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% Archived: mp$\_$arc@math.utexas.edu 97-472 or
% chao-dyn@xyz.lanl.gov 9709004 
% (version following the 2-d referee's report \#3). 
% Original title: {\it Pendulum: separatrix splitting}
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\*
\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
}}

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%**end of header
\fiat

%\headline{\nota\hss Draft \#14}
\null
\vskip2.truecm
\centerline{\titolo Separatrix splitting for systems
with three time scales}
\*\*

\centerline{\bf G. Gallavotti, G. Gentile, V. Mastropietro}
\*
\centerline{Universit\`a di Roma 1,2,3 }
\centerline{\Di}

\vskip1.truecm

\0{\bf Abstract.} {\it An exact expression for the
determinant of the splitting matrix is derived for three degrees of
freedom systems with three time scales: 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 asymptotic
validity of Melnikov's integrals is proved for the class of models
considered, which are polynomial perturbations. The technique for
exhibiting cancellations is inspired by renormalization theory in
quantum electrodynamics and uses an analogue of Dyson's equations to
prove an infinite family of identities, due to symmetries, that remind
us of the Ward's identities.}
\*\*


\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 Arnol'd
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.

Here we shall consider a pendulum subject both to 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}). So, if $(I,A,B)$ are the conjugate momenta of the
angles $(\f,\a,\l)$, the systems will
be described
%\ifnum\mgnf=1\pagina\fi\0
by Hamiltonians of the form:
%
$$ H={\h}^a A+\fra1{\h^{1/2}}B+\h^{2a} \fra{A^2}2+\fra {I^2}{2}+\,g^2\,
(\cos \f -1)+ \e\,f(\f,\a,\l) \Eq(1.1)$$
%
where $g$ and $\h$ are constants,
$\e$ is the perturbative parameter and the
function $f$ is an even trigonometric polynomial in its arguments.
The Hamiltonian in which the isochrony breaking term
$\fra{1}2\h^{2a} A^2$ is missing will also be studied,
essentially as a preliminary approach to more interesting
case \equ(1.1).

Our analysis deals mainly with systems with {\it three} different time
scales with the ratio between the smallest to the largest being
$\ll1$: the largest is $\h^{-1/2}$, the intermediate is $g=O(1)$ and
the smallest is $\h^a$, $a\ge0$ (the $a=0$ case being a limiting two
scales problem).

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
$\aa\defi(\a,\l)$ of the rotators there are well defined
values of the pendulum variables $(I,\f)$ as well as of the
rotators actions $\AA\defi(A,B)$. 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 $\ps_j\to\ps_j+\o_j t$, $j=1,2$,
for a suitable $\oo=(\o_1,\o_2)$.

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 $\det D$; see Appendix R1. 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
Mel'nikov integral} and coinciding with the ``first order
perturbation theory result''.

The splitting problem is to find under which conditions {\it
Mel'nikov integral} holds when {\it both} the perturbation and the
shortest forcing period are sent to $0$.

The main result of the paper will be called the {\it large angles
theorem} (theorem 2 in \S5, proved in
\S6,\S7 for the systems defined by \equ(1.1)
without and with the anisochronous term; see \equ(2.1), \equ(7.1) below).
The motivation for the name is that, even though the splitting
is found to be exponentially small in the parameter $\h$,
the gaps between the tori still have size which is narrower
than the splitting; see remark (1) after theorem 2 in \S 5.

Informally the result is (see theorem 2 in \S5 for a formal
statement):
\*

\0{\it Suppose that one of the forcing frequencies is very large, say
$\h^{-1/2}$ times the pendulum frequency, with $\h$ very small, and
suppose also that the second characteristic frequency is relatively
very small, say $\h^a$ times that of the pendulum, with $a\ge0$; see
the Hamiltonian \equ(1.1). 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 Mel'nikov integral}, \ie it is
given by $\g\h^{-b}\exp[-(\pi/2)\h^{-1/2}]$, with $\g\neq0$, for some
positive constant $b$, 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; remarkably the correct answer was, in the early cases, found
in experimental (\ie nu\-me\-ri\-cal) works, (see [S] and [BCF] for the
experimental part and, for the theory, see [DGJS] and the
related later paper [RW]). But the ratios between the various forcing
frequencies are different in the mentioned papers.

Obtaining 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 (see
Appendix R2). 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
study 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 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 [DGJS]
and in subsequent papers.

The recent works [DGJS], [RW] considerably overlap with ours: even
when $a=0$ (a two time scales problem) it cannot be used to achieve
all our results about the upper bound on the splitting 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 anisochronous).  We also require the
perturbations to be trigonometric polynomials, while the examples of
[DGJS], [RW] require, as an essential assumption to obtain the lower
bound for systems in which all rotators are equally fast, the
perturbation to have infinitely many non vanishing modes in the
rotators angles, although still finitely many in the pendulum angle
variable; see eqs. (16) and (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 [DGJS] (and the similar [RW]) lies in
the techniques: here we show that the techniques of [G3] do apply
immediately to the problem. See concluding remarks in \S 9 for a more
technical comparison. 

\0[{\it During the refereeing process our view of the
results in [RW] and [DGJS] has considerably changed, see Appendix R9.]}

\vskip1.truecm

\0{\bf\S2. Isochronous clock--pendulum system.}
\numsec=2\numfor=1\*

Our main result concerns anisochronous systems, \equ(1.1).
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 ${\bf T}^3=[0,2\p]^3$) and $(I,A,B)\in {\bf R}^3$.

The Hamiltonian will be:

$$H= \h^a A+\fra {I^2}{2}+\h^{-1/2}B+\,g^2\, (\cos \f -1)
+ \m\, \h^c\,f(\f,\a,\l),\qquad \e\defi\m\h^c\Eq(2.1)$$
%
where $\h,\m>0$ and $f$ is an {\it even trigonometric
polynomial} in the angles $(\f,\a,\l)$; 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}\/:

$$|\V \o\cdot\nn|> C^{-1}\h^d{|\nn|^{-\t}}
\defi C(\h)|\nn|^{-\t} \Eq(2.2)$$
%
for some {\it Diophantine constant} $C(\h),\t$ 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
$\fra12\h^{2a}A^2$ (see \equ(1.1)):
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 assumption 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).

\vskip1.truecm

\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 motions 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 $\V\a_0\defi(\a_0,\l_0)$ 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 hyperbolic tori, with rotation vector
$\oo\defi (\h^{a},\h^{-1/2})$: the energy of such motions is $
\h^a A+\h^{-1/2} B$).

The torus is unstable and its unstable manifold $W^-$ is
$3$--dimensional and contains the set parameterized by $(\f,\a,\l)$ via
the equations:

$$A=B=0, \qquad I=I(\f), \qquad (\a,\l)\in [0,2\p]^2,\qquad
\f\in(0,2\p)\Eq(3.1)$$
%
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
(canonical) {\it Jacobi 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 $t\to-\io$, the evolution
of initial 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 condition
\equ(2.2) and $\m_0>0$ is small enough. One can define functions
$\X,\G,\L,\Th$ 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 $|\m|\le\m_0$,
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 $\X,\G$ 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 estimate $\m_0$ for the radius of convergence is positive and
$\h$--independent for $\h$ small enough, 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 $\X,\G$ 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. About the proof of the theorem, see Appendix R4.

(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, for more general
(action dependent) perturbations, by an amount analytic in
$\m$ and divisible by $\m$ (\ie of order $O(\m)$).

(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], see Appendix R3.

(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 $\oo=(\h^a+\h^{2a} A_0,\h^{-1/2})$
verifies a Diophantine condition 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 Arnol'd'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). Anyway the proof of theorem 1
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].

\vskip1.truecm

\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 the stable and
unstable manifolds of the torus into which the unperturbed torus
(\ie $A=B=0,I=0,\m=0$) 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 {\bf 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 order $O(\m)$ 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 matrix}
(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 $\m=0$ one
has $I^+_0=I^-_0=I^0(\f)=g \sqrt{2(1-\cos\f)}, \AA^\pm_0=\V0$.  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
``Poincar\'e 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) 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:
%
$$X^\s_\m(t;\aa)=X^\s_\m(\oo t,t;\aa)\Eq(4.4)$$
%
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 $j$,
$j=0,1,\ldots,5$, write them ({\rm notation used in [G3]})
with the convention:
%
$$X_0\defi X_-,\quad (X_1,X_2)\defi \V X_\giu,\quad
X_3\defi X_+,\quad (X_4,X_5)\defi \V X_\su\Eq(4.6)$$
%
and we write first the angle variables ($(\f,\a,\l)=(X_-,\V X_\giu)$),
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 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 $L_{30}=g^2\cos\f^0(t)$; in the anisochronous case
discussed in \S 7 below also the matrix element $L_{14}$ is
not vanishing: $L_{14}=\h^{2a}$. 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 $\f(t)\defi\f^0(t),\aa(t)\defi
\aa+\oo t$. More generally, $F^{k}$ depend upon $X^0,...,X^{k-1}$
{\it but not on $X^{k}$}. It turns out to be
$F^k_-=0$ and $\V F^k_\giu\=\V 0$ 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 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 matrix 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
$W(t)$ 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 matrix:
%
$$\pmatrix{X^{k\s}_-\cr X^{k\s}_+\cr}=W(t)
\pmatrix{0\cr X^{k\s}_+(0)\cr} +
W(t)\ig_0^t{W\,}^{-1}(\t)\pmatrix{0\cr F^{k\s}_+(\t)\cr}\ d\t
\Eq(4.12)$$
%
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 $\fra12
\h^{2a}A^2$ the first component of the first of \equ(4.14) becomes:

$$X^{k\s}_1(t)=\h^{2a} \,\big(t X^{k\s}_1(0)+\ig_0^t (t-\t)
F_1^{k\s}(\t)\,d\t\big)\Eq(4.15)$$
%
where $X_1^{k\s}$ is the first component of $\V X_\giu^{k\s}$, 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 $w_{03}(t),w_{33}(t)$ 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 terms in \equ(4.13) proportional to $w_{30}(t)$ or $w_{33}(t)$
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]).

We learnt of the importance and relevance of improper integrals in the
present context from the works of the series leading to [HMS], whose
{\it importance we want to stress}.

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 a class $\MM$, see definition 1 of \S3 of [G3], 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 \S3 of [G3] and we proceed to a quick discussion of the
determination of the the initial constants.

\vskip1.truecm

\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:
%
$$X^h_-(t)=\OO F^h_+(t),\quad X^h_+(t)=\OO_+ F^h_+,\quad
\V X^h_\giu(t)=\h^{2a} E\lis\II^2 F^h_\su(t),
\quad \V X^h_\su(t)=\IG \V F^h_\su(t)\Eq(5.5)$$
%
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
$h'<h$.

Since, in this section, we consider an isochronous model the above
formulae are slightly simpler as $\V X^h_\giu(t)\=\V0$, \ie one must take
$E=0$ in the third of \equ(5.5). In \S7 we shall need, however,
\equ(5.5).

There is no difficulty in setting up a general recursive
scheme for the computation. We just give the result using the
convenient notations ($k^i_j\ge 1$, $\V m=(m_0,m_1,m_2)$, $m_i>0$):
%
$$
(k^i_j)_{\V m,p}\=(k^0_1,\ldots,k^0_{m_0},k^1_1,\ldots,k^1_{m_1},
\ldots,k^{2}_1,\ldots,k^{2}_{m_{2}})\qquad {\rm
s.t.\ }\sum k^i_j=p\Eq(5.6)$$
%
referring to \S2 of [G3] for more details (if needed). If $f_1\defi f,
f_0\defi g^2\cos\f$, we find:
%
$$\eqalignno{
& F_-^{k\s}\=0\ ,\  \V F_\giu^{k\s}\=\V 0\ ,\
\V F_\su^{k\s} = - \sum_{|\V m|\ge0\atop \d=1}\fra1{\V m!}
(\dpr^{\V m}\dpr_\aa f_\d)
\sum_{(k^i_j)_{\V m,k-1}} \prod_{i=0}^{2}\prod_{j=1}^{m_i}
X^{k^i_j\s}_i\cr
& F_+^{k\s} \=- {\sum_{|\V m|\ge 0\atop\d=0,1}}^* \fra1{\V m!}
(\dpr^{\V m}\dpr_\f f_\d)
%\sum_{|\V m|\ge0} (\dpr_0 f_\d)_{\V m}
\sum_{(k^i_j)_{\V m,k-1}} \prod_{i=0}^{2}\prod_{j=1}^{m_i}
X^{k^i_j\s}_i&\eq(5.7)\cr}$$
%
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 $i>0$),
while no sum with $\d=0$ appears for $\V F_\su^{k\s}$. The
functions are evaluated at $\f(t)\defi\f^0(t),\aa(t)\defi \aa+\oo t$.
The indices in \equ(5.7) are mutually contracted with a natural rule
that we leave to the reader to work out.

The relations $\V F^k_\giu=\V0$ and $F_-^k\=0$ 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 $N$:
%
$$\eqalign{
&f(\aa,\f)=f_S(\f,\a,\l)+f_F(\f,\a,\l)\cr
& f_j(\f,\a,\l)=
\sum_{n,\nn} f_{j,(n,\nn)}\,\cos(\n_1\a+\n_2\l+ n\f),\quad
j=S,F\cr}\Eq(5.9)$$
%
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 {\it only} on
slowly rotating angles and $f_F$ depends on fastly rotating
angles.  A nontrivial example can be:
%
$$f(\aa,\f)=\big(\cos(\a+\f)+\cos(\l+\f)\big)\Eq(5.10)$$
%
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], and the identity:
%
$$\ig_{-\io}^{+\io} dt\, F(t)\lis\II^2(H)(t)=\ig_{-\io}^{+\io} dt\,
H(t)\lis\II^2(F)(t)\Eq(5.15)$$
%
which can be proven as \equ(5.13); see again [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(1.1)
with $f$ an even trigonometric polynomial of degree $N$. Let $\m$ be small
($|\m|<\m_0$) and $c$ large, enough.  Consider an invariant torus with
Diophantine rotation vector $\oo$ 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 (``Mel'nikov integral''), \ie it is
$\g\h^{-b}e^{-\fra{\pi}2\h^{-1/2}}$, with $\g\neq0$ and some positive
constant $b>0$ (depending on $N$). The choice given by
\equ(5.10) is a concrete example of this result, generically holding
for \equ(5.9).}
\*

\noindent{\it Remarks}: (1)
The name of the theorem is due to the fact that the splitting, despite
its exponential smallness, is nevertheless far larger than the tori
separation, which is the natural scale over which to measure the
splitting size. In the anisochronous case the average actions of the tori
do not fill phase space (they do in the isochronous case as well as in
the case of [A]): in fact the above theorem will imply immediately,
along the lines of [CG] the existence of heteroclinic chains and
therefore of Arnol'd diffusion, see [GGM2].

