\magnification=1200
%\magnification=1000
\voffset=-1.5truecm
\def\vru{\vrule height0.15truecm width0.2truecm depth0.15truecm}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\D{{\cal D}}
\def\E{{\cal E}}
\def\F{{\cal F}}
\def\I{{\cal I}}
\def\J{{\cal I}}
\def\IIN{I^{(N+1)}}
\def\K{{\cal K}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\N{{\cal N}}
\def\S{{\cal S}}
\def\T{{\cal T}}
\def\VN{V^{(N+1)}}
\def\R{{\cal R}}
\def\V{{\cal V}}
\def\W{{\cal W}}
\def\J{{\cal J}}
\def\bac{\backslash}
\def\cv{campo $\spadesuit$ vettoriale$\spadesuit$}
\def\der{\partial}
\def\derv{\der_{x_i}}
\def\derw{\der_{\bf x}}
\def\dis{\displaylines}
\def\2qquad{\qquad\qquad}
\def\3qquad{\2qquad\qquad}
\def\4qquad{\3qquad\qquad}
\def\5qquad{\4qquad\qquad}
\def\6qquad{\5qquad\qquad}
\def\7qquad{\6qquad\qquad}
\def\8qquad{\7qquad\qquad}
\def\9qquad{\8qquad\qquad}
\def\dov#1{\dot{\overline#1}}
\def\edif{equazioni differenziali}
\def\eequiv{\,\equiv\,}
\def\ab#1{V^{(#1)}_{\vett x^\prime}\left(\bolt+\seu,
\theta^\prime\right)-V^{(#1)}_{\vett x^\prime}\left(\bolt+
D^{-2}_{\bolomo^\prime}V^{(#1)}_{\vett x}\left(
\bolt+\seu,\theta^\prime\right),
\theta^\prime+D^{-2}_{\bolomo^\prime}V^{(#1)}_{x^\prime}\left(
\bolt+\seu,\theta^\prime\right)\right)}
\def\ac#1{D^{-2}_{\bolomo^\prime}\,V^{(#1)}_{\vett x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)}
\def\ad#1{D^2_{\bolomo^\prime}\vett v\,-\,V^{(#1)}_{\vett x^\prime}
\left(\bolt^\prime+\prv\right)}
\def\add#1{D^2_{\bolomo^\prime}\teu\,-\,V^{(#1)}_{\vett x^\prime}
\left(\bolt^\prime+\quu\right)}
\def\af#1{D^2_{\bolomo^\prime}\teu\,-\,V^{(#1)}_{\vett x^\prime}
\left(\bolt^\prime+\teu\right)}
\def\ag#1{D^2_{\bolomo}\pru\,-\,V^{(#1)}_{\vett x}\left(\bolt
+\seu\right)}
\def\ah#1{\vett E(\pv)\=D^2_{\bolomo^\prime}\pv\,-\,V^{(#1)}
_{\vett x^\prime}\left(\bolt^\prime+\pv\right)}
\def\aii#1{D^2_{\bolomo}\pv\,-\,V^{(#1)}
_{\vett x}\left(\bolt+\pv\right)}
\def\ai#1{D^{-2}_{\bolomo^\prime}\,V^{(#1)}_{x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)}
\def\al#1{D^{-2}_{\bolomo^\prime}\,V^{(#1)}_{\vett x}\left(
\bolt+\seu\,,\,\theta^\prime\right)}
\def\ist{\Delta_{\xi_\ini}}
\def\ust{\Delta_{\xi_o}}
\def\dst#1{\Delta^{#1}_{\xi}}
\def\tst#1{\Delta^{#1}_{\xi_{_*}}}
\def\qst#1{\Delta^{#1}_{\xi^\prime}}
\def\Dst#1{{\xi^{(#1)}}}
\def\Tst#1{{\xi^{(#1)}_{_*}}}
\def\ast#1{\Delta^{#1}_{\xi^{(#1)}_{_*}}}
\def\Qst#1{\Delta^{#1}_{\xi^{(#1)}}}
\def\csi*{\xi_{_*}}
%%%%%%%%%%%%%Errore, u, v  con M%%%%%%%%%%%%%%%%
\def\pu#1{\pv^{(#1)}}
\def\pru{\vett u}
\def\pv{\vett v}
\def\qp{quasi-periodiche}
\def\ru#1{\pru^{(#1)}}
\def\seu{\pru(\bolt)}
\def\teu{\vett u^\prime}
\def\quu{\teu(\bolt^\prime)}
\def\prv{\vett v({\bolt}^\prime)}
\def\uu#1{{\teu}^{(#1)}}
\def\Du{D^2_{\bolomo^\prime}\teu\,-\,V^{(M)}_{\vett x^\prime}
\left(\bolt^\prime+\quu)\right)}
\def\DM{D^2_{\bolomo^\prime}\teu\,-\,V^{(M)}_{\vett x^\prime}
\left(\bolt^\prime+\teu)\right)}
\def\D2u{D^2_{\bolomo}\pru\,-\,V^{(M-1)}_{\vett x}\left(\bolt
+\seu)\right)}
\def\vu{D^{-2}_{\bolomo^\prime}\,V^{(M)}_{\vett x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)}  
\def\Ev{D^2_{\bolomo^\prime}\vett v\,-\,V^{(M)}_{\vett x^\prime}
\left(\bolt^\prime+\prv)\right)}
\def\EEv{ V^{(M)}_{\vett x^\prime}\left(\bolt+\seu\,,
\,\theta^\prime\right)\,-\,V^{(M)}_{\vett x^\prime}\left(\bolt+
D^{-2}_{\bolomo^\prime}V^{(M)}_{\vett x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)\,,\,
\theta^\prime+D^{-2}_{\bolomo^\prime}V^{(M)}_{x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)\right)}
%%%%%%%%%%%%%%%%%%%Errore, v,u, con N+1%%%%%%%%%%%
\def\DuN{D^2_{\bolomo^\prime}\teu\,-\,V^{(N+1)}_{\vett x^\prime}
\left(\bolt^\prime+\quu)\right)}

\def\DN{D^2_{\bolomo^\prime}\teu\,-\,V^{(N+1)}_{\vett x^\prime}
\left(\bolt^\prime+\teu\right)}


\def\D2uN{D^2_{\bolomo}\pru\,-\,V^{(N)}_{\vett x}\left(\bolt+\seu\right)}

\def\EN{\vett E(\pv)\=D^2_{\bolomo^\prime}\pv\,-\,V^{(N+1)}_{\vett x^\prime}
\left(\bolt^\prime+\pv\right)}

\def\vuN{D^{-2}_{\bolomo^\prime}\,V^{(N+1)}_{\vett x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)}  

\def\EvN{D^2_{\bolomo^\prime}\vett v\,-\,V^{(N+1)}_{\vett x^\prime}
\left(\bolt^\prime+\prv\right)}

\def\EEvN{ V^{(N+1)}_{\vett x^\prime}\left(\bolt+\seu\,,
\,\theta^\prime\right)\,-\,V^{(N+1)}_{\vett x^\prime}\left(\bolt+
D^{-2}_{\bolomo^\prime}V^{(N+1)}_{\vett x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)\,,\,
\theta^\prime+D^{-2}_{\bolomo^\prime}V^{(N+1)}_{x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)\right)}

\def\vvn{ V^{(N+1)}_{\vett x^\prime}\left(\bolt+\seu\,,
\,\theta^\prime\right)}

%%%%%%%%%%%%%%%%%%%%%%%%%%idem come sopra per K %%%%%%%%%%%%%%%%
\def\Duk{D^2_{\bolomo^\prime}\teu\,-\,V^{(K)}_{\vett x^\prime}
\left(\bolt^\prime+\quu)\right)}
\def\vuu{D^{-2}_{\bolomo^\prime}\,V^{(k)}_{\vett x^\prime}\left(
\bolt+\seu\,,\,\theta^\prime\right)}
\def\Evk{D^2_{\bolomo^\prime}\vett v\,-\,V^{(k)}_{\vett x^\prime}
\left(\bolt^\prime+\prv\right)}

