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\def\={{\ \equiv\ \ }}\def\ie{{\it i.e. }}\def\dpr{{\partial}}
\def\eg{{\it e.g.}}
\def\OO{{\cal O}}\def\TT{{\cal T}}\def\LL{{\cal L}}
\def\EE{{\cal E}}\def\AA{{\cal A}}\def\PP{{\cal P}}
\def\O{{\Omega}}\def\D{{\Delta}}
\def\t{{\tau}}\def\n{{\nu}}\def\z{{\zeta}}\def\k{{\kappa}}
\def\m{{\mu}}\def\e{{\varepsilon}}\def\a{{\alpha}}
\def\b{{\beta}}\def\o{{\omega}}\def\g{{\gamma}}
\def\p{{\pi}}\def\th{{\vartheta}}\def\s{{\sigma}}\def\Si{{\Sigma}}
\def\se{{ \sqrt{\varepsilon} }}
\def\Im{{\rm \ Im\ }}\def\Re{{\rm \ Re\ }}
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\def\giu{{\vskip.7truecm}}
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\vglue1.truecm
{\bf On the stability problem for nearly--integrable
Hamiltonian systems}
\giu
Luigi Chierchia\footnote{${}^1$}{Dip. di Matematica, $II^a$
Universit\`a di Roma, ``Tor Vergata",
via della Ricerca Scientifica, 00133 Roma, Italia.
{\it Lecture delivered at the S. Petersburg Conference ``Dynamical
Systems", November 1991 describing joint work in collaboration with
Giovanni Gallavotti}, Dip. di Fisica,
Universit\`a di Roma, ``La Sapienza", P. Moro 5, 00185 Roma, Italia}
\vskip1.5truecm
{\bf Abstract:} {\it The problem of stability of the action variables in
nearly--integrable (real--analytic) Hamiltonian systems is considered.
Several results (fully described in {\rm [CG2]}) are discussed;
in particular: (i) a generalization of Arnold's method ({\rm [A]})
allowing to prove instability
(i.e. drift of action variables by an amount of order $1$, often called
``Arnold's diffusion") for general perturbations of ``a--priori unstable
integrable systems" (i.e. systems for which the integrable structure
carries separatrices); (ii) Examples of perturbations of ``a--priori stable
sytems" (i.e. systems whose integrable part can be completely described by
regular
action--angle variables) exhibiting instability. In such examples, inspired
by the ``D'Alembert problem" in Celestial Mechanics (treated, in full
details, in {\rm [CG2]}),
the splitting of the asymptotic manifolds is not exponentially
small in the perturbation parameter.}
{\bf Abstract:} {\it The problem of stability of the action variables in
nearly--integrable (real--analytic) Hamiltonian systems is considered.
Several results (fully described in {\rm [CG2]}) are discussed;
in particular: (i) a generalization of Arnold's method ({\rm [A]})
allowing to prove instability
(i.e. drift of action variables by an amount of order $1$, often called
``Arnold's diffusion") for general perturbations of ``a--priori unstable
integrable systems" (i.e. systems for which the integrable structure
carries separatrices); (ii) Examples of perturbations of ``a--priori stable
sytems" (i.e. systems whose integrable part can be completely described by
regular
action--angle variables) exhibiting instability. In such examples, inspired
by the ``D'Alembert problem" in Celestial Mechanics (treated, in full
details, in {\rm [CG2]}),
the splitting of the asymptotic manifolds is not exponentially
small in the perturbation parameter.}
\giu\giu
{\bf \S 1 A Theorem and a Conjecture by V. I. Arnold}
\giu
Consider a Hamiltonian system with $N$ degree of freedom
with a Hamiltonian of the form $H_\m$ $\=H_0+\m f$,
where $H_0,f$ are real--analytic functions on the phase space
$V\times T^N$ ($V$ being a bounded region in $R^N$,
$T\=R/2\p Z$) and $\m$ is a real parameter. For $\m=0$,
$H_0$ is assumed to be {\it integrable} \ie there exist
$N$ integrals, $I_1,...,I_N$, independent and in involution
(see [A1] or [G] for the standard terminology). Typically,
for $\m\neq0$ the system will be no longer integrable,
however the time for a possible variation of $\OO(|\m|^c)$ of one of
the above integrals is (again: in typical situations) extremely long as
dictated by a result of Nekhoroshev [N].
In 1964 V. I. Arnold conjectured that,
{\it in general, for $\m\neq 0$ small enough, there exist
initial data $z\in V\times T^N$ and a time $T>0$ such that
$|I(\phi_\m^T(z))-I(z)|=\OO(1)$};
here $I\=(I_1,...,I_N)$ and $t\to \phi_\m^t$ is the flow
generated by the Hamiltonian $H_\m$.