(2) The result in the case $f$ is given by
\equ(5.10) is $32 \p \h^{-1/2} \e^2 e^{-\fra\p2 \h^{-1/2}}$
in both cases (isochronous, \equ(2.1), and anisochronous, \equ(1.1))
with $a>0$.

(3) This theorem has an analogue when $a=-\fra12$, \ie
only two time scales: an early review is in [G3]. In the latter case
it has been extended to (special) 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.

(4) 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 $\fra12 \h^{2a} A^2$ to
\equ(2.1), see \equ(1.1) (or \equ(7.1) below).
Both cases are treated in full detail
in \S6,\S7.

\vskip1.truecm

\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, by using
\equ(5.11). 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 on (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).

In this section we confine ourselves to the Hamiltonian \equ(2.1);
we shall see in \S 7 how to extend the graphical construction to the
anisochronous case.

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 $v$: if $\d_v=1$
it represents $f\defi f_1$ while if $\d_v=0$ it represents
$f_0=g^2\cos\f$. 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
$v$ 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''), where $p_v$ is the number of lines entering $v$;
for $v=v_0$ we write $-\dpr_j\dpr_0^{p_v} f_{\d_{v_0}}$.
Here and henceforth $\dpr_0\defi \dpr_\f$ and $\dpr_j\defi
\dpr_{\a_j}$, $j=1,2$ (this implies that $\d_{v_0}=1$): the functions
have to be evaluated at $(\f^0(t),\aa+\oo t)$.

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
$-\io$.

\ifnum\mgnf=1\pagina\fi\0
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\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 $k=12$,
and some labels. 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.}

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%\ifnum\mgnf=1\pagina\fi\0
per\-for\-med
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 $v$ lower than the first a label $\b_v=O,D, R$
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.

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
$v<v_0$) the node following $v$.

By \equ(5.12) we see that a circle with a $D$ or $R$ label encircling
a node $v$ linked to the higher node $v'$ (external to the circle)
represents just a function $|w_{03}(\t_{v'})|$ or $w_{00}(\t_{v'})$
\ifnum\mgnf=1\pagina\fi
\0multiplied 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.

Fixed $\th$ and setting $w(\t',\t)=w_{03}(\t') w_{00}(\t)-w_{03}(\t)
w_{00}(\t')$ define the function $A(\{\t_v\})$ of the time labels
$\t_v$ associated with the nodes by:

$$A\=
A(\{\t_v\})=\fra{(-1)^n}{2^{n} n!}\Big(\prod_{v<v_0} w(\t_{v'},\t_v)
\,\e^{\d_v}\,\dpr_0^{p_v+1} f_{\d_v}(\t_v)\Big) \,\dpr_0^{p_{v_0}}
f_{\d_{v_0}}(\t_{v_0})\Eq(6.2)$$
%
where $p_v$ is the number of lines entering the node $v$;
$f_\d(\t)\=f_\d(\f^0(t),\aa+\oo \t)$ and $n$ is the number of nodes of
the graph $\th$; $\dpr_0^qf_\d(\t)$ means $(\dpr_\f^q f_{\d})
(\f^0(t),\aa+\oo \t)$.

We define also $A^{i,j,m,\ldots}_{x,y,w,\ldots}$ by the same
expression as $A$ with extra derivatives $\dpr_i,\dpr_j,\dpr_m,\ldots$
acting on the function $f_{\d_x},f_{\d_y},f_{\d_m},\ldots$ in
\equ(6.2).  We shall say that $i,j,m,\ldots$ are {\it marks} on the
nodes $x,y,w,\ldots$. In terms of graphs we simply represent
$A^{i,j,m,\ldots}_{x,y,w,\ldots}$ by affixing {\it also} ``marks''
$i,j,m,\ldots$ to the nodes $x,y,w,\ldots$. Then let, for $j=1,2$:
\ifnum\mgnf=0\pagina\fi
$$\eqalign{
\G^{(1)}(\aa)=&\sum_{\th} \ig|w_{03}(\t_{v_0})|
A^0_{v_0}(\{\t_v\}),\cr
\ \G^{(0)}(\aa)=&\sum_{\th} \ig w_{00}(\t_{v_0})
A^0_{v_0}(\{\t_v\}),\qquad\ \lis\G_j(\aa)=2\sum_\th \ig
A_{v_0}^j(\{\t_v\})\cr}\Eq(6.3)$$
%
the integrals ranging over $\t_{v_0}\in (+\io,-\io)$ and over
$\t_v\in (-\io,\t_{v'})\cup (\io,\t_{v'})$ for $v<v_0$.

Given the above definitions, $\V G=(G^{(0)},G^{(1)})$ and $\V
\G=(\G^{(0)},\G^{(1)})$ verify the following (analogues of the well
known field theoretic) ``Dyson equations'' (see also [GGM1]):

\ifnum\mgnf=1\hsize=16.5truecm\fi

$$\eqalign{\textstyle &\textstyle{\textstyle
G^{(0)}(\aa)=\G^{(0)}(\aa)+\kern-5pt {\displaystyle\sum_{\th; y\in
\th}
\ig_{\io}^{-\io}} \kern-0.5cm A_{v_0,y}^{0,0} w_{00}(\t_{v_0})\,
(|w_{03}(\t_{y})| G^{(0)}(\aa)+
w_{00}(\t_y) G^{(1)}(\aa))+\ldots}\cr
&\textstyle{\textstyle
G^{(1)}(\aa)=\G^{(1)}(\aa)+\kern-5pt
{\displaystyle\sum_{\th; y\in \th}
\ig_{\io}^{-\io}}\kern-0.5cm A_{v_0,y}^{0,0} |w_{03}(\t_{v_0})|\,
(|w_{03}(\t_{y})| G^{(0)}(\aa)+
w_{00}(\t_y) G^{(1)}(\aa))+\ldots}\cr}\Eq(6.4)$$
%
\ifnum\mgnf=1\hsize=16truecm\fi
where the r.h.s. represents the contributions of the trees with no
fruit or with just one fruit and the $\ldots$ denote contributions
from the trees with more than just one fruit, see Appendix R5.
Equation \equ(6.4) is obvious if one writes the r.h.s. as a sum of
graphs and sees its graphical interpretation: note that \equ(6.4) is
a ``non perturbative'' identity since $\V\G,\V G$ are infinite series
in $\e$ (\ie they contain all the contributions from graphs of any
order). Then:
%
$$Q_{j}(\a)=\lis\G_j(\a)+2\sum_{\th;w\in \th}\ig_{\io}^{-\io}
A_{v_0,w}^{j,0}\Big(|w_{03}(\t_{w})| G^{(0)}(\aa)+ w_{00}(\t_w)
G^{(1)}(\aa)\Big)+\ldots\Eq(6.5)$$
%
with the above meaning of the $\ldots$ (\ie up to sums involving at
least two fruits).

The convergence of all the above series and the estimates about their
remainders are nontrivial and an {\it open problem}: they are, however,
finite order by order as it follows from the analysis of Appendix A1
of [G3]. Therefore what follows can be interpreted as a collection of
identities valid order by order in the expansion and as a way of
deriving and expressing compactly infinitely many identities. This
will be sufficient for all our conclusions; nevertheless one would get
better bounds if one could prove convergence.

Consider $G^{(0)}_{i},\G^{(0)}_{i},G^{(1)}_i,\G^{(1)}_i,
\lis\G_{ij}$ defined as $\dpr_i$
derivatives of $G^{(0)},\G^{(0)},G^{(1)},\G^{(1)},\lis\G_j$, see
\equ(6.3), \equ(6.4), {\it evaluated at $\aa=\V0$}. The terms hidden
in the $\ldots$ contribute $0$ since {\it at the
homoclinic point by parity $G^{(0)}(\V0)=0$ and, as well,
$G^{(1)}(\V0)=0$.}  Differentiating \equ(6.4) and evaluating
at $\aa=\V0$ we get the remarkable {\it exact} relations:

$$G^{(0)}_{j}=\G^{(0)}_{j}+ M_{01}\,G^{(0)}_{j}+ M_{11}\,G^{(1)}_j,\quad
G^{(1)}_j=\G^{(1)}_j+ M_{00}\,G^{(0)}_{j}+ M_{10}\,G^{(1)}_j\Eq(6.6)$$
%
with the matrix $M=\pmatrix{M_{00}&M_{01}\cr M_{10}&M_{11}\cr}$
defined by:

$$\pmatrix{
\sum_{\th;\,y\in \th}\ig_{\io}^{-\io}
A_{v_0,y}^{0,0} |w_{03}(\t_{v_0})|\,|w_{03}(\t_{y})|&
\sum_{\th;\,y\in \th}\ig_{\io}^{-\io}
A_{v_0,y}^{0,0} w_{00}(\t_{v_0})\,|w_{03}(\t_{y})|\cr
\sum_{\th;\,y\in \th}\ig_{\io}^{-\io}
A_{v_0,y}^{0,0} |w_{03}(\t_{v_0})|\,w_{00}(\t_{y})&
\sum_{\th;\,y\in \th}\ig_{\io}^{-\io}
A_{v_0,y}^{0,0} w_{00}(\t_{v_0})\,w_{00}(\t_{y})}\Eq(6.7)$$
%
$M$ is symmetric: this is implied by the symmetry of the operator
$\OO_0$, which follows immediately from that of $\OO$
(see \equ(5.13), and Appendix A2 of [G3]: or proceed
inductively). More generally the symmetry of $\OO_0$, repeatedly
applied (as done with $\OO$ in [G3]), implies that
$\sum_{\th,y}\ig A^{0,0}_{v_0,y}F(\t_{v_0}) H(\t_y)=\sum_{\th,y}\ig
A^{0,0}_{v_0,y}H(\t_{v_0}) F(\t_y)$ for all $F,H$: {\it used
several times below}.

Going back to \equ(6.5), in order to evaluate the derivatives $\dpr_i
Q_j(\aa),\, i,j=1,2$ at $\aa=\V0$ we need only the linear terms and
the $\ldots$ contribute exactly $0$. If we set:
%
$$\lis \G_{ij}=2\sum_{\th;\,y\in \th}\ig A_{v_0,y}^{j,i}\Eq(6.8)$$
%
we have that the matrix $2\times2$ $\lis\G$ is symmetric
(again use the symmetry of $\OO_0$).

Noting that $\V G=(1-\s M)^{-1}\V\G$ with $\s=\pmatrix{0&1\cr1&0\cr}$,
then the intersection matrix $D$ is:
%
$$D_{ij}=\lis\G_{ij}+2 (\G^{(1)}_j G^{(0)}_{i}+\G^{(0)}_{j}
G^{(1)}_{i})\Eq(6.9)$$
%
which, setting $C=\s(1-\s M)^{-1}$, becomes:
%
$$D=\pmatrix{\lis\G_{11} + 2(\V\G_1, C \V\G_1)& \lis\G_{12}+
2(\V\G_2, C \V\G_1)\cr
\lis \G_{21}+2(\V\G_1, C \V\G_2)&\lis\G_{22}+
2(\V\G_2, C \V\G_2)\cr}\Eq(6.10)$$
%
The matrix $C$ is symmetric, because $M$ is symmetric; as also
$\lis\G$ is symmetric, then $D$ is a
symmetric matrix (to all orders as \equ(6.9) is {\it exact}): this
agrees with the mentioned result (see comment following \equ(4.2)) of
Lochak and Sauzin on the symmetry of $D$.  Setting
$\G^{(0)}_{i}=\sum_{n=1}^\io \e^n \G^{(0)(n)}_{i},\,
\lis\G_{ij}=\sum_{n=1}^\io \e^n \lis\G^{(n)}_{ij}$,
an explicit calculation yields in the case \equ(5.10):

$$\G^{(0)(1)}_{2}=\fra14{\h^{-1/2}} \lis\G^{(1)}_{22}=2\p{\h^{-1}}
e^{-\fra\p2{\h^{-1/2}}}(1+O(e^{-2\p {\h^{-1/2}}}))\Eq(6.11)$$
%
An elementary (and {\it somewhat surprising}) calculation of the
determinant yields a remarkably simple expression in which the
(generically) leading first non trivial order of $\det D$ (\ie the second)
and the possibly non
exponentially small contributions are evaluated {\it without any
approximation}:

$$\det D= \lis\G^{(1)}_{11}\lis\G_{22}^{(1)}\e^2-2\,(\lis\G_{11}
M_{11}-2{\G^{(0)}_{1}}^2){\G^{(1)}_2}^2\,\det C\,+\,
\ldots\Eq(6.12)$$
%
the last $\ldots$ collects contributions to $\det D$ from the products
in $\det D$ involving $\lis \G_{12},\lis\G_{22}$ (for instance
$-4\lis\G_{12}(\V\G_1,C\V\G_2)$ or $2\lis\G_{22}(\V\G_1,C\V \G_1)$,
$-\lis\G^2_{12}$ and a few others, see Appendix R6) or the higher
orders of $\lis\G_{11}\lis\G_{22}$; we used that $\lis\G_{12}^{(1)}$
and $\lis\G_{22}^{(1)}$ are in general equally exponentially small,
while $\lis\G_{11}^{(1)}$ is not.

The terms not written in \equ(6.12) contain as factors (see Appendix
A2 below) derivatives $\dpr_2$ (``fast angle derivatives'') of
integrals of an analytic function of the $\t_v$'s: if we imagine
developing the kernels $A$ as Fourier series in $\aa$ we see that only
Fourier components $\nn=(\n_1,\n_2)$ with $\n_2\ne0$ can
contribute. This means that the dependence on the $\t_v$'s will be via
the matrix elements $w(\t_{v'},\t_v)$ of the pendulum Wronskian matrix
and via $e^{i\sum_v \t_v \oo\cdot\nn_v}$ with $\sum_v \nn_{v2}\ne0$
(we say that ``the dependence on time is via a total fast rotation'').

