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\title{Almost invariant elliptic manifolds\\
in a singularly perturbed Hamiltonian system}

\author{V. Gelfreich$^*$, L. Lerman$^\dag$\\[12pt]
\normalsize
$^*$ Institut f\"ur Mathematik I, Freie Universit\"at Berlin\\[-6pt]
\normalsize
Arnimallee 2-6, D-14195 Berlin, Germany\\[-6pt]
\normalsize
E-mail: gelf@math.fu-berlin.de\\[6pt]
\normalsize
$^\dag$ Institute for Applied Math. \& Cybernetics\\[-6pt]
\normalsize
 10 Ul'yanov St., Nizhny 
Novgorod, 603005 Russia}
\date{April 4, 2001}

\begin{document}
\maketitle

\begin{abstract} 
We consider a singularly perturbed Hamiltonian system, which loses one
degree of freedom at $\eps=0$. Assume the slow manifold to be normally
elliptic. In the case of an analytic Hamilton function it is shown
that the slow manifold persists up to an exponentially small error
term. 
\end{abstract}


\section{Introduction}

The main object of this paper is a singularly perturbed Hamiltonian
system 
\begin{equation}
\label{Eq:slow}
\begin{array}{rclrcl}
\eps\dot x&=&\;\displaystyle\frac{\partial H}{\partial y},
\qquad\qquad
&
\dot u_i&=&\;\displaystyle\frac{\partial H}{\partial v_i},
\\[12pt]
\eps\dot y&=&-\displaystyle\frac{\partial H}{\partial x},
&
\dot v_i&=&-\displaystyle\frac{\partial H}{\partial u_i},
\end{array}
\qquad i=1,\dots,n.
\end{equation}
Here $x$ and $y$ are scalar variables, and $u,v\in {\mathbb R}^n$.
Many model equations from applications can be transformed to this
form. For example, see \cite{AmickK1989}, where the equation was
derived to describe water waves in the presence of surface tension; 
or \cite{AlfimovEL1998}, where a system of two second order equations
was used to describe soliton solutions to non-local sine-Gordon
equation; or \cite{Br}, where a so-called Hamiltonian elliptic Hopf
bifurcation was studied. 

If we formally put $\eps=0$, the system (\ref{Eq:slow})
is reduced to a pair of algebraic equations,
\begin{equation}
\label{Eq:algebraic}
\frac{\partial H}{\partial y}=
\frac{\partial H}{\partial x}=0,
\end{equation}
and a Hamiltonian system in $(u,v)$ variables, which is obtained from 
(\ref{Eq:slow}) after exclusion of $(x,y)$ by means of (\ref{Eq:algebraic}). 
One can say the system (\ref{Eq:slow}) loses one degree of freedom at
$\eps=0$. The system (\ref{Eq:algebraic}) defines the {\em slow}
manifold. Here we assume that (\ref{Eq:algebraic}) can be solved at
$\eps=0$  with respect to $x$ and $y$,
\begin{equation}
\label{Eq:fg}
x=f(u,v),\qquad
y=g(u,v)
\end{equation}
in some domain $D_0$ of variables $(u,v)$. Our goal is to study the
dynamics  near the slow manifold. Here one of the main questions is:
does the slow manifold persists for small $\eps$ as an invariant
manifold or a manifold  sufficiently close to invariant one, and if
yes, then in which sense? The  answer is positive provided the slow
manifold is {\it normally  hyperbolic}. In order to explain, what
normal hyperbolicity means here,  it is convenient to rewrite the
system (\ref{Eq:slow}) using the fast time $\tau=\eps^{-1} t$:
\begin{equation}
\label{Eq:fast}
\begin{array}{rclrcl}
x'&=&\;\displaystyle\frac{\partial H}{\partial y}\qquad\qquad
&
u'&=&\;\eps\displaystyle\frac{\partial H}{\partial v}
\\[12pt]
y'&=&-\displaystyle\frac{\partial H}{\partial x}
&
v'&=&-\eps\displaystyle\frac{\partial H}{\partial u}.
\end{array}
\end{equation}
If we formally put $\eps=0$, then we obtain an one-degree-of-freedom 
Hamiltonian system in $(x,y)$ variables ({\em fast} system) which 
parametrically depends on $u$ and $v$. In this representation 
each point of the slow manifold corresponds to an equilibrium of the
fast system.  The slow manifold is normally hyperbolic if any of these
equilibria is a saddle. In the whole $(2n+2)$-dimensional phase space
of the system  (\ref{Eq:fast}) we get the following picture: at $\eps
=0$ the system  has an invariant manifold (namely, the slow manifold)
filled with equilibria, and this manifold is normally hyperbolic in
the usual sense, see, for example, \cite{HPSh,Fen}.
The normal hyperbolicity implies that this submanifold persists for 
$\eps >0$ small enough being a symplectic (with respect to the 2-form  
$\d y\wedge \d x + \eps^{-1}\sum_{i=1}^n\d v_i\wedge \d u_i$) 
invariant normally hyperbolic submanifold, 
and the restriction of the system (\ref{Eq:fast}) to this submanifold
is an $n$-degrees-of-freedom Hamiltonian system with a Hamiltonian
being the restriction of $H$ to this submanifold.
 
