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Hamiltonian systems, homoclinic orbits, variational methods.
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\documentclass[11pt]{article}
\usepackage{amsfonts,amssymb,amsmath,a4wide,epic,epsfig}
\def\Zm{{\mathbb Z}}
\def\Xm{{\mathbb X}}
\def\Rm{{\mathbb R}}
\def\Tm{{\mathbb T}}
\def\Nm{{\mathbb N}}
\def\Cm{{\mathbb C}}
\def\lto{\longrightarrow}
\def\lmto{\longmapsto}
\def\eq{\Longleftrightarrow}
\def\leq{\leqslant}
\def\geq{\geqslant}
\newtheorem{PROP}{Proposition}
\newtheorem{DEF}{Definition}
\newtheorem{THM}{Theorem}
\newtheorem{LEM}{Lemma}
\newtheorem{COR}{Corollary}
\newtheorem{CONJ}{\mdseries \scshape Conjecture}
\newtheorem{hyp}{Hypothesis}
\author{Patrick Bernard}
\title{Homoclinic Orbits in Families of Hypersurfaces with
hyperbolic Periodic Orbits.}
\begin{document}
%
\maketitle
%
\section*{Introduction.}
%
%
%
Variational methods provide interesting existence results on
homoclinic orbits to hyperbolic fixed points of Hamiltonian systems
under global conditions. The early result of Bolotin \cite{Bo} about 
Lagrangian flows has been extended to Hamiltonian  
systems in $\Cm^n$ in \cite{CZES}, the hypothesis has been lowered
in \cite{HW} and  \cite{Tanaka}, and finally in  \cite{Sere}
for autonomous systems. 
A natural generalization is the existence of homoclinic orbits to
hyperbolic periodic motions of autonomous Hamiltonian systems. 
Very interesting results have been obtained by Bolotin 
in \cite{Bol} and other papers,
for Lagrangian systems on compact Riemannian manifolds, 
but there are no global results available for systems in $\Cm^n$
where the lack of topology makes Bolotin's methods inefficient.
This paper is a first attempt in that direction.

A periodic motion of an autonomous Hamiltonian system always has
at least two Floquet multipliers equal to 1. As a consequence it
cannot be hyperbolic in the whole phase space, but only with
respect to its energy shell, and it is not
isolated, but included in a 1-parameter family of periodic 
motions, one motion on each energy shell.
The union of the orbits of the family is an invariant two dimensional 
manifold, we call it the central manifold.
It is normally hyperbolic in phase space 
and for that reason it is an easier problem to 
look for orbits homoclinic 
to that manifold than to look for  orbits homoclinic to a prescribed 
periodic motion. An orbit homoclinic to the central 
manifold is homoclinic to one of the periodic motions, by energy
conservation.
 
We study a model class of  systems in $\Cm^n$
where the central manifold is a plane with harmonic
oscillations on it.
We prove that the periodic orbits having a homoclinic orbit are 
dense in the central manifold ouside of a compat set.
We obtain the homoclinics  as accumulation points
of sequences of periodic orbits. These periodic orbits are
subharmonics of perturbed systems.
Convergence of subharmonics has already been used
to find homoclinics to hyperbolic fixed points, see \cite{Tanaka}.

One of the main features of homoclinic orbits is that they induce
 chaotic behavior.
 It is well-known indeed
that a Bernoulli shift with positive entropy exists in systems containing
a transverse homoclinic.
Our result provides classes of autonomous systems in $\Cm^n$ 
enjoying such a behavior at many energy levels.
This is in contrast with the systems with homoclinic orbits to fixed
points, where the chaotic region is found only near the energy shell of
 the fixed point.
We can study for example couplings between stable systems and unstable
ones, and obtain chaotic regions at high energy.

%
%
%
\section{Results and examples.}
%
%
%
In the following, $C$ will always be a positive constant,
possibly different from one line to the other. The $L^p$ norm of $f$
will be noted $\|f\|_p$. We will often use technical 
results from \cite{Tanaka}
without proof.
Let us define 
$$
J_2=\left[
\begin{array}{c c}
0&1\\-1&0
\end{array} \right]
\; ,\; J_{2n}=
\left[
\begin{array}{c c c c }
J_2& 0 &\cdots& 0\\ 
0& J_2& &0 \\
\vdots& & \ddots& \vdots\\
0& 0&\cdots&J_2
\end{array}
\right],
$$
and the associated symplectic form $\Omega_{2n}$ on $\Rm^{2n}$:
 $$\Omega_{2n}(X,Y)=\langle J_{2n}X,Y\rangle.$$
We will omit the subscript $2n$.
There is a splitting 
$$ (\Rm^{2n},\Omega)=(\Rm^2,\Omega)\oplus(\Rm^{2n-2},\Omega),$$
the subspaces 
$\Rm^2$ and $\Rm^{2n-2}$ are $\Omega$-orthogonal  and symplectic.
%
%
%
\subsection{Main result.}
%
%
%
We consider the Hamiltonian system
$$ \dot{X}=J\nabla H(X)$$
associated to the autonomous Hamiltonian
\begin{equation}\label{H}
H(X)=H(x,z)=\frac{1}{2} \omega |x|^2
+ \frac{1}{2}\langle Az,z\rangle +W(x,z),
\end{equation}
$$
X=(x,z)\in \Rm^2 \times \Rm ^{2n-2},
$$
where the pulsation $\omega$ is a positive number,
\begin{description}
\item[HA] A is a $(2n-2) \times (2n-2)$ real symmetric matrix such that
$$\sigma(JA)\cap i\Rm = \emptyset,$$
and $W$ is a $C^2$ function satisfying:
\item[HW1]  there is a $\alpha>2$ and a continous  function
$C:\Rm^2\lto \Rm^+$ such that  
$ W(x,z)\leq C(x)|z|^{\alpha}$ and 
$\nabla _z W(x,z)\leq C(x)|z|^{\alpha-1}$
in a neighborhood of $\Rm^2\times \{0\}\subset \Rm^2\times \Rm^{2n-2},$
\item[HW2]  there is a $\mu \in (2,\alpha]$ such that 
$$\mu W(X)\leq \langle\nabla W(X),X\rangle, $$
\item[HW3] there is $B>0$ such that
$$ B|z|^{\alpha} \leq W(x,z).$$
\end{description}
We will introduce in the proof auxiliary systems satisfying 
\begin{description}
\item[HW4] there exists a compact set outside of which
$$W(x,z)=a|z|^{\alpha}.$$
\end{description}
We obtain the useful inequalities 
\begin{equation}\label{maj}
W(x,z)\leq C|z|^{\alpha}\;\;, \nabla_z W(x,z)\leq C|z|^{\alpha-1}
\end{equation}
from [HW1] and [HW4].
The hypotheses [HA] and [HW1-3] are satisfied for example by the Hamiltonian
$$H(x_1,x_2,z_1,z_2)=x_1^2+x_2^2+z_1^2-z_2^2+
\left( 1+x_1^2\right) \left( z_1^2+z_2^2 \right)^2.
$$
More examples are given below.  

A system satisfying [HA] and [HW1] 
has a two dimensional invariant space $\Rm^2 \times \{0\}$
which is foliated by periodic orbits $O_r$ with equation 
$$
O_r(t)=(e^{J\omega t}(r,0),0)
$$
all having the same period $T_0=2\pi/\omega$.