So that by shifting the integration over $\t_v$ to $\sim \pm i
(\p/2-\h^{1/2})$, \ie almost as far out in the complex as the
singularities of the $w$--functions allow, one sees that such terms
contribute a quantity bounded to all orders $h$ proportionally to
$e^{-\fra\p2 \h^{-1/2}}$ because $|\oo\cdot\nn|>\h^{-1/2}-Nh\h^{a}$ if
$\n_2\ne0$: this gives a good bound for $h$ not too large, \eg
$h<\h^{-1/2}N^{-1}$ (actually even for $h\simeq \h^{-1}N^{-1/2-a}$,
see Appendix A2).

For $h>\h^{-1/2}N^{-1}$ one can invoke the convergence of the series
for $\det D$ and obtain, for some constant $\lis c$, a bound
$(\h^{-\lis c}\e)^{\h^{-1/2}N^{-1}}$ (much smaller than $e^{-\fra\p2
\h^{-1/2}}$ because $\e=\m\h^c$, provided $c$ is large enough); see
Appendix A2 for details.  The conclusion is that the terms not written
in \equ(6.12) can be bounded by $\e^3
e^{-\fra\p2\h^{-1/2}}O(\h^{-3\b})$, provided $c$ is large enough so
that the sum of the bounds of the orders from $3$ to $\h^{-1/2}N^{-1}$
is dominated by the third order bound; see Appendix A2 for details.

The value of $\b$ can be taken $2(N+1)+4d$ in terms of the
degree $N$ of $f$ and of the constant $d$ in \equ(2.2): it
is explained by the singularities of the elementarily computable Fourier
transforms of $\cos N\f(t)$ and $\sin N\f(t)$, see [GR]. A similar
argument is in \S8 of [G3].

%The above argument is good but it gives a rather poor bound: see
%Appendix A2 for some improvements.

Hence the leading value of $\det D$ {\it is given, as $\h\to0$, by its
first order expression} $\lis\G_{11}^{(1)}\lis\G_{22}^{(1)}$ plus the
{\it apparently much larger} $-2(\lis\G_{11}
M_{11}-2{\G^{(0)}_{1}}^2)\,{\G_2^{(1)}}^2\det C$. But we shall show that:
%
$$(\lis\G_{11} M_{11}-2{\G_{1}^{(0)}}^2)\,{\G_2^{(1)}}^2\det C=
O(\h^{-4\b})\e^4 e^{-\fra\p2 \h^{-1/2}}\Eq(6.13)$$
%
because of the first factor being of order $O(\e^2\h^{-2\b}e^{-\fra\p2
\h^{-1/2}})$, \ie essentially $0$,
again assuming (temporarily) convergence of the series for $\V
G,\V\G,\lis\G,M$ (see comments after \equ(6.5)).

The discussion of the convergence for the series for $\V\G, \lis\G,M$
is {\it very non trivial}, while one could show the convergence of the
series for $\V G$, following [Ge1]. However the series for $\det D$
converges and its convergence, {\it which is absolutely essential},
follows immediately from theorem 1 above.

It is remarkable that we can {\it avoid} proving the convergence of the
power series for $\V G,\V \G,\lis\G,M$ and get away with only the
easily established convergence of $\det D$: in fact we can just use
the above series as formal power series and that is all we really need
({\it together} with the convergence of $\det D$).

After all the identity \equ(6.12) as an identity between formal power
series and the formal bound \equ(6.13) show that {\it to all orders the
$\det D$ is exponentially bounded} and this is almost enough.
%In fact
%up to order $\h^{-1/2}N^{-1}$ one needs only bounds on finitely many terms
%($O(\h^{-1/2}N^{-1})$ in the series for $\V G,\V \G,\lis\G,M$, as explained
%above. For the remainder we use that it is bounded by the much smaller
%quantity $(\e\h^{-\lis c})^{\h^{-1/2}N^{-1}}$ derived from
%the convergence of the series for $\det D$ (if $\m$ in $\e=\m\h^c$
%is small enough and $c$ is large enough).

We suggest to proceed however by, {\it at first}, assuming convergence
of the series for $\V G,\V \G,\lis\G,M$
for $\e\h^{-\lis c}$ small enough,
and only on a second reading check that formal power series
considerations (plus analyticity of $\det D$) suffice: a technique
used in [G3], \S8. Here {\it the difficulty is not the bounds but the
cancellations} and assuming convergence removes unessential worries
and clarifies the algebra.

For instance in the case \equ(5.10), with $a>0$, one has
$\lis\G_{11}^{(1)}=4+O(\h^{a})$, by direct computation, and, by
\equ(6.11), $\lis\G_{22}^{(1)}=8\p\h^{-1/2} e^{-\fra\p2\h^{-1/2}}$,
so that the leading term in \equ(6.12) is found to be:
%
$$\det D=32\,\p\h^{-1/2}\e^2 e^{-\fra\p2\h^{-1/2}}\Eq(6.14)$$

{\it We also take $a=\fra12$ in the following calculations} to
simplify notations: the general case is obtained by replacing the
coefficients {\it explicitly} appearing in the following formulae $\o$
by ${\h^a}$ and ${\o^2}$ by $\h^{\fra12+a}$ respectively. We recall
that we set $\f(t)\=\f^0(t)$.

%
The reason why \equ(6.13) holds is that if
$\o\defi\h^{1/2}$:

$$\lis\G_{11}=\fra4\o \G^{(0)}_{1}-\fra1{\o^2}\lis\G_{12},\qquad
M_{11}=\fra\o2 \Big(\G^{(0)}_{1}+\fra1{\o^2}\G^{(0)}_{2}\Big)\Eq(6.15)$$

To prove \equ(6.15) note that $\sum_{\th,y}\ig A^{10}_{v_0
y}w_{00}(\t_y)=\sum_{\th,y}\ig A^{01}_{v_0 y}w_{00}(\t_{v_0})$ by the
symmetry of $\OO_0$; hence:

$$\lis\G_{11}=2\sum_{\th,y}\ig A^{11}_{v_0y},\quad
\G^{(0)}_{1}=\sum_{\th, y}\ig A^{10}_{v_0y}w_{00}(\t_y)\Eq(6.16)$$
%
Thinking \equ(6.16) as sums of graphs we see that to each graph with $n$
nodes contributing to $\lis\G_{11}$ with the node $y$ marked $1$ (see
definition preceding \equ(6.3)) as in \equ(6.16) there correspond {\it
two} graphs contributing, if $y<v_0$, to $\G^{(0)}_{1}$. Namely the one
with the node $y$ marked $0$ and the one obtained by deleting the mark
$0$ on $y$, adding a new node $y'$ marked $0$ on the line coming out of
$y$ and with index $\d_{y'}=0$.  If $y=v_0$ we associate with it only
the first of the above graphs.  Note that the second graph has $n+1$
nodes.

Suppose that $y$ is an endnode in a graph $\th$ associated with
$\lis\G_{11}$: by $A$--kernels definition we must evaluate, when
computing the contribution to $\lis\G_{11}$ from such graph, the
quantity:

$$\OO_0(\dpr_{10} f)=\fra1\o\OO_0(\dpr_\t\dpr_0
f)+\fra1\o\OO_0(-\dot\f\,\dpr_{00} f)-\fra1{\o^2}\OO_0(\dpr_{20}
f)\Eq(6.17)$$
%
where $\o\defi \h^{1/2}$(recall also that we take $a=\fra12$) and
having used $\dpr_1\=\fra1\o(\dpr_\t-\dot\f\,\dpr_0)-\fra1{\o^2}\dpr_2$
(because the derivatives act on functions of special form $F(\f(t),\oo
t)$ with $F(\f,\aa)$ suitable).

Since $\dot\f\,=-2w_{00}$ we see that the second term is
$\fra2\o\OO_0(w_{00}\dpr_{00} f)$: hence it appears in the
evaluation of the first among the two corresponding graph
contributions to $\fra4\o \G^{(0)}_{1}$. The second corresponding graph
contribution to $\fra4\o\G^{(0)}_{1}$ will require computing:

$$-\fra4\o\OO_0(w_{00}\dpr^3_0 f_0\OO_0(\dpr_0 f))=
\fra2\o\OO_0(\dot\f\, \dpr_0^3f_0 \OO_0(\dpr_0 f))\Eq(6.18)$$
%
The combinatorial coefficient associated with this graph is
$(n+1)!$ rather than $n!$: but one checks that this is what is
necessary to verify that the difference between the contributions to
$\lis \G_{11}$ and $\fra4\o \G^{(0)}_{1}$ due to the three graphs
considered requires computing:

$$\fra1\o\OO_0\big(\dpr_{\t_y}\dpr_0 f-\dot\f\,\dpr_0^3 f_0 \OO_0(\dpr_0
f)\big)-\fra1{\o^2}\OO_0\big(\dpr_{02}f\big)\Eq(6.19)$$
%
In Appendix A3 we prove for all odd $F$
the following commutation relation:
%
$$\dpr_{\t_{y'}}\OO_0(F)=\OO_0\big(\dpr_{\t_y} F -\dot\f\,\dpr_0^3
f_0\OO_0(F)\big)\Eq(6.20)$$
%
which, noting that below we only apply \equ(6.20) for odd  $F$'s,
allows us to rewrite \equ(6.19) as:

$$\fra1\o\dpr_{\t_{y'}}\OO_0(\dpr_0
f)-\fra1{\o^2}\OO_0(\dpr_{02}f)\Eq(6.21)$$

We proceed by picking a new endnode (if any) and by repeating the
construction until all endnodes are exhausted. The difference between
the contributions to $\lis\G_{11}$ and $\fra4\o\G^{(0)}_{1}$ considered so
far will therefore be given by a collection of graph values
contributing to $\lis\G_{11}$ with {\it no marked node} and with one
of the endnodes requiring the evaluation of:

$$\fra1\o\dpr_{\t_{y'}}\OO_0(\dpr_0 f)-\fra1{\o^2}\OO_0(\dpr_{20}f)
\Eq(6.22)$$
%
Consider now a node $y$ marked 1, which is next to an endnode; and
suppose that there are $p$ lines linking it to the endnodes.  The
differences between the contribution of the considered graphs to
$\lis\G_{11}$ and of the corresponding ones to $\fra4\o\G^{(0)}_{1}$
{\it plus} the already evaluated differences due to the previously
considered graphs will be, proceeding as before:

$$\eqalign{
&\fra1\o\OO_0\Big((\dpr_\t\dpr^{p+1}_0f)
\OO_0(\dpr_0f)^p\Big)+\fra1\o\OO_0\Big(
\dpr^{p+1}_0 f\sum_{j=1}^p\OO_0(\dpr_0f)^{p-1}
\dpr_{\t_y}\OO_0(\dpr_0 f)\Big)+\cr
&+\fra1\o\OO_0\Big(-\dot\f\,\dpr_0^3 f_0\OO_0\big(\dpr^{p+1}_0
f\OO_0(\dpr_0 f)^p\big) \Big)-
\fra1{\o^2}\OO_0\Big(\dpr_{2}\dpr^{p+1}_0f\OO_0(\dpr_0 f)^p\Big)-\cr
&-\fra1{\o^2}\OO_0\Big(\dpr^{p+1}_0
f\sum_{j=1}^p\OO_0(\dpr_0 f)^{p-1}\OO_0(\dpr_{20} f)\Big)\cr}\Eq(6.23)$$
%
which is, by \equ(6.20):
%
$$\fra1\o\dpr_{\t_{y'}}\OO_0(\dpr^{p+1}_0 f\OO_0(\dpr_0 f)^p)
-\fra1{\o^2}\dpr_2\OO_0(\dpr^{p+1}_0 f \OO_0(\dpr_0 f)^p)\Eq(6.24)$$
%

We proceed in the same way until we reach the first node and, in fact,
we can treat in the same way {\it also} the first node except that in
this case there is only one corresponding graph in $\G^{(0)}_{1}$. No
need, this time, of using the commutation relation \equ(6.20)
because the $\t_{v_0}$--integral is a simple integral not
involving $\OO_0$ operations any more.  Hence eventually we find that
the differences between the contributions to $\lis \G_{11}$ and
$\fra4\o\G^{(0)}_{1}$ add up to:
%
$$\Big(\fra2\o\ig \dpr_t A^1_{v_0}\Big)-\Big( \fra2{\o^2}\dpr_2\ig
A^1_{v_0}\Big)=-\fra2{\o^2}\dpr_2 \ig A^1_{v_0}=-\fra1{\o^2}\lis\G_{12}
\Eq(6.25)$$
%
where one should note that the first term in the l.h.s. vanishes: this
does not immediately follow from it being an integral of a derivative
because the integrals are improper: one checks this by
\equ(5.1) and the fact that $A^1_{v_0}$ is even (at $\aa=\V0$).

In a similar way one shows that $M_{11}=\fra\o2(\G^{(0)}_{1}
+\fra1{\o^2}\G^{(0)}_{2})$.  One proceeds by writing $\G^{(0)}_{1}$ as
$\G^{(0)}_{1}=\sum_{\th, y}\ig A^{01}_{v_0y}w_{00}(\t_{v_0})$, and
marking $0$ the first node for both $\G^{(0)}_{1}$ and $M_{11}$.  Then
note that to each graph contributing to $\G^{(0)}_{1}$ with the node
$y$ marked $1$ there correspond {\it two} graphs contributing to
$M_{11}$ also for $y=v_0$, when $\d_{v_0}=1$, and they are constructed
by following the same prescription given after \equ(6.16); moreover to
each graph contributing to $\G^{(0)}_{1}$ with $\d_{v_0}=0$ there
correspond two graphs contributing to $M_{11}$, namely the one with
the node $v_0$ marked $0$ twice and the one obtained by adding a new
node $v_0'$ (which becomes the new first node), marked $0$ twice and
with $\d_{v_0'}=0$, on the line coming out of $y$.  Then proceed as
before, with the only difference that, at the last step, when dealing
with the contributions arising from the graphs with $\d_{v_0}=0$, one
has to use the identity $\OO_0(w_{00}^2\sin\f)=-\fra12\fra{\sinh
t}{\cosh^2 t}=\fra12 \fra{d}{dt}\fra1{\cosh t}=-\fra14 \ddot\f$. This
can be proved in the same way as the commutation relation \equ(6.20);
see Appendix A3.