In this paper we will study the opposite case, when each equilibrium
is elliptic instead. In this case there is not rigorous persistence
results for the slow manifold, as we are aware of. 
We assume the Hamiltonian $H=H(x,y,u,v;\eps)$ to be analytic in all
its arguments and show that there is a slow manifold being invariant
one up to an error exponentially small in $\eps$.
This follows from the following theorem, which additionally describes 
dynamics in a neighborhood of the slow manifold.

Let us assume the functions $f$ and $g$ in (\ref{Eq:fg}) are analytic
in a complex $\delta$-neighborhood $D_0+\i\delta$ of the real domain
$D_0$. Consider the slow manifold
$$
W_0^{\mathrm{slow}}=\{\,(x,y,u,v): x=f(u,v),\ y=g(u,v),\
(u,v)\in D_0\,\}
$$
and denote
$$
\Delta(u,v)=\left.\det\left(\begin{array}{cc}
\partial_{xx}^2H&
\partial_{xy}^2H\\[6pt]
\partial_{yx}^2H&
\partial_{yy}^2H
\end{array}\right)
\right|_{x=f(u,v),y=g(u,v),\eps=0}.
$$
Denote $D_1(\delta)=W_0^{\mathrm{slow}}+\i\delta$.


\begin{theorem}\label{Th:main} \it
Let the Hamiltonian $H$ be analytic in $D_1(\delta)$ for some
$\delta>0$. Assume also $\Delta(u,v)>C>0$ for all $(u,v)\in D_0$ and
some $C$. Then there is a canonical change of coordinates 
$\Phi:(x,y,u,v)\mapsto (X,Y,U,V)$ and constants $\tilde{\delta}, c>0$
such that in the new coordinates the Hamiltonian $H$ takes the form
\begin{equation}
H\circ\Phi^{-1}= H_0(I,U,V;\eps)+R(X,Y,U,V;\eps),
\end{equation}
where $I=\frac12(X^2+Y^2)$ and
\begin{equation}
R=O\left(\e^{-\frac c\eps}\right).
\end{equation}
The change of coordinates and the new Hamiltonian are analytic with
respect to $(X,Y,U,V)$ in $D_1(\delta/2)$, and bounded in $\eps$.
\end{theorem}

In particular, the theorem implies that $I$ is almost an integral of
the system: $I'=O\left(\e^{-\frac c\eps}\right)$. The equation $X=Y=0$
defines an almost  invariant (normally) elliptic slow manifold. 

In particular, in the case $n=1$, the system (\ref{Eq:slow}) has two 
degrees of freedom and in some neighborhood of the slow manifold it is
exponentially close to an integrable system. This allows to use
powerful methods of Hamiltonian perturbation theory to a further study
of the system. 

\smallskip

The rest of the paper is devoted to a proof of Theorem 1. 
In Sections \ref{Se:ct} and \ref{Se:she} we prepare the necessary 
technical tools. Then we prove the theorem in two steps in Sections  
\ref{Se:ssm} and \ref{Se:ma}. The main goal is to straighten the slow 
manifold and separate the rapid motions by means of canonical
transformations. This is a common approach in the theory of
Hamiltonian systems. Of course, a complete separation of variables is
not possible, and we will show  that it can be done up to an
exponentially small remainder.
The system (\ref{Eq:fast}) formally has two fast variables.
In the elliptic case a trajectory in
a neighborhood of the slow manifold rapidly rotates, we will see that
it is possible to introduce new coordinates in such a way that the
phase of this rotation remains the only rapid variable. We use an averaging
procedure similar to \cite{Neishtadt1981, Neishtadt1984} to eliminate
the dependence of the Hamiltonian on the fast  phase up to an
exponentially small remainder.  


\section{Canonical transformations\label{Se:ct}}

The system (\ref{Eq:fast}) is Hamiltonian with respect to the
symplectic form 
$$
\Omega=\d y\wedge \d x+\eps^{-1}\sum_{i=1}^n\d v_i\wedge \d u_i.
$$
So the symplectic form $\Omega$ is not the standard one but reflects
the  singular nature of the perturbation instead. For this reason we
remind the definition and some basic properties of canonical
transformations related to our $\Omega$. A transformation is called
{\em canonical\/} one if it preserves the symplectic form  or, equivalently,
the form (\ref{Eq:fast}) of a Hamiltonian system with an arbitrary
$H$. It is convenient to define a canonical transformation using
a generating function $ S=S(x,Y,u,V)  $, so the transformation
$(x,y,u,v)\mapsto(X,Y,U,V)$ is defined implicitly by the system
\begin{equation}
\label{canon}
X=\frac{\partial S}{\partial Y},\qquad
y=\frac{\partial S}{\partial x},\qquad
U=\eps\frac{\partial S}{\partial V},\qquad
v=\eps\frac{\partial S}{\partial u} .
\end{equation}
This systems (locally) defines a coordinate change provided in
addition 
$$
\frac{\partial^2 S}{\partial x \partial Y}\,
\frac{\partial^2 S}{\partial u \partial V}-
\frac{\partial^2 S}{\partial x \partial V}\,
\frac{\partial^2 S}{\partial u \partial Y}\ne 0.
$$
The domain, where the transformation $(x,y,u,v)\mapsto(X,Y,U,V)$ is
correctly defined, may depend on the parameter $\eps$. 