The conjecture was based on the following theorem and on its
proof:
\giu
{\bf Theorem} ([A2]).
{\sl
Fix $\e>0$ and let $H_\m\=H_0+\m f$ with:
$$
H_0\= {a^2\over 2} + b+ \ {p^2\over 2} +
\e (\cos q -1)\ ,\qquad
f\= \e (\cos q - 1) (\sin \a + \cos \b)
\eqno(1.1)$$
where $(a,\a),(b,\b),(p,q)$ $\in$ $(R\times T)^3$ are standard canonical
(symplectic) coordinates \ie the equations of motion are:
$$
\eqalign{&\dot \a=a \cr &\dot a=\e \m (1-\cos q) \cos \a \cr}
\quad
\eqalign{&\dot \b=1 \cr &\dot b=\e \m (\cos q -1 ) \sin \b\cr}
\quad
\eqalign{&\dot q=p \cr &\dot p=\e \sin q [1 + \m (\sin \a + \cos \b)]\cr}
$$
For each $00$ such that
$a(0) a''$, provided $\m>0$ is small enough.
}
\giu
Before discussing the method of proof let us comment on the presence
of the {\it two parameters} $\e,\m$.
In the theory of nearly--integrable
Hamiltonian systems one considers Hamiltonians of the form:
$$
H\=h_0(J)+\e F_0(J,\psi)$$
$(J,\psi)\in V\times T^N$
being the so--called action--angle variables and $\e$ a small parameter.
Under suitable non--degeneracy assumptions, in a neighborhood of a {\it
simple resonance} (a ``simple resonance" is a hypersurface in $V$ of the
form $\{ J\in V: \dpr_J h_0\cdot \n_0=0$ for some $\n_0\in Z^N
\backslash\{ 0\}$ and $\dpr_J h_0\cdot \n\neq 0$ for all $\n$ not parallel
to $\n_0$ $\}$)
one can find, for any $Q>0$, a canonical transformation
$(J,\psi)\to(a,\a),(p,q)$ $\in R^{N-1}\times T^{N-1} \times R \times T$
such that in the new variables $H$ takes the form:
$$
\tilde h(a;\e) + P(p,q;a,\e) + \e^{Q+1} \tilde F(a,\a,p,q;\e)
$$
In general, the integrable Hamiltonian $P$ ({\it parametrized}
by $a\in V \subset R^{N-1}$ and $\e$) generates a structure similar to that
of the standard pendulum (\ie isolated unstable equilibria,
separatrices, etc.). The relation of Arnold's Hamiltonian
(1.1) to the general problem of nearly--integrable motions near simple
resonances is transparent.
The proof of the above Theorem is based on the idea of
{\it transition chains of whiskered tori}.
For $\m=0$ the
(integrable) system with Hamiltonian $H_0$ carries a
``partially--hyperbolic structure": there are
{\it lower dimensional invariant tori}
$$
\TT(\o) \=\{p=q=0\} \times \{a=\o,b=b_0\}
\times \{(\a,\b)\in T^2\}$$
on which the motion is {\it quasi--periodic}:
$(\a,\b)\to(\a+\o t,\b + t)$. These tori are linearly unstable
and admit asymptotic manifolds called by Arnold ``whiskers":
$$\eqalign{
W^{\pm}(\TT(\o)) & \= \{a=\o,b=b_0\}\times \{(\a,\b)\in T^2\}
\times \{(p,q): {p^2\over 2} + \e (\cos q -1)=0\}\cr
&= \{ {\rm phase \ points\ } z: {\rm \ dist.\ }
(\phi_0^t(z),\TT(\o))\to 0
\ {\rm as}\ t\to \pm \infty\}\cr}
$$
The whiskers $W^\pm(\TT(\o))$
are 3--dimensional manifolds
lying in the same 5--dimensional ambient space (energy level).
In general two 3--dimensional surfaces in a 5--dimensional space intersect
in a line; however in the integrable (and hence degenerate)
situation ($\m=0$) it is $W^+\=W^-$. The perturbation $f$ in Arnold's
Theorem is taken so that {\it all whiskered tori $\TT(\o)$ are preserved
for $\m\neq 0$}: this, of course, is a highly non--generic property.
The perturbation removes the degeneracy of the integrability and
persistent whiskers will, in general, intersect transversally
along a curve $\z$, which is a trajectory such that
dist.$(\z(t),\TT_\m(\o))\to0$ as $t\to\pm \infty$,
where $\TT_\m$ denotes the continuation in $\m$ of the
persistent torus $\TT(\o)$. Trajectories of this
type were called by Poincar\`e {\it homoclinic}.
