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\begin{document}
\title{On the Use of the Melnikov Integral in the Arnold Diffusion
Problem}
\author{Michael Rudnev
\\
Applied Mathematics 217-50 \\ Caltech \\ Pasadena, CA
91125
\hfill \\
\hfill \\
Stephen Wiggins \\
Applied Mechanics and
\\ Control and Dynamical Systems 116-81 \\ Caltech \\ Pasadena, CA
91125
\thanks{This research was partially supported by NSF Grant
DMS-9704759.}}
\date{\today}
\maketitle
\begin{abstract}
In this note we want to point out a number of difficulties of arithmetic
nature
with the the so-called Melnikov integral (i.e., first order perturbation
theory)
as a measure of the splitting distance between the stable and unstable
manifolds
of tori in perturbations of a-priori stable integrable Hamiltonian systems
with
three or more degrees-of-freedom. We do this by considering a specific
example
which illustrates a number of the issues. We show that it is possible to
introduce additional assumptions on the frequencies of the tori so that
the Melnikov integral is the dominant term in the perturbation series
for the distance between the stable and unstable manifolds of the torus.
However, even when the Melnikov integral can be used to estimate the
splitting distance, we show that even more difficulties arise when one
uses
it to determine if the manifolds intersect transversely, which is a key
ingredient for constructing transition chains in the Arnold diffusion
problem.
\end{abstract}
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\section{Introduction}
In 1964 Arnold published an example of a three degree of freedom
Hamiltonian system containing a mechanism for a global instability of the
action variables
which has come to be known as {\em "Arnold diffusion"}. The purpose of
this note is to highlight one of the main difficulties in proving that
this mechanism occurs in some generic sense. The difficulties are best
motivated by considering Arnold's original example.
Arnold [1964] considered a Hamiltonian of the form
\begin{equation}
H = \frac{I_1^2}{2} + I_2 + \frac{y^2}{2}
+ \varepsilon \left( \cos x -1 \right)
+ \mu \varepsilon \left( \cos x -1 \right)
\left( \sin \varphi_1 + \cos \varphi_2 \right),
\label{Arny}
\end{equation}
\noindent
where $x \in T^1, \, y \in \bbR, \, (I_1, I_2) \in \bbR^2, \,
(\varphi_1, \varphi_2) \in T^2$, and $\varepsilon$ and $\mu$ are
parameters. The Hamiltonian obtained by setting $\mu =0$ is
referred to as the {\em unperturbed Hamiltonian}, which one recognizes
as the Hamiltonian for three uncoupled systems: a pendulum and two
{\em rotors} described by two action-angle pairs of variables.
Hamilton's canonical equations for this system are given by
\begin{equation}
\begin{array}{l}
\dot{x} = \frac{\partial H}{\partial y} =y, \\
\hfill \\
\dot{y} = - \frac{\partial H}{\partial x} = \varepsilon \sin x
+ \mu \varepsilon \sin x \left( \sin \varphi_1 + \cos \varphi_2 \right),
\\
\hfill \\
\dot{\varphi}_1 = \frac{\partial H}{\partial I_1} = I_1, \\
\hfill \\
\dot{I}_1 = -\frac{\partial H}{\partial \varphi_1} = -\mu \varepsilon
\cos \varphi_1 \left( \cos x -1 \right), \\
\hfill \\
\dot{\varphi}_2 = \frac{\partial H}{\partial I_2} = 1, \\
\hfill \\
\dot{I}_2 = -\frac{\partial H}{\partial \varphi_2} =
\mu \varepsilon \sin \varphi_2 \left( \cos x -1 \right).
\end{array}
\end{equation}
\noindent
We want to describe the geometry of the phase space for $\mu=0$.
In this case the system has a four dimensional normally hyperbolic
invariant manifold given by
\[
{\cal M}_0 = \left\{ (x, y, \varphi_1, I_1, \varphi_2, I_2 ) \,
\vert \, x=y=0 \right\}.
\]
\noindent
whose five dimensional stable and unstable manifolds coincide along
\[
W^s \left({\cal M}_0 \right) = W^u \left({\cal M}_0 \right)
= \left\{ (x, y, \varphi_1, I_1, \varphi_2, I_2 ) \, \vert \,
\frac{y^2}{2} + \varepsilon \left( \cos x -1 \right) =0 \right\}.
\]
\noindent
${\cal M}_0$ is foliated by a family of two-tori, and the trajectories
on the two-tori are given by
\[
\begin{array}{l}
\varphi_1 (t)= I_1 t + \varphi_{10}, \\
\hfill \\
\varphi_2 (t) = t + \varphi_{20}.
\end{array}
\]
\noindent
We denote these two-tori by ${\cal T}(I_1)$.
Recall that the dynamics is restricted to lie in the five
dimensional energy surface given by
\[
h = \frac{I_1^2}{2} + I_2 + \frac{y^2}{2}
+ \varepsilon \left( \cos x -1 \right),
\]
\noindent
and the four dimensional ${\cal M}_0$ intersects the five dimensional
energy surface in a three dimensional set. This three dimensional set,
denoted by ${\cal M}_0^h$, is filled out by a family of two tori,
and each two-torus has three-dimensional stable
and unstable manifolds that coincide and are given by
\[
W^{s}\left({\cal T}(I_1) \right) = W^{u}\left({\cal T}(I_1) \right) =
\left\{ (x, y, \varphi_1, I_1, \varphi_2, I_2 ) \, \vert \, \frac{y^2}{2}
+ \varepsilon \left( \cos x -1 \right) =0,\ I_1 \ \mbox{fixed}
\right\}.
\]
\noindent
The stable and unstable manifolds of a given torus are often
referred to as the {\em whiskers} of the torus.
Moreover, $W^{s}\left({\cal T}(I_1) \right)$
intersects $W^{u}\left({\cal T}(I_1) \right)$
non-transversely, which is a situation that we do not expect to
persist for $\mu$ small, but nonzero, and it is to this
situation that we now turn.
The particular form of the perturbation in (\ref{Arny}) is such that it
does
not affect ${\cal M}_0$ or any of the tori on it. Nevertheless,
for any given torus the perturbation may cause the stable and
unstable manifolds to intersect transversely. Moreover, because
the tori are arbitrarily close on ${\cal M}_0^h$,
{\em and are not isolated by the energy integral for
systems with three or more degrees-of-freedom},
the stable and
unstable manifolds of nearby tori may intersect and,
consequently, give rise to a mechanism for a global drift in
the trajectories. Following Lochak [1995], we refer to this as the
{\em Arnold mechanism} and we now describe it in a bit more detail.
The tori can be denoted by the parameter $I_1$. Arnold showed
that one could find a sequence of tori, $\{I_1^i \}, \, i=1,
\ldots , N$, with neighboring tori
sufficiently close, i.e. $\vert I_1^i - I_1^{i+1} \vert =
O(\vert \mu \vert^2)$, such that the unstable manifold of
${\cal T}_\mu (I_1^i)$ transversely intersects the stable
manifold of ${\cal T}_\mu (I_1^{i+1}), \, i=1, \ldots, N-1$.
This sequence of whiskered tori is said to form a {\em
transition chain}, and Arnold uses it to prove the following
theorem.
\begin{theorem}[Arnold]
For every $\varepsilon , \, r >0$ there exists a $\mu_0 >0$ such
that for all $0 < \mu \le \mu_0$ there are invariant tori
${\cal T}_\mu (I_1)$ and ${\cal T}_\mu (I'_1)$, with $\vert
I_1 - I'_1 \vert >r$, which are connected by a transition chain.
\end{theorem}
\noindent
He then shows that the transition chain is shadowed by a true
orbit of the system and in this way one obtains an $O(1)$ drift
in the $I_1$ variable over some finite time interval.
In carrying out this program in general there are two main
difficulties that must be overcome. One is concerned with the problem of
the splitting of separatrices, and the other is concerned with the
construction of transition chains and shadowing orbits. It is the former
problem with which this note is concerned.