This completes the proof that the cancellation discussed in \S5 is in
fact taking place to all orders of perturbation theory.

\vskip1.truecm

\0{\bf\S7. The splitting in anisochronous cases.}
\numsec=7\numfor=1
\*

We now proceed to the analysis of the anisochronous case \equ(1.1).
We recall the form of the Hamiltonian:
%
$$H={\h}^a A+\fra1{\h^{1/2}}B+\h^{2a} \fra{A^2}2+\fra {I^2}{2}+\,g^2\,
(\cos \f -1)+ \e\,f(\f,\a,\l)\Eq(7.1)$$
%
where $\e=\m\, \h^c$ and $c$ will be chosen large enough.  This model
belongs, see [T],[G2], to a well studied class of models (``Thirring
models'') and it is a simplified version of the Hamiltonian considered
in [CG] in \S12. If we added \equ(7.1) a further ``{\it monochromatic}''
term $f_0(\f,\a,\l)$ which has Fourier harmonics $\nn$, for $(\a,\l)$,
multiples of a given {\it fast} harmonic $\nn^0$ (\ie a harmonic with
the component $\n^0_2$ different from $0$) it would offer all the
difficulties of that case in spite of being much simpler
analytically. We shall not attempt the analysis of such more general
Hamiltonian: the error in [CG] eliminated almost completely this
problem which, hence, has to be studied again if one wants to recover
the results of that paper.

Consider, {\it as an example}, the case \equ(5.10).  This time the first
problem is to find for how many values of $A$ one can have, for $\e$
small, an invariant hyperbolic torus with rotation number:
%
$$\oo=(\h^{a}+\h^{2a} A,\h^{-1/2})\Eq(7.2)$$
%
and $\e$--close to an unperturbed one.

For Hamiltonians like \equ(7.1) the average position in the $A$
variables of the torus with rotation \equ(7.2) will be exactly $A$
(this is a general property of ``twistless tori'', as discussed in
[G3],[G4]). The average position in the $B$ variables will be chosen
so that the energy of the torus is a fixed value, \eg $0$: this can be
done because the fast variable $\l$ conjugate to $B$ is still
isochronous (see \S5 in [CG] for the more general case in which also
$\l$ is anisochronous when the analysis is somewhat more involved).

Of course $\oo$ must verify a Diophantine condition: but in view of
using the result to show that the tori resisting the perturbation, \ie
the ones described by remark (7) to theorem 1 above, are dense enough
to build long chains of tori joined by heteroclinic trajectories,
``{\it heteroclinic chains}'', we must consider tori that verify a
{\it very generous} Diophantine condition, compared to \equ(2.2).

Since we shall not discuss the ambitious application attempted in
[CG] we just allow rotations vectors verifying \equ(2.2) with a
fixed, possibly very large, $d$.

 From lemmata 1,1' in \S5 of [CG], all tori with rotation
$\oo\defi\oo(A)\defi(\h^{a}+\h^{2a} A, \h^{-1/2})$ verifying:

$$|\oo\cdot\nn|> C^{-1}\h^d {|\nn|^{-3}}\defi
{C(\h)}{|\nn|^{-3}}\Eq(7.3)$$
%
will survive the perturbation if the parameter $c$ in the definition
$\e=\m \h^c$ of the coupling constant is large enough: so that
$\e C(\h)^{-q}<\e_0$ for some $\e_0$ and some $q>0$

The splitting theory is ``insensitive'', at given $\oo$, to the
presence or absence of the isochrony breaking term $\fra12\h^{2a} A^2$ in
\equ(7.1).
%provided $\h^{1-a}\tende{\h\to0}0$.
We discuss this delicate point below, {\it for general perturbation
$f$,} see \equ(5.9).  The homoclinic splitting is given by
\equ(6.14) {\it with no extra leading terms}.  The only effect of the
anisochrony is to introduce a few gaps in the foliations of phase
space into stable and unstable manifolds: but it has also the advantage
that we no longer must be careful about the values of $\h$. {\it
Anisochrony guarantees that the Diophantine conditions holds for
``many'' values of $A$}.
\*

Turning to the {\it main point} of this section (and of the whole
paper) {\it we prove that usually the first order (``Mel'nikov
integral'') dominates the splitting}. Again the technique will be
based on the general theory of [G3].

In the anisochronous case the graph labels have to be extended, see
\equ(5.7), \equ(5.14) and [G3]. On each node $v$ one adds a
further node label $j_v=0,1$ (which in the isochronous case would be
$j_v\=0$) and this has the effect that in the definition of $A$ one
replaces:

$$\eqalign{
&\dpr_0^{p_v+1} f_{\d_v}\ \to\ \dpr_{j_v}\dpr_{j_{v_1}}\ldots
\dpr_{j_{v_{p_{v}}}} f_{\d_v}\qquad{\rm if} \ v< v_0\cr
&\dpr_0^{p_{v_0}} f_{\d_{v_0}}\ \to\ \dpr_{j_{v_1}}\ldots
\dpr_{j_{v_{p_{v_0}}}}f_{\d_{v_0}}\qquad{\rm if} \ v= v_0\cr}\Eq(7.4)$$
%
where $p_v$ is the number of nodes $v_1,\ldots,v_{p_v}$ preceding
$v$. Furthermore the kernels $w(\t_v',\t_{v})$ become node dependent
$w_v(\t_v',\t_{v})$ and equal to $w(\t_v',\t_{v})$ if $j_v=0$ and
$\h^{2a}(\t_{v'}-\t_v)$ if $j_v=1$.

The first component $X_1(t;\aa)$ of $\V X_\giu(t;\aa)$ will not
vanish. Let us define $\a(t)$ as $\a+\o t+\sum_{k=1}^\io \e^k
X^{k}_1(t;\aa)$ where $X_1^k$ is the first (and only non
vanishing) component of $\V X_\giu^k$.

The contributions to the splitting $Q_j(\aa)$ due to fruitless trees will
be $2\sum_\th A^j_{v_0}$, with the same notations of \S6. The full
splitting will be:

$$Q_j(\aa)=2\sum_\th \ig A^j_{v_0}+2\sum_{\th;y,r}\ig A^{j,[r]}_{v_0,y}
w_r(\t_y) G^{(r)}(\aa)+\ldots \Eq(7.5)$$
%
here $[r]=0$ if $r=0,1$ and $[r]=1$ if $r=2,3$, and:

$$w_0(\t)=w_{00}(\t),\quad w_1(\t)=|w_{03}(\t)|,\quad
w_2(\t)=\h^{2a} , \quad w_3(\t)=\h^{2a}|\t| \Eq(7.6)$$
%
with $\V G=(G^{(0)},G^{(1)},G^{(2)},G^{(3)})$ representing the fruit
values, defined as in \S6, for fruits which now can carry also a label
$2,3$ on the first node: the latter values correspond to the fruits
carrying label $j_v=1$ ($2$ corresponds to a dry fruit and $3$ to a
ripe fruit): the choices of the $w_2,w_3$ arise from the form of the
operator corresponding to $\OO$ for the nodes with the new labels,
called $\lis\II^2$ as in [G3].

In complete analogy with \S6 the $\V G$ verify Dyson equations. If
we set:

$$\G^{(r)}(\aa)=\sum_\th \ig A^{[r]}_{v_0} w_r(\t_{v_0}),\qquad
M_{rs}=\sum_{\th;y}\ig A^{[r],[s]}_{v_0,y} (\s \V w)_r(\t_{v_0})
(\s\V w)_s(\t_y)\Eq(7.7)$$
%
where the matrix $\s$ is defined to be $\s_{rs}=0$ except for the matrix
elements $\s_{01}=\s_{10}=\s_{23}=\s_{32}=1$, then:

$$G^{(r)}(\aa)=\G^{(r)}(\aa)+\sum_{\th;y,s}\ig (\V w)_r(\t_{v_0})
A^{[r],[s]}_{v_0,y} (\s \V w)_s(\t_y) G^{(s)}(\aa)+\ldots\Eq(7.8)$$
%
where the dots represent contributions from the graphs with more than
one fruit, while the terms explicitly written represent the
contributions from the graphs with no fruits or with just one fruit.

At the homoclinic point the derivatives $\V G_j=\dpr_j G,\V \G_j
=\dpr_j \V \G$ verify exactly:

$$\V G_j=\V\G_j+\s M \V G_j,\qquad \V G_j=(1-\s M)^{-1}\V\G_j\Eq(7.9)$$
%
(compare with \equ(6.6)).

The intersection matrix will be, setting as in \S6,
$\lis\G_{ij}=2\sum_{\th;y}\ig A^{j,i}_{v_0,y},\ i,j=1,2$ and $C=\s(1-
M\s)^{-1}$:

$$D_{ij}=\lis\G_{ij}+2(\V \G_j,\s\V
G_i)=\lis\G_{ij}+2(\V\G_j,C\V\G_i)\Eq(7.10)$$
%
The convergence of the above series, \equ(7.5)$\div$\equ(7.8) and the
estimate of their remainders is discussed as in \S6: see Appendix A2.

The above equation is not sufficient this time: there are in fact too
many variables. There are however several relations between the matrix
elements of $M$ and $\V \G,\V G$.  In fact $M$ is symmetric for the
same reasons as the corresponding matrix in \S6: \ie by using the
symmetry of $\OO_0$ and of $\lis\II^2_0$ operators (see
\equ(5.13) and \equ(5.15)). If $\l\defi\fra{2\h^{2a}}\o,
\o=\h^{a}$ the relations are, {\it up to terms
exponentially small in $\h$}:
%
$$\eqalign{ &M_{11}=\fra\o2\G^{(0)}_1,\quad
\G^{(2)}_1= \l\G^{(0)}_1,\quad G^{(2)}_1=Z\,\l\,
G^{(0)}_1\cr & M_{03}= \l M_{01}-\l \L_0,\quad M_{23}=\l M_{12}-\l \L_1\cr
&\quad M_{33}=\l^2M_{11},\quad
M_{31}=\l M_{11}\cr}\Eq(7.11)$$
%
where $\L_0= (2\h^{1/2})^{-1}\dpr_2\sum\ig |w_{03}(\t_{v_0})| A^0_{v_0}$ and
$\L_1= (2\h^{1/2})^{-1}\dpr_2\sum\ig \h^{2a}|\t_{v_0}| A^0_{v_0}$ and
$Z=\fra{1-\L_0}{1+\l\L_1}$.  The relations among the $M$ elements are
proved by the same argument discussed in \S6 for the first of
\equ(7.11); one should use also the relation $\OO_0(w_{00}
|w_{03}|\sin\f)=|\dot w_{03}|/2$, proven in  Appendix A3.
The constant $Z$ arises solving by iteration
\equ(7.9): the structure of the matrix $\s M$
and of its powers is, given the relations between the $M_{ij}$ in
\equ(7.11), such that the first and third components of $\V G$ are
proportional via the constant $\l Z$; see Appendix A4 for details.

Equation \equ(7.11) allows us to reduce the size of the
vectors $\V G, \V\G$ and of the matrix $M$. We shall denote with a
tilde the new vectors and matrices. Introduce $\V{{\tilde G}}=(G^{(0)},
G^{(1)}, G^{(3)})$, $\V{{\tilde \G}}=(\G^{(0)},
\G^{(1)}, \G^{(3)})$, and $\tilde M, N$ as:

$$\tilde M=\pmatrix{M_{10}+Z\l M_{12}& M_{11}&\l M_{11}\cr
M_{00}+Z\l M_{02}& M_{01}& \l M_{01}-\l \L_0\cr
M_{20}+Z\l M_{22}& M_{21}& \l M_{21}-\l \L_1\cr},\quad
N=\pmatrix{0&1&\l\cr1&0&0\cr Z \l & 0&0\cr}\Eq(7.12)$$
%
The equations \equ(7.9), \equ(7.10) become respectively:
%
$$\V {{\tilde G}}=(1-\tilde M)^{-1}\V{{\tilde \G}},\qquad
D_{ij}=\lis\G_{ij}+2(\V{{\tilde \G}}_i,N(1-\tilde M)^{-1}
\V{{\tilde \G}_j})\Eq(7.13)$$
%
noting that the matrix $\tilde C= N(1-\tilde M)^{-1}$ is symmetric
(because $C$ in \equ(7.10) is symmetric) and that it has the second and
third row proportional one deduces, analogously to \equ(6.12), that:
%
$$\eqalign{
&\det D=\lis\G_{11}\lis\G_{22}+
2(\G^{(1)}_2)^2\big( \lis\G_{11} \tilde C_{11}+ 2(\G^{(0)}_1)^2
\D_{00,11}\big)+\cr
&+2(\G^{(3)}_2)^2\big( \lis\G_{11} \tilde C_{33}+2 (\G^{(0)}_1)^2
\D_{00,33}\big)+4\G^{(1)}_2\G^{(3)}_2
( \lis\G_{11} \tilde C_{13}+2 (\G^{(0)}_1)^2 \D_{00,13})\cr}\Eq(7.14)$$
%
where $\D_{00,11},\D_{00,33},\D_{00,13}$ denote the determinants of
the matrices:

$$\pmatrix{\tilde C_{00}&\tilde C_{01}\cr
\tilde C_{10}&\tilde C_{11}\cr},\quad
\pmatrix{\tilde C_{00}&\tilde C_{03}\cr
\tilde C_{30}&\tilde C_{33}\cr},\quad
\pmatrix{\tilde C_{00}&\tilde C_{03}\cr
\tilde C_{10}&\tilde C_{13}\cr}\Eq(7.15)$$
%
To compute all the above quantities we note that if we set
(see \equ(7.12)) $a=M_{11}, b=M_{01}, c=M_{12}$ and
$x=M_{10}+Z\l M_{12}=b+Z \l c$, $y=M_{00}+Z\l M_{02}$,
$z=M_{20}+Z\l M_{22}$, $\L_0=-b',\L_1=-c'$, $Z=(1+b')(1-\l c')^{-1}$:

$$1-\tilde M=\pmatrix{1-x&-a&-\l a\cr
-y& 1-b& -\l b-\l b'\cr
-z&-c&1-\l c-\l c'\cr}\Eq(7.16)$$
%
hence $\det (1-\tilde M)= -\Big((y+\l z Z)a-(1-x)^2\Big)(1-\l c')
\defi\D$, and $(1-\tilde M)^{-1}$ is:

$$\eqalign{
&\fra1\D \pmatrix{
(1-b)(1-\l \tilde c)-\l\tilde b c& a(1-\l c')&\l a (1+b')\cr
y(1-\l \tilde c)+\l z \tilde b& (1-x)(1-\l \tilde c)-\l a z
& (1-x)\l \tilde b+\l a y\cr
y c +(1-b) z& (1-x) c+ a z& (1-x)(1-b) -a y\cr}\cr}\Eq(7.17)$$
%
where $\tilde b\defi b+b',\,\tilde c=c+c'$.  Thus the matrix $\tilde
C= N (1-\tilde M)^{-1}$ is:
%
$$\tilde C=\fra1\D \pmatrix{
y(1-\l c')+\l z(1+b')& (1-x)(1-\l c')& \l (1-x)(1+b')\cr
(1-x)(1-\l c') & a (1-\l c') & \l a (1+b')\cr
\l(1-x)(1+b')& \l a (1+b')& \l^2 a (1+b') Z\cr}\Eq(7.18)$$
%
Noting that
$\D_{00,11}=-\D^{-1}(1-\l c '),\ \D_{00,33}=Z^2\l^2 \D_{00,11},\
\D_{00,13}=Z\l \D_{00,11}$ and $\tilde C_{11}=\fra{a(1-\l c')}\D,\,
\tilde C_{33}=(Z\l)^2 \tilde C_{11},
\tilde C_{13}=Z\l \tilde C_{11}$ with $a=M_{11}$ (not to be confused
with $a$ in \equ(7.1)) we get, for a suitable $\b>0$, for $\det D$:
%
$$\eqalign{
&\lis\G_{11}^{(1)}\lis \G^{(2)}_{22}+
2\fra{(\G^{(1)}_2+Z\l \G^{(3)}_2)^2}{(1-\l c')^{-1} \D }
\Big(\lis\G_{11} M_{11}-2(\G^{(0)}_1)^2\Big)+ O(\e^3\h^{-3\b}
e^{-\fra\p2\h^{-1/2}})=\cr
&=\lis\G_{11}^{(1)}\lis \G^{(2)}_{22}+ O(\e^3\h^{-3\b}
e^{-\fra\p2\h^{-1/2}})=\det D\cr}\Eq(7.19)$$
%
by the argument leading to \equ(6.12), \equ(6.14):
%and having taken the
%parameter $a$ in \equ(7.1) equal to $\fra12$ (not to be confused with
%the $a=M_{11}$ in \equ(7.18))
this completes the analysis of the
remarkable cancellations for separatrices splitting in the
anisochronous case. The leading order remains {\it exactly the same}
as in the isochronous case: anisochrony only alters the final result
by a factor of order $(1+O(\h^a))$, as it should have been expected \ap
once understood the isochronous case.
\*

The proof of the domination of the first order now follows the same
path as the corresponding in \S8 of [G3], see Appendix R7: one uses
the above results to treat the first $\h^{-1/2}N^{-1}$ orders of
perturbation theory and for the remainder one just uses that the
series for the splitting is convergent (see also \S 6 and Appendix
A2).

\vskip1.truecm

\0{\bf\S8. Heteroclinic chains.}
\numsec=8\numfor=1
\*
For completeness we give the argument for the existence of
heteroclinic chains, following [CG], {\it in the easy case} of
isochronous systems.  Below we imagine to have fixed $\m$ and to take
$\h\to0$ (so that $\e=O(\h^c)$).

It is worth noting that {\it no gaps} (\ie {\it all actions} $\AA$ are
the average actions of an invariant torus) are present in the
isochronous case \equ(2.1), which is, therefore, very similar to the
original example proposed by Arnol'd (also gapless).

Let $A_0=0< A_1<\ldots< A_\NN=\lis A_0$ and choose correspondingly
$B_0,\ldots, B_\NN$ so that the sequence of action variables
$(A_j,B_j)$ describes the time averaged location of invariant tori for
\equ(2.1) with energy $0$ (say).

We consider a perturbation like \equ(5.9) for which the splitting is
given by {\it Mel'nikov integral} $\s=O(\h^{-b}e^{-\fra\p2\h^{-1/2}})$
for some $b>0$.  Since there are no gaps the sequence
$A_i$ can be chosen so that $A_{i+1}-A_i< e^{-\d\fra\p2 \h^{-1/2}}$ for a
prefixed $\d>1$ and for all $i$'s. Hence the number $\NN$ has size
$O(\lis A_0 e^{+\d\fra\p2\h^{-1/2}})$.

We want to show that there are heteroclinic intersections $H_i$ between
the unstable manifold of the torus $\V A_i$ and the stable manifold of
$\V A_{i+1}$. Since by construction the tori have the same energy this
simply means finding a solution for the equations: $\V X^-_\su(\p,\aa; \V
A_i)-\V  X^+_\su(\p,\aa; \V A_{i+1})=\V 0$ (the energy being equal, this
equality then implies $X^-_+(\p,\aa; \V A_i)- X^+_+(\p,\aa; \V
A_{i+1})=0$, \ie also the pendulum momenta match).

The tori equations depend linearly on their average actions, \ie
$\V X^\pm_\su(\p,\aa,\AA)\defi \AA+ \V Y^\pm_\su(\aa)$ (see theorem 1)
where $Y_\su^\pm$ is defined here. We can regard the equation for the
heteroclinc intersection $\V X^-_\su(\p,\aa; \V A_i)- \V
X^+_\su(\p,\aa; \V A_{i+1})=\V 0$ as an implicit function problem
which for $\V A_{i+1}=\V A_i$ has $\aa=\V0$ as a solution.

The linearization of the equation at $\V A_i$ involves the
intersection matrix $D$ at $\V A_i,\aa=\V0$ (which in the isochronous
case is $\AA$--independent):
\ifnum\mgnf=1\pagina\fi

$$ D\,\aa=\AA_i-\AA_{i+1}\Eq(8.1)$$
%
showing that the implicit functions problem of determining the
heteroclinic point $\aa$ can be solved for $\h$ small enough because
$\det D= O(\h^{-b} e^{-\fra\p2\h^{-1/2}}) $ and $|\V
A_{i+1}-\V A_i|=O(e^{-\d\fra\p2\h^{-1/2}})$, with $\d>1$.

It might be surprising, at first, that the equation for $\aa$ can be
solved {\it without an explicit estimate of the $\aa$--derivatives} of
the $\V Y_\su(\aa)$ at points $\aa$ near $\V0$. Such estimates can be
made directly from the existence theorem: however they give bounds on
derivatives values that are {\it much larger} than $\s$,
\ie they have size $O(1)$.
%for some $\b>0$.
This may seem to
undermine the foundations of the implicit functions methods, that rely
on the solubility of the linear equation.

However the corrections to \equ(8.1) are bounded by
$O(\aa^2)$; and $|\AA_{i+1}-\AA_i|\le
e^{-\d\fra\p2\h^{-1/2}}$. The solution of the linear equation
\equ(8.1) has size $O(\h^{b} e^{-\d\fra\p2\h^{-1/2}}\cdot
e^{+\fra\p2\h^{-1/2}})$. Hence near such $\aa$ the higher order
corrections have roughly still size $O(e^{-(\d-1)\p\h^{-1/2}})$: much
smaller than the linear contribution, if $\d>1$.
This shows that our knowledge of
the smoothness of $X$ {\it suffices}, together with the basic estimate
on the homoclinic angles, to deduce that the linear approximation
dominates and to claim that the solutions for the heteroclinic point
do exist and are very close to those of \equ(8.1).

Therefore there is a chain of heteroclinic points $H_0, H_1,\ldots,
H_{\NN-1}$ ``connecting'' a neighborhood of $\AA_0$ to one of
$\AA_\NN$. The ``length'' of the chain is
$\NN=O(e^{+\d\fra\p2\h^{-1/2}})$, \ie in some sense it is the inverse
of the splitting.
\*
In the more interesting anisochronous case \equ(7.1) there are
gaps (\ie not all actions are the average positions of
an invariant torus), but one can show that the average actions fill
action space within a distance much smaller than
the splitting size, see [GGM2]. So one can prove immediately
the existence of heteroclinic chains.
\*
We now consider the case in which the system in \equ(2.1)
is further perturbed by a {\it monochromatic} perturbation $\b
f_0(\l,\f)$. The radius of convergence of the whiskers series in
$\b,\m$ (recall that $\e=\m \h^c$) can be shown to have size of order
$|\b|<O(\h^{-\fra12})$, $|\m|< O(1)$, see Appendix A10 of [CG].

The splitting $\det D$ is analytic in $\b,\m$ for $\b\ll \h^{-1/2}$,
see Appendix A10 in [CG], and it is different from $0$ for $|\b|\ll
O(\h^c), \m\ne0$, provided the generically non zero splitting at
$\b=0$ is not zero. In the latter case the splitting can only vanish
finitely many times, at $\m$ fixed, in the domain of convergence
of the series if $\m\ne0$.

Hence for all values of $\b$ close to $1$ (including $\b=1$
generically in $f_0$), there exist heteroclinic chains as long as we
wish. However not having an estimate for the size of the splitting
{\it we cannot infer from the above argument how many tori build the
chains}.

The remark is interesting in view of the general fact, [A2], that
heteroclinic chains imply diffusion in phase space, \ie existence of
motions starting near the torus at one end of the chain and reaching
the one at the other hand in due (finite) time; see [G5] for the
discussion of the method of proof in [CG].

\vskip1.truecm

\0{\bf\S9. Other results. Comments.}
\numsec=9\numfor=1\*

\0{\it(A) General theory of splitting}.
\*
We call {\it Diophantine} with Diophantine constants $C_0,\t>0$ a vector
$\oo\in R^\ell$ such that:

$$|\oo\cdot\nn|> {C_0^{-1}|\nn|^{-\t}},\qquad
\forall\nn\in Z , \; \nn\ne\V0\Eq(9.1)$$
%
(compare with \equ(2.2); here there is no extra parameter $\h$).  We
use the notations of \S2 for the other symbols that are not redefined.
\*
%\pagina
\0{\it {\bf Theorem 3.} Suppose that $\oo\in R^{\ell-1}$ is
Diophantine and consider the Hamiltonian:

$$H=\fra{\AA^2}{2J_1} + \oo\cdot\AA+\fra{I^2}{2J_0}+ g^2 J_0(\cos\f
-1)+\m f(\aa,\f)\Eq(9.2)$$
%
with $(\AA,\aa)$, $(I,\f)$ being $\ell$ pairs of canonically
conjugate action--angle variables.  Let $f$ be a even trigonometric
polynomial of degree $N$ and, for simplicity, $J_0\le J_1$.  Then:

(1) The separatrix splitting, for the torus with rotation
vector $\oo$ into which the unperturbed torus $\AA=\V0$ evolves with
$\m$, is analytic in $\m$ near $\m=0$.

(2) The power series expansion of the splitting vector
$\V Q(\aa)$ in powers
of $\m$ has coefficients with $0$--average; their Fourier components
$\V Q^{k}_\nn$ are bounded, for any $\d<1$, by:

$$|\V Q^{k}_\nn|\le \cases {J_0 g D \d^{-\b}(B\d^{-\b})^{k-1} k!^p
\e(k,\nn) \cr
J_0 g D \h^{-\b}(B\h^{-\b})^{k-1}\cr}\qquad \nn\ne\V0\Eq(9.3)$$
%
where $D,B $ are suitable constants and $\b,p$ can be taken
$\b=4(N+1)$, $p=4\t+4$, if $\t$ is the Diophantine constant of $\oo$, and:
%
$$\e(k,\nn)=\max_{h\le k;\, \{\nn_j\}_{j=1,\ldots,h}\atop
\V0\ne \nn'=\sum_{j=1}^h \nn_j ;\,|\nn'|\le|\nn|}
\Big(\big[\prod_{j=1}^h|f_{\nn_j}|\big]
e^{-g^{-1}|\nn'\cdot\oo|(\fra\p{2}-\d)}\Big)\Eq(9.4)$$}

\0{\it Remarks:} (1) This theorem is the main result of [G3];
note that \equ(9.4) is stronger than the form in which it is quoted in
eq. (6) of [RW] which refers to the theorem stated in [G3] but not to
its proof (which gives in fact \equ(9.4)), see Appendix R8. The method
of proof in [G3] could yield in fact the result for $f$ depending
analytically on the rotators angles $\aa$, by using the ideas in [GM],
but extra work in necessary. See [DGJS],[RW] for alternative proofs;
see also [BCG], where the stronger form (with $f$ analytic) was
derived, in a similar problem.

(2) There are many instances in which the first order expression
(called {\it Mel'nikov integral}) of the splitting vector $\V
Q^{1}(\V \a)$ gives in fact the {\it leading behavior} (as
$\m\to0$) in the calculation of the splitting. In the case of fixed
$\oo$, \ie for the {\it one time scale problem}, this follows from the
classical results of Mel'nikov, [Me].

(3) Another interesting question arises when $\oo=\g\oo_0$ with
$\oo_0$ Diophantine and $\g$ a parameter that we let to $\io$: this is
a {\it two time scales} problem. In the case $\ell=2$ (hence $\oo_0$
is a constant $\o_0$) and with $f$ a trigonometric polynomial the
above theorem proves that the splitting is (generically)
$O(e^{-\fra\p2 g^{-1}\o_0\g})$: in fact this result was the main purpose of
the theory in [G3] (see \S8 in [G3] and, in particular, (8.6) and the
related discussion). It should be stressed that the latter
reference simply provides a new proof of a result already
obtained, in a slightly different case, by [HMS] or, in the same case,
by [Gl], [GLT], LST]. The interest of [G3] lies in the technique.