The orbit $O_r$ has energy $H=\omega r^2/2$ and is hyperbolic
with respect to its energy shell. It has $n-1$ dimensional
stable and unstable manifolds which in the $2n-1$ dimensional
energy shell may intersect along a homoclinic orbit.
In this paper we study this phenomenon, and prove:
%
%
\begin{THM}
Let us consider the Hamiltonian system (\ref{H}) satisfying [HA], [HW1-3].
Let  
$$\mathcal{R}=\left\{ r>0  \text{ such that } O_r 
\text{ has a homoclinic orbit} \right\}.$$ 
There is a positive number $M$ depending 
only on $A,B$ and $\alpha$ 
such that
$$\left[\sqrt{\frac{M}{\pi}},\infty \right)\subset \overline{\mathcal{R}},$$ 
where $ \overline{\mathcal{R}}$ is the closure of $\mathcal{R}$.
\end{THM}
\textbf{Remark 1 : }There is an estimate for $M$, see (\ref{M}) in
the proof, which is enough to obtain  that for fixed $A$ and $\alpha$
$$\lim_{B\lto \infty}M=0.$$
\textbf{Remark 2 : }We do not know whether
$\overline{\mathcal{R}}=\Rm.$ 
Since the origin  does not have any homoclinic in general, it is not
surprising that  we can not find easily homoclinics close to the
origin, but they may well exist.\newline
\textbf{Remark 3 : }The result cannot be
improved to $[C,\infty)\subset \mathcal{R}$ without additional
assumption, see example below.
%
%
%
\subsection{Coupling  stable and  unstable systems.}
%
%
%
Let us consider the unstable system in $\Rm^2$ associated to the
Hamiltonian
$$G(z)=\frac{1}{2}\langle Az,z\rangle + R(z),$$
where the matrix $A$ satisfies [HA] and the nonlinearity $R$
is superquadratic:
$$R(z)=o(|z|^2) \text{ near 0,}$$
$$R(z)\geq C|z|^{\alpha} \text { with } \alpha>2,$$
$$\langle \nabla R(z),z\rangle\geq \mu R(z),  \text{ with } \mu>2.$$
The origin is  a hyperbolic fixed point and has a homoclinic
orbit.
It is well-known from Melnikov theory that a generic time-dependent
perturbation creates transversal homoclinic orbits, which  implies a
chaotic behavior with positive topological entropy. A new  way to
introduce a chaotic behavior  is to couple the system with a 
harmonic oscillator.
Consider the system in $\Rm^4$ associated to the
Hamiltonian
$$H(x,z)=|x|^2+ \frac{1}{2}\langle Az,z\rangle+(1+F(x))G(z),$$
with a positive function $F$ such that 
$ \langle \nabla F(x),x\rangle \geq 0.$
We can apply theorem 1 to that system, this provides homoclinics 
to many of  the periodic motions $z=0$ at high energy,
these orbits are generically transversal and induce chaotic behavior
in fast regions of phase space, that is in regions that contain
no rest point.
%
%
%
%
%
\subsection{Hypersurfaces of $\Rm^{2n}$.}
%
%
%
We now interpret our result in terms of hypersurfaces of
$\Rm^{2n}$.
Let $\Sigma$ be a compact starshaped (with respect to the origin) hypersurface
of $\Rm^{2n}$, let $U_{\Sigma}$ be the bounded connected component of
$\Rm^{2n}-\Sigma$, the notation $\Sigma'\preccurlyeq \Sigma$ means
that $\Sigma' \subset \overline U_{\Sigma}.$
It is well-known that a hypersurface carries a canonical direction
field $D(x)$ satisfying
$$J\nabla H(x)\in D(x)\,\,\,\,\, \forall x \in \Sigma$$ 
for any function $H$ having $\Sigma$ as a regular level hypersurface.
Let us fix a matrix $A$ satisfying [HW1], for any $B>0$ and
$\alpha>2$, we define the compact hypersurface 
$$\Sigma(B,\alpha)= \bigg\{(x,z)\text{ such that }
|x|^2+\frac{1}{2}\langle Az,z \rangle +B|z|^{\alpha}=1\bigg\}.$$
Let $\Sigma$ be a starshaped hypersurface of $\Rm^{2n}$ such that
there exist $0<B\leq D$ and $\alpha$  with
$$\Sigma(D,\alpha)\preccurlyeq \Sigma \preccurlyeq \Sigma(B,\alpha),$$
it is not hard to see that $S=\Sigma \cap \{z=0\}$ is an  invariant
circle of the canonical direction field.
We can define the function
$$R(x,z)=1-|x|^2-\frac{1}{2}\langle Az,z\rangle\geq B|z|^{\alpha}$$
on $\Sigma$.
Since  $\Sigma$ is starshaped there is an $\alpha$-homogeneous function 
$W:\Rm^{2n}\lto \Rm$
extending $R$.
The surface $\Sigma$ is then the regular level  $H=1$ of the
Hamiltonian
$$ H(x,z)=|x|^2+\frac{1}{2}\langle Az,z \rangle +W(x,z),$$
and [HW2] holds.
For any $(x,z)\in \Rm^{2n}-0$, there is a $t>0$ such that 
$(x/t,z/t)\in \Sigma$.
We  check [HW3] writing 
$$W(x,z)=t^{\alpha}W(x/t,z/t)=t^{\alpha}R(x/t,z/t)
\geq t^{\alpha}B\left| z/t \right| ^{\alpha}\geq B|z| ^{\alpha},$$
[HW1] is also easily seen to hold.
We can  apply theorem 1 (with remark 1) to obtain:\newline
\textbf{theorem 1' }
\begin{itshape}
If $B\geq B_0$ there is a sequence $l_n\lto 1$ such that the
hypersurface
$$\Sigma_n=\big\{H=l_n\big\}$$ 
carries an orbit homoclinic to the 
periodic hyperbolic trajectory $\Sigma_n\cap \{z=0\}$,
where $B_0$ is a constant depending only on $A$ and $\alpha$.
\end{itshape}\newline
Using local perturbation techniques  of Hayashi,  Xia 
has proved in a much more general setting that 
a homoclinic orbit can be created by a $C^1$ small perturbation,
(\cite{Hayashi},\cite{Xia}).
We see here that under our additional hypothesis,
a $C^{\infty}$ perturbation is enough.
Of course, it would be very interesting to find sufficient conditions
for the existence of a homoclinic on a given surface.
Our conditions are not sufficient, as shown by the following example.
%
%
%
\subsection{Example.}
%
%
%
Let $F:\Rm^2\lto \Rm$ be a smooth function satisfying
$$z_2^2-z_1^2+C|z|^4\leq F(z_1,z_2)\leq z_2^2-z_1^2+D|z|^4 \,\, \,
\forall z,   $$
$$F(z)=|z|^4\, \text{ outside of a compact set,} $$
the zero level of $F$ having the shape shown in the figure below.
%
%
%
\begin{center}
\epsfig{file=example.ps}
\end{center}
%
%
%
Consider the function 
$$F_{\lambda}(z)=\frac{1}{\lambda^2}F(\lambda z),$$
the Hamiltonian $H_{\lambda}:\Rm^4\lto\Rm$
$$H_{\lambda}(x,z)=|x|^2+F_{\lambda}(z)$$
and the surface
$$\Sigma_{\lambda}=\Big\{H_{\lambda}(x,z)=1\Big\}.$$
The vector field associated to $H_{\lambda}$ has a product
structure, its trajectories satisfy 
$\dot z= J\nabla F_{\lambda}(z)$.
