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%% Modified February 5, 1997
\controldates{25-FEB-1997,25-FEB-1997,25-FEB-1997,25-FEB-1997}
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%Definitions of numbers
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\Z}{\mathbb{Z}}
% Math macros
\newcommand{\abs}[1]{\left | #1 \right |}
\newcommand{\Or}{\operatorname{O}} %\mathop{\rm O}\nolimits}
\newcommand{\const}{\operatorname{const}} %{\rm const}\,}
\newcommand{\dif}{\mathrm{d}} %{\rm d}}
\newcommand{\e}{\,\mathrm{e}} %{\rm e}}
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\issueinfo{3}{1}{January}{1997}
\dateposted{March 12, 1997}
\pagespan{1}{10}
\PII{S 1079-6762(97)00017-6}
\copyrightinfo{1997}{American Mathematical Society}
\commby{Jeff Xia}
\date{July 9, 1996}
\begin{document}
\title[Separatrix splitting for fast quasiperiodic forcing]
{Lower and upper bounds for the splitting of separatrices of the
pendulum under a fast quasiperiodic forcing}
\author[A. Delshams]{Amadeu Delshams}
\address{Departament de Matem\`atica Aplicada I\\
Universitat Polit\`ecnica de Catalunya\\
Diagonal 647, 08028 Barcelona, Spain}
\email{amadeu@ma1.upc.es}
\author[V. Gelfreich]{Vassili Gelfreich}
\address{Departament de Matem\`atica Aplicada i An\`alisi\\
Universitat de Barcelona\\
Gran via 585, 08007 Barcelona, Spain}
\curraddr{Chair of Applied Mathematics\\
St.Petersburg Academy of Aerospace Instrumentation\\
Bolshaya Morskaya 67, 190000, St. Petersburg, Russia}
\email{gelf@maia.ub.es, gelf@misha.usr.saai.ru}
\author[A. Jorba]{\`Angel Jorba}
\address{Departament de Matem\`atica Aplicada I\\
Universitat Polit\`ecnica de Catalunya\\
Diagonal 647, 08028 Barcelona, Spain}
\email{angel@tere.upc.es}
\author[T. M. Seara]{Tere M. Seara}
\address{Departament de Matem\`atica Aplicada I\\
Universitat Polit\`ecnica de Catalunya\\
Diagonal 647, 08028 Barcelona, Spain}
\email{tere@ma1.upc.es}
\subjclass{Primary 34C37, 58F27, 58F36; Secondary 11J25}
\keywords{Splitting of separatrices, quasiperiodic forcing,
homoclinic orbits, normal forms.}
\begin{abstract}
Quasiperiodic perturbations with two frequencies
$(1/\varepsilon ,\gamma /\varepsilon )$ of a pendulum are considered,
where $\gamma $ is the golden mean number.
We study the splitting of the three-dimensional invariant manifolds
associated to a two-dimensional invariant torus in a neighbourhood
of the saddle point of the pendulum.
Provided that some of the Fourier coefficients of the perturbation
(the ones associated to Fibonacci numbers) are separated from zero,
it is proved that the invariant manifolds split for $\varepsilon $
small enough.
The value of the splitting, that turns out to be
$\Or \left(\exp\left(-\const /\sqrt{\varepsilon }\right)\right)$,
is correctly predicted by the Melnikov function.
\end{abstract}
\maketitle
\section{Introduction}
The rapidly (and periodically) forced pendulum has been widely used
as a model for the motion near a resonance of Hamiltonian systems
with two degrees of freedom.
As is well known in several
situations~\cite{DelshamsS92},
\cite{Gelfreich93}, the separatrices of the
perturbed system do not coincide, giving rise to the so-called
splitting of separatrices, which seems to be the main cause of
stochastic behaviour in Hamiltonian systems.
