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\begin{document}

\title[Overcoming the large gap problem]{A geometric mechanism for 
diffusion in Hamiltonian systems
       overcoming the large gap problem: Announcement of results}

\author{Amadeu Delshams}
\address{Departament de Matem\`{a}tica Aplicada I,
           Universitat Polit\`{e}cnica de Cata\-lunya,
           Diagonal 647, 08028 Barcelona, Spain
}
\email{Amadeu.Delshams@upc.es}

\author{Rafael de la Llave}
\address{Department of Mathematics,
           University of Texas,
           Austin, TX 78712-1802}
\email{llave@math.utexas.edu}

\author{Tere M. Seara}
\address{Departament de Matem\`{a}tica Aplicada I,
           Universitat Polit\`{e}cnica de Cata\-lunya,
           Diagonal 647, 08028 Barcelona, Spain
}
\email{tere.m-seara@upc.es}


\date{March 9, 2003 and, in revised form, September 19, 2003}


\commby{Svetlana Katok}

\subjclass[2000]{Primary 37J40; Secondary 70H08, 37D10, 70K70}

\keywords{Nearly integrable Hamiltonian systems, normal forms, slow variables,
normally hyperbolic invariant manifolds, KAM theory, Arnold diffusion}

\begin{abstract}
We present a geometric mechanism for diffusion in
Hamiltonian systems.
 We also present tools that allow us to verify it
in a concrete model.  In particular, we verify it in
a system  which
presents the \emph{large gap problem}.
\end{abstract}

\maketitle

\section{Background and preliminaries}
About 40 years ago, the paper \cite{Arnold64} introduced the remarkable
Hamiltonian
\begin{equation}  \label{eq:arnoldexample}
\begin{split}
H_{\ep, \mu}(A_1, A_2,  \vp_1, \vp_2, t)
& =
H_0 + \ep H_\ep + \mu H_\mu \\
& = \frac{1}{2} A_1^2 + \frac{1}{2} A_2^2 +\ep( \cos \varphi_1 -
1)\\
&\quad + \mu ( \cos\vp_1 - 1 )\, (\sin \varphi _{2}+\cos t)
\end{split}
\end{equation}
and showed that for $ 0 < \mu \ll \ep \ll 1$, the flow of
Hamiltonian \eqref{eq:arnoldexample} contains orbits such that,
for some $T > 0$, $\abs{A_2(T) - A_2(0)}\ge 1$.

This result is in marked contrast with the behavior in the case of $\ep = \mu
= 0$  for which $A_1, A_2$ are conserved quantities, and it is all
the more surprising since the KAM theorem (proved slightly earlier
than \cite{Arnold64}) showed that the phase space of the
system---except for a set of  measure $\Or( \ep + \mu)^{1/2}$---is
covered by quasi-periodic solutions for which $\abs{\bA(t)-\bA(0)}
\le K ( \ep + \mu)^{1/2}$ for all real $t$, where $\bA
=(A_{1},A_{2})$ and $K$ is a constant independent of $\ep$, $\mu$.



Motivated by applications, related phenomena of instability were
studied numerically and a wealth of heuristic understanding was
accumulated (see e.g. \cite{Chirikov79}).

An essential ingredient in the geometric description of the
argument in \cite{Arnold64} is that the flow of
Hamiltonian~\eqref{eq:arnoldexample}  preserves for all $\ep, \mu$
a one-parameter family  of $2$-dimensional tori
\[
\Tau_{a} = \{ (\bA, \bvp, t):\ A_1 = 0, \  A_2 = a, \ \vp_1 = 0, \
(\vp_2, t) \in \torus^2 \}.
\]
This is a consequence of the fact that all the terms
in~\eqref{eq:arnoldexample} which involve $\ep, \mu$ contain as a
factor $(\cos \vp_1 - 1)$ which vanishes up to second order in
$\vp_1$.

When $\ep > 0$, the tori $\Tau_{a}$ have $3$-dimensional stable
and unstable manifolds, so they are called \emph{whiskered tori}.
By performing first order perturbative calculations in $\mu$, it
is possible to show that the stable and unstable manifolds of
nearby tori cross transversally.


