\font\tenmath=cmbx10 \font\sevenmath=cmbx7 \font\fivemath=cmbx5 \def\crs{\cr\noalign{\smallskip}} \def\crm{\cr\noalign{\medskip}} \font\eightrm =cmr8 \font\sixrm =cmr6 \font\fiverm =cmr5 \font\eighti =cmti8 \font\sixi =cmti10 scaled 600 \font\fivei =cmti10 scaled 500 \font\eightsy =cmsy8 \font\sixsy =cmsy6 \font\fivesy =cmsy5 \font\eightit =cmitt10 scaled 800 \font\eightsl =cmsl8 \font\eighttt =cmtt8 \font\eightbf =cmbx8 \font\sixbf =cmbx6 \font\fivebf =cmbx5 \font\sc =cmssbx10 \font\ssc =cmcsc10 \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam0\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\oldstyle{\fam1\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\itfam =\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam =\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam =\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \abovedisplayskip =9pt plus 2pt minus 6pt \belowdisplayskip =\abovedisplayskip \abovedisplayshortskip =0pt plus 2pt \belowdisplayshortskip =5pt plus 2pt minus 3pt \smallskipamount =2pt plus 1pt minus 1pt \medskipamount =4pt plus 2pt minus 2pt \bigskipamount =9pt plus 4pt minus 4pt \setbox\strutbox =\hbox{\vrule height 7pt depth 2pt width 0pt}% \normalbaselineskip =9pt \normalbaselines\rm} \def\m@th{\mathsurround=0pt} \def\matrice#1{\left[\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,\right]} \def\matrices#1{\left[\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} \noalign{\smallskip} #1\crcr \mathstrut\crcr\noalign{\kern-\baselineskip}}}\,\right]} \def\rmatrices#1{\left[\,\vcenter{\normalbaselines\m@th \ialign{\hfil $##$&&\quad\hfil $##$\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} \noalign{\smallskip} #1\crcr \mathstrut\crcr\noalign{\kern-\baselineskip}}}\,\right]} \def\matricem#1{\left[\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} \noalign{\smallskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,\right]} \def\build#1_#2^#3{\mathrel{\mathop{\kern 0pt#1}\limits_{#2}^{#3}}} \def\Rotx#1{\left(\matrix{1& 0& 0\cr 0& \cos#1& -\sin#1\cr 0& \sin#1& \cos#1\cr}\right)} \def\Roty#1{\left(\matrix{\cos#1& 0&\sin#1\cr 0& 1& 0\cr -\sin#1& 0& \cos#1\cr}\right)} \def\Rotz#1{\left(\matrix{\cos#1&-\sin#1&0\cr\sin#1&\cos#1&0\cr 0& 0& 1\cr}\right)} \def\Vecd#1#2{\left[\matrix{#1\cr#2\cr}\right]} \def\Vect#1#2#3{\left[\matrix{#1\cr#2\cr#3\cr}\right]} \def\Vects#1#2#3{\left[\matrix{\noalign{\smallskip}#1\crs#2\crs#3\crs}\right]} \def\Vectm#1#2#3{\left[\matrix{\noalign{\medskip}#1\crm#2\crm#3\crm}\right]} \def\VEC#1{\matrices{#1_0\crs#1_1\crs#1_2\crs\vdots\crs\hfill#1_{n-1}\crs#1_n\crs}} \def\no{$\hbox{{\bf n}}^\circ$} % pour les \'equations multiples (faire crm ) \def\EqM#1{\vcenter{\normalbaselines\m@th \ialign{$##$\hfil&&\ $##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} \noalign{\smallskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}} % pour les \'equations multiples (faire crm ) \def\EQM#1{\vcenter{\normalbaselines\m@th \ialign{${\displaystyle ##}$\hfil&&\ ${\displaystyle ##}$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} \noalign{\smallskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}} \def\diagram#1{\def\normalbaselines{\baselineskip=0pt \lineskip=10pt\lineskiplimit=1pt} \matrix{#1}} \def\hfl#1#2{\smash{\mathop{\hbox to 12mm{\rightarrowfill}}\limits^{\scriptstyle#1}_{\scriptstyle#2}}} \def\vfl#1#2{\llap{$\scriptstyle #1$}\left\downarrow\vbox to 6mm{}\right.\rlap{$\scriptstyle #2$}} \def\mapright#1{\ \smash{\mathop{\longrightarrow}\limits^{#1}}\ } \catcode`\@=11 \def\system#1{\left\{\null\,\vcenter{\openup1\jot\m@th \ialign{\strut\hfil$##$&$##$\hfil&&\enspace$##$\enspace& \hfil$##$&$##$\hfil\crcr#1\crcr}}\right.} \catcode`\@=12 \def\sysvert#1{\left\vert\null\,\vcenter{\openup1\jot\m@th \ialign{\strut\hfil$##$&$##$\hfil&&\enspace$##$\enspace& \hfil$##$&$##$\hfil\crcr#1\crcr}}\right.} \def\overrightarrow#1{\vbox{\ialign{##\crcr \rightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip} $\hfil\displaystyle{#1}\hfil$\crcr}}} % macro pour les dessins \def \picture #1 by #2 (#3){ \vbox to #2{ \hrule width #1 height 0pt depth 0pt \vfill \special{picture #3} }} \def \scaledpicture #1 by #2 (#3 scaled #4){ \dimen0=#1 \dimen1=#2 \divide \dimen0 by 1000 \multiply \dimen0 by #4 \divide \dimen1 by 1000 \multiply \dimen1 by #4 \picture \dimen0 by \dimen1 (#3 scaled #4)} % dessins de l'article %\def \Fig1{\scaledpicture 108truemm by 114truemm (Fig1 scaled 1000)} %\def \Fig2{\scaledpicture 167ruemm by 160truemm (Fig2 scaled 700)} %\def \Fig3{\scaledpicture 167truemm by 160truemm (Fig3 scaled 700)} \def\tpar#1{{\bf \vskip 1truecm \goodbreak \noindent #1 \vskip 0.7truecm \par \nobreak\noindent} } \def\titre#1{\vskip 2cm\goodbreak{\bf #1 }\vskip 1cm \par\nobreak} \def\trait{\noalign{\smallskip\hrule\smallskip}} \def\tag #1${\eqno(#1)$} \def\text#1{\hbox{#1}} \def\cases{\eqalign} \font\tit=TimesB at 13 pt \def\soul#1{{\parindent=0pt $\underline {\it {\hbox{#1}}}$}} \def\soulr#1{{\hskip 0pt\bf {\hbox{#1}}}} \def\soulb#1{{\parindent=20pt $\underline {\bf {\hbox{#1}}}$}} \def\soulbis#1{{\parindent=20pt $\underline {\bf {\hbox{#1}}}$}} \def\soult#1{{\parindent=30pt $\underline {\tit {\hbox{#1}}}$}} \def\souli#1{{\hskip 0pt {\it {\hbox{#1}}}}} \def\soulc#1{{\hskip 0pt {\sc {\hbox{#1}}}}} \def\centersoul#1{{\centerline{${\tit {\hbox{#1}}}$}}} \def\gresp{\medskip\bigskip} \def\pesp{\bigskip} \def\ppesp{\medskip} \def\norm#1{\left\Vert#1\right\Vert} \def\abs#1{\left\vert#1\right\vert} \def\Frac#1#2{{{\displaystyle\strut#1}\over{\displaystyle\strut#2}}} \def\Dron#1#2{\Frac{\partial#1}{\partial#2}} \def\Der#1#2{\Frac{d#1}{d#2}} \def\DDer#1#2{\Frac{d^2#1}{d#2^2}} \def\Dt#1{\Frac{d#1}{dt}} \def\dtt#1{\Frac{d^2#1}{dt^2}} \def\summ#1{\sum_{#1=0}^{\infty}} \def\mmu#1#2{{m_{#1}\over\mu_{#2}}} \def\mumu#1#2{{\mu_{#1}\over\mu_{#2}}} \def\Vec#1{\overrightarrow{#1}} \def\tra{\vphantom{0}^t} \def\pha{\phantom{0}} \def\X{{\cal X}} \def\po{p_{\cal O}} \def\M{{\cal M}} \def\tM{{\tilde {\cal M}}} \def\B{{\cal B}} \def\RR{{\bf R}} \def\NN{{\bf N}} \def\CC{{\bf C}} \def\ZZ{{\bf Z}} \def\tB{{\tilde {\cal B}}} \def\mb{{{\cal M}\setminus {\cal B}'}} \def\fm{{{\tilde {\cal M}}\setminus \tilde {\cal B}}} \def\bi{{\cal E}^{n,m}_k} \def\ens{\lbrace 1, \cdot \cdot \cdot , n\rbrace} \def\y{\cal Y} \def\PM{{\cal M}^\prime} \def\O{{\cal O}} \def\U{{\cal U}} \def\V{{\cal V}} \def\sys{$(M,X)$} \def\prol{(M',X',\varphi,\xi)} \def\rd{\rm r\acute egularisation\ dynamique} \def\rs{\rm r\acute egularisation\ statique} \def\G{{\cal G}} \def\DPN{\partial ^+ N} \def\DMN{\partial ^- N} \def\DN{\partial N} \def\taub{\overline{\tau}} \def\DPNM{\DPN\setminus a^+} \def\DMNM{\DMN\setminus a^-} \def\S{{\cal S}} \def\Sp{\S^+} \def\Sm{\S^-} \def\tal{t_{\alpha(x)}} \def\tom{t_{\omega(x)}} \def\tX{\tilde X} \def\Rta{\R ta} \def\Rtr{\R tr} \def\cinf{C^{\infty}(M,\RR^+)} \def\Reg{{\rm Reg}} \def\Regm{\Reg\sys} \def\Rp{\R^+} \def\Rm{\R^-} \def\sing{{\rm Sing}} \def\tPsi{\tilde\Psi} \def\tU{\tilde\U} \def\Int{{\rm Int}} \def\Ap{{\cal A}^+} \def\Am{{\cal A}^-} \def\DPT{\DPN\setminus T} \def\DMT{\DMN\setminus T} \def\Extp{{\rm Ext_+}} \def\Extm{{\rm Ext_-}} \def\Orbp{{\rm Orb_+}} \def\om{\O\setminus \Ap} \def\op{\O\setminus \Am} \def\tY{\tilde {\cal Y}} \def\Y{{\cal Y}} \def\sh{{\rm sh}} \def\ch{{\rm ch}} \def\RE{\mathop{\Re e}\nolimits} \def\IM{\mathop{\Im m}\nolimits} \def\Sup{\mathop{\rm Sup}\nolimits} \def\Inf{\mathop{\rm Inf}\nolimits} %\font\tenmath=msym10 %\font\sevenmath=msym7 %\font\fivemath=msym5 \newfam\bbfam \scriptscriptfont\bbfam=\fivemath% \textfont\bbfam=\tenmath \scriptfont\bbfam=\sevenmath% \def\bb{\fam\bbfam\tenmath}% \def\N{{\bb N}} \def\Z{{\bb Z}} \def\Q{{\bb Q}} \def\R{{\bb R}} \def\C{{\bb C}} \def\K{{\bb K}} \def\T{{\bb T}} % macros pour les dessins \def \picture #1 by #2 (#3){ \vbox to #2{ \hrule width #1 height 0pt depth 0pt \vfill \special{picture #3} }} \def \scaledpicture #1 by #2 (#3 scaled #4){ \dimen0=#1 \dimen1=#2 \divide \dimen0 by 1000 \multiply \dimen0 by #4 \divide \dimen1 by 1000 \multiply \dimen1 by #4 \picture \dimen0 by \dimen1 (#3 scaled #4)} % on l'utilise de la maniere suivante: exemple: %\def \dessin{\scaledpicture 165mm by 95.2mm (Euler scaled 500)} % puis on appelle la macro a l'endroit voulu. \parindent=0pt \def\tvi{\vrule height 12pt depth 5pt width 0mm} \def\tv{\vrule height 12pt depth 5pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hoffset=0,45cm \magnification 1000 \hsize=15cm \parindent=0,8truecm \overfullrule=0mm \def\TL{\widetilde{L}} \font\petit=TimesI at 8 pt \font\tit=TimesB at 13 pt \centerline{\tit Dynamic around a chain of simple resonant tori} \centerline{\tit in nearly integrable Hamiltonian systems} {\bigskip} \centerline{\bf Laurent Niederman\footnote{(*)}{{\it Universit\'e Paris XI --- Topologie et Dynamique --- URA D1169 du CNRS --- B\^at. 425 --- 91405 ORSAY Cedex France ;}} \footnote{}{\ \ \ {\it E-mail~: niederma@topo.math.u-psud.fr}}}\footnote{} {$\!\!\! \!\!\!\!\!\!{\rm and}$}\footnote{}{{\it Bureau des Longitudes --- Astronomie et Syst\`emes Dynamiques --- URA D707 du CNRS --- 3 rue Mazarine --- 75006 PARIS France.}} \bigskip {\bf R\'esum\'e~:} Grace \`a une application de la th\'eorie K.A.M., nous construisons une forme normale permettant une \'etude fine de la dynamique au voisinage des tores invariants partiellement hyperboliques qui apparaissent dans les r\'egions simplement r\'esonantes associ\'ees \`a un syst\`eme hamiltonien int\'egrable faiblement perturb\'e. Ceci donne des estimations pr\'ecises sur les temps de transition au voisinage des vari\'et\'es stable et instable li\'ees \`a ces tores. Ainsi, ces formes normales fournissent un moyen de calculer la vitesse de d\'erive d'orbites qui $\ \!\grave{}\!\grave{}$~ombrent~$\acute{}\!\acute{}\ \!$ une cha\^\i ne de tores hyperboliques associ\'ee \`a une courbe simplement r\'esonante dans l'espace des actions. \medskip {\bf Abstract~:} By an application of the K.A.M. theory, we derive an accurate normal form valid in the vicinity of partially hyperbolic tori which arise close to simple resonances in nearly integrable Hamiltonian systems. This allows precise estimates on the times of transition around the stable and unstable manifolds of these tori. Hence, these normal forms provide an efficient tool to compute the speed of drift of orbits shadowing a chain of hyperbolic tori associated to a simple resonant curve in the action space. \medskip {\bf Mots cl\'es~:} Syst\`emes Hamiltoniens -- Stabilit\'e -- Flots Quasip\'eriodiques -- Perturbations -- Formes Normales -- Structures Hyperboliques. \medskip {\bf Key words~:} Hamiltonian systems -- Stability -- Quasi periodic flows -- Perturbations -- Normal forms -- Hyperbolic structures. \medskip AMS classification numbers : 58F05, 58F10, 58F27, 58F30, 58F36, 58F15. \bigskip \soulr{I Introduction~:} \medskip We look at perturbed integrable Hamiltonian systems which are governed by the classical Hamiltonian~: ${\cal H}(I,\varphi )=h(I)+\varepsilon f(I,\varphi )$ with the action-angle variables $(I,\varphi )\in\R^n\times\T^n$ where $\T =\R /\Z$. As it is well known, K.A.M. theory states that under the (sufficient) assumption of analyticity of $H$ and non-degeneracy of $h$ ($\vert\nabla^2 h\vert\not= 0$), there remain many $n$-dimensional Lagrangian tori invariant for the perturbed flow which are slight deformations of the initial tori ($I= I_0$) located in non resonant area. Nevertheless, in case $n\geq 3$, this does not allow to prevent from a drift of the orbits over a large part of the phase space in the perturbed system because the complement of the invariant tori is a connected set. Actually, according to a theorem of Nekhorochev [N, 1977], this possible instability can only occur with a speed which is at most exponentially small with respect to the inverse of the size of the perturbation ($\varepsilon$). Hence, two questions arise : to prove the existence of unstable orbits and to estimate their times of instability. In a famous paper of 1964, Arnold [A, 1964] (see also [A, 1992]) has given an example of a three degrees of freedom nearly integrable Hamiltonian system where a global instability of the action variables occurs. The mechanism which generate this instability is based on the existence in the perturbed system of arbitrary long chains of hyperbolic tori connected by heteroclinic orbits and Arnold finds orbits which drift along these lines of tori. Nevertheless, it should be pointed out that the existence of these chains is ensured by very specific properties of the Hamiltonian studied by Arnold. The generalisation of this result in a wider class of nearly integrable Hamiltonian systems is a difficult problem, a precise survey on this question has recently been written by Lochak ([L, 1996], see also the introduction of [RW2, 1997]). In the general case, the first step to prove instability along the lines of Arnold's reasoning is to ensure the existence of enough hyperbolic tori invariant under the perturbed flow. The study of the lower dimensional tori which survive under perturbation was initiated by Moser [Mo, 1967], Brjuno [Br, 1971] and then by numerous authors. A refined result was obtained by Graff [Gr, 1973] (see also Zehnder [Ze, 1976]) in the case where the phase space is foliated by tori with a hyperbolic structure (stable/unstable manifolds) invariant under the unperturbed flow. In this initially hyperbolic setting, he showed that the tori with a strongly non resonant frequency and the associated manifolds persist in the perturbed problem. This result has been extended in the general case of nearly integrable Hamiltonian systems by Treschev [Tr, 1991] who showed that many tori of dimension $(n-d)$ with $n$-dimensional Lagrangian hyperbolic manifolds (the $\grave{}\!