%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LaTeX
\documentstyle{article}
\title{On Higherorder Melnikov Method for the Exponentially
Small Splitting of Separatrices}
\author{V.Gelfreich
\thanks{Partially supported by the INTAS grant 93339ext, and by the grant
from the Russian Foundation for Basic Research. The author especially
thanks the Alexander von Humboldt foundation.}
}
\date{December 1, 1998}
\newcommand\E{{\rm e}}
\newcommand\I{{\rm i}}
\newcommand\tfrac[2]{{\textstyle\frac{{#1}}{{#2}}}}
\renewcommand\d{{\rm d}}
\begin{document}
\maketitle
\begin{center}
St.Petersburg Department of Steklov Mathematical Institute,\\
Fontanka 27, St.Petersburg, Russia\\[3pt]
current address: Institut f\"ur Mathematik I,\\
Freie Universit\"at Berlin,\\ Arnimallee 26, \\14195 Berlin, Germany\\[3pt]
email: gelf@math.fuberlin.de
\end{center}
This note has appeared as an answer to a question posed by Dr.~Gallavotti,
who asked about the way of calculating higher order (with respect to the
Melnikov method) corrections to the splitting of separatrices, which would
be compatible with the methods presented in
\cite{Gelfreich97a,DelshamsGJS97b}. In fact, the method
of these papers do not require these calculations, but they may be
useful if the main term ``vanishes". Moreover I feel that
these calculations are useful for a better understanding of the
method and are important for comparisons with the results
provided by alternative approaches.
In the considered class of problems
the splitting of separatrices is exponentially small, and the possibility
of using the Melnikov method is not evident at all. In some sense the
present paper is a step backwards with respect to the results of
\cite{Gelfreich97b,Gelfreich98a}, where the unique small parameter was
related to the perturbation period. Compared with these papers the present
note makes use of an additional small parameter, which is proportional to
the perturbation amplitude. A method, similar to the present one, may be
used to compute Taylor coefficients of the correction factor defined in
\cite{Gelfreich97b}.
\section{Parametrization of separatrices}
For the sake of brevity we restrict the consideration by the Hamiltonian,
\begin{equation}\label{Eq:mainH}
H=\frac{\omega I}{\varepsilon}+\frac{y^2}{2}
+(1+\mu m(\theta))(\cos x1)\;,
\end{equation}
which was already considered in \cite{DelshamsGJS97b}. This example
affords to illustrate all main features of the method.
The generalization is straightforward.
We assume that $x$ and $y$ are scalar variables, and
$\theta\in{\bf T}^n$, $I\in D\subset{\bf R}^n$,
$\omega$ is a fixed frequency vector. In \cite{DelshamsGJS97b}
we restricted our consideration to the case $2$ and
$\omega=(1,\gamma)$, where $\gamma$ is the golden mean number.
The system depends on two small parameters $\varepsilon$ and $\mu$.
The origin $x=y=0$ is a partially hyperbolic invariant torus. Like
\cite{DelshamsGJS97b} we study its whiskers in a parametric form.
The parametrization is a
function of the parameters $s$ and $\theta$, which satisfies
the following system:
\begin{equation}\label{Eq:basic}
\begin{array}{rclrcl}
D_tx^\sigma&=&y^\sigma\;,\qquad&
D_ty^\sigma&=&(1+\mu m(\theta))\sin x^\sigma\;,\\[3pt]
D_t\theta&=&\frac\omega\varepsilon\;,&
D_tI^\sigma&=&\mu\frac{\partial m}{\partial\theta}(\cos x^\sigma1)\;,
\end{array}
\end{equation}
where the operator
$D_t=\lambda\frac{\partial}{\partial s}+\frac\omega\varepsilon
\frac{\partial}{\partial\theta}$,
the constant $\lambda$ is the Lyapunov exponent of the torus, and
$\sigma\in\{\,{},{+}\,\}$.
The symbol ``$+$'' refers to the stable whisker and ``$$'' to the
unstable one.