The origin is a hyperbolic rest point for this equation,
but its stable and unstable manifolds are 
heteroclinic orbits
connecting this fixed point to the two other ones
and not homoclinic orbits.
It follows that 
$\Sigma_{\lambda}\cap \{z=0\}$ has no homoclinic 
orbit for the vector field, and thus no homoclinic either for the
canonical
direction field.
Yet we now prove that for $\lambda$ large enough it
satisfies all hypotheses of theorem 1' with $\alpha =4$.
We first note that 
$$F_{\lambda}=\frac{1}{2} \langle Az,z\rangle +O(|z|^4),$$
where 
$$A=\left[
\begin{array}{l r} 2 & 0 \\ 0 &-2 \end{array} \right]$$
satisfies [HA].
To prove that $\Sigma_{\lambda}$ is starshaped for $\lambda$
large enough
we  observe  that 
$$
\langle \nabla H_{\lambda}(x,z),(x,z)\rangle 
=2|x|^2+\langle \nabla F_{\lambda}(z),z\rangle =
\langle \nabla F_{\lambda}(z),z\rangle-2 F_{\lambda}(z)+2$$
on $\Sigma_{\lambda}$ since $|x|^2+F_{\lambda}(z)=1$. 
This gives 
$$
\langle \nabla H_{\lambda}(x,z),(x,z)\rangle
= \frac{1}{\lambda^2}\langle \nabla F(\lambda z),\lambda z\rangle
-\frac{2}{\lambda^2}F(\lambda z) +2 =
\frac{1}{\lambda^2}\Big( \langle \nabla F(y),y\rangle
-2F(y)+2\lambda^2\Big).
$$
This is positive when  $\lambda$ is 
large enough, because $\langle \nabla F(y),y\rangle-2F(y)$ has a lower
bound, the surface is thus starshaped in this case.

There remains to estimate
$$W_{\lambda}(x,z)=F_{\lambda}(x,z)-z_2^2+z_1^2 $$
on $\Sigma_{\lambda}$.
From
$$C|z|^4\leq W(x,z)\leq D|z|^4$$
we get
$$
\lambda^2 C |z|^4   \leq W_\lambda(x,z)\leq \lambda^2 D|z|^4
$$
and thus the condition
$$D|z|^4 \geq W_{\lambda}(z)\geq B_0|z|^4\; \Rightarrow
\Sigma(D,4)\preccurlyeq \Sigma_{\lambda} \preccurlyeq \Sigma(B_0,4)$$
is satisfied for $\lambda$ large.
%In fact, when we build the homogeneous Hamiltonian associated
%to $\Sigma_{\lambda}$ (that is not $H_{\lambda}$), the surface
%$\Sigma_{\lambda}$ appears as a bifurcation between its
%inside and its outside, where the homoclinics drawn on the figure with
%dotted lines exist.
\hfill $\Box$
%
%
%
\section{Convergence of periodic orbits.}
%
%
We prove theorem 1 in the sequel of this paper.
We obtain the homoclinic orbits as limits of sequences of periodic 
orbits of $H$.
It is useful to define the action of a $T$-periodic $C^1$ loop:
$$I_T(X)=\int_0^T \frac{1}{2}\langle JX(t),\dot X(t)\rangle-H(X(t))\,dt.$$ 
We have the following existence result, that will be proved in 
section 3.
\begin{THM}
There is a constant $M$ depending only on $A,B$ and $\alpha$ such that
for any 
$$R_0\geq \sqrt{M/\pi}\,\,,H_0=\frac{1}{2}\omega R_0^2$$
and any $\epsilon>0$
there is a $N(\epsilon)>0$ and a
sequence 
$X_k$ of $T_k$-periodic orbits
satisfying
\begin{align}\label{t1}
&T_k \lto \infty,\\\label{t2}
&0 \leq I_{T_k}(X_k)\leq N(\epsilon),\\\label{t3}
&|H(X_k)-H_0|\leq \epsilon,\\\label{t4}
&z_k\not \equiv 0.
\end{align}
\end{THM}
%
%
That $N$ has to depend of $\epsilon$ in this lemma is what 
makes it impossible to obtain a homoclinic orbit on a given energy 
surface: we can not control in the same time the closeness and the action.
We now prove that theorem 2 implies theorem 1,
that is 
we study the convergence of the sequence $X_k=(x_k,z_k)$ obtained 
by theorem 2. 
%
%
\begin{LEM}
The sequences $\|z_k\|_{\alpha}$ and $\|X_k\|_{C^1}$ are bounded.
\end{LEM}
%
%
Proof:
Since the function $H$ is proper, it follows from (\ref{t3}) that
$\|X_k\|_{\infty}$ is bounded, as well as  $\|X_k\|_{C^1}$ since
$X_k$  satisfies the equation
$$\dot X_k=J\nabla H(X_k).$$
To prove the first part of the lemma, let us write (\ref{t2}) 
and use [HW2,3]:
\begin{eqnarray*}
N \geq I(X_k) & = &\int_0^{T_k} \frac{1}{2}
\langle\nabla H(X_k),X_k\rangle  -H(X_k)\,dt\\
& = & \int_0^{T_k} \frac{1}{2}
\langle\nabla W(X_k),X_k\rangle  -W(X_k)\,dt\\
& \geq & \int_0^{T_k} \left(\frac{\mu}{2}-1\right)W(X_k)\, dt\\
& \geq & B\left(\frac{\mu}{2}-1\right)\|z_k\|^{\alpha}_{\alpha}.
\end{eqnarray*}
\hfill$\Box$\newline
We are now in a position to use Ascoli's theorem to obtain a limit.
Yet we first have to insure non triviality of the limit. It will result
from
\begin{LEM}
There is a $\delta>0$ such that any periodic orbit of $H$
staying in 
$$V_{\delta}=\left\{|z|\leq \delta\right\}\cap 
\left\{H\leq H_0+1\right\}$$ must satisfy $z\equiv 0$.
\end{LEM}
Proof:
This lemma is a consequence of the fact that $z=0$ is a normally 
hyperbolic manifold for $H$. To be more precise, 
let $F_s$ and $F_u$ be the stable and unstable spaces of $JA$,
$\Rm^{2n-2}=F_s\oplus F_u$ by [HA]. We denote the projections by
$$P_s:\Rm^{2n-2}\lto F_s \,\text{ and }\, 
P_u:\Rm^{2n-2}\lto F_u.
$$
There are Euclidean structures $|.|_s$ on $F_s$ and $|.|_u$ 
on $F_u$ and a $\lambda>0$ such that 
$ \langle JAz,z\rangle_s \leq -\lambda |z|_s^2$ when $z\in F_s$ and 
$\langle JAz,z\rangle_u \geq \lambda |z|_u^2 $ when $z\in F_u$. 
From [HW1],
we obtain a $\delta>0$ such that 
$$\langle P_s
J\nabla H_l (x,z),P_s(0,z)\rangle_s \leq -\frac{\lambda}{2} |P_s(z)|^2
$$
and 
$$\langle P_u
J\nabla H_l (x,z),P_u(0,z)\rangle_u \geq \frac{\lambda}{2}
|P_u(z)|^2$$
when $|z|\leq \delta$ and $H\leq H_0+1$.