\grave{}$~whiskers~$\acute{}\!\acute{}$~) arise in the perturbed problem close to the manifolds associated to resonances of multiplicity $d$ ($d\lambda >0$, $\vert\vert{\bar{P}}\vert \vert_{\bar{\kappa}}\leq{\bar{C}}E_1$, $\vert\vert{\bar{P}}-\lambda\left({\bar {I}}\right){\bar{s}}{\bar{u}}\vert\vert_{\bar{\kappa}}\leq{\bar{C}}E_1{\bar {\kappa}}^3$ where ${\bar{C}}$ is a positive constant and $\vert\vert .\vert \vert_{\bar{\kappa}}$ is the supremum norm over $V_{{\bar{\kappa}}}$. Then, if $${\bar \kappa}\leq C\Frac{\lambda}{E_1}\eqno (1)$$ where $\lambda$ is the uniform bound on the Liapounov exponent over the considered domain and $C$ is a positive constant (we will assume that $C\leq 1$), there exists a real analytic canonical transformation $$\eqalign{{\cal T}:\ V_{{\bar{\kappa}}/2}\ \ \ &\longrightarrow\ \ \ \ \ V_{ {\bar{\kappa}}}\ \ \ {\it with}\ {\bar{I}}=I,\ {\cal T}\left( I,\varphi ,s,u \right) -\left( I,\varphi ,s,u\right) ={\cal O}_2 (s,u;I,\varepsilon )\cr \left({\bar{I}},{\bar{\varphi}},{\bar{s}},{\bar{u}}\right) &\longmapsto\left( I, \varphi ,s,u \right)\cr}$$ such that the unperturbed Hamiltonian becomes in the new variables : $$H_0 (I,s,u)=h\left( I;\varepsilon\right) +\varepsilon P\left( I,su;\varepsilon \right) =h\left( I\right) +\varepsilon\lambda\left( I\right) su+\varepsilon {\cal O}_2\left( su;I,\varepsilon\right) .$$ } \vskip0,3truecm A proof of this theorem based on an iterative quadratic scheme can be found in [CG, 1994] (annex A3), this allows simplifications in comparison with Moser's original reasoning which uses majoring series. We also note that the previous transformations leave the actions $I$ invariant and are defined on a domain whose size is independent of $\mu$. Hence, they are fitted for the study of Arnold's instability where we want to estimate the speed of drift of orbits shadowing lines of hyperbolic tori whose lengths are $\mu$-independent. \medskip At present, the original Hamiltonian is properly reduced and, in order to apply the K.A.M. theory, we should select an origin $I_0$ in the action space linked to a torus whose associated frequency vector is strongly non resonant, that is $\omega_0 =\nabla h(I_0)$ satisfies a Diophantine condition : $$\omega_0\in\Omega_{\gamma ,\tau}=\left\{\omega\in\R^n\ {\rm such\ that}\ \vert k.\omega\vert\geq{{\gamma}\over{\vert\vert k\vert \vert_\infty^\tau}}\ {\rm for\ all}\ k\in\Z^n\right\}$$ where $\gamma$ is an arbitrary positive constant and $\tau\in ]n-1,+\infty [$. We recall that for $\tau >n-1$, the measure of the complementary set of $\Omega_{\gamma ,\tau}$ is of the order of ${\cal O}(\gamma )$ With a translation of the origin at $\left( I_0,0,0,0\right)$, the Taylor expansion of the Hamiltonian can be written : $${\cal H}(I,\varphi ,s,u)=\omega_0 .I +\varepsilon\lambda_0 su+\varepsilon f(I,su)+\varepsilon\mu g(I,\varphi ,s,u)\eqno(2)$$ where (i) $\lambda_0 =\lambda\left( I_0\right)$, hence $\lambda_0 >\lambda >0$ (ii) $f(I,su)= {\cal O}_2 (I,su;\varepsilon )$ and $g(I,\varphi ,s,u)$ is an arbitrary function. (iii) With the previous notations, the total Hamiltonian ${\cal H}$ is supposed analytical over the complex domain $V_K$ where $K$ is a positive constant (iv) We set $\vert\vert\omega_0\vert\vert =\nu_0$ where $\vert\vert.\vert\vert$ denotes the Euclidean norm (v) With the operator norm induced by the Euclidean norm, we have $\vert\vert\partial^2_I f(0,0)\vert\vert\leq p^{-1}$ where $p$ is the constant defined previously for $h$ and the unperturbed Hamiltonian $H$ satisfies a condition of isoenergetic non degeneracy with the same constant $p$ : $${\rm for\ all}\ {\bf V}\in \R^{n+1},\ {\rm we\ have}\ \left\vert\left\vert \pmatrix{\varepsilon\partial^2_I f(0,0) &\omega_0\cr \omega_0 & 0\cr}{\bf V} \right\vert\right\vert\geq\varepsilon p\vert\vert{\bf V}\vert\vert$$ (vi) With the sup norm ($L^{\infty}$) over $V_K$, we have $\vert\vert {\cal H}\vert\vert_K < E_1$ and $\vert\vert g\vert\vert_K < E_1$. Here, we also assume that $E_1\!\geq\!\sup\left( 1,\nu_0\right)$ \medskip Now, we can state the main result : \medskip \soulc{Theorem 2.2. (main result)} {\it Let $m_*\! =\!\inf\left({\Frac{p}{2}},{\Frac{\lambda}{2}}\right)$, $\sigma\!=\! 2^{4n+1}\left( \Frac {n+1}{e}\right)^{n+1}$ with $1=\ln (e)$ ; under the previous assumptions and also $$n-1<\tau\leq n\ ;\ K\leq C\Frac{m_*}{E_1}\ ;\ \gamma\leq\Frac{\varepsilon} {30}\ ;\ \Frac{\varepsilon\mu}{\gamma^4}<\left(\Frac{m_*^3}{3\sigma^3 K}\right)^2 \left(\Frac{K}{240}\right)^{4(2n+5)}\eqno(3)$$ there exist a symplectic transformation ${\cal T} : V_{K/3}\longrightarrow V_K$ such that : $${\cal H}\circ{\cal T}=(1+\eta )\omega_0{\widetilde{I}}+\varepsilon\lambda_1 {\widetilde{s}}{\widetilde{u}}+\varepsilon{\widetilde{f}}\left({\widetilde{I}}, {\widetilde{\varphi}},{\widetilde{s}},{\widetilde{u}}\right)\ {\it with}\ {\widetilde{f}} \left({\widetilde{I}},{\widetilde{\varphi}},{\widetilde{s}},{\widetilde{u}} \right) ={\cal O}_2\left({\widetilde{I}},{\widetilde{s}}{\widetilde{u}}; {\widetilde{\varphi}},{\widetilde{s}},{\widetilde{u}},\varepsilon ,\mu\right)$$ with $\Frac{1}{2}<1+\eta <\Frac{3}{2}$ , $\lambda_1>\Frac{\lambda_0}{2}$ and $$\left\vert\left\vert\eta\omega_0{\widetilde{I}}+\varepsilon\left(\lambda_1 - \lambda_0\right){\widetilde{s}}{\widetilde{u}}+\varepsilon\left[{\widetilde{f}} \left({\widetilde{I}},{\widetilde{\varphi}},{\widetilde{s}},{\widetilde{u}} \right) -f({\widetilde{I}},{\widetilde{s}}{\widetilde{u}})\right]\right\vert \right\vert_{K/3}\leq\Frac{E_1}{2^{4n+6}}\sqrt{\varepsilon\mu}$$ Hence, $\{{\widetilde{I}}=0\}$ becomes an hyperbolic invariant set for the perturbed flow with linear motions.} \medskip \souli{Remark :} The previous estimates are valid for $n-1<\tau\leq n$, in the case of $\tau >n$ the main result is valid with the thresholds : $$n<\tau\ ;\ K\leq C\Frac{m_*}{E_1}\ ;\ \gamma\leq\Frac{\varepsilon}{30}\ ;\ \Frac{\varepsilon \mu}{\gamma^4}<\left(\Frac{m_*^3}{3\sigma^3 K}\right)^2\left(\Frac{K}{240} \right)^{4(2\tau +5)}\ {\rm with}\ {\widetilde{\sigma}}\!=\! 2^{4\tau +1}\left( \Frac{\tau +1}{e}\right)^{\tau +1}\eqno({\widetilde{3}})$$ This follows from the estimates derived in the lemma A.3. and A.4. in the case of a Diophantine condition with $\tau >n$. \medskip \soulr{II.2 Formal scheme of the proof} \medskip We use the classical method to build a Kolmogorov normal form in the vicinity of an invariant torus. Actually, we follow the lines of the proof of Kolmogorov theorem given by Benettin, Galgani, Giorgilli and Strelcyn [BGGS, 1984]. One first expand the total Hamiltonian in the vicinity of $(I,s,u)=(0,0,0)$, hence $${\cal H}(I,\varphi ,s,u)=H^{(0)}(I,\varphi ,s,u)+\varepsilon\mu g^{(0)}(I,\varphi ,s,u)$$ $$\eqalign{{\rm with}\ :\ H^{(0)}(I,\varphi ,s,u)=\omega^{(0)}I+\varepsilon \lambda^{(0)}su+\varepsilon b^{(0)}(\varphi,s,u)Isu+&\varepsilon c^{(0)}( \varphi ,s,u) (su)^2\cr +&\varepsilon d^{(0)}(\varphi ,s,u)I^2 +\varepsilon e^{(0)}(I,\varphi ,s,u) I^3\cr}$$ $$\eqalign{{\rm and}\ g^{(0)}(I,\varphi ,s,u)=\alpha^{(0)}_0 &(\varphi )+ \alpha^{(0)}_{10}(\varphi ,s)s+\alpha^{(0)}_{01} (\varphi ,u)u+\alpha^{(0)}_{11} (\varphi )su\cr +&\alpha^{(0)}_{21} (\varphi ,s)s^2 u+\alpha^{(0)}_{12} (\varphi ,s)su^2+ \beta^{(0)}_0 (\varphi )I+\beta^{(0)}_{10} (\varphi ,s)Is+\beta^{(0)}_{01} (\varphi ,u)Iu\cr}$$ where $\omega^{(0)}=\omega_0$, $\lambda^{(0)}=\lambda_0$ and (i) For all ${\bf V}\in \R^{n+1}$, we can write with $p_0 =p-\Frac{4 E_1}{K^2} \mu$ : $$\left\vert\left\vert\!\pmatrix{2\varepsilon{\overline{d^{(0)}}}(0,0)&\omega^{(0)} \cr \omega^{(0)} & 0\cr}\!\!{\bf V}\right\vert\right\vert\!\geq\!\left\vert\left \vert\!\pmatrix{\varepsilon\partial^2_I f(0,0,0)&\omega_0\cr \omega_0 & 0\cr}\!\! {\bf V}\right\vert\right\vert\! -\!\left\vert\left\vert\!\pmatrix{\varepsilon\mu {\overline{\partial^2_I g}}(0,0,0) & 0\cr 0 & 0\cr}\!\!{\bf V}\right\vert\right \vert\!\geq\!\varepsilon p_0\vert\vert{\bf V}\vert\vert$$ where the averaging over the angles is denoted by a bar : for any function ${\rm f}$ defined over $V_K$, we denote $${\overline{{\rm f}}}(I,s,u)=\Frac{1}{\left( 2\pi\right)^n}\int_0^{2\pi}\!\!\! \ldots\int_0^{2\pi}{\rm f}(I,\varphi ,s,u)d\varphi_1\!\ldots d\varphi_n$$ and the thresholds (1) gives $m_*\leq\Frac{p}{2}0$, $r_0 >0$ and the distance to ${\cal P}$ given by the Euclidean norm in $\C^{n+1}$. Denoting $\vert\vert .\vert\vert_{r_0 ,s_0}$ the sup norm ($L^{\infty}$) for a numerical or a vector valued function defined over $V_{r_0 ,s_0}{\cal P}$, we assume that $\vert\vert{\cal H}\vert\vert_{r_0 ,s_0}\leq E$ and $\vert\vert f\vert\vert_{r_0 , s_0}\leq E$ with $E>1$. For the norm on the operators induced by the Euclidean norm, we assume that the Hessian matrix $\partial^2_J h$ satisfies $\left\vert \left\vert\partial^2_J h\right\vert\right\vert_{r_0 ,s_0}\leq p^{-1}$ with $0

0$. Let $\alpha >0$, a subset ${\cal D}\subset{\cal P}\subset\R^{n+1}$ is said to be $\alpha$, $L$ non resonant modulo $\Lambda_0$, if we have : $$\left\vert k.\partial^2_J h(J)\right\vert =\left\vert k.\omega (J)\right\vert \geq\alpha\ {\rm for\ all}\ J\in {\cal D}\ {\rm and}\ k\in\Z_L^{n+1}\backslash \Lambda_0$$ For the time being, $\alpha$ and $L$ will be considered as independent parameters but they will be chosen as suitable functions of $\varepsilon$ in lemma 3.2. In the vicinity of such a set ${\cal D}$, the perturbed Hamiltonian is transformed in a $\Lambda_0$-resonant normal form $h+f_* +g$ up to $g$ which means that the expansion $f_* (J,\varphi )={\displaystyle{\sum_{k\in\Lambda_0}}}{f_*}_k(J)\exp(ik. \theta )$ contains only harmonics in $\Z_L^{n+1}\cap\Lambda_0$ while $g$ is a general term. \medskip \soulc{Lemma 3.1. (normal form)} {\it Suppose that ${\cal D}\subset{\cal P}$ is $\alpha$, $L$-non resonant modulo $\Lambda_0$. If $$6\leq L s_0\ ;\ \varepsilon\leq C_1\Frac{\alpha r}{L}\ ;\ r\leq\inf\left( C_2 \Frac{p\alpha}{L}, r_0\right)\ {\it where}\ C_1 >0\ {\it and}\ C_2 >0$$ then, there exist a real analytic, symplectic coordinate transformation $\Phi : V_{r_*,s_*}{\cal D}\longmapsto V_{r, s_0}{\cal D}$, where $r_* ={{r}\over{2}}$ and $s_* ={{s_0}\over{6}}$, such that ${\cal H}\circ\Phi =h+f_* +g$ is in $\Lambda_0$-resonant normal form up to $g$ with $$\left\vert\left\vert f_* -f^{(0)}\right\vert\right\vert_{r_* ,s_*}\leq C_3{{ \varepsilon^2 L}\over{\alpha r}}\ (C_3 >0)\ ;\ \left\vert\left\vert g\right \vert\right\vert_{r_* ,s_*}\leq\varepsilon E\exp\left( -{{L s_0}\over{6}}\right)$$ where $f^{(0)}$ is the partial sums of the Fourier expansion of $f$ containing only harmonics in $\Z_L^{n+1}\cap\Lambda_0$. Moreover : $$\left\vert\left\vert\Pi_J\Phi -Id_J\right\vert\right\vert_{r_* ,s_*}\leq C_3{{L} \over{\alpha}}\varepsilon$$ uniformly on $V_{r_* ,s_*}P$ where $\Pi_J$ denotes the projection onto the action space and $Id_J$ is the identity in the action space.} \vskip0,3truecm Now, we specify a domain ${\cal D}_0$ which is $\alpha$, $L$ non resonant modulo $\Lambda_0$. This set is first constructed in the frequency space and then pulled back in the action space via the frequency map. Let $\omega\in\R^{n+1}$ and $k\in\Z_L^{n+1}$, we assume that $$\left\vert\left\vert\omega -\Pi_0\left(\omega\right) -\left< k\right>^\perp \cap\left< k_0\right>^\perp\right\vert\right\vert\geq\Frac{2\alpha}{\left\vert \left\vert k-\Pi_0\left( k\right)\right\vert\right\vert}\ {\rm and}\ \left\vert \left\vert\Pi_0\left(\omega\right)\right\vert\right\vert\leq\Frac{\alpha}{L}$$ where $\Pi_0$ denotes the orthogonal projection over the vectorial line $\left< k_0\right>\subset\R^{n+1}$ generated by $k_0$ and $\left\vert\left\vert\omega -\Pi_0\left(\omega\right) -\left< k\right>^\perp\cap\left< k_0\right>^\perp\right\vert\right\vert$ is the distance to the set $\left< k\right>^\perp\cap\left< k_0\right>^\perp$ given by the Euclidean norm, then : $$\eqalign{\vert k.\omega\vert\geq &\left\vert k-\Pi_0 (k).\omega -\Pi_0(\omega )\right\vert -\left\vert\Pi_0 (k).\Pi_0 (\omega )\right\vert\cr \geq &\left\vert\left\vert k-\Pi_0 (k)\right\vert \right\vert\left\vert\left\vert\omega -\Pi_0\left(\omega\right) -\left< k \right>^\perp\cap\left< k_0\right>^\perp\right\vert\right\vert -\left\vert \left\vert\Pi_0 (k)\right\vert\right\vert\left\vert\left\vert\Pi_0 (\omega ) \right\vert\right\vert\cr \geq &\left\vert\left\vert k-\Pi_0 (k)\right\vert \right\vert\Frac{2\alpha}{\left\vert\left\vert k-\Pi_0 (k)\right\vert\right \vert}-\left\vert\left\vert\Pi_0 (k)\right\vert\right\vert\Frac{\alpha}{L}\geq \alpha\ {\rm since}\ \left\vert\left\vert\Pi_0 (k)\right\vert\right\vert\leq L. \cr}$$ We denote ${\cal Z}_L^{n+1}$ the set of vectors in $\Z_L^{n+1}$ which generate a maximal $L$-lattice in $\Z^{n+1}$, that means a lattice not properly contained in any other lattice of dimension one. The precedent reasoning ensure that the following set defined with the previous notations : $${ D}_0\! =\!\left\{\!\omega\!\in\!\R^{n+1}\!\!\ {\rm such\ that}\!\ \left \vert\!\left\vert\Pi_0 (\omega )\right\vert\!\right\vert\!\leq\!\Frac{\alpha}{L} \!\ {\rm and}\!\ \left\vert \left\vert\omega\! -\!\Pi_0\left(\omega\right)\! -\! \left< k\right>^\perp\!\!\cap\!\left< k_0\right>^\perp\!\right\vert\right\vert\! \geq\!\Frac{2\alpha}{\left\vert\left\vert k\! -\!\Pi_0 (k)\right\vert\right \vert}\!\ {\rm for\ all}\!\ k\!\in\!{\cal Z}_L^{n+1}\!