Mel'nikov's ``approximation'', \ie the dominance of the first order
value of the splitting, is more delicate if $\ell \ge3$. The
techniques of [G3] are inadequate to deal with this case and they only
show that the splitting is smaller than any power of $\g^{-1}$ while
the first order value is $O(e^{c\g})$ for some (computable) $c>0$: in
the case $\ell=3$ this has been studied in [DGJS], [RW], where the
\equ(9.2) is improved by replacing $\e(k,\nn)$ by the much better
$e^{-\fra\p2g^{-1}\g|\oo_0\cdot\nn|}$. Several examples of first order
dominance are provided in the latter references: see however Appendix
R9.  However all examples are constructed with $f$ analytic: it would
be nice to find a model with a trigonometrical polynomial $f$ for
which the first order theory gives the asymptotic result; see the
recent work [G6] for a precise general conjecture on the result.
\*
\0{\it (B) Three time scales. Anisochrony strength. Homoclinic scattering.}
\*
The three time scales condition for the first order dominance
(Mel'nikov integral) includes the case $a=0$ which is in fact a $2$
time scales problem: denoting always $\h^{-1/2}$ the fast velocity
scale, from the analysis of \S7 we see that the slow scale could be
$\h^a$ with $a\ge0$.

This means that the above theory provides a class of models in which
Mel'nikov integral gives the exact asymptotics as $\h\to0$ {\it and
the perturbation is a trigonometric polynomial}, of which \equ(5.10)
is a concrete example. This does not seem to contradict the results of
[DGJS] and [RW] who show that Mel'nikov integral does not necessarily
give the leading asymptotics as $\h\to0$ in cases corresponding to
their $n=3,s=2$.

In the only almost overlapping case $a=0$, however, the above question
is not treated in [DGJS] and [RW] (they consider the {\it very}
different case $n=3,s=2$ in which $a=-\fra12$, \ie two fast rotators
and a pendulum). This illustrates also that there are several ``$2$
time scales problems'', depending on which pair among the three time
scales is identified.

The value $a=0$ is a case considered with other techniques in the
paper [RW] (it corresponds to their $n=3,s=1$): there the attention is
dedicated to a wider question (namely the leading order of the
splitting {\it everywhere} on the section $\f=\p$ rather than just at
the homoclinic point), see Appendix R9. Our asymptotic result is
consistent with their theorem 2.1. We also get the complete
asymptotics in the case of trigonometric polynomial perturbations: but
they do not seem interested in this point and deal only with other
cases ($n=3,s=2$ and non trigonometric peturbations); their technique
seems to apply to our (special) case $a=0$ as well (in fact a simpler
case).

The case $a>0$ is not considered in [RW] except, perhaps, for a remark
at the end of the abstract and following eq.(15): we do not know
whether this case, that is explicitly excluded in the paper, can be
treated with their techniques. In the end the main difference between
our work and that of [DGJS], [RW] might just lie in the technique, see
Appendix R9: we have shown that the work in [G3] provides all the
necessary technical tools for the analysis of the splitting and even
leads to an ``exact'' expression for it. It is however limited to the
splitting at the homoclinic point $\f=\p,\aa=\V0$ (unlike [G3] and
[RW], where the splitting is measured at $\aa$ arbitrary, on the
section $\f=\p$).

The work [RW] is the last in a series of papers (like [DGJS])
which are inextricably linked with each other (and with [G3], [BCG]). The
above comments therefore are easily presented in connection with [RW]:
but we are aware of the role of the other papers quoted in [RW].

Fixing $a=\fra12$ the anisochrony coefficient (of $A^2$) in \equ(7.1)
is $\h^\b$ with $\b=2a$. The value $\b>a$ is necessary if one
wants that the anisochronous and the isochronous splittings coincide
to leading order as $\h\to0$ (at given rotators velocities): however
the analysis above does not seem yet sharp enough for such an
improvement (\ie taking $\b<1$). Finally the physical interpretation
of the precession problem (\ie diffusion in presence of a double
resonance for a \ap stable system) requires $\b=1, a=\fra12$.

Extensions of the cancellations theory of \S7 to $\ell>3$ seem only a
matter of patience. And they would be interesting as they can be
conceivably used to treat a variety of systems and one should expect
that the results will be quite different when the number of fast
scales exceeds $1$, as shown in the ``maximal case'' in which it is
$\ell-1$ ([DGJS], [RW], see Appendix R9). However a general theory of
\ap stable systems, with a free Hamiltonian without free parameters
and a perturbation consisting of terms of equal order of magnitude
{\it seems to require substantial new ideas}.

In \S10 of [CG] there is also a statement about the {\it homoclinic
scattering}: the techniques of this paper apply to its theory as
well. We have not worked out, however, the corresponding details ({\it
the statement was not used anywhere in [CG]}) and at the moment it is
still an open question for us whether the homoclinic phase shifts are
exponentially small or not at the homoclinic point (as claimed in
[CG] on the basis of the computational error mentioned above).

\vskip1.truecm

\0{\bf Appendix A1. Computation of the pendulum Wronskian matrix.}
\numsec=1\numfor=1\*

The pendulum Hamiltonian: $H=I^2/2J_0+g_0^2J_0(\cos\f-1)$ generates a
separatrix motion $t\to\f^0(t)$ which is exactly computable. One finds,
starting at $\f=\p$ at $t=0$:
%
$$\eqalign{
\sin\f^0(t)/2=&1/\cosh g_0t,\cr
\cos\f^0(t)/2=&\tanh g_0t,\cr}\qquad
\eqalign{
\sin\f^0(t)=&2 \sinh g_0t\,(\cosh g_0t)^{-2}\cr
\cos\f^0(t)=&1-2\,(\cosh g_0t)^{-2}\cr}$$
%
A further elementary discussion of the pendulum quadratures near $E=0$
allows us to find the $E$ derivatives of the separatrix motion and leads
to:
%
$$\eqalign{
&I^0={-2g_0J_0\over{\,\rm cosh\,}g_0t}=-2g_0J_0
\sin{\f^0\over2},\kern.2truecm
\dpr_EI^0=J_0(I^0)^{-1}\Bigl(1+J_0g_0^2(\dpr_E\f^0)\sin\f^0\Bigr)\cr
&\f^0=4{\,\rm arctg\,}e^{-g_0t},\qquad
\dpr_E \f^0={-1\over8g_0^2J_0}\,\bigl(2g_0t+{\rm\, sinh\,}2g_0t\bigr)
\sin{\f^0\over2}\cr}\Eqa(A1.1)$$
%
exhibiting the analyticity properties in the complex $t$ plane that are
useful in discussing the size of the homoclinic angles. The \equ(A1.1)
allows us to compute the Wronskian matrix of the above separatrices, \ie
the solution of the pendulum equation, namely $\dot\f=\fra{I}{J_0}$,
$\dot I=J_0 g^2\sin\f$, linearized on the separatrices:
%
$$\dot{  W}=  L(t)  W,\quad  W(0)=1,\quad
L(t)=\pmatrix{0&J_0^{-1}\cr
J_0g_0^2\cos\f^0(t)&0\cr}\Eqa(A1.2)$$
%
and we get:
%
$$  W(t)=\pmatrix{
\dot\f^0/c_2& \dpr_E\f^0/c_1\cr \dot I^0/c_2&\dpr_EI^0/c_1\cr},\qquad
\matrix{c_1=\dpr_EI^0(0)\cr c_2=\dot\f^0(0)=-2\cr}\Eqa(A1.3)$$
%
where the $E$ derivative is computed by imagining motions close to the
separatrix (which has energy $E=0$) and with the same initial $\f=\p$.
This becomes:
%
$$ W(t)=\pmatrix{
{1\over\cosh g_0t}&
{{\lis w}\over4J_0g_0}\cr
-J_0g_0{\sinh g_0t\over\cosh^2 g_0t}&
(1-{{\lis w}\over4}{\sinh g_0t\over\cosh^2g_0t})\cosh g_0t\cr},
\qquad{\lis w}\={2g_0t+\sinh 2g_0t\over\cosh g_0t}\Eqa(A1.4)$$
%

The theory of the Jacobian elliptic functions shows how to perform a
complete calculation of the functions $R_0,S_0$ in \equ(3.2): see
[CG], Appendix A9.

\vskip1.truecm

\0{\bf Appendix A2. Convergence of the ``form factors'' $\V
\G,\lis\G,M$ and remainders bounds.}
\numsec=2\numfor=1
\*

Integrals for $\V \G,\lis\G,M$, see
\equ(6.3),\equ(6.7),\equ(6.8) and the analogous ones in \S7,
are precisely the object of the analysis of Appendix A1 of [G3].
Hence we adhere closely to it.

Consider any of the form factors, \ie any of the series in \S6 or \S7.
Following [G3], word by word, we obtain a bound on the sum of the
contributions of the values of all trees $\th$ with $m$ nodes and
order $h$, $m\le 2h$, as $D_0 B_0^{h-1} \e^h m!^2
\max_{0<|\nn|\le Nh}|\oo\cdot\nn|^{-4h}$ with $D_0,B_0$ constants
(see (A1.13) and (A1.4) of [G3]: noting that in (A1.4) $m!^2$ is
missing due to a typo). This factorial becomes $h!^4$ in the bound
(8.2) of [G3] (and consequently on the form factors and $\det D$ at
order $h$, which interest us here and which are expressed essentially
by the same integrals) because $m\le 2h$ and it is responsible of the
first value $4$ in the variable called $p$, in [G3], $p=4+4\t$.

We know from theorem 1 above, that $\det D$ is given by a convergent
series but we do not know whether the series for $\V \G, \lis \G$ and
for the matrix $M$ converge. For the ``dressed form factors'' $\V G$
one could in fact show that the series in $\e$ converge for small $\e$
by the method developed in Appendix A1 of [G3] (consisting in
reexpressing the $\V G$ in terms of the unsplit operators $\OO$): but
a lot of work can be saved because as we show now we do not need to
know such convergence properties and the analyticity of $\det D$
suffices.

One can find some improvements over the bounds (8.2) of [G3] by using the
fact that one has a better bound as long as $Nh<\g\h^{-1/2-a}$,
with any prefixed $\g<1$, because it is
only for $Nh\ge\g\h^{-1/2-a}$ that one can find really small divisors: for
$h<\g\h^{-b}N^{-1}$, with $0<b\le 1/2+a$, we can bound $|\oo\cdot\nn|$
below by $\h^{a}$, if $\h$ is small enough. Hence for such values of
$h$ we have the bound: $\lis D_1 \lis B_1^{h-1} (\e\h^{-4a})^hh!^{4}$,
with $\lis D_1,\lis B_1$ constants and for
$h<N^{-1}\h^{-b}\g$. This bound holds for all form factors and also
for $\det D$ (note that it does not imply any convergence).

On the other hand we know that $\det D$ is analytic in $\e$ for
$\e<\h^c$ and $c$ large enough. Hence, to order $h$, $\det
D$ (but {\it a priori not} individual form factors) can be
bounded by:

$$D_2 B_2^{h-1} (\e\h^{-\lis c})^h, \qquad {\rm for\ all}\ h\ge1
\Eqa(A2.1)$$
%
for some $\lis c >0$.  It was conjectured in [G3], see Appendix A1,
that this bound could be obtained directly from the graphical
expansions.  This has been proved in [Ge1] (getting $\lis c=4d$ if $d$
is the constant in \equ(2.2) and \equ(7.3));
but we are showing that such stronger result is not needed here.

Finally {\it in the case of} $\G^{(0)}_2,\G^{(2)}_2, \lis
\G_{i2}$, \ie in the case of the {\it ``bare'' or ``analytic form factors'' },
which are expressed as integrals of analytic functions, one can
further improve the bound by the usual $\t_v$--variables integrations
shift to $\Im\t_v=\pm
i(\fra\p2-\h^{1/2})$, choosing the quantity called $d$ in [G3] as
$\h^{1/2}$, a natural but quite arbitrary choice.

One checks directly (as explained in [G3], Appendix A1) that this
simply introduces a factor $\h^{-\b'}$ wth $\b'=2(N+1)$ due to
closeness of the singularities of the  functions
appearing in the Wronskian matrix or of the
$f_\d(\f^0(\t_v),\aa+\oo\t_v)$ (located at the same places because $f$ is
assumed to be a trigonometric polynomial); {\it it introduces also an
exponentially small factor}: $\e_h=\min_{0<|\nn|\le Nh,\,0< |\n_2|}
e^{-(\fra\p2 -\h^{1/2})|\oo\cdot\nn|}$
so that the bound of the $h$--th order, for
such form factors (and hence for $\det D$ by \S7), becomes :
%
$$ D_3 \, h!^4\,B_3^{h-1} (\e\h^{-\b})^h\max_{0<|\nn|\le Nh, 0<|\n_2|}
e^{-(\fra\p2-\h^{1/2})|\oo\cdot\nn|}, \qquad {\rm for\ all}\ h<\g
(N\h^b)^{-1}\=h_0 \Eqa(A2.2) $$
%
with $\b=2(N+1)+2$ and $B_3,D_3$ constants. Note that in order to
deduce this all we really need is that the {\it exact} expressions
\equ(6.12) and \equ(7.10), regarded as formal power series, be true
order by order in the expansion.  We do not really need the (yet unknown)
convergence of the series for the $\V G$'s and for the matrices
$\lis \G$ and $M$, which might be not true.

Since $(\e\h^{-\lis c})^h=(2\e\h^{-\lis c})^h\,2^{-h}$ and for $\h$ small
enough and $h>h_0$ the quantity $2^{-h}$ is $< e^{-\fra\p2
\h^{-1/2}}$, provided $b\ge1/2$, the bounds \equ(A2.1) and
\equ(A2.2) can be combined into the bound for the $h$--th order value
of $\det D$:

$$ D_4 (B_4\e\h^{-c})^h e^{-\fra\p2 \h^{-1/2}}\qquad {\rm for\ all}\
h\ge1\Eqa(A2.3)$$
%
with $c>\max\{2(N+1)+2+4b,\lis c\}$ and $B_4,D_4$ suitable constants.
The bound \equ(A2.3) is an immediate consequence of
\equ(A2.1) for $h>h_0$ and the fact that, as
$|\oo\cdot\nn|\ge \h^{-1/2}- Nh\h^{a}$, for $\n_2\ne0$,
\equ(A2.2) can be used to give
%
$$ \sum_{h=3}^{h_0} D_3 h!^4 B_3^{h-1} (\e\h^{-\b})^h
e^{-\fra\p2 \h^{-1/2}+1+\fra{\pi}2Nh\h^a}
\le D_4 B_4^2  (\e\h^{-\b})^3
e^{-\fra\p2 \h^{-1/2}} \Eqa(A2.4) $$
%
provided that in \equ(A2.2) one has $1/2\le b \le 1/2+a$;
the best choice is $b=1/2$, which gives
$c>\max\{2(N+1)+4,\lis c\}$. For instance, if
$f$ is given by \equ(5.10), supposing also
$\lis c = 4d < 2(N+1)+4$, one finds $c>8$.