It follows that if $X(t)=(x(t),z_s(t)+z_u(t)) $ is a solution of the
Hamiltonian equation lying in $V_{\delta}$, $|z_u|_u $ is increasing
 or $0$, and $|z_s|_s$ is decreasing or $0$, 
thus the solution can not be periodic
unless $z\equiv 0$.\hfill $\Box$\newline
Since the equation is autonomous, we can change the time origin of $X_k$
to obtain 
$$z_k(0)\geq \delta/2.$$
For any fixed $\tau$, the sequence 
$\left. X_k \right|_{[-\tau,\tau]}$ has a uniform
limit (up to taking a subsequence) and  by diagonal extraction we can
find a subsequence of $X_k$ converging pointwise and uniformly on any
compact set to a limit $X_{\infty}$ satisfying
$$ \dot X_{\infty}=J\nabla H(X_{\infty}).$$
We also see using Fatou's lemma that $\|z_{\infty}\|_{\alpha}$ is
finite and since $\dot z_{\infty}$ is bounded,
\begin{align*}
&z_{\infty}(t)\lto 0 \text { as }t\lto \pm \infty,\\
&z_{\infty}(0)\geq \delta/2.
\end{align*}
The energy $H_{\infty}=H(X_{\infty})$ satisfies
$$H_{\infty}\in\,[H_0-\epsilon,H_0+\epsilon]$$
because of (\ref{t3}), and the equation
$$\frac{1}{2}\omega|x_{\infty}|^2-H_{\infty}=
-\frac{1}{2} \langle Az_{\infty},z_{\infty}\rangle  
-W(x_{\infty},z_{\infty})$$
implies that $x_{\infty}(t)$ must go to 
$$r=\sqrt{\frac{2H_{\infty}}{\omega}}$$
when $z_{\infty}(t)$ goes to $0$.
Thus the trajectory $X_{\infty}$ is homoclinic to
$\{z=0\}\cap \{H=H_{\infty}\}$.
This proves theorem 1, since $\epsilon>0$ can be chosen as small as needed.
\hfill$\Box$\newline
%
%
%
%
\section{Existence of periodic orbits.}
%
%
%
We prove theorem 2 in this section using variational methods.
Let us fix a radius $R_0$ and the associated energy 
$H_0=\omega R_0^2/2$.
The functional $I$ does not satisfy PS condition
because the oscillations on the central manifold form a non-compact
family of critical points of zero action.
Moreover, we have to find a way to specify around which
energy surface we are working.
For these reasons, it will be useful to introduce a perturbation
that will turn PS condition on, and that will confine critical points
around the fixed energy surface.

Before we perturb the system, let us notice that
since we are looking for phenomena taking place around a fixed
energy surface, it is harmless to change the Hamiltonian 
at infinity. We use this remark following a well-known
trick, see \cite{Rab} for example.
Let $K>0$ be a large number, let $\chi\in C^{\infty}(\Rm,\Rm)$ 
be a smooth increasing
function such that $\chi(x)=0$ for $x\leq K$, $\chi(x)=1$ if 
$x\geq K+1$ and $\chi'\leq 2$.
We introduce the function
$$ 
\tilde W(X)=(1-\chi(|X|))W(X)+\chi(|X|)a|z|^{\alpha} 
$$
where 
$$
a=\max_{K\leq |X|\leq K+1} \frac{W(X)}{|z|^{\alpha}}.$$
It is not hard to check that this function satisfies 
[HW1-3], with the same constants.
If $K$ is large enough
the hamiltonian have not been changed for $H(X)\leq H_0+1$ and
it is the same to prove theorem 2 for $\tilde W$ or for $W$.
In the following we will work with $\tilde W$ instead of $W$,
but for simplicity we will still call it $W$, that is we will suppose 
that [HW4] holds.
We are now in a position to introduce the perturbed hamiltonian we are
going to study.

Let us take a function
\begin{eqnarray*}
S:\Rm^{2n}& \lto & \Rm\\
(x,z)&\lmto&(H(x,z)-H_0)^4 \,\text{ when $H(x,z)\leq H_0+1,$}\\
(x,z)&\lmto& C(|x|^3+|z|^{\alpha})\, \text{ outside of a compact set.}
\end{eqnarray*}
We moreover assume that 
$$H\geq H_0+1 \Rightarrow  S\geq 1$$
and
$$ S(x,0)=f(|x|^2)$$
where $f$ is a smooth and convex function.
It is not hard to see that the 
above class of functions is not empty.
We  consider the Hamiltonian 
$$
H_l(x,z)=H(x,z)+lS(x,z).
$$
where  l will always be chosen small enough so that the equation
$H_l=E+l(E-H_0)^4$ has only one solution $E(H_l)\geq \min H$.
The shell $H_l=h_l$ of $H_l$ is the shell $H=E(h_l)$ of $H$
when $h_l\leq H_0+1 $ thus  the local structure of the flow has
not been changed by the perturbation in this region, where there holds
\begin{equation}\label{grad}
\nabla H_l= (1+4l(H-H_0)^3)\nabla H.
\end{equation}
Although $H$ and $H_l$ have the same periodic solutions
in the region under interest, we will  look 
for $T$-periodic trajectories of $H_l$, that are easier to be found
as critical points of 
\begin{eqnarray*}
I_l(x,z)& = & \int_0^T \langle -\frac{1}{2}J\dot X,X \rangle -H_l(X)\,dt
\end{eqnarray*}  
on a suitable function space.
We will prove the following proposition,
that leads to theorem 2.
\begin{PROP}
There exists a constant $M$ depending only on $A,B$ and $\alpha$,
such that if
$$ \pi R_0^2\geq M$$
there holds:
For any $\Delta>0$ and any 
$$
T\in \frac {2\pi}{\omega} \Nm \cap [1,\infty)$$
there exists $ l(T)$ in the interval $(0,\Delta/T)$
and a $T$-periodic trajectory $(x_T,z_T)$ of $H_{l(T)}$ such that
\begin{align}\label{p1}
&0< I_{l(T)}(x_T,z_T)\leq M,\\ \label{p2}
&\int_0^T S(x_T,z_T)\,dt \leq \frac{TM}{\Delta} +1,\\ \label{p3}
&z_T \not\equiv 0.
\end{align}
\end{PROP}
Before we prove this proposition, let us see that it implies 
theorem 2.
Set
$$h_T=H_{l(T)}(X_T).$$
If $\Delta$ has been chosen large enough,
(\ref{p2}) implies that $S$ must take a value below one 
when $T$ is large enough, thus $X_T$ is contained in $H\leq H_0+1$
and has a fixed energy $E_T=H(X_T)$.
We apply (\ref{p2}) once again and get 
$$E_T-H_0\leq \left(2\frac{M}{\Delta}\right) ^{\frac{1}{4}}
$$
and 
$$
0\leq h_T-E_T= l(E_T-H_0)^4\leq 2\frac{M}{T}
$$
when $T$ is large enough.
Let us now define the curve
$$\tilde X_T (t)=X_T\Bigl(\bigl(1+4l(E_T-H_0)^3\bigr)^{-1}t\Bigr),$$ 
it comes directly from (\ref{grad}) that $\tilde X_T$ is a 
trajectory of $H$, the period of which 
$$\tilde T= \bigl(1+4l(E_T-H_0)^3\bigr)T$$
satisfies
$$\tilde T\geq T+4l(E_T-H_0)^3T\geq T-5M^{\frac{3}{4}}\Delta^{\frac{1}{4}}.$$
We can estimate its action
$$I(\tilde X_T)=I_{l(T)}(X_T)+Th_T-\tilde T E_T=
I_{l(T)}(X_T)+ T(h_T-E_T)+(T-\tilde T)E_T$$
and obtain
\begin{equation}\label{p4}
 0\leq I(\tilde X_T)\leq 3M+
5 M^{\frac{3}{4}}\Delta^{\frac{1}{4}}
(H_0+1).