\def\xtr{\Delta^k_{\xi}}
\def\jtr{\Delta^k_{\xi_*}}
\def\ktr{\Delta^k_{\xi^\prime}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\espT#1{{\cal T}^{#1}}
\def\fb{\hfil\break}
\def\ggra{{\it g-gradiente }}
\def\ini{\infty}
\def\iin{\,\in\,}
\def\la{\lambda}
\def\lle{\,\le\,}
\def\limo#1{\lim_{#1\to+\infty}}
\def\moler#1{\left\vert{\rm #1}\right\vert}
\def\mw#1{\left\vert{\bf #1}\right\vert}
\def\mv#1{\vert{\bf #1}\vert}
\def\mov#1{\left\vert{#1}\right\vert}
\def\num#1#2{\eqno\hbox{(#1--#2)}}
\def\nv#1{\left\Vert\vett #1\right\Vert}
\def\om{\omega}
\def\ov#1{\overline{#1}}
\def\pr{^\prime}
\def\qpm{{\it quasi-periodiche massimali }}
\def\qpam{{\it quasi-periodica massimale }}
\def\ra{\rightarrow}
\def\rz{{\it razionalmente-indipendente }}
\def\ro#1#2{\sum_{i\in Z}\rho(#1_i,#2_i)w_i}
\def\sedif{sistema di equazioni differenziali}
\def\sied{sistema infinito di equazioni differenziali}
\def\spa{\spadesuit}
\def\th{\theta}
\def\ustr{\Delta_{\xi_o}}
\def\dstr{\Delta^N_{\xi}}
\def\tstr{\Delta^{N+1}_{\xi_{_*}}}
\def\qstr{\Delta^{N+1}_{\xi^\prime}}
\def\dtr{\Delta^M_{\xi}}
\def\ttr{\Delta^{M+1}_{\xi_{_*}}}
\def\qtr{\Delta^{M+1}_{\xi^\prime}}
\def\={\,=\,}
\def\a{\alpha}
\def\ao{\alpha_o}
\def\lr{{\rm L}}
\def\mr{{\rm m}}
\def\thp{\th^\prime}
\def\TTi{{\cal T}^\infty}
\def\u1{\vett u^{[k]}}
\def\vett#1{{\bf #1}}
\def\xp{x^\prime}
\font\mathbold=cmmib10
\def\bolt{\hbox{\mathbold\char 18}}
\def\bolom{\hbox{\mathbold\char 33}}
%\font\tenbm=msbm10
%\def\comp{\hbox{\tenbm\char 67}}
%\def\re{\hbox{\tenbm\char 82}}
%\def\tor{\hbox{\tenbm\char 84}}
%\def\nat{\hbox{\tenbm\char 90}}
\font\mathboldo=cmmib10 at 8pt
\def\bolto{\hbox{\mathboldo\char 18}}
\font\mathboldc=cmmib10 at 5pt
\def\boltc{\hbox{\mathboldc\char 18}}
\def\bolomo{\hbox{\mathboldo\char 33}}
\def\bolomc{\hbox{\mathboldc\char 33}}
\def\bolcnu{\hbox{\mathboldc\char 23}}
\def\bolonu{\hbox{\mathboldo\char 23}}
%\font\mathma=cmr10
%\def\matm{\hbox{\mathma\char 77}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    	Il seguente file di macros contiene:
%
%       (2) formattazione pagina;
%       (3) pagina iniziale: titolo, indirizzi e abstract;
%       (4) numerazione & riferimento automatico formule;
%       (5) riferimento automatico bibliografia;
%       (6) definizioni di simboli matematici;
%       (7) file NUOVISIMBOLI (Benettin)
%       (1000) ESEMPI di come usare queste macro si trovano in fondo al file.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%               (2) FORMATTAZIONE PAGINA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification=\magstep1
%\magnification=\magstep0
%\magnification=\magstephalf
\hoffset=0.cm
\lineskip=4pt\lineskiplimit=0.1pt      
%%%%%%%
%%%%%%%
\tolerance=1000
\hfuzz=1pt
\vsize=23.truecm
\voffset=0.truecm
\hsize=15.8 truecm
\hoffset=0.4 truecm
\normalbaselineskip=5.25mm
%\baselineskip=5.25mm
\baselineskip=14pt plus0.1pt minus0.1pt \parindent=19pt
\parskip=0.1pt plus1pt
\font\titlefont=cmbx10 scaled\magstep1
\font\sectionfont=cmbx10 scaled\magstep1
\font\subsectionfont=cmbx10
\font\small=cmr7
\font\grandibold=cmbx10 scaled\magstep1
\font\Grandibold=cmbx10 scaled\magstep2
\font\GRandibold=cmbx10 scaled\magstep3
\font\sectio=cmr10 scaled\magstep3
\font\section=cmr10 scaled\magstep2
\font\sectioo=cmr10 scaled\magstep1
% Le due seguenti istruzioni producono la numerazione delle pagine in alto
%\nopagenumbers
%\headline={\ifnum\pageno>1 {\hss\tenrm-\ \folio\ -\hss} \else {\hfill}\fi}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%                   FONTS %%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%define sym
\font\tenmsy=msym10
\font\sevenmsy=msym7
\font\fivemsy=msym5
\newfam\msyfam
\font\ninerm=cmr9
\font\ninei=cmmi9
\font\ninesy=cmsy9
\font\ninebf=cmbx9
\font\ninett=cmtt9
\font\ninesl=cmsl9
\font\nineit=cmti9
\font\eightrm=cmr8
\font\eighti=cmmi8
\font\eightsy=cmsy8
\font\eightbf=cmbx8
\font\eighttt=cmtt8
\font\eightsl=cmsl8
\font\eightit=cmti8
\font\sixrm=cmr6
\font\sixbf=cmbx6
\font\sixi=cmmi6
\font\sixsy=cmsy6
\def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type
\textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\eighti \scriptfont1=\sixi   \scriptscriptfont1=\fivei
\textfont2=\eightsy \scriptfont2=\sixsy   \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex   \scriptscriptfont3=\tenex
\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}%
\tt \ttglue=.5em plus.25em minus.15em
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselineskip=9pt
\let\sc=\sixrm  \let\big=\eightbig  \normalbaselines\rm
}
%
%%%%% constant subscript positions %%%%%
%
\fontdimen16\tensy=2.7pt
%\fontdimen13\tensy=2.7pt
\fontdimen13\tensy=4.3pt
\fontdimen17\tensy=2.7pt
\fontdimen14\tensy=4.3pt
\fontdimen18\tensy=4.3pt
\fontdimen16\eightsy=2.7pt
\fontdimen13\eightsy=4.3pt
\fontdimen17\eightsy=2.7pt
\fontdimen14\eightsy=4.3pt
\fontdimen18\eightsy=4.3pt
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% (4) (5) macros  for cross reference %%%%%%
%
%
%%%% Per far apparire il nome simbolico a sinistra inserire  \DRAFT
%%%% all'inizio del testo
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%  counters %%%
%%
\newcount\EQNcount \EQNcount=1
\newcount\CLAIMcount \CLAIMcount=1
\newcount\SECTIONcount \SECTIONcount=0
\newcount\SUBSECTIONcount \SUBSECTIONcount=1
%%
%% defining the symbolic value
%%
\def\NEWDEF #1,#2,#3 {
 \ifundefined{#1#2}
 \expandafter\xdef\csname #1#2\endcsname{#3}
 \else
 \write16{!! WARNING: doubly defined #1,#2}
 \fi
}
\def\actualnumber{\number\SECTIONcount}
\def\EQ(#1){\lmargin(#1)\eqno\tag(#1)}
\def\NR(#1){&\lmargin(#1)\tag(#1)\cr}  %the same as &\tag(xx)\cr in eqalignno
\def\tag(#1){\lmargin(#1)(\actualnumber.\number\EQNcount)
 \NEWDEF e,#1,(\actualnumber.\number\EQNcount)
\global\advance\EQNcount by 1\write16{ EQ \equ(#1) has symbol: #1  }}
\def\SECT(#1)#2\par{\lmargin(#1)
\SECTION #2\par
\NEWDEF s,#1,{\actualnumber}
}
\def\SUBSECT(#1)#2\par{\lmargin(#1)
\SUBSECTION #2\par
\NEWDEF s,#1,{\actualnumber.\number\SUBSECTIONcount}
}
%%%% the actual macro %%%%%%
\def\CLAIM #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\lmargin(#2)\bf #1~\actualnumber.\number\CLAIMcount.} {\sl #3}\par
\NEWDEF c,#2,{#1~\actualnumber.\number\CLAIMcount}
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\CLAIMNONR #1(#2) #3\par{
\vskip.1in\medbreak\noindent
{\lmargin(#2)\bf #1~#2} {\sl #3}\par
\global\advance\CLAIMcount by 1
\ifdim\lastskip<\medskipamount
\removelastskip\penalty55\medskip\fi}
\def\SECTION#1\par{\vskip0pt plus.3\vsize\penalty-75
    \vskip0pt plus -.3\vsize\bigskip\bigskip
    \global\advance\SECTIONcount by 1
    \immediate\write16{^^JSECTION \actualnumber:#1}\noindent
     {\sectionfont \actualnumber.\ #1}
    \EQNcount=1
    \CLAIMcount=1
    \SUBSECTIONcount=1
    \nobreak\smallskip\noindent}
\def\SECTIONNONR#1\par{\vskip0pt plus.3\vsize\penalty-75
    \vskip0pt plus -.3\vsize\bigskip\bigskip
    \global\advance\SECTIONcount by 1
    \immediate\write16{^^JSECTION:#1}\noindent
     {\sectionfont  #1}
     \EQNcount=1
     \CLAIMcount=1
     \SUBSECTIONcount=1
     \nobreak\smallskip\noindent}
\def\SUBSECTION#1\par{\vskip0pt plus.2\vsize\penalty-75
    \vskip0pt plus -.2\vsize\bigskip\bigskip
    \immediate\write16{SECTION:#1}\noindent{\subsectionfont
    \actualnumber.\number\SUBSECTIONcount.\ #1}
    \global\advance\SUBSECTIONcount by 1
    \nobreak\smallskip\noindent}
\def\SUBSECTIONNONR#1\par{\vskip0pt plus.2\vsize\penalty-75
    \vskip0pt plus -.2\vsize\bigskip\bigskip
    \immediate\write16{SECTION:#1}\noindent{\subsectionfont
     #1}
    \nobreak\smallskip\noindent}
%%
%%  referring to something
%%
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\equ(#1){\ifundefined{e#1}$\spadesuit$#1\else\csname e#1\endcsname\fi}
\def\clm(#1){\ifundefined{c#1}$\spadesuit$#1\else\csname c#1\endcsname\fi}
\def\sec(#1){\ifundefined{s#1}$\spadesuit$#1\else Section \csname s#1\endcsname\
   fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%   (3)  TITLE PAGE     %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\endarg=\par
\def\finish{\def\endarg{\par\endgroup}}
\def\start{\endarg\begingroup}
\def\getNORMAL#1{{#1}}
\def\TITLE{\beginTITLE\getTITLE}
 \def\beginTITLE{\start
   \titlefont\baselineskip=1.728
   \normalbaselineskip\rightskip=0pt plus1fil
   \noindent
   \def\endarg{\par\vskip.35in\endgroup}}
 \def\getTITLE{\getNORMAL}
\def\AUTHOR{\beginAUTHOR\getAUTHOR}
 \def\beginAUTHOR{\start
   \vskip .25in\rm\noindent\finish}
 \def\getAUTHOR{\getNORMAL}
\def\FROM{\beginFROM\getFROM}
 \def\beginFROM{\start\parskip=0pt\vskip\baselineskip
\def\finish{\def\endarg{\egroup\par\endgroup}}
  \vbox\bgroup\obeylines\eightpoint\sl\finish}
 \def\getFROM{\getNORMAL}
\def\ENDTITLE{\endarg}
\def\ABSTRACT#1\par{
\vskip 1in {\noindent\sectionfont Abstract.} #1 \par}
\def\ENDABSTRACT{\vfill\break}
\def\TODAY{\number\day~\ifcase\month\or January \or February \or March \or
April \or May \or June
\or July \or August \or September \or October \or November \or December \fi
\number\year\timecount=\number\time
\divide\timecount by 60
}
\newcount\timecount
\def\DRAFT{\def\lmargin(##1){\strut\vadjust{\kern-\strutdepth
\vtop to \strutdepth{
\baselineskip\strutdepth\vss\rlap{\kern-1.2 truecm\eightpoint{##1}}}}}
\font\footfont=cmti7
\footline={{
{\eightbf\actualnumber}:{\eightrm\folio}
\footfont \hfil File:\jobname, \TODAY,  \number\timecount h}}
}
%%%subitem an item in a vbox%%%%
\newbox\strutboxJPE
\setbox\strutboxJPE=\hbox{\strut}
\def\subitem#1#2\par{\vskip\baselineskip\vskip-\ht\strutboxJPE{\item{#1}#2}}
\gdef\strutdepth{\dp\strutbox}
\def\lmargin(#1){}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%    (5)   BIBLIOGRAPHY    %%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\period{\unskip.\spacefactor3000 { }}
%
% ...invisible stuff
%
\newbox\noboxJPE
\newbox\byboxJPE
\newbox\paperboxJPE
\newbox\yrboxJPE
\newbox\jourboxJPE
\newbox\pagesboxJPE
\newbox\volboxJPE
\newbox\preprintboxJPE
\newbox\toappearboxJPE
\newbox\bookboxJPE
\newbox\bybookboxJPE
\newbox\publisherboxJPE
\newbox\inprintboxJPE
\def\refclearJPE{
   \setbox\noboxJPE=\null             \gdef\isnoJPE{F}
   \setbox\byboxJPE=\null             \gdef\isbyJPE{F}
   \setbox\paperboxJPE=\null          \gdef\ispaperJPE{F}
   \setbox\yrboxJPE=\null             \gdef\isyrJPE{F}
   \setbox\jourboxJPE=\null           \gdef\isjourJPE{F}
   \setbox\pagesboxJPE=\null          \gdef\ispagesJPE{F}
   \setbox\volboxJPE=\null            \gdef\isvolJPE{F}
   \setbox\preprintboxJPE=\null       \gdef\ispreprintJPE{F}
   \setbox\toappearboxJPE=\null       \gdef\istoappearJPE{F}
   \setbox\inprintboxJPE=\null        \gdef\isinprintJPE{F}
   \setbox\bookboxJPE=\null           \gdef\isbookJPE{F}  \gdef\isinbookJPE{F}
     
   \setbox\bybookboxJPE=\null         \gdef\isbybookJPE{F}
   \setbox\publisherboxJPE=\null      \gdef\ispublisherJPE{F}
     
}
\def\ref{\refclearJPE\bgroup}
\def\no   {\egroup\gdef\isnoJPE{T}\setbox\noboxJPE=\hbox\bgroup}
\def\by   {\egroup\gdef\isbyJPE{T}\setbox\byboxJPE=\hbox\bgroup}
\def\paper{\egroup\gdef\ispaperJPE{T}\setbox\paperboxJPE=\hbox\bgroup}
\def\yr{\egroup\gdef\isyrJPE{T}\setbox\yrboxJPE=\hbox\bgroup}
\def\jour{\egroup\gdef\isjourJPE{T}\setbox\jourboxJPE=\hbox\bgroup}
\def\pages{\egroup\gdef\ispagesJPE{T}\setbox\pagesboxJPE=\hbox\bgroup}
\def\vol{\egroup\gdef\isvolJPE{T}\setbox\volboxJPE=\hbox\bgroup\bf}
\def\preprint{\egroup\gdef
\ispreprintJPE{T}\setbox\preprintboxJPE=\hbox\bgroup}
\def\toappear{\egroup\gdef
\istoappearJPE{T}\setbox\toappearboxJPE=\hbox\bgroup}
\def\inprint{\egroup\gdef
\isinprintJPE{T}\setbox\inprintboxJPE=\hbox\bgroup}
\def\book{\egroup\gdef\isbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\sl}
\def\publisher{\egroup\gdef
\ispublisherJPE{T}\setbox\publisherboxJPE=\hbox\bgroup}
\def\inbook{\egroup\gdef\isinbookJPE{T}\setbox\bookboxJPE=\hbox\bgroup\sl}
\def\bybook{\egroup\gdef\isbybookJPE{T}\setbox\bybookboxJPE=\hbox\bgroup}
\def\endref{\egroup \sfcode`.=1000
 \if T\isnoJPE  \item{[\unhbox\noboxJPE\unskip]}
     \else  \noindent    \fi
 \if T\isbyJPE    \unhbox\byboxJPE\unskip: \fi
 \if T\ispaperJPE \unhbox\paperboxJPE\unskip\period \fi
 \if T\isbookJPE {\it\unhbox\bookboxJPE\unskip}\if T\ispublisherJPE, \else.
\fi\fi
 \if T\isinbookJPE In {\it\unhbox\bookboxJPE\unskip}\if T\isbybookJPE,
\else\period \fi\fi
 \if T\isbybookJPE  (\unhbox\bybookboxJPE\unskip)\period \fi
 \if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if
T\isyrJPE \  \else\period \fi\fi\fi
 \if T\istoappearJPE (To appear)\period \fi
 \if T\ispreprintJPE Pre\-print\period \fi
 \if T\isjourJPE    \unhbox\jourboxJPE\unskip\ \fi
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%
%\SECTIONNONR References
%     
%\ref
% \no A
% \by Arnold, V.
% \book Ordinary Differential Equations
% \publisher MIT Press; Cambridge, MA
% \yr 1973
%\endref
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% .....
%\endref
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%\DRAFT

\centerline{\titlefont An infinite-dimensional extension of 
a Poincar\'e's} 
\centerline{\titlefont result concerning the continuation of periodic orbits}
\vskip2.0truecm
{\centerline{\bf Paolo Perfetti}}
\vskip0.2truecm
\centerline{\ninerm Dipartimento di Matematica II Universit{\accent"12 a} di 
Roma} 
\centerline{\ninerm via della Ricerca Scientifica 00133 Roma, Italy}

\vskip1.5truecm
\noindent
{\bf Abstract}{\ninerm \quad We study the existence of periodic 
solutions for  
the infinite-dimensional second order system $\ddot x=V_{x},\ 
x\in{\picctorus}^{{\integer}_+}.$ 
Using the Implicit-Function-Theorem we prove the 
existence of time-periodic solutions at \lq\lq high frequencies"; no \lq\lq 
smallness condition" on $V(x)$ is required. 
}
\vskip1.0truecm
\noindent
{\bf Contents}
\vskip0.2truecm
{\small{
\noindent
\mathhexbox2780 Introduction\dotfill p.2\quad

\noindent
\mathhexbox2781 Setup and definitions\dotfill p.3\quad
 
\noindent
\mathhexbox2782 The model\dotfill p.5\quad

\noindent
\mathhexbox2783 Main results\dotfill p.6\quad

\noindent
\mathhexbox2784 Proofs\dotfill p.9\quad

\noindent
\mathhexbox2785 Applications\dotfill p.14\quad

\noindent
Appendix 1: Existence of the dynamics\dotfill
p.17}}
\vfill\eject
%\baselineskip=9pt plus0.1pt minus0.1pt
\noindent{\bf \mathhexbox2780. Introduction}

