h+\m f\ ,\qquad f\=\cos(\f_1+\f_3)+\cos(\f_2+\f_3)\ , $$ so that $\widetilde H(I,\f;\e,\e^n)=H(I,\f;\e)$. For $\m=0$, $\e>0$ and $\o\in\real^2$, consider the $h$--invariant set $$ \cT(\e,\o)\=\{ I_1=\o_1\ ,\quad I_2=\o_2\ ,\quad I_3=\f_3=0\ ,\quad (\f_1,\f_2)\in\torus^2\}\ , $$ on which the flow is quasi--periodic with frequencies $\e\o$: $$ \phi_h^t(I_1,I_2,0,\f_1,\f_2,0)=(I_1,I_2,0,\f_1+\e\o_1 t, \f_2+\e\o_2 t,0)\ .$$ These tori admit ``asymptotic manifolds", called by Arnold ``whiskers", given by $$ W(\e,\o)\=\{ I_1=\o_1\ ,\quad I_2=\o_2\ ,\quad {1\over 2} I_3^2 +\e(\cos\f_3-1)=0 \ ,\quad (\f_1,\f_2)\in\torus^2\}\ . $$ Each $h$--trajectory $z(t)$ starting on $W(\e,\o)$ approaches, as $t\to \pm\io$, the torus $\cT(\e,\o)$; note that $W(\e,\o) \supset \cT(\e,\o)$ and that if $z(0)\in W(\e,\o)\backslash \cT(\e,\o)$, then $\dist\big(z(t),\cT(\e,\o)\big)=O\big(\exp(-|t|\sqrt{\e})\big)$. As a set, $\cT(\e,\o)$ is {\sl independent of $\e$} (hence, in particular, $\cT(\e,\o)$ is continuous in $\e\in\real$); what changes with $\e$ are the frequencies $\e\o$. In the limit case $\e=0$, $\cT(0,\o)$ is still an invariant 2--torus made up by fixed points. On the other hand, in the limit case $\e=0$, the asymptotic manifolds $W(0,\o)$ cease to exist in the sense that the set $W(0,\o)\=\{I_1=\o_1$, $I_2=\o_2$, $I_3=0$, $\f\in\torus^3\}$ is, again, an invariant set made up by fixed points hence a trajectory starting on $W(0,\o)\backslash \cT(0,\o)$ {\sl does not} approach $\cT(0,\o)$. The proof is based on three separate steps, which we proceed to describe. {\bf (1)} Let, again, $\e>0$. Roughly speaking, the first step consists in constructing the analytic continuation in $\m$ of the tori $\cT(\e,\o)$ together with their ``local" whiskers, for ``many" values of the ``rescaled frequencies" $\o$; by ``local" we mean ``in a ($\e$--dependent) neighbourhood of $\cT(\e,\o)$". If $\e>0$ is small enough, the $\m$--analyticity domain contains a complex disk of radius $\m_0\=\e^m$ (for $m$ suitable). The set of $\o$'s for which the continuation holds (i.e. the set of ``persistent tori") is a Cantor set of ``local arbitrarly high density" and the whiskers can be smoothly interpolated. A somewhat more precise formulation is given in the following: \thm{Proposition}{ Given $R>0$ and an integer $s\ge 1$, there exist $\e_0>0$, an integer $d\ge 2$, and an open set $B\subset B_R^2$ (the bidimensional sphere of radius $R$) with the following properties. For all $0<\e<\e_0$ and for all $m>\max\{2,d/s\}$ there exists a Cantor set $\O_\e\subset B$ such that: \newline (i) $\O_\e\subset \O_{\e'}$ whenever $\e'\le\e$; if $\o\in\O_\e$ then the ray $(1+a)\o\in \O_\e$ for all $0\le aFrom the above analysis it follows that $\d(\m)$ is analytic for $|\m|\le \m_0\=\e^m$ and that $\d(0)=0$ i.e. $\d(\m)=\d_1\m+\d_2\m^2 +\cdots$, the series being convergent for $|\m|\le \m_0$. It is clear that, in principle, one could compute $\d_k$ using the strategy outlined above, which is based on KAM techniques. In practice, however, it is much simpler to derive a {\sl direct algorithm}, which we shall briefly describe in \S 6.2 below. {\sl The upshot is that $\d_1$ is not exponentially small in $\e$}, contrary to what happens in Arnold's example [where $\d_1\sim \exp(-\e^{-1/2})$], but, as we shall see in \S 6.2, it is\nota{**}{Here $a\sim x$ means that there exist two ($x$--independent) constants $c_1,c_2$ such that $c_1x\le a\le c_2 x$ for all $x$ small enough.