\end{equation}
The first inequality above holds because any critical point
has positive action.
The sequence $\tilde X_T$ satisfies all the conclusions of theorem 2
which is finally proved.
We remark that $\Delta$ appears in this estimate, so that 
we must fix it before passing to the limit, and that's why
we can't reach the surface $H=H_0$ itself.
\hfill $\Box$\newline
We now have to prove proposition 1.
Let us fix a period
$T=\tau 2\pi  / \omega$, $\tau\in \Nm$,
and define the following functionals on smooth $T$-periodic arcs:
\begin{eqnarray}\label{e}
e(x(t)) & = & \int_0^T-\frac{1}{2}\langle J\dot x(t) +\omega x(t),
x(t)\rangle dt ,\\\label{h}
h(z(t)) & = & \int_0^T-\frac{1}{2}\langle J\dot z(t) +A z(t),
z(t)\rangle dt,\\\label{b}
b(x(t),z(t)) & = & \int_0^T W(x(t),z(t))dt,\\ \label{p}
p(x(t))& = & \int_0^T  S(x(t),z(t))dt.
\end{eqnarray}
We are going to obtain $T$-periodic orbits of $H_l$ as critical points
of 
$$ I_l(x(t),z(t))=e(x(t))+h(z(t))-b(x(t),z(t))-lp(x(t),z(t)). $$
The proof of proposition 1
goes  along the following line. We first take a good
function space on which the above functional can be studied. We
see that this functional has a ''universal'' linking structure,
this allows us to define a critical level $c_T(l)$
which is a nonincreasing function of $l$.
It will appear from the construction that $0<c_T(l)\leq M$ for a constant $M$
independent of $l$ and $T$, this is (\ref{p1}).
Since $l$ is allowed to take values in 
the interval $(0,\Delta/T)$, there must 
be a $l$ such that $c'_T(l)$ exists and $|c'_T(l)|\leq MT/\Delta$.
Using Struwe's monotony  method (see \cite{Struwe}, II.9), it can be deduced
that there is a critical point $X$ at level $c_T(l)$ with
$$p(x(t)) = \left|\frac{\partial}{\partial l}H_l(X)\right|\leq 
1+MT/\Delta,$$
this is (\ref{p2}).
To obtain (\ref{p3}) we just have to check that no $T$-periodic
solution of $H_l$
on $\Rm^2\times \{0 \}$ has its action in $(0,M]$ if
$\pi R_0^2\geq M$.
let us start with this program.
%
%
%
%
\subsection{The analytical setting.}
%
%
%
%
We use Fourier series 
$$x(t)=\sum_{k\in \Zm} e^{Jk\omega t / \tau}x_k, \,\, x_k\in \Rm^2$$
to compute $e$:
$$ e\left(\sum_{k\in \Zm} e^{Jk\omega t / \tau}x_k\right)
=\sum _{k\in \Zm}
\frac{\omega}{2}\left(\frac{k}{\tau}-1\right) |x_k|^2.$$
We define the  inner product 
$$ \langle x,y\rangle_e = 
\frac{\omega}{\tau}\langle x_{\tau},y_{\tau}\rangle +
\sum _{k\in \Zm}
\omega\left|\frac{k}{\tau}-1\right| \langle x_k ,y_k\rangle,
$$
its associated norm $\|x\|^2_e=\langle x,x \rangle _e$ 
and the space
$$E_e = \left\{ x\in L^2(0,T;\Rm^2) \text{ such that } 
\|x\|_e<\infty\right\}.$$
It is classical that $e$ can be extended to $E_e$ as a continuous
quadratic form, and there is an orthogonal splitting
$$E_e=E_e^+ \oplus E_e^0 \oplus E_e^-,$$
with
\begin{eqnarray*}
E_e^+ & = & \{x \text{ such that } x_k=0 \text{ for k } \leq 0\},\\
E_e^0 & = & \{x \text{ such that } x_k=0 \text{ for k } \neq 0\},\\
E_e^- & = & \{x \text{ such that } x_k=0 \text{ for k } \geq 0\},
\end{eqnarray*}
$P^{\pm}_e$, $P^0_e$ are  the associated projections.
We then obtain the nice expression
$$ e(x)= \frac{1}{2}\|P^+_e(x)\|^2- \frac{1}{2}\|P^-_e(x)\|^2,$$
and we can sum up some important properties:
\begin{LEM}
The space $E_e$ is the standard $H^{1/2}_T$ space,
the norm $\|x\|_e$ is equivalent to the standard $\|x\|_{H^{1/2}}$
though non uniformly with respect to $\tau$, thus for any $p>1$
the embedding 
$$ j_e^p : E_e\longrightarrow L^p_T(\Rm^2)$$
is compact. Moreover for any $x\in E_e$
there holds
\begin{equation}\label{mprep}
\|x\|_e^2\geq \frac{\omega}{\tau}\|x\|^2_2.
\end{equation}
\end{LEM}
The proof is well-known, see \cite{HZ} for a clear exposition. The
last inequality follows directly from expressions in Fourier series.
\hfill $\Box$\newline
The quadratic form $h$ can also be extended as
$$h(z)=\frac{1}{2}\|P^+_h(z)\|^2_h-\frac{1}{2}\|P^-_h(z)\|^2_h$$
on a Hilbert space $E_h$,
where $P^{\pm}_h $ are the projections  on $E^{\pm}_h$
associated with the orthogonal splitting
$E_h=E_h^+\oplus E_h^-$. 

\begin{LEM}
The space $E_h$ is the standard $H^{1/2}_T(\Rm^{2n-2})$ and the 
norm $\|z\|_h$ is uniformly equivalent to the standard
$\|z\|_{H^{1/2}}$,
that is there are constants $C$ and $C'$ independent of $T$
such that
$$C\|z\|_{H^{1/2}}\leq \|z\|_h \leq C'\|z\|_{H^{1/2}}. $$
As a consequence,  the embeddings 
$$ j_h^p : E_h\longrightarrow L^p_T(\Rm^{2n-2})$$
are compact for any $p>1$, moreover for $p\geq 2$ there are constants
$C_p$ and   $P_p$ independent of $T$ such that 
\begin{equation}\label{zp}
\|z\|_p \leq C_p \|z\|_h
\end{equation}
and 
\begin{equation}\label{proj}
\|P^{\pm}_hz\|_p\leq P_p \|z\|_p.
\end{equation}
\end{LEM}
Proof: This is proposition 1.1 of \cite{Tanaka}.\hfill $\Box$\newline
We can now define the total function space 
$$E_T=E_e\times E_h\,,\,\,\,\, \|(x,z)\|^2=\|x\|_e^2+\|z\|_h^2,$$
which is nothing but $H^{1/2}_T(\Rm^{2n})$ with an equivalent inner
product (not uniformly in $T$).
We have seen that $e$ and $h$ are continuous, and thus $C^{\infty}$,
quadratic forms. Let us now study the non quadratic parts.