\right\}$$ is $\alpha$, $L$ non resonant modulo $\Lambda_0$. The use of Dirichlet's theorem on Diophantine approximation (see [LM, 1988] and the section 4 of [P${\ddot{\rm o}}$, 1993]) shows that $\alpha$ must be at most of the order of $L^{-n+1}$. In this case, measure theoretical considerations exposed in [P${\ddot{\rm o}}$, 1993] ensure that the complementary of ${ D}_0$ in the resonant area linked with $\Lambda_0$ (i.e. : $\Pi_0 (\omega )=0$) is of the order of $\alpha L^{n-1}$. Gathering everything together, we arrive at the following : \medskip \soulc{Lemma 3.2. (simple resonance normal form)} {\it We consider the subset ${\cal D}_0 =\left\{ J\in{\cal P}\ {\it such\ that}\ \omega (J)\in D_0\right\}\subset{\cal P}$ where ${D}_0$ is defined with $\alpha =\Frac{A} {L^{n-1}}$ and $A$ is a strictly positive function of $\varepsilon$. If $$6\leq L s_0\ ;\ \varepsilon <\Frac{C_1 r_0^2}{C_2 p}\ {\it and}\ r=\sqrt {\Frac {C_2}{C_1}p\varepsilon}\ ;\ {L^{n}}=A\sqrt{\Frac{C_1 C_2 p}{\varepsilon} }\ {\it with}\ C_1 >0,\ C_2 >0 \eqno(6)$$ then there exist a real analytic, symplectic coordinate transformation $\Phi : V_{r_*,s_*}{\cal D}_0\longmapsto V_{r, s_0}{\cal D}_0$, where $r_* ={{r}\over{2}}=\sqrt{{{pC_2} \over{4C_1}}\varepsilon}$ and $s_* ={{s_0}\over{6}}$, such that ${\cal H}\circ\Phi =h+f_* +g$ is in $\!\! -$resonant normal form up to $g$ with $$\left\vert\left\vert f_* - f^{(0)}\right\vert\right\vert_{r_* ,s_*}\leq {\widetilde{C}}_3\varepsilon\ ({\widetilde{C}}_3 >0)\ {\it and}\ \left\vert \left\vert g\right\vert\right\vert_{r_* ,s_*}\leq\varepsilon E\exp\left( -{{L s_0}\over{6}}\right)$$ where $f^{(0)}$ is the partial Fourier expansion of $f$ containing only harmonics in $\Z_L^{n+1}\cap\Lambda_0$. Moreover $\left\vert\left\vert\Pi_J\Phi -Id_J\right\vert\right\vert_{r_* ,s_*}\leq {\widetilde{C}}_3\sqrt{\varepsilon}$ uniformly on $V_{r_* ,s_*}{\cal D}_0$ with the notations of lemma 3.1.} \medskip \souli{Remarks :} 1) By means of an unimodular transformation in the phase space, we can always assume that $\Lambda_0$ is generated by the first vector of the basis in the frequency space, then we can write in the new variables : $${\widehat{{\cal H}}}\left({\widehat{J}},{\widehat{\theta}}\right) ={\cal H}\circ\Phi \left({\widehat{J}},{\widehat{\theta}}\right) =h\left({\widehat{J}}\right) +\varepsilon{\widehat{f_*}}\left({\widehat{J}},{\widehat{\theta}}_1\right)+ \varepsilon\mu{\widehat{g}}\left({\widehat{J}},{\widehat{\theta}}\right)\ {\rm with}\ \mu =\exp\left( -{{L s_0}\over{6}}\right)\eqno(7)$$ 2) If we choose $A=A_0$ where $A_0$ is a positive constant, then $L=\left( A_0 \sqrt{C_1 C_2 p}\right)^{{{1}\over {n}}}\!\!\varepsilon^{-{{1}\over{2n}}}$ and we find the normal form of P${\ddot{\rm o}}$schel for a simple resonance with a remainder of the order of $\exp\left(-C\varepsilon^{-{{1}\over {2n}}}\right)$. \medskip \soulr{III.2 Reduction to an initially hyperbolic Hamiltonian and main result.} \medskip Now, we reduce the normalized Hamiltonian of the previous paragraph to an initially hyperbolic Hamiltonian studied in section II. Hence, we start with the expression $(7)$ of the averaged Hamiltonian $${\cal H}(J,\theta ) = h\left( J_1 ,{\widetilde{J}}\right) + \varepsilon f_*\left( J_1 ,{\widetilde{J}},\theta_1\right) +\varepsilon\mu g\left( J_1 ,{\widetilde{J}},\theta_1 ,{\widetilde{\theta}}\right)\ ;\ \left( J_1 , {\widetilde{J}}\right)\in\R\times\R^n\ ;\ \left(\theta_1 ,{\widetilde{\theta}} \right)\in\T\times\T^n$$ with ${\cal H}(J,\theta )$ analytical over $V_{r_*,s_*}{\cal D}_0$ where $r_*$ is of the order of $\sqrt{\varepsilon}$. \medskip We consider $\left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)}\right)\in {\cal D}_0\times\T$ such that : $$\left( J_1^{(0)},{\widetilde{J}}^{(0)}\right)\ {\rm is\ on\ the\ resonant\ manifold}\ :\ \Dron{h}{J_1}\left( J_1^{(0)},{\widetilde{J}}^{(0)}\right) =0 \hskip2truecm\eqno\left({\cal C}_1\right)$$ $$\theta_1^{(0)}\ {\rm is\ a\ critical\ point\ of}\ f_*\left( J_1^{(0)}, {\widetilde{J}}^{(0)},\theta_1\right) ,\ {\rm hence}\ :\ \Dron{f_*}{\theta_1} \left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)}\right) =0\hskip-0,2truecm \eqno\left({\cal C}_2\right)$$ $$\eqalign{\left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)}\right)\ {\rm satisfy}\ &{\rm the \ following\ hypothesis\ of\ hyperbolicity}\ :\hskip1,3truecm \cr -\lambda_{(0)}^2= &\Frac{\partial^2 h}{\partial J_1^2}\left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)}\right)\Frac {\partial^2 f_*}{\partial\theta_1^2}\left( J_1^{(0)},{\widetilde{J}}^{(0)}, \theta_1^{(0)}\right) <0\cr}\eqno\left({\cal C}_3\right)$$ \medskip We also assume that the unperturbed Hamiltonian $h$ satisfies a condition of isoenergetic nondegeneracy with the constant $p$ considered in the previous paragraph : $${\rm for\ all}\ {\bf V}\in \R^{n+2},\ {\rm we\ have}\ \left\vert\left\vert \pmatrix{\partial^2_{J_1} h\left( J_1 ,{\widetilde{J}}\right) & \partial^2_{{\widetilde{J}}J_1} h\left( J_1 ,{\widetilde{J}}\right) & \partial_{J_1} h\left( J_1 ,{\widetilde{J}}\right)\cr \partial^2_{J_1{\widetilde{J}}} h\left( J_1 ,{\widetilde{J}}\right) & \partial^2_{{\widetilde{J}}{\widetilde{J}}} h\left( J_1 ,{\widetilde{J}}\right) & \partial_{{\widetilde{J}}} h\left( J_1 ,{\widetilde{J}}\right)\cr \partial_{J_1} h\left( J_1 ,{\widetilde{J}}\right) & \partial_{{\widetilde{J}}} h\left( J_1 ,{\widetilde{J}}\right) & 0\cr}{\bf V} \right\vert\right\vert\geq p\vert\vert{\bf V}\vert\vert$$ uniformly for $\left( J_1 ,{\widetilde{J}}\right)\in {\cal D}_0$. \medskip For the reduction of the averaged Hamiltonian $H_\varepsilon\left( J_1 , {\widetilde {J}},\theta_1\right) =h\left( J_1 ,{\widetilde{J}}\right) +\varepsilon f_*\left( J_1 ,{\widetilde{J}},\theta_1\right)$ to an initially hyperbolic Hamiltonian, we must find a family of points in ${\cal D}_0\times\T$ parametrized by ${\widetilde{J}}$ which satisfy the hypothesis $({\cal C}_1)$, $({\cal C}_2)$ and $({\cal C}_3)$ with $H_\varepsilon$ instead of $h$ and $f_*$. First, condition $({\cal C}_3)$ of hyperbolicity allows to use an analytical implicit function theorem on: $${{\cal F}}_0\left( J_1 ,{\widetilde{J}},\theta_1\right) =\left( \partial_{J_1} h\left( J_1 ,{\widetilde{J}}\right) ,\partial_{\theta_1}f_*\left( J_1 ,{\widetilde{J}},\theta_1\right)\right)$$ at the point $\left( J_1^{(0)},{\widetilde{J}}^{(0)}, \theta_1^{(0)}\right)$ ; hence, there exist a neighbourhood of ${\widetilde{J}}^{(0)}$ denoted ${\cal V}_{{\widetilde{J}}^{(0)}}$ and a function : $$\eqalign{{{\cal G}}_0\! :\!{\cal V}_{{\widetilde{J}}^{(0)}} & \!\!\longrightarrow\Pi_{J_1}\left({\cal D}_0\right)\times\T\hfill\cr {\widetilde{J}} &\!\!\longmapsto {{\cal G}}_0\left({\widetilde{J}}\right) =\left( J_{10}\left( {\widetilde{J}}\right) ,\theta_{10}\left({\widetilde{J}}\right)\right)\cr}$$ such that ${{\cal G}}_0\left({\widetilde{J}}^{(0)}\right) =\left( J_1^{(0)}, \theta_1^{(0)}\right)$ and ${{\cal F}}_0\left({\widetilde{J}},{\widetilde{{\cal G}}}_0\left({\widetilde{J}}\right)\right) =\left( 0,0\right)$ for all ${\widetilde{J}}\in{\cal V}_{{\widetilde{J}}^{(0)}}$. Then, we consider $${\cal F}\left( J_1 ,{\widetilde{J}},\theta_1,\varepsilon ,x,y,z,s\right) = \pmatrix{\partial_{J_1} H_\varepsilon (x,y,z)-(1+s)\partial_{J_1} h\left( J_1 , {\widetilde{J}}\right)\cr \partial_{{\widetilde{J}}} H_\varepsilon (x,y,z)-(1+s) \partial_{{\widetilde{J}}} h\left( J_1 ,{\widetilde{J}}\right)\cr\partial_{ \theta_1}f_*(x,y,z)-(1+s)\partial_{\theta_1}f_*\left( J_1 ,{\widetilde{J}},\theta_1 \right)\cr H_\varepsilon (x,y,z)-h\left( J_1 ,{\widetilde{J}}\right)\cr}$$ the isoenergetic non-degeneracy condition and the assumption that $\partial^2_{\theta_1} f_*\left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)} \right)\not= 0$ allow to use the implicit function theorem on ${\cal F}$ at the point $\left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)},0, J_1^{(0)}, {\widetilde{J}}^{(0)},\theta_1^{(0)},0\right)$. Hence, we find a neighbourhood ${\cal V}$ of $\left( J_1^{(0)}, {\widetilde{J}}^{(0)},\theta_1^{(0)}\right)$ and $\varepsilon_0 >0$ such that : $$\eqalign{{\cal G}\! :\!{\cal V}\times\left[0,\varepsilon_0\right[ & \longrightarrow {\cal D}_0\times\T\times\R\hfill\cr \left( J_1 ,{\widetilde{J}},\theta_1,\varepsilon\right) &\longmapsto (x,y,z,s)=\left({\cal G}_\varepsilon\left( J_1 ,{\widetilde{J}},\theta_1\right) ,s\right)\cr}$$ with $\left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)},0\right) ={\cal G} \left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)},0\right)$ and ${\cal F} \left({\cal G}\left( J_1 ,{\widetilde{J}},\theta_1 ,\varepsilon\right) , J_1 , {\widetilde{J}},\theta_1 ,\varepsilon\right) =(0,0,0,0)$, moreover $\left\vert \left\vert{\cal G}_\varepsilon -Id\right\vert\right\vert_{{\cal V}}$ is of the order of $\varepsilon$. We denote ${\cal G}_\varepsilon\left( J_{10}\left({\widetilde{J}}\right) , {\widetilde{J}},\theta_{10}\left({\widetilde{J}}\right)\right) =\left( J_{1\varepsilon}\left({\widetilde{J}}\right) ,{\widetilde{J}}_\varepsilon\left( {\widetilde{J}}\right) ,\theta_{1\varepsilon}\left({\widetilde{J}}\right)\right) ={\cal M}_\varepsilon\left({\widetilde{J}}\right)$ for ${\widetilde{J}}\in{\cal V}_{{\widetilde{J}}^{(0)}}$ such that $\left( J_{10}\left({\widetilde{J}}\right) ,{\widetilde{J}},\theta_{10}\left({\widetilde{J}}\right)\right)\in{\cal V}$, then : $$\matrix{(i)\ {\cal M}_\varepsilon\left({\widetilde{J}}^{(0)}\right) = \left( J_1^{(0)},{\widetilde{J}}^{(0)},\theta_1^{(0)}\right) ,\hskip2,1truecm (ii)\ \Dron{H_\varepsilon}{{\widetilde{J}}}\left({\cal M}_\varepsilon \left({\widetilde{J}}\right)\right) =(1+s)\Dron{h} {{\widetilde{J}}}\left( J_{10}\left({\widetilde{J}}\right) ,{\widetilde{J}}\right) ,\hfill\cr (iii)\ \Dron{H_\varepsilon}{J_1}\left({\cal M}_\varepsilon\left( {\widetilde{J}}\right)\right) =\Dron{H_\varepsilon}{\theta_1}\left({\cal M}_\varepsilon\left( {\widetilde{J}}\right)\right) =0,\hskip0,4truecm (iv)\ H_\varepsilon\left({\cal M}_\varepsilon\left({\widetilde{J}} \right)\right) =h\left( J_{10}\left({\widetilde{J}}\right) ,{\widetilde{J}} \right) .\hfill\cr}$$ Now, as in [CG, 1994], we make the change of variables generated by the function $${\cal S}_\varepsilon\left( J'_1 ,\theta_1 ,{\widetilde{J}}' ,{\widetilde {\theta}}\right) =\left(\theta_1 -\theta_{1\varepsilon}\left({\widetilde{J}}' \right)\right) J'_1 +J_{1\varepsilon}\left({\widetilde{J}}'\right)\sin\left( \theta_1 -\theta_{1\varepsilon}\left({\widetilde{J}}'\right)\right) +{\widetilde {\theta}}.{\widetilde{J}}_\varepsilon\left({\widetilde{J}}'\right)$$ and the averaged Hamiltonian becomes : $$H'_\varepsilon\left( J'_1 ,{\widetilde{J}}' ,\theta'_1\right) = H_\varepsilon \left( J'_1 + J_{1\varepsilon}\left({\widetilde{J}}'\right)\cos\left(\theta'_1 \right) ,{\widetilde{J}}_\varepsilon\left({\widetilde{J}}'\right) ,\theta'_1 + \theta_{1\varepsilon}\left({\widetilde{J}}'\right)\right) .\eqno(8)$$ Hence, for ${\widetilde{J}}'\in{\cal V}_{{\widetilde{J}}^{(0)}}$ such that $\left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}',\theta_{10}\left( {\widetilde{J}}'\right)\right)\in{\cal V}$, the point $\left( 0,{\widetilde{J}}', 0\right)$ correspond to ${\cal M}_\varepsilon\left({\widetilde{J}}'\right)$ in the old variables ; moreover, the total Hamiltonian ${\cal H}'$ is analytical over the complex domain $V_{r'\sqrt{\varepsilon},s'}{\cal D}'_0$ where ${\cal D}'_0$ is a neighbourhood of ${\cal M}_\varepsilon\left({\cal V}_{{\widetilde{J}}^{(0)}} \right)$. Then, we look at the derivatives of $H'_{\varepsilon}$, condition (iii) implies : $$\Dron{H'_\varepsilon}{J_1'}\left( 0,{\widetilde{J}}' ,0\right) =\Dron {H'_\varepsilon}{\theta_1'}\left( 0,{\widetilde{J}}' ,0\right) =0,$$ and the conditions (iii) and (iv) gives : $$\Dron{H'_\varepsilon}{{\widetilde{J}}'}\left( 0,{\widetilde{J}}' ,0\right) = \Dron{h}{J_1}\left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right) \Dron{J_{10}}{{\widetilde{J}}}\left({\widetilde{J}}'\right) +\Dron{h} {{\widetilde{J}}}\left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}' \right) =\Dron{h}{{\widetilde{J}}}\left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right) .$$ For the second derivatives, condition (iv) yields : $$\partial^2_{{\widetilde{J}}' {\widetilde{J}}'}H'_\varepsilon\left( 0, {\widetilde{J}}' ,0\right) =\partial^2_{{\widetilde{J}}{\widetilde{J}}}h \left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right) -\Frac{{}^t\!\partial^2_{{\widetilde{J}}J_1}h\left( J_{10}\left({\widetilde{J}}' \right),{\widetilde{J}}'\right)\partial^2_{{\widetilde{J}}J_1} h\left( J_{10} \left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right)}{\partial^2_{J_1 J_1}h \left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right)}$$ $$\Longrightarrow\eqalign{\pmatrix{\partial^2_{{\widetilde{J}}'{\widetilde{J}}'} H'_\varepsilon\left( 0, {\widetilde{J}}',0\right) &{}^t\!\partial_{{\widetilde{J}}'} H'_\varepsilon\left( 0,{\widetilde{J}}' ,0 \right)\cr\partial_{{\widetilde{J}}'}H'_\varepsilon\left( 0,{\widetilde{J}}' , 0\right) &0\cr}=\hskip10truecm\cr}$$ $$\hskip1truecm\pmatrix{\partial^2_{{\widetilde{J}} {\widetilde{J}}}h\left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right) -\Frac{{}^t\!\partial^2_{{\widetilde{J}}J_1} h\left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right) \partial^2_{{\widetilde{J}}J_1} h\left( J_{10} \left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right)}{\partial^2_{J_1 J_1}h \left( J_{10}\left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right)} &{}^t\!