\section{Classes of Systems Studied}
Before proceeding it is important to clarify the nature of the types of
near-integrable Hamiltonian systems under consideration and what one means
by the "Arnold diffusion"
problem itself (see Lochak [1995]). We feel obliged to do so, for lately a
number of papers
have appeared, namely Xia [1993], [1994], and Moeckel [1996], which claim
to study and
establish "Arnold's diffusion" in certain celestial mechanics problems.
However,
these papers do not really deal with the phenoma under consideration here
(nor do they
prove the existence of any global instabilities). In particular, the very
first passage of Arnold [1964]
emphasizes that the unperturbed part of the system under consideration is
expressed in action-angle
variables, with the actions being first integrals, and no hyperbolicity
present. The three papers,
mentioned above consider systems that are not of this type.
In addition, we want to make the following point, implicit in Arnold
[1964]:
\begin{itemize}
\item The Hamiltonian under consideration is real-analytic in all its
variables, and the unperturbed
part satisfies certain non-degeneracy properties.
\end{itemize}
\noindent
Broadly speaking, there are two types of systems which, following
terminology
introduced by Gallavotti, we refer to as {\em a-priori stable} or
{\em a-priori unstable}. This classification is with respect to the
structure
of the {\em unperturbed system}. The coordinates with respect to which
the unperturbed system is characterized are also used to characterize
the perturbation.
\medskip
\paragraph{A-Priori Stable Systems}
\hfill
\medskip
\noindent
This is the perturbation theory for these systems, that was called
by Poincar\'{e} "the general problem of dynamics"
\footnote{Poincar\'{e}, H. New Methods of Celestial Mechanics,
{\em History of Modern Physics and Astronomy}, {\bf 13}, I23, 22, American
Institute of Physics.}
We can describe {\em a-priori stable systems} in several ways.
\begin{enumerate}
\item All trajectories of the unperturbed system are quasiperiodic and
Lyapunov stable.
\item The unperturbed system is expressed in terms of action-angle
variables.
\item There is no ``hyperbolicity'' in the unperturbed system. All
orbits are purely ``elliptic'' in stability type.
\end{enumerate}
\noindent
Hence, the Hamiltonian for an a-priori stable system is given by
\begin{equation}
H(\vec{I}, \vec{\varphi}, \mu) = H_0 (\vec{I}) + \mu H_1 (\vec{I},
\vec{\varphi}, \mu), \quad (\vec{I}, \vec{\varphi}) \in \bbR^n \times T^n,
\label{apstab}
\end{equation}
\noindent
with associated Hamiltonian vector field
\begin{eqnarray*}
\dot{\vec{I}} & = & -\mu \frac{\partial H_1}{\partial \vec{\varphi}}
(\vec{I}, \vec{\varphi}, \mu), \\
\dot{\vec{\varphi}} & = & \frac{\partial H_0}{\partial \vec{I}} (\vec{I})
+
\mu
\frac{\partial H_1}{\partial \vec{I}} (\vec{I}, \vec{\varphi}, \mu),
\end{eqnarray*}
\noindent
where $\mu$ is the perturbation parameter.
\medskip
\paragraph{A-Priori Unstable Systems}
\hfill
\medskip
We can also describe {\em a-priori unstable systems} in several ways.
\begin{enumerate}
\item Some trajectories of the unperturbed system are saddle-type,
with exponentially growing components.
\item The unperturbed system cannot be expressed
entirely in terms of action-angle
variables. There must be auxiliary variables that describe the saddle
type trajectories.
\item ``Hyperbolicity'' is present in the unperturbed system.
\end{enumerate}
\noindent
Hence, an example of a Hamiltonian for an a-priori unstable system would
be
\begin{equation}
H(y, x, \vec{I}, \vec{\varphi}, \mu) = H_0 (\vec{I}) +
\tilde{H}_0 (y, x)
+ \mu H_1(y, x, \vec{I}, \vec{\varphi}, \mu),
\quad (y, x) \in \bbR^2, \quad (\vec{I}, \vec{\varphi})
\in \bbR^{n-1} \times T^{n-1},
\label{apstab2}
\end{equation}
\noindent
with associated Hamiltonian vector field
\begin{eqnarray}
\dot{y} & = & -\frac{\partial \tilde{H}_0}{\partial x} (y, x) -
\mu \frac{\partial H_1}{\partial x}(y, x, \vec{I},
\vec{\varphi}, \mu), \nonumber \\
\dot{x} & = & \frac{\partial \tilde{H}_0}{\partial y} (y, x) -
\mu \frac{\partial H_1}{\partial y}(y, x, \vec{I},
\vec{\varphi}, \mu), \nonumber \\
\dot{\vec{I}} & = & -\mu \frac{\partial H_1}{\partial \vec{\varphi}}
(y, x, \vec{I}, \vec{\varphi}, \mu), \nonumber \\
\dot{\vec{\varphi}} & = & \frac{\partial H_0}{\partial \vec{I}} (\vec{I})
+
\mu
\frac{\partial H_1}{\partial \vec{I}} (y, x, \vec{I}, \vec{\varphi}, \mu).
\end{eqnarray}
\noindent
Notice that for the unperturbed system ($\mu =0$) the
$x-y$ component of the vector field decouples from the rest.
One introduces hyperbolicity into the unperturbed
system by imposing an assumption, such as the following.
\begin{assumption}[Hyperbolicity in the Unperturbed System]
For the system
\begin{eqnarray*}
\dot{y} & = & -\frac{\partial \tilde{H}_0}{\partial x} (y, x), \\
\dot{x} & = & \frac{\partial \tilde{H}_0}{\partial y} (y, x),
\end{eqnarray*}
\noindent
has a hyperbolic equilibrium point at $(y, x) = (y_0, x_0)$
that is connected to itself by a homoclinic orbit $(y^h (t),
x^h (t))$, i.e.,
\[
\lim_{\vert t \vert \rightarrow \infty} (y^h (t),
x^h (t)) = (y_0, x_0).
\]
\end{assumption}
\noindent
The consequences of this assumption for the full phase space of
the unperturbed system is that the set
\[
{\cal M}_0 = \left\{ (y, x, \vec{I}, \vec{\phi}) \, \vert \,
y=y_0, x=x_0 \right\},
\]
\noindent
is a $2n-2$ dimensional normally hyperbolic invariant manifold
that coincide along a branch of their $2n-1$ dimensional stable
and unstable manifolds given by
\[
W^s ({\cal M}_0) \cap W^u ({\cal M}_0) =
\left\{ (y, x, \vec{I}, \vec{\phi}) \, \vert \,
y=y^h (t), x=x^h (t), \, t \in \bbR \right\}.
\]
\noindent
Moreover, ${\cal M}_0$ is foliated by a $n-1$ parameter family of
$n-1$-tori, denoted $\tau (\vec{I})$, each having $n$
dimensional stable and unstable manifolds (the ``whiskers'')
that coincide along a branch. (A number of other examples of a-priori
unstable systems
can be found in Wiggins [1988].)
With respect to the separatrix splitting problem, the main
difference between the two classes of systems are that the whiskers are
part of the {\em unperturbed} problem for a-priori unstable
systems, and they are {\em created} by the perturbation in
a-priori stable systems (which are often referred to as
{\em weakly hyperbolic}). In other words, for a-priori stable systems, one
has to deal with the singular perturbation theory versus non-singular for
a-priori unstable systems.
Consequentially, in the homoclinic splitting problem for the former, one
has to develop the perturbation
theory in all orders, whereas for the latter the first order will
generically suffice.
We will address this point more fully in the following sections. Not
surprisingly, a-priori
unstable systems are much easier to deal with, so it becomes important to
develop certain
perturbative technique for them and further hope that at least part of it
will digest the singular weak
hyperbolicity of a-priori stable systems. The following idea goes back to
Poincar\'{e}.
\subsection{The Two-Parameter Stratagem}
Many problems have more than one small parameter, but are such that if
all the parameters are set equal to zero then the problem is a-priori
stable. Poincar\'{e} used a two-parameter setting to study the
periodically forced pendulum. Arnold's original example is also of this
form. One can
sometimes make such problems appear as a-priori unstable problems by
selecting a
particular parameter as the perturbation parameter ($\mu$ in Arnold's
example), and let all the remaining parameters remain fixed, but nonzero.