In this announcement we consider a quasiperiodic high-frequency
perturbation of the pendulum (it can be regarded as a model near
a resonance of a Hamiltonian system with three
degrees of freedom), described by the Hamiltonian function
\begin{equation}
\frac{\sprod{\omega}{I}}{\varepsilon }+h(x,y,\theta,\varepsilon ),
\label{Eq:h}
\end{equation}
where
\[
\sprod{\omega}{I}=\omega_1 I_1+\omega_2 I_2,\qquad
h(x,y,\theta,\varepsilon )
=\frac{y^2}{2}+\cos x +\varepsilon ^p m(\theta_1,\theta_2)\cos x,
\]
with symplectic form
$\dif x \wedge \dif y + \dif \theta_1\wedge\dif I_1
+\dif \theta_2\wedge\dif I_2$.
We assume that $\varepsilon $ is a small positive parameter and that $p$
is a positive parameter. Mainly due to a technical limitation imposed by the
Extension Theorem (Theorem~\ref{Th:extension}),
we will restrict ourselves to the case $p>3$.
We also assume that the frequency is of the form $\omega/\varepsilon $
for $\omega=(1,\gamma )$, where $\gamma =(1+\sqrt{5})/2$ is the
golden mean.
The equations of motion related to Hamiltonian~(\ref{Eq:h}) are:
\begin{xalignat}{2}
\dot x&=y,&\dot y&=(1+\varepsilon ^p m(\theta_1,\theta_2))\sin x,
\nonumber\\
\dot\theta_1&=\frac{1}{\varepsilon },&\dot I_1&=-\varepsilon ^{p}
\cos x\frac{\partial m}{\partial \theta_1}(\theta_1,\theta_2),
\label{Eq:motion}\\
\dot\theta_2&=\frac\gamma \varepsilon ,&\dot I_2&=-\varepsilon ^{p}
\cos x\frac{\partial m}{\partial \theta_2}(\theta_1,\theta_2).
\nonumber
\end{xalignat}
The function $m$ is assumed to be a $2\pi$-periodic function of two
variables $\theta_1$ and $\theta_2$. Thus it can be represented
as a Fourier series:
\[
m(\theta_1,\theta_2)
=\sum_{k_1,k_2} m_{k_1 k_2}\e ^{i(k_1\theta_1+k_2\theta_2)} .
\]
We assume that, for some positive numbers $r_1$ and $r_2$,
\begin{equation}
\sup_{k_1,k_2}\abs{m_{k_1k_2}\e ^{r_1\abs{k_1}+r_2\abs{k_2}}}<\infty,
\label{Eq:supmkk}
\end{equation}
and that there are positive numbers $a$ and $k_0$ such that
\begin{equation}
\abs{m_{k_1k_2}}\ge a \e ^{-r_1\abs{k_1}-r_2\abs{k_2}}
\label{Eq:mfib}
\end{equation}
for all $k_1$, $k_2$ such that $\abs{k_1}=F_{n+1}$ and $\abs{k_2}=F_n$,
where $F_{n}$ and $F_{n+1}$ are Fibonacci numbers, which are defined by
the following recurrent formula:
\begin{equation}
F_0=1,\qquad F_1=1,\qquad
F_{n+1}=F_n+F_{n-1},\quad n\ge 1.
\label{Eq:Fibonacci}
\end{equation}
We call the corresponding terms in the perturbation
{\em resonant\/} or {\em Fibonacci terms\/}.
For example, the function
\[
m(\theta_1,\theta_2)=\frac{\cos\theta_1\cos\theta_2}
{(\cosh r_1-\cos\theta_1)(\cosh r_2-\cos\theta_2)}
\]
satisfies these conditions.
The upper bound~(\ref{Eq:supmkk}) implies that the function $m$
is analytic on the strip
$\{\abs{\Im \theta_1}3$ and small
$\varepsilon >0$ the invariant manifolds split, and the value of the
splitting (i.e., the distance function between these invariant
manifolds) is correctly predicted by the Melnikov function, which is
$\Or (\e ^{-\const /\sqrt{\varepsilon }})$.
\begin{remark}
Our model~(\ref{Eq:h}) is based on a previous work by
C.~Sim\'o~\cite{Simo94}, where Neishtadt's Averaging
Theorem~\cite{Neishtadt84} was generalized to quasiperiodic systems,
giving rise to upper estimates of the splitting which are exponentially
small with respect to the parameter of perturbation $\varepsilon $.