Later on we mainly use coordinate transformations with generating
functions of the following special form  
\begin{equation}\label{Eq:genfun}
S=xY+\frac{uV}{\eps}+ \eps s(x,Y,u,V),
\end{equation}
which generate close-to-identity transformations 
\begin{equation}\label{Eq:implicit}
X=x+\eps\frac{\partial s}{\partial Y},\quad
y=Y+\eps\frac{\partial s}{\partial x},\quad
U=u+\eps^2\frac{\partial s}{\partial V},\quad
v=V+\eps^2\frac{\partial s}{\partial u} .
\end{equation}
Note, that despite the singular form of the generating function $S$ 
the transformation is nonsingular at $\eps=0$.

For an analytic function $s$ given in a bounded domain $G$, and
continuous in  the closure $\bar{G}$ we denote $\|s\|=\sup_G|s|$.
For a vector-valued function the norm equals to the maximal norm among 
its partial components.

\begin{lemma} \it\label{Le:change} Let $0<\eps<1$, $\delta>0$,
$0<\xi<\delta$, $D\subset{\mathbb R}^{2n+2}$. Let the function $s$ be
analytic in $D+\i\delta$ and
\begin{equation}\label{Eq:condition}
\|s\|_\delta\le \frac{\xi^2}{8\eps}.
\end{equation}
Then {\rm (\ref{Eq:implicit})} defines a canonical transformation
$D+\i(\delta-\xi)\to D+\i(\delta-\xi/2)$ of the form
$$
\begin{array}{rclrlc}
x&=&X+\eps\alpha_1(X,Y,U,V;\eps),\qquad
&u&=&U+\eps^2\beta_1(X,Y,U,V;\eps),\\
y&=&Y+\eps\alpha_2(X,Y,U,V;\eps),\qquad
&v&=&V+\eps^2\beta_2(X,Y,U,V;\eps),
\end{array}
$$
where
$$
\|\alpha_k\|_{\delta-\xi},\ \|\beta_k\|_{\delta-\xi}
\le \frac{2\|s\|_\delta}{\xi}\,,      \qquad k=1,2.
$$
\end{lemma}

\noindent{\it Proof.} We prove the lemma  using a convergent iteration
scheme. It is identical to the usual proof of Implicit Function
Theorem, and we repeat it only to check the quantitative estimates, 
which will be essential for the proof of the main theorem.