\noindent
In this paper we study the following (infinite) second order ordinary 
differential equation  
$$
\ddot x=V_x(x), \quad x\in{\torus}^{{\integer}_+}\equiv
\bigotimes_{i=1}^\infty\torus_i 
\EQ(0.10)
$$ 
Over $\picctorus^{\integer_+}$ we introduce the 
topology induced by the quadratic metric, see (1.1), that makes it 
locally homeomorphic to $l_2$ (the space of the 
 sequences $\{a_i\}$ such that $\sum a^2_i<\infty)$ 
so we indicate $\picctorus^{\integer_+}$ as ${\cal T}_2.$ 
$V(x)$ is a real function (sometimes called 
{\it interaction} or {\it potential}) of class 
$C^2$ over ${\cal T}_2$ in the sense of Fr\'echet 
and we look for {\it time-periodic} solutions at \lq\lq high frequencies". 
The idea of studying solutions with 
\lq\lq high frequencies" has been already used in [CP] but while there we 
find solutions with infinite energy 
(besides {\it maximal almost-periodic}), here the energy of each solution is 
finite. 
This difference sets a severe restriction on the potentials $V(x)$ that can 
be considered here. Now in fact 
$V(x)$ must be a true function of an infinite (countable) set of variables 
\footnote{$^{(1)}$}{
in [CP] the system of equations $\ddot x_i=f_i(x)\ \ (i\ge1)$ is studied and 
what must be well defined is $f_i\colon\picctorus^{\integer_+}\to\picreal;$ 
the function of an infinite countable set of variables 
$V\colon\picctorus^{\integer_+}\to\picreal$ 
such that ${\partial V\over \partial x_i}(x)=f_i(x)$ may not even exist.}
and this causes 
a strong decay of the interaction among the variables $x_j$ and 
$x_{j^\prime}$ with  
$\vert j-j^\prime\vert$ large (the bigger is $\vert j-j^\prime\vert$ the 
smaller 
is the size of the interaction between $x_j$ and $x_{j^\prime}).$ Here in 
fact we 
will consider the following potentials well defined over ${\cal T}_2$ 
($g_j\colon\picctorus^{2L_o+1}
\to\picreal,$ $x^{(L_o)}\equiv(x_{j-L_o},\ldots,x_{j+L_o}),$ 
$L_o$ is a positive integer)
$$
V(x)=\sum_{j=1}^\infty g_j(x^{(L_o)}),\qquad\qquad 
V(x)=\sum_{i,j=1}^\infty e^{-\vert i-j\vert}(1-\cos(x_i-x_j)).
\EQ(00.0)
$$
The first $V(x)$ is called {\it short-range interaction} while the second 
one is a kind of {\it long-range interaction} and in both of them the decay 
announced before is transparent.
\noindent
Technically speaking we apply the Implicit-Function-Theorem and look for 
solutions that are {\it continuation} of {\it unperturbed solutions}. 
The unperturbed solutions arise when all the variables $x_i$ are decoupled 
and in this way the result contained in this paper can be 
considered an infinite-dimensional extension of a result of Poincar\'e, see
[Po],[SM],[M]. 

\noindent
>From a physical point of view the system 
\equ(0.10) represents a 
collection of identical rotators, whose mass can be different from zero and 
interacting 
via the potential $V(x).$ Roughly speaking we find solutions in which the 
first rotator $x_1$ rotates so fast that its motion \lq\lq almost decouples" 
from the others one that instead give rise to librations at
\lq\lq high frequency" and of definitively small amplitude (in the 
asymptotic regime the frequency of rotation of $x_1$ goes to 
infinity and the variables are decoupled). Hence 
we can consider the previous dynamic generated by \equ(0.10) 
as representative of a 
one-dimensional dynamical system on which acts an infinite-dimensional 
perturbation. This is quite clear writing the   
solutions as 
$$
\eqalign{x_1(t)&=[\omega t+u_1(\omega t)],\quad\quad \omega+
\omega{du_1\over d(\omega t)}\ne0\quad\forall t\cr
x_i(t)&=[u_i(\omega t)]\qquad\qquad (i\ge2)\cr
}
$$
$\omega$ ({\it the frequency}) is a real (\lq\lq large") number, 
$u_i:{\real}\rightarrow {\real}$ is a $2\pi$-periodic, smooth enough, 
function while 
$[\cdot]$ is the standard projection of 
${\real}$ over ${\torus}.$ Practically we find periodic, parameterized by 
$\omega,$ solutions such that when $\omega\to\infty$ they 
reduces to $x_1(t)=[ct+x^o_1], \ \ x_j(t)\equiv [x^o_j]
\ \ (j\ge2)$ and this will 
be the unperturbed solution. The introduction of the quadratic metric over 
$\picctorus^{\integer_+}$ makes quite natural the search of 
$\lq\lq l_2"$ solutions $x(t);$ briefly this means that 
$\sup_{t\in[0,2\pi]}\vert u_j(t)\vert\to0$ as $j\to\infty$ in such a way that 
$\sum_j\sup_{t\in[0,2\pi]}\vert u_j(t)\vert^2<\infty.$

Equations like \equ(0.10) often arise in the study of energy propagation 
(sound, heat) in crystal lattices (see [FSW]). In an other context the 
equation \equ(0.10) can be 
considered as the discretization of a partial differential equation and 
some features of the solutions of the partial equation can be 
recovered performing some suitable limits ({\it Lagrange limit} for example; 
see [G] and [Ga]) 
starting from the solutions of the discrete model. 

The existence of periodic solutions in systems of differential 
equations (ordinary as well as partial but both infinite), is an argument 
extensively studied. From the 
\lq\lq local methods" point of view (as it is the {\it continuation method} 
used in the present paper) we mention [AF], [AFS], [Ku], [Wa], [CW], [P\"o]. 
We just point out that most of the models studied in the given references 
involve the presence of small divisors while in our case such a kind of 
problem is absent. 
\vskip1.0truecm
\noindent
{\it Acknowledgements} 
It is a pleasure to thank Prof. Luigi Chierchia for having proposed to me the 
problem, for useful suggestions and advises.
\vskip2.0truecm
\SECTIONcount=1\EQNcount=1
\noindent
{{\bf\mathhexbox2781.  Setup and definitions}}

\noindent
Let ${\torus}^{{\integer}_+}$ denotes the Cartesian product of infinitely 
many copies of the one-dimensional flat torus 
$$
{\torus}^{{\integer}_+}
=\bigotimes_{i=1}^\infty\torus_i,\qquad
\torus_i=\torus=\real/2\pi\integer.
$$
On ${\torus}^{{\integer}_+}$ we introduce the distance between two points 
$[x],[x^\prime]\in\torus^{\integer_+}$
\footnote{$^{(2)}$}{$[x]\in\picctorus^{\integer_+}$ is the equivalence class 
mod.$2\pi$ of $x\in\picreal^{\integer_+}$
} 
$$
\rho([x],[x^\prime])=\bigl(\sum_{i=1}^\infty
\rho^2([x_i],[x^\prime_i])\bigl)^{1\over2},\qquad\quad
\rho([x_i],[x^\prime_i])=\inf_{k\in{\integer}}
\{\vert x_i-x^\prime_i+2\pi k\vert\}
$$ 
and consider the following metric and vector spaces respectively 
\footnote{$^{(3)}$}{For notational convenience we suppress the square 
parenthesis understanding $x\in\picctorus^{\integer_+}$ 
}
$$
{\cal T}_2\doteq\bigl\{x\in{\torus}^{{\integer}_+}\bigm\vert\sum_{i=1}^\infty
\rho^2(x_i)<\infty\bigl\},
\qquad{\it l}_2\doteq
\bigl\{y\in{\real}^{{\integer}_+}\bigm\vert\sum_{i=1}^\infty y_i^2
<\infty\bigl\}
$$ 
The space ${\cal T}_2\times
{\it l}_2$ is a complete metric with distance
$$
\rho\bigl((x,y),(x^\prime,y^\prime) 
\bigl)=
\bigl(\sum_{i=1}^\infty\rho^2(x_i,x^\prime_i)\bigl)^{1\over2}+
\bigl(\sum_{i=1}^\infty{(y_i-{y^\prime_i}})^2\bigl)^{1\over2}
\EQ(01.11)
$$
and denoting $x\equiv(x_1,\hat x)$ we define $S(\hat x^o,\delta)=\{x
\in{\cal T}_2\vert(\sum_{i=2}^\infty\rho^2(x_i,x^o_i))^{1\over2}
\le\delta,\ 0\le\delta<\infty\}$
\footnote{$^{(4)}$}{In the following we will 
sometimes write $\Vert\hat p\Vert$ which is $\rho((\hat0,\hat p),0)=
({\sum_{i=2}^\infty p_i^2})^{1\over2}$}.

\noindent
Let $V\colon{\cal T}_2\to{\real}$ be a continuous function 
$(V(x)\equiv V(x_1,x_2,x_3,\ldots)\equiv
V(x_1,\hat x))$ that satisfies 

\noindent
i) $V(x)\in C^2({\cal T}_2,\real),$ 
$\sup_{x\in{\cal T}_2}\Vert V_{xx}\Vert\doteq V_2<\infty$

\noindent
The second of i) implies that:

\noindent
1) $V_x$ a 
{\it uniform Lipschitz} gradient: 
$\Vert V_x(x)-V_x(x^\prime)\Vert^2\le L^2\rho^2(x,x^\prime)\ \forall 
x,x^\prime\in{\cal T}_2$ ($L$ is a constant). 
 
\noindent
2) $\{\sum_{i=1}^\infty
V^2_{x_i}(x)\}^{1\over2}\le a\rho(x,0)+b$ \quad $a$ and $b$ constants. 
\footnote{$^{(5)}$}{
Being $V(x)\in C^2({\cal T}_2,\picreal)$ there exists a 
continuous map $V_{xx}\colon{\cal T}_2\to{\cal L}(l_2)$ 
(${\cal L}$ is the space of linear bounded maps over 
$l_2$) defined by means of the expression 
$\Vert V_x(x+h)-V_x(x)-V_{xx}(x)h\Vert<\epsilon\Vert h\Vert$ for 
$\Vert h\Vert
<\delta_{\epsilon}.$ In the following we will write  
$\{\sum_{i=1}^\infty\sup_{S(\hat x^o,\delta)}
V^2_{x_i}(x)\}^{1\over2}\doteq V^1(x^o,\delta)$
} 

\vskip0.2truecm
\noindent
In this paper we will consider two kind of interactions 
$V\colon{\cal T}_2\to\real$: {\it \lq\lq short-range interaction"} and  
{\it \lq\lq long-range interaction"}. In the first case we have  
$$
V(x)=\sum_{j=1}^\infty g_j(x^{(L_o)})
$$
where $g_j\colon\torus^{2L_o+1}\to\real$ depends only on the variables 
$x_{j-L_o},\ldots, x_{j+L_o}$ ($L_o$ positive integer). In the second case the 
bound represented by $L_o$ is relaxed and each variable $x_i$ can interacts 
with every other variable without any constraint on the distance of their 
sites e.g.
$$
V(x)=\sum_{i,j=1}^\infty e^{-\vert i-j\vert}(1-\cos(x_i-x_j))
$$
\vskip0.5truecm
\noindent
{{\bf\mathhexbox2782. The model}}
\SECTIONcount=2\EQNcount=1

\noindent
Consider the following real hamiltonian 
$H\colon{\cal T}_2\times l_2\to\real$ 
$$
H(x,y)={1\over2} y^2-V(x)
$$ 
where $V$ satisfies i) of {\bf\mathhexbox2781}. 
As we look for {\it time-periodic} solutions of the Newton's equations 
$$
{d^2x_i\over dt^2}\equiv\ddot x_i={\partial V\over\partial x_i}\equiv V_{x_i},
\quad i\in{\integer}_+
\EQ(00.1)
$$  
\noindent
we write them as 
$$
\left\{\eqalign{x_1(t)&=[\omega t+u_1(\omega t)],\quad\quad \omega+
\omega{du_1\over d(\omega t)}\ne0\quad\forall t\cr
x_i(t)&=[u_i(\omega t)]\qquad\qquad (i\ge2)\cr
}\right.
\EQ(1.0)
$$
where $\omega\in{\real}$ is the {\it frequency}, 
$u_i:{\real}\rightarrow {\real}$ is a $2\pi$-periodic function and 
$[\cdot]$ is the standard projection of 
${\real}$ over ${\torus}$ (from now on this fact will be omitted). If we 
for a moment suppose (as it will be) that $u\in C^2(\torus,{\cal T}_2)$ 
then \equ(1.0) 
represents a one-dimensional torus $\torus_\omega$ {\it embedded} in the 
ambient-space ${\cal T}_2\times l_2$ i.e. 
$\torus_\omega=\{(x,y)\in{\cal T}_2\times l_2\vert x_1=
\tau+u_1(\tau),\ x_j=u_j(\tau),\ y_1=\omega(1+{du_1\over d\tau}),\ 
y_j=\omega{du_j\over d\tau},\ \ \tau\in\torus\}.$
Inserting \equ(1.0) into \equ(00.1) we get the system 
\footnote{$^{(6)}$}{We define $\tau=\omega t$}
$$
{{d^2u_i}\over d\tau^2}=\varepsilon V_{x_i}(\tau+u_1,\hat u)\quad 
i\in\integer_+,\qquad(\varepsilon\equiv\omega^{-2},\quad u\equiv(u_1,\hat u))
\EQ(1.2)
$$
and in order to apply the Implicit-Function-Theorem we look at 
\equ(1.2) 
as a system of equations of the first order and then we rewrite it as 
$$
\bigl(u_i\doteq Q_i,\quad {du_i\over d\tau}\doteq P_i,\quad
\dot Q_i={dQ_i\over d\tau},\quad\dot P_i={dP_i\over d\tau}
\bigl)
\Rightarrow
\left\{\eqalign
{
\dot Q_i&=P_i\qquad\cr
\dot P_i&=\varepsilon V_{x_i}(\tau+Q_1,\hat Q).\cr
}
\right. 
\EQ(1.3)
$$
Now \equ(1.3) can be viewed as a 
{\it non-autonomous hamiltonian} system with hamiltonian function  
$H={1\over2}P^2-\varepsilon V(\tau+Q_1,\hat Q)$ for which, of course, 
$\dot H\not\equiv0.$ Instead the function 
$$
G(\tau,P,Q)=P_1+{1\over2}P^2-\varepsilon V(\tau+Q_1,\hat Q)
\EQ(0.1)
$$ 
is a first integral for \equ(1.3). 