As shown in a classic calculation in \cite[\S\S403--409]{Poincare99}
(modern expositions emphasizing the Lagrangian character of the
manifolds and its consequences are 
\cite{DelshamsR97a,DelshamsG00}) the distance between the modified manifolds
associated to $\Tau _a$ can be expressed as $ \mu \nabla_{\vp_2,t}
\LL(a, \vp_2, t)+ \Or(\mu^2)$.

In the case of \eqref{eq:arnoldexample}, the function $\LL$,
usually called Melnikov potential, has the form
\[
\LL(a, \vp_2, t) =  2 \pi \left[ \frac{a^{2} \sin\vp_2} {\sinh(
\frac{ \pi a}{2 \sqrt{\ep} } )} - \frac{ \cos t}{\sinh(
\frac{\pi}{2\sqrt {\ep}})} \right].
\]
Note that the function $\LL$ is exponentially small in $\ep$, but
if we choose
\begin{equation}\label{range}
\mu \ll  \max \left( \exp( - \frac{\pi}{2} a \ep^{-1/2} ),
\exp( - \frac{\pi}{2}  \ep^{-1/2} )\right),
\end{equation}
the first order in $\mu$ of the perturbative calculation  of the
intersection  is much larger than the remainder and it allows us
to conclude that the stable and unstable
 manifolds  of a torus $\Tau_{a}$ intersect
transversally. Hence, the unstable manifold of $\Tau_{a}$
intersects transversally the stable manifold of $\Tau_{a +
\delta}$ for $\delta \ll |\mu \nabla \LL|$.


Therefore, it is possible to construct a sequence of tori
$\{\T_{a^{i}}\}_{i = 0}^{N} $
such that the unstable manifold of one intersects
transversally the stable manifold of the next and $a^1 = 0, a^N = 1$.

Then, an argument---which can be made essentially
topological---shows that there is an orbit following all the
intersections. More details on the paper \cite{Arnold64} can be
found in \cite{ArnoldA68},
\cite{DelshamsLS03},
\cite{FontichM03}.


In \cite[p.~176]{Arnold63} one can find the formulation of the
unsolved problem  of establishing that the  mechanism in
\cite{Arnold64} happens for \emph{many} Hamiltonian systems. The
formulation of the problem already includes the hint that one
would need to consider resonances more carefully.


If instead of the term containing $\mu$ in
\eqref{eq:arnoldexample} we would have included a more general
function which does not happen to vanish to second order at $\vp_1
=  0$, the family $\Tau_{a}$ would not have been preserved, since
the tori $\Tau_{a}$ for which the frequency $A_{2}=a$ is rational
are destroyed (see \cite[\S 81]{Poincare99}, \cite{Treschev89}).

At resonances (regions where the frequency $A_2$ is rational), the
family of tori is interrupted by gaps whose size is at most of
order $\Or(\mu^{1/2})$. This is much larger than the distance that
the manifolds move in the range of parameters \eqref{range} that
we required to ensure that the first order perturbation theory in
$\mu$ gave conclusive results. The fact that the gaps on the
family of tori are larger than the size of the intersections of
the stable and unstable manifolds---for the ranges of
perturbations considered---is what in the literature is called
\emph{the large gap problem}.


The heuristic discoveries of numerical exploration  show that
\emph{the phenomenon of diffusion is strongest precisely on
resonances}. Indeed this is the most important guide in the
numerical exploration and in applications. See 
\cite{Chirikov79},
\cite{Tennyson82}, 
\cite{ChirikovLSV85}. In contrast, the diffusion exhibited
in \cite{Arnold64} happens precisely---by design---in a zone which
is devoid of resonances since the resonant tori are preserved.

Hence, there seems to have been a divergence between the
mathematical litera\-ture---that has aimed to verifying the
mechanism of  \cite{Arnold64}---and the physical literature that
has explored the role of resonances. See 
\cite{Douady88},
\cite{DouadyC83}, 
\cite{FontichM01}, 
\cite{Gallavotti98}, 
\cite{Moeckel96} for rigorous
mathematical verifications of the existence of the mechanism of
\cite{Arnold64}
in some specially constructed models
as well as \cite{BertiB02},
\cite{BertiBB03} and references there for
a functional analysis approach. See \cite{DelshamsLS00} and
\cite{BolotinT99} for rigorous verification variations of the mechanism in
perturbed geodesic flows, which has also been considered in
\cite{Mather95-96}. In all the cases above, the gaps between KAM
tori are smaller than the splitting of separatrices.
(A more extensive bibliography is available in \cite{DelshamsLS03}.)