\partial_{{\widetilde{J}}}h\left( J_{10} \left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right)\cr \partial_{{\widetilde{J}}}h\left( J_{10} \left({\widetilde{J}}'\right) ,{\widetilde{J}}'\right) &0\cr}$$ The previous estimate, the analyticity of $H'_\epsilon$ and the lemma B.1. of the appendix B ensure that, for a small enough neighbourhood ${\cal V}_{{\widetilde{J}}^{(0)}}$ and $\varepsilon <\varepsilon_0$, the Hamiltonian $H'_\varepsilon\left( 0,{\widetilde{J}}' ,0\right)$ satisfies uniformly an isoenergetic non degeneracy condition over ${\cal V}_{{\widetilde{J}}^{(0)}}$ with a constant $p'$ of the order of $p\partial^2_{J_1 J_1}h\left(J_{10}\left( {\widetilde{J}}'\right) ,{\widetilde{J}}'\right)$. Finally $\Dron{H'_\varepsilon}{J'_1}\left( 0,{\widetilde{J}}' ,0\right) =\Dron{ H'_\varepsilon}{\theta'_1}\left( 0,{\widetilde{J}}',0\right) =0\Longrightarrow \partial^2_{{\widetilde{J}}' J'_1}H'_\varepsilon\left( 0,{\widetilde{J}}' ,0 \right) =\partial^2_{{\widetilde{J}}' \theta'_1}H'_\varepsilon\left( 0, {\widetilde{J}}' ,0\right) =0$ and $$\partial^2_{J'_1 J'_1}H'_\varepsilon\left( 0,{\widetilde{J}}' ,0\right) = \partial^2_{J_1 J_1}H_\varepsilon\left({\cal M}_\varepsilon\left({\widetilde {J}}'\right)\right) =\alpha ({\widetilde{J}}')\ ;\ \partial^2_{J'_1\theta'_1} H'_\varepsilon\left( 0,{\widetilde{J}}',0\right) =\varepsilon\partial^2_{J_1 \theta_1}f_*\left({\cal M}_\varepsilon\left({\widetilde{J}}'\right) \right) =\varepsilon\beta ({\widetilde{J}}')$$ $$\partial^2_{\theta'_1 J'_1}H'_\varepsilon\left( 0,{\widetilde{J}}' ,0\right) = \varepsilon\partial^2_{\theta_1 J_1}f_*\left({\cal M}_\varepsilon \left({\widetilde{J}}'\right)\right) =\varepsilon\beta ({\widetilde{J}}')\ ;\ \partial^2_{\theta'_1\theta'_1}H'_\varepsilon\left( 0,{\widetilde{J}}' ,0\right) =\varepsilon\partial^2_{\theta_1\theta_1}f_*\left({\cal M}_\varepsilon \left({\widetilde{J}}'\right)\right) =\varepsilon\gamma ({\widetilde{J}}')$$ Now, the Taylor expansion of $H'_\varepsilon$ at $\left( 0,{\widetilde{J}}' ,0\right)$ can be written : $$H'_\varepsilon\left( J'_1 ,{\widetilde{J}}',\theta'_1\right) = H'_\varepsilon \left( 0 ,{\widetilde{J}}' ,0\right) +{{1}\over{2}}^t\!\!\!\pmatrix{ J'_1\cr\theta'_1\cr}\pmatrix{\alpha ({\widetilde{J}}') &\varepsilon\beta ({\widetilde {J}}')\cr \varepsilon\beta ({\widetilde{J}}') &\varepsilon\gamma ({\widetilde {J}}')\cr}\pmatrix{J'_1\cr\theta'_1\cr}+{\cal O}_3 ( J'_1 ;{\widetilde{J}}')+ \varepsilon {\cal O}_3 ( J'_1 ,\theta'_1 ;{\widetilde{J}}').$$ The eigenvalues of the Hessian matrix with respect to $J'_1$ and $\theta'_1$ are : $$\lambda_1 ({\widetilde{J}}')=\alpha ({\widetilde{J}}')+{\cal O}_1(\varepsilon ;{\widetilde{J}}')\ {\rm and}\ \varepsilon\lambda_2({\widetilde{J}}')= \varepsilon\gamma ({\widetilde{J}}') +{\cal O}_2(\varepsilon ;{\widetilde{J}}')$$ and we assume that ${\cal V}_{{\widetilde{J}}^{(0)}}$, the neighbourhood around ${\widetilde{J}}^{(0)}$, is small enough to ensure that the hyperbolicity condition $\alpha ({\widetilde{J}}')\gamma({\widetilde{J}}')=\partial^2_{J'_1 J'_1}H_\varepsilon\left({\cal M}_\varepsilon\left({\widetilde{J}}'\right)\right) \partial^2_{\theta_1\theta_1}g_\varepsilon\left({\cal M}_\varepsilon\left( {\widetilde{J}}'\right)\right) <-{{\lambda_{(0)}^2}\over{2}}<0$ is uniformly satisfied over ${\cal V}_{{\widetilde{J}}^{(0)}}$. \medskip Hence, we see that the original Hamiltonian is nicely amenable to hyperbolic case studied in the second section except for the dissymetry in the eigenvalues of the Hessian (since $\lambda_1$ and $\lambda_2$ have respective sizes of the order of 1 and $\varepsilon$). Consequently, the reduction made in section II cannot be carried out in a suitable way. This difficulty is overcomed by a rescaling of the action variables but, now, we have to restrict our study to an area whose size is $\varepsilon -$dependent. More specifically, we first select an action linked to an hyperbolic torus with a Diophantine frequency as the origin in the action space. Hence, we consider ${\widetilde{J}}'^{(1)}\in{\cal V}_{{\widetilde {J}}^{(0)}}$ such that $$\omega_1 =\Dron{{{H}}'_\varepsilon}{{\widetilde{J}}'}\left( 0,{\widetilde{J}}'^{(1)},0\right) =\Dron {h}{{\widetilde{J}}}\left( J_{10}\left({\widetilde{J}}'^{(1)}\right) ,{\widetilde{J}}'^{(1)}\right)\in\Omega_{ \gamma ,\tau}\eqno\left({\cal C}_1^{(1)}\right)$$ where $\Omega_{\gamma ,\tau}$ still denotes the set $\left\{\omega\in\R^n\ {\rm such\ that}\ \vert k. \omega\vert\geq{{\gamma}\over{\vert\vert k\vert \vert_\infty^\tau}}\ {\rm for\ all}\ k\in\Z^n\right\}$ with $\gamma >0$ and $\tau\in ]n-1,+\infty [$. Then, the energy is settled to zero $${{H}}'_\varepsilon\left(0,{\widetilde{J}}'^{(1)},0\right) =h\left( J_{10}\left({\widetilde{J}}'^{(1)}\right) ,{\widetilde{J}}'^{(1)}\right)=0\eqno \left({\cal C}_2^{(1)} \right)$$ and the origin is translated to $\left( 0,{\widetilde{J}}'^{(1)},0,0\right)$. The change of variables : $$\left( J'_1,{\widetilde{J}}',\theta'_1,{\widetilde{\theta}}'\right) ={\widehat {\cal T}}\left({\widehat{J}},{\widehat{\theta}},{\widehat{p}},{\widehat{q}}\right) \ {\rm where}\ J'_1=\sqrt{\varepsilon}{\widehat{p}},\ {\widetilde{J}}'=\sqrt {\varepsilon}{\widehat{J}},\ \theta'_1 ={\widehat{q}},\ {\widetilde{\theta}}' = {\widehat{\theta}},$$ is symplectic up to a multiplier equal to $\sqrt{\varepsilon}$ and, with a fitted time scaling by $\sqrt{\varepsilon}$, the Hamiltonian becomes in the new variables ${\widehat{\cal H}}\left({\widehat{J}},{\widehat{\theta}},{\widehat {p}},{\widehat {q}}\right) ={\widehat{H}}_\varepsilon\left({\widehat{J}},{\widehat{p}},{\widehat {q}}\right) +\sqrt{\varepsilon}\mu{\widehat{g}}\left({\widehat{J}},{\widehat {\theta}},{\widehat{p}},{\widehat{q}}\right)$ where the averaged Hamiltonian can be written ${\widehat{H}}_\varepsilon\left({\widehat{J}},{\widehat{p}},{\widehat{q}}\right) ={\widehat{h}}\left({\widehat{J}};\sqrt{\varepsilon}\right) +\sqrt{\varepsilon} {\widehat{P}}\left({\widehat{J}},{\widehat{p}},{\widehat{q}}\right)$ with $${\widehat{h}}\left({\widehat{J}}\right) ={{1}\over{\sqrt{\varepsilon}}}{{H}}'_\varepsilon\left( 0, \sqrt{\varepsilon}{\widehat{J}},0\right) =\omega_1{\widehat{J}}+{{\sqrt {\varepsilon}}\over{2}}{}^t\!\!{\widehat{J}}\Gamma_1{\widehat{J}}+\varepsilon{\cal O}_3\left({\widehat{J}};\sqrt{\varepsilon}\right)$$ and $${\widehat{P}}\left({\widehat{J}},{\widehat{p}},{\widehat{q}}\right) ={{1}\over {2}}^t\!\!\!\pmatrix{{\widehat{p}}\cr{\widehat{q}}\cr}\pmatrix{{\widehat{\alpha}}( {\widehat{J}})&{\widehat{\beta}}({\widehat{J}})\cr{\widehat{\beta}}({\widehat{J}}) &{\widehat{\gamma}}({\widehat{J}})\cr}\pmatrix{{\widehat{p}}\cr{\widehat{q}}\cr}+ {\cal O}_3\left({\widehat{p}},{\widehat{q}};{\widehat{J}},\sqrt{\varepsilon} \right)$$ where $\Gamma_1 =\partial^2_{{\widetilde{J}}'{\widetilde{J}}'}H'_\varepsilon\left( 0,{\widetilde{J}}'^{(1)},0\right)$, ${\widehat{\alpha}}({\widehat{J}})=\alpha \left(\sqrt{\varepsilon}{\widehat{J}}\right)$, ${\widehat{\beta}}({\widehat{J}})= \beta\left(\sqrt{\varepsilon}{\widehat{J}}\right)$, ${\widehat{\gamma}}({\widehat {J}})=\gamma\left(\sqrt{\varepsilon}{\widehat{J}}\right)$. Moreover, using the notations of section II, we can ensure that the total Hamiltonian ${\widehat{\cal H}}$ is analytical on the complex domain $V_{\widehat{K}}$ where ${\widehat{K}}= \inf\left( r',s'\right)$ (one should notice that, in the old variables, the size of ${\widehat{\cal T}}\left( V_{\widehat{K}}\right)$ is of the order of $\sqrt {\varepsilon}$). Finally, the lemma B.2 of the appendix B ensures that, for a small enough neighbourhood $V_{\widehat{K}}$ of the origin and $\varepsilon<\varepsilon_0$, the Hamiltonian ${\widehat{h}}$ satisfies uniformly an isoenergetic non degeneracy condition over $V_{\widehat{K}}$ with a constant ${\widehat p}=p_0\sqrt {\varepsilon}$. We see that, now, we can directly apply the results of section II on ${\widehat {\cal H}}$ and, just replacing $\varepsilon$ by $\sqrt{\varepsilon}$, $K$ by ${\widehat{K}}$, $E_1$ by $E$, $\lambda$ by ${{\lambda_{(0)}}\over{\sqrt{2}}}$ and $\nu_0$ by $\nu_1=\vert\vert\omega_1\vert\vert$ in theorem 2.2., we obtain : \medskip \soulc{Theorem 3.3. (Main result in the nearly integrable case).} {\it Let $E\!\geq\!\sup\left( 1,\nu_1\right)$, $m_*\! =\!\inf\left({\Frac{p_0}{2}}, {\Frac{\lambda_{(0)}}{\sqrt{2}}}\right)$, $\sigma\!=\! 2^{4n+1}\left(\Frac{n+1} {e}\right)^{n+1}$ with $1=\ln (e)$ and assume that~: $$n-1<\tau\leq n\ ;\ {\widehat K}\leq C\Frac{m_*}{E}\ ;\ \gamma\leq\Frac{\sqrt {\varepsilon}}{30}\ ;\ \Frac{\sqrt{\varepsilon}\mu}{\gamma^4}<\left(\Frac {m_*^3} {3\sigma^3{\widehat K}}\right)^2\left(\Frac{{\widehat K}}{240} \right)^{4(2n+5)}\eqno(9)$$ then there exist a symplectic transformation ${\cal T}^{(1)} : V_{{\widehat K}/ 6}\longrightarrow V_{{\widehat K}/2}$ such that : $${\cal H}\circ{\cal T}^{(1)}=(1+\eta^{(1)} )\omega_1 I^{(1)}+\sqrt{\varepsilon}\lambda_1 s^{(1)}u^{(1)}+\sqrt{\varepsilon}f^{(1)}\left( I^{(1)},\varphi^{(1)},s^{(1)},u^{(1)} \right)$$ with $f^{(1)}\left( I^{(1)},\varphi^{(1)},s^{(1)},u^{(1)}\right) = {\cal O}_2\left( I^{(1)},s^{(1)}u^{(1)}; I^{(1)},\varphi^{(1)},s^{(1)},u^{(1)}\right)$ and $$\Frac{1}{2}<1+\eta^{(1)} <\Frac{3}{2}\ ;\ \lambda_1>\Frac{\lambda_{(1)}}{2}>\Frac {\lambda_{(0)}}{4}$$ moreover $$\eqalign{\left\vert\left\vert\eta^{(1)}\omega_1 I^{(1)}+\sqrt{\varepsilon}\left(\lambda_1 -\lambda_{(1)}\right) s^{(1)}u^{(1)} +\sqrt{\varepsilon}\left[ f^{(1)} ( I^{(1)}, \varphi^{(1)},s^{(1)},u^{(1)} ) -f(I^{(1)},s^{(1)}\right.\right.\right. \!\!\! & \left.\left.\left. u^{(1)})\right]\right\vert\right\vert_{{\widehat K}/6}\cr &\leq\Frac{E}{2^{4n+6}}\root 4\of{\varepsilon}\sqrt{\mu}\cr}$$ Hence, $\{ I^{(1)}=0\}$ becomes an hyperbolic invariant set for the perturbed flow with linear motions.} \medskip \souli{Remark :} The extension in the case of $\tau >n$ is still valid with the modifications given in section II. \medskip \soulr{IV Implementation on a transition chain} \medskip In [CG, 1994] and [RW1, 1996], many hyperbolic tori and their invariant manifolds are straightened by means of a {\it single chart} built by applying the K.A.M. theory with Arnold's scheme to an initially hyperbolic Hamiltonian. This allowed Chierchia and Gallavotti ([CG, 1994]) to prove, in the initially hyperbolic case, the existence of transition chains (i.e. : families of tori invariant under the perturbed flow and linked by heteroclinic orbits) which are fitted for the construction of Arnold's mechanism. Rudnev and Wiggins ([RW2, 1997]) have obtained the same result in the (much more delicate) case of a nearly integrable Hamiltonian with some specific perturbations. On the other hand, as it was said in the introduction, the charts considered in the previous studies is convergent only on a Cantor set and cannot be used for the study of the dynamics of orbits shadowing these lines of tori. On the contrary, the changes of variables given in this paper are defined on an open set and, consequently, gives the desired control around the invariant manifolds of a partially hyperbolic torus. The price to pay for this extension is the non-integrability of the normalized Hamiltonian whose associated flow is linear only on a {\it single} torus and its linked hyperbolic directions. Now, using the results of [CG, 1994] and [RW2, 1997], we assume the existence of a transition chain and study the problem of building our normal forms around each torus of the considered family. We first restrict our study to an initially hyperbolic Hamiltonian, the equivalent results in the general nearly-integrable case will be specified at the end of this paragraph. \medskip We denote ${\bf T}$ the set of tori and their invariant manifolds which can be straightened with the use of theorem 2.2. A crude application of the latter shows that the tori in ${\bf T}$ admit a frequency which is Diophantine with a constant $\gamma$ of the order of $\root 4\of{\varepsilon\mu}$. Since we assume $\tau > n-1$, the complementary of $\Omega_{\gamma ,\tau}$ have a measure of the order of ${\cal O}(\gamma )={\cal O}(\root 4\of{\varepsilon\mu})$. Consequently, because of the isoenergetic nondegeneracy condition and the local invertibility of the frequency map on a constant energy manifold, we can ensure that the width of the gaps between tori in ${\bf T}$ is ${\cal O} (\root 4\of{\mu /\varepsilon^3})$. Actually, in Arnold's instability we look for orbits around a family of tori associated to a curve ${\cal C}$ in the non resonant action space $\R^n$ (see [L, 1996]). Hence, the average distance between two tori in ${\bf T}$ with a linked action in ${\cal C}$ is given by the measure of $\omega ({\cal C})\cap \Omega^c_{\gamma ,\tau}$ where $\omega$ is the frequency map in the unperturbed problem and $\Omega^c_{\gamma ,\tau}$ is the complement of $\Omega_{\gamma , \tau}$ in $\R^n$. If ${\cal C}$ has a full torsion (i.e.