\noindent
The system \equ(1.3) is naturally defined over the phase-space
$\torus\times{\cal T}_2\times{\it l}_2\doteq{\cal M}$ which is a 
manifold (see [LA] for a treatment of 
infinite-dimensional manifolds). 
Its tangent space ${\cal M}_x$ at $x\doteq(\tau,P,Q)\in{\cal M}$  is 
$\real\times{\it l}_2\times{\it l}_2.$ 
Over ${\rm T}{\cal M}=\bigcup_{x\in{\cal M}}(x,{\cal M}_x)$ 
the {\it non-singular} (see [A]) 2-form $\omega^2\equiv\sum_{j=1}^\infty 
dP_j\wedge dQ_j-dH(\tau,P,Q)\wedge d\tau$ is defined
\footnote{$^{(7)}$}{\noindent 
$\omega^2=d\omega^1$ where 
$\omega^1\equiv\sum_{j=1}^\infty P_jdQ_j-H(\tau,P,Q)d\tau;$ 
at every point of ${\cal M}$ the 
tangent vector 
$\xi(\tau,P,Q)=\alpha(1,\varepsilon V_x(\tau+Q_1,\hat Q),P)$ 
($\alpha\in\picreal)$ is defined; 
$\xi(x)\in{\cal M}_x\ (x=(\tau,P,Q))$ and the following relation holds 
$\omega(\xi,\eta)=0\forall \eta\in{\cal M}_x.$ The integral curves, lying in 
${\cal M},$ of the field of directions associated to the set of 
$\xi-$vectors are the solutions of the system \equ(1.3).

\noindent
The manifold ${\cal M}$ is locally homemomorphic to 
$\picreal\times l_2\times l_2$ through a 
homeomorphism $\varphi\colon\M\to\picreal\times l_2\times l_2.$ 
A function $F\colon\M\to\picreal$ is said to be 
Fr\'echet differentiable if $F\circ\varphi^{-1}$ is 
Fr\'echet differentiable as a function defined over the Banach space 
$\picreal\times l_2\times l_2.$ 
}.
  
\vskip0.3truecm
\noindent
{\bf Definition}{\sl\  A {\it periodic-solution} of 
\equ(1.3) will be a $2\pi-$periodic map 
$(P,Q)\in C^1(\real,l_2\times{\cal T}_2)$.} 
\vskip0.2truecm
\noindent
Global existence and uniqueness for the Cauchy problem associated to 
\equ(1.3) are standard applications of the contraction techniques 
(see Appendix 1):

\noindent
{\bf Theorem 2.1}\ {\sl Let 
$V\colon{\cal T}_2\to{\real}$ be a continuous function  
that verifies the property i) in {\bf\mathhexbox2781}. 
Then there exists a unique 
solution $(P(t),Q(t))\in C^1(\real,{\it l}_2\times{\cal T}_2)$
of the Cauchy problem 
$$
\left\{\eqalign
{
\dot Q&=P\cr
\dot P&=\varepsilon V_x(\tau+Q_1,\hat Q)\qquad
\bigl(P(0),Q(0)\bigl)=\bigl(P^o,Q^o\bigl)\in{\it l}_2\times{\cal T}_2.\cr
}
\right. 
\EQ(1.4)
$$
}
\noindent
Remark

\noindent
In Appendix 1, besides the statement of Theorem 2.1 we will show that 
$(P,Q)$ are $C^1$ functions of $\varepsilon$ and of the initial data 
$(P^o,Q^o).$ This fact will be used in {\bf\mathhexbox2784} where it will 
proved the statement of the next Theorem  
\vskip0.75truecm
\SECTIONcount=3\EQNcount=1
\noindent
{{\bf\mathhexbox2783. Main Results}}
\vskip0.3truecm
\noindent
Following Poincar\'e and Lyapunov 
(see [Po], [LY], [SM] and references therein) 
we look for solutions that are {\it continuation} ($\varepsilon\ne 0$) of 
{\it periodic solutions} with $\varepsilon=0.$  
At $\varepsilon=0$ the most general solution of the system \equ(1.4) is

\centerline{$Q(\tau)=Q^o+P^o\tau,\quad P(\tau)=P^o$}

\noindent
(we denote it as 
$\gamma(\tau;P^o,Q^o,0)\doteq \bigl(\phi(\tau;P^o,Q^o,0),\psi(\tau;P^o,Q^o,0)
\bigl),$ the {\it flow} of the system, where $0$ refers 
to $\varepsilon=0$) and if we want it periodic it is necessary that  
$$
\left\{\eqalign
{
&\phi(2\pi;P^o,Q^o,0)-\phi(0;P^o,Q^o,0)=2\pi P^o\cr
&\psi(2\pi;P^o,Q^o,0)-\psi(0;P^o,Q^o,0)=0.\cr
}
\right.
\EQ(1.7)
$$
Now two facts are clear:
\noindent
i) the equations \equ(1.7) imply that $P^o_n\in\integer$ so that 
$2\pi P^o=0$ (mod.$2\pi$). We indicate with $\bar P^o_n$ a particular 
value of $P^o_n\in{\integer}.$

\noindent
ii) $P^o\in{\it l}_2,$ see \equ(1.4) and i) implies that there 
must exist an 
integer $N$ such that $\bar P^o_n=0$ 
when $n>N$ i.e.$\left\{\bar P^o_n\right\}=\bigl(\bar P^o_1,\bar P^o_2,
\bar P^o_3
\ldots,\bar P^o_N,0,0,\ldots\bigl).$ This fact imposes a severe restriction 
on the periodic solutions that can be continuated at $\varepsilon\ne0$.

\vskip0.3truecm\noindent
When $\varepsilon\ne0$ the system of equations \equ(1.7) becomes
$$
\eqalign
{
&\phi(2\pi;P^o,Q^o,\varepsilon)-\phi(0;P^o,Q^o,\varepsilon)=
\int_0^{2\pi}d\tau\,{\dot\phi}(\tau;P^o,Q^o,\varepsilon)
=\cr
&\qquad\qquad\qquad\qquad\qquad\qquad=\int_0^{2\pi}d\tau\psi
(\tau;P^o,Q^o,\varepsilon)=0(mod.2\pi)\cr
&{1\over\varepsilon}(\psi(2\pi;P^o,Q^o,\varepsilon)-\psi
(0;P^o,Q^o,\varepsilon))=
{1\over\varepsilon}\int_0^{2\pi}d\tau\,{\dot\psi}
(\tau;P^o,Q^o,\varepsilon)=\cr
&\qquad\qquad\qquad\qquad\qquad\qquad=\int_0^{2\pi}d\tau V_Q
(\tau+\phi_1,\hat\phi)=0\cr
}
\EQ(1.8)
$$
where $(P^o,Q^o)\equiv (Q(0),P(0))
=\gamma(0;P^o,Q^o,\varepsilon)$ is the 
initial datum of the flow with $\varepsilon\ne0.$

\noindent
Of course if we had a solution $(\bar P^o,\bar Q^o,
\bar \varepsilon)$ of \equ(1.8) 
$\gamma(\tau;\bar P^o,\bar Q^o,\bar \varepsilon)$ would be the looked for 
2$\pi$-periodic solution.

\noindent
Hence we want to view \equ(1.8) as a 
system of equations for the variables $(P^o,Q^o,\varepsilon)$ in 
the sense 
that there exist a real number $\varepsilon_o$ (sufficiently small) and 
two functions 
$(P^o(\varepsilon),Q^o(\varepsilon))$ such that  
$(P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)$ identically 
solve \equ(1.8) for any value $\varepsilon\in(-\varepsilon_o,\varepsilon_o)$ 
\footnote{$^{(8)}$}{
It may be useful to say something about the notations. When we write 
$Q^o$ we mean the variable $Q^o\in{\cal T}_2$ that can vary over 
some domain of initial data. Instead when we write $\bar Q$ 
we mean some fixed initial datum. Moreover we will write, sometimes, 
$Q^o=(Q^o_1,\hat Q^o).$ An analogous argument is valid for the 
variables $P^o.$
}.

\noindent
In order to formulate our main result we must introduce a linear map 
$\Gamma\colon l_2\times l_2\to l_2\times l_2$ whose entries are 
(see (4.10)):
$$
\eqalign{
&\Gamma_{2i-1,2i}=2\pi\ (i\ge2),\qquad
\Gamma_{2i,2j}=\int_0^{2\pi}d\theta \theta V_{\hat Q_i,\hat Q_j}
(\pm\theta\vert1+\bar P_1\vert,\hat{\bar Q})\ (i,j\ge2),\cr
&\Gamma_{2i,2j-1}=\int_0^{2\pi}d\theta V_{\hat Q_i,\hat Q_j}
(\theta,\hat{\bar Q})\ (i,j\ge2),\qquad 0\ \ {\rm (otherwise)}\cr
}
\EQ(nnd)
$$
(about the meaning $\bar P_1$ and $\hat{\bar Q}$ see Theorem 3.1).

\noindent
A critical point of the map $\hat Q^o\to
\int_0^{2\pi}d\tau V_{\hat Q}(\tau,\hat Q^o)$ is said to be 
{\it non-degenerate} iff the map $\Gamma$ has bounded inverse. 
The entries of the map $\Gamma$ are given by the 
derivatives of a suitable set of equations obtained after that \equ(1.8) 
has gone through some coordinate-transformations which will be explained 
in {\bf\mathhexbox2784}. We then have 

\noindent
{\bf Theorem 3.1} {\sl Let $(\bar Q,\bar P)\in{\cal T}_2\times l_2$ 
such that: 
$\bar P_1\in{\integer}\backslash\{-2,-1,0\}$, $\hat{\bar P}=0,$ 
$\hat{\bar Q}$ is a {\it non-degenerate} critical point of the map
$$
\hat Q^o\to\int_0^{2\pi}d\tau V_{\hat Q}
(\tau,\hat Q^o)
\EQ(2..)
$$ 
(where $V(x)$ satisfies i) of {\bf\mathhexbox2781}). 

\noindent
Then there exist an $\varepsilon_o>0$ and two 
functions $(P^o,Q^o)\colon(-\varepsilon_o,\varepsilon_o)\to l_2
\times{\cal T}_2,$ continuous with the first derivative in 
$\varepsilon=0,$ solutions of \equ(1.8), such that: 
$\gamma(\tau;P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)\in l_2
\times{\cal T}_2\ \forall \tau\in\real$ is a $2\pi-$periodic solution of 
\equ(1.3), $Q^o(0)=\bar Q, P^o(0)=\bar P,$ and 
$$
\max_{\tau\in[0,2\pi],p=0,1}\bigl\{\rho\bigl(
{d^p\over d\tau^p}\gamma(
\tau;P^o(\varepsilon),Q^o(\varepsilon),\varepsilon),
{d^p\over d\tau^p}\gamma(\tau;\bar P,\bar Q,0)
\bigl)\bigl\}\le C(\varepsilon,L)
\EQ(3..) 
$$
\footnote{$^{(9)}$}{Being $\phi\in{\cal T}_2,$ for $p=0$ the distance is 
given in \equ(01.11) while for $p=1$ it is the usual distance of 
$l_2\times l_2$}
}
with $C(\varepsilon,L)\to0$ as $\varepsilon\to 0.$


\noindent
Remarks

\noindent
i) The proof will be such that 
$P^o_1(\varepsilon)\equiv\bar P_1$ $Q^o_1(\varepsilon)\equiv\bar Q_1.$

\noindent
ii) It could be taken a slightly more general $\bigl(Q^o(0),P^o(0)\bigl)$ 
i.e. $\bar Q$ 
as in the statement of the theorem but $\bar P_1\in\integer
\backslash\{-2,-1,0\},$ $\bar P_i\in\integer\ 2\le i\le N,$ 
$\bar P_i=0\ i>N$ for some integer $N.$ 
However it can be easily shown that with a suitable canonical
change of coordinates the former initial datum can be reduced to that one 
given in the statement of the theorem. The proof will be not given but it is 
an easy matter in the framework of canonical mappings
\footnote{$^{(10)}$}{
Let's define an operator 
${\cal A}\colon l_2\times{\cal T}_2\to l_2\times{\cal T}_2,
\  {\cal A}(P,Q)=(P^\prime,Q^\prime)$ that satisfies the following requests: 
1) it is linear and bounded  
(over ${\cal T}_2$ the linearity is restricted to the vector sum)\ \ 
2) preserves the structure of ${\cal T}_2$ (i.e. $Q^\prime_i\in\picctorus$)\ \ 
3) preserves the 2-form $\omega=\sum_{j=1}^\infty dP_j\wedge dQ_j$\ \ 
4)${\cal A}(\bar P^o_1,0,\bar P^o_2,0,\bar P^o_3,\ldots,
\bar P^o_N,0,0,0,\ldots)^T=(\bar P^o_1,0,0,0,\ldots)^T$ ($T$ denotes the 
transposition). 
As a consequence we can take initial data for which $\hat P^o=0.$ 
}

\noindent
iii) $P_1^o$ is different from $-1$ 
because, otherwise, we would not have a one-dimensional torus {\it embedded} 
in 
the ambient space ${\cal T}_2\times l_2.$ Instead the values $-2,0$ must 
be avoided in order to have a well defined Poincar\'e-map (see 
(4.5) and in particular the equation for $q_1(\theta)$ of 
{\bf\mathhexbox2784}).