In this way
the unperturbed problem obtained by setting only the selected small
parameter be zero may have the form of an a-priori {\em unstable} system,
although the hyperbolicity in this case is weak. An example of such a
system
would be:
\begin{equation}
H( y, x, \vec{I}, \vec{\varphi}) = H_0 (\vec{I}) +
\varepsilon \tilde{H}_0 (y, x)
+ \mu H_1(y, x, \vec{I}, \vec{\varphi}, \mu, \varepsilon),
\quad (y, x) \in \bbR^2, \quad (\vec{I}, \vec{\varphi})
\in \bbR^{n-1} \times T^{n-1},
\label{apstab3}
\end{equation}
\noindent
with associated Hamiltonian vector field
\begin{eqnarray*}
\dot{y} & = & -\varepsilon \frac{\partial \tilde{H}_0}{\partial
x} (y, x) -
\mu \frac{\partial H_1}{\partial x}(y, x, \vec{I},
\vec{\varphi}, \mu, \varepsilon), \nonumber \\
\dot{x} & = & \varepsilon \frac{\partial \tilde{H}_0}{\partial y} (y, x) -
\mu \frac{\partial H_1}{\partial y}(y, x, \vec{I},
\vec{\varphi}, \mu, \varepsilon), \nonumber \\
\dot{\vec{I}} & = & -\mu\frac{\partial H_1}{\partial \vec{\varphi}}
(y, x, \vec{I}, \vec{\varphi}, \mu, \varepsilon), \nonumber \\
\dot{\vec{\varphi}} & = & \frac{\partial H_0}{\partial \vec{I}} (\vec{I})
+ \mu
\frac{\partial H_1}{\partial \vec{I}} (y, x, \vec{I}, \vec{\varphi},
\mu, \varepsilon),
\label{2param}
\end{eqnarray*}
\noindent
where we view $\mu$ as the perturbation parameter, and $\varepsilon$ as
small and fixed. The terms multiplied by $\varepsilon$ are assumed to
introduce hyperbolicity
into the unperturbed system in the same way that the analogous terms
account for hyperbolicity in
the Hamiltonian (\ref{apstab2}).
This type of two-parameter problem has been considered by a number
of people (e.g. Arnold [1964], Xia [1993], [1994], Chierchia and
Gallavotti [1994]), and is motivated by
Arnold's original example.
Naturally, carrying out proofs and estimates in this setting will result
in some smallness condition on
$\mu$ in terms of the parameters of what is considered as the "unperturbed
problem", which now includes
$\varepsilon$. In particular, Arnold [1964] requires that for
$\varepsilon>0$ one has
$|\mu|\sim \exp \left( -{1\over\sqrt{\varepsilon}}\right)$ or less, which
makes it severely non-generic.
Nevertheless, if it is possible to extend his results for the case
$\mu\sim\varepsilon^p$ for a finite power
$p$, the model becomes much more general, for the reasons that we will
further expose.
Now we explain how hyperbolicity can be created by the perturbation in
the a-priori stable system (\ref{apstab}). Arnold's example is
considered to be a model for the dynamics near a multiplicity one
resonance surface, and it is believed by some that global motion along
these multiplicity one resonance surfaces is influenced by
transition chains.
For the unperturbed Hamiltonian (\ref{apstab}) consider a {\em
multiplicity
one} or {\em simple} resonance, i.e. there exists {\em one}
independent integer vector, $\vec{k}$, such that
\begin{equation}
\vec{k}\cdot \partial_{\vec{I}} H_0 (\vec{I}) =0.
\label{multoneres}
\end{equation}
\noindent
For nondegenerate $H_0 (\vec{I})$ (\ref{multoneres}) defines a
codimension one surface in the action space or, equivalently, a
hyperplane in the frequency space. Near this {\em resonance
surface} the Hamiltonian can be transformed to a normal form of
the following form
\begin{equation}
H( y, x, \vec{I}, \vec{\varphi}, \sqrt{\varepsilon})
= h_0(\vec{I}, \varepsilon) + P(y,x,\vec{I}, \varepsilon)
+ \mu F(y,x,\vec{I}, \vec{\varphi}, \sqrt{\varepsilon}),
\label{resnf}
\end{equation}
\noindent
where $(y, x) \in \bbR^2$, $(\vec{I}, \vec{\varphi})
\in \bbR^{n-1} \times T^{n-1}$,
and $|\mu| \leq \mu_0= {\cal O}(\varepsilon^d)$,
the latter being such that $d\geq{\frac{3}{2}}$.
Precise statements on the
size of domains and analyticity conditions can be found in
Rudnev and Wiggins [1997a]. We regard $\mu$ as the
perturbation parameter in (\ref{resnf}). Then
the important point in
relation to Arnold's example is that under certain
non-degeneracy conditions, for fixed $\vec{I}$ and $\varepsilon$,
$P(y,x,\vec{I},\varepsilon)$ is the Hamiltonian of a one
degree-of-freedom Hamiltonian system whose phase portrait is
qualitatively that of a simple pendulum. In particular, it has
hyperbolic equilibria connected by homoclinic orbits. Thus,
we see that (\ref{resnf}) has the structure of the Hamiltonian
of a pendulum coupled to $n-1$ action-angle pairs, or {\em rigid
rotors}. This is the way in which one can view Arnold's example as a
model
for
the global dynamics near a multiplicity one resonance surface. This
naturally
raises the question ``is Arnold's mechanism of transition
chains a generic mechanism causing the drift of trajectories
along a multiplicity one resonance surface?'', to which the answer is
not known. This example also brings up two other points.
\begin{itemize}
\item The two-parameter stratagem is often useful for artificially
decomposing the problem into various components in a way that allows one
to examine separately the role that each component plays with respect to
some phenomenon of interest, e.g., separatrix splitting. However, the
parameters may be functionally related, or one may want to consider a
functional relationship in order to estimate the size in parameter space
where a certain phenomenon occurs. The nature of this functional
relationship will play a crucial role in the justification of the
Melnikov integral as the dominant term in the splitting distance.
\item In this particular example (i.e., dynamics near a multiplicity one
resonance surface) the perturbation contains an infinite number of
harmonics (the perturbation in Arnold's example contained two). These
multifrequency perturbations will lead to small divisors in the
perturbation series for the splitting distance which can make it very
difficult to estimate the relative sizes of various terms in the
perturbation series.
\end{itemize}
\subsection{Arnold's diffusion versus Arnold's mechanism}
In this section we will continue further establishing the terminology, the
task, which,
however didactic, we feel obliged to pay heed to, although it has already
been pointed
out by Arnold [1964] and further by Chierchia and Gallavotti [1994],
Lochak [1995],
Rudnev and Wiggins [1997b].
In the Introduction we have described Arnold's mechanism, and clearly it
requires
the presence of some hyperbolicity, however weak (a-priori stable systems
near simple
resonances) or strong (a-priori unstable systems). Nevertheless, "Arnold's
diffusion" will
pertain to the former class of systems only.
Recall, that with a certain geometric assumption, such as convexity about
the unperturbed
energy surface, the Nekhoroshev theorem will guarantee that the "speed of
diffusion", or the
quantity inverse to the time it takes for the actions, which used to be
the first
integrals for the unperturbed problem, to evolve by $O(1)$, be
exponentially small in the
perturbation parameter, with the exponent depending upon the number of
degrees of freedom. In other
words, at least one aspect of the "Arnold diffusion" problem shall go
beyond all orders of
canonical perturbation theory. Arnold's mechanism appears to be so far the
easiest way, at
least, to imagine such an evolution of the action variables. Besides,
different aspects of
it can be treated analytically reasonably well.
This is undoubtedly one of the most important and difficult problems in
the theory of
near-integrable Hamiltonian systems. Furthermore, using the term
"Arnold's diffusion"
we will imply "Arnold's mechanism for a-priori stable systems". The rest
of this section
will delineate what we consider the main problems.