Transversality is
checked by a first order (in $\m$) computation by means of
``Poincar\`e [P]--Melnikov [Me]--Arnold [A2] integrals"
(we shall come back on this point).
Persistent whiskers
are arbitrarly close one to the other, therefore transversality
yields {\it heteroclinic trajectories} \ie
trajectories $\z(t)\=\z(t;\o,\o')$ such that $\z$ $\=\{\z(t): t\in R\}$
$= W^+(\TT_\m(\o))\cap W^-(\TT_\m(\o'))$ with $\o\neq \o'$.
Since
in Arnold's example {\it all tori} are preserved, one can construct {\it
long} chains of whiskered tori $\TT_i\=\TT_\m(\o_i)$ for which
$|\o_{i+1}-\o_i|=o(|\m|)$ (say, $|\o_{i+1}-\o_i|=c|\m|^2$)
and $\emptyset \neq W^+(\TT_i)\cap W^-(\TT_{i+1})=\z_i$;
$\o_1a''$. Here $M=\OO( (a''-a')/c \m^2)$
and the reason for taking $|\o_{i+1}-\o_i|=o(\m)$ is related to
the fact that transversality is measured by a suitable determinant,
which is, in general, of $\OO(|\m|)$ (see below).
The idea is now to
``shadow" the ``pseudo--orbit" $\PP\=\cup_{i=1}^M \z_i$
with a true orbit starting near $\TT_1$, staying close for $0\le t\le T$
to $\PP$ and passing at time $T$ near $\TT_M$.
In [A2] it is claimed that this is possible as long as all $\o_i$'s are
irrational. Below we shall sketch the argument under stronger (\ie
``Diophantine") assumptions on the frequencies $\o_i$: the argument will
rely upon a ``strong" KAM linearization around the whiskered tori.
\vskip1.truecm
{\bf \S 2 Three classes of model problems}
\giu
We shall now discuss a few generalizations of the results and methods
outlined above. Rather than introducing the general setting to which our
theory applies, we shall restrict here on the following three {\it model
problems}.
Consider a {\it real--analytic} Hamiltonian of the form:
$$
H_\m\=H_0(a,p,q;\e)+\m\e^Q f(a,\a,p,q)$$
with $(p,q)\in R\times T$, $(a,\a)\in R^{N-1}\times T^{N-1}$
standard symplectic coordinates, $Q\ge 0$, and
the integrable part $H_0$ given by:
$$
H_0\=h(a)+P(p,q;a,\e)\ ,\qquad P\={p^2\over 2J_0}+\e g_0^2J_0
(\cos q -1)
\eqno(2.1)$$
where $g_0,J_0$ may depend on $a$.
We shall then consider the following three models:
$$\eqalignno{
&h\=\sum_{i=1}^{N-1} {a_i^2\over 2 J_i}\ ,
\quad \e>0\quad {\rm fixed}\ ,\quad
|\m|\ll 1\ ,\quad (Q\ {\rm arbitrary}) &{\bf (M1)}\cr
&h\=\sum_{i=1}^{N-1} {a_i^2\over 2 J_i}\ ,
\quad Q> Q_0,\quad 0<\e<\e_0\ll 1, \quad |\m|\le 2,
\quad ({\rm suitable}\ Q_0,\e_0) &{\bf (M2)}\cr
&h\= \o_1 a_1 + \e {a_2^2\over 2 J}\ ,\quad
Q, \e,\m \ {\rm as\ in\ {\bf (M2)}} \qquad (N=3) &{\bf(M3)}}$$
%
\giu
{\bf Remarks:} (i) To cover Arnold's example one should substitute
in {\bf (M1)} $h$ with
$\sum_{i=1}^{N-2}(a_i^2/2J_i)+\o a_{N-1}$, to which,
with the due (and well known) modifications all the theory below
applies (see [CG2]).
(ii) In the models {\bf (M2)} and {\bf (M3)} $\m$ is just an
{\it auxiliary parameter} as its large domain of definition allows to set
it equal to $1$.
(iii) The Hamiltonian $H_0$ in the third model {\bf (M3)} (for which
the above comment (ii) holds as well) is an example of {\it degenerate
system}: for $\e=0$, $H_0$ is independent of the action variable $a_2$.
Such systems are important as they are common in Celestial Mechanics.