~: the tangent vector of ${\cal C}$ and its first derivatives span the action space $\R^n$), Pyartli ([Py, 1969]) has given estimates for the size of $\omega ({\cal C})\cap \Omega^c_{\gamma ,\tau}$ which are explicitly computed in [CG, 1994] (section 3). They give a measure of the considered set of the order of ${\cal O}\left(\gamma^{1/n}\right)$ for $\tau =n^2$ and, here, the width of the gaps between tori in ${\bf T}$ associated to the curve ${\cal C}$ is ${\cal O}\left(\varepsilon^{-1}\left(\varepsilon\mu\right)^{{{1}\over{4n}}} \right)$. On the other hand, the results of [CG, 1994] show that hyperbolic tori can be connected by heteroclinic orbits only if their mutual distance is of the order of ${\cal O}\left( (\varepsilon\mu )^\delta\right)$ with $\delta >1$ and, consequently, we cannot directly apply the theorem 2.2. on a transition chain in a general case since the gaps between invariant tori are too important. As it can be seen in [CG, 1994] (section 7), this difficulty can be overcomed by making $p$ steps of perturbation theory which reduce the studied Hamiltonian in section II to ${\breve h}\left({\breve I},{\breve s}{\breve u};\varepsilon\right) + (\varepsilon\mu)^{p}{\breve g}\left({\breve I},{\breve \varphi},{\breve s}, {\breve u}\right)$ and the distance between invariant tori is small enough for a high exponent $p$ but this procedure imposes to consider perturbations given by trigonometric polynomial of finite order in order to get a global result. \medskip For an arbitrary perturbation, the results derived in this paper are still relevant because they allow the construction of a Birkhoff normal form around each persistent torus. Studies of stability in the vicinity of an invariant torus provided by the K.A.M. theory have been made by Morbidelli and Giorgilli ([MG, 1995]) in the Lagrangian case, by Jorba and Villanueva ([JV2, 1996]) in the case of a lower-dimensional elliptic torus and these results can be easily extended here in the partially hyperbolic case. Indeed, by application of the theorem 2.2. we can locally consider the Hamiltonian $${{\cal H}^\flat}={\omega^\flat}{I^\flat}+\varepsilon{\lambda^\flat}{s^\flat} {u^\flat}+\varepsilon{ f^\flat}\left({I^\flat},{s^\flat}{u^\flat}\right) +\sqrt {\varepsilon\mu}{g^\flat}\left({I^\flat},{\varphi^\flat},{s^\flat},{u^\flat} \right)\ {\rm where}\ {\omega^\flat}\in\Omega_{\gamma,\tau},\ {\lambda^\flat} >0$$ and ${f^\flat}\left({ I^\flat},{ s^\flat}{ u^\flat}\right)$ (resp. ${g^\flat} \left({I^\flat},{\varphi^\flat},{s^\flat},{ u^\flat}\right)$) is on the order of ${\cal O}_2\left({ I^\flat},{ s^\flat}{ u^\flat};\varepsilon \right)$ (resp. on the order of ${\cal O}_2\left({ I^\flat},{ s^\flat} {u^\flat };{I^\flat},{\varphi^\flat},{ s^\flat},{ u^\flat},\varepsilon\right)$). Since, the small denominators in the construction of the Birkhoff's normal form with the elliptic-saddle Hamiltonian ${\omega^\flat}{ I^\flat}+\varepsilon{ \lambda^\flat}{ s^\flat}{ u^\flat}$ can be written $i\left({k^\flat}. {\omega^\flat}\right) +\varepsilon{l^\flat}.{\lambda^\flat}$ for ${k^\flat}\in\Z^n$ and ${ l^\flat}\in \Z$, the convergence of the normalizing transformation is only prevented by the arithmetical properties of ${\omega^\flat}$. Using the estimates given by P${\ddot{\rm o}}$schel ([P${\ddot{\rm o}}$, 1993], theorem 5), we can normalize the Hamiltonian up to an exponentially small remainder and obtain the following : \medskip \soulc{Theorem 4.1.} {\it Let ${{\cal H}^\flat}={\omega^\flat_0}{ I^\flat}+\varepsilon{\lambda^\flat} { s^\flat}{ u^\flat}+\varepsilon{ f^\flat}\left({ I^\flat},{ s^\flat}{ u^\flat} \right) +\sqrt{\varepsilon\mu}{ g^\flat}\left({ I^\flat},{\varphi^\flat},{ s^\flat},{ u^\flat}\right)$ be the Hamiltonian obtained after the application of theorem 2.2. around a first torus ${\cal T}_0$ with a linked frequency ${\omega^\flat_0}$ in $\Omega_{\gamma_0,\tau}$ where $\tau >n-1$. We denote $V_{ K^\flat}$ the complex domain of analyticity of ${{\cal H}^\flat}$, then there exist ${ K}^\flat_0 >0$ such that, provided $\varepsilon{ K}^\flat < \gamma_0{K}^\flat_0$, we can find a symplectic transformation ${\cal T}:V_{{ K^\flat}'}\longrightarrow V_{ K^\flat}$ for some ${ K^\flat}'=\rho{ K^\flat}$ ($0<\rho <1$) with $\left( { I^\flat},{\varphi^\flat},{ s^\flat},{u^\flat} \right) ={\cal T}(I,\varphi ,s, u)$ which cast the original Hamiltonian ${{\cal H}^\flat}$ to the following : $${{\cal H}^\flat}\circ{{\cal T}^\flat}={\omega^\flat_0}I+\varepsilon{ \lambda^\flat}su+\varepsilon f(I,su)+\sqrt{\varepsilon\mu}\exp\left( -C\left( {{\gamma_0 K^\flat_0}\over{\varepsilon{K^\flat}}}\right)^{{{1}\over{1+\tau}}} \right) g(I,\varphi ,s,u)$$ with $g(I,\varphi ,s,u) ={\cal O}_2 (I,su;I,\varphi,s,u)$. } \vskip0,3truecm Hence, in these new coordinates, the normalized Hamiltonian agrees with the expression (2) of section II and the theorem 2.2. can be applied on a torus with a linked frequency in $\Omega_{\gamma,\tau}$ if : $$\sqrt{\varepsilon\mu}\exp\left( -C\left({{\gamma_0 K^\flat_0} \over{\varepsilon{K^\flat}}}\right)^{{{1}\over{1+\tau}}}\right) < C_1\gamma^4\ {\rm with}\ C_1 >0.$$ Consequently, if $\gamma_0$ is $\varepsilon$-independent, we find a neighbourhood of ${\cal T}_0$ whose size is of the order of ${\cal O}\left(\left\vert\ln (\mu ) \right\vert^{-(\tau +1)}\right)$ where the hyperbolic tori with a linked Diophantine frequency in $\Omega_{\gamma ,\tau}$ for $\gamma$ of the order of ${\cal O}\left( (\varepsilon\mu )^{4n\delta}\right)$ can be straightened by means of theorem 2.2. In that way, we obtain open sets where our normal forms can be used around two tori connected by heteroclinic orbits in the cases studied by Gallavotti-Chierchia ([CG, 1994]). \medskip In the general nearly-integrable case, the perturbation in P${\ddot{\rm o}}$schel's normal form is exponentially small with respect to $\varepsilon^{{1} \over{2n}}$ (c.f. lemma 3.2.) and, at least in the cases studied by Rudnev and Wiggins ([RW2, 1997]), the mean distance between two tori connected by heteroclinic orbits is also of the order of ${\cal O}\left(\exp (-C \varepsilon^{-{{1}\over{2n}}})\right)$. Consequently, the problem encountered previously with an initially hyperbolic Hamiltonian is still present in a nearly-integrable system where we replace $\varepsilon$ and $\mu$ by $\sqrt{\varepsilon}$ and $\exp\left( -C\varepsilon^{-{{1}\over{2n}}}\right)$. Hence, if the frequency associated to the initial torus ${\cal T}_0$ is Diophantine with a constant $\gamma_0$ independent of $\varepsilon$, we can straighten close enough tori in a neighbourhood of ${\cal T}_0$ whose size is of the order of ${\cal O}\left( \varepsilon^{{{\tau +1}\over{2n}} -{{1}\over{2}}}\right)$ for $\tau >n-1$. Consequently, in the original coordinates, our theorems can be applied to study the dynamic around transition chains close to ${\cal T}_0$ whose lengths are of the order of ${\cal O}\left(\varepsilon^{{{1}\over{2}}+\zeta}\right)$ where $\zeta >0$. \medskip At this step, we can also specify the minimal distance to multiple resonances needed to derive our results in the nearly-integrable case. An examination at the lower bound for resonances $\alpha$ given in lemma 3.2. and at the size of the perturbation $\mu ={\cal O}\left(\varepsilon^{3/2}\right)$ needed to apply theorem 3.3. (this comes from the last two thresholds in (9)) shows that our study is valid in a strip of width $\sqrt{\varepsilon}$ around a manifold in the action space linked to a simple resonance where we exclude gaps around multiple resonances whose total measure is of the order of ${\cal O}\left(\sqrt {\varepsilon}\vert\ln (\varepsilon )\vert^n\right)$. \medskip \soulr{V Conclusion and prospects.} \medskip Despite the restrictions given at the end of the previous section, estimating the speed of drift of orbits contained in the domain of application of our theorems is already relevant to look at the optimality of the bounds on instability obtained in certain Nekhorochev-like results. Indeed, assuming the existence of transition chains, we give tools for an accurate analysis of the dynamics of orbits shadowing relatively long lines of tori. First, P${\ddot{\rm o}}$schel ([P${\ddot{\rm o}}$, 1993], theorem 2) has shown that, except on a set of small measure, each action is confined over exponentially long times in an ${\cal O}(\sqrt{\varepsilon})$-area and this is almost the length of the chains considered at the end of section IV. More generally, the theorems of exponential stability given by Lochak [LN, 1992] and P${\ddot{\rm o}}$schel [P${\ddot{\rm o}}$, 1993] rely on the properties of resonant normal forms which, under suitable assumptions, ensure a drift of the actions along the considered resonant manifold at most of the order of ${\cal O}(\sqrt{\varepsilon})$ over exponentially long times. Here, the combination of our theorems and of Marco's geometrical methods ([Ma, 1996]) might ensure (in certain cases) the existence of orbits moving over exponentially long times along pieces of resonant manifolds whose lengths are also of the order of ${\cal O}(\sqrt{\varepsilon})$. Hence, one can expect results on the limits of the previous reasonings to confine the orbits in nearly-integrable Hamiltonian systems. Moreover, we should notice that the simple resonant case studied here allows only to prove the $\grave{}\!\grave{}$~worst~$\acute{} \!\acute{}$ estimate of stability over exponentially long times (see Lochak [L, 1996]) and, consequently, it seems to be the best setting to look for orbits with a speed of drift comparable with the upper bound on the rate of instability predicted in Nekhorochev-like theorems. On the other hand, the initial Arnold's conjecture (i.e. : the existence of orbits in a generic nearly-integrable systems connecting two actions separated by an arbitrary distance) cannot be tackled without accurate informations on the dynamic close to multiple resonances. First, we should mention that Chierchia [Ch, 1996] have recently studied instability of orbits around but not too close from double resonances by means of a reduction to a perturbation of the Hamiltonian of section II with a single hyperbolic degree of freedom ; hence, the results obtained here are still applicable in this setting. Close to arbitrary multiple resonances, following the lines of Treschev's reasonings [Tr, 1991], we can still reduce a generic nearly-integrable Hamiltonian to the initially hyperbolic case of section II (now, with several degrees of freedom) but, unfortunately, we do not have anymore tools like the Moser's transformation used here because of the generic divergence of Birkhoff's normal forms in higher dimension. Actually, denoting by $T$ an hyperbolic torus and its linked manifolds, the results of Brjuno ([Br, 1971] and [Br, 1989]) ensure the existence of a normal form with a linear flow on $T$ but the normalizing transformation is convergent only on $T$. On the other hand, Graff ([Gr, 1973], theorem 7) has given a formal scheme for the construction of the desired normalizing transformation but the question of convergence is not discussed. A first track for a generalisation of our results is the use of the work of Jorba and Villanueva [JV1, 1996] who proved the persistence under quasi-periodic perturbation of lower-dimensional elliptic tori. Here, we can focus our attention on tori associated to resonances of order $k$ where the averaged Hamiltonian admits $n-k-1$ elliptic degrees of freedom and one hyperbolic degree of freedom when we carry out a reduction similar to Treschev's reasoning in the complete hyperbolic case. Hence, we are almost in the setting of this study and similar results can be expected in some cases of multiple resonances. Another clue for a generalisation of the theorems obtained here is the possibility of making our K.A.M. iterative scheme in the case where the unperturbed flow is only linear {\it on} the hyperbolic tori and their linked invariant manifolds (i.e. : the unperturbed Hamiltonian can be written $H(I,s,u)=\omega .I +\varepsilon\lambda su+\varepsilon P(I,s,u)$ where $P(I,s,u)={\cal O}_2 (I,su;s,u,\varepsilon )$). Hence, the integration of the initially hyperbolic Hamiltonian by means of Moser's theorem is not necessary and, along the lines of the reasoning in section II, we only need a symplectic transformation which straighten and linearize the flow {\it on} the invariant manifolds associated to an hyperbolic fixed point in $\R^{2n}$. Actually, the conditions needed to prove the existence of such a transformation are much less stringent than those imposed to ensure the convergence of Birkhoff's normal forms (see Brjuno [Br, 1989]). Then, in certain cases, we can certainly carry out a multidimensional K.A.M. scheme analogous to the unidimensional K.A.M. scheme made in section II. \bigskip \soulr{Appendix A : Proof of the iterative lemma 2.3.} \medskip We omit the indexes ($n$) in this paragraph and use the same notations as in the section $II.2$. \medskip First, we need some preliminary analytical lemmas. Let $\left({\cal A}_K ;\vert\vert . \vert\vert_K\right)$ be the normed space of numerical or vector valued functions which are analytical over $V_K$. \medskip \soulc{Lemma A.1.} {\it Let $(f,g)\in{\cal A}_K^2$, $0<\delta <1$ and two integers $(k,l)\in\N^2$, then : 1) $\left\vert\left\vert\Frac{\partial^{k+l}f}{\partial s^k\partial u^l}(I, \varphi ,s,u)\right\vert\right\vert_{(1-\delta )K}\leq\Frac{k!l!}{\left(\delta K \right)^{k+l}}\vert\vert f\vert\vert_K$ and $\left\vert\left\vert\Frac{\partial^{k +l}f}{\partial s^k\partial u^l}(0,\varphi ,0,0)\right\vert\right\vert_K\leq\Frac{k !l!}{K^{k+l}}\vert\vert f\vert\vert_K$ 2) $\left\vert\left\vert\Frac{\partial f}{\partial I}(I,\varphi ,s,u)\right\vert \right\vert_{(1-\delta )K}\leq\Frac{\vert\vert f\vert\vert_K}{\delta K}$ and $\left \vert\left\vert\Frac{\partial^{1+k+l}f}{\partial I\partial s^k\partial u^l}f(0, \varphi ,0,0)\right\vert\right\vert_{(1-\delta )K}\leq\Frac{k!l!