As it is easy to see from the first and third equations of
(\ref{Eq:implicit})
\begin{eqnarray*}
\alpha_1&=&-\frac{\partial s}{\partial Y}
(X+\eps\alpha_1,Y,U+\eps^2\beta_1,V;\eps),\\
\beta_1&=&-\frac{\partial s}{\partial V}
(X+\eps\alpha_1,Y,U+\eps^2\beta_1,V;\eps).
\end{eqnarray*}
We solve this system by iterations 
\begin{eqnarray*}
\alpha^{(n+1)}&=&-\frac{\partial s}{\partial Y}
(X+\eps\alpha^{(n)},Y,U+\eps^2\beta^{(n)},V;\eps),\\
\beta^{(n+1)}&=&-\frac{\partial s}{\partial V}
(X+\eps\alpha^{(n)},Y,U+\eps^2\beta^{(n)},V;\eps).
\end{eqnarray*}
starting with
$\alpha^{(0)},\beta^{(0)}\equiv0$. The first iteration gives 
\begin{eqnarray*}
\alpha^{(1)}&=&-\frac{\partial s}{\partial Y}
(X,Y,U,V;\eps),\\
\beta^{(1)}&=&-\frac{\partial s}{\partial V}
(X,Y,U,V;\eps).
\end{eqnarray*}
Using Cauchy estimates for the derivatives we obtain:
$$
\|\alpha^{(1)}\|_{\delta-\xi}\,,\ 
\| \beta^{(1)}\|_{\delta-\xi}
\le\frac{\|s\|_\delta}{\xi} .
$$
We continue by induction. Obviously,
\begin{eqnarray*}
\alpha^{(n+1)}-\alpha^{(n)}&=&-\frac{\partial s}{\partial Y}
(X+\eps\alpha^{(n)},Y,U+\eps^2\beta^{(n)},V;\eps)\\&&+
\frac{\partial s}{\partial Y}
(X+\eps\alpha^{(n-1)},Y,U+\eps^2\beta^{(n-1)},V;\eps),\\
\beta^{(n+1)}-\beta^{(n)}&=&-\frac{\partial s}{\partial U}
(X+\eps\alpha^{(n)},Y,U+\eps^2\beta^{(n)},V;\eps)\\&&+
\frac{\partial s}{\partial U}
(X+\eps\alpha^{(n-1)},Y,U+\eps^2\beta^{(n-1)},V;\eps).
\end{eqnarray*} 
Let us assume that 
$$
\eps\|\alpha^{(i)}\|_{\delta-\xi},\ 
\eps\|\beta^{(i)}\|_{\delta-\xi}\le\xi/2 
$$
for all $i\le n$. We estimate the differences using the maxima of the 
derivatives, and bound the derivatives by Cauchy estimates. We obtain 
$$
\| \alpha^{(n+1)}-\alpha^{(n)} \|_{\delta-\xi}
\le
\frac{4\|s\|_\delta}{\xi^2}\left(\eps
\|\alpha^{(n)}-\alpha^{(n-1)}\|_{\delta-\xi}
+\eps^2\|\beta^{(n)}-\beta^{(n-1)}\|_{\delta-\xi}\right).
$$
The same upper bound holds for 
$\| \beta^{(n+1)}-\beta^{(n)} \|_{\delta-\xi}$. From
these upper bounds we conclude inductively that
$$
\| \alpha^{(n+1)}-\alpha^{(n)} \|_{\delta-\xi}\,,\
\| \beta^{(n+1)}-\beta^{(n)} \|_{\delta-\xi}
\le
a_0q^n,
$$
where
$$
a_0=
\frac{\|s\|_\delta}{\xi}\le\frac{ \xi}{8\eps},\qquad
q=\frac{4\eps\|s\|_\delta}{\xi^2}(1+\eps)\le\frac12.
$$
In particular, $\eps\|\alpha^{(n+1)}\|_{\delta-\xi}$,
$\eps\|\beta^{(n+1)}\|_{\delta-\xi}\le\xi/2$, which justifies
the induction assumption. 

\smallskip
 
Consequently, the sequence $\alpha^{(n)}$, $\beta^{(n)}$
converges uniformly in $D+\i(\delta-\xi)$ and the limit functions
$\alpha_1$ and $\beta_1$ are bounded from above by $2a_0$.

Finally the functions $\alpha_2$ and $\beta_2$ can be bounded directly 
from the second and fourth equations of (\ref{Eq:implicit}) since they
are explicitly given after $\alpha_1$ and $\beta_1$ have been found. 
\hfill
\qed


\section{Solutions of the homological equation\label{Se:she}}

In this section we study the homological equation
\begin{equation}\label{Eq:homol1}
x\partial_y s-y\partial_x s= f(x,y),
\end{equation}
which we have to solve at each step of the averaging procedure described
in Section~\ref{Se:ma}. Let us describe the suitable class of functions.
Let $f$ be an analytic function of two complex variables in a 
ball $B_R =\{\, |x|^2 +|y|^2 < R^2\,\}$.
We connect with $f$ a function $\bar f$ of a single complex
variable $I=\frac12(x^2 + y^2)$. We call $\bar f$ the {\em average\/} 
of $f$. This $\bar{f}$ is defined as follows.
We introduce symplectic polar coordinates in
$B_R\setminus \{\,x^2+y^2=0\,\}$,
\begin{equation}
\label{Eq:polar}
x=\sqrt{2I}\cos\varphi,\qquad y=\sqrt{2I}\sin\varphi,
\end{equation}
where we use the main branch of the square root.
In the new coordinates $B_R$ is determined by the inequality 
$$
|x|^2 +|y|^2 =
 2|I|\left(|\cos\varphi|^2 +|\sin\varphi|^2\right)
=2|I|\cosh(2\Im\varphi) <R^2.
$$
The intersection of $B_R$ with the level set $\{\,I=I_0\,\}$,
where $0<|I_0|<\frac12R^2$, is a cylinder 
$\{\,|\Im\varphi\,|<\rho\,\}$, where
$\rho\equiv\rho(I_0)=\frac12\cosh^{-1}(R^2/2|I_0|)$.
This cylinder shrinks into a circle as $|I_0|\to\frac12R^2$.
For a fixed $I$ from the punctured disk $0<|I|<\frac12R^2$ 
the function $f(x,y)=f(\sqrt{2I}\cos\varphi, \sqrt{2I}\sin\varphi)$
is analytic 
and $2\pi$-periodic in $\varphi$ in the strip $|\Im \varphi|<\rho(I)$
and continuous in its closure. We call its mean value 
\begin{equation}\label{Eq:average}
\bar f(I)=
\frac{1}{2\pi}\int_{-\pi}^{\pi}
f(\sqrt{2I}\cos\varphi, \sqrt{2I}\sin\varphi)\,d\varphi
\end{equation}
the average of the function $f$. It is an analytic function of $I$.
The analyticity is evident for $I\ne0$. 
In principle, $\bar f$ might have a branching at $I=0$.
But, expanding $f(x,y)$ in the power series in
$(x,y)$, inserting polar coordinates and integrating, we get a
convergent power  series in $\sqrt{2I}$ with coefficients being
averages of the related  trigonometric monomials. The average of an
odd monomial vanishes, so in the power series only even degrees of
$\sqrt{2I}$ remain. Thus, one gets a convergent power series in $I$,
that is, the average is an analytic function at $I=0$ as well.
Thus, $\bar f$ is analytic in the disk $|I|<\frac12R^2$.