Consider the {\it D'Alembert model of the Earth precession}:
a planet assimilated to a rigid rotational ellipsoid with small
``flattness" $\e$ (\ie with ``equatorial radius" $R$ and ``polar radius"
$R/(1+2\e)^{1/2}$), revolving on a given Keplerian orbit of eccentricity
$e\=\e^Q$, around a fixed star and subject only to Newtonian gravitational
forces. One can show that the model just described is a suitable
generalization of {\bf (M3)} above: see [CG2] where ``Arnold's diffusion"
for such a model is proved.
(iv) The model {\bf (M1)} will be called {\it a--priori unstable} as the
integrable part ($\m=0$) contains separatrices on which the motion is
partially hyperbolic as already explained above, The models {\bf (M2)},
{\bf (M3)} are, instead, examples of {\it a--priori stable systems} as
the perturbative (small) parameter is $\e$ [see (ii)] and
separatrices may be introduced only at resonances by the perturbation
(see also the comment after the Theorem in \S 1).
(v) In fact the theory sketched below can be extended so as to cover cases
where $h(a)$ is a rather arbitrary function verifying
standard non--degeneracy conditions and where $J_i,g_0$ could also depend
on $p,q$ (besides depending on the actions $a$ as assumed here):
this extensions allow to cover nearly--integrable
situations in a neighborhood of a simple resonance (included the D'Alembert
model).
\vskip1.truecm
{\bf \S 3 Instability mechanism}\penalty10000
\giu\penalty10000
Roughly speaking the mechanism for instability (or ``drift" or
``diffusion") of the perturbed integrals is based on the following
idea.
Fixed a reference energy $E$, consider a path $\LL$
$\subset \{a: H_0(a,0,0;\e)=E\}$, piecewise analytic.
Attached to
each $a_0\in \LL$ we shall think a lower dimensional
invariant torus
$$
\TT(\o) \=\{\a\in T^{N-1},\
a=a_0,\ p=q=0\} \ ,\quad \o\=\dpr_a h(a_0)
$$
(which is actually independent of $\e$) together with its whiskers
$$
W^\pm(\TT(\o))
\=\{\a\in T^{N-1}; a=a_0;(p,q): P(p,q;a_0,\e)=0\}
$$
According to (partially hyperbolic) KAM theory
([M],[Gr],[Z]), for general $\LL$, ``most" of the whiskered tori
persist for $\m\neq 0$ but small enough. However, {\it gaps}
of size as big as $\OO(\sqrt{|\m|\e^Q})$ where
none of the above tori persist, have to be expected.
It is therefore clear which are the problems to be overcome:
{\bf (P1)} Persistence of whiskered tori emerging from the
unperturbed ones attached to the path $\LL$; smoothness
properties of the whiskers (with respect to parameters,
smooth interpolation properties, etc.); behaviour
of the trajectories around the whiskered tori; analytic continuation for
large values of $\m$ for the models {\bf (M2)}, {\bf (M3)};
{\bf (P2)} Transversal splitting and creation of homo/heteroclinic
orbits;
{\bf (P3)} Quantitative relation between the size of gaps and
the ``size of transversality" (location of high--density zones of
persistent tori and admissible choises of ``diffusion" paths $\LL$).
\vskip1.truecm
{\bf \S 4 Persistence of whiskered tori and their analyticity
properties}
\giu
Regarding problem {\bf (P1)} the following result holds. Let $\LL\=$
$\{a_\s:\s\in [0,1]\}$ be a piecewise analytic (in $\s$) path
on a fixed energy level $\EE_E\=\{a:H_0(a,0,0;\e)=0\}$
and, if $\o_\s\=\dpr_a h(a_\s)$,
let
$$
\tilde \LL \=\{a\=a_{\s \g}:\dpr_a h(a)=\o_\s (1+\g)\ ,\
\s\in [0,1], \g\in [-\bar \g,\bar \g]\}
$$
Clearly if $\bar \g$ is small enough,
$\tilde \LL$ is $|\g|$--close to $\LL$. Finally, let
$$
\Sigma(C,\t)\=
\{ \s\in [0,1]: |\o_\s\cdot \n|^{-1} \le C|\n|^\t , \
\forall \ \n\in Z^{N-1}, \n\neq 0\}
$$
and denote by $B^n_r\=\{x\in R^n:|x_i|\le r\ , i=1,...,n\}$.
\giu
{\bf Theorem 1} ([CG2]). {\bf (i):}
{\sl
Consider the model {\bf (M1)} and fix $k>0$.