Related upper estimates can be found
in~\cite{ChierchiaG94},
\cite{Gallavotti94},
\cite{BenettinCG95},
\cite{Benettin96}.
In contrast to these results, Theorem~\ref{Th:main} provides {\em both}
lower and upper bounds for our model.
\end{remark}
\begin{remark}
As an example in~\cite{DelshamsGJS96a} shows, the splitting can be of
the order of some power of $\varepsilon $ if the function $m$ is not
analytic. This makes a first qualitative difference between periodic and
quasiperiodic perturbations. Indeed, in the periodic case, only the
$C^1$ dependence with respect to $\theta $ of the perturbed Hamiltonian
is needed to prove that the splitting is $\Or (\e ^{-c/\varepsilon })$,
where $c$ is the width of the analyticity strip of the unperturbed
separatrix.
(In both cases, the analyticity of the unperturbed system is essential.)
\end{remark}
\begin{remark}
In the case of an entire function $m$, we think that the method used in
the present paper can be modified in order to improve the estimate of
the error and to prove that the Melnikov function gives the actual
asymptotics at least when the resonant terms decrease not much faster
than $1/k!$.
\end{remark}
\section{The Melnikov function}
As is well known, the Melnikov function
\begin{equation}
\label{Eq:melfun}
M(\theta_1,\theta_2;\varepsilon )=\int_{-\infty}^{\infty}
\{h_0,h\}(x_0(t),y_0(t),
\theta_1+t/\varepsilon ,\theta_2+\gamma t/\varepsilon )\,dt
\end{equation}
gives a first order approximation of the difference between the values
of the unperturbed pendulum energy $h_0$ on the stable and unstable
manifolds.
Using the Fourier series of $m(\theta _{1},\theta _{2})$, one can
compute the Fourier coefficients of $M(\theta _{1},\theta _{2})$ as
\[
M_{k_{1}k_{2}}(\varepsilon )=
-\frac{2\pi i\varepsilon ^{p}(k_{1}+\gamma k_{2})^{2}}
{\varepsilon ^{2}\sinh
\left(\pi (k_{1}+\gamma k_{2})/(2\varepsilon )\right)}
\cdot m_{k_{1}k_{2}}.
\]
In order to bound the Melnikov function, it is important to know for
each fixed $\varepsilon $ which are the indices $(k_{1},k_{2})$
corresponding to the biggest Fourier coefficient
$M_{k_{1}k_{2}}(\varepsilon )$.
From the expression above, $M_{k_{1}k_{2}}(\varepsilon )$ is a product
of two factors.
For $\varepsilon $ fixed and small enough, the first factor of
$M_{k_{1}k_{2}}(\varepsilon )$ behaves as
$\varepsilon^{p-2}(k_{1}+\gamma k_{2})^{2}
\* \e ^{ -\pi (k_{1}+\gamma k_{2})/(2\varepsilon ) }$,
and it turns out that it becomes bigger for small $k_{1}+\gamma k_{2}$,
i.e., just for the resonant terms where the second factor
$m_{k_{1}k_{2}}$ decreases with respect to $(k_{1},k_{2})$ according to
the behaviour~(\ref{Eq:supmkk}) and~(\ref{Eq:mfib}).
Here we have the main difference between the quasiperiodic case and
the periodic one: for a periodic perturbation, the first Fourier
coefficients $M_{\pm 1}(\varepsilon )$ of the Melnikov function
$M(\theta;\varepsilon)$ give generically the main contribution
to the Melnikov function $M(\theta;\varepsilon)$, since
$M_{k}(\varepsilon )=\Or(\e ^{-\abs{k}/\varepsilon})$.
However,
in the quasiperiodic case the biggest Fourier coefficient depends
strongly on the exponent $k\cdot (1,\gamma )=k_{1}+\gamma k_{2}$
of the Fourier coefficient, i.e., on the rational approximations of
$\gamma $.