See also 
\cite{ChierchiaG94},
\cite{ChierchiaG98} for a rather complete set of tools. See
\cite{Tennyson82}, 
\cite{ChirikovLSV85} as well as
 \cite{Chirikov79} for
heuristic discussions of the importance  of resonances in
diffusion. Among conferences which include papers with the
mathematical and physical points of view on diffusion we mention
\cite{Lakeway}, 
\cite{Simo99}.

The goal of this note is to describe a geometric mechanism that
overcomes  the large gap problem. We  also describe a rather
detailed verification of the existence of this mechanism in a
concrete problem. A more detailed manuscript with full
proofs is available in
\cite{DelshamsLS03}.

Similar problems have been considered in
\cite{Treschev03}, 
\cite{ChengY03}, which we received after finishing
this paper.


We also mention that there are detailed announcements of results 
using variational methods 
\cite{Mather02} as well as shorter announcements
\cite{Xia98}.


\section{The two dynamics mechanism for diffusion (heuristic)} \label{twodynamics}

The first main  idea of our proposed mechanism is that,
at the same time that resonances destroy KAM whiskered tori, they
generate some other objects that can also be used
as part of transition chains that generate diffusion.

An example which is well known is that perturbations of integrable
twist maps break resonant tori but leave hyperbolic periodic
orbits---with their stable and unstable manifolds---and elliptic
islands. Note that neither of those objects have analogues in the
integrable map. (See \cite[\S\S74, 79, 81]{Poincare99},
\cite{LlaveW89} for some higher-dimensional extensions.) All these
objects generated by the resonances will take up the role of the
invariant tori that are destroyed by the resonances.



One convenient way of organizing the many different objects that
we encounter is to identify normally hyperbolic invariant
manifolds $\tilde \Lambda$ that are present in the system. It
happens frequently that the stable and unstable manifolds of
$\tilde \Lambda$ intersect transversally. This is commonly
described as saying that there are transverse homoclinic
intersections to $\tilde \Lambda$. An easy dimension count shows
that the intersection of these stable and unstable manifolds will
typically contain a manifold that has the same dimension as
$\tilde \Lambda$.


More precisely, given a normally hyperbolic invariant manifold
$\tilde \Lambda$ whose stable and unstable manifolds
$W^{\st,\un}_{\tilde \Lambda }$ intersect transversally, we can
associate to it two dynamical systems:

\textbf{ The inner map:} This is simply the time one map
restricted to $\tilde \Lambda$, and we will denote it by $F$.


\textbf{The scattering map or outer map:} We also associate
another dynamical system to the homoclinic excursions along the
prescribed connection.


More precisely, given a family of orbits $\gamma \subset
W^\st_{\tilde \Lambda }\cap W^\un_{ \tilde \Lambda}$ such  that
the intersection of the stable and unstable manifolds is
transversal along $\gamma$---hence $\gamma$ should be a manifold
of the same dimension of $\tilde \Lambda$---we  define the scattering map
$S$ associated to $\gamma$ by setting $x_+ = S(x_-)$ when we can
find $z \in \gamma$ such that $d( \Phi_t(z), \Phi_t(x_\pm)) \le C
e^{-\lambda | t|}$ as $ t \to \pm \infty$. Here $\Phi_t$ denotes
the dynamics of the system and $\lambda >0$ is the exponent of the
normal rate of contraction of the normally hyperbolic manifold.

That is, when performing a homoclinic excursion, the system goes
from resembling the orbit of $x_-$ to resembling the orbit of $x_+
= S(x_-)$.


Since the map $S$ is formulated in terms of normally hyperbolic
invariant manifolds, it is possible to compute it using
perturbation theory of normally hyperbolic manifolds. It admits a
rather explicit expansion $S = S_0 + \ep S_1 + \ep^2 S_2 +
\cdots$.



We note that  the scattering map can be regarded as an
alternative to the standard methods in Melnikov theory.
Conceptually, it relies less on coordinates, so that it can be
computed quite effectively and can work for all points in $\tilde
\Lambda$ independently of the motion they experience.