\noindent
iv) The inequality \equ(3..) means that 
$\gamma(\tau;P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)$ is a 
{\it continuation} of the solution with $\varepsilon=0$ and initial datum 
$(\bar P,\bar Q).$

\noindent
v) The statement of Theorem 3.1) resembles that one relative to 
the finite-dimensional theory but it is different in two points.
The first one is the necessity of avoiding the values $-2,0$ of $\bar P_1.$ 
The second one is that with a finite number 
of degrees of freedom 
the {\it non-degenerateness} condition would become the non-singularity 
of the matrix $\int_0^{2\pi}d\theta V_{\hat Q,\hat Q}(\theta,\hat{\bar Q}).$ 
Of course our {\it non-degenerateness} 
condition implies that the submatrix $\Gamma_{2i,2j-1}$ is 
non-singular.

\noindent
vi) We will show how actually the functions 
$(P^o(\varepsilon),Q^o(\varepsilon))$ are continuous up to the derivative of 
order one in $(-\varepsilon_o,\varepsilon_o)$ and this depends on the fact 
that $V\in C^2({\cal T}_2,\real).$
\vskip1.0truecm
\SECTIONcount=4\EQNcount=1
\noindent
{\bf \mathhexbox2784. Proofs}
\vskip0.3truecm
\noindent
The proof splits in two parts. In the first one we show how  
$\bigl(P^o(\varepsilon),Q^o(\varepsilon),\varepsilon\bigl)$ solve \equ(1.8) 
for all $\varepsilon\in(-\varepsilon_o,\varepsilon_o)$ and this fact 
implies that 
$\gamma(\tau,P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)$ is a 
$2\pi-$periodic solution. In the second part of the proof we get
\equ(3..).

\noindent
We begin with the second part. It is a straightforward computation of the 
following three quantities 
$$
\eqalign{
&\rho\bigl(\phi(t;P^o(\varepsilon),Q^o(\varepsilon),\varepsilon),
\phi(t;\bar P,\bar Q,0)\bigl)\cr
&\Vert\psi(t;P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)-
\psi(t;\bar P,\bar Q,0)\Vert,\qquad 
\Vert\dot\psi(t;P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)-
\dot\psi(t;\bar P,\bar Q,0)\Vert.\cr
}
$$
Using i) of {\bf\mathhexbox2781}
we obtain the following bounds 
(we will introduce some numerical constants $B_i\ i\ge1$)
$$
\eqalign{
&\max_{t\in[o,2\pi]}\rho\bigl(\phi(t;P^o(\varepsilon),Q^o(\varepsilon),
\varepsilon),\phi(t;\bar P,\bar Q,0)\bigl)\le B_1
\bigl[\rho\bigl((\hat P^o(\varepsilon),\hat Q^o(\varepsilon)),
(\hat0,\hat{\bar Q})\bigl)+\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+\varepsilon V^1(Q^o(\varepsilon),\bar\Delta)\bigl]<\infty\cr
&\max_{t\in[o,2\pi]}\Vert\psi(t;P^o(\varepsilon),Q^o(\varepsilon),
\varepsilon)-\psi(t;\bar P,\bar Q,0)\Vert
\le\Vert\hat P^o(\varepsilon)\Vert+
\varepsilon 2\pi V^1(Q^o(\varepsilon),\bar\Delta)
<\infty\cr
&\max_{t\in[o,2\pi]}\Vert\dot\psi(t;P^o(\varepsilon),Q^o(\varepsilon),
\varepsilon)-\dot\psi(t;\bar P^o,\bar Q^o,0)\Vert\le
\varepsilon V^1(Q^o(\varepsilon),\bar\Delta)<\infty\cr
}
$$
\noindent    
$\bar\Delta=\Delta(\varepsilon,L,T,P^o(\varepsilon),Q^o(\varepsilon))
\vert_{T=2\pi}$ and 
$\Delta(\varepsilon,L,T,P^o(\varepsilon),Q^o(\varepsilon))$ is an upper 
bound of 

\noindent
$\sup_{t\in[-T,T]}\rho\bigl(Q(t;P^o(\varepsilon),
Q^o(\varepsilon),\varepsilon),Q^o(\varepsilon)\bigl)$ 
(see (6.3)) while for the definition of 

\noindent
$V^1(Q^o(\varepsilon),\bar\Delta)$ and $L$ see 
{\bf\mathhexbox2781}.  
Being $Q^o(\varepsilon),$ $P^o(\varepsilon)$ and $V_Q$ 
continuous functions of their arguments we could compute $C(\varepsilon,L).$

The discussion about the smoothness of $(P^o(\varepsilon),Q^o(\varepsilon))$ 
of Remark vi) will be postponed at the end of Appendix 1.
\vskip0.5truecm

\noindent
Now we show the proof of the first part of Theorem 3.1. 

\noindent
First of all we rewrite the system \equ(1.4) as autonomous
\footnote{$^{(11)}$}{\noindent
from now on $Q\equiv(Q_o,Q_1,\hat Q)$} 
$$
\dot Q=P\qquad\dot P=\varepsilon V_x(Q_o+Q_1,\hat Q)\qquad\dot Q_o=1
\quad(Q_o=Q_o(\tau)\equiv\tau)
\EQ(1.13)
$$
\noindent
then we make the following change of coordinates
\footnote{$^{(12)}$}{\noindent$G(P,Q)=G(P,Q_o,Q_1,\hat Q)$; 
the integral of motion $G(\tau,P,Q)$, see \equ(0.1), is assumed as new 
coordinate in order to eliminate a Floquet multiplier that inevitably is 
equal to 1
} 
$(p,q)=U(P,Q)\colon\torus\times{\cal T}_2\times l_2
\to\torus\times{\cal T}_2\times l_2$
$$
q=Q,\qquad p_1=G(P,Q)
\qquad \hat p=\hat P. 
\EQ(cv)
$$
\noindent
The map $U(P,Q)$ is Fr\'echet differentiable 
for every $(P,Q)\in l_2\times\torus\times{\cal T}_2$. It is 
non-singular if $1+P_1\ne0$ so we require 
$P_1(\tau;Q^o,P^o,\varepsilon)\ne-1\forall \tau\in[0,2\pi]$ 
which is possible for $\varepsilon$ sufficiently small. 
In the new $(p,q)$ coordinates the system \equ(1.13) becomes
$$
\eqalign{
&\dot q_o=1\qquad
\dot q_1=-1\pm\bigl(1-\hat p^2+2\varepsilon V(q_o+q_1,\hat q)
+2G^o\bigl)^{1\over2}\cr
&\dot p_1=0\qquad\dot {\hat q}=\hat p\qquad
\dot {\hat p}=\varepsilon V_{\hat q}(q_o+q_1,\hat q)\cr
}
\EQ(1.81)
$$
$G^o=G(P^o,Q^o)\doteq G(P(0),Q(0)).$ Setting $\varepsilon=0$ in \equ(1.81) 
we obtain

\centerline{
$\dot q_o=1\quad
\dot q_1=-1\pm\bigl(1-\hat p^2+2G^o\bigl)^{1\over2}
\quad\dot p_1=0\ \ \quad\dot{\hat q}=\hat p\ \ \quad
\dot{\hat p}=0$}
i.e.
$$
\left\{\eqalign{
&q_o(\theta)=\theta+q_o^o\quad\quad q_1(\theta)=\bigl(-1\pm
(1-\hat p^2+2p^o_1)^{1\over2}\bigl)\theta+q^o_1\cr
&p_1(\theta)=p_1^o\quad\quad\hat q(\theta)=\hat q^o+\hat p^o\theta\quad\quad
\hat p(\theta)=\hat p^o\cr
}
\right.
$$
and taking into account the change \equ(cv) we get 
$$
\left\{\eqalign{
&q_o(\theta)=\theta+Q^o_o,\quad\qquad p_1(\theta)=P_1^o+{1\over2}(P^o_1)^2,
\quad\qquad 
\hat q(\theta)=\hat Q^o+\hat P^o\theta\cr
&q_1(\theta)=\bigl(-1\pm
\bigl(1+2P^o_1+(P^o_1)^2-\hat P^2\bigl)^{1\over2}\bigr)\theta+Q^o_1,
\qquad\quad 
\hat p(\theta)=\hat P^o,\cr
}
\right.
\EQ(Rita..)
$$    
then (recall that $\hat P^o=0$ and 
$P^o_1=\bar P_1\in{\integer}\backslash\{-2,-1,0\}$ and that $\hat{\bar Q}$ is 
the critical point of the map \equ(2..))
$$
\left\{\eqalign{
&q_o(\theta)=\theta+\bar Q_o\quad\qquad \hat q(\theta)=\hat{\bar Q}
\qquad\quad
p_1(\theta)=\bar P_1+{1\over2}\bar P_1^2\cr
&q_1(\theta)=\bigl(-1\pm\vert1+\bar P_1\vert\bigl)\theta+Q^o_1,\quad\qquad 
\hat p(\theta)=0.\cr
}
\right.
\EQ(.Rita)
$$
The equations \equ(.Rita) represent the periodic solution when 
$\omega\to\infty$ 
(the \lq\lq unperturbed solution") and consequently 
the interaction among the variables disappears. We look for 
$2\pi$-periodic 
solutions $\gamma(\tau,P^o,Q^o,\varepsilon)$ near the unperturbed solution 
in the sense that when $\omega\to\infty\ \ (\varepsilon\to0)$ 
$\gamma(\tau,P^o,Q^o,\varepsilon)$ tends to the unperturbed one. Moreover 
we note that taking the plus sign in the equation for $q_1(\th)$ we are 
forced to exclude $\bar P_1=0,-2.$

Now denoting the flux generated by the system of equations \equ(1.81) as  
$\lambda(\theta;p^o,q^o,\varepsilon)\equiv\bigl(
\xi(\theta;p^o,q^o,\varepsilon),\eta(\theta;p^o,q^o,\varepsilon)\bigl)$ 
the equations that have to be solved are 
$$
\eqalign{
&\xi_o(2\pi;p^o,q^o,\varepsilon)-\xi_o(0;p^o,q^o,\varepsilon)=2\pi(=0
\,mod.\,2\pi);\cr
&\xi(2\pi;p^o,q^o,\varepsilon)-\xi(0;p^o,q^o,\varepsilon)=
\int_0^{2\pi}d\theta\,\dot{\xi}(\theta;p^o,q^o,\varepsilon)=\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\int_0^{2\pi}d\theta
\eta(\theta;p^o,q^o,\varepsilon)=0\ 
(mod.2\pi)\cr
&{1\over\varepsilon}(\eta(2\pi;p^o,q^o,\varepsilon)-\eta(0;p^o,q^o,
\varepsilon))=
{1\over\varepsilon}\int_0^{2\pi}d\theta\,\dot{\eta}(\theta;p^o,q^o,
\varepsilon)=\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
=\int_0^{2\pi}d\theta V_Q(\xi_o+\xi_1,\hat\xi)=0.\cr
}
\EQ(1.88)
$$

\noindent
Now we do the {\it Poincar\'e-map} taking as {\it surface-map}  

\centerline{
$\Sigma=\bigl\{(p,q)\in l_2\times{\torus}\times{\cal T}_2\vert
q_o=0,\, q_1=0,\, p_1=\bar P_1+{1\over2}\bar P_1^2
\ ,\bar P_1\ne-2,-1,0\bigl\}.$}

\noindent
and in this way the first three equations of \equ(1.88) are excluded 
giving
$$
\eqalign{
&\hat\xi(2\pi;\hat p^o,\hat q^o,\varepsilon)-\hat\xi
(0;\hat p^o,\hat q^o,\varepsilon)=
\int_0^{2\pi}d\theta\hat\eta(\theta;\hat p^o,\hat q^o,\varepsilon)=0\ 
(mod.2\pi)\cr
&{1\over\varepsilon}(\hat\eta(2\pi;\hat p^o,\hat q^o,\varepsilon)-\hat\eta
(0;\hat p^o,\hat q^o,\varepsilon))=
\int_0^{2\pi}d\theta V_{\hat Q}(\xi_o+\xi_1,\hat\xi)=0.\cr
}
\EQ(1.888)
$$
We indicate \equ(1.888) as\footnote{$^{(13)}$}{
$\Gamma_{2i-1}=\Phi_i,$ $\Gamma_{2i}=\Psi_i$ $i\ge2,$ $\Phi_i=
\int_0^{2\pi}d\theta\hat\eta_i,$ $\Psi_i=
\int_0^{2\pi}d\theta V_{\hat Q_i}$
} 
$\Gamma(\hat p^o,\hat q^o,\varepsilon)\doteq
(\Phi(\hat p^o,\hat q^o,\varepsilon),\Psi(\hat p^o,\hat q^o,\varepsilon))=0$ 
and to solve it means to show that there exist two functions
$(\hat p^o(\varepsilon),\hat q^o(\varepsilon))$ so that 
$\Gamma(\hat p^o(\varepsilon),\hat q^o(\varepsilon),\varepsilon)\equiv0$ 
for every 
$\varepsilon$ sufficiently small. The resolution is a standard application of 
the Implicit-Function-Theorem whose statement is: 