For $\gamma =(1+\sqrt{5})/2$, it is very well known that its best
approximation by rational numbers is given by the quotient of
successive Fibonacci numbers~(\ref{Eq:Fibonacci}).
Indeed, it is easy to check that for large values of $n$ one has
the following approximation of $\gamma $ by Fibonacci numbers:
\[
F_n-\gamma F_{n-1}=(-1)^n
\frac{C_F}{F_{n-1}}+
\Or \left(\frac1{F_{n-1}^3}\right),\qquad
C_F=\frac{1}{\gamma +\gamma ^{-1}},
\]
whereas for the other integers one has the following result.
\begin{lemma}
If $N\in\N$ is not a Fibonacci number, then for all integers $k$
\[
\abs{k-\gamma N}>\frac{\gamma C_F}{N}.
\]
\end{lemma}
Using this lemma one can see that the indices $(k_{1},k_{2})$
corresponding to the leading Fourier coefficients
$M_{k_{1}k_{2}}(\varepsilon )$ depend on $\varepsilon $.
In fact, the largest terms correspond to
$(k_1,k_2)=\pm \left(F_{n(\varepsilon )+1},-F_{n(\varepsilon )}\right)$,
where $F_{n(\varepsilon )}$ is the Fibonacci number closest to
$F^*(\varepsilon )=\sqrt{{\phi_0}/{\varepsilon }}$,
where $\phi_0=\pi/(2(\gamma +\gamma ^{-1})(r_{1}\gamma + r_{2}))$.
Except for a small neighbourhood of
$\varepsilon =\varepsilon ^*\gamma ^{-n}$, with $\varepsilon ^{*}$
given in~(\ref{Eq:Constants}), there is a unique Fibonacci number
closest to $F^*(\varepsilon )$, and then only the two corresponding terms
dominate in the Fourier series.
Studying the size of this term (which also depends on $r_1$ and
$r_2$), one can observe that it is
$\Or (\e ^{-c/\sqrt{\varepsilon }})$.
With a more detailed analysis, one can see that a better estimate is
provided by taking $c$ to be not a constant function but a bounded
oscillating one:
\begin{equation}
\label{Eq:expconst}
c(\delta)=C_0\cosh\left(\frac{\delta-\delta_0}{2}\right)
\quad \hbox{for} \quad
\delta\in[\delta_0-\log\gamma ,\delta_0+\log\gamma ],
\end{equation}
where
\begin{equation}
C_0=\sqrt{\frac{2\pi(\gamma r_1 +r_2)}{\gamma +\gamma ^{-1}}},
\qquad \delta_0=\log\varepsilon ^*,
\quad \varepsilon ^*
=\frac{\pi(\gamma +\gamma ^{-1})}{2\gamma ^2(r_1\gamma +r_2)},
\label{Eq:Constants}
\end{equation}
continued periodically onto the whole real axis.
In this way, the function $c$ is piecewise-analytic, continuous, and
$2\log\gamma$-periodic.
This is summarized in the following lemma.
\begin{lemma}[Properties of the Melnikov function]
The Melnikov function defined by\/~{\rm(\ref{Eq:melfun})}
is a $2\pi$-periodic function of $\theta_1$ and $\theta_2$, such that
\begin{itemize}
\item[1)]
$M\left(\theta_1-T/\varepsilon ,\theta_2-\gamma T/\varepsilon ;
\varepsilon \right)$ is analytic in the product of strips
\[
\{\abs{\Im \theta_1}\max c(\delta)=C_0\cosh(\log\sqrt{\gamma })$.
\end{itemize}
\end{lemma}
Since we have established that for most small values of $\varepsilon $ only
the terms with
$(k_1,k_2)=\pm (F_{n(\varepsilon )+1},-F_{n(\varepsilon )})$
are important, the Melnikov function is essentially
\[
M(\theta_1,\theta_2;\varepsilon )\approx
2\abs{M_{F_{n(\varepsilon )+1},-F_{n(\varepsilon )}}}
\sin\left(F_{n(\varepsilon )+1}\theta_1
-F_{n(\varepsilon )}\theta_2+\varphi(\varepsilon )\right).