By intermingling the two dynamics, we construct sequences of
points   $\{x_N\}_{N \in \nat}$, where $ x_{N} = F^{n_N}\circ
S\circ \cdots\circ S\circ F^{n_1}(x)$, which diffuse.

Such a sequence can be described heuristically  as a pseudo-orbit
which stays  \emph{``parked''} near the invariant manifold $\tilde
\Lambda$ along some convenient times $n_i$ and, at certain times
suitably chosen, performs homoclinic excursions.

Under many circumstances, there are variants of the obstruction
property which ensure that there are orbits of the original system
that track $x_N$ closely. Notably if all the $x_N$ lie in
non-resonant tori invariant under $F$, we can use  the obstruction
argument in \cite{ArnoldA68} (a very careful implementation can be
found in \cite{FontichM01}).

As a consequence, if $x_N$ experiences diffusion---and is such
that the obstruction property can be applied---then there is an
orbit $\{ \Phi _{t}(y)\}$ in the original system which also
experiences diffusion.


Note that in Example \eqref{eq:arnoldexample}
the manifold $A_1 = 0$, $\vp_1 = 0$
is normally hyperbolic for $|\ep| > 0$.
Hence it persists for $|\mu| \ll 1$.

We emphasize that the persistence of the normally hyperbolic
invariant  manifold is true for any perturbation $H_\mu$, not just
the one used in \eqref{eq:arnoldexample}.

If we take the perturbation $H_\mu$ to be a generic one we can
expect that the dynamics in the manifold $\tilde \Lambda$ includes
not only the tori that survive from $H_0 + \ep H _{\ep}$ but also
the dynamics associated to the resonances.



\section{Rigorous verification in a model}

In \cite{DelshamsLS03} we have undertaken a very detailed rigorous
verification of the existence of the mechanism described above for
models of the form
\begin{equation} \label{verification}
H_{\ep} = \pm \left( \frac{1}{2} A_1^2 +  V(\vp_1) \right)
 + \frac{1}{2} A_2^2
+ \ep h( A_1, A_2, \vp_1, \vp_2, t;\ep).
\end{equation}

The system \eqref{verification} presents the large
gap problem and has been a very popular model
for the study of diffusion since \cite{HolmesM82}.

A particular case of the results of \cite{DelshamsLS03} is:

\begin{theorem} \label{main}
Assume that a Hamiltonian of the form \eqref{verification}
satisfies:
\begin{itemize}
\item[\Hyp{1}.] The terms $V$ and $h$ in
\eqref{verification} are uniformly $C^r$ for
$r \ge r_0$, which is sufficiently large.

\item[\Hyp{2}.] The potential $V: \torus \to \real$
has a global  maximum at $0$ which is non-degenerate (i.e., the
second derivative is not zero). We denote by $( A_{1}^{0}(t),
\vp_{1}^{0}(t))$ an orbit of the pendulum $\pm ( \frac{1}{2} A_1^2
+ V(\vp_1) )$ homoclinic to $(0,0)$.
\item[\Hyp{3}.] $h$ is a trigonometric polynomial in
$\vp$ and $t$:
\[
h( A_1, A_2, \vp_1, \vp_2, t;\ep)= \sum_{k,l \in \N } \hat
h_{k,l}(A_1, A_2, \vp_1;\ep) e^{  i ( k \vp_{2} + l t)},
\]
where $\N \subset \integer^2$ is a finite set.
\item[\Hyp {4}.]
Consider the Poincar\'e function, also called Melnikov potential,
associated to $h$ (and to the homoclinic orbit $( A_{1}^{0},
\vp_{1}^{0})$ mentioned in \Hyp{2}):
\begin{eqnarray}
\LL(A_{2},\vp_{2},t) &=& -\int
_{-\infty}^{+\infty}\Bigl(h\left(A_{1}^{0}(\sigma),A_{2},
\vp_{1}^{0}(\sigma),\vp_{2} +A_{2}\sigma,t+\sigma;0\right)\label{potential} \\
&&\mbox{}\qquad\qquad\qquad - h(0,A_{2},0,\vp_{2}
+A_{2}\sigma,t+\sigma;0)\Bigr)\,d\sigma . \nonumber
\end{eqnarray}
Assume that, for any value of $A_{2} \in (a^{-},a^{+})$
there exists an open set $\J_{A_{2}}\subset \torus ^{2}$, with the
property that when $(A_{2},\vp_{2},t)\in H_{-}$, where
\begin{equation} \label{dominiS}
H_{-}= \bigcup _{A_{2}\in (a^{-},a^{+})} \{A_{2}\} \times \J_{A_{2}}
\subset
(a^{-},a^{+})\times\torus ^{2},
\end{equation}
 the map
\begin{equation} \label{Melnikov}
\tau \in \real \mapsto \LL (A_{2},\vp _{2}-A_{2}\tau,t-\tau)
\end{equation}
has a non-degenerate critical point $\tau$ which is locally given,
by the implicit function theorem, in the form $\tau = \tau^*
(A_{2},\vp _{2},t) $ with $\tau^*$ a smooth function.