\smallskip

Let ${\cal X}$ be the Banach space of all functions analytic in $B_R$, 
continuous in its closure, and having zero average. The space is
equipped with supremum norm.

\begin{lemma} \it\label{Le:homol}
The operator
$
L=x\partial_y -y\partial_x 
$
has a continuous left inverse $L^{-1}:{\cal X}\to{\cal X}$,
defined on all ${\cal X}$, and $\|L^{-1}\|\le2\pi$.
\end{lemma}

\noindent{\em Proof.}
Let $f\in{\cal X}$.
We check that (\ref{Eq:homol1}) has a unique solution $s\in{\cal X}$ 
and establish the following upper bound
\begin{equation}
\|s\|_{B_R}\le 2\pi\,\|f\|_{B_R} .
\end{equation}
In the polar coordinates (\ref{Eq:polar}) Equation
(\ref{Eq:homol1}) takes the following simple form
\begin{equation}\label{Eq:hompolar}
\partial_\varphi s=f(\sqrt{2I}\cos\varphi,\sqrt{2I}\sin\varphi).
\end{equation}
This equation can be easily solved. Existence, uniqueness and the
upper bound for the solution are consequences of the following lemma. 

\begin{lemma}\label{lm5} \it
Let a $2\pi$-periodic function $f$ be real-analytic in a strip
$\Pi_\rho=\{\,|\Im z|<\rho\,\}$ and continuous  in its closure,
and  $\int_0^{2\pi}f(z)\,\d z=0$.
Then there is a unique primitive $F$, $F'=f$, with
$\int_0^{2\pi}F(z)\,\d z=0$. Moreover, $F$ is real analytic in
$\Pi_\rho$ and 
$$
\sup_{\Pi_\rho}|F|\le 2\pi \sup_{\Pi_\rho}|f|.
$$
If $f$  analytically depends on a complex parameter $I$ at $I=I_0$,
and $\rho=\rho(I)$ is continuous, then $F$ is analytic in $I$ at
$I=I_0$. 
\end{lemma}
\noindent{\it Proof.}
Existence and uniqueness of the primitive with zero mean is obvious.
We have to establish the upper bound only. 
It is remarkable that the upper bound does not depend on $\rho$.
Actually, it is sufficient to estimate $F$ on the boundary
$\partial\Pi_\rho$, where it reaches its maximum value, due to
analyticity and continuity. The restriction of $F$ onto the boundary
$\partial\Pi_\rho$ is  a continuous complex function of the real
variable $t$, if we let  $z=t+\i\rho$. It is sufficient to
consider only the upper component of the boundary
since $F$ is symmetric with respect to complex conjugation
due to its reality.

Let $u(t)$ be a real $2\pi$-periodic continuous function of a real
variable $t$ with zero mean value $\int_0^{2\pi}u(t)\,\d t=0$. Then
there is the unique primitive $U$, $\dot U=u$, with zero mean value.
It is easy to see that $|U(t)|\le\pi\|u\|$, where $\|u\|:=\sup|u|$.
Now we consider a complex-valued function $g(t)=u(t)+\i v(t)$,
where real functions $u(t)$ and $v(t)$ have the above-mentioned
properties. Then $g$ has a unique primitive $G$ with zero mean, and 
$$
|G(t)|^2=|U(t)|^2+|V(t)|^2\le
 \pi^2\left(\|u\|^2+\|v\|^2\right)\le2\pi^2\|g\|^2.
$$
Finally, we apply the last estimate to the function $g(t)=f(t+\i\rho)$, 
and obtain 
$$
|F(t+\i\rho )|=|G(t )|\le2\pi\|g\|=2\pi\|f\|,
$$
which implies the desired estimate.
\hfill\qed