We remark that establishing Arnold's mechanism for a-priori {\em unstable
systems} is considerably
much easier, for it is almost entirely the application of canonical
perturbation theory to a
finite (first) order. It was successfully studied in detail by
Chierchia and Gallavotti [1994]. Bernard [1996] studied Arnold's example
with $\varepsilon=1$ and showed that the drift along the orbits shadowing
the transition chains
can be as fast as the square of the perturbation parameter $\mu$. In
general, due to the two
above mentioned works, Arnold's mechanism for the a-priori unstable
systems can be deemed rather
well understood, although certain questions pertaining to its
establishment for general systems
still remain open. Notably, the ``gap problem'', as described below, is
still an impediment
to constructing transition chains of $O(1)$ length in the action space.
Another remark will be that real analyticity is indeed one of the main
sources of difficulty.
For in the $C^\infty$ case, by using suitable perturbations with compact
support,
it's reasonably well understood how one can construct transition chains,
due to
Douady [1988]. Apparently, in this case the Nekhoroshev theorem does not
apply.
The obvious strategy to attack the Arnold diffusion problem has been to
attempt to adapt
to it or slightly modify certain things, known about a-priori unstable
systems. Since the
geometric picture behind it still remains the same, construction of a
transition chain would
in principle suffice to vindicate "Arnold's diffusion", as far as the
existence is concerned.
This is undoubtedly the most important direction of research in this area
today.
In this connection, the main difficulty, as is mostly regarded, is the
splitting size versus the
gaps between the surviving whiskered tori caused by the presence of
resonances of higher
multiplicity for the progenitor a-priori stable system (or, resonances one
order lower multiplicity,
if the system is viewed as a-priori unstable). The latter have a maximum
size of the square
root of the perturbation parameter, whereas the former are either
exponentially small for
a-priori stable systems, or of the order of the perturbation parameter for
a-priori unstable systems.
In both cases, in principle, for one wants the splitting size to exceed
the size of the gaps,
one can try to restrict the analysis by considering non-resonant regions
in the action space
(in particular, with respect to the perturbation spectrum), where the gaps
are considerably
smaller, and go fairly far with a-priori unstable systems, as did
Chierchia and
Gallavotti [1994]. However, such a domain will become uninterestingly
small (of the characteristic
size, given by some power of the perturbation parameter, larger than
${1\over2}$) in
the a-priori stable case. Practically, it can be achieved by fulfilling a
number of
normalization steps, as it is done in the analytic part of the proof of
the Nekhoroshev theorem.
For a-priori unstable systems it would require a finite fixed number of
such steps
(one or two) to make sure that in the new coordinates the splitting size
exceeds the remaining
gaps that have not been cut out in the process of construction of the
Normal form. Whereas in
the a-priori stable case, it would depend on the perturbation parameter,
being its certain
inverse power, which is another incarnation of the singular nature of the
problem. Hence, as a
price to pay, the characteristic size of the domain in the action space
where such a Normal
form is valid will never exceed the radius of confinement in the
Nekhoroshev theorem.
Therefore, Arnold's mechanism for the instability is really interesting if
one can establish
the existence of transition chains which are longer than the Nekhoroshev
confinement
radius, and at least longer than $\sqrt{\varepsilon}$, with $\varepsilon$
being a perturbation
parameter for an a-priori stable system. The fact that the values of the
actions can
"oscillate" rather rapidly, but with a small amplitude of the order
$\sqrt{\varepsilon}$ near
resonances is quite well known. Furthermore, the exhaustive treatment of
Arnold's mechanism,
even for a-priori unstable
systems, will not be complete until one attempts to understand the
dynamics involving the
resonances of higher multiplicity, or the resonance junctions, and how
they can be incorporated into
the overall picture. This part is clearly missing in the original recipe
due to Arnold.
On the other hand, both technically and from the point of view of
optimality of the estimates,
construction of a real orbit shadowing the transition chain has so far
been very little
attended and is likely to turn out to be very hard just as well.
Traditionally, this has been
considered an easier aspect of the matter. Moeckel [1996] (considering an
a-priori unstable
system and using the first order of perturbation theory to construct a
very short transition
chain) says that the diffusion rate has the same order as the perturbation
parameter, which is
a very optimistic statement (Bernard [1996] believes that for a-priori
unstable systems,
perturbed by order $\mu$ the fastest possible drift rate will not exceed
$O(\mu\log^{-1}\mu^{-1})$, whereas he himself ends up getting $O(\mu^2)$).
It's not clear how it
can be neatly done in the general setting for a-priori unstable systems,
and whether there is
any principal difference between a-priori stable and unstable systems,
provided that the
transition chain has been constructed. So far,
the constructive estimates, obtained by Chierchia and Gallavotti [1994]
and Marco [1996] seem to be anything but optimal. As for the variational
methods used by
Bessi [1996], and after him Bernard [1996], which produced much better
results,
they seem to have heavily needed the fact that the perturbation vanished
on the tori, thus
being free to choose arbitrary frequencies for the transition tori.
Eventually, there is another source of severe difficulties which pertain
in our opinion
exclusively to a-priori stable systems, and so far have displayed
themselves in our attempts
to study the splitting, but will almost surely be around on any further
step of the program
above outlined. These problems come from the necessity to fulfill certain
uniform
(with respect to the tori connected into a transition chain) linear
approximations of the
frequency vectors on these tori, with the error going to zero as the
perturbation parameter
goes to zero. This, in particular, implies a certain non-uniformity in the
constructions,
for as the perturbation size goes to zero the approximation properties of
the frequency
vectors in consideration may change. The last chapter in Rudnev and
Wiggins [1997b]
(see also Rudnev and Wiggins [1997c]) deals
with such, when the tori have dimension $2$, in which case the well
developed tools of Continued
fractions theory can be employed. Even then the obstacles are numerous,
and the problem
solved is, perhaps, the simplest of this nature. In higher dimension,
though, the situation is
more dour, for no approximation tools that compare in efficiency with the
theory of Continued
fractions are available. The rest of this note demonstrates one such
problem for which we can make some progress.
\section{The Generalized Arnold Model}
We will illustrate the main issues concerning the use of the Melnikov
vector as as measure of the splitting
distance in a-priori stable systems by considering a generalization of
Arnold's model given in (\ref{Arny}) that we described earlier. We will
refer to this as the {\em generalized Arnold model}, which is given by the
following ``two-parameter'' system:
\begin{equation}
H(y,x,\vec{I},\vec{\varphi})=\sum^{n-1}_{j=1}{I_j^2\over 2}+{y^2\over 2}
+\varepsilon(\cos{x}-1)+\mu F(x,\vec{\varphi}),
\label{Aham}
\end{equation}
\noindent
where
\[
(y,x)\in \bbR\times T,\ \ (\vec{I},\vec{\varphi})\in
\bbR^{n-1}\times T^{n-1},\ \ T=\bbR/\bbZ,
\]
\noindent
and $\varepsilon$ and $\mu$ are (small) parameters. Later, we will take
$\mu$ as a specific
algebraic function of $\varepsilon$ so that (\ref{Aham}) is indeed an
a-priori stable system.
\medskip
\begin{quotation}
\noindent
{\em Since the purpose of this note is to point out the difficulties with
justifying the Melnikov integral as the dominant term in the
perturbation series for the distance between the stable and unstable
manifolds of a given torus with diophantine frequency, we will refer the
reader to Rudnev and Wiggins [1997b] for proofs of many of the technical
estimates, as well as discussion and proofs of the analyticity
properties of the functions and clarification of the sizes of
certain expressions with respect to various norms.}
\end{quotation}
\medskip
\noindent
We refer to the Hamiltonian (\ref{Aham}) with
$\mu =0$ as the unperturbed Hamiltonian, and we will discuss the
assumptions on the perturbation $F(x,\vec{\varphi})$ in the next
subsection.
If $\mu=0$,
the Hamiltonian (\ref{Aham}) describes the uncoupled motion of
a pendulum in the $y-x$
plane (these variables we will further call {\it hyperbolic}) and
$n-1$ rotors, rapidly (compared to the time-scale of the pendulum)
gyrating with constant frequencies.