\]
The zeros of the Melnikov function correspond to homoclinic
trajectories. The above formula implies that the zeros of the Melnikov
function form two lines on the torus. As already noticed by
C.~Sim\'o~\cite{Simo94}, the averaged slopes of those lines approach
$\gamma $ when $\varepsilon\to 0$.
\begin{remark}
We note that the Melnikov function is not invariant with respect to
canonical changes of variables. After a change, e.g., after a step of
the classical averaging procedure, a lot of nonzero harmonics, which
were not present in the original system, can appear.
If in the original system the Fibonacci terms were not big enough,
these new harmonics may give larger contribution to the splitting.
This idea was used in~\cite{Simo94} to detect the splitting for a
system with only four perturbing terms.
\label{Rem:averaging}
\end{remark}
\begin{remark}
The hypothesis that $\omega $ in the frequency vector is just
$(1,\gamma )$ can be relaxed.
The generalization of the present result to the case when $\gamma $
is a quadratic number is straightforward, with a similar
expression~(\ref{Eq:ExpSize}) for the size of the Melnikov function.
The case in which $\omega =(\omega_1,\omega_2)$, with the ratio
$\omega_1/\omega_2$ being of constant type (the continued fraction
expansion has bounded coefficients), but not quadratic,
can be similarly analyzed, but in this case $c(\delta )$ is no longer a
periodic function.
In some sense one can say, properly speaking, that there are no asymptotics.
But it seems that there still exist upper and lower bounds, with
the factor $\sqrt{\varepsilon}$ in the denominator of the exponential
term.
The case of two frequencies whose ratio $\omega_1/\omega_2$ is not of
constant type, as well as the case of more than two perturbing
frequencies, is more complicated.
\end{remark}
In the following sections we sketch the method used to justify that
the prediction given by the Melnikov function is correct.
The method used here is a generalization to the quasiperiodic case of
the method used in~\cite{Lazutkin84},
\cite{DelshamsS92},
\cite{Gelfreich93}.
\section{Normal form and local manifolds}
The first step is to give a description of the dynamics near the
$2D$-dimensional invariant torus $\mathcal T$.
So, we will show the existence of a convergent normal form in a
neighbourhood of $\mathcal T$.
As we have seen during the analysis of the Melnikov function, the size
of the splitting depends essentially on the widths of the analyticity
strip $(r_1,r_2)$ of the angular variables $\theta _{1}$, $\theta _{2}$,
as well as on the width of the analyticity strip of the separatrix
$(x_{0}(t),y_{0}(t))$.
Therefore, to detect the splitting in the quasiperiodic case the loss of
domain in the angular variables must be very small
(i.e., $\Or(\varepsilon^{\alpha})$, where $\alpha$ depends on the
Diophantine properties of the frequencies).
This makes another difference with the periodic case, where the size of
the splitting does {\em not} depend on the width of the analyticity
strip of the angular variable $\theta $, but only on the width of the
analyticity strip of the separatrix $(x_{0}(t),y_{0}(t))$.
When dealing with the frequencies $(1,\gamma)$ one needs a
reduction of $\Or(\sqrt{\varepsilon })$ at most. Hence, during the
proof of the convergence of the normal form one has to bound carefully
the loss of domain (with respect to the angular variables) in order to
achieve such a small reduction.
Finally, we want to stress that if the amount of reduction is
something bigger, one can only produce upper bounds for the splitting
of separatrices.
\begin{theorem}[Normal Form Theorem]
\label{Th:NF}
Let $\varepsilon \in(0,\varepsilon _0)$.
In a neighbourhood of the hyperbolic torus $\mathcal T$ there is a
canonical change of variables $(x,y)\to(X,Y)$, which depends
$2\pi$-periodically on $\theta_1$ and $\theta_2$, such that the
Hamiltonian\/~{\rm(\ref{Eq:h})} takes the form
\[
H(XY,\varepsilon )=H_0(XY)+\varepsilon ^{p-1} H_1(XY,\varepsilon ),
\]
where $H_0$ is the normal form Hamiltonian for the unperturbed pendulum.