\bigskip

Our problem is complicated by the fact that the complex polar coordinates
are not well defined in the whole $B_R$. 
Lemma~\ref{lm5} provides a solution $s$ of Equation (\ref{Eq:hompolar}),
which is analytic in 
$$
\tilde B_R=
\{\,(\varphi,I):I\ne 0, \ |I|\cosh(2\Im\varphi) < {\textstyle \frac12R^2} \,\}.
$$
Moreover, it does not imply single-valuedness of $s$ as a function of $I$.
For this reason, we have to check analyticity of $s=s(x,y)$ in the original
Cartesian coordinates. Consider the Taylor series
$$
f(x,y)=\sum_{j,k}a_{j,k}x^jy^k,
$$
and construct the Taylor series for $s$. We have
$$
f(\sqrt{2I}\cos\varphi,\sqrt{2I}\sin\varphi)=
\sum_{j,k=0}^\infty a_{j,k}(2I)^{(j+k)/2}\cos^j\varphi\sin^k\varphi
$$
or equivalently
\begin{equation}\label{Eq:f_series}
f= \sum_{l=0}^\infty \sum_{m=-l}^l b_{lm}I^{l/2}\e^{\i m\varphi},
\end{equation}
where
$$
b_{lm}=\sum_{k=0}^l\frac{a_{l-k,k}}{2^l\i^k}\sum_{j_1+k_1=\frac{m+l}2}
\binom{l-k}{j_1}\binom{k}{k_1}.
$$
It is easy to see that $ b_{lm}=0$ for all odd $m+l$.
The assumption $\bar f=0$ is equivalent to $b_{l0}=0$ for all $l$.
And the only solution of the equation $\partial_\varphi s=f$ with
zero average has the form:
\begin{equation}\label{Eq:s}
s=\sum_{l=1}^\infty \sum_{m=-l,\, m\ne0}^l
 \frac{b_{lm} }{m}\,I^{l/2}\e^{\i m\varphi}.
\end{equation}
At inner points of $\tilde B_R$ this series obviously converges 
(since the series (\ref{Eq:f_series}) is convergent
in $\tilde B_R$),
and we have only to check that they define an analytic function at
$(x,y)=(0,0)$ in spite of the apparent presence of the non-integer powers
of $I$.
Indeed, we have
$$
\e^{\i \varphi}=\frac{x+\i y}{\sqrt{2I}},\qquad
\e^{-\i \varphi}=\frac{x-\i y}{\sqrt{2I}},
$$
for all $(x,y)\in B_R$ with $x^2+y^2\ne0$.
Thus
$$
s=\sum_{l=1}^\infty
 \sum_{m=1}^{l}
\frac{(x^2+y^2)^{(l-m)/2}}{2^{l/2}m}
\Bigl( b_{l,m} (x+\i y)^m - b_{l,-m} (x-\i y)^m \Bigr),
$$
where all $b_{lm}$ with odd $l-m$ vanish.
Consequently, the solution $s$ defined by (\ref{Eq:s})
is single-valued and analytic outside of the set $x^2 + y^2 =0$. In fact,
this function has a unique analytic continuation on
the planes $x=\pm\i y$. Indeed, according to the estimate of Lemma \ref{lm5} 
$s$ is bounded near the set $K$ being the union of two disks
$x=\pm \i y$. Furthermore, set $K$ is fine (see, \cite{GuRo}),  that
is, it is locally represented as a zero level of some analytic
function, here $I=x^2 +y^2$. So, by the theorem Osgood-Brown on the
erasing singularities, this function is uniquely extended onto the set
K as the analytic function.  


Since the solution is unique this finishes the proof
of Lemma~\ref{Le:homol}. 
\hfill \qed


\section{Straightening of the slow manifold\label{Se:ssm}}

Now we are ready to prove Theorem~\ref{Th:main}.
As a first step we straighten the slow manifold of the Hamiltonian
system (\ref{Eq:fast}) using a change of coordinates defined
by the generating function (\ref{Eq:genfun}) with
$$
\eps s=x g(u,V)-Yf(u,V).
$$
Here $f$ and $g$ define the slow manifold by (\ref{Eq:fg}).
Then in the new variables the Hamiltonian takes the form
$$
H=H_0(x,y,u,v)+\eps R_0(x,y,u,v;\eps),
$$
where
$$
\left.\frac{\partial H_0}{\partial x}\right|_{x=y=0}=
\left.\frac{\partial H_0}{\partial y}\right|_{x=y=0}=
0.
$$
As it is usual we denote  the new variables by the same
letters as the old ones after the change of coordinates. 
We hope this will not lead to a confusion.
In the new coordinates the slow manifold is given by $x=y=0$.

Let us fix $u$ and $v$ for a moment and make a
coordinate change $(x,y)\mapsto(X,Y)$ defined by a generating
function $s=s(x,Y,u,v,\eps)$. Then the generating function
$$
S=\frac{uV}{\eps}+s(x,Y,u,V,\eps)
$$
defines due to (\ref{canon}) a symplectic change of coordinates 
$(x,y,u,v)\mapsto(X,Y,U,V)$, which is  close to identical
transformation in  $u$ and $v$. Moreover, the transformations defined
by $s$ and $S$ coincide  on the $(x,y)$ component.