The Hamiltonian vector field generated by (\ref{Aham}) is given by:
\begin{equation}
\begin{array}{l}
\dot{x}=y, \\
\hfill \\
\dot{y}=\varepsilon\sin{x}-\mu g(x,\vec{\varphi}), \\
\hfill \\
\dot{\vec{\varphi}}=\vec{I}, \\
\hfill \\
\dot{\vec{I}}=-\mu \vec{f}(x,\vec{\varphi}).
\end{array}
\label{flow}
\end{equation}
\noindent
Here $g(x,\vec{\varphi})=\partial_x F(x,\vec{\varphi}),\
\vec{f}(x,\vec{\varphi})
=\partial_{\vec{\varphi}} F(x,\vec{\varphi})$.
This system was originally considered by Lochak [1990].
For $\mu=0$ the motion, described by the Hamiltonian (\ref{Aham}),
is very simple.
There is a $2n-2$ dimensional invariant manifold
\[
{\cal M}_0\equiv\{(x,y,\vec{\varphi},\vec{I}):(x,y)=(0,0)\},
\]
\noindent
which is foliated by $(n-1)$-tori with frequencies
$\vec{\omega}(\vec{I})=\vec{I}$. The manifold ${\cal M}_0$ is
normally hyperbolic. Its
$2n-1$ dimensional stable and unstable
manifolds coincide and form an invariant manifold ${\cal M}^h_0$
which is foliated by the coinciding
stable and unstable manifolds (whiskers)
of each individual torus. The separatrix
of the pendulum is defined by the relation
\begin{equation}
{y^2\over 2}
+\varepsilon(\cos{x}-1)=0.
\label{sep}
\end{equation}
\noindent
The unperturbed trajectories in ${\cal M}^h_0$ are given by
\begin{equation}
\begin{array}{lll}
x_0(t) & = & 4\arctan\left(e^{-\sqrt{\varepsilon}t}\right), \\
\hfill \\
y_0(t) &= & {2\sqrt{\varepsilon}\over\cosh(\sqrt{\varepsilon}t)},
\\
\hfill \\
\vec{\varphi}_0(\vec{\omega},t,\vec{\alpha}) &=& \vec{\alpha} +
\vec{\omega} t, \\
\hfill \\
\vec{I}_0(\vec{\omega}) &= &\vec{\omega}.
\end{array}
\label{Aunp}
\end{equation}
\noindent
where $\vec{\omega}\in D\subseteq \bbR^n$,
and $\vec{\alpha}$ corresponds
to the initial condition at
$t=0$.
Our analysis will simultaneously apply to any of the
tori of the normally hyperbolic
manifold ${\cal M}_0$,
whose frequencies $\vec{\omega}\equiv\partial_{\vec{I}} H|_{\mu=0}$ enjoy
the classical diophantine condition, formulated in terms of two parameters
$\gamma, \tau$, $0<\gamma\leq 1,\ \tau \geq n-2$,
the latter will be considered fixed. In other words, we require that
$\vec{\omega}\in\Omega_\gamma$ for
\begin{equation}
\Omega_\gamma\equiv\{\vec{\omega}:|\vec{\omega}\cdot\vec{k}|\geq\gamma
|\vec{k}|^{- \tau},\
\forall \vec{k}\neq \vec{0}\}.
\label{dio}
\end{equation}
\subsection{The Perturbation}
There are a number of key features that we require the perturbation to
possess.
\begin{enumerate}
\item {\bf Parity.} We choose the perturbation to be time-angle even, or
equivalently,
even in the variables $(x,\vec{\varphi})$. This is, as we already
mentioned,
necessary to be able to ascertain the existence of zeroes of the splitting
distance function to all orders.
\item{\bf Analyticity.} We require that the perturbation be a
trigonometric polynomial of
finite order in the $x$ variable. The (technical) reason why it is
necessary is as follows.
To develop canonical perturbation theory for the whiskers, the derivatives
of the perturbations
shall be evaluated on the unperturbed homoclinic orbit of the pendulum,
which has a singularity
away from the real axis. We need to have quantitative control over the
perturbation in the whole
complex domain for the unperturbed homoclinic manifold, and at the
aforementioned singularity of
the unperturbed solution, the perturbation will have a pole of the order,
proportional to its degree
in the $x$ variable. Thus, if this degree is not bounded, we will lose
the desired control.
\item{\bf Full spectrum.} In order to guarantee the dominance of the
Melnikov vector for the splitting
distance, expressed as a convergent Fourier series in $\vec{\alpha}$, the
former shall possess all the harmonics.
The reason for it is the appearance of the expressions
${\vec{k}\cdot\vec{\omega}\over\sqrt{\varepsilon}}$ in the
exponents further in (\ref{Melnikov}) in the first, and generally in all,
orders of perturbation theory. Therefore,
depending upon the approximation properties of $\vec{\omega}$, a certain
(depending on $\varepsilon$ and changing
as the latter goes to zero) mode is likely to dominate the rest, and we
must be sure, it is present
in the Melnikov function. The precise analysis of the way it happens can
be done for three degrees of
freedom. In higher dimensions, one has to be content with most elementary
estimates.
\item{\bf Control over Fourier coefficients.} Since we are not going to
compute the higher orders of perturbation
theory, all we will know about the remainder to the splitting distance
function will be its norm estimate and analyticity properties,
in particular in the angle variables, which will bound from above its
Fourier coefficients. Therefore, to insure that the
Melnikov function dominates, we will need to know that its Fourier
coefficients do not vanish too fast in the ultraviolet
region, but stick in their behavior to the upper bound. That's why
(\ref{special}) below has an exponential factor
$\exp(-|\vec{k}|\sigma_0)$. So, not only shall the perturbation have a
full
spectrum, but all its Fourier coefficients must be in a way maximum
possible (allowed by the norm).
One can compare this with the previous passage; one can also see that in
(\ref{Melnikov}) there are
two competing quantities in the exponents: $|\vec{k}|\sigma_0$ which grows
as $|\vec{k}|$ grows,
and $\frac{\vert \vec{k}\cdot\vec{\omega} \vert}{\sqrt{\varepsilon}}$,
which can become smaller. Where the balance
is achieved depends on the approximation properties of $\vec{\omega}$, and
appears to be fairly tractable
when the number of degrees of freedom equals $3$ (see Rudnev and Wiggins
[1997b], [1997c]).
\end{enumerate}
\noindent
{\em Remark 1.} We remark that the last two requirements above can be
interpreted in terms of analyticity
properties of the perturbation in the $\vec{\varphi}$ variable. Delshams
et al. [1996] for a similar purpose
considered a meromorphic perturbation with a known location of the nearest
pole in the complex extension of a torus.
Indeed, our Fourier series for the perturbation (\ref{special}) will
diverge if $|\Im\vec{\varphi}|\geq\sigma_0$.
Besides, the presence of a singularity in the $\vec{\varphi}$ variables
certainly makes the spectrum infinite, and is,
perhaps, enough to insure that the Fourier coefficients vanish not too
fast as we require it.
\medskip
\noindent
{\em Remark 2.} We also remark that if the perturbation does not have a
certain Fourier mode, it can be made
nonzero in a certain order of canonical perturbation theory by either
computing the Linstedt series or doing
the averaging. If the perturbation has "gaps" in its spectrum, one such
step will probably suffice. If it is a
finite trigonometric polynomial in $\vec{\varphi}$, the number of such
steps will depend on $\varepsilon$ as its
some inverse power.
Clearly, this renders the task to estimate the norm of the splitting
distance function in this case much more difficult.