Assume moreover that for every $(A_{2},\vp_{2},t)\in H_{-} $, the
function
\begin{equation}\label{signs}
\frac {\partial \LL}{\partial \vp_{2}}(A_{2},\vp_{2}-A_{2}\tau ^{*},t-\tau ^{*})
\end{equation}
is non-constant and negative (respectively, positive).





\item[\Hyp{5}.] The perturbation terms $h(A_1, A_2, \vp_1, \vp_2, t;0)$,
$\frac{\partial h}{\partial \ep}(A_1, A_2, \vp_1, \vp_2, t;0)$,
satisfy some non-degeneracy conditions that can be stated quite explicitly.
\end{itemize}

Then, for $0 < |\ep| \le \ep^*$, the system \eqref{verification}
has orbits  such that $A_2(0) \le a^- + \Or(\ep)$, $A_2(T) \ge
a^+ + \Or(\ep)$ (respectively, $A_2(0) \ge a^+ + \Or(\ep)$, $A_2(T) \le
a^- + \Or(\ep)$).
\end{theorem}

\begin{remark}
Since \cite{ChierchiaG94}, it is sometimes customary to
distinguish between \emph{a priori stable} and
\emph{a priori unstable} systems.
Following this terminology, our model \eqref{verification}
is an a priori unstable system because the
unperturbed Hamiltonian has no global action angle variables (in
fact, it has partially hyperbolic tori with homoclinic trajectories).

This distinction makes sense only for one-parameter families.
When one considers two-parameter families as in
\cite{Arnold64} or generic results, the results for
a priori unstable maps automatically imply results
for perturbations of an a priori stable
Hamiltonian in a cusp residual set.


For example, if we consider the a priori stable
system with two parameters
\begin{equation} \label{twoparameter}
H_{\delta, \ep} = \pm \left( \frac{1}{2} A_1^2 + \delta V(\vp_1) \right)
 + \frac{1}{2} A_2^2
+ \ep h( A_1, A_2, \vp_1, \vp_2, t;\ep)
\end{equation}
satisfying non-degeneracy conditions,
we obtain similar results as in
Theorem \ref{main}  if we assume, as in
\cite{Arnold64}, that $\ep$ is (exponentially) small with respect to
$\delta$. Note, that in \eqref{twoparameter} the gaps
between KAM tori at a non-degenerate primary resonance
are $  B \ep^{1/2} + \Or(\ep)$,   the splitting  of separatrices is
$ \Or\left(\exp( - C \delta^{-1/2} )\right)  + \Or(\ep^2)$
and the distance between secondary and primary tori is,
again, smaller than $\Or(\ep^{3/2})$.
\qed
\end{remark}

The verification undertaken in \cite{DelshamsLS03} is very
explicit so that, given  $h$,  $V$, a finite number of
calculations can establish that \eqref{verification} satisfies the
mechanism outlined heuristically above, mainly \Hyp{4} and
\Hyp{5}.