There exist $\m_0,\k_0,\bar \g>0$ and $C^k$--functions
$$\eqalign{
& \z: \ (\m,\psi,y,x,\s,\g)\in B^1_{\m_0}\times T^{N-1}
\times B^2_{\k_0} \times [0,1]\times B^1_{\bar \g}
\to \z_\m(\psi,y,x;\s,\g) \in R^N\times T^N\cr
& g: \ (\m,z,\s,\g) \in B^1_{\m_0}\times B^1_{\k_0^2}
\times [0,1] \times B^1_{\bar \g}
\to g_\m(z;\s,\g) \in R\cr}$$
which, for $\s\in \Sigma(C,\t)$ and $\g\in B^1_{\bar \g}$
fixed, are analytic in the remaining arguments, and are such that:
$$
\phi_\m^t\ \z_\m(\psi,y,x;\s,\g) =
\z_\m(\psi + \o_\s(1+\g) t, y e^{-g_\m t}, x e^{g_\m t};\s,\g)\ ,
\quad (\s\in\Sigma(C,\t), |\g|\le\bar \g)
$$
where $\phi_\m^t$ denotes the flow generated by $H_\m$.
In particular
$$
\TT_{\s \g}\=
\{\z_\m(\psi,0,0;\s,\g) : \psi \in T^{N-1}\}
$$
is a $H_\m$--invariant torus with local whiskers given by:
$$\eqalign{
& W^+_{loc}(\TT_{\s \g})\=\{\z_\m(\psi,y,0;\s,\g):\psi\in T^{N-1},
y\in [-\k_0,\k_0]\} \cr
& W^-_{loc}(\TT_{\s \g})\=\{\z_\m(\psi,0,x;\s,\g):\psi\in T^{N-1},
x\in [-\k_0,\k_0]\}\cr}\eqno(4.1)
$$
Finally, the value $\g$ can be chosen to be a smooth function of $\m,\s$
and so that the energy $H_\m(\z_\m(\psi,0,0;\s,\g))$ has a preassigned
value $E$ within the range of the possible unperturbed energy values.
{\bf (ii):} Consider the models {\bf (M2)}, {\bf (M3)}: There exist
$Q_0>1$, $\e_0>0$ such that for $Q>Q_0$, $0<\e<\e_0$ all the above
results hold with $\m_0=2$. More precisely:
Let $\O\=\sup_{\s,\g}|\o_{\s \g}|$; then
there exist a (universal) number $Q_0>1$ and a positive constant $K$
(depending on $\t,\k_0$ and other natural parameters associated to
the functions $H_0,f$) such that if:
$$
K \m_0 \e_0^{Q-Q_0} \ (C \O)^{14} < 1
\eqno(4.2)$$
then the results in {\bf (i)} hold for all $|\m|\le\m_0$,
$0<\e\le\e_0$, $Q>Q_0$.
}
\giu
>From (4.2) it follows that the radius of analyticity in $\m$ is in fact
proportional to $\e_0^{-m}$ for a suitable $m$.
Clearly for $\m_0$ (respectively, $\e_0$) small enough, the
above functions defining the local whiskers can be analytically
extended, using the flow, to ``long stretches": \eg,
if $\p_q$ denotes the projection on the $q$--variable,
the $y,x$--domain of $\z$ can be extended so as to contain
a segment $[-\k,\k]$ so that
$|\p_q \z_\m(\psi,\pm \k,0;\s,\g)|$ and
$|\p_q \z_\m(\psi,0,\pm \k;\s,\g)|$ are larger than $3\p/2$;
of course, in such an extended region, the local whiskers are
$\OO(\e^{Q-Q_0}|\m|)$--close to the unperturbed ones.
The existence of whiskered tori was first established by [M]
and [Gr], and, more recently, it has been re--examined
by [LW]. For the joint smoothness of the parametric representation
(which yields smooth interpolation of the whiskered tori)
compare with the analogous results for maximal
($N$--dimensional) tori by [L1], [S], [P\"o], [CG1].
One can show (see \S 8 of [CG2]) that
from the above linearization it follows that the tori $\TT_{\s\g}$
are {\it transition tori} in the sense of [A2].
Finally, notice that the $\m$ analyticity allows to give a precise
meaning to $\m$--expansions also for the models {\bf (M2)}, {\bf (M3)}
and yields an algorithm to compute (``perturbatively") various
objects of interest.
\vskip1.truecm
{\bf \S 5 Transversal homoclinic points}
\giu
Here we discuss {\bf (P2)}. Consider first the model {\bf (M1)}.