}{\left(\delta K\right)^{1+k+l}}\vert\vert f\vert\vert_K$ 3) $\left\vert\left\vert\Frac{\partial^2 f}{\partial I^2}(I,\varphi ,s,u)\right \vert\right\vert_{(1-\delta )K}\leq\Frac{4\vert\vert f\vert\vert_K}{\left(\delta K\right)^2}$ 4) $\vert\vert\{ f,g\}\vert\vert_{(1-\delta )K}\leq\Frac{\vert\vert f\vert\vert_K \vert\vert g\vert\vert_K}{\left(\delta K\right)^2}\ {\it and}\ \vert\vert\{ f,g\} \vert\vert_{(1-\delta )K}\leq\Frac{2\vert\vert f\vert\vert_K\vert\vert g\vert \vert_{\left( 1-{{\delta}\over{2}}\right) K}}{\left(\delta K\right)^2}$} \vskip0,3truecm \souli{Proof}\ \ 1) is a direct application of the Cauchy inequality (see [BGGS, 1984]). To obtain 2), we write $\left\vert\left\vert{{\partial f}\over{\partial I}}(I, \varphi ,s,u)\right\vert\right\vert_{(1-\delta )K}=\Sup_{\vert\vert e\vert\vert =1} \left\vert\left\vert {{df}\over{dt}}_{\vert t=0}f(I+te ,\varphi ,s,u)\right\vert \right\vert_{(1-\delta )K}$ and apply Cauchy formula to the function $t\mapsto f (I+te ,\varphi ,s,u)$ of the complex variable $t$ defined for $\vert t\vert\leq \delta K$ when $(I,\varphi ,s,u)\in V_{(1-\delta )K}$, this yields the desired estimate. It also allows to deduce the second part of 2). For 3), we write $$\left\vert\left\vert{{\partial^2 f}\over{\partial I^2}}(I,\varphi ,s,u)\right\vert\right\vert_{(1-\delta )K}\!\!\!\!\! =\Sup_{\vert \vert e\vert\vert =1,\vert\vert f\vert\vert =1}\left\vert\left\vert{{\partial f} \over{\partial s\partial t}}_{\vert (s,t)=(0,0)}f(I+se+tf ,\varphi ,s,u)\right \vert\right\vert_{(1-\delta )K}$$ and apply Cauchy formula to the function of the complex variables $s$ and $t$ in the right hand side, defined for $\vert s\vert$ and $\vert t\vert\leq{{\delta K} \over{2}}$ when $(I,\varphi ,s,u)\in V_{(1-\delta )K}$, this gives the estimate 3). To obtain the size of the Poisson brackets for two analytical functions on $V_K$ given in 4), we write in a similar way that $\{ f,g\} (I,\varphi ,s,u)\! =\!{{d} \over{dt}}_{\vert t=0}\!\left[\! g\! \left(\! I\! -\! t {{\partial f}\over{\partial \varphi}},\varphi\! +\! t {{\partial f}\over{\partial I}} , s\! -\! t{{\partial f} \over{\partial u}} , u\! +\! t{{\partial f}\over{\partial s}}\right)\right]$ and the function of the complex variable $t$ in the right hand side is defined for $\vert t\vert\leq{{\left(\delta K\right)^2}\over{\vert\vert f\vert\vert_K}}$ (we apply the Cauchy formula to $f$), this yields the estimates 4). \medskip \soulc{Lemma A.2.} {\it 1) Let $f(I,\varphi ,s,u)\in{\cal A}_K$ with the Fourier expansion $f(I,\varphi ,s,u)={\displaystyle\sum_{k\in\Z^n}}f_k (I,s,u)\exp (ik.\varphi )$ then $$\left\vert\left\vert f_k\right\vert\right\vert_K <\vert\vert f\vert\vert_K\exp \left[ -\left(\left\vert k_1\right\vert +\ldots +\left\vert k_n\right\vert\right) K \right]$$ 2) Let $f_k (I,s,u)\in{\cal A}_K$ where $k\in\Z^n$ such that $\left\vert\left \vert f_k\right\vert\right\vert_K 0$, we have the following inequality : $x^s\leq\left(\Frac{s} {ey}\right)^s\exp(xy)$ where $1=\ln (e)$.} \vskip0,3truecm The proof of this lemma is given in the appendix of [BGGS, 1984]. \medskip \soulc{Lemma A.3.} {\it $\!\!\!\!\!\!\!\!\!$ Let $0<\delta <1$, $g(I,\varphi ,s,u)\in{\cal A}_K$ such that ${\overline{g}} (I,s,u)={{1}\over{\left( 2\pi\right)^n}}\int_0^{2\pi}\!\!\!\ldots\int_0^{2\pi} g(I,\varphi ,s,u)d\varphi_1\!\ldots d\varphi_n=0$ and $\omega\in\Omega_{\gamma, \tau}$ where $\tau\leq n$, there exist $f\in{\cal A}_{(1-\delta )K}$ solution of the equation : $$(1+\eta )\sum_{l=1}^n\omega_l\Dron{f}{\varphi_l}(I,\varphi ,s,u)=g(I,\varphi , s,u)\ {\it where}\ 1+\eta >m_*$$ such that : $$\left\vert\left\vert f\right\vert\right\vert_{(1-\delta)K}\leq\Frac{\sigma}{ m_*\gamma}\Frac{\vert\vert g\vert\vert_K}{\left(\delta K\right)^{2n}}\ ;\ \left \vert\left\vert\Dron{f}{\varphi}\right\vert\right\vert_{(1-\delta)K}\leq\Frac {\sigma}{m_*\gamma}\Frac{\vert\vert g\vert\vert_K}{\left(\delta K\right)^{2n+1}} \ {\it with}\ \sigma = 2^{4n+1}\left(\Frac{n+1}{e}\right)^{n+1}$$} \vskip0,3truecm \souli{Proof}\ \ We are looking for $f$ solution of the equation ${\widetilde{ \omega}}\partial_\varphi f=g$ where ${\widetilde{\omega}}=(1+\eta )\omega\in \Omega_{m_*\gamma,\tau}$ and the existence of a function which satisfies the desired properties is the content of the lemma 2 in [BGGS, 1984]. \medskip \soulc{Lemma A.4.} {\it We consider $g(x,\varphi )\in{\cal A}_K$ where $x=s$ or $u$ ; $0<\delta <1$ and $0< m_* <{{\lambda_0}\over{2}}<\lambda$ ; $\omega\in\R^n$. There exist $f^{\pm}(x,\varphi )\in{\cal A}_{(1-\delta )K}$ solution of the equation $$\partial^\pm_{\varepsilon\lambda ,\omega}f^{\pm}(x\varphi )=g(x,\varphi )\ {\it with\ the\ operator}\ \partial^\pm_{\varepsilon\lambda ,\omega}\ {\it defined\ in\ section\ II.2}$$ such that $$\left\vert\left\vert f^{\pm}\right\vert\right\vert_{(1-\delta)K}\leq\Frac{\sigma} {\varepsilon m_*}\Frac{\vert\vert g\vert\vert_K}{\delta^{n+1}K^n}\ ;\ \left\vert \left\vert\Dron{f^{\pm}}{\varphi}\right\vert\right\vert_{(1-\delta)K}\leq\Frac{\sigma} {\varepsilon m_*}\Frac{\vert\vert g\vert\vert_K}{\delta^{n+2}K^{n+1}}\ {\it with \ the\ previous\ constant}\ \sigma$$} \vskip0,3truecm \souli{Proof}\ \ We denote the expansion $g(x,\varphi )={\displaystyle\sum_{k\in\N}} {\displaystyle\sum_{l\in\Z^n}}g_{kl} x^k\exp(il.\varphi )={\displaystyle\sum_{k \in\N}} g_k (\varphi )x^k$, then : $$f^{\pm}(x,\varphi )={\displaystyle\sum_{k\in\N}}{\displaystyle\sum_{l\in\Z^n}}\Frac{ g_{kl}}{(k+1)\varepsilon\lambda\pm i(\omega .l)}x^k\exp(il.\varphi )= {\displaystyle\sum_{k \in\N}} f_k^{\pm} (\varphi )x^k$$ $${\rm Lemma\ A.2.}\!\Longrightarrow\!\left\vert f^\pm_{kl}\right\vert\!\leq\!\Frac {\left\vert g_{kl}\right\vert}{\varepsilon\lambda}\!\leq\!\Frac{\left\vert\left \vert g_k\right\vert\right\vert_K}{\varepsilon m_*}\exp\left[ -\left(\left\vert l_1\right\vert +\ldots +\left\vert l_n\right\vert\right) K\right]\! \Longrightarrow\!{\left\vert\left\vert f_k^{\pm}\right\vert\right\vert_{(1- \delta )K}}\!\leq\!\Frac{\left\vert\left\vert g_k\right\vert\right\vert_K} {\varepsilon m_*}\!\left(\Frac{4}{\delta K}\right)^n$$ then $\left\vert g_k\right\vert_K\leq\Frac{\left\vert\left\vert g\right \vert\right\vert}{K^k}$ implies ${\left\vert\left\vert f^\pm\right\vert\right \vert_{(1-\delta )K}}\leq\Frac{\left\vert\left\vert g\right\vert\right\vert_K} {\varepsilon m_*}\Frac{4^n}{\delta^{n+1} K^n}\leq\Frac{\sigma}{\varepsilon m_*} \Frac{\left\vert\left\vert g\right\vert\right\vert_K}{\delta^{n+1} K^n}$ For the derivatives $\partial_{\varphi_j} f^{\pm}(x,\varphi )={\displaystyle \sum_{k\in\N}}{\displaystyle\sum_{l\in\Z^n}}f^\pm_{kl} x^ki l_j\exp(il.\varphi ) ={\displaystyle\sum_{k\in\N}}\partial_{\varphi_j} f^{\pm}_k (\varphi )x^k$, we have : $$\eqalign{\left\vert\left\vert f^\pm_{kl}l\right\vert\right\vert =&\left\vert f^\pm_{kl}\right\vert\vert\vert l\vert\vert\ {\rm with\ the\ Euclidian\ norm\ for}\ l\in\Z^n\cr \leq &\Frac{\left \vert\left\vert g_k\right\vert\right\vert_K}{\varepsilon m_*}\exp\left[ -\left( \left\vert l_1\right\vert +\ldots +\left\vert l_n\right\vert\right)K\right] \left(\left\vert l_1\right\vert +\ldots +\left\vert l_n\right\vert\right)\cr {\rm Lemma\ A.2.}\!\Longrightarrow\!\left\vert\left\vert f^\pm_{kl}l\right\vert \right\vert\leq &\Frac{\left\vert\left\vert g_k\right\vert\right\vert_K} {\varepsilon m_*}\Frac{1}{e\delta K}\exp\left[\left(\left\vert l_1\right\vert + \ldots +\left\vert l_n\right\vert\right)\delta K\right]\exp\left[ -\left(\left \vert l_1\right\vert +\ldots +\left\vert l_n\right\vert\right) K\right]\cr \leq &\Frac{\left\vert\left\vert g_k\right\vert\right\vert_K}{\varepsilon m_*} \Frac{\exp\left[ -\left(\left\vert l_1\right\vert +\ldots + \left\vert l_n\right \vert\right) (1-\delta )K\right]}{e\delta K}\cr}$$ $$\eqalign{{\rm Lemma\ A.2.}\!\Longrightarrow\!\left\vert\left\vert\partial_\varphi f^\pm_k\right\vert\right\vert_{(1-\delta )K}\leq &\Frac{\left\vert\left\vert g_k \right\vert\right\vert_K}{\varepsilon m_*}\Frac{1}{e\delta K}\left(\Frac{4} {\delta K}\right)^n\cr \left\vert\left\vert g_k\right\vert\right\vert_K\leq\Frac{\vert\vert g\vert\vert_K}{K^k}\!\Longrightarrow\!\left\vert\left\vert\partial_\varphi f^\pm \right\vert\right\vert_{(1-2\delta )K}\leq &\Frac{\left\vert\left\vert g\right \vert\right\vert_K}{\varepsilon m_*}\Frac{4^n}{e\left(\delta K\right)^{n+1}} \sum_{k\in\N}\Frac{\left( 1-2\delta\right)^k K^k}{K^k}\cr \Longrightarrow\!\left\vert\left\vert\partial_\varphi f^\pm\right\vert\right \vert_{(1-2\delta )K}\leq &\Frac{\left\vert\left\vert g\right\vert\right\vert_K} {\varepsilon m_*}\Frac{2^{2n-1}}{e\delta\left(\delta K\right)^{n+1}}\cr}$$ We replace $\delta$ by $\Frac{\delta}{2}$ and obtain the given estimate : $$\left\vert\left\vert\partial_\varphi f^\pm\right\vert\right\vert_{(1-\delta )K} \leq\Frac{\left\vert\left\vert g\right\vert\right\vert_K}{\varepsilon m_*} \Frac{2^{3n+1}}{e\delta\left(\delta K\right)^{n+1}}\leq\Frac{\sigma}{\varepsilon m_*}\Frac{\left\vert\left\vert g\right\vert\right\vert_K}{\delta^{n+2} K^{n+1}}$$ \bigskip Now, we recall the statement of the iterative lemma 2.3. : \soulc{Lemma A.5. (iterative lemma).} {\it We consider an Hamiltonian ${\cal H}$ analytical over $V_{K}$, where $0< K_* m_*\eqno(A_2)$$} \vskip0,3truecm \souli{Proof}\ \ We use the same notations and the same transformations as in section II. \medskip First, the size of the coefficients in the perturbation $g$ is estimated. We write the expansion $g(I,\varphi ,s,u)={\displaystyle\sum_{k\in\N^n}} {\displaystyle\sum_{(l,m)\in\N^2}} g_{klm} (\varphi ) I^k s^l u^m$ where $g_{klm} (\varphi )={{1}\over{k!l!m!}}{{\partial^{k+l+m}g}\over{\partial I^k \partial s^l\partial u^m}}(0,\varphi ,0,0)$ and : $$\left\vert\left\vert\alpha_0(\varphi )\right\vert\right\vert_K =\vert \vert g(0,\varphi ,0,0)\vert\vert_K\!\! < E_1$$ $$\left\vert\left\vert\alpha_{10}(\varphi ,s)\right\vert\right \vert_{(1-\delta )K}\!\leq\left\vert\left\vert{\displaystyle\sum_{l\geq 1}}g_{0l0} (\varphi ) s^{l-1}\right\vert\right\vert_{(1-\delta )K}\!\!\!\!\!\leq{\displaystyle\sum_{l \geq 1}}\Frac{\vert\vert g\vert\vert_K} {K^l}((1-\delta )K)^{l-1}\!\!\! <\Frac{E_1} {\delta K}\ {\rm and}\ \left\vert\left\vert\alpha_{01}(\varphi ,u)\right\vert \right\vert_{(1-\delta )K}\! <\!\!\Frac{E_1}{\delta K}$$ $$\left\vert\left\vert\alpha_{11}(\varphi )\right\vert\right\vert_K\! =\left \vert\left\vert\partial_{su}g(0,\varphi ,0, 0)\right\vert\right\vert_K\! <\Frac {E_1}{K^2}$$ $$\left\vert\left\vert\alpha_{21}(\varphi ,s)\right\vert\right \vert_{(1-\delta )K}\!\leq\!\left\vert\left\vert{\displaystyle\sum_{l\geq 2}}g_{0l1} (\varphi ) s^{l-2}\right\vert\right\vert_{(1-\delta )K}\!\!\!\!\!\leq\! {\displaystyle\sum_{l\geq 2}}\Frac{\vert\vert g\vert\vert_K}{K^{l+1}}((1-\delta )K )^{l-2}\!\!<\!\Frac{E_1}{\delta K^3}\ {\rm and}\ \left\vert\left\vert\alpha_{12} (\varphi ,u)\right\vert\right\vert_{(1-\delta)K}\!\! <\!\!\Frac{E_1} {\delta K^3}$$ $$\left\vert\left\vert\beta_0(\varphi )\right\vert\right\vert_K =\vert \vert\partial_I g(0,\varphi ,0,0)\vert\vert_K\!\leq\Frac{\vert\vert g\vert\vert_K}{K} \! <\Frac{E_1}{K}$$ $$\left\vert\left\vert\beta_{10}(\varphi ,s)\right\vert\right\vert_{ (1-\delta )K}\!\leq\!\left\vert\left\vert{\displaystyle\sum_{l\geq 1}}g_{1l0} (\varphi ) s^{l-1}\right\vert\right\vert_{(1-\delta )K}\!\!\!\!\!\leq\!{ \displaystyle\sum_{l\geq 1}}\Frac{\vert\vert g\vert\vert_K}{K^{l+1}}((1-\delta )K )^{l-1}\!\!\! <\!\Frac{E_1}{\delta K^2}\ {\rm and}\ \left\vert\left\vert \beta_{01}(\varphi ,u)\right\vert\right\vert_{(1-\delta )K}\!\! <\!\!\Frac {E_1}{\delta K^2}$$ \medskip With $\omega.\partial_\varphi X_0 =\alpha_0 -{\overline{\alpha_0}}$, the lemma A.3. of this appendix and $1+\eta > m_*$, we find : $$\left\vert\left\vert X_0\right\vert\right\vert_{(1-\delta )K}\leq{{\sigma E_1}\over{m_*\gamma\left(\delta K\right)^{2n}}}\ {\rm and}\ \left\vert \left\vert\partial_\varphi X_0\right\vert\right\vert_{(1-\delta )K}\leq{{\sigma E_1}\over{m_*\gamma\left(\delta K\right)^{2n+1}}}$$ then, with the assumption $(A_1)$, one can write $$\left\vert\left\vert{\overline{\beta_0}} -2\varepsilon{\overline{\left( d_0 .\partial_\varphi X_0\right)}}\right\vert\right\vert\leq\left\vert \left\vert\beta_0\right\vert\right\vert_K +2\varepsilon{{\left\vert\left\vert \partial_\varphi X_0\right\vert\right\vert_{(1-\delta )K}}\over{m}}\leq{{E_1 }\over{K_*}}+{{2\varepsilon\sigma E_1}\over {m_*^2\gamma\left(\delta K_*\right)^{2n+1}}}\leq{{3\varepsilon\sigma E_1}\over{m_*^2\gamma\left(\delta K_*\right)^{2n+1}}}$$ In the same way, we find $\left\vert\left\vert {\overline{\alpha_0}}\right\vert\right\vert_K\!\!\leq E_1\!\!\leq{{\sigma E_1 \varepsilon}\over{m_*^2\gamma\left(\delta K_*\right)^{2n+1}}}$ and $$\left\vert\left\vert\pmatrix{2\varepsilon{\overline{d\ \!}}(0,0)& \omega\cr\omega & 0\cr}^{-1}\right\vert\right\vert\!\leq\!{{1}\over{\varepsilon m_*}} \!\Longrightarrow\!\!\left\vert\eta_A\right\vert\ {\rm and}\ \left\vert\left\vert\xi\right\vert\right\vert\!\!