The verification undertaken in \cite{DelshamsLS03} consists of  a sequence
of perturbation theories. Most of them are somewhat standard in
geometric theory of diffusion. They need to be
developed with very quantitative statements so that they  form a
coherent toolkit. We highlight here the main steps.
\begin{enumerate}
\item

Using the theory of normally hyperbolic invariant manifolds, we
show:
\begin{enumerate}
\item The manifold $\tilde \Lambda = \{ A_1 = 0, \vp_1 = 0\}$,
which is invariant and normally hyperbolic for $\ep = 0$, persists
for $|\ep| \ll 1$ giving rise to $\tilde \Lambda_\ep$. We  compute
expansions for $\tilde \Lambda_\ep$ and for the Hamiltonian
restricted to it.
\item
Under the non-degeneracy conditions in \Hyp{4}, the stable and
unstable manifolds $W^{\st,\un}_{\tilde \Lambda_{\ep}}$ of the
normally hyperbolic invariant manifold $\tilde
\Lambda_\ep$---which agree for $\ep = 0 $---have transverse
intersections.
\item
Given the transverse intersections of the previous point, the
scattering map $S$ can be computed in first order perturbation
theory.

One of the conclusions of the calculations is that the scattering
map $S$ is close to the identity in a $C^{r'}$ sense, where $r'$
can be taken  arbitrarily large if $r$ is sufficiently large.
\end{enumerate}
\item
The motion restricted to  the normally hyperbolic invariant
manifold $\tilde \Lambda_\ep$  is analyzed as follows:
\begin{enumerate}
\item
Using that the system restricted to $\tilde \Lambda_\ep$ is a
periodic perturbation of  a one degree of freedom Hamiltonian
system, averaging theory shows that the system can be transformed
up to small errors into a time independent system. Near the
resonances, the system can be accurately described by systems
similar to a pendulum.

\item
Far away from the resonances,  using that a time independent one
degree of freedom Hamiltonian is integrable, we can use the
standard KAM theorem to show that there are closely spaced KAM
tori.
\item
Near the resonances we switch  to singular action-angle variables
and show that, under appropriate non-degeneracy conditions,  we
can find
\begin{itemize}
\item[i)]
KAM tori close to the separatrices of the pendulum.
\item[ii)]
Secondary KAM tori---tori which are contractible to a periodic
orbit---close to the separatrices of the pendulum.
\item[iii)]
Stable and unstable
manifolds of periodic orbits close to the separatrices of the pendulum.
\end{itemize}

Note that the objects ii), iii) above are not present in the
unperturbed system but are generated by the resonances. We refer
to these objects as \emph{secondary} objects.

We emphasize that the objects in i), ii), iii)  can be made to be
very close to each other. In particular, the secondary tori of ii)
are very close to the KAM tori of i) and the stable and unstable
manifolds of iii) are in between.

These secondary objects dovetail in the gap produced by the
resonances among the KAM tori. See  Figure \ref{fig1}.


\end{enumerate}


\item
Under appropriate non-degeneracy conditions---which are reflected
in the non-vanishing of certain terms of the perturbation theory
of the map $S$---it will happen that the image under the
scattering map $S$ of some of the objects considered in (b) and
(c) of step~2 intersect transversally in $\tilde \Lambda_\ep$
other such objects.



A moment's reflection shows that if $S(\V_1)$, $\V_2$ intersect
transversally as submanifolds of $\tilde \Lambda_\ep$, then
$W^\un_{\V_1}$ intersects $ W^\st_{\V_2}$ transversally as
submanifolds of the phase space.

Since the objects we constructed in step~2 are closer than
$\ep^{3/2}$, and the scattering map moves objects by an amount
$\Or(\ep)$, some non-degeneracy assumptions will allow us to
establish that they intersect transversally under the action of
the scattering map. Therefore, all these objects form what is
called a transition chain.

 Note that the method allows us to
establish without problem the existence of transition chains
containing objects that have different topological type and which
do not have analogues in the unperturbed problem.

\item
The obstruction methods can be adapted to show that there
are orbits that track the escaping pseudo-orbits.