} $$\d_1\sim \e^{d_0}$$ % for a suitable number $d_0$ {\sl depending only on $h$ and $f$}. >From such relation and the analyticity properties in $\m$, it follows easily that {\sl $\d$ can be exatly evaluated and is of the size of $\e^{m_2}$ for a suitable $m_2$}, which we now compute. In fact, since $\d(\m)$ is analytic in a disc of radius $\m_0=\e^m$ and can be bounded by an $\e$--independent constant, one has that $|\d_k|\le$ $\const\e^{-km}$. Thus, if one takes $$ \m=\e^{m_1}\ ,\qquad {\rm with}\ m_1> \max\{2m+d_0, m\}$$ one has, for all $k\ge 2$, $$ |\d_k\m^k|\le \const \e^{k(m_1-m)}\le \const \e^{2(m_1-m)}< \const \e^{d_0+m_1}\le |\d_1\m|$$ (as above $\const$'s denote positive constants, which depend only on $R$). Thus $$ \d(\e^{m_1},\e,\o)\sim \e^{m_2}\ ,\qquad {\rm with}\quad m_2\=d_0+m_1\ .$$ We finally point out that {\sl for the eccentric D'Alembert problem $\d_1$ is indeed exponentially small in $\e$}, $\d_1\sim\exp(-\e^{-1/2})$, {\sl while it is $\d_2$ which is a power of $\e$} (allowing to repeat the above argument). {\bf (3)} To construct transition chains we have to study {\sl heteroclinic connections} i.e. orbits $z(t)$ approaching, as $t\to\pm\io$, {\sl different} invariant tori. In other words, we have to study intersections of $W^+_{\m,{\rm loc}}(\e,\o)$ with $W^-_{\m,{\rm loc}}(\e,\o')$ with $\o\neq \o'$. To do this, we fix $0<\e<\e_0$, $\m\=\e^{m_1}$, $\bar\o\in\O_\e$, and let $\o$ and $\o+\a$ belong to $\O_\e(\bar\o,\m)$. Now, define, for $i=1,2$, $$ F_i(\f_1,\f_2,\a)\=I^-_i(\f_1,\f_2,\p,\e^{m_1},\e,\o+\a)- I^+_i(\f_1,\f_2,\p,\e^{m_1},\e,\o)\ , $$ so that $F(\f_1,\f_2,0)=\tF(\f_1,\f_2;\e^{m_1})$. Since, by our choice of $\o$ and $\a$, the tori $\cT_\m(\e,\o)$ and $\cT_\m,(\e,\o+\a)$ lie in the same energy manifold [which is that of $\cT_\m(\e,\bar\o)$], heteroclinic connections correspond to solutions $\f_1(\a)$, $\f_2(\a)$ satisfying $F(\f_1(\a),\f_2(\a),\a)=0$. By the Implicit Function Theorem\nota{*}{We use a standard Implicit Function Theorem, which guarantees continuous solutions $\a\in B^2_a\to\f(\a)\in B^2_b$ of $F(\f,\a)=0$ [here $\f=(\f_1,\f_2)$] provided $F(0,0)=0$, $F$ and ${\partial F\over \partial \f}$ are continuous in $B^2_b\times B^2_a$, the Jacobian ${\partial F\over \partial\f}(0,0)$ has an inverse $T$ and $$ \sup_{|\a|\le a}|F(0,\a)|\le {1\over 2} \|T\|^{-1}\ b\ ,\qquad \sup_{|\f|\le b,|\a|\le a} \|I- T{\partial F\over \partial \f}\|\le {1\over 2}\ . $$} and the properties listed in {\bf (1)} and {\bf (2)}, it follows that one has (transversal) heteroclinic intersections whenever $$ \o,\o+\a\in\O_\e(\bar\o,\e^{m_1})\ ,\qquad {\rm and}\quad |\a|\le \const \e^{m_2}\ . $$ To insure the existence of heteroclinic connections, we have to guarantee that the gaps on $\O_\e(\bar\o,\e^{m_1})$ are smaller that $\e^{m_2}$. In view of point (v) of the Proposition in step {\bf (1)}, it is sufficient to require that $(ms-d)/ 2>m_2$. All conditions on $m$ and $s$ (which are still free) may be collected in $$ m>\max\{2,{d\over s}\}\ ,\qquad ms-d>2 \max\{m+d_0, 2(m+d_0)\} $$ and it is clear that such conditions may always be met choosing suitably $m$ and $s$ in terms of $d$ and $d_0$. The number $n_0$ in Theorem 3, can be taken to be equal to $m_1$ which , in turn, may be fixed so that $$ \max\{m,2m+d_0\}a R\ ,\ \forall\ n\in \integer^2\ , \ 0<|n|\le s-1\}\cr \O_\e &\=\{\o\in B:\ |\o\cdot n|\ge a\ {\e^{(ms-d)/2}\over |n|^2}\ ,\ \forall\ n\in \integer^2\ ,\ n\neq 0\}\ ,\cr} $$ where $a>0$ is a suitable constant (depending on $s$) small enough and `` $\cdot$ " detotes scalar product in $\real^2$. The set $B$ is a disk of radius $R$ minus a finite number of strips crossing the origin; the constant $a$ may be chosen so that the measure of $B$ is close as we like to $\p R^2$. $\O_\e$ is a Cantor set in $B$ (excluding, for all $\e$, the dense set of ``resonant" $\o$'s for which $\o\cdot n=0$ for some $n\in\integer^2$) that becomes of full measure as $\e$ approaches 0. The definition of $B$, as we shall see below, allows to perform $(s-1)$ steps of classical (Poincar\'e) perturbation theory, in a neighbourhood of ``resonant" tori $\cT(\e,\o)$ for which $\o\in B$. The definition of $\O_\e$ is the classical (Kolmogorov) way of defining the frequencies ``simultaneously accessible" by KAM techniques (the $\e$--dependent constant appearing in the definition of $\O_\e$ will be explained below). Next, we introduce symplectic ``hyperbolic coordinates" for the pendulum $P\= {1\over 2} I_3^2+\e(\cos \f_3 -1)$. Such a pendulum has the origin $I_3=0=\f_3$ as unstable (hyperbolic) fixed point. The quadratic part of $P$, given by ${1\over 2}\big(I_3^2-(\sqrt{\e}\f_3)^2\big)$, is transformed, via the symplectic map $$\pmatrix{I_3\cr \f_3\cr}={1\over \sqrt{2}} \pmatrix{ \e^{1/4} & \e^{1/4}\cr -{1\over \e^{1/4}} & {1\over \e^{1/4}}\cr} \pmatrix{y\cr x\cr} \eqno(9)$$ into the ``normal hyperbolic" form $\sqrt{\e} yx$. It is therefore clear that, near the origin, there exists a symplectic (real--analyitc) transformation $S$, whose linear part is given by Eq. (9), transforming the pendulum $P$ into the normal form $$ P\circ S= g(yx, \sqrt{\e})=\sqrt{\e}(yx)+ O\big( (yx)^2\big)\ . $$ The domain of definition of $S$ (and hence of $g$) can be taken to be a 2--sphere centered at the origin of radius, say, $4b$ for a suitable $b\sim\e^{1/4}$. In the $(I_3,\f_3)$--coordinates, the sphere $B^2_{4b}$ correspond, in a first approximation, to an ellipse with $I_3$--semiaxis of length $\sim \sqrt{\e}$ and with $\f_3$--semiaxis of length $\sim 1$. In the symplectic coordinates $$ \bI\=(I_1,I_2)\ ,\quad \baf\=(\f_1,\f_2)\ ,\quad (y,x)\ , $$ the Hamiltonian $\wH$ takes the form $$ H_1(\bI,\baf,y,x,\e,\m) \= h_1+\m f_1 \= {\e\over 2}\bI^2 + g(yx,\sqrt{\e})+ \m f_1(\baf, y,x, \sqrt{\e})\ , $$ where $f_1$ is a linear combination of $\cos \f_i$ and $\sin \f_i$, with $i=1,2$, with coefficients which are real--analytic functions of $\x=yx$. By classical perturbation theory, one can find a symplectic transformation $\m$--close to the identity, conjugating $H_1$ to a Hamiltonian $$\eqalign{ H_2(\bI',\baf',y',x',\e,\m) &\= h_2+\m^s f_2\cr &\= h_1(\bI', y'x', \sqrt{\e})+ \m g_1(y'x',\sqrt{\e},\m)+ \m^s f_2(\bI', \baf', y',x',\sqrt{\e},\m)\ . \cr} $$ The ``small divisors" appearing in such a symplectic transformation\nota{\dag}{The transformation can be found, e.g., by solving up to order $\m^{s-1}$ the Hamilton--Jacobi equation for a generating function of the form $\bI'\cdot \baf+y'x+$ polynomial in $\m$ of order $(s-1)$ vanishing at $\m=0$.}, have the form $i\e \bI'\cdot k+ j \sqrt{\e} y'x'$ with $k\in\integer^2$, $j\in\natural$, $|k|+j>0$ and $|k|\le s-1$. Thus the domain of definition (and analyticity) of $H_2$ is given, for $0<\e<\e_0$, by $B\times\torus^2\times B^2_{2b}\times B^1_{\bar\m}$ and $\bar \m$ may be choosen proportional to $\e^2$. In these coordinates, the tori $\cT(\e,\o)$ are given by $\bI'=\o$, $\baf'\in\torus^2$, $y'=x'=0$ and the stable/unstable local whiskers are given by, respectively, $\bI'=\o$, $\baf'\in\torus^2$, $|y'|\le 2b$, $x'=0$, and $\bI'=\o$, $\baf'\in\torus^2$, $|x'|\le 2b$, $y'=0$. At this point one can use a KAM ``fast--converging" scheme to construct, simultaneously for $\o\in\O_\e$, invariant tori with flow conjugated to $t\to\e\o t$, and their local whiskers, which will be $\m^s$ close to the local whiskers for $h_2$. As it is well known, the smallness requirement (or ``KAM condition") necessary in order to carry out the scheme, is of the form $$ \const\ {\m_0^s \over \e^d \g^2}<1\ , \eqno(10)$$ where the analyticity domain in $\m$ will contain the disk $|\m|\le \m_0$ and $\g$ is the ``rescaled Diophantine constant" associated to the persistent tori $\cT(\e,\m)$, i.e., $\o$ is such that $$ |\o\cdot n|\ge {\g\over |n|^2}\ ,\qquad \forall\ n\neq 0\ , \eqno(11) $$ [hence the small divisors appearing in the KAM scheme satisfy $|\e\o\cdot n|\ge (\e\g)/|n|^2$]; $d$ is a constant (depending only on the structure of $h$) which is greater or equal than two since, as already mentioned, the actual (``non--rescaled") Diophantine constant is $\e\g$ and such a constant appears quadratically in KAM conditions. All the frequencies $\o$ satisfying the classical condition Eq. (11), may be taken care of simultaneously and the complementary set $\{\o\in B:$ Eq. (11) does not hold $\}$ has a measure of the order of $\g$. Thus, choosing $\m_0=\e^m$, with $m>\max\{2,d/s\}$, we see that $\g$ may be taken to be [in order to meet the KAM condition Eq. (10)] $\g=\const \e^{(ms-d)/2}$. The conclusion is that, with the above choices, one can construct $C^3$ functions $\z$ and $\l$, $$ \eqalign{ & \z: (\th,(y,x),\o,\m)\in\torus^2\times B^2_b\times B^2_R \times B^1_{\m_0} \to \z(\th,y,x,\o,\m)\in\cM\ ,\cr & \l:(\x,\o,\m)\in B^1_b\times B^2_R\times B^1_{\m_0}\to \l(\x,\o,\m)\in\real\ , \cr}$$ such that the following holds. For $\o\in \O_\e$, $\z$ and $\l$ are real--analytic in all their remaining variables; $\l(0,\o,0)=\sqrt{\e}$; $\z(\th,0,0,\o,0)=(\o,0,\th,0)$; $$ \sup_{\torus^2\times B^2_b\times B^2_R \times B^1_{\m_0}}\ |\partial^j_\th \partial^k_\o \z|\le \const\ ,\qquad\ \forall\ |j|\le 2\ ,\ |k|\le 1\ ; $$ for $\o\in \O_\e$ and $|\m|\le \m_0$ one has $$\eqalign{ &\cT_\m(\e,\m)= \{z=\z(\th,0,0,\o,\m): \ \th\in \torus^2\}\ ,\cr &W^+_{\m, {\rm loc}}(\e,\o)= \{z=\z(\th,y,0,\o,\m);\ \th\in\torus^2\ , |y|0\}$ crosses any manifold which intersects transversally $W^-_{\m, {\rm loc}}$ in $z_-$]. As for the final point (v) of the KAM Proposition, observe that from the definition of the set $\O_\e$, it follows that the ray $(1+a)\o$, with $0\le a< R-|\o|$, belongs to $\O_\e$, for any $\o\in \O_\e$. Also, the set $\O_\e(\bar\o,\m)$ coincides with $$ \O_\e(\bar\o,\m)\=\big\{\o\in\O_\e:\ \wH\big(\z(0,0,0,\o,\m)\big) = \wH\big(\z(0,0,0,\bar\o,\m)\big)\big\} $$ and the set $$\G_\m= \big\{\o\in B:\ \wH\big(\z(0,0,0,\o,\m)\big) = \wH\big(\z(0,0,0,\bar\o,\m)\big)\big\} $$ is a finite union of $C^1$ curves, which are $|\m|$--close [by the smoothness of $\z$ in $\o\in B^2_R$ and the fact that $p_1\big(\z(0,0,0,\bar\o,0)\big)=\bar\o$] to $\{|\o|=|\bar\o|\}$ intersected with $B$. Thus, since $\O_\e$ has a measure not smaller than, approximately, $R^2 \big(1-\e^{(ms-d)/2}\big)$, it is clear that, in general, $\O_\e\cap \G_\e(\bar\o,\e)$ cuts on $\G_\e(\bar\o,\e)$ a region of length not smaller than $|\bar\o| \big(1-\e^{(ms-d)/2}\big)$. \vglue0.4cm \subsection{6.2}{Generalized Poincar\'e Integrals} \medskip Here we discuss with some detail {\sl a recursive algorithm that allows to compute all the $\m$--derivatives of the function $\d(\m)$} introduced in Eq. (8). In particular, we shall prove that, for the Hamiltonian in Eq. (5), it is $$\d_1= {4\over \e}\big(1+O(\e)\big)\ ,\qquad {\rm i.e.}\quad d_0=-1\ . \eqno(13)$$ Actually, to prove Eq. (13) is rather easy, but we shall spend some time in order to discuss, in some generality, the above mentioned algorithm, which, we believe, is quite interesting in itself. As already mentioned in the previous section, the derivatives of $\d$ could be computed by using the KAM results, however we prefer, following seminal ideas of Poincar\'e\nota{**}{See also Ref.\ref{2} and, for generalities and further references, Ref.\ref{1}.