The parametrization is defined by this system, the boundary conditions,
\[
\lim_{s\to\infty}x^(s,\theta)=0\;,
\qquad
\lim_{s\to+\infty}x^+(s,\theta)=2\pi\;,
\]
\[
\lim_{s\to\sigma\infty}y^\sigma(s,\theta)=0\;,
\qquad
\lim_{s\to\sigma\infty}I^\sigma(s,\theta)=0\;,
\]
and the following analytic properties: the parametrizations are
$2\pi$periodic in $\theta$ and $2\pi\I$periodic in $s$. The last
property assumes that the functions are analytic in the halfplanes
$\sigma\Re t>R$ for some positive constant $R$.
We describe the expansions for the parametrizations and $\lambda$ in power
series with respect to the parameter $\mu$,
\begin{equation}\label{Eq:expan}
\begin{array}{rcl}
x^\sigma(s,\theta)&=&x_0(s)+\mu x_1^\sigma(s,\theta)
+\mu^2 x_2^\sigma(s,\theta)+\ldots\;,\\
y^\sigma(s,\theta)&=&y_0(s)+\mu y_1^\sigma(s,\theta)
+\mu^2 y_2^\sigma(s,\theta)+\ldots\;,\\
I^\sigma(s,\theta)&=&I_0(s)+\mu I_1^\sigma(s,\theta)
+\mu^2 I_2^\sigma(s,\theta)+\ldots\;,\\
\lambda&=&\lambda_0+\mu\lambda_1+\mu^2\lambda_2+\ldots\;.
\end{array}
\end{equation}
We dedicate special attention to the terms of the first and second order,
which we write explicitly. We use
\begin{equation}\label{Eq:xyi0}
x_0(s)=4\arctan \E^s\;,\qquad
y_0(s)=\frac{2}{\cosh s}\;,
\qquad
I_0(s)=0\;,
\qquad
\lambda_0=1\;,
\end{equation}
as the zero order approximation. At this order the stable and
unstable separatrices coincide. We substitute the series (\ref{Eq:expan})
into the equations (\ref{Eq:basic}) and collect the terms of the same order
in $\mu$. We denote
\[
D_t^0=\frac{\partial}{\partial s}+\frac{\omega}{\varepsilon}
\frac{\partial}{\partial\theta}\;.
\]
We obtain the following system for the first order terms:
\begin{eqnarray}
\label{Eq:x1eq}
D^0_tx_1^\sigma+\lambda_1\frac{\partial x_0}{\partial s}&=&y_1^\sigma\;,\\
\label{Eq:y1eq}
D^0_ty_1^\sigma+\lambda_1\frac{\partial y_0}{\partial s}&=&
\cos x_0 x_1^\sigma+m(\theta)\sin x_0\;,\\
\label{Eq:I1eq}
D^0_t I_1&=&\frac{\partial m}{\partial\theta}(\theta)
(\cos x_01)\;.
\end{eqnarray}
The analysis of the first order approximation leads to
the almost standard Melnikov
theory. We are mainly interested in the second order, which
is defined by
\begin{eqnarray}
D^0_tx_2^\sigma+\lambda_1\frac{\partial x_1}{\partial s}
+\lambda_2\frac{\partial x_0}{\partial s}
&=&y_1^\sigma\;,\\
D^0_ty_2^\sigma+\lambda_1\frac{\partial y_1}{\partial s}
+\lambda_2\frac{\partial y_0}{\partial s}
&=&
\cos x_0 x_1^\sigma+m(\theta)\cos x_0 x_1\tfrac12\sin x_0 x_1^2\;,\\
\label{Eq:I2eq}
D^0_t I_2
+\lambda_1\frac{\partial I_1}{\partial s}
&=&\frac{\partial m}{\partial\theta}(\theta)
\sin x_0 x_1\;.
\end{eqnarray}
The procedure may be continued. At the $n^{\rm th}$ order in $\mu$
we obtain a linear nonhomogeneous system for the $n^{\rm th}$ order
approximation for the separatrices. All these equations have to be
supplied with the boundary conditions. We provide the explicit
solutions later after explaining the way of solving these equations.
\subsection*{Homological equations}
The basic homological equation has the form
\[
D^0_tf=g\;,
\]
where $g=g(s,\theta)$ is assumed to be periodic in $\theta$.
This equation appears at each order of the $I$ component.