It is well-known that 
\begin{eqnarray*}
\tilde p:L^3(\Rm^2)\times L^{\alpha}(\Rm^{2n-2}) & \longrightarrow &\Rm \\
(x(t),z(t)) & \lmto & \int_0^T S(x(t),z(t))dt
\end{eqnarray*}
is $C^1$, and 
$$p=\tilde p \circ j_T$$ also is, where
 $$j_T(x,z)=\left(j_e^3(x),j_h^{\alpha}(z)\right)\in 
L^3\times L^{\alpha}.$$ 
In the same line,
\begin{eqnarray*}
\tilde b:L^3(\Rm^2) \times L^{\alpha}(\Rm^{2n-2}) & \longrightarrow &\Rm \\
\left(x(t),z(t)\right) & \lmto & \int_0^T W(x(t),z(t))dt
\end{eqnarray*}
is $C^1$ thanks to (\ref{maj}), and
$$b=\tilde b \circ j_T$$ 
also is.
\begin{LEM}
The functional $I_l$ is well defined and $C^1$ on $E_T$,
it can be written
$$ I_l(x,z)= \frac{1}{2} \|P^+_e(x)\|^2 -\frac{1}{2} \|P^-_e(x)\|^2+
\frac{1}{2} \|P^+_h(z)\|^2 -\frac{1}{2} \|P^- _h (z)\|^2
-(\tilde b+l\tilde p)\circ j_T(x,z),
$$
and its gradient is
\begin{eqnarray*}
\nabla I_l (x,z)  & = &  P^+_e(x)-P^-_e(x)
+P^+_h(z)-P^-_h(z)+j_T^*\left( \nabla (\tilde b+
l\tilde p)\circ j_T (x,z)\right),\\
 & = & L(x,z) +K(x,z),
\end{eqnarray*}
where $K$ is continuous and maps bounded sets
into relatively compact ones . The solutions of 
$$\nabla I_l(X)=0$$
are precisely the $C^1$ T-periodic trajectories of the system $H_l$.
\end{LEM}
The proof is classical, see \cite{HZ}.\hfill $\Box$\newline
There remains to study the behavior of Palais Smale sequences.
The unperturbed functional $I_0$ does not satisfy PS condition, but
%
%
%
\begin{LEM}
the functional $I_l$ satisfies the PS condition for any $l>0$.
\end{LEM}
%
%
%
Proof: The proof follows the line of \cite{Rab}, chapter 6. We go in
it since many details are different.
Let $X_m$ be a bounded PS sequence,
$$\nabla I_l(X_m)=L(X_m)+K(X_m) \longrightarrow  0$$
implies that 
$L(X_m)=(P^+_e(x_m)-P^-_e(x_m),P^+_h(z_m)-P^-_h(z_m))$ 
has a convergent subsequence, but 
then $P^0_e (X_m)=X_m-P^+_e (X_m)-P^-_e (X_m)-P^+_h (X_m)-P^-_h(X_m)$
is bounded and thus has a convergent subsequence since $E^0_e$ is
finite dimensional.
Thus any bounded PS sequence has a convergent subsequence.
There remains to prove that all PS sequences are bounded.
It will be useful to estimate 
\begin{align*}
I_l(X) -\frac{1}{2}\langle \nabla I_l(X),X\rangle 
& = \int_0^T \frac{1}{2}\langle\nabla W_l(X),X\rangle -W_l(X)\,dt\\
\intertext{where $W_l=W+lS= A|z|^{\alpha}+D|x|^3$ at infinity,
  thus}
I_l(X) -\frac{1}{2}\langle \nabla I_l(X),X\rangle 
& \geq \int_0^T A\left(\frac{\alpha}{2}-1\right)|z|^{\alpha}
+\frac{D}{2}|x|^3-C\,dt\\
&\geq C(\|z\|_{\alpha}^{\alpha}+\|x\|_3^3-1)
\end{align*}
%
%
%\frac{1}{2}\langle\nabla W(X),X\rangle -W(X)\,dt\\
%& +\int_0^T l(H-H_0)\langle\nabla H(X),X\rangle-l(H-H_0)^2\, dt\\
%& = \int_0^T \frac{1}{2}\langle\nabla W(X),X\rangle -W(X)\,dt\\
%& +\int_0^T l(H-H_0)\left(\langle\nabla H(X),X\rangle+H_0-H\right)\,dt\\
%&\geq C\int_0^T (|x|^2+|z|^{\alpha}-1)^2\,dt.
%\end{align*}
Applying the above to a PS sequence $X_m$ gives
\begin{equation}\label{PS}
\|z_m\|_{\alpha}^{\alpha}+\|x_m\|_3^3
\leq C(1+\epsilon_m\|X_m\|),
\end{equation}
with $\epsilon_m\lto 0$.
Next 
$$
|\langle \nabla I_l (X_m),z_m^+\rangle |=
\left| 2\|z_m^+\|^2_e - \int_0^T \langle \nabla W_l(X_m), z_m^+\rangle \,dt
\right|
\leq \epsilon_m \|z_m^+\|_e$$
gives 
\begin{align*}
2\|z_m^+\|_e^2 
 & \leq \left| \int_0^T \langle  \nabla_z  W_l(X_m), z_m^+\rangle \,
  dt\right|
+ \epsilon_m\|z_m^+\|_e\\
 &\leq C \int_0^T (1+|z|^{\alpha-1})|z_m^+|\, dt+ \epsilon_m\|z_m^+\|_e\\
&\leq C \left\|1+|z|^{\alpha-1}\right\|_{\frac{\alpha}{\alpha-1}}
\, \|z_m^+\|_{\alpha} + \epsilon_m \|z_m^+\|_e\\
& \leq C(1+\|z_m\|^{\alpha-1}_{\alpha})\|z_m^+\|_{\alpha}     
 + \epsilon_m \|z_m^+\|_e\\
& \leq C(1+\|z_m\|^{\alpha}_{\alpha})\,\|z_m^+\|_e\,,
\end{align*}
combining this  with (\ref{PS}) yields
$$\|z^+_m\|_e \leq C(1+\epsilon_m\|X_m\| ).$$
The same can be written for $z_m^-$ and $x_m^{\pm}$,
there just remains to deal with $x^0_m$, which is done noticing that
(\ref{PS}) gives 
$$\|x^0_m\|_e=\frac{\omega}{\tau}\|x^0_m\|_2 \leq 
\frac{\omega}{\tau}\|x_m\|_2\leq C\|x_m\|_3
\leq C(1+\epsilon_m\|X_m\|).$$
All this together
$$\|X_m\|^2=\|x^0_m\|^2+\|x^+_m\|^2+\|x^-_m\|^2+\|z^+_m\|^2+\|z^-_m\|^2\leq
C(1+\epsilon_m \|X_m\|)^2$$
implies that $\|X_m\|$ is bounded.\hfill$\Box$\newline
We are now ready to apply classical variational methods to $I_l$.
%
%
%
\subsection{ The topology. }
%
%
%
The topological argument is inspired from the one in \cite{Tanaka}.
Yet the center of our linking is not the origin as usual, but
the distinguished orbit $O_{R_0}(t)$. It is not hard to check that 
$O_{R_0}(t)$ is a critical point of our
variational problem.
As usual (see \cite{EH})
, we introduce a group $\Gamma$ of homeomorphisms of $E_T$:
\begin{DEF}
A homeomorphism $\gamma : E_T\rightarrow E_T $ belongs to $\Gamma$
iff it can be written  in the form
$$
\gamma(x,z)= e^{a^+_e(x)}P^+_e(x) + e^{a^-_e(x)}P^-_e(x) +P^0_e(x)
+ e^{a^+_h(z)}P^+_h(x) + e^{a^-_h(z)}P^-_h(z)
+k(x,z)
$$
where $a^{\pm}_{e,h}: E_T\rightarrow \Rm $ are continuous and map
bounded sets into bounded sets, and $k: E_T\rightarrow E_T$ is
continuous and maps bounded sets into relatively compact ones. 