\vskip0.3truecm
\noindent
{\it Let us consider a map $F\colon X\times Y\to Z$ 
($X, Y, Z$ are Banach spaces) such that 1) $F(x^o,y^o)=0,$ 2) $F$ is 
Fr\'echet differentiable respect to $y$ and $x$ in $(x^o,y^o);$ $F_y(x,y)$ 
and $F_x(x,y)$ are continuous in $(x^o,y^o)$ 
3) $F_{y}(x^o,y^o)$ has bounded inverse. Then 
there exists a function 
$f\colon U\to U^\prime,$ $x_o\in U\subset X,$ $y_o\in U^\prime\subset Y,$ 
continuous and differentiable in $(x_o,y_o),$ such that $f(x_o)=y_o$ and 
$F(x,f(x))\equiv0$ $\forall x\in U.$}  
\vskip0.3truecm
\noindent
Doing the suitable analogies\footnote{$^{(14)}$}{We should consider 
$X=l_2=\{\hat p_i\}_{i=2}^\infty$, $Y=l_2=\{\hat\phi_i\circ
\hat q_i\}_{i=2}^\infty$ and $Z=X\times Y$. $\phi_i$ is the homeomorphism of 
$\picctorus$ into $\picreal$ such that $\otimes_{i=1}^\infty\phi_i\doteq\phi$
makes ${\cal T}_2$ locally homeomorphic to $l_2$ (see footnote (7)). 
Actually we consider $X={\cal T}_2$ because it is a linear manifold 
respect to vector sum. Moreover what in the statement of the Implicit-
Function-Theorem is $x$ in our case is $\varepsilon$ while $F$ is 
represented by the system \equ(1.888)} 
we have to find under what conditions the following relations are true: 
there exist $\hat{\bar p},\hat{\bar q},\bar\varepsilon$ such that 

\noindent
{\it1)}$\Gamma(\hat{\bar p},\hat{\bar q},\bar\varepsilon)=0$

\noindent
{\it2)}$\Gamma(\hat p^o,\hat q^o,\varepsilon)$ is 
Fr\'echet differentiable respect to $(\hat p^o, \hat q^o)$ and $\varepsilon$ 
in $(\hat{\bar p},\hat{\bar q},\bar\varepsilon);$ 
$\Gamma_{\hat p^o,\hat q^o}(\hat{\bar p},\hat{\bar q},
\bar\varepsilon)$ and $\Gamma_\varepsilon(\hat{\bar p},\hat{\bar q},
\bar\varepsilon)$ continuous in $(\hat{\bar p},\hat{\bar q},\bar\varepsilon);$

\noindent
{\it3)}$\Gamma_{\hat p^o,\hat q^o}(\hat{\bar p},\hat{\bar q},
\bar\varepsilon)$ has bounded inverse
\vskip0.5truecm
\noindent
{\it1)} We solve $\Gamma(\hat p^o,\hat q^o,\varepsilon)=0$ 
for $\varepsilon=\bar\varepsilon=0,\ 
\hat q^o=\hat{\bar q}=\hat{\bar Q},\ \hat p^o=0=\hat{\bar p}=\hat{\bar P}$
$$
\left\{\eqalign{
&\Phi(0,\hat{\bar Q},0)=0:\quad \hat\eta(2\pi;0,\hat{\bar Q},0)-
\hat\eta(0;0,\hat{\bar Q},0)=0\cr
&\Psi(0,\hat{\bar Q},0)=0:\quad {1\over\varepsilon}
\bigl(\hat\eta(2\pi;0,\hat{\bar Q},0)-\hat\eta(0;0,\hat{\bar Q},0)\bigl)
=0\Rightarrow\cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\Rightarrow\int_0^{2\pi}d\theta 
V_{\hat Q}(\pm\vert1+\bar P_1\vert\theta,\hat{\bar Q})=0.\cr
}
\right.
\EQ(2.tana)
$$
The periodicity of $V(x)$ and $\bar P_1\in{\integer}\backslash\{-2,-1,0\}$ 
imply that \equ(2.tana) is satisfied if (see \equ(2..))
$$
\int_0^{2\pi}d\theta 
V_{\hat Q}(\theta,\hat{\bar Q})=0
\EQ(2.1)
$$
\vskip0.5truecm
\noindent
{\it3)}
$\Gamma_{\hat p^o,\hat q^o}(\hat{\bar p},\hat{\bar q},0)$ with 
bounded inverse 
means that the linear map whose representation is given by the following 
matrix has to be non-singular and the inverse of it bounded
$$
{\partial\Gamma\over\partial(\hat p^o,\hat q^o)}
\vert_{\varepsilon=0,\hat p^o=0,\hat q^o=\hat{\bar q}}=
\left(\matrix{
        0&\Gamma_{34}&0&0&\ldots\cr
        \Gamma_{43}&\Gamma_{44}&\Gamma_{45}&\Gamma_{46}&\ldots\cr
        0&0&0&\Gamma_{56}&0&0&0&\ldots\cr
        \Gamma_{63}&\Gamma_{64}&\Gamma_{65}&\Gamma_{66}&\Gamma_{67}&\Gamma_{68}&\ldots\cr
        0&0&0&0&0&\Gamma_{78}&0&0&\ldots\cr
        \vdots&\vdots&\ldots&\cr}\right)
\EQ(ma)
$$
$$
\Gamma_{2i-1,2i}=2\pi,\ 
\Gamma_{2i,2j}=\int_0^{2\pi}d\theta \theta V_{\hat Q,\hat Q}
(\pm\theta\vert1+\bar P_1\vert,\hat{\bar Q}),\  
\Gamma_{2i,2j-1}=\int_0^{2\pi}d\theta V_{\hat Q_i,\hat Q_j}
(\theta,\hat{\bar Q}).
$$
A necessary condition is the non-singularity of the matrix \equ(ma) and a 
short meditation shows as this is equivalent to the injectivity of the 
following matrix
$$
M=\left(\matrix{\Gamma_{43}&\Gamma_{45}&\Gamma_{47}&\Gamma_{49}&\Gamma_{4,11}&\ldots\cr
\Gamma_{63}&\Gamma_{65}&\Gamma_{67}&\Gamma_{69}&\Gamma_{6,11}&\ldots\cr
\vdots&\vdots&\ldots\cr}\right)
\EQ(ma1)
$$
Being the problem infinite-dimensional, we have to show that the inverse 
of the matrix \equ(ma) is bounded.
This explains the {\it non-degenerateness} condition of the critical point 
$(0,\hat{\bar q},0)$ for the map in \equ(2..). In the next section two 
examples at which to apply this scheme.
\vskip0.5truecm

\noindent
{\it2)} Being $V\in C^2({\cal T}_2,\real),$ by \equ(1.888) 
we realize that the continuity of 
$\Gamma(\hat p^o,\hat q^o,\varepsilon)$ and its 
derivatives involved in point {\it 2)} are consequences of the same 
properties of the flow $\lambda(\theta;p^o,q^o,\varepsilon).$
Then by means of the following relation 
$$
\lambda(\theta;p^o,q^o,\varepsilon)=
U(\gamma(\theta;P^o,Q^o,\varepsilon))
\vert_{U^{-1}(p^o,q^o)}
$$ 
(which is the 
transformation law of the flows $\lambda$ and $\gamma$ 
through the map \equ(cv)) we reduce 
ourselves to study the flow 
$\gamma(\theta;P^o,Q^o,\varepsilon)$ 
(note that if $\vert\varepsilon\vert$ and $\Vert\hat p^o\Vert$ are small 
enough then $\vert 1+2p^o_1-\hat p^2+2\varepsilon V(q_o+q_1,\hat q)\vert
\ge\vert{1\over2}
(1+2p^o_1)\vert$ and this makes the change $U$ well defined for all times, 
differentiable and invertible).

Now using standard techniques in the 
theory of ordinary differential equations we will show that $\gamma,$ 
${\partial\gamma\over\partial\varepsilon}$ and 
${\partial\gamma\over\partial(P^o,Q^o)}$ are continuous as functions of 
$(P^o,Q^o,\varepsilon)$ and a fortiori in $P^o=(\bar P_1,\hat 0)\ \ 
Q^o=(\bar Q_1,\hat{\bar Q}),\ \ \varepsilon=0.$ In fact supposing that 
$\gamma(\cdot;P^o,Q^o,\varepsilon)$ is continuous (the proof will be given 
in Appendix 1) we have 
\vskip0.3truecm
\noindent
{\bf Corollary 4.1} {\sl Let be $\gamma(\theta;y^o,\varepsilon)$ 
the flow of the system \equ(1.4). Then 
$$
{\partial\gamma\over\partial\varepsilon}(\theta;P^o,Q^o,\varepsilon)\quad 
{\rm and}\quad 
{\partial\gamma\over\partial y^o}(\theta;P^o,Q^o,\varepsilon)
$$ 
are continuous as functions of $P^o,Q^o,\varepsilon.$
}

\noindent
Proof 

\noindent
We introduce the so called {\it linearized equation}
\footnote{$^{(15)}$}{we set $(\hat P^o,\hat Q^o)=y^o$ i.e. 
$y^o_{2i}=\hat P^o_i,$ $y^o_{2i-1}=\hat Q^o_i$ $(i\ge2)$} 
$$
{d\over d\theta}{\partial\gamma\over \partial y^o}(\theta;y^o,\varepsilon)=
{\partial f\over\partial y}(\gamma(\theta;y^o,\varepsilon),\varepsilon)
{\partial\gamma\over\partial y^o}(\theta;y^o,\varepsilon)
\EQ(888)
$$
whose solution is the series
$$
{\partial\gamma\over \partial y^o}(\theta;y^o,\varepsilon)
=\sum_{k=0}^\infty\int_0^{\theta}d\theta_1\int_0^{\theta_1}d\theta_2\ldots
\int_0^{\theta_{k-1}}d\theta_k f_y^{(1)}\ldots f_y^{(k)}
\EQ(88)
$$
$f_y^{(i)}\doteq f_y(\gamma(\theta_i;y^o,\varepsilon),\varepsilon)$ and 
$f(y,\varepsilon)$ is symbolically the right hand side of \equ(1.4).

\noindent
First of all we prove that 
${\partial\gamma\over \partial y^o}$ is a bounded (and of course linear) 
operator over 
$\real\times l_2\times l_2.$ Suppose in fact we know that $f_y^{(i)}$ maps 
$\real\times l_2\times l_2$ into itself, then the sequence 
$\bigl\{\sum_{k=0}^n\int_0^{\theta}d\theta_1\int_0^{\theta_1}d\theta_2\ldots
\int_0^{\theta_{k-1}}d\theta_k f_y^{(1)}\ldots f_y^{(k)}\bigl\}$ is a Cauchy 
sequence in ${\cal L}(\real\times l_2\times l_2)$ (the space of the linear 
bounded operators on  $\real\times l_2\times l_2$) which is 
complete and then it converges to a well defined bounded linear operator that 
is ${\partial\gamma\over \partial y^o}.$ So all the work is reduced to show 
that $f_y^{(i)}$ maps $\real\times l_2\times l_2$ into itself and this can be 
achieved by means of the following bound 
$$
\sup_{\theta\in[0,2\pi]}
\Vert f_y(\gamma(\theta;P^o,Q^o,\varepsilon),\varepsilon)\Vert\le
2\max\{1,\vert\varepsilon\vert V_2\}.
$$
Indeed, via the continuity of $V_{xx}$ in $x$ and of $\gamma$ in 
$P^o, Q^o, \varepsilon,$ $f_y$ is continuous too  
$$
\sup_{\theta\in[0,2\pi]}\Vert f_y(\gamma(\theta_i;y^o,\varepsilon),
\varepsilon)-f_y(\gamma(\theta_i;\tilde y^o,\tilde\varepsilon),
\tilde\varepsilon)\Vert\le
A(\vert\varepsilon-\tilde\varepsilon\vert,\rho(y^o,\tilde y^o))
\EQ(13)
$$
and $A(\vert\varepsilon-\tilde\varepsilon\vert,\rho(y^o,\tilde y^o))\to0$ 
as $\vert\varepsilon-\tilde\varepsilon\vert+\rho(y^o,\tilde y^o)\to0.$ 
The uniform convergence of the series \equ(888) makes the 
operator ${\partial\gamma\over\partial y^o}$ continuous.