It is easy to solve this equation, provided the righthand side
decreases at infinity ($s\to+\infty$ or $\infty$).
The desired solution is given by
\[
f(s,\theta)=\int_{\sigma\infty}^0g(s+t,\theta+\tfrac{\omega t}{\varepsilon})
\,\d t\;.
\]
This integral preserves the periodicity
and gives a function, which decreases at infinity too.
The equations for the $x$ and $y$ components have the following form:
\begin{eqnarray}\label{Eq:homx}
D^0_tx&=&y+g_1(s,\theta)\;,\\
\label{Eq:homy}
D^0_ty&=&
\cos x_0 x + g_2(s,\theta)\;.
\end{eqnarray}
We know two basic solutions of the corresponding homogeneous system:
\begin{eqnarray}
\vec u_1(s)&=&
\left(\begin{array}{c}
\xi_1\\ \eta_1
\end{array}\right)
=
\left(\begin{array}{c}
\frac{2}{\cosh s}\\[3pt] \frac{2\sinh s}{\cosh^2s}
\end{array}\right)
\;,\\[3pt]
\vec u_2(s)&=&
\left(\begin{array}{c}
\xi_2\\ \eta_2
\end{array}\right)
=
\left(\begin{array}{c}
\sinh s+\frac{s}{\cosh s}\\[3pt]
\cosh s+\frac{1}{\cosh s} \frac{s\sinh s}{\cosh^2s}
\end{array}\right)\;.
\end{eqnarray}
The first solution, $\vec u_1$, is the derivative of $(x_0,y_0)(s)$.
The Wronskian of these tow solutions is constant,
\[
\left\begin{array}{cc} \xi_1&\xi_2\\ \eta_1&\eta_2\end{array}\right=4\;.
\]
We solve the nonhomogeneous system using the method of variation of
parameters.
Indeed, if $C_1$ and $C_2$ satisfy
\begin{eqnarray}
\label{Eq:Cvar}
D_t^0C_1=\frac14\left\begin{array}{cc} g_1&\xi_2\\ g_2&\eta_2\end{array}
\right
\;,\qquad
D_t^0C_2=\frac14\left\begin{array}{cc} \xi_1&g_1\\ \eta_1& g_2\end{array}
\right
\;,
\end{eqnarray}
then the functions,
\begin{eqnarray}\label{Eq:xc}
x&=&C_1(s,\theta)\xi_1(s)+C_2(s,\theta)\xi_2(s)\;,\\
\label{Eq:yc}
y&=&C_1(s,\theta)\eta_1(s)+C_2(s,\theta)\eta_2(s)\;,
\end{eqnarray}
solve the nonhomogeneous system (\ref{Eq:homx}), (\ref{Eq:homy}).
\subsection*{First order terms}
Now we can calculate the first order terms.
We have from (\ref{Eq:I1eq})
\[
I_1^\sigma(s,\theta)=
\int_{\sigma\infty}^0 (\cos x_0(s+t)1)
\frac{\partial m}{\partial \theta}(\theta+\tfrac{\omega t}{\varepsilon})\,
\d t\;.
\]
The first order splitting is
\[
\Delta I_1=I_1^I_1^+=
\int_{\infty}^{\infty} (\cos x_0(s+t)1)
\frac{\partial m}{\partial \theta}(\theta+\tfrac{\omega t}{\varepsilon})
\d t\;.
\]
The last integral can be easily calculated using residues. This integral
coincides with the usual Melnikov function. It has some special
analytical properties to be discussed later.
To obtain the $x$ and $y$ components we use the method of variation of
parameters, which was described in the previous subsection.
We put the following terms onto the righthand side of
the equations (\ref{Eq:x1eq}), (\ref{Eq:y1eq}):
\[
\vec g=
\left(\begin{array}{c} 0\\m(\theta)\sin x_0\end{array}\right)
\lambda_1\left(\begin{array}{c}\frac{\partial x_0}{\partial s}\\[3pt]
\frac{\partial y_0}{\partial s}
\end{array}\right)\;.