Let us fix $u$ and $v$ and transform the Hamiltonian $H_0$ 
to its angle-action variables near $x=y=0$. This is always possible
since the corresponding system has one degree of freedom only.
Then we extend this change of variables to the  $(2n+2)$-dimensional
space. Since it is $\eps$-close to identity in  $u$ and $v$, we have:
\begin{equation}
\label{Eq:newham}
H(x,y,u,v;\eps)= h_0(I,u,v)+\eps r_0(x,y,u,v;\eps),
\end{equation}
where $I=\textstyle\frac12(x^2+y^2)$. 
It is easy to see that $I'=O(\eps)$, thus we found a
hidden slow variable in our system.


\section{Averaging method\label{Se:ma}}

Now we try to decrease the remainder term $r_0$ in (\ref{Eq:newham})
by a sequence of close-to-identity canonical transformations
$\Phi_n:\tilde D_{\delta_{n-1}}\to\tilde D_{\delta_{n}}$,
where 
$$
\tilde D_{\delta_n}=B_{\delta_n}
\times(D_0+\i\delta_n).
$$
The sequence of $\tilde D_{\delta_n} \subset{\mathbb C}^{2n+2}$ is
strictly decreasing.
After the $n^{\mathrm{th}}$ step the Hamiltonian 
$H=H\circ \Phi_1\circ\dots\circ\Phi_n$ takes the form
\begin{equation}\label{Eq:hn}
H= h_n(I,u,v;\eps)+ \eps r_n(x,y,u,v;\eps),
\qquad
I=\textstyle\frac12(x^2+y^2).
\end{equation}

First we study a single step and then continue recurrently.
On the $n^{\mathrm th}$ step we use a coordinate change defined 
by a generating function of the form (\ref{Eq:genfun}),
where we define $s$ to be the unique (see Lemma~\ref{Le:homol}) 
solution of the equation
\begin{equation}\label{Eq:homol}
y\partial_x s-x\partial_y s=\frac{r_n-\overline r_n}{\partial_I h_n}
\end{equation}
with $\overline s=0$, where the bar denotes
an average of a function defined by (\ref{Eq:average}).  Let
\begin{eqnarray}
h_{n+1}&=&h_{n}+\eps\overline r_{n}\;,\\
\eps r_{n+1}&=&H-h_{n+1}\;.\label{Eq:rnplus1}
\end{eqnarray}
Then the Hamiltonian $H$ takes the form (\ref{Eq:hn})
with $n$ replaced by $n+1$.
 
\medskip

This procedure is similar to the classical averaging
\cite{BogolyubovM1961}.  The classical estimates give upper bounds 
$r_n=O(\eps^n)$ and $h_n=h_{n-1}+O(\eps^n)$.
These estimates are not  uniform in $n$.
It is usually possible to make more averaging steps for smaller
$\eps$.  But the classical approach does not give indications on the
optimal choice for $n=n(\eps)$. Technically the upper bound on the
maximum number of steps comes from the reduction of the domain on each
step, which is necessary to apply Cauchy estimates for derivatives.


In a similar situation Neishtadt observed that it is possible to
make up to $\sim\eps^{-1}$ averaging steps.
We will use Neishtadt-type estimates
\cite{Neishtadt1981,Neishtadt1984} to show that 
$$
\|r_{n+1}\|_{\delta_{n+1}}\le \|r_1\|_{\delta_1}\e^{-n}
$$ 
for all $n<c\eps^{-1}$. Thus after $n=[c\eps^{-1}]$ steps the
remainder is exponentially small in $\eps$. Here we denote by
$\|\cdot\|_{\delta}$ the supremum norm of a function on 
$\tilde D_\delta$. 


Both approaches use the same changes of coordinates. The only
difference between the classical and Neishtadt approaches is related
to the choice of $\delta_n$. In both cases the domain reduction
$\xi=\delta_n-\delta_{n-1}$ is independent from $n$. The classical
choice for $\xi$ is a small constant, and Neishtadt used $\xi=A\eps$
with sufficiently large constant $A$. In this way Neishtadt showed
that a larger number of averaging steps is possible before the domain
of the change becomes empty. 

\begin{proposition}[classical-type estimates] \it
Let $\xi\in(0,\delta_0)$ and $\delta_n=\delta_0-n\xi$, 
$1\le n\le \delta_0/(2\xi)$.
There is $\eps_0>0$ and $C_1,C_2>0$ such
that for $0<\eps<\eps_0$
the generating functions {\rm (\ref{Eq:homol})} define
a sequence of canonical changes of coordinates
$\Phi_n:\tilde D_{\delta_n}\to \tilde D_{\delta_{n-1}}$, such that 
$H\circ\Phi_1\circ\ldots\circ\Phi_{n}$ has the form 
{\rm  (\ref{Eq:hn})} with
$$
\|h_n-h_{0}\|_{\delta_n}\le C_1\eps,\qquad \|r_n\|\le C_2\eps^n.
$$
\end{proposition}

It is convenient to make a single averaging step before making
``exponential'' averaging. Namely we may assume $r_0=O(\eps)$.