}, a more direct approach. Keeping fixed $0<\e<\e_0$ and $\o\in\O_\e$ [and using the above conventions: $\bI\=(I_1,I_2)$, $\baf\=(\f_1,\f_2)$], we denote $\wH$--trajectories starting at $W^\pm_\m\cap\{\f_3=\p\}$ by $$\eqalign{ z^\pm(t;\baf,\m)&\= \big( I^\pm(t;\baf,\m),\f^\pm(t;\baf,\m)\big) \=\big( \bI^\pm(t;\baf,\m),p^\pm(t;\baf,\m),\baf^\pm(t;\baf,\m), q^\pm(t;\baf,\m)\big)\cr &\=\phi_\wH^t\big(I^\pm(\baf,\p,\m),\baf,\m\big) \cr} $$ where $I^\pm(\f,\m)$ are the functions introduced in {\bf (2)} of \S 5.2 (in the first line of this definition, we have introduced different symbols for the same object, which is then defined in the second line). In particular, $I^\pm(0;\baf,\m)=I^\pm(\baf,\p,\m)$, $\f^\pm(0;\baf, \m)=(\baf, \p)$. An important remark, based on the KAM analysis, is that, since trajectories on $W^\pm_\m$ approach (exponentially fast) the invariant tori $\cT_\m(\e,\o)$ on which the flow is conjugated to quasi--periodic, {\sl the functions}\nota{*}{If not needed explicitly, we shall drop the dependence on $\baf$ and $\m$.} $I^\pm(t)$, $\f^\pm(t)$ {\sl are ``asymptotically (as $t\to\pm\io$) quasi--periodic functions" with frequencies $\e\o$}. Here by ``$f^\s$ asymptotically quasi--periodic", we mean that $f^\s=\psi+g$ with $\psi$ quasi--periodic with frequencies $\e\o$ and $g$ converging to 0 {\sl exponentially fast}, as $t\to\s\io$. The function $z^\pm(t)\=\big(I^\pm(t), \f^\pm(t)\big)$ is analytic in $|\m|\le \m_0$ and has a convergent power--series representation $$ I^\pm(t;\baf,\m)=\sum_{k=0}^\io I^{k\pm}(t,\baf)\m^k\ ,\qquad \f^\pm(t;\baf,\m)=\sum_{k=0}^\io \f^{k\pm}(t,\baf)\m^k\ , $$ where $\Big(I^{0\pm}(t),\f^{0\pm}(t)\Big)\=z^0(t)$ is the unperturbed $h$--trajectory given by $$ z^0(t)\=\big(\o,p^0(t),\baf+\e\o t, q^0(t)\big)\ , $$ $(p^0,q^0)$ being the separatrix motion of the pendulum ${1\over 2}I_3^2+\e(\cos\f_3-1)$ starting at $\f_3=\p$: $$q^0(t)=4\arctan e^{-\sqrt{\e}t}\ ,\quad p^0(t)={2\sqrt{\e}\over \cosh \sqrt{\e} t}\ ,\quad \cos q^0=1-{2\over (\cos \sqe t)^2}\ , \quad \sin q^0=2{\sinh \sqe t\over (\cos \sqe t)^2}\ . $$ (and, for definitess, we have choosen the upper branch, $I_3>0$). Letting $(\bI^{k\pm},p^{k\pm})\=I^{k\pm}$, $(\baf^{k\pm},q^{k\pm})\= \f^{k\pm}$ and inserting the power series into the $\wH$--Hamilton equations and equating the coefficients of $\m^k$ we get, for $k\ge 1$, the hierarchy of equations $$ \eqalign{ &\dot \bI^{k\pm}=\bar\F^{k\pm}\ ,\qquad\qquad\qquad\ \dot\baf^{k\pm}=\e\bI^{k\pm}\ ,\cr & \dot p^{k\pm}=(\e\cos q^0) q^{k\pm}+\Phi_3^{k\pm}\ ,\quad \dot q^{k\pm}=p^{k\pm}\cr} \eqno(14)$$ where the functions $\Phi^{k\pm}(t;\baf) \=(\Phi_1^{k\pm},\Phi_2^{k\pm},\Phi_3^{k\pm}) \=(\bar\Phi^{k\pm},\Phi_3^{k\pm})$ can be easy computed by Taylor expansion and depend only on $z^{j\pm}$ with $0\le j\le k-1$. For example, $\bar\Phi^{1\pm}$, $\Phi_3^{1\pm}$, which are actually independent of the $+$ or $-$ sign, are given by $$ \eqalign{ \bar\Phi^1(t,\baf)&\=-{\partial f\over \partial \baf}\big(\baf+\e\o t,q^0(t)\big)\= \Big(\sin\big(\f_1+\e\o_1 t+q^0(t)\big), \sin\big(\f_2+\e\o_2 t+q^0(t)\big)\Big)\ ,\cr \Phi^1_3(t,\baf)&\=-{\partial f\over \partial \f_3}\big(\baf+\e\o t,q^0(t)\big)\= \sin\big(\f_1+\e\o_1 t+q^0(t)\big)+ \sin\big(\f_2+\e\o_2 t+q^0(t)\big) \ .