\noindent
About the continuity of 
${\partial\gamma\over\partial\varepsilon}(\theta;P^o,Q^o,
\varepsilon)$ we have to solve the equation 
$$
{d\over d\theta}{\partial\gamma\over \partial\varepsilon}
(\theta;y^o,\varepsilon)=
{\partial f\over\partial y}(\gamma(\theta;y^o,\varepsilon),\varepsilon)
{\partial\gamma\over\partial\varepsilon}(\theta;y^o,\varepsilon)+
{\partial f\over\partial\varepsilon}(\theta;y^o,\varepsilon)
$$
whose solution is 
$$
{\partial\gamma\over \partial\varepsilon}(\theta;y^o,\varepsilon)
=\sum_{k=0}^\infty\int_0^{\theta}d\theta_1\int_0^{\theta_1}d\theta_2\ldots
\int_0^{\theta_{k-1}}d\theta_k f_y^{(1)}\ldots f_y^{(k-1)}f_\varepsilon^{(k)}
$$
and repeating the argument used for \equ(888) we can prove that the series 
converges uniformly and being each term continuous in $P^o,Q^o,\varepsilon$ 
it defines a continuous vector function i.e.
${\partial\gamma\over \partial\varepsilon}.$

 
\vskip0.75truecm
\SECTIONcount=5\EQNcount=1
\noindent
{{\bf\mathhexbox2785. Applications}

\noindent
Consider the following potential function (it represents an infinite countable 
set of point masses all rotating over a circle of radius one and positioned 
on a vertical plane; $\kappa$ is a positive constant, $m$ is the mass and $g$ 
is the gravity constant)
$$
V(Q)=-\sum_{i=1}^\infty mg(1-\cos Q_i)+\sum_{i=1}^\infty {\kappa}
(1-\cos(Q_{i+1}-Q_i))
\EQ(3.1)
$$
$$
\eqalign{
V_{Q_i}&=-mg\sin Q_i+{\kappa}\bigl(\sin(Q_i-Q_{i-1})-\sin(Q_{i+1}-Q_i)\bigl)
\quad i\ne1\cr
V_{Q_1}&=-mg\sin Q_1+{\kappa}\bigl(\sin(Q_1-Q_2)\bigl)
}
$$
$$
V_{Q_jQ_i}=\left\{\eqalign{
&i=j\qquad -mg\cos Q_j+{\kappa}\bigl(\cos(Q_j-Q_{j-1})+\cos(Q_{j+1}-Q_j)
\bigl)\cr
&i=j+1\qquad -{\kappa}\cos(Q_j-Q_{j+1})\cr
&i=j-1\qquad -{\kappa}\cos(Q_j-Q_{j-1})\cr
}
\right.
$$
It is easy to check the property i) of {\bf \mathhexbox2781} 
($V_2\le 6(2\kappa+mg)$) 
and then Theorem 2.1 applies together with the smoothness property of the 
flow $\gamma.$

\noindent
$\hat Q^o\equiv0$ is a critical point of the map \equ(2..)
while the matrix associated to \equ(ma1) and calculated in $0$ 
gives
$$
\matrix{
(-mg+{\kappa})2\pi&-2{\kappa}\pi&0&0&\ldots&\ldots&\ldots\cr
-2{\kappa}\pi&(-mg+2{\kappa})2\pi&-2{\kappa}\pi&0&\ldots&\ldots&\ldots\cr
0&-2{\kappa}\pi&(-mg+2{\kappa})2\pi&-2{\kappa}\pi&0&\ldots&\ldots\cr
0&0&-2{\kappa}\pi&(-mg+2{\kappa})2\pi&-2{\kappa}\pi&0&\ldots\cr
\vdots&\vdots&\ldots&\ldots&\cr
}
$$
and if $2{\kappa}$ is small compared to $\vert mg\vert$ then the 
matrix is non-singular proving that the matrix \equ(ma) is non-singular to.
The next step consists in showing that \equ(ma) has bounded inverse. 
In fact a short calculation gives 
$\Vert N\tilde v\Vert\ge C\Vert\tilde v\Vert$ with 
$C=(2\pi)^2(\vert2\kappa-mg\vert)$
hence $\Vert N^{-1}\Vert\le C^{-1}$
\footnote{$^{(16)}$}{recall that for the non-singularity of 
the matrix \equ(ma1) we have imposed ${2\kappa\over\vert mg\vert}\ll1$; 

\noindent
with $N$ we indicate the matrix \equ(ma) 
}.

Now for the potential \equ(3.1) we compute the first order in 
$\varepsilon$ of the 
functions $\hat P^o(\varepsilon),\hat Q^o(\varepsilon).$ As it should be 
clear, these functions 
are the initial values that originates {\it periodic solutions} of the 
problem  with potential function \equ(3.1). 
$0\equiv \Gamma(y^o(\varepsilon),\varepsilon)=\Gamma(\bar y,0)+
\bigl({\partial \Gamma\over\partial y^o}
(y^o(\varepsilon),\varepsilon){d y^o\over d\varepsilon}+
{\partial \Gamma\over\partial\varepsilon}\bigl)\vert_{\varepsilon=0}\varepsilon
+\circ(\varepsilon)$
and then
${d y^o\over d\varepsilon}\vert_{\varepsilon=0}=
-\bigl({\partial \Gamma\over\partial y^o}(y^o(\varepsilon),\varepsilon)
\vert_{\varepsilon=0}\bigl)
^{-1}{\partial \Gamma\over\partial \varepsilon}\vert_{\varepsilon=0}
=-\bigl({\partial \Gamma\over\partial y^o}(\bar y,0)\bigl)^{-1}
{\partial \Gamma\over\partial \varepsilon}(\bar y,0).$

\noindent
>From the last equality we derive two relations. The first one is 
$$
{\partial\over\partial\varepsilon}\Psi
(y^o(\varepsilon),\varepsilon)=
{\partial\over\partial\varepsilon}\int_0^{2\pi}d\theta V_{\hat Q}
(\theta,\hat y^o)=0
$$
(as $\varepsilon$ is not explicitly present but only 
implicitly in $\hat y^o$) while the second one is 
$$
\eqalign{
&{\partial\over\partial\varepsilon}\Phi(y^o(\varepsilon),\varepsilon)
\vert_{\varepsilon=0}=
{\partial\over\partial\varepsilon}\int_0^{2\pi}d\theta\  
\hat\psi\bigl(\theta,y^o(\varepsilon)\bigl)\vert_{\varepsilon=0}=
{\partial\over\partial\varepsilon}\int_0^{2\pi}d\theta\int_0^{\theta}
d\theta^\prime
\ \dot{\hat\psi}\bigl(\theta^\prime,y^o(\varepsilon))\vert_{\varepsilon=0}\cr
&={\partial\over\partial\varepsilon}\int_0^{2\pi}d\theta\int_0^{\theta}
d\theta^\prime\ 
\varepsilon V_{\hat Q}(\theta^\prime,\hat q^o(\varepsilon))
\vert_{\varepsilon=0}
=\int_0^{2\pi}d\theta\int_0^{\theta}d\theta^\prime V_{\hat Q}
(\theta^\prime,\hat{\bar q})\doteq \hat b.\cr
}
$$
Consequently we have ${\partial \Gamma\over\partial\varepsilon}
\vert_{\varepsilon=0}=
-(b_3,0,b_5,0,b_7,0,b_9,0,\ldots)$ and considering the potential \equ(3.1) 
we have $b_3=-2\kappa\pi$ while $b_j=0$ for $j\ne3;$ 
${\partial \Gamma\over\partial\varepsilon}\vert_{\varepsilon=0}=
-(b_3,0,\ldots)$ and then it results 
$$
{d y^o\over d\varepsilon}
\vert_{\varepsilon=0}=(0,2\kappa\pi,0,\ldots)\quad{\rm i.e.\ }
{d Q^o\over d\varepsilon}\vert_{\varepsilon=0}=0\quad
{d P^o_2\over d\varepsilon}\vert_{\varepsilon=0}=2\kappa\pi
\quad{d P^o_i\over d\varepsilon}\vert_{\varepsilon=0}=0\ i\ge3
\EQ(3.2)
$$

\noindent
Now we briefly investigate the meaning of the latter computation.

With the Theorem 3.1 we have shown that on the surface $\Sigma$ there are 
initial data that originate periodic solutions. The local nature of the 
Implicit Function Theorem implies that these initial data are located close 
with $\varepsilon,$ to the point $\bar Q,\bar P$ with 
$\bar Q_1=0 (mod.2\pi),$ $\bar P_1\in{\integer}\backslash\{-2,-1,0\},$ 
$\hat{\bar P}=0$ 
$\hat{\bar Q}$ is a non degenerate critical point of the map \equ(2..).

Suppose now to expand a component $\hat Q^o_j(\varepsilon)$ or 
$\hat P^o_j(\varepsilon)$ in power series of $\varepsilon$; 
for example we have 
$Q_j^o(\varepsilon)=\bar Q_j +\varepsilon {d Q^o_j\over d\varepsilon}
\vert_{\varepsilon=0}+\circ(\varepsilon)$ and it follows that 
the unique contribution to first order in $\varepsilon$ comes from 
$P^o_2(\varepsilon)$ i.e. ${d P^o_2\over d\varepsilon}
\vert_{\varepsilon=0}=-2\pi\kappa$ (the others one are all equal to $0$). The 
velocities $\dot x_i,$ where $x_i$ is the generic angle of \equ(1.0), assume 
the form 
$$
\left\{\eqalign{
\dot x_1&=\omega +\omega{d u_1\over d(\omega t)}=
\omega+\omega P_1(\tau,P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)
=\omega+\omega\bar P_1(\tau)+\omega O(\varepsilon)\equiv\cr
&=\omega+\omega\bar P_1+O(\omega^{-1})\cr
\dot x_2&=\omega{d u_2\over d(\omega t)}=
\omega P_2(\tau,P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)=
\omega(\bar P_2+2\pi\kappa\varepsilon+\circ(\varepsilon))=
{2\pi\kappa\over\omega}+\circ(\omega^{-1})\cr
\dot x_j&=\omega{d u_j\over d(\omega t)}=
\omega P_j(\tau,P^o(\varepsilon),Q^o(\varepsilon),\varepsilon)=
\omega(\bar P_j+\circ(\varepsilon))=\circ(\omega^{-1})\quad j>2.\cr
}
\right.
\EQ(3.00)
$$
The various $\circ(\cdot)$ and $O(\cdot)$ 
are all $2\pi$ periodic functions of $\omega t.$ 
We then conclude that the rotator $x_1$ moves  
very fast while the velocities of $x_j$ with $j\ge2$ are small and tend to 
zero as $\omega\to\infty.$

\noindent
Remarks

\noindent
i) $\hat Q^o=0$ is not the only critical point of the map 
\equ(2..). For example also the point 
$Q^o_2=\pm\pi,\ \{Q^o_i\}_{i=3}^\infty=0$ is a good one. In general, for the 
potential \equ(3.1) we could take as critical point any one in the set 
${\cal Q}=\{x\in{\cal T}_2\vert\exists N\in\natural 
s.t.\ x_j=0,\pm\pi\ 1\le j\le N,\ x_j=0\ j>N\}.$
\vskip0.2truecm
\noindent
For the {\it long-range potential} 
$V(Q)=\sum_{i,j=1}^\infty e^{-\vert i-j\vert}(1-\cos(Q_i-Q_j))$ 
we have 
$$
V_{Q_k}=2\sum_{j=1}^\infty e^{-\vert k-j\vert}\sin(Q_k-Q_j),\quad
\left\{
\eqalign{
V_{Q_kQ_k}=&2\sum_{j=1}^\infty e^{-\vert k-j\vert}\cos(Q_k-Q_j)\cr
V_{Q_kQ_l}=&-2e^{-\vert k-l\vert}\cos(Q_k-Q_l)\cr
}
\right.
$$
and property i) of {\bf\mathhexbox2781} is verified ($V_2\le2\sqrt2
{e+1\over e-1}$. 
$\hat Q^o=0$ is a critical point \equ(2..) and the matrix \equ(ma1) 
can be written as $4\pi{e+1\over e-1}I+B$ where $I$ is the identity and 
$B_{kk}=-4\pi{e^{2-k}\over e-1}$ while 
$B_{k,l}=-4\pi e^{-\vert k-l\vert}\ k\ne l.$ Being $\Vert B\Vert<
4\pi{e+1\over e-1}$ the operator \equ(ma1) is non-singular; the matrix 
\equ(ma) has a bounded inverse and Theorem 
3.1 applies in this case of {\it long-range potential} too. A calculation 
analogous to \equ(3.2) and consequently \equ(3.00) is less available now 
because of the long-range feature of the model.

\vskip1.0truecm
\SECTIONcount=6\EQNcount=1
\noindent{\bf Appendix 1. Existence of the dynamics}

\noindent
In this Appendix we prove the Theorem 2.1 and some useful bounds relative 
to Corollary 4.1.

\noindent
Let us define the space of continuous function  
${\cal Q}_T=\{Q:[-T,T]\subset{\real}\to{\cal T}_2\};$ 
on ${\cal Q}_T$ 
the following metric is given
$$
\rho(Q,Q^\prime)=\bigl(\sum_{i=1}^\infty\sup_{t\in[-T,T]}
\rho^2(Q_i(t),Q_i^\prime(t))\bigl)^{1\over2}
$$
\noindent
and makes ${\cal Q}_T$ 
a Fr\'echet space. 
For what concerns the existence of the dynamic (i.e. the existence of the 
functions $(P,Q)$ of Theorem 2.1) we study the system \equ(1.4) as a second 
order system
$$
\ddot Q=\varepsilon V_x(\tau+Q_1,\hat Q)\qquad\qquad
\bigl(\dot Q(0),Q(0)\bigl)=\bigl(P^o,Q^o\bigl)\in l_2\times{\cal T}_2
$$
\noindent
and we show that this system admits a unique solution $Q\in C^2([-T,T],
{\cal T}_2)$ so the $P(t)$ of theorem 2.1 is simply 
$\dot Q\in C^1([-T,T],l_2).$ Then using point 2) of i) in 
{\bf \mathhexbox2781} we can affirm that the solution exists for all 
$t\in\real.$

Let us define the operator $F\colon{\cal Q}_T\to{\cal Q}_T$ by means 
of the sequence of functions $Q^{(n)}(\tau)$ as follows  
$$
\left\{
\eqalign{
Q^{(n)}(\tau)\doteq &F(Q^{(n-1)})(\tau)
\equiv Q^{(o)}(\tau)+\varepsilon\int_0^{\tau}dt\int_0^t d t^\prime 
V_x(t^\prime+Q_1^{(n-1)}(t^\prime),\hat Q^{(n-1)}(t^\prime))\cr
Q^{(o)}(\tau)\doteq&Q^o+P^o\tau\qquad n\ge1.\cr
}
\right.
$$
\noindent
First we impose that 