\begin{proposition}[Neishtadt-type estimates] \it 
Let $\|r_0\|_{\delta_0}\le C_0\eps$.
There is $\eps_0>0$,  
and a sufficiently large positive constant $A$,
such that for $0<\eps<\eps_0$, $\delta_n=\delta_0-nA\eps$,
$1\le n\le \delta_0/(2A\eps)$
the generating functions {\rm (\ref{Eq:homol})} define
a sequence of canonical changes of coordinates 
$\Phi_n:\tilde D_{\delta_n}\to \tilde D_{\delta_{n-1}}$, such that 
$H\circ\Phi_1\circ\ldots\circ\Phi_{n}$ has the form 
{\rm  (\ref{Eq:hn})} with
$$
\|h_n-h_{0}\|_{\delta_n}\le C_1\eps^2,\qquad  
\|r_n\|\le\|r_0\|\,\e^{-n}.
$$
\end{proposition}

The last proposition directly implies Theorem~\ref{Th:main}.

\medskip

\noindent{\em Proof of the propositions.} The proof is done
inductively. According to the condition of normal ellipticity
$\partial_I h_0|_{I=0}$ is positive and uniformly bounded away from
zero. Then by Lemma~\ref{Le:homol}
Equation (\ref{Eq:homol}) uniquely  defines the generating  
function $s$ with $\overline s=0$, and 
$$
\|s\|\le C_3 \|r_n\|.
$$
Then Lemma~\ref{Le:change} says
that the corresponding change of coordinates is well-defined
and
$$
|x-X|,\ |y-Y|\le\frac{2\eps\|s\|_{\delta_n}}{\xi}\le\frac{\xi}{2}
\quad\mbox{and}\quad
|u-U|,\ |v-V|\le\frac{2\eps^2\|s\|_{\delta_n}}{\xi}\le\frac{\eps\xi}{2},
$$
provided $\epsilon_0$ is sufficiently small.
 
We estimate the new Hamiltonian. The upper bound for
$h_{n+1}$ is straightforward. Let us estimate $r_{n+1}$. From
(\ref{Eq:rnplus1}) and (\ref{Eq:hn}) we have
\begin{eqnarray*}
\eps r_{n+1}(X,Y,U,V)
&=&h_n({\textstyle\frac12}(x^2+y^2),u,v)-
h_n({\textstyle\frac12}(X^2+Y^2),U,V)\\
&&+\eps r_n(x,y,u,v)-\eps \overline r_n(X,Y,U,V),
\end{eqnarray*}
where we suppressed the explicit dependence  on $\eps$. It is easy to 
see that 
\begin{eqnarray*}
h_n({\textstyle\frac12}(x^2+y^2),u,v)&=&
h_n + Y(\partial_I h_n) \eps\partial_x s+E_1,\\
h_n({\textstyle\frac12}(X^2+Y^2),U,V)&=&
h_n + x(\partial_I h_n) \eps\partial_Y s+E_2,\\
 r_n(x,y,u,v)&=& r_n+E_3,\\
 \overline r_n(X,Y,U,V)&=& \overline r_n+E_4,
\end{eqnarray*} 
where all functions in the right hand sides depend on the ``mixed''
variables $(x,Y,u,V)$. These equalities may be considered as
definitions for $E_k$, $k=1,\dots,4$.
Taking into account (\ref{Eq:homol}) we obtain
$$
\eps r_{n+1}(X,Y,U,V)=E_1-E_2+\eps E_3-\eps E_4.
$$
Using Cauchy estimates we obtain
\begin{eqnarray*}
\|E_1\|_{\delta_{n+1}},\ \|E_2\|_{\delta_{n+1}}& \le& 
\eps^2 \|h_n''\|_{\delta_n-\xi/2}\frac{\|s\|_{\delta_n}^2}{\xi^2}+
\eps^2\|h_n'\|_{\delta_n-\xi/2}\frac{\|s\|_{\delta_n}}{\xi},\\
\|E_3\|,\ \|E_4\| &\le& 
\eps\,\frac{2\|r_n\|_{\delta_n}\|s\|_{\delta_n}}{\xi^2}.
\end{eqnarray*}
We check the uniform boundedness of the derivatives of $h_n$ using
Cauchy estimates and obtain
$$
\| r_{n+1}\|_{\delta_{n+1}}\le 
\frac{\eps C_3}{\xi}\;\|r_n\|_{\delta_n}.
$$
The proposition follows immediately from this estimate and
the corresponding choices of $\xi$ and $\eps_0$.
\hfill\qed


\section*{Acknowledgments}

The authors thank D. Turaev for his helpful remarks.
This research was partially supported by the INTAS grant 00-221.
L. Lerman is thankful for a partial support to the Russian Foundation of 
Basic Research (grant 01-01-00905) and the program 
``Universities of Russia'' (grant N 1905).
V.G. work was also supported by the Deutsche Forschungsgemeinschaft.


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