\]
After the substitution of this $\vec g$
the equation (\ref{Eq:Cvar}) takes the form,
\begin{eqnarray}
D_t^0C_1&=&
\tfrac14 m(\theta)\sin x_0(s)\xi_2(s)\lambda_1\;,\\
D_t^0C_2&=&\phantom{}
\tfrac14 m(\theta)\sin x_0(s)\xi_1(s)\;,
\end{eqnarray}
or, equivalently,
\begin{eqnarray}\label{Eq:c11}
D_t^0C_1&=&
\frac {m(\theta)}2\lambda_1+\frac{m(\theta)}{2\cosh^2s}\left(
\frac{s\sinh s}{\cosh s}1\right)\;,\\
D_t^0C_2&=&
\frac{m(\theta)\sinh s}{\cosh^3s}\;.
\end{eqnarray}
The absence of secular terms (the periodicity properties)
proposes $\lambda_1={<}\tfrac m2{>}=0$.
We have to take into account that the righthand side of (\ref{Eq:c11})
is not absolutely summable at infinity. The equation is linear,
and we may represent the solution in the form
\begin{eqnarray}
C_1^\sigma&=&\frac{\tilde m(\theta)}{2}
\frac14
\int_{\sigma\infty}^0
m(\theta+\tfrac{\omega t}{\varepsilon})
\left(\sin x_0(s+t)\xi_2(s+t)2\right)\,\d t\;,\\
C_2^\sigma&=&
\frac14
\int_{\sigma\infty}^0
m(\theta+\tfrac{\omega t}{\varepsilon})\sin x_0(s+t)\xi_1(s+t)\,\d t\;,
\end{eqnarray}
where $\tilde m$ is a periodic solution of the following equation:
\[
\frac{\omega}{\varepsilon}\frac{\partial\tilde m}{\partial \theta}(\theta)
=m(\theta)\;.
\]
We subtracted $2$ under the first integral to ensure its
convergence. An alternative approach can be based on the improper integrals
technics of \cite{Gallavotti94}.
Then the desired solutions, $x_1^\sigma$ and $y_1^\sigma$,
may be obtained by (\ref{Eq:xc}) and (\ref{Eq:yc}),
\begin{eqnarray}\label{Eq:x1c}
x_1^\sigma&=&C_1^\sigma(s,\theta)\xi_1(s)+C_2^\sigma(s,\theta)\xi_2(s)\;,\\
\label{Eq:y1c}
y_1^\sigma&=&C_1^\sigma(s,\theta)\eta_1(s)+C_2^\sigma(s,\theta)\eta_2(s)\;.
\end{eqnarray}
For the splitting we have:
\begin{eqnarray}\label{Eq:Deltax1}
\Delta x_1&=&\xi_1(s)\Delta C_1(s,\theta)+ \xi_2(s)\Delta C_2(s,\theta)\;,\\
\label{Eq:Deltay1}
\Delta y_1&=&\eta_1(s)\Delta C_1(s,\theta)+ \eta_2(s)\Delta C_2(s,\theta)\;,
\end{eqnarray}
where
\begin{eqnarray}\label{Eq:DeltaC1}
\Delta C_1&=&

\frac14
\int_{\infty}^\infty m(\theta+\tfrac{\omega t}{\varepsilon})
\left(\sin x_0(s+t)\xi_2(s+t)2\right)\,\d t\;,\\
\label{Eq:DeltaC2}
\Delta C_2&=&
\frac14
\int_{\infty}^\infty
m(\theta+\tfrac{\omega t}{\varepsilon})\sin x_0(s+t)\xi_1(s+t)\,\d t\;.
\end{eqnarray}
\subsection*{Second order terms}
We consider the second order approximation for the $I$ components only.
>From the equation (\ref{Eq:I2eq}) we obtain
\begin{equation}
I_2^\sigma(s,\theta)=
%\lambda_1\int_{\sigma\infty}^0 \frac{\partial I_1^\sigma}{\partial s}
%(s+t,\theta+\tfrac{\omega t}{\varepsilon})\,\d t+
\int_{\sigma\infty}^0
\frac{\partial m}{\partial\theta}(\theta+\tfrac{\omega t}{\varepsilon})
\sin x_0(s+t) x_1^\sigma(s+t,\theta+\tfrac{\omega t}{\varepsilon})\,\d t\;,
\end{equation}
where we used $\lambda_1=0$.