\medskip
\noindent
We choose the perturbation as follows:
\begin{equation}
F(x,\vec{\varphi})=
\sum_{\vec{k}\neq \vec{0}}
P_{\vec{k}}(x)\exp(-|\vec{k}|\sigma_0)\cos(\vec{k}\cdot\vec{\varphi}),
\label{special}
\end{equation}
\noindent
where $\sigma_0 >0$ is a parameter (which is unimportant for the
purposes of this discussion, but which actually describes the size of
the extension of the domain of the perturbation to a complex
neighborhood of the torus with respect to the angle variables) and
$P_{\vec{k}}(x)=P_{-\vec{k}}(x)$ is given explicitly by
\begin{equation}
P_{\vec{k}}(x)=1+\sum_{j=1}^{\nu_{\vec{k}}}2^{-j}(2j-1)!A_{j\vec{k}}(1-\cos
x)^j,
\label{Pk}
\end{equation}
\noindent
where the integers $\nu_{\vec{k}}$ are such that
$0\leq\nu_{\vec{k}}\leq\nu_0$, with $\nu_0$ a finite, nonnegative integer,
and $0\leq A_{j,-\vec{k}}=A_{j\vec{k}}$
(this latter {\em parity property} is
important for certain technical matters, besides, it allows to
ascertain the existence of homoclinic points to all orders of perturbation
theory;
this parity property will further not play a visible role in our
discussion).
The factors $2^{-j}(2j-1)!$
merely play a normalizing role that serves to simplify certain
expressions.
The following assumptions are crucial for the Melnikov integral
to ``work'' for this example.
\begin{assumption}[Property of the Perturbation]
We assume that
\begin{equation}
\forall\vec{k}\neq\vec{0},\quad \mbox{and} \quad \forall
j\in\{1,..,\nu_{\vec{k}}\}\quad A_{j\vec{k}}\geq0
\quad \mbox{and}\quad \exists j_{\vec{k}}\in\{1,..,\nu_{\vec{k}}\}:\
A_{j_{\vec{k}}\vec{k}}\geq C_1>0
\label{sumAk}
\end{equation}
\noindent
for some positive constant $C_1$.
\label{pertassump}
\end{assumption}
The significance of this assumption is that it is one of the
ways to insure that all the Fourier modes in the Melnikov function
(\ref{Melnik}) are nonzero.
We will be measuring the splitting between the stable and unstable
manifolds of a torus labeled by the frequency $\omega$, for which we
will need the following property to hold.
\begin{assumption}[Assumption on the Frequencies of the Torus]
For a given torus, labeled by the frequency $\omega$, we assume that
any $l$-dimensional subvector of $\omega$, $l=2, \ldots , n-1$,
satisfies the diophantine property (\ref{dio}) with $\tau = l-
2\delta$, for some $0<\delta << \frac{1}{2}$.
\label{freqassump}
\end{assumption}
\section{The Splitting of Separatrices: Justification of the
Melnikov Vector as the Dominant Term for the Splitting Distance}
For a whiskered torus with the diophantine
frequency $\vec{\omega}$ we will define the homoclinic splitting distance
function as follows:
\begin{equation}
\begin{array}{lll}
\Delta(\vec{\omega},\vec{\alpha},\mu,\varepsilon)&=&(\Lambda,\vec{\Delta})
(\vec{\omega},\vec{\alpha},\mu,\varepsilon),
\\ \hfill \\
\mbox{where}
\\ \hfill \\
\Lambda(\vec{\omega},\vec{\alpha},\mu,\varepsilon)&
=&y^u(\vec{\omega},\vec{\alpha},\mu,\varepsilon)-
y^s(\vec{\omega},\vec{\alpha},\mu,\varepsilon),
\\ \hfill \\
\vec{\Delta}(\vec{\omega},\vec{\alpha},\mu,\varepsilon)
&=&\vec{I}^u(\vec{\omega},\vec{\alpha},\mu,\varepsilon)-
\vec{I}^s(\vec{\omega},\vec{\alpha},\mu,\varepsilon),
\end{array}
\label{dstnc}
\end{equation}
\noindent
where $\vec{\alpha}\in T^{n-1}$ is an angular parameter, whose choice up
to a near identity transformation depends upon the way one chooses to
parameterize the whiskers.
The measure of transversality of the splitting will be the quantity
\begin{equation}
\Upsilon=|\det \partial_{\vec{\alpha}}\vec{\Delta}|,
\label{msr}
\end{equation}
\noindent
evaluated at a zero of $\vec{\Delta}$, which is sufficient if one recalls
that $\Lambda$ and $\vec{\Delta}$ are
dependent via energy conservation.
{\em This is the measure of transversality, or the size of the splitting
which
is meant to be exponentially small when we say that splitting is
exponentially small and
in which one is interested in the Arnold diffusion problem}.
Using standard perturbation arguments (see, e.g., Wiggins [1988]), one
can express the splitting distance as a perturbation series in $\mu$ of
the following form ($\mu_0$ will be discussed shortly):
\begin{equation}
\vec{\Delta}(\vec{\omega},\vec{\alpha},\mu, \varepsilon)
=\mu\vec{M}(\vec{\omega},\vec{\alpha}, \varepsilon))+
\left({\mu\over\mu_0}\right)^2 \vec{N}(\vec{\omega},
\vec{\alpha},\mu, \varepsilon),
\label{splittingdistance}
\end{equation}
\noindent
where the leading order term
\begin{equation}
\vec{M}(\vec{\omega},\vec{\alpha}, \varepsilon)=-\int^{+\infty}_{-\infty}
\vec{f}(x_0(t),\vec{\psi})dt
\label{Melnikov}
\end{equation}
\noindent
is the Melnikov (vector) integral with
$\vec{\psi}=\vec{\alpha}+\vec{\omega}t$, and $\vec{N}(\vec{\omega},
\vec{\alpha},\mu, \varepsilon)$ denotes the remainder.
Generally, $\vec{M}(\vec{\omega},\vec{\alpha}, \varepsilon)$ will be
exponentially small with respect to $\varepsilon$, i.e.,
\[
\vert \vec{M}(\vec{\omega},\vec{\alpha}, \varepsilon) \vert = {\cal
O}(\exp (-
\varepsilon^{-a})), \quad a>0.
\]
\noindent
However, if $\mu$ is independent of $\varepsilon$ (with $\vec{N}$
bounded), then a simple implicit function theorem argument allows one to
conclude that simple zeros of $\vec{M}(\vec{\omega},\vec{\alpha},
\varepsilon)$
(with respect to $\vec{\alpha}$) imply
the existence of simple zeros of $\vec{\Delta}(\vec{\omega},\vec{\alpha}
,\mu, \varepsilon)$.
A very difficult situation arises if $\mu = \mu (\varepsilon)$ (i.e., the
a-priori stable case). In this
case if we do not have detailed knowledge of the size of
$\vec{N}(\vec{\omega},\vec{\alpha},\mu, \varepsilon)$ then we can only
assume that the Melnikov integral ``dominates'' the distance measurement
provided we take
\[
\mu = {\cal O}(\exp (-
\varepsilon^{-b})), \quad b > a>0.
\]
\noindent
This was clearly pointed out by Arnold [1964] in his original paper.
In practical applications these issues are important for determining the
size of the region in parameter space where Arnold diffusion exists.
Recently, Xia [1993], [1994] has published two papers concerned with
Arnold diffusion in versions of the three body problem. Each of these
papers also uses the ``two-parameter stratagem'' that we
described above. In Xia [1994] his $e$ (corresponding to our
$\varepsilon$)
denotes eccentricity and his $m_3$ (corresponding to our $\mu$) denotes
the mass of a ``third'' body, and he requires $m_3 << e$.
The two parameters in Xia [1993] are a reduced mass $\mu < 1$
(corresponding to our $\varepsilon$) and the eccentricity $e$
(corresponding to our $\mu$), where $e << \mu$.
We emphasize that choosing $\mu$ exponentially small with
respect to $\varepsilon$ is the most specific feature of Arnold's example,
this is this assumption that allowed to treat the Hamiltonian (\ref{Arny})
as a-priori unstable,
for from the point of view of canonical perturbation theory in
$\varepsilon$, $\mu$ is simply zero,
for it is smaller than any power of $\varepsilon$; conversely, from the
point of view of canonical
perturbation theory in $\mu$, $\varepsilon$ is order one, for
$\varepsilon^{-1}\mu^{1/p}\rightarrow0$
for any finite $p$.