Even if some of the earlier proofs of the obstruction properties
use coordinate systems, we point out that there are more recent
proofs which make it clear that the obstruction property is true
independently of a common system of coordinates in the tori.
\end{enumerate}

The motion on the normally hyperbolic invariant manifold is
depicted in Figure \ref{fig1}, where we indicated the objects and
their relative positions and distances.
\begin{figure}[t]
\noindent
\begin{minipage}{.45\textwidth}
\includegraphics[scale=.33]{era121el-fig-3}
\end{minipage}
\begin{minipage}{.45\textwidth}
\includegraphics[scale=.33]{era121el-fig-4}
\end{minipage}
\caption{Surface
of a section of $\tilde \Lambda_\ep$ illustrating the main invariant
objects. The primary KAM tori and the secondary tori are on the left.
The primary tori and the stable and unstable manifolds of periodic
orbits are on the right. } \label{fig1}
\end{figure}

In Figure \ref{fig2}, we indicate the
effect of the scattering map on the objects previously found.
\begin{figure}[t]
\begin{minipage}{.45\textwidth}
\includegraphics[scale=.33]{era121el-fig-1}
\end{minipage}
\begin{minipage}{.45\textwidth}
\includegraphics[scale=.33]{era121el-fig-2}
\end{minipage}
\caption{
Schematic illustration of the main invariant objects in $\tilde
\Lambda_\ep$ as well as their images (dashed) under the scattering
map. } \label{fig2}
\end{figure}



\section{Final remarks}

The  proof of   Theorem \ref{main} shows that   there exists a
transition chain with a torus $\Or(\ep)$ close to $\T_{a
^{-}}$ and another torus  $\Or(\ep)$ close to $\T_{a^{+}}$
and so that the tori composing  this transition chain are closely
spaced. Note that under hypothesis \Hyp{2} there are indeed two
homoclinic orbits of the pendulum $\pm ( \frac{1}{2} A_1^2 +
V(\vp_1) )$. Moreover, for each of these orbits, the function
given in~\eqref{Melnikov} will have several critical points, which
are typically non-degenerate. If two of these choices give us
intervals $[a^-_1, a^+_1]$, $[a^-_2, a^+_2]$ and $a^+_1 > a^-_2$,
then, the result produces a transition chain that starts near
$a^-_1$ and ends near $a^+_2$.


Besides orbits that diffuse along the action $A_{2}$, the
geometric mechanism used in the proof of Theorem~\ref{main}
provides orbits that visit the tori in a somewhat arbitrary order.


Note that the transition chains produced in \cite{Arnold64}
increase the action  $\Or(\ep)$ per step in the transition chain.
The transition chains produced here take steps $\Or(\ep^{1/2})$ at
resonances. This agrees with the numerical and heuristic intuition,
which  suggests that the diffusion is faster precisely at
resonances.


We note that the verification in \cite{DelshamsLS03} uses mainly
tools which are rather standard in the geometric approach to
diffusion. Of course, they require adaptations so that they can
work together and form an efficient toolkit. We hope that this
toolkit can be used for other problems. We also hope that new
tools can be incorporated to the toolkit. Notably, variational
methods and more topological methods. Indeed, a very similar
mechanism to the one described in Section \ref{twodynamics} was
implemented in \cite{DelshamsLS00} to give a geometric proof of
the results in \cite{Mather95-96}. In turn, ingredients of the
geometric proof were used to simplify some of the variational
arguments.

A similar toolkit has been used in \cite{DelshamsLS03a} to verify
the existence of orbits of unbounded energy of quasi-periodic
perturbations of geodesic flows---for Riemannian, Lorentz or
Finsler metrics---in many manifolds. We think it would be
interesting to find variational analogues of the later result.



It seems to us that, if one does not insist on verifying the
results for concrete systems, but rather show that the
mechanism happens for quasi-integrable generic systems,
 the verification could be somewhat simplified.
We hope that the modularity of the method could encourage the use of
topological or variational methods for some of the steps. We also
note that the method presented here admits several variants with
different quantitative properties. Diffusion seems to include all
the variety of possible mechanisms.



\section{Acknowledgments}
The work described in this announcement
has been supported by the \emph{Comisi\'on Conjunta
Hispano Norteamericana de Cooperaci\'on Cient\'{\i}fica y
Tecnol\'gica} and Spanish MCyT (I+D+I) grants.
 A first version was prepared while R.L. was enjoying a
\emph{C\'{a}tedra de la Fundaci\'{o}n FBBV}, and A.D. was visiting
the \emph{Centre de Recerca Mate\-m\`{a}tica}. A.D. and T.S. have
also been partially supported by the Catalan grant 2001SGR-70 and
the INTAS grant 00-221, and R.L. by NSF grants.

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\end{document}