\noindent
$\sup_{t\in[-T,T]}\rho(Q^{(n)}(\tau),Q^o)\le\varepsilon{T^2\over2}
V^1\bigl(Q^o,\bar\rho\bigl)+T\Vert P^o\Vert\le\bar\rho$ which gives an 
upper bound over $T;$ then consider the subset ${\cal Q}_{T,\bar\rho}=
\{Q:[-T,T]\subset{\real}\to{\cal T}_2\ \vert\rho(Q,Q^o)\le\bar\rho\}\subset
{\cal Q}_T.$ ${\cal Q}_{T,\bar\rho}$ is a closed subset in a complete 
metric space and then it is complete. We then look for the solution 
$Q(t)\in{\cal Q}_{T,\bar\rho}.$ 
$$
\eqalign{
&\rho(Q^{(n+1)}(\tau),Q^{(n+1)}(\tau^\prime))\le\cr
&\le\vert\tau-\tau^\prime\vert\Vert P^o\Vert+\varepsilon
\int_{\tau^\prime}^\tau dt\int_0^t dt^\prime
\bigl[\sum_{i=1}^\infty
V^2_{x_i}(t^\prime+Q^{(n)}_1(t^\prime),\hat Q^{(n)}(t^\prime))\bigl]^{1\over2}
\le\cr
&\le\vert\tau-\tau^\prime\vert\Vert P^o\Vert+
\varepsilon{1\over2}\vert\tau^2-{\tau^\prime}^2\vert
V^1\bigl(Q^o,\bar\rho\bigl)\cr
}
\EQ(A.aa)
$$
\noindent
so $Q^{(n)}\in{\cal Q}_{T,\bar\rho}\Rightarrow Q^{(n+1)}\in
{\cal Q}_{T,\bar\rho}.$ 
\vskip0.4truecm
\noindent
{\it Local Existence}

\noindent
We will show that  
the sequence $\{Q^{(n)}(\tau)\}$ is uniformly convergent 
to a function $Q\in C^2([-T,T],{\cal T}_2)$ and being $T$ arbitrary the 
existence is established for $\tau\in\real.$
$$
\eqalign{
&\rho(Q^{(n+1)}(\tau),Q^{(n)}(\tau))\le\cr
&\le\varepsilon\Bigl(\sum_{i=1}^\infty\bigl
\vert\int_0^\tau dt_1\int_0^{t_1}dt_2
V_{x_i}(t_2+Q^{(n)}_1(t_2),\hat Q^{(n)}(t_2))-\cr
&\qquad\qquad\qquad\qquad\qquad\qquad
-V_{x_i}(t_2+Q^{(n-1)}_1(t_2),\hat Q^{(n-1)}(t_2))\bigl\vert^2
\Bigr)^{1\over2}\le\cr
&\le\varepsilon 
\int_0^\tau dt_1\int_0^{t_1}dt_2\Bigl[\sum_{i=1}^\infty\bigl\vert
V_{x_i}(t_2+Q^{(n)}_1(t_2),\hat Q^{(n)}(t_2))-\cr
&\qquad\qquad\qquad\qquad\qquad\qquad
-V_{x_i}(t_2+Q^{(n-1)}_1(t_2),\hat Q^{(n-1)}(t_2))\bigl\vert
\Bigr]^{1\over2}\cr
&\le\varepsilon L\int_0^\tau dt_1\int_0^{t_1}dt_2
\rho(Q^{(n)}(t_2),Q^{(n-1)}(t_2))\cr
&\vdots\cr
&\le(\varepsilon L)^n
\int_0^{\tau}dt_1\int_0^{t_1}dt_2\ldots\int_0^{t_{2n-1}}dt_{2n}
\rho(Q^{(1)}(t_{2n}),Q^{(o)}(t_{2n}))\cr
&\le{(\varepsilon LT^2)^n\over(2n)!}{\varepsilon T^2\over2}
\sup_{S(Q^o,T\Vert \hat P^o\Vert)}
\bigl(\sum_{i=1}^\infty
V^2_{x_i}(t_{2n}+Q^o_1+P^o_1t_{2n},\hat Q^o+\hat P^o_1t_{2n})
\bigl)^{1\over2}\cr
&\le{(\varepsilon LT^2)^n\over(2n)!}{\varepsilon T^2\over2}
V^1(Q^o,T\Vert \hat P^o\Vert).\cr
}
\EQ(A.3)
$$
\noindent 
\equ(A.3) makes uniform in $[-T,T]$ the existence of the following limit 
defining a function $Q\in{\cal Q}_{T,\bar\rho}$

\centerline{
$\lim_{n\to\infty}Q^{(n)}(\tau)=
\lim_{n\to\infty}\sum_{k=1}^n \bigl(Q^{(k)}(\tau)-Q^{(k-1)}(\tau)\bigl)
+Q^{(o)}(\tau)\doteq Q(t);$}

\noindent
{\it Uniqueness}

\noindent
Let $Q(\tau)$ $\bar Q(\tau)$ be two different solutions of the 
problem \equ(1.4). We can easily prove the bound 
$$
\rho\bigl(Q(\tau),\bar Q(\tau)\bigl)
\le({\varepsilon L})^{n-1}{T^{2n}\over(2n)!}\rho(Q,\bar Q)
$$
which is a contradiction for $n$ big enough unless $\rho(Q,\bar Q)=0.$ 

\noindent
About the global existence and uniqueness we can say that a solution of the 
system \equ(1.4) cannot be extended over $\real$ if and only if it diverges 
in a finite time: there exists $t_o$ such that $\lim_{t\to t_o^-}
\vert\rho(Q(t),Q^o)\vert=+\infty.$ But the upper bound on $V_x$ in 
{\bf \mathhexbox2781} forbids the divergence at $t_o.$ 

\noindent
Now we check that $Q\in C^2([-T,T],{\cal T}_2).$ 

Let us consider (of course $\dot Q^{(k)}$ is continuous for 
$\tau\in[-T,T]$) 
$$
\lim_{n\to\infty}\dot Q^{(n)}(\tau)=
\lim_{n\to\infty}\sum_{k=1}^n \bigl(\dot Q^{(k)}(\tau)-\dot Q^{(k-1)}(\tau)
\bigl)+\dot Q^{(o)}(\tau)\doteq\dot Q(\tau)
$$
and by means of the following bound 
$$
\rho(\dot Q^{(n)}(\tau),\dot Q^{(n-1)}(\tau))\le
{(\varepsilon LT^2)^{n-1}\over(2n-2)!}{\varepsilon^2 T^3L\over2}
V^1(Q^o,T\Vert \hat P^o\Vert)
$$
the limit, by definition $\dot Q,$ exists uniformly in $[-T,T]$ defining 
a continuous function. 

\noindent
The last point of Theorem 2.1 is to show  that  
$\dot Q\in C^1([-T,T],l_2).$ We then consider the series 
$$
\eqalign{
\ddot Q(\tau)&=\lim_{n\to\infty}\ddot Q^{(n)}(\tau)=
\lim_{n\to\infty}\sum_{k=1}^n \bigl(\ddot Q^{(k)}(\tau)-\ddot Q^{(k-1)}(\tau)
\bigl)\cr
&=\lim_{n\to\infty}\sum_{k=1}^n
\varepsilon\bigl(V_x(\tau+Q_1^{(k-1)},\hat Q^{(k-1)})-
V_x(\tau+Q_1^{(k-2)},\hat Q^{(k-2)})
\bigl)\cr
}
\EQ(A.444)
$$
and because of (see footnote of {\bf \mathhexbox2781}) 
$$
\rho\bigl(V_x(\tau+Q_1^{(k-1)},
\hat Q^{(k-1)}),V_x(\tau+Q_1^{(k-2)},\hat Q^{(k-2)})\bigl)
\le L\rho\bigl(Q^{(k-1)},Q^{(k-2)}\bigl)
$$ 
the limit in 
\equ(A.444) exists uniformly in $[-T,T]$ determining 
the continuity of $\ddot  Q(\tau)$ too.

\vskip0.5truecm
\noindent
Here we prove the continuity of $\gamma$ respect to $\varepsilon.$
The continuity respect to $P^o,Q^o$ can be achieved exactly in the same way.
$$
\sup_{t\in[-T,T]}
\rho(Q^{n+1}(t;P^o,Q^o,\varepsilon),Q^{n+1}(t;P^o,Q^o,\tilde\varepsilon))
\le\vert\varepsilon-\tilde\varepsilon\vert{T^2\over2}V^1(Q^o,\Delta)
\sum_{k=0}^n{(\tilde\varepsilon LT^2)^{2k}\over(2k)!}
$$
and then 
$$
\sup_{t\in[-T,T]}
\rho(Q(t;P^o,Q^o,\varepsilon),Q(t;P^o,Q^o,\tilde\varepsilon))\le
\vert\varepsilon-\tilde\varepsilon\vert{T^2\over2}V^1(Q^o,\Delta)
\cosh\sqrt{\tilde\varepsilon LT^2)}.
$$ 
\noindent
We can also perform 
the following estimates 
$$
\eqalign{
&\sup_{t\in[-T,T]}\rho(Q(t),Q^o)\le\sum_{n=1}^\infty
{(\varepsilon T^2L)^{n-1}\over(2n-2)!}
{\varepsilon\over2}T^2 V^1(Q^o,T\Vert \hat P^o\Vert)+
T\Vert P^o\Vert=\cr
&={\varepsilon\over2}T^2V^1(Q^o,T\Vert \hat P^o\Vert)
\cosh\sqrt{T^2L\varepsilon}+T\Vert P^o\Vert\doteq
\Delta(\varepsilon,L,T,Q^o,P^o).\cr}
\EQ(A.41)
$$
and
$$
\eqalign{
&\rho(\dot Q(\tau),P^o)\le\sum_{n=1}^\infty 
{(\varepsilon LT^2)^{n-1}\over(2n-2)!}{\varepsilon^2 T^3L\over2}
V^1(Q^o,T\Vert \hat P^o\Vert)=\cr
&={\varepsilon^2 T^3L\over2}
V^1(Q^o,T\Vert \hat P^o\Vert)\cosh\sqrt{\varepsilon LT^2}
\equiv\Xi(Q^o,P^o,T,\varepsilon).\cr
}
\EQ(A.a)
$$
\vfill\eject
\noindent
\centerline{\bf References}
\vskip0.5truecm
\noindent
[A] V.I.ARNOL'D, \lq\lq Mathematical Methods of Classical Mechanics", 
      Springer-Verlag, New York second ed. 1980 (chapter 9 p.233-235) 

\noindent
[AF]  C.ALBANESE and J.FR\"OLICH, {\sl Periodic Solutions of Some Infinite-
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\noindent
[AFS] C.ALBANESE, J.FR\"OLICH and T.SPENCER {\sl 
Periodic Solutions of Some Infinite-
Dimensional Hamiltonian Systems Associated with Non-Linear Partial Difference 
Equations II}, Commun. Math. Phys., {\bf 119} (1988), 677-699

\noindent
[CP] L.CHIERCHIA and P.PERFETTI, {\sl Second Order Hamiltonian Equations 
on $\torus^\infty$ and Almost-Periodic Solutions}, Journal of 
Differential Equations Vol.{\bf 116} No.1 1995, 172-201

\noindent
[CW] W.CRAIG and C.E.WAYNE, {\sl Newton's method and periodic solutions of 
nonlinear wave equations}, Commun. Pure Appl. Math. {\bf 46}(1993) 1409-1498 


\noindent
[FSW] J.FR\"OLICH, T.SPENCER and C.E.WAYNE, {\sl Localization in Disordered 
Nonlinear Dynamical Systems}, J. Stat. Phys. {\bf 42} (1986), 247-274

\noindent
[G] G.GALLAVOTTI, \lq\lq The Elements of Mechanics", Springer-Verlag, New York
1983

\noindent
[Ga] F.GANTMACHER, \lq\lq Lectures in Analytical Mechanics", Mir Publishers 
Moscow, second printing 1975 (chapter 6 p.227)

\noindent
[Ku] S.B.KUKSIN, {\sl Nearly Integrable Infinite-Dimensional Hamiltonian 
Systems}, Lecture Notes in Mathematics 1556, Springer-Verlag, 
New York 1993

\noindent
[LA] S.LANG, \lq\lq Differential Manifolds", Addison-Wesley Eds., 1972

\noindent
[LY] A.LYAPUNOV, \lq\lq Probl{\accent"12 e}me G\'en\'eral de La Stabilit\'e 
du Movement", Princeton Univ.Press, 1947

\noindent
[M] J.K.MOSER, \lq\lq Dynamical Systems", Courant Institute of Mathematical 
Sciences, 1979-1980

\noindent
[Po] H.POINCAR\'E, \lq\lq Les M\'ethodes Nouvelles de La M\'ecanique 
C\'eleste" Tome I No.36-42, Gauthier-Villars, 1892 (Reprinted by 
\lq\lq Librairie Albert Blanchard", rue de M\'edicis, 75006 Paris (1987))

\noindent
[P\"o] J.P\"OSCHEL, {\sl Some Recent Results concerning Quasi-periodic 
Solution for a Nonlinear String Equation}, Preprint 95-1 Universit\"at 
Stuttgart.

\noindent
[SM] C.L.SIEGEL and J.K.MOSER, \lq\lq Lectures on Celestial Mechanics" 
\mathhexbox27821, Springer-Verlag, New York, 1971 

\noindent
[Wa] C.E.WAYNE, {\sl Periodic and Quasi-Periodic Solutions of Nonlinear 
Wave Equations via KAM Theory}, Commun. Math. Phys. {\bf 127} (1990) 479-528
\vskip0.5truecm

\noindent{\ninerm E-mail: perfetti@mat.utovrm.it}
\vfill\eject
\end