\subsection{Splitting Distance in the Generalized Arnold Model}
This Melnikov function can be explicitly computed, and is found to be:
\begin{equation}
\vec{M}(\vec{\omega},\vec{\alpha}, \varepsilon)= \sum_{\vec{k}\neq
\vec{0}}
\vec{k}\sin(\vec{k}\cdot\vec{\alpha})\exp(-|\vec{k}|\sigma_0)
{\pi(\mz)\over\varepsilon\sinh\left({\pi\over2}
{\mz\over\sqrt{\varepsilon}}\right)}
\sum_{j=1}^{\nu_{\vec{k}}} A_{j\vec{k}}
\prod^{j-1}_{l=1}\left({(\mz)^2\over\varepsilon}+4l^2\right),
\label{Melnik}
\end{equation}
\noindent
and is clearly exponentially small in $\varepsilon$.
The remainder $\vec{N}$ can be shown to be exponentially small in
$\varepsilon$ also (Rudnev and Wiggins [1997b]).
We fix an ultraviolet cutoff parameter
$K(\varepsilon)=[\varepsilon^{-{1\over2}+\delta}]$
(where $[ \cdot ]$ denotes the integer part of a number).
It is shown in Rudnev and Wiggins [1997b] that the Fourier coefficients of
$\vec{N}$ obey the following bounds:
\begin{equation}
\begin{array}{llll}
|\vec{N}_{\vec{k}}|\varepsilon^{\tau\over2} & \leq & \exp\left(
-|\vec{k}|\sigma_0-{\pi\over 2}
{ | \mz |\over\sqrt{\varepsilon}}
\right), & |\vec{k}|1$ is an integer. Then there exists an integer $l$-vector $\vec{q}$
and an integer $p$ such that
\begin{equation}
1\leq\max(|q_1|,..,|q_l|)\leq M^{1\over l}\ \mbox{and}\
|\vec{\beta}\cdot \vec{q}-p|\leq {1\over M}.
\label{sapr}
\end{equation}
\label{Dir1}
\end{theorem}
>From the statement of the theorem it should be clear how the Dirichlet
theorem
can be used to find a {\em lower bound} for the
splitting distance function. It allows us to estimate
the dot products $\mz$ in the exponents from above. Hence, one can argue
that
there exists an integer vector $\vec{k}^*$ with its norm bounded as
${\cal O}(M^{1\over n-2})$, such that $|\vec{k}^*\cdot\vec{\omega}|\leq
M^{-1}$,
and then use a simple fact that the Fourier norm, which is the sum of
the absolute values of all the Fourier components, will be greater than
the absolute value of the particular Fourier coefficient
in the Fourier series for the
splitting distance function, corresponding to $\vec{k}^*$.
Unfortunately, for an arbitrary $\vec{\omega}$ it is not clear that all
the
$q's$ and $p$
in the Dirichlet Theorem, or equivalently all the components of the
aforementioned vector $\vec{k}^*$, are going to be nonzero,
which one must necessarily have to be able to perform the estimate from
below
following this scheme
because of the presence of
$\vec{k}$ in the formula (\ref{Melnik}) for the Melnikov
function.
One cannot totally avoid this difficulty, and given $\vec{\omega}$, the
strategy for getting the lower bound may vary. The approach that we
follow
can be applied to all the frequency vectors for which the ratio of the
absolute value of the maximum modulus component and the absolute
value of the second maximum modulus component is independent of
$\varepsilon$.
We denote
\[
\omega_+=\max_{j=1,..,n-1}|\omega_j| = |\vec{\omega}|,
\]
\noindent
and without loss of generality we can assume that the maximum is achieved
on the first component: $\omega_+=|\omega_1|$.
Then for the dot
products $\mz$ we can write
\begin{equation}
|\mz| = \omega_+|p+\vec{q}\cdot\vec{\beta}|,
\label{lapr1}
\end{equation}
\noindent
where $p=\pm k_1$ - the first component of the integer $(n-1)$-vector
$\vec{k}$,
which is multiplied by $-1$ if $\omega_1$ is negative (the latter
cannot be zero by the diophantine condition),
$\vec{q}=(\vec{k}_2,..,k_{n-1}) \in \bbZ^{n-2}$, and $\vec{\beta}=
({\omega_2\over \omega_+ },..,{\omega_{n-1}\over \omega_+ }) \in
\bbR^{n-2}$.
We will also denote
\[
\omega_-=\max_{j=2,..,n-1}|\omega_j|,
\]
\noindent
namely the "second maximum", then again, without loss of generality
we can assume that $\omega_-=|\omega_2|$.
(In the standard notation $[\cdot]$ will stand for an integer part and
$\{\cdot\}$
for a fractional part of a real number.)
Then it is easy to establish the following Proposition, which is proved in
Rudnev and Wiggins [1997b]
by using the standard box argument due to Dirichlet.
\begin{proposition}
Given an $(n-1)$-vector $(\pm 1,\vec{\beta})$,
where $\vec{\beta}\in \bbR^{n-2}$ has rationally
independent components, with $|\vec{\beta}|=\max_{j=1,..,n-2}|\beta_j|$,
and a real number $M>2$ such that also
\begin{equation}
M > \left ({(n-3)(n-2)\over2}\right)^{n-2},
\label{mcond}
\end{equation}
\noindent
there exists an $(n-1)$-integer vector $\vec{k}^*$, whose all components,
except for possibly the first one, are nonzero, and such that
\begin{equation}
|\vec{k}^*|< 2(n-1)M^{1\over n-2}\max(1,|\vec{\beta}|),
\ \ \ |\vec{k}^*\cdot\vec{\omega}|<{1\over M}.
\label{kstar}
\end{equation}
\label{smalldivisors}
\end{proposition}
\noindent
Using this Proposition we can argue that given $M>2$ and satisfying
(\ref{mcond})
there exists an integer vector $\vec{k}^*$ with no component, except for
possibly the first one, equal
to zero, such that the corresponding
exponential term in (\ref{exponent}) will be bounded
from
below as follows:
\begin{equation}
\begin{array}{lll}
\exp\left(-|\vec{k}^*|\sigma_0-{\pi\over2}
{|\vec{k}^*\cdot\vec{\omega}|\over\sqrt{\varepsilon}}\right)
&\geq&
\exp\left(-2(n-1)M^{1\over n-2}\sigma_0-{\pi\over2}
{\omega_+\over M\sqrt{\varepsilon}}\right)
\\
\hfill & \hfill & \hfill \\
&\geq&
\exp\left(-2(n-1)M^{1\over n-2}\sigma_0{\omega_+\over\omega_-}-{\pi\over2}
{\omega_+\over M\sqrt{\varepsilon}}\right).
\end{array}
\label{exponent1}
\end{equation}
\noindent
Unfortunately, we don't know whether $k^*_1$ is nonzero. If it is, then
we repeat what we have just done, representing in the same notation
$\mz=\omega_- \vec{k}\cdot(\pm1, \vec{\beta})$, where now
$\vec{\beta}=({\omega_1\over\omega_-},{\omega_3\over\omega_-},..,
{\omega_{n-1}\over\omega_-})$, and again apply Proposition
\ref{smalldivisors}
to establish the existence of an integer vector $\vec{k}'$ which satisfies
(\ref{kstar}) and has all the nonzero components except for possibly
the second one, yielding the following estimate for the corresponding
exponential term:
\begin{equation}
\begin{array}{lll}
\exp\left(-|\vec{k}'|\sigma_0-{\pi\over2}
{|\vec{k}'\cdot\vec{\omega}|\over\sqrt{\varepsilon}}\right)
& \geq &
\exp\left(-2(n-1)M^{1\over n-2}\sigma_0{\omega_+\over\omega_-}-{\pi\over2}
{\omega_-\over M\sqrt{\varepsilon}}\right)
\\ \hfill \\
&\geq &
\exp\left(-2(n-1)M^{1\over n-2}\sigma_0{\omega_+\over\omega_-}-{\pi\over2}
{\omega_+\over M\sqrt{\varepsilon}}\right).