The analyticity
properties established in Theorem 1 imply that $W^\pm\cap \{q=\p\}$
is a {\it graph} over the angles $\a\in T^{N-1}$. By the Implicit
Function Theorem there exist real--analytic functions
$x_\m,y_\m,\psi^\pm_\m$ such that (dropping the $\s,\g$ parameters
in the notation and denoting $\p_q$,...,$\p_a$ the projections on the
coordinates $q$,...,$a$):
$$\eqalign{
& \p_q\ \z_\m(\psi^+_\m(\a),y_\m(\a),0) = \p =
\p_q \ \z_\m(\psi^-_\m(\a),0,x_\m(\a)) \cr
& \p_\a \ \z_\m(\psi^+_\m(\a),y_\m(\a),0) = \a =
\p_\a \ \z_\m(\psi^-_\m(\a),0,x_\m(\a)) \cr}
$$
Define:
$$\eqalign{
& a^+_\m(\a)\= \p_a\ \z_\m(\psi^+_\m,y_\m,0) \ ,\qquad
p^+_\m(\a)\= \p_p\ \z_\m(\psi^+_\m,y_\m,0) \cr
& a^-_\m(\a)\= \p_a\ \z_\m(\psi^-_\m,0,x_\m) \ ,\qquad
p^-_\m(\a)\= \p_p\ \z_\m(\psi^-_\m,0,x_\m) \cr}
$$
Then:
$$
W^\pm\cap \{q=\p\} = \{(a^\pm_\m(\a),\a,p^\pm_\m(\a),\p):\a\in
T^{N-1}\}
$$
Since the $p$--variable can be eliminated by conservation of energy, we see
that finding non--degenerate homoclinic intersections is equivalent to
find $\a_0$ such that:
$$
\Delta(\a_0;\m)\=a^+_\m(\a_0)-a^-_\m(\a_0) = 0\ ,
\quad D(\a_0;\m)\=\det[\partial_\a\Delta(\a_0)]\neq 0
\eqno(5.1)$$
The quantity $D(\a_0;\m)$ is {\it a measure of the transversality}
of the intersection between $W^+$ and $W^-$;
in fact, the eigenvalues of $\partial_\a \Delta$
are related to the geometric angles between $W^+\cap\{q=\p\}$ and
$W^-\cap\{q=\p\}$ (of course such geometry depends upon our coordinates).
Notice that $\D(\a_0;0)\=0$ by the degeneracy of the unperturbed system.
Theorem 1 and a perturbative study of the linearized (in $\m$)
Hamiltonian equations yield easily the following first--order evaluation
of $\Delta$ (compare [P], [Me], [A2]):
\giu
{\bf Proposition 1.}
{\sl In all the above model {\bf (M1)} $\div$ {\bf (M3)},
$\Delta$ is analytic both in $\a$ and $\m$
and if $\Delta\=\m \Delta_1(\a)$ $+\m^2 \Delta_2(\a)$ $+...$ then:
$$
\Delta_1(\a) = \partial_\a m_f
$$
with
$$
m_f(\a;\s,\g)\= -\sum_{0\neq \n\in Z^{N-1}}
e^{i \n\cdot \a}
\int_{-\infty}^{+\infty}
{ e^{i \o_{\s \g}\cdot \n t} \over i \o_{\s \g}\cdot \n}
\partial_t[e^{i\n\cdot \th_t} \hat f_\n(a_{\s \g},p_t,q_t)]dt
\eqno(5.2)$$
where $\hat f_\n$ are the Fourier coefficients of
$\a\to f(a,\a,p,q)$,
$(p_t,q_t)$ is the unperturbed separatrix motion starting
at $q=\p$ and $\th_t$ is such that
$\p_q$ $\phi^t_0(a_{\s \g},\a,p_0,\p)$
$=\a+\o_{\s \g}t + \th_t$.
}
\giu\noindent
Thus, for the model {\bf (M1)} (where $\m$ is small), by the standard
Implicit Function Theorem, {\it homoclinic points}
(in the transversal section $\{ q=\p\}$ ) correspond to
{\it non--degenerate critical points} of the periodic function
$\a\to m_f(\a)$. And it is easy to see that, {\it generically},
$m_f$ has at least $2^{N-1}$ non--degenerate critical points.
The determinant $D$ will be (again: in general) of
$O(|\m|)$ and therefore in order to establish {\it drift along
a path $\bar\LL\=\{\bar a_\s:$ $\s\in[0,1]\}$ }
one will have to check that such a path has a density
of persistent whiskered tori of $o(|\m|)$; see next section.
Here and below $\bar\LL$ will denote a path
$\{a_{\s \g(\s)}:$ $\s\in[0,1]\}$ where the value
$\g(\s)$ has been fixed so that $\bar\LL$ belongs to the same
energy level (compare {\bf (i)} in Theorem 1).
For {\bf (M2)} the situation is more delicate as, in general,
$\partial_\a \D_1$ is {\it exponentially small in $\e$}.