\leq{{4\sigma E_1}\over{m_*^3\gamma\left(\delta K_*\right)^{2n+1}}}$$ which give $\left\vert\left\vert\partial_\varphi\chi_A\right\vert\right\vert_{(1- \delta )K}\leq\varepsilon\mu\left(\left\vert\left\vert\partial_\varphi X_0\right \vert\right\vert_{(1-\delta )K}+\vert\vert\xi\vert\vert\right)\leq\Frac{5\sigma E_1\varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2n+1}}$ $$\Longrightarrow V_{(1-3\delta )K}\subset{\cal T}_A\left( V_{(1-2\delta )K} \right)\subset V_{(1-\delta )K}\ {\rm under\ the\ assumption}\ (A_1)$$ To estimate the size of the new perturbation $g_A$, we use $\vert \vert H\vert\vert_K\leq\vert\vert{\cal H}\vert\vert_K +\varepsilon\mu\vert\vert g\vert\vert_K\leq{{6 E_1}\over{5}}$ and, under the assumption $(A_1)$, one can write in the new variables : $$\left\vert\left\vert\left\{\chi_A,H\right\}\right\vert\right\vert_{(1-2\delta )K}\leq\left\vert\left\vert\partial_\varphi\chi_A\right\vert\right\vert_{(1-2 \delta )K}\left\vert\left\vert\partial_I H\right\vert\right\vert_{(1-2\delta )K}\leq\Frac{5\sigma E_1\varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2n+1} }\Frac{6 E_1}{5\delta K_*}$$ then $$\left\vert\left\vert\alpha^A_0(\varphi )\right\vert\right\vert_{(1-2\delta )K} \leq\left\vert\left\vert\alpha_0(\varphi )\right\vert\right\vert_{(1-2\delta)K} +\Frac{1}{\varepsilon\mu}\left\vert\left\vert\left\{\chi_A ,H\right\} (0, \varphi ,0,0)\right\vert\right\vert_{(1-2\delta )K}\leq\Frac{7\sigma E_1^2} {m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}}$$ The same reasonings give the other coefficients of $g_A$ and, using $\delta <{{ 1}\over{12}}$, we can write $$\left\vert\left\vert g_A\right\vert\right\vert_{(1-3\delta )K}\leq\vert\vert g \vert\vert_{(1-2\delta )K} +3\left( 1+{{2}\over{\delta}}\right)\Frac{1} {\varepsilon\mu}\left\vert\left\vert\left\{\chi_A ,H\right\}\right\vert\right \vert_{(1-2\delta )K}\leq \Frac{50\sigma E_1^2}{m_*^3\gamma\delta\left(\delta K_*\right)^{2(n+1)}}$$ Finally, we must estimate the size of the remainder ${\cal R}_A$, we have $$\left\vert\left\vert\left\{\chi_A,g\right\}\right\vert\right\vert_{(1-2\delta ) K}\leq\left\vert\left\vert\partial_\varphi\chi_A\right\vert\right\vert_{(1-2 \delta )K}\left\vert\left\vert\partial_I g\right\vert\right\vert_{(1-2\delta )K}\leq\Frac{5\sigma E_1 \varepsilon\mu}{m_*^3\gamma\delta^{2n+1}K_*^{2n+1}}\Frac{E_1}{\delta K_*}\leq \Frac{5\sigma E_1^2\varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}} $$ and the use of second order Taylor formulas along solutions of the system linked to $\chi_A$ gives $$\eqalign{\left\vert\left\vert{\cal H}\!\circ\!{\cal T}_A -{\cal H}\! -\!\left\{\chi_A ,{\cal H}\right\}\right\vert\right\vert_{(\!1\!-\!3\delta\!)\!K}\! \leq &\!\Frac{1}{2}\left\vert\left\vert\left\{\chi_A ,\left\{\chi_A ,{\cal H}\right\}\right\}\right\vert\right\vert_{(\!1\!-\!2\delta\! )\!K}\!\leq\!\Frac{1}{2}\left\vert\left\vert\left(\partial_\varphi \chi_A\right)^2\right\vert\right\vert_{(\!1\!-\!2\delta\! )\!K}\!\left\vert\left\vert \partial^2_I{\cal H}\right\vert\right\vert_{(\!1\!-\!2\delta\! )\!K}\cr \leq &\Frac{1}{2}\left(\Frac {5\sigma E_1\varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2n+1}}\right)^2 \Frac{4 E_1}{\delta^2 K_*^2}=\Frac{50\sigma^2 E^3_1\varepsilon^2\mu^2}{m_*^6 \gamma^2\left(\delta K_*\right)^{4(n+1)}}\cr}$$ hence, the assumption $(A_1)$ yields $$\left\vert\left\vert{\cal R}_A\right\vert\right\vert_{(1-3\delta)K}\leq\Frac{55 \sigma^2 E^3_1\varepsilon^2\mu^2}{m_*^6 \gamma^2\left(\delta K_*\right)^{4(n+1)} }$$ \medskip We should also estimate the variations of the frequency ($\omega -{\widetilde{\omega}}$) and the exponent ($\lambda -{\widetilde{\lambda}}$) : (i) Since $1+{\widetilde{\eta}}=\left( 1+\varepsilon\mu\eta_A\right) (1+\eta )$, with $\varepsilon\mu\left\vert\eta_A\right\vert\leq\Frac{4\sigma E_1 \varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2n+1}}$ and $\Frac{1}{2}< 1+\eta <\Frac{3}{2}$, we find $$1+\eta -\Frac{3}{2}\Frac{4\sigma E_1\varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2n+1}}< 1+{\widetilde{\eta}}<1+\eta +\Frac{3}{2}\Frac{4\sigma E_1 \varepsilon\mu}{m_*^3\gamma\left(\delta K_*\right)^{2n+1}}$$ which gives the bounds on the variation of the frequency in the iterative lemma A.5. (ii) Since ${\widetilde{\lambda}}=\lambda +\mu{\overline{\alpha^A_{11}}}$ with $\left\vert\left\vert\alpha_{11}(\varphi )\right\vert\right\vert_K\! =\left\vert \left\vert\partial_{su}g(0,\varphi ,0, 0)\right\vert\right\vert_K\! <\Frac{E_1} {K^2}$ and $$\left\vert\left\vert\alpha^A_{11}-\alpha_{11}\right\vert\right\vert_{(1-2 \delta)K}={{1}\over{\varepsilon}}\left\vert\left\vert\partial_{su}\left\{\chi_A , H\right\}(0,\varphi ,0,0)\right\vert\right\vert_{(1-2\delta )K}\leq{{6\sigma E_1^2 (1-2\delta )^{-2}\mu}\over{m_*^3\gamma\delta^{2(n+1)} K_*^{2(n+2)}}}$$ the assumption $(A_1)$ and $\left\vert\left\vert\alpha^A_{11}\right \vert\right\vert_{(1-2\delta)K}\leq\left\vert\left\vert\alpha_{11}\right\vert \right\vert_{(1-2\delta)K}+\left\vert\left\vert\alpha^A_{11}-\alpha_{11}\right \vert\right\vert_{(1-2\delta)K}$ allows to write $$\left\vert\left\vert\alpha^A_{11}\right\vert \right\vert_{(1-2\delta )K}\leq\Frac{E_1}{K^2}+\Frac{6\sigma E_1^2\mu (1-2\delta )^{-2}}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}K^2}\leq\Frac{7\sigma E_1^2} {m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}K^2}$$ hence we find the lower bound on the exponent in the iterative lemma A.5. \medskip Now, we look at the second transformation. First, we have ${\widetilde{\omega}}.\partial_\varphi X_{11}=\alpha^A_{11} - {\overline{\alpha^A_{11}}}$ then the above estimate of $\left\vert\left\vert \alpha^A_{11}\right\vert\right\vert_{(1-2\delta )K}$ and the lemma A.3. gives : $$\left\vert\left\vert X_{11}\right\vert\right\vert_{(1-3\delta)K}\leq\Frac{7 \sigma^2 E_1^2}{m_*^4\gamma^2\left(\delta K_*\right)^{2(2n+1)} K^2}\ {\rm and}\ \left\vert\left\vert\partial_\varphi X_{11}\right\vert\right\vert_{(1-3\delta)K} \leq\Frac{7\sigma^2 E_1^2}{m_*^4\gamma^2\left(\delta K_*\right)^{4n+3}K^2} \eqno(A)$$ In the same way, we have ${\widetilde{\omega}}.\partial_\varphi Y_0 =\beta^A_0$ and $\left\vert\left\vert\beta^A_0\right\vert\right\vert_{(1-2\delta )K}\leq\Frac{7\sigma E_1^2}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}K}$ which yields~: $$\left\vert\left\vert Y_0\right\vert\right\vert_{(1-3\delta)K}\leq\Frac{7 \sigma^2 E_1^2}{m_*^4\gamma^2 \left(\delta K_*\right)^{2(2n+1)}K} \ {\rm and}\ \left\vert\left\vert\partial_\varphi Y_0\right\vert\right\vert_{(1-3\delta )K} \leq\Frac{7\sigma^2 E_1^2}{m_*^4\gamma^2\left(\delta K_*\right)^{4n+3}K}\eqno(B) $$ Now, $\partial^-_{\varepsilon{\widetilde{\lambda}},{\widetilde{\omega}}} X_{10}=-\alpha^A_{10}$ and $\left\vert\left\vert\alpha^A_{10}\right\vert \right\vert_{(1-3\delta )K}\leq\Frac{7\sigma E_1^2}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}\delta K}$ in the lemma A.4. give $$\left\vert\left\vert X_{10}\right\vert\right\vert_{(1-4\delta)K}\leq\Frac{7 \sigma^2 E_1^2}{m_*^4\varepsilon\gamma\left(\delta K_*\right)^{3n+2}\delta^2 K}\ {\rm and}\ \left\vert\left\vert\partial_\varphi X_{10}\right\vert \right\vert_{(1-4\delta) K}\leq\Frac{7\sigma^2 E_1^2}{m_*^4\varepsilon\gamma\left(\delta K_*\right)^{3(n+1)}\delta^2 K}\eqno(C)$$ the same reasonings give also these estimates for $\left\vert\left\vert X_{01} \right\vert\right\vert_{(1-4\delta )K}$ and $\left\vert\left\vert \partial_\varphi X_{01}\right\vert\right\vert_{(1-4\delta )K}$. We should also estimate $\partial_{10} X_{10}$ and $\partial_{10} X_{01}$, with $$X_{10}(\varphi ,s)={\displaystyle\sum_{k\in\N}}(k+1) X_{10}^{(k)}(\varphi )s^k {\rm and}\ \left\vert\left\vert X_{10}^{(k)}\right\vert\right\vert_{(1-4\delta ) K}\leq{{\left\vert\left\vert X_{10}\right\vert\right\vert_{(1-4\delta )K}}\over {((1-4\delta )K)^k}}$$ and the analogous formulas for $X_{01}$, we find : $$\left\vert\left\vert\partial_{10} X_{01}\right\vert\right\vert_{(1-5\delta)K} \leq\Frac{7\sigma^2 E_1^2}{m_*^4\varepsilon\gamma\left(\delta K_*\right)^{3n+2}\delta^4 K} \ {\rm and}\ \left\vert\left\vert\partial_{10} X_{10}\right\vert\right\vert_{(1- 5\delta)K}\leq\Frac{7\sigma^2 E_1^2}{m_*^4\varepsilon\gamma\left(\delta K_*\right)^{3n+2} \delta^4 K}$$ $\!\!\!\!\!$ Then $\partial^-_{\varepsilon{\widetilde{\lambda}},{\widetilde{\omega}}} X_{21} (\varphi ,s) =-\alpha^A_{21}(\varphi ,s)+\varepsilon\left( b_A (\varphi ,s,0). \partial_\varphi X_{10}(\varphi ,s)\right) -2\varepsilon c_A (\varphi ,s,0) \partial_{1,0} X_{10}(\varphi ,s)={\cal D}$, we have $$\left\vert\left\vert\alpha^A_{10}\right\vert \right\vert_{(1-3\delta )K}\leq\Frac{7\sigma E_1^2}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}\delta K}$$ and $\left\vert\left\vert b_A (\varphi ,s,0)\right\vert\right\vert_{(1-3\delta )K}\leq\left\vert\left\vert b(\varphi ,s,0)\right\vert\right\vert_{(1-3\delta ) K}+\left\vert\left\vert b_A (\varphi ,s,0)-b (\varphi ,s,0)\right\vert\right \vert_{(1-3\delta )K}$ with $$\left\vert\left\vert b(\varphi ,s,0)\right\vert\right\vert_{(1-3\delta )K}\! \leq\!\left\vert\left\vert{\displaystyle\sum_{k\geq 1}}H_{1k1}(\varphi ) s^{k-1} \right\vert\right\vert_{(1-3\delta )K}\!\!\!\!\!\leq\!{\displaystyle\sum_{k\geq 1}}\Frac{6 E_1}{5((1-2\delta )K)^{k+2}}((1-3\delta )K)^{k-1}\!\!\!\leq\!\Frac {6 E_1}{5(1-2\delta )^2\delta K^3}$$ where we write the expansion $H(I,\varphi ,s,u)={\displaystyle\sum_{k\in\N^n}} {\displaystyle\sum_{(l,m)\in\N^2}} H_{klm}(\varphi ) I^k s^l u^m$ ; in the same way $$\left\vert\left\vert b_A (\varphi ,s,0)-b(\varphi ,s,0)\right\vert\right \vert_{(1-3\delta )K}\!\leq\!\Frac{1}{(1-2\delta )^2\delta K^3}\Frac{6\sigma E_1^2\mu}{ m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}}$$ Thus, under the assumption $(A_1)$ and $\delta <{{1}\over{12}}$, we can write $$\left\vert\left\vert b_A (\varphi ,s,0)\right\vert\right\vert_{(1-3\delta )K} \!\leq\!\Frac{6 E_1}{(1-2\delta )^2\delta K^3}\left(\Frac{1}{5}+\Frac{\sigma E_1\mu}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}}\right)\leq\Frac{2 E_1}{ \delta K^3}$$ hence $\left\vert\left\vert b_A (\varphi ,s,0).\partial_\varphi X_{10}(\varphi , s)\right\vert\right\vert_{(1-4\delta )K}\leq\Frac{14\sigma^2 E_1^3}{m_*^4 \varepsilon\gamma\left(\delta K_*\right)^{3n+4}\delta^2 K^3}$ For the third term of ${\cal D}$, we have : $$\left\vert\left\vert c(\varphi ,s,0)\right\vert\right\vert_{(1-3\delta )K}\! \leq\!\left\vert\left\vert{\displaystyle\sum_{k\geq 2}}H_{0k2}(\varphi ) s^{k-2} \right\vert\right\vert_{(1-3\delta )K}\!\!\!\!\!\leq\!{\displaystyle\sum_{k\geq 2}}\Frac{6 E_1}{5((1-2\delta )K)^{k+2}}((1-3\delta )K)^{k-2}\!\!\!\leq\!\Frac {6 E_1}{5(1-2\delta )^3\delta K^4}$$ and $\left\vert\left\vert c_A (\varphi ,s,0)-c(\varphi ,s,0)\right\vert\right \vert_{(1-3\delta )K}\!\leq\!\Frac{1}{(1-2\delta )^3\delta K^4}\Frac{6\sigma E_1^2\mu}{ m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}}$. Under the assumption $(A_1)$ and $\delta <{{1}\over{12}}$, we can write $$\left\vert\left\vert c_A (\varphi ,s,0)\right\vert\right\vert_{(1-3\delta )K} \!\leq\!\Frac{6 E_1}{(1-2\delta )^3\delta K^4}\left(\Frac{1}{5}+\Frac{\sigma E_1\mu}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}}\right)\leq\Frac{3 E_1}{ \delta K^4}$$ hence $\left\vert\left\vert c_A (\varphi ,s,0).\partial_{10} X_{10}(\varphi ,s) \right\vert\right\vert_{(1-5\delta )K}\leq\Frac{21\sigma^2 E_1^3}{m_*^4 \varepsilon\gamma\left(\delta K_*\right)^{3n+4}\delta^3 K^3}$ Finally, we obtain $\left\vert\left\vert{\cal D}(\varphi ,s)\right\vert\right \vert_{(1-5\delta )K}\leq\Frac{63\sigma^2 E_1^3}{m_*^4\gamma\left(\delta K_* \right)^{3n+4}\delta^3 K^3}$ and the lemma A.4. gives $$\left\vert\left\vert X_{21}\right\vert\right\vert_{(1-6\delta)K}\leq\Frac{63 \sigma^3 E_1^3}{m_*^5\varepsilon\gamma\left(\delta K_*\right)^{4(n+1)}\delta^4 K^3}\ {\rm and}\ \left\vert\left\vert\partial_\varphi X_{21}\right\vert\right \vert_{(1-6\delta )K}\leq\Frac{63\sigma^3 E_1^3}{m_*^5\varepsilon\gamma\left( \delta K_*\right)^{4n+5}\delta^4 K^3}\eqno(D)$$ the same reasonings give also these estimates for $\left\vert\left\vert X_{12} \right\vert\right\vert_{(1-6\delta )K}$ and $\left\vert\left\vert \partial_\varphi X_{12}\right\vert\right\vert_{(1-6\delta )K}$. $${\rm In\ \!the\ \! same\ \! way}\ \partial^-_{\varepsilon{\widetilde{\lambda}}, {\widetilde{\omega}}}Y_{10}(\varphi ,s) =-\beta^A_{10}(\varphi ,s)+2\varepsilon \left( d_A (\varphi ,s,0).\partial_\varphi X_{10}(\varphi ,s)\right) -\varepsilon b_A (\varphi ,s,0)\partial_{1,0} X_{10}(\varphi ,s)={\cal E},$$ we have $\left\vert\left\vert\beta^A_{10}\right\vert\right\vert_{(1-3\delta )K}\leq \Frac{7\sigma E_1^2}{m_*^3\gamma\left(\delta K_*\right)^{2(n+1)}\delta K^2}$ and $$\left\vert\left\vert d_A (\varphi ,s,0)\right\vert\right\vert_{(1-2\delta ) K}\leq\left\vert\left\vert d(\varphi ,s,0)\right\vert\right\vert_{(1-2\delta )K} +\left\vert\left\vert d_A (\varphi ,s,0)-d (\varphi ,s,0)\right\vert\right \vert_{(1-2\delta )K}$$ $$\eqalign{{\rm with}\ \left\vert\left\vert d_A (\varphi ,s,u)-d (\varphi ,s,u) \right\vert\right\vert_{(1-3\delta )K}\leq{{1}\over{2\varepsilon }}\left\vert \left\vert\partial^2_{II}\left\{\chi_A ,H\right\}(0,\varphi ,s,u) \right\vert\right\vert_{(1-2\delta )K}\leq &\Frac{12\sigma E_1^2 (1-2\delta )^{-2} \mu}{m_*^3\gamma\delta^{2(n+1)}K_*^{2(n+2)}}\cr \leq &\Frac{18\sigma E_1^2\mu}{m_*^3\gamma\delta^{2(n+1)}K_*^{2(n+2)}}\cr}$$ and $\left\vert\left\vert d(\varphi ,s,u)\right\vert\right\vert_{(1-3\delta )K}\! \leq\!\Frac{1}{{\rm p}}$ ; the assumption $(A_2)$ yields $\left\vert\left\vert d_A (\varphi , s,u)\right\vert\right\vert_{(1-2\delta )K}<\Frac{1}{m_*}$, hence $$\left\vert\left\vert d_A (\varphi ,s,0).\partial_\varphi X_{10}(\varphi ,s) \right\vert\right\vert_{(1-4\delta )K}\leq\Frac{7\sigma^2 E_1^2}{m_*^5 \varepsilon\gamma\left(\delta K_*\right)^{3(n+1)}\delta^2 K}$$ then $\left\vert\left\vert b_A (\varphi ,s,0)\right\vert\right\vert_{(1-3\delta )K}\!\leq\!\Frac{2 E_1}{\delta K^3}$ yields $$\left\vert\left\vert b_A (\varphi ,s,0).