\end{array}
\label{exponent11}
\end{equation}
\noindent
The absolute value of the exponent in the coinciding rightmost sides of
(\ref{exponent1}) and (\ref{exponent11})
as a function of $M$ has a minimum when
\begin{equation}
M=M^*=\left({\pi\omega_-(n-2)\over
4(n-1)\sigma_0\sqrt{\varepsilon}}\right)^{n-2\over n-1},
\label{Mstar}
\end{equation}
\noindent
which for $\varepsilon$ small enough is certainly bigger than $2$
and satisfies (\ref{mcond}), provided that the quantity $\omega_-$
is $\varepsilon$-independent.
Substituting (\ref{Mstar}) into the lower bound at (\ref{exponent1}) or
(\ref{exponent11}), we can rewrite it as:
\begin{equation}
\exp\left(-\omega_+(n-1)\left({2\sigma_0(n-1)\over
\omega_-(n-2)}\right)^{n-2\over n-1}
\left({\pi\over2\sqrt{\varepsilon}}\right)^{1\over n-1}\right).
\label{exponent2}
\end{equation}
\noindent
So in the Fourier series for (\ref{splittingdistance}) there are at least
two
terms below the ultraviolet cutoff (labeled by $\vec{k}^*$ and
$\vec{k}'$), whose exponentially small part is estimated
from below by (\ref{exponent2}). That's why we can give an estimate {\it
from
below} for the Fourier norm
of the splitting distance function. Moreover, it is in applying
Proposition \ref{smalldivisors} to construct $\vec{k}^*$ and
$\vec{k}'$ where we use Assumption \ref{pertassump}. This is because we
have no real control over the construction of $\vec{k}^*$ and
$\vec{k}'$ (Proposition \ref{smalldivisors} is an existence proof), and
once constructed, we need to know that the corresponding Fourier
amplitudes are nonzero. This is guaranteed by Assumption
\ref{pertassump}.
Therefore, the Fourier norm (see Rudnev and Wiggins [1997a], [1997c] for
definition and properties)
for the $i$th component of the homoclinic
splitting-distance
function $(i=1,..,n-1)$ will be bounded from below by the following
expression (we neglect the exponentially small ultraviolet part of the
remainder from (\ref{Gestimates}), thinking that $\varepsilon$ is small
enough,
and the exponentially small error has also been taken care of in the
quite rough estimates for the norm of $\vec{k}^*$ or $\vec{k}'$,
given by Proposition \ref{smalldivisors}.
We use the notation
\[
\vec{k}^{\delta_{i1}}=\left[
\begin{array}{l}
\vec{k}' \ \mbox{for}\ i=1 \\
\vec{k}^*\ \mbox{for}\ i\neq1,
\end{array}
\right.
\]
\noindent
which means that for $i=1$ we take the mode, indexed by $\vec{k}'$, since
we know for sure
that $k'_1\neq0$, and for the rest of the
components of the splitting distance function we take the mode indexed by
$\vec{k}^*$,
because we also know that $k^*_2,..,k^*_{n-1}\neq 0$. This can be used
to obtain the following estimate for a lower bound on the sup-norm of
each component of the splitting
distance:
\begin{equation}
\begin{array}{lll}
|\Delta_i|
&\geq& \left( |\mu|\varepsilon^{-1+{\tau \over2}}
\sum_{j=1}^{\nu_{\vec{k}^{\delta_{i1}} }}
A_{j\vec{k}^{\delta_{i1}}}
\prod^{j-1}_{l=1}\left({(\vec{k}^{\delta_{i1}}\cdot\vec{\omega})^2\over\varepsilon}+4l^2\right)
- \left|{\mu\over\mu_0}\right|^2
\varepsilon^{1-{\tau\over2}} \right)
\\ \hfill \\
& \times &\exp\left(-\omega_+(n-1)\left({2\sigma_0(n-1)\over
\omega_-(n-2)}\right)^{n-2\over
n-1}\left({\pi\over2\omega_-\sqrt{\varepsilon}}\right)^{1\over n-1}\right)
\\ \hfill \\
&\geq&{|\mu|\over\mu_0} \exp\left(-\omega_+(n-1)\left({2\sigma_0(n-1)\over
\omega_-(n-2)}\right)^{n-2\over n-1}
\left({\pi\over2\omega_-\sqrt{\varepsilon}}\right)^{1\over n-1}\right).
\end{array}
\label{lowersd}
\end{equation}
\noindent
The last thing to show is that the Melnikov function really dominates
over the remainder. This basically follows from (\ref{Gestimates}) if we
manage to
compare the lower bound for the norm of the Melnikov function with the
upper bound for
the sum of all the terms in $N_i$ such that $k_i=0$ for
$|\vec{k}|3$.
Indeed, for $\varepsilon$ small enough and positive constants $K_1, K_2$,
such that $K_1>K_2$ and exponents
$a,p$ such that $00$, we always have
\[
\varepsilon^p \exp\left( -K_2{1\over\varepsilon^a}\right)\gg
\exp\left( -K_1{1\over\varepsilon^a}\right),
\]
\noindent
which means that if $\mu$ is a finite power of $\varepsilon$, to prove the
fact that the
first order of perturbation theory dominates in the quantity $\Upsilon$
we shall have to compare the constants $K_1$ in the first order and $K_2$
in the remainder,
which is practically almost impossible in a broad setting when $n>3$.
Nevertheless, when $n=3$, namely when there are two rotators in the
Generalized Arnold Model,
it's fairly easy, see Rudnev and Wiggins [1997b], [1997c].
\section*{References}
\begin{verse}
Arnold, V.I. [1964] Instability of dynamical systems with several
degrees of freedom. {\em Sov. Math Dokl.}, {\bf 5}, 581-585.
Bernard, P. [1996] Perturbation d'un Hamiltonien partiellement
hyperbolique. {\em C. R. Acad. Sci. Paris}, {\em 323}, 189-194.
Bessi, U. [1996] An approach to Arnold's diffusion through the calculus of
variations. {\em Nonlinear Analysis}, {\bf 26}(6), 1115-1135.
Chierchia, L. and Gallavotti, G. [1994] Drift and diffusion in
phase space. {\em Ann. de la Inst. H. Poincar\'e Phys. Th.}, {\bf
60}(1), 1-144.
Delshams, A., Gelfreich, V., Jorba, A., Seara T. [1996] Exponentially
small splitting of separatrices under fast quasiperiodic forcing. {\em
Preprint}.
Douady, R. [1988] Stabilit\'{e} ou Instabilit\'{e} des Points Fixes
Elliptiques.
{\em Ann. scient. \'{E}c. Norm. Sup.}, {\bf 21}(4), 1-46.
Lochak, P. [1990] Effective speed of Arnold's diffusion and
small denominators.
{\em Phys. Lett. A}, {\bf 143}(1,2), 39-42.
Lochak, P. [1995] "Arnold diffusion"; a compendium of remarks and
questions.
To appear in {\em Proceedings of 3DHAM, s'Agaro 1995.}
Moeckel, R. [1996] Transition Tori in the Five-Body Problem. {\em J. Diff.
Eqns.},
{\bf 129}, 290-314.
Rudnev, M. and Wiggins, S. [1997a] KAM theory near multiplicity one
resonant
surfaces in perturbations of a-priori stable Hamiltonian systems.
{\em Journal of Nonlinear Science}, {\bf 7}(2), 177-209.
Rudnev, M. and Wiggins, S. [1997b] Existence of Exponentially Small
Separatrix Splittings and Homoclinic Connections Between Whiskered Tori
in Weakly Hyperbolic Near Integrable Hamiltonian Systems. submitted to
{\em Physica D}.
Rudnev, M. and Wiggins, S. [1997c] On the Dominant Fourier Modes
in the Separatrix Splitting Distance Function for an
A-Priori Stable, Three Degree-of-Freedom Hamiltonian System. Caltech
preprint.
Wiggins, S. [1988] {\em Global Bifurcations and Chaos: Analytical
Methods}. Springer-Verlag: New York.
Xia, Z. [1993] Arnold diffusion in the elliptic restricted three-body
problem. {\em J. Dyn. Diff. Eqns}, {\bf 5}(2), 219-240.
Xia, Z. [1994] Arnold diffusion and oscillatory solutions in the planar
three-body problem. {\em J. Diff. Eqns.}, {\bf 110}, 289-321.
\end{verse}
\end{document}