More precisely, assume (for simplicity) that $f$ is a {\it trigonometric
polynomial} in the angular variables $\a,q$ and recall the analyticity
properties of the unperturbed separatrix motion:
$$
p_t= \pm 2 \se \bar J \bar g (\cosh \se \bar g t)^{-1}\ ,
\quad
q_t = 4 \arctan e^{\pm \se \bar g t}\ ,
\quad
\th_t= - 2 \partial_a (\se \bar g \bar J) \tanh \se \bar g t
$$
where $\bar J\=J_0(\bar a_{\s})$, $\bar g \= g_0(\bar a_{\s})$.
In particular, the integrand in (5.2) is holomorphic in $t$ in
a strip $|\Im t|<\p/(2\se\bar g)$.
Therefore, shifting the contour of integration to
$$
\Im t= \ {\rm sign}(\o_{\s \g}\cdot \n) \big( {\p\over 2 \se\bar g}-r\big)
$$
(with any $r>0$) one obtains the estimate:
$$
\sup_{\a\in T^{N-1}} |\partial_\a^j m_f|\le k_j
\sup_{\n\neq 0}[e^{-|\o_{\s \g}\cdot \n| \p/(2\se\bar g)} e^{-|\n|}]
\eqno(5.3)$$
where $k_j$ is a suitable constant. Under our Diophantine assumption on
the frequencies $\o_{\s \g}$ the supremum in the right hand side of (5.3)
can be bounded by (or, better: is essentially equal to)
$c_1 \exp(-c_2/\e^{{(\t+1)\over 2}})$, for suitable positive constants
$c_1,c_2$.
In the case {\bf (M3)} the frequency vector has the form $\o=(\o_1,\e \bar
\o_2)$, for a suitable $\bar \o_2(a)$; thus the {\it supremum in}
(5.3) {\it is of order $1$}. However what is really important here is to
estimate from below $D\=\det \D$ and one can show that, even in the case
{\bf (M3)}, $D|_{\m=1}$ is {\it exponentially small in $\e$}. But this
phenomenon is an {\it accident}: indeed, in general, a second order
(in $\m$) computation shows that $D|_{\m=1}=\OO(\e^M)$ for a suitable
$M>1$; see \S 7 below for a precise statement.
\giu
{\bf \S 6 Density of whiskered tori}
\giu
The strategy at this point is clear: it remains to find , on the energy
surface $\EE_E$, ``diffusion paths" where the gaps ($\=$ ``intervals of
non--persistent whiskered tori") are smaller than the size of $D$
(\ie intuitively, of the ``smallest" non--trivial angle formed by vectors
tangent to the whiskers).
High--density zone of persistent tori may be detected by using the theory
of normal forms in the way it is used in Nekhoroshev's Theorem
([N], [BG]). Then, choosing suitably the Diophantine constant $C$
(recall the definition of $\Si(C,\t)$ in \S 4) as a function of $\m$
or $\e$, one can prove the following results concerning {\bf (M1)}
and {\bf (M3)}.
Before we need a {\bf definition}: {\sl we say that a path $\bar \LL$
$\subset \EE_E$ has ``residual measure" $\rho$ if the Lebesgue
measure of $[0,1]$ $\backslash \Si_{\rm persistent}$, with
$\Si_{\rm persistent}$ $\=$ $\{\s\in [0,1]$ s.t. to $a_\s$ it can be
associated
a persistent whisker (2.1) with the properties described in Theorem 1 $\}$,
is bounded above by $\rho$}. The residual measure gives an upper bound on the
maximal size of gaps in $\bar \LL$.
Now, assume (for simplicity) that $f$ is a trigonometric polynomial in
the angle variables $\a,q$.
\giu
{\bf Proposition 2.}
{\sl
[Case {\bf (M1)}] Let $m>1$. There exist a $\m_1<\m_0$, a $\t_0\ge N-1$,
an open $(N-2)$--dimensional
set $\AA_{f,m}\subset\EE_E$ and a function $C=C_{\m,m}$ such that
$\forall$ $|\m|<\m_1$, $\t\ge \t_0$
any two points of $\AA_{f,m}$ can be joined by a path $\bar \LL$
with $\Si(C_{\m,m},\t_0)$ $\subset$ $\Si_{\rm persistent}$ and
residual measure of $\OO(|\m|^m)$. Furthermore, the set
$\EE_E\backslash$ $\AA_{f,m}$ itself has ($N-2$ dimensional) measure
of $\OO(|\m|^m)$.