Moreover, the change of variables has the form
\begin{equation}
\label{Eq:xnf}
\begin{array}{rcl}
x=\xn(X,Y)+\varepsilon ^{p-1}\xnu(X,Y,\theta_1,\theta_2;\varepsilon ),
\\
y=\yn(X,Y)+\varepsilon ^{p-1}\ynu(X,Y,\theta_1,\theta_2;\varepsilon ),
\end{array}
\end{equation}
where $(\xn ,\yn )$ are normal form coordinates for the unperturbed
pendulum.
The functions $H_0$, $H_1$, $\xn$, $\yn$, $\xnu$, and $\ynu$ are analytic
and uniformly bounded in the complex domain defined by
\[
\abs{X}^2+\abs{Y}^20$.
\end{theorem}
The proof of this theorem can be found in~\cite{DelshamsGJS96b}.
The Normal Form Theorem provides a convenient parametrization
for the local invariant manifolds. Let $\lambda$ be $H'(0,\varepsilon)$.
Then,
\begin{equation}
\begin{array}{l}
x=x^s(T,\theta_1,\theta_2)
\equiv x(0,\e ^{-\lambda T },\theta_1,\theta_2),\\
y= y^s(T,\theta_1,\theta_2)
\equiv y(0, \e ^{-\lambda T },\theta_1,\theta_2),
\end{array} \quad\mbox{ for } T\ge T_0,
\end{equation}
and
\begin{equation}
\label{eq:lum}
\begin{array}{l}
x= x^u(T,\theta_1,\theta_2)
\equiv x(\e ^{\lambda T },0,\theta_1,\theta_2),\\
y= y^u(T,\theta_1,\theta_2)
\equiv y( \e ^{\lambda T },0,\theta_1,\theta_2),
\end{array} \quad\mbox{ for } T\le -T_0,
\end{equation}
where we have used the change~(\ref{Eq:xnf}).
Theorem~\ref{Th:NF} also implies that, in the domains above,
\[
\begin{array}{l}
\abs{x^\beta (T,\theta_1,\theta_2)-x_0(T)}\le C\varepsilon ^{p-1},\\
\abs{y^\beta (T,\theta_1,\theta_2)-y_0(T)}\le C\varepsilon ^{p-1},
\end{array}\quad\mbox{ for } \beta =s,u.
\]
\section{Extension Theorem}
The Normal Form Theorem provides a local approximation for the unstable
manifold in terms of the unperturbed separatrix, which is
$\Or{\left(\varepsilon ^{p-1}\right)}$.
The following theorem extends this local approximation for solutions
of system~(\ref{Eq:motion}) to a global one.
Since the unperturbed separatrix $(x_0(T),y_0(T))$ has a singularity on
$T=\pm \pi/2$, we will restrict ourselves to
$\abs{\Im T}\leq \pi/2 -\sqrt{\varepsilon }$, i.e., up to a distance to
the singularity $T=\pm \pi/2$ of the same order as the loss of domain
in the angular variables.
Besides, the extension time $t+T$ will be chosen big enough in order
that the unperturbed separatrix reaches again the domain of convergence
of the normal form. This procedure follows the same ideas as in the
Extension Theorem of~\cite{DelshamsS92}, and its complete proof can
be also found in~\cite{DelshamsGJS96b}.