\vskip1.truecm

\0{\bf Appendix A3. The commutation relation \equ(6.20)}
\numsec=3\numfor=1\*

Let us denote by $\f\=\f_0(t)$ the motion of the
unperturbed pendulum on the separatrix (see Appendix A1).
Let $\V F=\pmatrix{0\cr F\cr}$ and $\V T=\pmatrix{\OO(F)\cr\dpr_t
\OO(F)\cr}$: the vector $\V T$ verifies $\dot{{\V T}}=
L(t)\V T+\V F$ where $L(t)=\pmatrix{0&1\cr\cos\f&0\cr}$. By
differentiating with respect to $t$ both sides we get $\ddot{{\V T}}
=L(t)\dot{{\V T}}+\dot{{\V F}}+\pmatrix{0&0\cr-\dpr_0^3
f_0&0\cr}\dot\f\, \V T$ which means
%
$$\dpr_{t}\OO(F)=\OO\big(\dpr_{t} F -\dot\f\,\dpr_0^3
f_0\OO(F)\big)\Eqa(A3.1)$$
%
up to a homogeneous solution $t\to W(t) X, \, X\in {\bf R}^2$
of the latter linear equation, when written for the first component;
but the only function $X$ such that $\OO(F)$ and $\dpr_t\OO(F)$
are both bounded uniformly in time is $X=0$.

Choosing $\V T=\pmatrix{\dot\f\cr \ddot\f\cr}$
or $\V T=\pmatrix{\dpr_E\f\cr \dpr_E\dot\f\cr}$
and $F=0$, and proceeding as before,
one gets in the same way the identities
%
$$ \OO(\dot\f^2\sin\f) = -\ddot\f \; , \qquad
\OO(\dot\f[\dpr_E\f]\sin\f) =-\dpr_E\dot\f \Eqa(A3.2) $$
%

Noting that $\OO_0(F)=\OO(F)$ for odd $F$, and that
$\OO_0({\rm sign}\,\t\, F)(t)={\rm sign}\,t\,\OO_0(F)(t)$,
from the above equalities it follows immediately
%
$$ \OO_0 (w_{00}^2\sin\f)=\fra12w_{30} \; , \qquad
\OO_0 (w_{00}|w_{03}|\sin\f) =\fra12|w_{33}| \Eqa(A3.3) $$
%
which are used in \S 6 and \S 7.

By definition of $G_0$ in \equ(5.12) and by the first
of \equ(A3.2), one has
%
$$ \eqalign{
G_0\big(\dpr_{t} F -\dot\f\,\dpr_0^3 f_0\OO_0(F)\big)
& = {1\over2} \int_{+\io}^{-\io} d\t \, w_{00}(\t)
\Big(\dpr_{\t} F - \dot\f\,\dpr_0^3 f_0\OO_0(F)\Big) \cr
& = {1\over4} \int_{+\io}^{-\io} d\t\Big[
F \OO_0(\dot\f^2\sin\f) - \dot\f \dot F \Big] = 0 \cr} \Eqa(A3.4) $$
%
A similar identity is obtained by considering
$G\big(\dpr_{t} F -\dot\f\,\dpr_0^3 f_0\OO_0(F)\big)$
and using the second of \equ(A3.2), with $|\dpr_E\f|$
replacing $\dpr_E\f$ (see comments after \equ(A3.2)).

The oddness of $F$ implies that $\dpr_t\OO_0(F)=\dpr_t\OO(F)$.
Then, as a consequence of \equ(A3.4) and
the analogous relation for $G$, \equ(6.20) follows.

\vskip1.truecm

\0{\bf Appendix A4. Proportionality between $G^{(0)}$ and $G^{(2)}$.}
\numsec=4\numfor=1\*

 From \equ(7.9) one has that, formally:
%
$$ \V G_1=\sum_{k=0}^\io (\s M)^k \V\G_1=
\lim_{N\to\io} \sum_{k=0}^N (\s M)^k \V\G_1 \=
\lim_{N\to\io} \V G_1(N) \Eqa(A4.1) $$
%
so that, by using the relations between the matrix elements of $M$
listed in \equ(7.11), one has
%
$$ \V G_1(N) = \left( \matrix{
M_{01} & M_{11} & M_{12} & \l M_{11} \cr
M_{00} & M_{01} & M_{02} & \l M_{01} - \l \L_0 \cr
\l M_{01} - \l \L_0 & \l M_{11} & \l M_{12} - \l \L_1 & \l^2 M_{11} \cr
M_{02} & M_{12} & M_{22} & \l M_{12}- \l \L_1 \cr} \right)
\V G_1(N-1) + \V \G_1\Eqa(A4.2) $$
%
which gives
%
$$ \eqalign{
G^{(0)}_1(N) & = M_{01} G^{(0)}_1(N-1) + M_{11} G^{(1)}_1(N-1)
\cr & + M_{12} G^{(2)}_1(N-1) + \l M_{11} G^{(3)}_1(N-1) + \G_1^{(0)} \cr
G^{(2)}_1(N) & = \l G^{(0)}_1(N) - \l \L_0 G^{(0)}_1(N-1)
- \l \L_1 G^{(0)}_1(N-1) \cr}\Eqa(A4.3) $$
%
as $\G_1^{(2)}=\l \G_1^{(0)}$ (see the second identity in \equ(7.11)).
Taking the limit $N\to\io$, one obtains the third equality
in \equ(7.11), defining $Z=(1-\L_0)/(1+\l\L_1)$.

\vskip1.truecm

\0{\bf Appendix R. After refereeing comments.}
\numsec=5\numfor=1\*

This appendix contains a few clarifications on the text and on the
comments on [RW], [DGJS], requested by the referee.
\*

\0R1 ({\it Why $\det D$ is a measure of transversality}).
The splitting vector clearly gives the distance between the two
manifolds at corresponding $\V\a$'s (because it is just the difference
between the two). Therefore if the angles are increased by an
infinitesimal amount $d\V\a$ away from a homoclinic point ($\V\a=\V0$
in our case) the vector changes by its derivative $D$ times $d\V\a$:
the derivative is the intersection matrix. Therefore $D d\V\a$ is the
increment of the splitting vector.  Hence this cannot vanish unless
the determinant of $M$ vanishes: hence the determinant measures
transversality.

Other evidence that the determinant measures transversality is that
the square root of its value is a bound on the lowest eigenvalue of
the matrix $M$ which gives the ``minimum transversality'' or the
minimum splitting angle (the arctangents of the ``principal angles''
can be identified with the eigenvalues). This is the second geometrical
interpretation.

That it is a quantity of physical interest also for the theory of
heteroclinic intersections is made clear by the whole content of the
paper. In any event in \S8 we repeat the argument used in [CG]: the
determinant has the mathematical interpretation of Jacobian
determinant for the implicit equation problem that has to be solved
when one looks for heteroclinic intersections (hence for Arnol'd
diffusion, as an example). Further analysis can be found in [GGM2].

We briefly explain here how to find such intersections. The equations
for the stable and unstable manifolds at $\f=\p$ of an invariant torus
to which perturbation theory with $\e<\h^c$ with a suitably large $c$
can be applied have the form:

$$\V A^b(\aa)=\V A'+\V H^b(\V A',\aa),\qquad b=s,u \Eqa(R1.1)$$
%
where $\V A'$ is such that $\oo(\V A')=(\h^a+\h^{2a} A'_1,\h^{-\fra12})$
provided $|\oo(\V A')\cdot\nn| \ge e^{-\h^{-\g}}|\nn|^{-\t}$ for
$\g,\t>0$ and $\g<1$: this is a consequence of a detailed analysis of
the classical proofs (one can take for instance that in [CG], explicitly
and more carefully reworked out for the case of the present models in
[GGM2]).\annota1{\rm Furthermore the remarkable ``twistless'' property
of the models, see [G3], [GGM2], implies that the time average of $\V A$
on the invariant torus is $\V A'$ so that there is a simple relation
between the frequency spectrum of the motion on the invariant torus and
the parameter $\V A'$ in \equ(R1.1): there is ``no twist'' on the
frequencies due to the perturbation (one says that the tori are {\it
``twistless''}, [G3]): the dispersion relation (\ie the relation between
frequencies and average actions) remains {\it the same} in presence of
obsence of perturbation (at least for the non resonant tori that survive
the pertubation).}

The point is that the constant $c$ should be fixed suitably large once
and for all and then, fixed $\g<1$ and $\t$, for $|\e|<\h^c$ and $\h$
small, the function $H^b$ togheter with the invariant tori equations
and the splitting matrix determinant, can be evaluated by perturbation
theory and in particular the splitting determinant will be generically
of order $e^{-O(\h^{-\fra12})}$, [GGM2]. 

The relation between $\oo(\V A')$ and $\V A'$, after \equ(R1.1), says
that the spacing between the $\V A'$ that correspond to invariant tori
is essentially the same as the spacing between the frequencies of the
invariant tori: hence the resonance ``gaps'' around such $\V A'$'s are
of size at most $O(e^{-\h^{-\g}})$ by the ``generosity'' of the
Diophantine condition that we can use.

Hence the splitting is by far greater than the gaps if $\g>\fra12$
(this is of course, once more, the reason why we call our main result
the ``large splitting theorem''). The equation for a heteroclinic
point between (existing) tori with average actions $\V A'_1$ and $\V
A'_2$ is therefore, at $\f=\p$:
%
$$\V A'_1-\V A'_2=-(H^u(\V A'_1,\aa)-H^s(\V A'_2,\aa))\Eqa(R1.2)$$
%
which is an implicit equation for the point location $\aa$ which at the
trivial solution near $\V A'_1$ (namely ($\V A'_2=\V A'_1,\aa=\V0$) has
Jacobian determinant precisely given by the determinant of the splitting
matrix. Hence if $|\V A'_1-\V A'_2|< e^{-\h^{-\g}}$ with $1>\g>\fra12$
we see that, since $|\V A'_1-\V A'_2|$ is far smaller than the value of the
Jacobian determinant, the equation \equ(R1.2) has a solution
$\aa_{1,2}$. See [GGM2] for the proofs of the large density of the
invariant tori.
\*

\0R2 ({\it Why the result remains valid if the analysis is
marginal}). We use the word in a technical sense: this means that the
series that we study does not necessarily provide a good approximation
if truncated at a fixed order. To each order one gets a splitting
value that cannot be considered correct (although in the end it turns
out to be so) until an analysis to all orders is performed: a typical
situation that arises in problems in QFT for the expansions that
involve running couplings of marginal operators. Of course the purpose
of this paper is precisely to perform the analysis to all orders with
all rigor and clarity possible and the fact that the analysis is
called ``marginal'' is not meant at all to diminsh the result but to
stress that there is work to do.
\*
\0R3 ({\it What is the import of the reference to Appendix 9 of
[CG]\/}). Simply to exhibit the form of functions $R_0,S_0$; the
nontriviality refers to the fact that this work (of Jacobi) founded
the theory of elliptic functions which we consider non trivial even
though it deals with a one degree of freedom pendulum.
\*
\0R4 ({\it Theorem 1 is nowhere proved}).
A proof did not seem necessary to us, as many other consequences of
KAM theory that are surprisingly appearing and being published even on
prestigious journals, while deserving a place only as lecture notes or
as chapters in monographies or books.  In fact a proof can be found
{\it for instance} in the paper [G5] (p.2 and 3 plus half of p. 10),
which is in our opinion a trivial adaptation of the proofs in \S5 of
[CG] which deals with the harder anisochronous case. Further
developments on the subject can be found in [GGM2].
\*
\0R5. ({\it Is \equ(6.4) in [G3] and where?}). It is there and, in
any event, it does not require any proof because it simply expresses
in a formula what is discussed in words in \S6, A, of [G3]. One
considers the tree expansion and replaces in it (6.1) of [G3] with
(6.2) and (6.4) with (6.5), obtaining the result explained in the first
sentence of p. 375 of [G3] whose translation in formulae is \equ(6.5).
%(hence of \equ(6.4).
It is worth stressing that the quoted sentence
is one of the main results in [G3], whose contents are assumed
throughout the present work. The comment on the graphical meaning is
by no means {\it meant as a proof}, but as a further suggestion to the
reader on how to interpret it (it could, however, also be regarded as
a proof if [G3] is assumed).  Since Feynman, at least, graphs are
concise ways to write (and even derive) involved formulae and to
perform algebraic operations with them; all the results we present can
be conveniently interpreted graphically, and yet {\it perfectly
rigorously}.
\*
\0R6. ({\it What are the ``omitted terms'' in \equ(6.12)?}). They are a
few terms whose properties are described in words and whose expression
can be very easily derived by patiently evaluating the determinant of
the $2\times2$ matrix in \equ(6.10). That their contribution is
trivial for our purposes is immediate and the difficulty, the whole
difficulty, lies in bounding the second terms of \equ(6.12). The
reference to Appendix A2 is not meant as a proof of the properties of
the omitted terms, but rather as a place where it is discussed why the
terms omitted (which are integrals of functions which are analytic and
contain ``fast angle derivatives'') will provide neglegible
contributions to the leading asymptotics. %, stated.

In any event we write here the complete form of the determinant:
%\ifnum\mgnf=1\pagina\fi
$$ \eqalignno{
\det D = & \lis\G_{11}^{(1)}\lis\G_{22}^{(1)}\e^2
-2\big( \lis\G_{11} M_{11} - 2 (\G_1^{(0)})^2 \big)
(\G_2^{(1)})^2 \det C +
\cr &
+ \Big( \sum_{k_1+k_2\ge 3}
\lis\G_{11}^{(k_1)}\lis\G_{22}^{(k_2)} \e^{k_1+k_2} \Big) +
& \eqa(R6.1)\cr &
+ \Big[ - \lis\G_{12}^2-
4\lis\G_{12}(\V\G_1, C \V\G_2)+
2\lis\G_{22}(\V\G_1, C \V\G_1) \Big]+
\cr &
+ 4 \Big\{
(\G_2^{(0)})^2 (\G_1^{(1)})^2
- 2 (\G_1^{(0)}\G_1^{(1)}) (\G_2^{(0)}\G_2^{(1)}) \Big\} \det C
\; , \cr}$$
%
from which \equ(6.12) follows; in particular the terms in the last line
of \equ(R6.1) are the terms referred to as ``a few others'' after \equ(6.12).
They are just two terms.