In addition there
exists a $\rho >0$ such that the support of  $a^{\pm}_{e,h}$ and $k$
is contained in 
$$ \left\{ (x,z)\in E_T \text{ such that } e(x)+h(x)>0 
\text{ and } \|(x,z)\|\leq \rho\right\}.$$
The functionals $e$ and $h$ are defined in (\ref{e}) and (\ref{h}) above.
\end{DEF}
It is not hard to see that $\Gamma$ is a group, see \cite{HZ}, 5.3
for related material.
Let us now introduce the sphere
$$ S^+=\{(x,z)\in E^+_e+E^+_h \text{ such that } \|(x,z)\|=1\} .$$
We shall link $\tilde S^+=O_{R_0}+S^+$
 with an affine subspace of $E_T$ of the
form $O_{R_0}+E^-_e +E^0_e +E^-_h +\Rm z_T$, with $z_T\in E_h^+$. 
We follow Tanaka \cite{Tanaka} for the choice of $z_T$, and take
$z_T=P^+_h(\phi)$, where 
$\phi \in C_0^{\infty}((0,1),\Rm^{2n-2})$
is extended by $0$ to $[0,T]$ and satisfies
$$\int_0^{1} \langle J\dot {\phi}+A\phi,\phi \rangle dt <0.$$   
\begin{LEM}\label{zt}
There are constants $C_p$ and $C'_p$ independent of $T \geq 1 $ such that 
for all $p>1$
\begin{eqnarray*}
C_0\leq &\|z_T\|& \leq C'_0\\
C_p\leq &\|z_T\|_p & \leq C'_p.
\end{eqnarray*}
\end{LEM}
This is lemma 1.4 of \cite{Tanaka}.\hfill $\Box$\newline
Let 
$$V=E^-_e +E^0_e +E^-_h +\Rm z_T,$$
The spaces $V$ and $S^+$ link with respect to $\Gamma$:
\begin{LEM}[Intersection property]
For $\gamma \in \Gamma$, we have 
$$\gamma(S^+)\cap V\neq \emptyset.
$$
\end{LEM}
Proof: This is classical, see for example
\cite{EH},  proposition 1.\hfill $\Box$\newline 
It is therefore natural to define:
\begin{DEF}
$$c_T(l)=\sup_{\gamma \in \Gamma} 
\left(\inf_{S^+} I_l\circ \gamma\right).$$
\end{DEF}
Before we prove that $c_T(l)$ is a critical value, 
it is of interest for  us to estimate it.
\begin{PROP}\label{clev}
There is a constant $M$ that depends only on $A,B$ and $\alpha$
such that for all $l>0$
$$0<c_T(l)\leq M.$$
\end{PROP}
Proof: 
For all $\eta>0$ there exists $\gamma \in \Gamma$ such that 
$\gamma(S^+)= O_{R_0}+\eta  S^+$.
On the other hand, the intersection property above implies that
$c_T(l)\leq \sup_{V} I_l$.
For these reasons, proposition \ref{clev} follows from
lemma \ref{mlink} and \ref{Mlink} below.\hfill $\Box$
%
%
%
\begin{LEM}\label{mlink}
Let us fix all parameters.
There are  $\eta>0$ and $\delta>0$ such that 
$$I_l( O_{R_0} +x^+,z^+)>\delta $$
whenever $(x^+,z^+)\in E^+_e\times E^+_h$ satisfy 
$\|(x^+,z^+)\|=\eta$.
\end{LEM} 
%
%
%
Proof:
We first compute 
\begin{eqnarray*}
S(O_{R_0}(t)+x(t),z(t)) &\leq & C\Big(H(O_{R_0}(t)+x(t),z(t))-H_0\Big)^4\\
         &\leq & C\Big(|x(t)|+|x(t)|^2+|z(t)|^{\alpha}\Big)^4\\
         &\leq & C(|x(t)|^4+|x(t)|^8+|z(t)|^{4\alpha}).
\end{eqnarray*}
Noticing  that $O_{R_0}\in E^0_e$ this yields
\begin{align*}
I_l(O_{R_0}+x^+,z^+) & =  \frac{1}{2}\|x^+\|^2 +\frac{1}{2}\|z^+\|^2
-b(O_{R_0}+x^+,z^+)-lp(X)\\
               &\geq  \frac{1}{2}\|x^+\|^2 +\frac{1}{2}\|z^+\|^2\\ 
 &-C\|z^+\|_{\alpha}^{\alpha}
-C\|z^+\|_{4\alpha}^{4\alpha}-C\|x^+\|^4_2-C\|x^+\|^8_2\\
&\geq \frac{1}{2}\eta^2-C(\eta^{\alpha}+\eta^{4\alpha}
+\eta^4+\eta^8).
\end{align*}
We have used (\ref{mprep}) and (\ref{zp}) for the last inequality.
\hfill $\Box$
%
%
%
\begin{LEM}\label{Mlink}
There is a constant $M$ that depends only on $A,B$ and $\alpha $ 
such that for all $l>0$ 
$$\left. I_l \right|_V\leq M.$$
\end{LEM}
%
%
%
Proof:
Let $X=(x^- +x^0,z^- +  rz_T)\in V$, from [HW3] we get
\begin{eqnarray*}
I_l(X)& =&-\frac{1}{2} \|x^-\|^2 -\frac{1}{2}\|z^-\|^2 
+\frac{1}{2}\|rz_T\|^2 -b(X)-lp(X)\\
   & \leq & \frac{1}{2}\|z_T\|^2r^2 - B\|z^-+rz_T\|^{\alpha}_{\alpha}.
\end{eqnarray*}
Using (\ref{proj}) gives: 
$$\|rz_T\|^{\alpha}_{\alpha}=
\|P^+_h(z^-+rz_T)\|^{\alpha}_{\alpha}
\leq P_{\alpha}^{\alpha} \|z^-+rz_T\|^{\alpha}_{\alpha},
$$
combining these equations yields
$$
I_l(X)\leq \frac{1}{2}\|z_T\|^2r^2
-BP_{\alpha}^{-\alpha}\|z_T\|^{\alpha}_{\alpha}r^{\alpha},
$$
and we obtain the lemma setting 
\begin{equation}\label{M}
M= \sup _{T\in [1,\infty)} \sup_{r\in \Rm^+}
\left( \frac{1}{2}\|z_T\|^2r^2-
BP_{\alpha}^{-\alpha}\|z_T\|^{\alpha}_{\alpha}r^{\alpha}\right)
\end{equation}
which is finite according to lemma \ref{zt}.\hfill $\Box$




\subsection{The critical point.}
We will now prove that
there exists $l(T)\in ]0,\Delta/T[$ and
a critical point  $X_T$ of $I_{l(T)}$ at level $c_T(l(T))$ such that
$p(x_T)\leq 1+TM/\Delta$. 