\cr} $$ Observing that the fundamental solution (or ``Wronskian") of the linearized pendulum equation $\dot p=(\e\cos q^0)q$, $\dot q=p$, is given by $$ W(t)=\pmatrix{ (1-{w(t)\over4}{\sinh \sqrt{\e}t\over\cosh^2\sqe t}) \cosh \sqe t&-\sqe{\sinh \sqe t\over\cosh^2 \sqe t}\cr {w(t)\over4\sqe}&{1\over\cosh \sqe t}\cr},\qquad{w(t)}\={2\sqe t+\sinh 2\sqe t\over\cosh \sqe t}\ ,$$ integrating Eq. (14), one can express $z^{k\pm}(t)$ in terms of $z^0(t)$,...,$z^{(k-1)\pm}(t)$: $$ \eqalign{ &\bI^\kp(t)=\bI^\kp(0)+\int_0^t\bar\F^\kp(\t)d\t\ ,\quad \baf^\kp(t)=\e\bI^\kp(0)t + \e\int_0^t(t-\t) \bar\F^\kp(\t)d\t\ ,\cr & p^{k\pm}(t)= w_{11}(t)q^{k\pm}(0)+ w_{11}(t) \int^t_0 w_{22}(\t) \Phi^{k\pm}_3(\t) d\t- w_{12}(t)\int^t_0 w_{21}(\t) \Phi^{k\pm}_3(\t)\,d\t\ ,\cr & q^{k\pm}(t)= w_{21}(t)q^{k\pm}(0)+ w_{21}(t) \int^t_0 w_{22}(\t) \Phi^{k\pm}_3(\t) d\t- w_{22}(t)\int^t_0 w_{21}(\t) \F^{k\pm}_3(\t)\,d\t\ ,\cr} \eqno(15)$$ where $w_{ij}$ are the entries of $W(t)$ and we have used the fact that, for $k\ge 1$, $\baf^\kp(0)=0=q^\kp(0)$. Now, {\sl the key idea is to integrate from $0$ to $\pm\io$ and, using the fact that $z^\kp(t)$ is aymptotically (as $t\to\io$) a quasi--periodic function, to try to identify the ``initial value"} $I^\kp(0)\={1\over k!} \left.{d\over d\m}\right|_{\m=0}I^\pm(\baf,\p,\m)$ {\sl in terms of integrals of the $\Phi$'s functions}. In order to perform with ease improper integrations, we introduce the following {\sl generalization of standard improper integration}\nota{\dag}{ The operations discussed here are introduced in Ref.\ref{5}.}. Let $\s$ denote the $+$ or $-$ sign. Let $\cH_\s$ be the class of measurable functions $f:\real\to\complex$ such that there exist $a>0>b$ for which, given $t\in \real$, the function $$u\to F(u,t)\=\int_{\s\io}^t e^{-u\s \t} f(\t)d\t $$ is analytic on the complex domain $\{u\in\complex:\Re\ u>a\}$ and admits an analytic continuation to $\{u\in\complex:\Re\ u0$, then $f\in\cH_\s$ and $\cJ^t_\s f=\int_{\s\io}^t f$. It is also easy to check that $f\=t^j e^{c t}\in\cH_\s$ for any $j,\s$ and any {\sl non vanishing} complex number $c$. Instead, the polynomials are not contained in $\cH_\s$. Nevertheless, we extend, by linearity, the operator $\cJ$ on $\wcF_\s\=\cH_\s\oplus$ ring of polynomials in $t$, by defining $$ \cJ_\s^t\t^j= {t^{j+1}\over (j+1)}\ . $$ Such an extension implies the following useful property of $\cJ$: $$ \cJ_\s^tf-\cJ_\s^sf=\int_s^t f(\t)d\t\ , \eqno(16)$$ for any $f\in\wcF_\s$ and any $s,t$. An easy recursive argument (either based on the KAM analysis of \S 6.1 or direct) shows that {\sl $z^\kp$, and hence $\F^\kp$, belong to $\cH_\pm$ for all $k\ge 1$}. >From Eq. (15) and Eq. (16) it then follows $$ \bI^\kp(t)-\cJ_\pm^t\bar\F^\kp= \bI^\kp(0)-\cJ_\pm^0\bar\F^\kp\ . \eqno(17)$$ We, now, note the following two facts (which are easily checked): (i) The quasi--periodic average, $\lim_{T\to\io}(\s T)^{-1}\int_0^T f$, of a function $f\=\psi+g$, which is asymptotically (as $t\to\s \io$) quasi--periodic, coincides with the quasi--periodic average of the limiting function $\psi$, and we shall denote such a number by $\langle f\rangle$; (ii) if $f$ is asymptotically quasi--periodic with $\langle