Thus $\Delta I_2$ is described by a combination of two double integrals,
\begin{eqnarray}\nonumber
\Delta I_2(s,\theta)&=&
\int_{\infty}^0
\frac{\partial m}{\partial\theta}(\theta+\tfrac{\omega t}{\varepsilon})
\sin x_0(s+t) x_1^(s+t,\theta+\tfrac{\omega t}{\varepsilon})\,\d t\\
&&
\int_{\infty}^0
\frac{\partial m}{\partial\theta}(\theta+\tfrac{\omega t}{\varepsilon})
\sin x_0(s+t) x_1^+(s+t,\theta+\tfrac{\omega t}{\varepsilon})\,\d t
\;,
\end{eqnarray}
In a quite similar situation \cite{GallavottiGM98} it was shown, that the
properties of $\Delta I_2$ and $\Delta I_1$ are different. In our
context this difference comes from the fact, that the functions under these
two integrals are not analytical continuations of each other. This is a
very delicate point, because this difference is exponentially small in
$\varepsilon$.
The expressions for $\Delta I_2$ may be rewritten in the following form:
\begin{eqnarray}\nonumber
\Delta I_2(s,\theta)&=&
\int_{\infty}^\infty
\frac{\partial m}{\partial\theta}(\theta+\tfrac{\omega t}{\varepsilon})
\sin x_0(s+t) x_1^(s+t,\theta+\tfrac{\omega t}{\varepsilon})\,\d t\\
\label{Eq:DeltaI2m}
&&
4\left\begin{array}{cc}
\tfrac{\partial C_1^+}{\partial\theta}&\Delta C_1\\[3pt]
\tfrac{\partial C_2^+}{\partial\theta}&\Delta C_2
\end{array}\right\;.
\end{eqnarray}
To see this one should replace $x_1^+=x_1^ \Delta x_1$ and
use the explicit formula for $\Delta x_1$.
In the formula above the integral is not absolutely convergent,
a special care should be taken to understand this integral
in the proper way.
\section{Observables}
The choice of convenient observables may notably
simplify the study of the splitting. In
\cite{DelshamsGJS97b} it was shown that the Hamiltonian (\ref{Eq:mainH})
may be transformed to a especially simple form,
\[
H'=H\circ{\cal C}=\frac{\omega J}{\varepsilon}+E\;,
\]
by an analytic canonical coordinate change,
\[
{\cal C}:(s,\varphi,E,J)\mapsto(x,\theta,y,I)\;,
\]
in a neighborhood of the stable whisker.
The change is identical on the angles, $\theta=\varphi$,
but we use different notations for these angles, because
the canonically conjugate momenta are not the same, they are $I$ and $J$,
respectively. The normalization condition for the change is
\[
{\cal C}(s,\theta,0,0)=P^+(s,\theta)\;,
\]
where
\[
P^+(s,\theta)=(x^+(s,\theta),\theta,y^+(s,\theta),I^+(s,\theta))
\]
represents the stable separatrix. In other words, this means that the
stable whisker is given by $E=0$, $J=0$, and the $s$ and $\varphi$
coordinates coincide with the corresponding parameters on the stable
separatrix. Note, that the coordinate $s$, restricted onto the unstable
whisker, do not coincide with the parameter $s$ (such an error was
done in \cite{DelshamsS92}, but had no consequences for the main result
of that paper).
The coordinates $E$ and $J$ are local integrals
of motion. It is convenient to use them as observables to measure
the splitting. Let $\Delta$ be a splitting vector, we can define
\[
\Delta E=\Omega'(\Delta,\tfrac{\partial}{\partial s})\;,
\qquad
\Delta J=\Omega'(\Delta,\tfrac{\partial}{\partial \varphi})\;,
\]
where $\Omega'=\d E\wedge\d s+\d J\wedge \d\varphi$ is the symplectic form.
It is easy to see that $\Delta E$ and $\Delta J$ are projections
of the splitting vector $\Delta$ onto the corresponding coordinate
axes. These quantities are not independent: the relation,
\[
\frac{\omega\Delta J}{\varepsilon}+\Delta E=0\;,
\]
is due to the conservation of the energy $H'$ (the stable and unstable
whiskers belong to one energy level).