\begin{theorem}[Extension Theorem]
\label{Th:extension}
Assume $p>2$. Then, there exists $\varepsilon _{0}>0$ such that the
following extension property holds:
For any positive constants $C$ and $T_0$ there exists a constant $C_1$,
such that for any $\varepsilon \in(0,\varepsilon _{0})$, every solution
of system~\eqref{Eq:motion} that satisfies the initial conditions
\begin{eqnarray*}
&\abs{x(t_0)-x_{0}(t_0+T)}\leq C\varepsilon ^{p-1},\quad
\abs{y(t_0)-y_{0}(t_0+T)}\leq C\varepsilon ^{p-1},
\\[10pt]
&
\abs{\Im \theta_1(t_0)}\leq r_1-\sqrt{\varepsilon },\quad
\abs{\Im \theta_2(t_0)}\leq r_2-\sqrt{\varepsilon },
\nonumber
\end{eqnarray*}
for some $T\in\C $, $t_0\in\R $ with
\[
\abs{\Im T}\leq \pi/2 -\sqrt{\varepsilon },\qquad
-T_{0}\leq t_{0}+\Re T<0,
\]
can be extended for $-T_{0}\leq t+\Re T\leq T_{0}$ satisfying
\[
\abs{x(t)-x_{0}(t+T)}\leq C_{1}\varepsilon ^{p-2},\quad
\abs{y(t)-y_{0}(t+T)}\leq C_{1}\varepsilon ^{p-2}.
\]
\label{th:ExtThm}
\end{theorem}
In particular, Theorem~\ref{Th:extension} can be applied to the
local invariant unstable manifold given in~(\ref{eq:lum}).
As we will see in Lemmas~\ref{Le:fourier} and~\ref{Le:ese}, the
above approximation of these invariant manifolds in such a complex domain
will allow us to derive suitable bounds of the error on the real axis
to detect the splitting.
Before closing this section let us note that, as a direct consequence
of the Extension Theorem, the difference of unperturbed energies
along the invariant manifolds can also be estimated.
\begin{corollary}
\label{Co:1}
The following estimate holds:
\[
h_{0}(x^{u},y^{u})-h_{0}(x^{s},y^{s}) =
M(\theta_{1}-T/\varepsilon ,\theta_{2}-\gamma T/\varepsilon ) +
\Or \left(\varepsilon ^{2(p-2)}\right),
\]
where $h_0$ is evaluated on the invariant manifolds corresponding to
$T,\theta_1,\theta_2$,
\begin{equation}
\Re T\in (T_{0}-R,T_{0}), \quad \abs{\Im T}\le \pi /2-
\sqrt{\varepsilon }
, \quad \abs{\Im \theta_{k}}\le r_{k}-\sqrt{\varepsilon },
\quad k=1,2,
\label{dfn}
\end{equation}
for any positive constants $T_0$ and $R$, $R0$ and $r_2>0$.
\end{lemma}
Applying these two lemmas to the error function
$F=\Or \left(\varepsilon ^{2p-4}\right)$ in equation~(\ref{Eq:difer}),
we obtain the desired exponentially small estimates.
Now we can summarize the above results on the splitting function
$H^{u}(T,\theta _{1},\theta _{2})$ in the following theorem, which
is the main result of this paper.
\begin{theorem}[Main Theorem]
\label{Th:main}
There exist positive constants $T_0$ and $R$, $R3$ and real $T$, $\theta_1$ and $\theta_2$,
\[
\abs{H^u(T,\theta_1,\theta_2)
-M\left(\theta_1-T/\varepsilon ,\theta_2
-\gamma T/\varepsilon \right)}
\leq \const \varepsilon ^{2p-4}
\exp\left(-\frac{c(\log\varepsilon )}{\sqrt{\varepsilon }}\right),
\]
where $c(\delta)$ is defined in~\eqref{Eq:expconst}.
If condition\/~{\rm(\ref{Eq:mfib})} is fulfilled, then there exists
$\varepsilon_{0}>0$ such that, for $0<\varepsilon<\varepsilon_{0}$,
the maximum of the modulus of the Melnikov function is larger than
the right-hand side of the last upper bound.
\end{theorem}
\section*{Acknowledgments}
We are indebted to C.~Sim\'o for relevant discussions and remarks.
Three of the authors (A. D., A. J. and T. M. S.) have been partially
supported by the Spanish grant DGICYT PB94--0215, the EC grant
ER\-BCHRXCT\-940460, the NATO grant CRG950273,
and the Catalan grant CIRIT 1996SGR-00105.
One of the authors (V. G.) was supported by a CICYT grant.
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%\bibliographystyle{amsalpha}
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\end{document}
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