f\rangle=0$, then ($f\in\cH_\s$ and) so is $\cJ_\s^t f$. Thus, taking the quasi--periodic average in the first line of Eq. (14), one sees that both $\bar\F^\kp$ and $\bI^\kp$, which are asymptotically quasi--periodic, have vanishing quasi--periodic average. Therefore, taking the quasi--periodic average in Eq. (17), we see that $$ \bI^\pm(0)=\cJ_\pm^0\bar\F^\kp\ ,\qquad{hence}\quad \bI^\pm(t)=\cJ_\pm^t\bar\F^\kp\ ,\qquad \forall\ t\ . $$ With similar arguments one finds also that $$\eqalign{ & p^{k\s}(t)= w_{11}(t)\int_{\s\io}^t w_{22}(\t)\F^{k\s}_3(\t)d\t- w_{12}(t)\int^t_0 w_{21}(\t) \F^{k\s}_3(\t)d\t\ ,\cr & q^{k\s}(t)= w_{21}(t)\int_{\s\io}^t w_{22}(\t) \F^{k\s}_3(\t)d\t- w_{22}(t)\int^t_0 w_{21}(\t) \F^{k\s}_3(\t)d\t\ .\cr} $$ Let us now go back to the computation of $\d_1$ and let us introduce the following notation: if $f^\pm$ denotes {\sl two} functions $f^+\in\cH_+$ and $f^-\in\cH_-$, we let $$ \cJ f^\pm\=\cJ_-^0f^- - \cJ_+^0 f^+\ .$$ Thus, recalling the definition of $\bar\F^1$, we have $$\eqalign{ \bI^{1-}(\baf)-\bI^{1+}(\baf)&\=\left.{d\over d\m}\right|_{\m=0} \big(\bI^{1-}(\baf,\p,\m) -\bI^{1+}(\baf,\p,\m)\big)\cr &= \Big(\cJ \sin\big(\f_1+\e\o_1 t+q^0(t)\big), \cJ \sin\big(\f_2+\e\o_2 t+q^0(t)\big)\Big)\ .\cr} $$ A straightforward computation, based upon the definition of $\cJ$, symmetries\nota{**}{If $f\in\cH_\s$ is an even/odd function of $t$, then $\cJ^t_\s f$ is odd/even. } and residue theory, yields $$\eqalign{ \cJ \sin(\f_i+\e\o_i t+q^0)&= \sin\f_i\cJ\Big(\cos\e\o_i t\cos q^0-\sin \e\o_i t\sin q^0\Big)\cr &=\sin \f_i \int_{-\io}^\io\{\cos \e\o_i t (\cos q^0 -1)\}- \sin\f_i \int_{-\io}^\io \sin \e\o_i t\sin q^0\cr &= -2\sin \f_i \int_{-\io}^\io \Big\{ {\cos\e\o_i t\over (\cosh \sqe t)^2} + {\sin\e\o_i t \senh \sqe t\over (\cosh\sqe t)^2}\Big\}\cr &=-\p\o_i\sin\f_i \Big({1\over \sinh{\p\sqe \o_i\over 2} }+ {1\over \cosh{\p\sqe \o_i\over 2}}\Big)\cr} $$ from which Eq. (13) follows at once. \vglue 0.6cm \line{\elevenbf 7. References \hfil} \vglue 0.4cm \medskip \item{1.} V.I. Arnold (Ed.) , {\elevenit Encyclopaedia of Mathematical Sciences} {\elevenbf III} {\elevenit ``Dynamical Systems"} (Springer--Verlag, 1988) . \item{2.} V. I. Arnold, {\elevenit Instability of Dynamical Systems with Several Degrees of Freedom}, in {\elevenit Sov. Math. Dokl.} {\elevenbf 6} (1964) p. 581--585. %\item{3.} B. V. Chirikov, %{\elevenit A Universal Instability of Many--Dimensional %Oscillator Systems}, in {\elevenit Phys. Rep.} {\elevenbf 52} (1979), %p. 263--379. \item{3.} H. Poincar\'e, {\elevenit Les M\'ethodes Nouvelles de la M\'ecanique C\'eleste} Vol. 1--3 (Paris, Gauthier--Villars, 1892, 1893, 1899). \item{4.} Z. Xia, {\elevenit Arnold Diffusion in the Elliptic Restricted Three--Body Problem}, in {\elevenit J. Dyn. Differ. Equations} {\elevenbf 5}, No. 2, (1993), p. 219--240. \item{5.} L. Chierchia and G. Gallavotti, {\elevenit Drift and Diffusion in Phase Space}, in {\elevenit Ann. H. Poincar\`e} (Physique Theorique), {\elevenbf 60}, No. 1 (1994), p. 1--144. \bye