We may come back to the original variables. The normalizing condition
and the canonicity of the change imply
\begin{equation}\label{Eq:DeltaJorg}
\Delta E=\Omega(\Delta P,\tfrac{\partial P^+}{\partial s})
+O_2\;,
\qquad
\Delta J=\Omega(\Delta P,\tfrac{\partial P^+}{\partial \theta})
+O_2\;,
\end{equation}
where $\Delta P$ is the splitting vector in the original coordinates,
$\Omega=\d y\wedge\d x+\d I\wedge \d\theta$,
and $O_2$ denotes the quadratic in $\Delta P$ terms.
The last term appears because the splitting vector $\Delta P$
transforms as a tangent vector in the linear approximation only.
But since this vector is exponentially small, the quadratic
terms are not relevant for our study.
It is interesting to write explicitly the first terms of the expansion
for the $J$ components. Expanding the functions
we obtain from (\ref{Eq:DeltaJorg}):
\begin{equation}\label{Eq:DeltaJ}
\Delta J=\mu\Delta I_1+\mu^2\left(\Delta I_2+
\Delta y_1\tfrac{\partial x_1^+}{\partial\theta}
\Delta x_1\tfrac{\partial y_1^+}{\partial\theta}
\right)+\ldots\;,
\end{equation}
where we used (\ref{Eq:expan}) and (\ref{Eq:xyi0}).
Of course, the error term depends on $\varepsilon$ too.
The methods of \cite{DelshamsGJS97b} allow to prove that
this residue is exponentially small.
The expression for the quadratic in $\mu$ term may be considerably
simplified. We note, that
\[
\left\begin{array}{cc}
\tfrac{\partial x_1^+}{\partial\theta}&\Delta x_1\\[3pt]
\tfrac{\partial y_1^+}{\partial\theta}&\Delta y_1\end{array}\right=
\left\begin{array}{cc}
\tfrac{\partial C_1^+}{\partial\theta} \xi_1+
\tfrac{\partial C_2^+}{\partial\theta} \xi_2
&\Delta C_1 \xi_1+\Delta C_2 \xi_2\\[3pt]
\tfrac{\partial C_1^+}{\partial\theta} \eta_1+
\tfrac{\partial C_2^+}{\partial\theta} \eta_2
&\Delta C_1 \xi_1+\Delta C_2 \eta_2
\end{array}\right=
4\left\begin{array}{cc}
\tfrac{\partial C_1^+}{\partial\theta}&\Delta C_1\\[3pt]
\tfrac{\partial C_2^+}{\partial\theta}&\Delta C_2
\end{array}\right\;.
\]
Finally, we have
\begin{eqnarray*}
\Delta J_1&=&\Delta I_1\;,\\
\Delta J_2&=&\Delta I_2+ 4 \left\begin{array}{cc}
\tfrac{\partial C_1^+}{\partial\theta}&\Delta C_1\\[3pt]
\tfrac{\partial C_2^+}{\partial\theta}&\Delta C_2
\end{array}\right\;.
\end{eqnarray*}
Comparing with (\ref{Eq:DeltaI2m}) we obtain
\[
\Delta J_2=
\int_{\infty}^\infty
\frac{\partial m}{\partial\theta}(\theta+\tfrac{\omega t}{\varepsilon})
\sin x_0(s+t) x_1^(s+t,\theta+\tfrac{\omega t}{\varepsilon})\,\d t\;.
\]
Note that the last integral is not absolutely convergent.
\section{The Poincar\'e section $x=\pi$}
Sometimes it is interesting to study the splitting on the natural
Poincar\'e section $x=\pi$. For example, this may be necessary
to compare the results
obtained by different methods. Of course, this is not necessary for
establishing the transversality of the separatrices.
Here we describe how to derive the formulas
for the separatrix splitting on the Poincar\'e section $x=\pi$.
Our parametrizations may be normalized by the condition
$x^\sigma(0,0)=\pi$. On the other hand, the condition,
$x^\sigma(0,\theta)=\pi$ for all $\theta$, is too restrictive and it is
not compatible with some of the key analytical properties.