Let us first chose $l(T)$.
\begin{LEM}
There exists $l(T)\in (0,\Delta/T)$ such that 
$l\lmto c_T(l)$ is differentiable in $l(T)$ and
$$|c'_T(l(T))|\leq TM/\Delta.$$
\end{LEM}
Proof: From its definition, $c_T(l)$ is a nonincreasing function of 
$l$, it is thus differentiable almost everywhere in $]0,\Delta/T[$ and
there holds
$$\int_0^{\Delta/T} c'_T(l) dl \geq -M.$$ 
\hfill $\Box$\newline
We are now going to \newline
\begin{itshape}
suppose that there is no critical point at level  $c_T(l(T))$
satisfying $p(x_T)\leq 1-c'_T(l(T))$, \newline
\end{itshape}
and prove that this leads to a contradiction.
Let $l_n\lto l(T)$ be a decreasing sequence, $I_n=I_{l_n}$,
$c_n=c_T(l_n)$, $c'=|c'_T(l(T))|$ and $c=c_T(l(T))$.
Using the supposition above and the fact that PS is satisfied for
$I_{l(T)}$ we can prove the following lemma by a deformation argument:
\begin{LEM}
There is an $\epsilon$ in the interval 
 $(0,c/2)$ such that for any $K$ there is 
an homeomorphism $\gamma_K\in \Gamma$ satisfying 
%\begin{eqnarray*}
%I_{l(T)}(\gamma_K(X)) & \geq & I_{l(T)}(X),\\
% & \text{ and } &   \\
%I_{l(T)}(\gamma_K(X))\geq c+\epsilon & \text{ when }&
%p(X)\leq c'+ 1/2 \\
% & \text{ and } & I_{l(T)}(X)\geq c-\epsilon\\
% & \text{and} & \|X\|\leq K.\\
%\end{eqnarray*}
$$I_{l(T)}(\gamma_K(X))  \geq  I_{l(T)}(X)$$
for all $X\in E_T$, and such that 
$$I_{l(T)}(\gamma_K(X))\geq c+\epsilon$$
for all $X$ satisfying the following three inequalities
\begin{eqnarray*}
p(X)&\leq &c'+ 1/2, \\
I_{l(T)}(X)&\geq& c-\epsilon,\\
\|X\|&\leq &K.
\end{eqnarray*}
\end{LEM}
\hfill$\Box$\newline
From the definition of $c_n$, we can choose $\gamma_n\in \Gamma$ 
such that 
$$\inf_{S^+}I_n\circ \gamma_n \geq c_n - (l_n-l)/10.$$
Let us set $K_n=\sup_{S^+} \|\gamma_n\|$, 
$\varphi_n=\gamma_{_{K_n}}$. For $n$ large enough there holds
\begin{eqnarray*}
\left. I_{l(T)}\circ \gamma _n\right| _{S^+} \geq 
 \left. I_n\circ \gamma_n\right| _{S^+} & \geq &
c_n-(l_n-l)/10\\
& \geq & c- (c'+1/10)(l_n-l)-(l_n-l)/10\\
&\geq & c-(c'+1/5)(l_n-l).
\end{eqnarray*}
Take $X\in S^+$:\newline
\textit{Although }
$I_{l(T)}(\gamma_n(X)) \leq  c+(l_n-l)/5$,\newline
and since
\begin{eqnarray*}
(l_n-l)p(\gamma_n(X)) & =& I_{l(T)}(\gamma_n(X))- I_n(\gamma_n(X))
\leq c+(l_n-l)/5-c_n+(l_n-l)/5\\
& \leq& (c'+1/2)(l_n-l),
\end{eqnarray*}
we can apply the lemma for $n$ large enough and get
$$I_{l(T)}(\varphi_n \circ \gamma_n (X))\geq c+\epsilon;
$$
\textit{or}
$$I_{l(T)}(\varphi_n \circ \gamma_n (X))\geq 
I_{l(T)}(\gamma_n(X))\geq c+(l_n-l)/5.
$$
In both cases we have, for $n$ large enough,
$$I_{l(T)}(\varphi_n \circ \gamma_n (X))\geq 
 c+(l_n-l)/5,
$$
which means that there exists 
$\gamma=\varphi_n\circ \gamma_n \in \Gamma$
such that 
$$\inf_{S^+}I_{l(T)}\circ \gamma >c,$$
this is in contradiction with the definition of $c$. We have 
proved the existence of a critical point satisfying 
(\ref{p1}) and (\ref{p2}). There remains to prove (\ref{p3}).

\subsection{ Non triviality.}
In this subsection, we prove conclusion (\ref{p3}).
We point out that this is the only part in  the proof of proposition 1
where the condition $\pi R_0^2\geq M$ is used.
In fact, the critical point constructed always exists, but
it may be contained in the plane $z=0$. That it is not the case under
our hypotheses is a key ingredient for the non triviality of
the homoclinic obtained after convergence.
We first observe that
$$\nabla_z H_l(x,0)=0 \,\,\Rightarrow \,\,\nabla_zI_l(x,0)=0,$$ 
which means that the plane $z=0$ is left invariant by the flow,
and that the subspace $E_e\times \{0\}$ is transversally critical.
As a consequence, the critical points of $I_l$ that
are on the form $(x(t),0)$ are precisely the critical points of 
$\left. I_l\right|_{E_e\times\{0\}}$, and they are the 
$T$-periodic orbits of the flow contained in $z=0$.
\begin{LEM}
Let $(x,0)$ be a critical point, then the set $\{ x(t),t\in \Rm\}$
is a circle $S(r)$, where $r$ satisfies 
$$ lf'(r^2)\in \frac{\pi}{T}\Zm, 
$$
and we have 
$$I_{l(T)}(x,0)=T\left(r^2lf'(r^2)-lf(r^2)\right) \not\in\,
 (0,\pi R_0^2].$$
\end{LEM}
Proof: The plane $z=0$ is invariant, and the equation on it is
$$\dot x =J(\omega +2lf'(|x|^2))x,$$
the solutions of which   
$$ X_r(t)=re^{J(\omega+2lf'(r^2))t}$$
have period 
$$T(r)=\left|\frac{2\pi}{\omega+2lf'(r^2)}\right| \cdot$$
These solutions are  critical points 
only  if $T\in \Nm T(r)$ which gives the
first condition of the lemma after a short calculation. 
The computation of the action
is straightforward, that it can not take values in the 
forbidden interval when $r$ is critical is a consequence of the
convexity of $f$:
the function
$$x\lmto g(x)=xf'(x)-f(x) $$
is increasing and thus $g(x)\leq 0$ when $x\leq R_0^2$ since 
$f(R_0^2)=f'(R_0^2)=0$.
On the other hand, the function
$$x\lmto (x-R_0^2)f'(x)-f(x)$$
is increasing for $x\geq R_0^2$,
which implies that 
$$g(x)> R_0^2f'(x)$$
when $x> R_0^2$.
Either $r>R_0$ and we must have 
$$I=Tlg(r^2)>TlR_0^2 f'(r^2)\geq \pi R_0^2$$
or $r\leq R_0$
and $I\leq 0$.
\hfill$\Box$\newline
The proposition 1 follows from the fact that $c_T(l(T))$ is in the
hole if  $\pi R_0^2 \geq M$,
and thus can not be one of the ''bad'' critical points.\hfill $\Box$

\textit{Acknowledgments :}
I thank Professor E. S�r� for submitting me this problem, for many 
helpul and useful discussions, and for his patient encouragements. 
I thank Professor 
I. Ekeland for his interest to this work and for  fruitful comments.
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\begin{small}
Patrick BERNARD,
Universit� Cergy Pontoise,
pbernard@clipper.ens.fr
\end{small}
\end{document}
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