}
\giu
{\bf Proposition 3.}
{\sl
[Case {\bf (M3)}] There exist $a_2$--intervals $(a,b)$ of order $\OO(1)$,
constants $Q_1,\e_1$ with $Q_1>Q_0$, $0<\e_1<\e_0$ and
a function $C=C_{\e}$ such that $\forall$ $0<\e<\e_1$,
$|\m|\le 2$ the path $\bar \LL$ $\=$ $\{ (a_1(\s), a+ \s (b-a))\}$
($a_1$ being determined by $\bar \LL$ $\subset \EE_E$)
has $\Si(C_\e,1)$ $\subset$ $\Si_{\rm persistent}$ and
residual measure of $\OO(\exp(-c /\se))$,
$c$ being a suitable positive constant.
}
While Proposition 2 is easily obtained by performing a {\it finite number}
of ``classical perturbative steps", Proposition 3 involves the use of high
order normal forms as in the ``analytic part" of Nekhoroshev theory
(see [N] and [BG]).
We also remark that the sentence ``there exist" in the above propositions
really means ``one can construct", \ie it is possibly to decide wether
a certain region of phase space has a high density of persistent tori.
It is clear that the above results are sufficient in order to get for
{\bf (M1)} and {\bf (M3)} the so--called Arnold's diffusion \ie a drift of
order $1$ in action variables no matter how small the perturbation is
(as long as certain non--degeneracies are verified; see also next section).
The model {\bf (M2)} is more difficult as indeed the size of $D|_{\m=1}$
is smaller than any power in $\e$ (see [L2], [GLT], [HMS], [DS] for lower
dimensional discussions) and we refrain to formulate a density
result as further analysis is needed.
\vfill\eject
{\bf \S 7 Instabilities and Arnold's diffusion}
\giu
>From Proposition 2 and the results of \S 5, it follows easily,
for the model {\bf (M1)}, the existence of ``diffusion paths"
\ie of paths having a density of persistent whiskered tori
higher than the measure of the splitting of the whiskers;
generiticity being related to the condition:
$$
\inf_{\s\in \Si} |\det \partial_\a^2 m_f| \ge \delta > 0
\eqno(6.1)$$
If such a condition (easily checked on explicit examples including
Arnold's one, at least for subintervals of $\OO(1)$) holds,
any two points of $\AA_{f}$ can be joined by a path $\bar a_\s$.
Then using the linearization of Theorem 1,
one can prove that the time $T_\m$ needed for the $a$--variable to
{\it drift along the path $\bar \LL$} can be {\it bounded below} by:
$$
T_\m \le c_3 e^{c_4/|\m|^2}
$$
for suitable positive constants $c_3,c_4$; such constants depends of course
on $\e$ (which for {\bf (M1)} is fixed) and indeed $c_3$ is exponentially
small in $\e$.
We collect now the above results for the model {\bf (M3)} with
$$
f\= \cos(\a_1+q) + \cos(\a_2+q)
\eqno(7.1)$$
in the following
\giu
{\bf Theorem 2}. {\sl Consider {\bf (M3)} and set $\m=1$ \ie consider
$H_1\=H_0(a_1,a_2,p,q;\e)$ $+$ $\e^Q f$ with $f$ as in (7.1). Then the
above Theorem 1 and Proposition 3 apply for $Q>Q_1$ and $0<\e<\e_1$.
Furthermore $\a_0\=(0,0)$ (at $\phi=\p$) is a ``transversal homoclinic
point" \ie a
non--degenerate solution of (5.1) and $D(\a_0;1)$ $=$ $\OO(\e^m)$
for a suitable $m>1$. Along any $\OO(1)$--path $\bar \LL$ as in
Proposition 3 ``Arnold's drift or diffusion" takes place \ie there exist
initial data $z_\e(0)$ generating a trajectory $z_\e(t)$ which for $t=0$
has $a_2$--coordinate $\e$--close to one extreme of $\bar \LL$ while for a
suitable $t=T_\e$ it has $a_2$--coordinate $\e$--close to the other extreme
of $\bar \LL$ [recall that {\rm length} of $\bar \LL$ $=$ $\OO(1)$].
The time $T_\e$ can be bounded below by $c_5 \exp (1/\e^{c_6})$ for
suitable constants $c_5,c_6$.
}
\giu
As already mentioned above this theorem is a consequence of Theorem 1,
Proposition 3 and of a second order (in $\m$) analysis of the Melnikov
integrals showing that the ``homoclinic angles" (better: $D$) are {\it
polynomially small in $\e$} (the higher $\m$--orders being easily
controlled because of the large radius of $\m$--convergence).
\giu\giu
{\bf References}
\giu
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\bye