First, we define delay functions, $B^\sigma(\theta)$, by the equation,
\[
x^\sigma(B^\sigma(\theta),\theta)=\pi\;.
\]
Expanding the function in power series,
\[B^\sigma(\theta)=\mu B_1^\sigma(\theta)+
\mu^2 B_2^\sigma(\theta)+\ldots\;,\]
substituting to the previous equation and collecting the terms of the
first order in $\mu$, we obtain
\[
\frac{\partial x_0}{\partial s}(0) B_1^\sigma(\theta)+
x_1^\sigma(0,\theta)=0\;.
\]
Since $\frac{\partial x_0}{\partial s}(0)=2$,
this immediately gives us the following expression:
\[
B_1^\sigma(\theta)=\tfrac12 x_1^\sigma(0,\theta)=C_1^\sigma(0,\theta)\;,
\]
where we used (\ref{Eq:Deltax1}) and the equalities, $\xi_1(0)=2$
and $\xi_2(0)=0$. It is easy to see that
$B_1^\sigma$ is not exponentially
small in $\varepsilon$. On the other hand, the delay function
is related to the time, which is required for a point, which
starts on the section $x=\pi$, to reach the torus $s=0$ on the whisker.
Since the stable and unstable whiskers are exponentially close,
the difference $B^B^+$ has to be exponentially small too:
\[
\Delta B_1=C_1^+C_1^=

\frac14
\int_{\infty}^\infty
m(\theta+\tfrac{\omega t}{\varepsilon})
\left(\sin x_0(t)\xi_2(t)2\right)\,\d t\;.
\]
Let us compute the splitting on the section $x=\pi$:
\[
\left.\Delta I\right_{x=\pi}=
I^(B^(\theta),\theta)I^+(B^+(\theta),\theta)\;.
\]
We can expand in powers of $\mu$:
\begin{eqnarray*}
\left.\Delta I\right_{x=\pi}
&=&
\mu\Delta I_1(0,\theta)\\
&&
+\mu^2\left(
\Delta I_2(0,\theta)+\frac{\partial I_1^}{\partial s}(0,\theta)
B_1^(\theta)
\frac{\partial I_1^+}{\partial s}(0,\theta)B_1^+(\theta)\right)
+\ldots\;.
\end{eqnarray*}
All terms from the righthand side have been computed.
\section*{Discussion}
The described calculations are formal. Taking into account the analyticity
of the whiskers in $\mu$ it is not too difficult to see that the errors of
the approximation are of the first missing order of $\mu$. Of course, the
approximation errors also depend on $\varepsilon$. Studying this
dependence is the central point, since the main terms of the
splitting are exponentially small compared with $\varepsilon$.
We described the second order Melnikov method, which is compatible with
the methods used in \cite{Gelfreich97a,Gelfreich97b,DelshamsGJS97b}. That
is this method can be used to show that for $\mu=\mu_0\varepsilon^p$ the
estimates for the errors are exponentially small and to establish the
conditions, which guaranty that the Melnikov function dominates the
error. In fact, the exponentially small upper bounds for the error can be
directly obtained for the special observables $E$ and $J$ only. That is
the dots in (\ref{Eq:DeltaJ}) refer to an exponentially small residue.
These functions and their approximations are quasiperiodic functions
with fastdecreasing Fourier modes.
I say that
$F(s,\theta)=\sum_k F_k\E^{\I k\theta+\I sk\omega\varepsilon^{1}}$
is a {\em quasiperiodic function with fastdecreasing Fourier modes\/},
if $F_k\le A\exp(rk\rhok\omega\varepsilon^{1})$
for some positive vector $r$ and positive constants $\rho$ and $A$
(comp. with Lemma~3 from Sect.~7 of \cite{DelshamsGJS97b}).
This property can be derived from the study of the form of the analyticity
domain of the function $F$. The Fourier modes of the functions
$\Delta I_1=\Delta J_1$ and $\Delta J_2$ are fastdecreasing.
But most of the functions described in the present note do not possess
this property. In particular, the function $\Delta I_2$ does not have
this property.
\begin{thebibliography}{DGJS97}
\bibitem[DGJS97]{DelshamsGJS97b}
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\end{thebibliography}
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