That the above equation really follows by performing the calculation
of the determinant can be checked (explicitly) as follows. We compute
the determinant of the matrix \equ(6.10)
%
$$ D=\pmatrix{\lis\G_{11} + 2(\V\G_1, C \V\G_1)& \lis\G_{12}+
2(\V\G_2, C \V\G_1)\cr
\lis \G_{21}+2(\V\G_1, C \V\G_2)&\lis\G_{22}+
2(\V\G_2, C \V\G_2)\cr} \;  ,\Eqa(R6.2) $$
%
where
%
$$ C=\s(1-\s M)^{-1} \; , \Eqa(R6.3) $$
%
with $C,M,\lis\G$ symmetric. One has
%
$$ \eqalign{
\det D & = \lis\G_{11}\lis\G_{22} - \lis\G_{12}^2+
2\lis\G_{11}(\V\G_2, C \V\G_2)+
2\lis\G_{22}(\V\G_1, C \V\G_1) \cr
& + 4(\V\G_1, C \V\G_1)(\V\G_2, C \V\G_2)-
4\lis\G_{12}(\V\G_1, C \V\G_2)-
4(\V\G_1, C \V\G_2)^2 \; . \cr} \Eqa(R6.4) $$
%
By taking into account the fact that all contributions to \equ(R6.4)
containing factors $\lis\G_{12}$, $\lis\G_{22}$ and $\G_2^{(0)}$ are
trivially either exponentially small in $\o$ and of order $3$ at least
in $\e$ or they are ``exponentially smaller'' (like $\lis\G_{12}^2$)
than the first term in the r.h.s (which is of order $\e^2$ unless it
vanishes, which is not the case, generically), one has that the only
not {\it a priori} exponentially small terms in \equ(R6.4) are
%
$$ 2\lis\G_{11} (\G_2^{(1)})^2 C_{11} +
4(\V\G_1, C \V\G_1)\,(\G_2^{(1)})^2 C_{11}
- 4(\G_1^{(0)}C_{01}+\G_2^{(1)}C_{11} )^2 (\G_2^{(1)})^2 \; , \Eqa(R6.5) $$
%
where
%
$$ \eqalign{
& (\V\G_1, C \V\G_1) C_{11} = (\G_1^{(0)})^2C_{00}C_{11}
+ 2(\G_1^{(0)}\G_1^{(1)})C_{01}C_{11} + (\G_1^{(1)})^2 C_{11}^2 \; , \cr
& (\G_1^{(0)}C_{01}+\G_2^{(1)}C_{11} )^2 =
(\G_1^{(0)})^2 C_{01}^2 + 2(\G_1^{(0)}\G_1^{(1)})C_{01}C_{11} +
(\G_1^{(1)})^2 C_{11}^2 \; , \cr} \Eqa(R6.6) $$
%
so that \equ(R6.5) becomes
%
$$ (\G_2^{(1)})^2 \Big\{ 2\lis\G_{11} C_{11} +
4(\G_1^{(0)})^2 C_{00}C_{11}
- 4 (\G_1^{(0)})^2 C_{01}^2 \Big\}  \; . \Eqa(R6.7) $$
%
Then, by using also that,
%
$$ \det C = C_{00}C_{11}-C_{01}^2 \; , \qquad
C_{11} = - M_{11} \det C \; , \Eqa(R6.8) $$
%
one can write \equ(R6.7) as
%
$$ (\G_2^{(1)})^2 \Big\{ - 2\lis\G_{11} M_{11} +
4(\G_1^{(0)})^2 \Big\} \det C \; . \Eqa(R6.9) $$
%
so that, if we note also that the only terms
to second order in $\e$ are $\lis\G_{11}\lis\G_{22}$
and $\lis\G_{12}^2$ (and that the latter decays
twice faster than the first one), \equ(6.12) of [GGM] follows.
\*

The {\it skeptical reader} can find useful to check that all the other terms
(neglected so far) are really exponetially small, \ie they have the
properties claimed in \S6.

To see this, let us rewrite \equ(R6.4) as
%
$$ \det D = \lis\G_{11}\lis\G_{22} - \lis\G_{12}^2-
4\lis\G_{12}(\V\G_1, C \V\G_2)+
2\lis\G_{22}(\V\G_1, C \V\G_1) + \GG \; , \Eqa(R6.10) $$
%
where
%
$$ \GG \= 2\lis\G_{11}(\V\G_2, C \V\G_2)
+ 4(\V\G_1, C \V\G_1)(\V\G_2, C \V\G_2)-
4(\V\G_1, C \V\G_2)^2 \; . \Eqa(R6.11) $$
%
By performing explicitly the calculations one has
%
$$ \eqalign{
\GG = & 2\lis\G_{11} \Big\{ (\G_2^{(0)})^2 C_{00} +
(\G_2^{(1)})^2 C_{11} + 2(\G_2^{(0)}\G_2^{(1)}) C_{01} \Big\} \cr
& + 4 \Big\{
(\G_1^{(0)})^2 (\G_2^{(0)})^2 C_{00}^2
+ 4 (\G_1^{(0)}\G_1^{(1)}) (\G_2^{(0)}\G_2^{(1)}) C_{01}^2
+ (\G_1^{(1)})^2 (\G_2^{(1)})^2 C_{11}^2 \cr
& \qquad + 2\big[
(\G_1^{(0)})^2 (\G_2^{(0)}\G_2^{(1)})
+ (\G_2^{(0)})^2 (\G_1^{(0)}\G_1^{(1)}) \big] C_{00} C_{01}
\cr & \qquad
+ \big[
(\G_1^{(0)})^2 (\G_2^{(1)})^2
+ (\G_2^{(0)})(\G_1^{(1)}) \big] C_{00} C_{11} \cr
& \qquad + 2\big[
(\G_1^{(0)}\G_1^{(1)}) (\G_2^{(1)})^2
+ (\G_2^{(0)}\G_2^{(1)}) (\G_1^{(1)})^2 \big] C_{01} C_{11}
\Big\} \cr
& - 4 \Big\{
(\G_2^{(0)})^2 (\G_1^{(0)})^2 C_{00}^2
\cr & \qquad
+ \big[ (\G_2^{(0)})^2 (\G_1^{(1)})^2
+ 2(\G_2^{(0)}\G_2^{(1)}) (\G_1^{(0)}\G_1^{(1)})
+ (\G_2^{(1)})^2 (\G_0^{(1)})^2 \big] C_{01}^2
\cr & \qquad
+ (\G_2^{(1)})^2 (\G_1^{(1)})^2 C_{11}^2 \cr
& \qquad + 2\big[
(\G_1^{(0)})^2 (\G_2^{(0)}\G_2^{(1)})
+ (\G_2^{(0)})^2 (\G_1^{(0)}\G_1^{(1)}) \big] C_{00} C_{01}
\cr & \qquad
+ 2 \big[
(\G_2^{(0)}\G_2^{(1)}) (\G_1^{(0)}\G_1^{(1)}) \big]  C_{00} C_{11} \cr
& \qquad + 2\big[
(\G_1^{(0)}\G_1^{(1)}) (\G_2^{(1)})^2
+ (\G_2^{(0)}\G_2^{(1)}) (\G_1^{(1)})^2 \big] C_{01} C_{11}
\Big\} \; , \cr} \Eqa(R6.12) $$
%
which, by exploiting the fact that some terms cancel between
each other, becomes
%
$$ \eqalign{
\GG = & 2\lis\G_{11} \big\{ (\G_2^{(0)})^2 C_{00} +
(\G_2^{(1)})^2 C_{11} + 2(\G_2^{(0)}\G_2^{(1)}) C_{01} \Big\} \cr
& + 4 \Big\{ \Big[
2 (\G_1^{(0)}\G_1^{(1)}) (\G_2^{(0)}\G_2^{(1)}) -
(\G_2^{(0)})^2 (\G_1^{(1)})^2 - (\G_2^{(1)})^2 (\G_1^{(0)})^2
\Big] C_{01}^2 \cr
& \qquad - \Big[
2 (\G_1^{(0)}\G_1^{(1)}) (\G_2^{(0)}\G_2^{(1)}) -
(\G_2^{(0)})^2 (\G_1^{(1)})^2 - (\G_2^{(1)})^2 (\G_1^{(0)})^2
\Big] C_{00} C_{01} \Big\} \; . \cr} \Eqa(R6.13) $$
%

The two expressions in square brackets
in \equ(R6.13) are equal and (parenthetically) are the same as
%
$$ \Big[ \G_2^{(0)}\G_1^{(1)} - \G_2^{(1)}\G_1^{(0)} \Big]^2 \;,\Eqa(R6.14) $$
%
so that, by using also \equ(R6.8), one can write
%
$$ \eqalign{
\GG = & 2\lis\G_{11} \Big\{ (\G_2^{(0)})^2 C_{00} +
2(\G_2^{(0)}\G_2^{(1)}) C_{01} \Big\} \cr
& + 4 \Big\{
(\G_2^{(0)})^2 (\G_1^{(1)})^2
- 2 (\G_1^{(0)}\G_1^{(1)}) (\G_2^{(0)}\G_2^{(1)}) \Big\} \det C \cr
& \qquad -2\big( \lis\G_{11} M_{11} - 2 (\G_1^{(0)})^2 \big)
(\G_2^{(1)})^2 \det C \; . \cr} \Eqa(R6.15) $$
%
So one can conclude, combining \equ(R6.10) and \equ(R6.15), that the
determinant of the matxix \equ(R6.2) can be writen as \equ(R6.1) from
which \equ(6.12) follows.
\*

\0R7. ({\it Where is the proof of domination?}). As we say the proof is
in Appendix A2: the reference to [G3] is here only to say that the same
proof (\ie combining the two estimates for the $k$-th order term and
optimizing) appears there for the first time.
\*

\0R8. ({\it Is the remark 1 after \equ(9.4) a reproach to other
authors?}). This statement is by no means intended as a critique to
the authors of [RW] for not having read the proof of the result that
they quote. This is not what we mean here: it just seemed to us the
right place to point out that [G3] gave a more general result which,
while uninteresting for the purposes of [G3], was worth pointing
out. In fact it was exploited in [BCG] and, recently, in [GGM3]. It
is also true that reading the proof of the ``quasi flat'' bounds of
[G3] would not only have shown the validity of \equ(9.4), but also
would have shown why the result claimed in [RW] could not be right,
see [GGM4]: the work [GGM3] gives further improvements of \equ(9.4)
useful to derive results of the type considered in [RW].

\*

\0R9 ({\it Relation of the present results with [DGJS] and [RW]}).
The numerous discussions intervened since the present work appeared as
a preprint have considerably clarified the relations between the above
papers and ours. It is therefore useful to give our present view.

The results in [RW] are not correct as there are serious flaws in the
proof, [GGM4].

The work [DGJS] deals only with the isochronous case and, more
important, it uses a {\it very} different definition of splitting.

A definition that if used instead of the one in [CG] {\it would make
the error there disappear\/}! They do not study the determinant of the
intersection matrix but only the difference of the values of a certain
observable called ``the analytic integral'', closely related to the
energy of the free pendulum, evaluated on the two manifolds. This is
interesting (and it seems related to preexisting numerical
experiments) but it is different from the problems studied in [G3]
(and [CG]). This cannot be considered as a measure of the splitting,
see Appendix R1, in our sense.

On the other hand the results in [G3] can be seen to imply, in some
cases, already essentially all the results of [DGJS] and {\it
furthermore} bounds on the splitting determinant in the isochronous
{\it and} anisochronous cases. Essentially means that the results will
follow for even interactions which are polynomials of very high degree
and with non zero coefficients verifying the conditions of [DGJS] and
replacing the ``analytic integral'' with the energy of the free
pendulum (the relation between the two is also quite simple). This is
explained in [GGM3].

Furthermore the results in [RW] and [DGJS] deal with the splitting on
sections that {\it are not} the one we consider, namely $\f=\p$. This
is for us an important further difference as we are interested to show
existence of heteroclinic chains with the method of [CG] which deals
with the section at $\f=\p$; see [GGM5]. In fact an application of the
present work to heteroclinic chains and Arnol'd diffusion is analyzed
in [GGM2], and described above in Appendix R1: we do not see that the
results of [DGJS], in the isochronous case not to mention the
anisochronous ones that they do not treat, suffice to prove the
existence of heteroclininc chains as discussed in Appendix R1 above
and in [GGM2].
\*
\0{\bf Acknowledgments.} We are indebted to P. Lochak for many
discussions and for encouraging one of us to revise the previous work
[CG] in order to present a simplified version.  One of us (GiG) is
deeply indebted to V. Gelfreich for pointing out, in a meeting
organized and led by P. Lochak, the error in [CG] that is corrected in
the present paper.  We also thank C. Sim\'o and A. Jorba for
comments on the manuscript.  This work is part of the research program
of the European Network on: ``Stability and Universality in Classical
Mechanics", \# ERBCHRXCT940460.
\*

%\vskip 2cm
{\bf References}
\*
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\item{[A2] } Arnol'd, V.I.:
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\item{[ACKR] } Amick, C., Ching, E.S.C., Kadanoff, L.P., Rom--Kedar, V.:
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\item{[DGJS]} Delshams, S., Gelfreich, V.G., Jorba, A., Seara, T.M.:
{\it Exponentially small splitting of separatrices under fast
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\item{[E] } Eliasson, L.H.:
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\*\*
\ifnum\mgnf=1\pagina\fi
\FINE

\*
\0Addresses:\\
\0G.Ga.: Dipartimento di Fisica, Universit\`a di Roma 1, P.le Moro 2,
00185, Italy\\
\0G.Ge.: Dipartimento di Matematica, Universit\`a di Roma 3, Largo
S. Leonardo Murialdo 1, 00146, Roma, Italy\\
\0V.Ma.: Dipartimento di Matematica, Universit\`a di Roma 2, Viale
Ricerca Scientifica, 00133, Roma, Italy

Paper archived  in\\
mp$\_$arc@math.utexas.edu 97-472 and\\
chao-dyn@xyz.lanl.gov 9709004

\ciao