\partial_{10} X_{10}(\varphi ,s)\right \vert\right\vert_{(1-5\delta )K}\leq\Frac{14\sigma^2 E_1^3}{m_*^4\varepsilon \gamma\left(\delta K_*\right)^{3n+4}\delta^3 K^2}$$ and finally, we obtain $\left\vert\left\vert{\cal E}(\varphi ,s)\right\vert\right \vert_{(1-5\delta )K}\leq\Frac{35\sigma^2 E_1^3}{m_*^5\gamma\left(\delta K_* \right)^{3n+4}\delta^3 K^2}$ and the lemma A.4. gives $$\left\vert\left\vert Y_{10}\right\vert\right\vert_{(1-6\delta)K}\leq\Frac{35 \sigma^3 E_1^3}{m_*^6\varepsilon\gamma\left(\delta K_*\right)^{4(n+1)}\delta^4 K^2}\ {\rm and}\ \left\vert\left\vert\partial_\varphi Y_{10}\right\vert\right \vert_{(1-6\delta )K}\leq\Frac{35\sigma^3 E_1^3}{m_*^6\varepsilon\gamma\left( \delta K_*\right)^{4n+5}\delta^4 K^2}\eqno(E)$$ the same reasonings give also these estimates for $\left\vert\left\vert Y_{01} \right\vert\right\vert_{(1-6\delta )K}$ and $\left\vert\left\vert \partial_\varphi Y_{01}\right\vert\right\vert_{(1-6\delta )K}$. The assumption $(3)$ of the main result allows to write $$\left\vert\left\vert\chi_B\right\vert\right\vert_{(1-6\delta)K}\leq\Frac{14 \sigma^3 E_1^3\varepsilon\mu}{m_*^6\gamma\delta^4\left(\delta K_*\right)^{4(n+1) }}\left(\Frac{15}{\varepsilon}+\Frac{1}{\gamma}\right)\leq\Frac{21\sigma^3 E_1^3 \varepsilon\mu}{m_*^6\gamma^2\delta^4\left(\delta K_*\right)^{4(n+1)}}$$ and the assumption $(A_1)$ gives $$\left\vert\left\vert\partial\chi_B\right\vert\right\vert_{(1-7\delta)K}\leq \delta K\Longrightarrow V_{(1-9\delta )K}\subset{\cal T}_B\left( V_{(1-8\delta ) K}\right)\subset V_{(1-7\delta )K}$$ \medskip To estimate the size of the remainder ${\widetilde{\cal R}}$, we write in the new variables : $$\left\vert\left\vert\left\{\chi_B,g_A\right\}\right\vert\right\vert_{(1-9\delta )K}\leq\Frac{\left\vert\left\vert\chi_B\right\vert\right\vert_{(1-8\delta )K} \left\vert\left\vert g_A\right\vert\right\vert_{(1-8\delta )K}}{\left(\delta K_* \right)^2}\leq\Frac{1050\sigma^4 E_1^5}{m_*^9\gamma^3\delta^5\left(\delta K_* \right)^{6n+8}}\Frac{\varepsilon^2\mu^2}{\gamma^3}$$ the use of second order Taylor formulas along solutions of the system linked to $\chi_B$ gives $$\eqalign{\left\vert\left\vert{\cal H}\!\circ\!{\cal T}_A\!\circ\!{\cal T}_B - {\cal H}\!\circ\!{\cal T}_A -\!\left\{\chi_B ,{\cal H}\!\circ\!{\cal T}_A\right \}\right\vert\right\vert_{(\!1\!-\! 10\delta\!)\!K}\!\leq &\!\Frac{1}{2}\left \vert \left\vert\left\{\chi_B ,\left\{\chi_B ,{\cal H}\!\circ\!{\cal T}_A\right \}\right\}\right\vert\right\vert_{(\!1\!-\! 9\delta\! )\!K}\!\cr \leq &\!\Frac{4\left\vert\left\vert\chi_B\right\vert\right\vert^2_{(\!1\!-\! 8 \delta\!)\!K}\!\left\vert\left\vert{\cal H}\!\circ\!{\cal T}_A\right\vert\right \vert_{(\!1\!-\!8\delta\! )\!K}}{\delta^4 K^4_*}\cr \leq &\left(\Frac{42\sigma^3 E_1^3}{m_*^6\delta^4\left(\delta K_*\right)^{4n+6}} \right)^2\Frac{\varepsilon^2\mu^2}{\gamma^4}E_1\cr}$$ and the assumption $(A_1)$ yields $\left\vert\left\vert\left\{\chi_B ,{\cal R}_A \right\}\right\vert\right\vert_{(1-9\delta )K}\leq\left\vert\left\vert {\cal R}_A\right\vert\right\vert_{(1-8\delta )K}\left(\delta K_*\right)^2\leq\Frac{55 \sigma^2 E^3_1}{m_*^6\left(\delta K_*\right)^{4n+2}}\Frac{\varepsilon^2\mu^2}{ \gamma^2}$ hence $$\left\vert\left\vert{\widetilde{\cal R}}\right\vert\right\vert_{(1-10\delta )K} \leq\left(\Frac{54\sigma^3 E^3_1}{m_*^6\delta^4\left(\delta K_*\right)^{4n+6}} \right)^2\left(\Frac{\varepsilon\mu}{\gamma^2}\right)^2 E_1$$ and we find the estimate of the iterative lemma A.5. \medskip We should specify the variations of the Hessian matrix. We have computed previously the bound : $$\left\vert\left\vert d_A (\varphi ,s,u)-d (\varphi ,s,u)\right\vert\right \vert_{(1-8\delta )K}\leq\Frac{18\sigma E_1^2\mu}{m_*^3\gamma\delta^{2(n+1)} K_*^{2(n+2)}}$$ $$\eqalign{{\rm and}\ \left\vert\left\vert{\widetilde{d}} (\varphi ,s,u)-d_A (\varphi ,s,u)\right\vert\right\vert_{(1-9\delta )K}\leq &{{1}\over{2\varepsilon }}\left\vert\left\vert \partial^2_{II}\left\{\chi_B ,H_A\right\}(0,\varphi ,s,u)\right\vert\right \vert_{(1-9\delta )K}\cr \leq &\Frac{2\left\vert\left\vert\chi_B\right\vert\right \vert_{(1-8\delta )K}\left\vert\left\vert H_A\right\vert\right\vert_{(1-8\delta )K}}{\varepsilon (1-9\delta )^2\delta^2 K_*^4}\cr}$$ With $H_A -H=\varepsilon\mu\left( g- g_A\right) +\left\{\chi_A ,H\right\}$, we can write : $$\eqalign{\left\vert\left\vert H_A -H\right\vert\right\vert_{(1-8\delta )K}\leq &\varepsilon\mu\left\vert\left\vert g_A -g\right\vert\right\vert_{(1-8\delta )K} +\left\vert\left\vert\left\{\chi_A ,H\right\}\right\vert\right\vert_{(1-8\delta )K}\cr \leq &\left[ 3\left( 1+\Frac{2}{\delta}\right) +1\right]\left\vert\left\vert \left\{\chi_A ,H\right\}\right\vert\right\vert_{(1-8\delta )K}\cr \leq &\Frac{38\sigma E_1^2\varepsilon\mu}{m_*^3\gamma\delta \left(\delta K_*\right)^{2(n+1)}}\Longrightarrow\left\vert\left\vert H_A\right \vert\right\vert_{(1-8\delta )K}\leq 3 E_1\ ({\rm using\ the\ assumption}\ A_1).\cr}$$ and we find $\left\vert\left\vert{\widetilde{d}} (\varphi ,s,u)-d_A (\varphi ,s, u)\right\vert\right\vert_{(1-9\delta )K}\leq{{2016\sigma^3 E_1^3\mu}\over{m_*^6 \gamma^2\delta^2\left(\delta K_*\right)^{4(n+2)}}}$ $$\Longrightarrow\left\vert\left\vert{\widetilde{d}}(\varphi ,s,u)-d (\varphi ,s, u)\right\vert\right\vert_{(1-9\delta )K}\leq\Frac{\left( 13\sigma E_1\right)^3 \mu}{m_*^6\gamma^2\delta^2\left(\delta K_*\right)^{4(n+2)}}$$ Hence, for all ${\bf V}\in\R^n$, we have $$\eqalign{\left\vert\left\vert\pmatrix{2 \varepsilon{\overline{{\widetilde{d}}}}(0,0)&{\widetilde{\omega}}\cr {\widetilde{\omega}}& 0\cr}{\bf V}\right\vert\right\vert\!\geq &\left\vert\left \vert\pmatrix{2\varepsilon {\overline{d}}(0,0)&\omega\cr\omega &0\cr}{\bf V} \right\vert\right\vert -\left\vert\left\vert\pmatrix{2\varepsilon{\overline{{ \widetilde{d}} -d}}(0,0)&\varepsilon\mu\eta_A\omega\cr \varepsilon\mu\eta_A\omega & 0\cr}{\bf V}\right\vert\right\vert\cr \geq &\left[\varepsilon{\rm p}-2\left(\varepsilon\left\vert\left\vert\left( {\widetilde{d}}-d\right) (\varphi ,0,0)\right\vert\right\vert +\varepsilon\mu \left\vert\eta_A\right\vert\vert 1+\eta\vert\left\vert\left\vert \omega_0\right\vert\right\vert\right)\right]\vert\vert{\bf V}\vert\vert\cr}$$ then,using $\nu_0\leq E_1$ and $1+\eta <3/2$, we find the estimate for ${\widetilde{{\rm p}}}$ given in the assumption $(A_2)$. In the same way, we have $\left\vert\left\vert{\widetilde{d}}(\varphi ,0,0) \right\vert\right\vert_{(1-9\delta )K}\leq{\rm p}^{-1}+{{\left( 13\sigma E_1 \right)^3\mu}\over{m_*^6\gamma^2\delta^2\left(\delta K_*\right)^{4(n+2)}}}\leq {\widetilde{{\rm p}}}^{-1}$ and this gives the second part of the assumption $(A_2)$. There remains to estimate the variations on the main part of the Hamiltonian, we have previously derived : $$\left\vert\left\vert H_A -H\right\vert\right\vert_{(1-8\delta )K}\leq\Frac{38 \sigma E_1^2\varepsilon\mu}{m_*^3\gamma\delta\left(\delta K_*\right)^{2(n+1)}}\ {\rm and}\ \left\vert\left\vert H_A\right\vert\right\vert_{(1-8\delta )K}\leq 3 E_1$$ in the same way ${\widetilde{H}}- H_A =\varepsilon\mu g_A +\left\{\chi_B ,H_A \right\}$ allows to write : $$\eqalign{\left\vert\left\vert{\widetilde{H}}- H_A\right\vert\right\vert_{(1-10 \delta )K}\leq &\varepsilon\mu\left\vert\left\vert g_A\right\vert\right\vert_{(1- 10\delta )K}+\left\vert\left\vert\left\{\chi_B ,H_A\right\}\right\vert\right \vert_{(1-10\delta )K}\cr \leq &\Frac{113\sigma^3 E_1^4\varepsilon\mu}{m_*^6\gamma^2\delta^4\left(\delta K_*\right)^{4n+6}}\cr}$$ and we find the bound given in the lemma A.5. \medskip \soulr{Appendix B : Results on the isoenergetic nondegeneracy condition} \medskip This second appendix is composed of three basic lemmas of linear algebra. \medskip \soulc{Lemma B.0.} {\it Let $Q$ be a matrix in ${\cal M}_{n+1}(\R )$ which can be written : $$Q=\pmatrix{M & F\cr{}^t\! F & 0\cr}\ {\it where}\ M\in{\cal S}_n(\R )\ {\it is\ a\ symmetric\ matrix\ and}\ F\in\R^n\backslash\{ 0\} .$$ We also assume that $\vert\vert F\vert\vert\geq{\tilde{p}}$ where ${\tilde{p}} \in ]0,1[$ and that $\vert\vert M\vert\vert\leq k{\tilde{p}}$ ($k>0$) for the operator norm induced by the Euclidean norm. Then, we have the following properties : (i) If, for all ${\tilde{v}}\in ^\perp\subset\R^n$ and $\lambda\in\R$, we have $\left\vert\left\vert M{\tilde{v}}+\lambda F\right\vert\right \vert\geq{\tilde{p}}\vert\vert{\tilde{v}}\vert\vert$ then : $${\it for\ all}\ v\in\R^{n+1}\!\! ,\ {\it we\ can\ write}\ \left\vert\left\vert Qv\right \vert\right\vert\geq p\vert\vert v\vert\vert\ {\it where}\ p={{{\tilde{p}}}\over {2(2+k)^2}}.$$ (ii) Conversely, if for all $v\in\R^{n+1}$ we have $\left\vert\left\vert Qv\right \vert\right\vert\geq p\vert\vert v\vert\vert$, then : $${\it for\ all}\ {\tilde{v}}\in ^\perp\subset\R^n {\it and}\ \lambda\in\R ,\ {\it we\ can\ write}\ \left\vert\left\vert M{\tilde{v}}+\lambda F\right\vert\right \vert\geq{p}\vert\vert{\tilde{v}}\vert\vert .$$ } \eject \souli{Proof} \medskip $\!\!\!\!$(i) Let $e_{n+1}$ be the last vector of the canonical basis and ${\cal F}$ the vectorial line generated by~$\pmatrix{F\cr 0\cr}$. Then all $v\in\R^{n+1}$ can be decomposed in $v=v'+v''+\alpha e_{n+1}$ with $v'\in{\cal F}^{\perp}\cap ^{\perp}$, $v''\in{\cal F}$ and $\alpha\in \R$. We can write $Qv =X+Y$ with $X=M(v'+v'')+\alpha F$ and $Y=(v'+v''.F) e_{n+1}$, then $v'\in{\cal F}^{\perp}$ and $v''\in{\cal F}$ imply $\vert\vert v''\vert\vert ={{\vert\vert Y\vert\vert}\over{\vert\vert F\vert\vert}}\leq{{ \vert\vert Y\vert\vert}\over{{\tilde{p}}}}$, our hypothesis (i) yields $$\vert\vert X\vert\vert =\vert\vert M(v'+v'')+\alpha F\vert\vert\geq{\tilde{p}}\vert\vert v'\vert\vert -k{\tilde{p}}\vert\vert v''\vert\vert\ \Longrightarrow\vert\vert v'\vert\vert\leq{{\vert\vert X\vert \vert}\over{{\tilde{p}}}}+k{{\vert \vert Y\vert\vert}\over{{\tilde{p}}}},$$ finally $\alpha F=X-M(v'+v'')$ implies $\vert\alpha\vert\leq (1+k)\left[{{\vert \vert X\vert\vert}\over{{\tilde{p}}}}+k{{\vert\vert Y\vert\vert}\over{{\tilde{p}}}}\right]$. Then we find the majoration $$\vert\vert v\vert\vert^2 =\vert\vert v'\vert\vert^2 +\vert\vert v''\vert \vert^2 +\alpha^2\leq{{2}\over{{\tilde{p}}^2}}(1+k)^2\left( 1+2(1+k)^2\right) \left(\vert\vert X\vert\vert^2 +\vert\vert Y\vert\vert^2\right)$$ and $\vert\vert Qv\vert\vert^2 =\vert\vert X\vert\vert^2 +\vert\vert Y\vert\vert^2$ gives the isoenergetic nondegeneracy condition with the value $p$ computed in the lemma. (ii) For ${\tilde{v}}\in ^\perp\subset\R^n$ and $\lambda\in\R$ : $$\left\vert\left\vert\pmatrix{M & F\cr{}^t\! F & 0\cr}\pmatrix{{\tilde{v}}\cr \lambda\cr}\right\vert\right\vert =\left\vert\left\vert M{\tilde{v}}+\lambda F \right\vert\right\vert\geq p\sqrt{\vert\vert{\tilde{v}}\vert\vert^2 +\lambda^2} \geq p\vert\vert{\tilde{v}}\vert\vert .$$ \medskip \soulc{Lemma B.1.} {\it Let $Q$ be a matrix in ${\cal M}_{n+1}(\R )$ which can be written : $$Q=\pmatrix{{\tilde{M}} & A & F\cr{}^t\! A & a & 0\cr{}^t\! F & 0 & 0\cr}\ {\it where}\ {\tilde{M}}\in{\cal S}_{n-1}(\R ),\ A\in\R^{n-1},\ a\in\R\backslash\{ 0\}\ {\it and}\ F\in\R^{n-1}\backslash\{0\} .$$ We also assume that $\vert\vert F\vert\vert\geq{\tilde{p}}$ where ${\tilde{p}} \in ]0,1[$ and that $\left\vert\left\vert\pmatrix{{\tilde{M}}& A\cr{}^t\! A & a \cr}\right\vert\right\vert\leq k{\tilde{p}}$ ($k>0$). Finally, the matrix $Q$ is supposed isoenergetically non-degenerate, i.e. : $${\it for\ all}\ {\tilde{v}}\in {\cal F}^\perp\subset\R^n\ {\it and}\ \lambda\in \R,\ {\it we\ have}\ \left\vert\left\vert\pmatrix{{\tilde{M}}& A\cr{}^t\! A & a\cr}{\tilde{v}}+\lambda\pmatrix{F\cr 0\cr}\right\vert\right\vert\geq{\tilde{p}} \vert\vert{\tilde{v}}\vert\vert$$ where ${\cal F}$ is the vectorial line generated by $\pmatrix{F\cr 0\cr}$. Then, the matrix $\pmatrix{{\tilde{M}}-{{1}\over{a}}A{}^t\! A &F\cr{}^t\! F & 0 \cr}$ is also isoenergetically non-degenerate : $${\it for\ all}\ v\in ^\perp\subset\R^{n-1}\ {\it and}\ l\in \R,\ {\it we\ have}\ \left\vert\left\vert ({\tilde{M}}-{{1}\over{a}}A{}^t\! A) v+lF\right\vert \right\vert\geq {\tilde{p}}\vert\vert v\vert\vert$$ and $\left\vert\left\vert{\tilde{M}}-{{1}\over{a}}A{}^t\! A\right\vert\right \vert\leq{{k^2{\tilde{p}}^2}\over{\vert a\vert}}$. Hence, the previous lemma B.0. shows that, for all ${\bar v}\in\R^n$, we have $$\left\vert\left\vert\pmatrix{{\tilde{M}}-{{1}\over{a}}A{}^t\! A & F\cr{}^t\! F & 0 \cr}{\bar v}\right\vert\right\vert\geq C_1\vert a\vert{\tilde{p}}\vert\vert {\bar v}\vert\vert\ {\it where}\ C_1\ {\it is\ a\ positive\ constant.}$$} \vskip0,3truecm \souli{Proof} First, we consider $v=\pmatrix{v_1\cr v_2\cr}\in\R^n$ with $v_1 .F=0$ and $v_2 =-{{(A.v_1)}\over{\vert a\vert}}\in\R$ then $$\left\vert\left\vert ({\tilde{M}}-{{1}\over{a}}A{}^t\! A) v_1 +lF\right\vert \right\vert = =\left\vert\left\vert\pmatrix{{\tilde{M}} & A\cr{}^t\! A & a \cr}\pmatrix{v_1\cr v_2\cr}+l\pmatrix{F\cr 0\cr}\right\vert\right\vert\geq {\tilde{p}}\vert\vert v\vert\vert ={\tilde{p}}\sqrt{\vert\vert v_1\vert\vert^2 +v_2^2} \geq{\tilde{p}}\vert\vert v_1\vert\vert ,$$ and, since $\sqrt{a^2 +\vert\vert A\vert\vert^2}\leq k{\tilde{p}}$, we can write : $$\left\vert\left\vert\left({\tilde{M}}-{{1}\over{a}}A{}^t\! A\right) v_1 \right\vert\right\vert =\left\vert\left\vert\pmatrix{{\tilde{M}} & A\cr{}^t\! A & a \cr}\pmatrix{v_1\cr v_2\cr}\right\vert\right\vert\leq k{\tilde{p}}\sqrt{\vert\vert v_1\vert\vert^2 +v_2^2}\leq{{k^2{\tilde{p}}^2}\over{\vert a\vert}}\vert\vert v_1\vert\vert .$$ \medskip \soulc{Lemma B.2.} {\it Let $Q$ be a matrix in ${\cal M}_{n+1}(\R )$ which can be written : $$Q=\pmatrix{M & F\cr{}^t\! F & 0\cr}\ {\it where}\ M\in{\cal S}_n (\R )\ {\it and} \ F\in\R^n\backslash\{ 0\} ,$$ and satisfies $\left\vert\left\vert Mv+lF\right\vert\right\vert\geq{\tilde{p}}\vert\vert v\vert\vert$ for all $v\in ^\perp\subset\R^n$ and $l\in \R$. Then, the matrix $Q_\varepsilon =\pmatrix{\sqrt{\varepsilon}M & F\cr{}^t\! F & 0 \cr}$ is also isoenergetically non-degenerate : $${\it for\ all}\ v\in ^\perp\subset\R^n\ {\it and}\ l\in \R,\ {\it we\ have} \ \left\vert\left\vert\sqrt{\varepsilon}Mv+lF\right\vert\right\vert\geq\sqrt{ \varepsilon}{\tilde{p}}\vert\vert v\vert\vert .$$ Hence, the previous lemma B.0. shows that, for all $v\in\R^n$, we have $$\left\vert\left\vert\pmatrix{\sqrt{\varepsilon} M & F\cr{}^t\! 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