\magnification=1200

\def\phi{\varphi}
\def\eps{\varepsilon}
\def\om{\omega}
\def\la{\lambda}

\def\D{{\cal D}}
\def\C{{\cal C}}
\def\G{{\cal G}}
\def\T{{\cal T}}
\def\R{{\cal R}}

\def\toro{{\bf T}}
\def\rin{{\bf Z}}
\def\meno{\hskip1pt\backslash}


%%% notazioni cambiate %%%

\def\Fm{F} % era {\langle f\rangle}
\def\Gm{G} % era {\langle g\rangle}
\def\fm{f^-} % era f^{\leq K}
\def\fp{f^+} % era f^{>K}

\def\.#1{{\dot #1}}
\def\norm#1{{\vert \vert #1 \vert \vert}}


\parindent=0pt
\parskip=10pt

{\bf EXISTENCE OF INVARIANT TORI FOR NON HAMILTONIAN PERTURBATIONS OF
INTEGRABLE SYSTEMS}

\bigskip\bigskip\bigskip

Dario Bambusi,
{\it Dipartimento di Matematica, Universit\`a di Milano, via Saldini 50,
20133 Milano (Italy)}

Giuseppe Gaeta,
{\it Department of Mathematical Sciences, Loughborough University,
Loughborough LE11 3TU (England)}


\bigskip\bigskip\bigskip


{\bf Abstract} We extend known results on the existence of invariant tori
for non hamiltonian
perturbations of integrable systems, to the case where the unperturbed
system is anisocronus. The
result holds for perturbations admitting a definite finite number of
derivatives, and belonging to a
generic class.

\bigskip\bigskip\bigskip





We address here the problem of existence of invariant tori for systems of
the form
$$
\eqalign{
\dot I
&=\eps \ f(I,\phi,\eps)
\cr
\dot\phi
&=\omega(I)+\eps \ g(I,\phi,\eps)\cr
}
\eqno{(1)}
$$
where $I\in\G\subset\Re^n$ (with $\G$ open) are the slow variables,
$\phi\in\toro^m$ are the fast angular variables, and $\eps\in[0,\eta]$
is a (small) real parameter ($\eta>0$). All functions are assumed to be
at least $C^2$. We recall that this kind of systems arise naturally for
example as non hamiltonian perturbations of integrable hamiltonian
systems. Notice also that if the unperturbed system is resonant, then
there are also some angles that play the role of slow variables.

We remark that for $m=1$ the problem is of a different nature, as on the
one side we are dealing with periodic solutions, and on the other we
have no problems due to resonances. Thus, in the following we will
assume we have to deal with a proper torus $T^m$, i.e. we will assume
$m>1$ (for the case $m=1$ see e.g. [1]).

The search of invariant tori for system (1) is usually performed in
three steps: first one performs a first order averaging, i.e. a
coordinate transformation $(I,\phi)=\T(J,\psi,\eps)$ taking the system
(1) into the system
$$
\eqalign{
\dot J&=\eps \ \Fm (J) + \eps^{1+p} \
\R(J,\psi,\eps)\ ,
\cr
\dot\psi&=\omega(J) + \eps \tilde g (J,\psi,\eps)\ .}
\eqno{(2)}
$$
where
$$
\Fm (I) := {1 \over (2\pi)^m}
\int_{\toro^{m}} f (I,\phi,0) \ d \phi_1 ... d \phi_m\ , \eqno(3)
$$
and $p$ is a suitable positive number.
Then one considers the truncated system
$$
\eqalign{
\dot J &= \eps \ \Fm (J)
\cr
\dot\phi&=\omega(J) }\eqno{(4)}
$$
and looks for a zero $J_0$ of $\Fm$; obviously the manifold
$T_0:=J_0\times\toro^m$ is an invariant torus for the averaged systems
(4). Finally one has to prove that such a torus is only deformed by a
small perturbation, so that there exists an invariant torus also for
system (2) and therefore also for the original system (1).

The technical difficulties one meets in making the above
procedure rigorous rest essentially in justifying the averaging procedure
transforming (1) into (2), and in proving that the invariant tori of (4)
can be continued to invariant tori of (2).

It is particularly simple to solve the above thechnical problems when
the frequency vector is diophantine and constant, i.e. independent of
the slow variables $I$. Indeed, in such a case the averaging
transformation which maps (1) in (2) is globally defined (on $\G$), and
the problem of continuing a torus can be solved in different ways.
Bogolyubov et al. (see [2]) solved it by using a KAM type technique,
obtaining that there exists a large set of $\Re^m$ such that if the
frequency vector belongs to such a set, and if $\Fm$ has a zero in $\G$,
then the original system has an invariant torus. A different solution
was given by Yagasaki$^{[3]}$ who realized that, since the zero of $\Fm$
is generically hyperbolic (i.e. all the eigenvalues of the linearization
of $\Fm$ at such a point have non vanishing real part), the invariant
torus of (4) is a normally hyperbolic invariant manifold, and therefore
it persists under small perturbation$^{[4,5]}$.

On the other side, we were not able to find in the literature any result on
the general case where $\omega=\omega(I)$ is non constant. In the
present note we prove under generic conditions that an invariant torus of the
averaged system can be continued to an invariant torus of the original
system (1) even when the frequency is actually dependent on $I$.

Here the question is more delicate than in the case of constant
frequency. Indeed, the averaging transformation is defined only on the
set of slow variables on which the frequency vector is sufficiently non
resonant; therefore the system (1) is equivalent to (2) only on such a set.
It follows that an invariant torus of (4) is also an invariant torus of
(1) only if it is completely contained in such a nonresonant set.
Moreover, the nonresonant set depends on $\eps$, and when $\eps=0$ it
has a dense complement; therefore the question of existence of invariant
tori for (1) requires some care.

The rest of the paper is devoted to the proof of existence of an
invariant torus for the general case.

In order to obtain such a proof we have to look more in detail at the
averaging procedure. Indeed it is well known that the averaging method
holds as long as the frequencies are nonresonant enough. The whole
problem consists in specifying quantitatively what does it means
``enough''.

To this aim, fix a positive $K$, and split $f$ as
$f=\fm + \fp$, where
$$
\fm (I,\phi):=\sum_{|k|\leq K}f_{k}(I) \ e^{ik\cdot \phi}\ , \eqno(5)
$$
and $f_{k}(I)$ is the $k-th$ Fourier coefficient
of $f(I,\phi,0)$. The key
remark is that, due to the decay of the Fourier coefficients of a smooth
function, taking $K=O(\eps^{-b})$ with some suitable $b>0$, one can
consider $\fp$ as being a higher order perturbation. Therefore the
construction of the averaging transformation involves only $\fm$ which, by
definition, contains only Fourier components of order less than $K$. This
allows in turn to show that the system (2) is equivalent to (1) on an open
set, of size of the order of some (negative) power of $K$. In order to make
quantitative the above procedure let us consider the distinguished subset of
points $I\in\G$ characterized by a diophantine frequency $\omega(I)$.
Formally, we fix two positive real constants $\gamma$ and $\tau$, and consider
the set $\Gamma(\gamma,\tau)$ defined as
$$
\Gamma(\gamma,\tau)
=\left\{I\in\G\>:\>|k\cdot\omega(I)|\geq \gamma|k|^{-\tau}\ {\rm for\ all}
\ k\in\rin^m\meno\left\{0\right\}
\right\}\ .
\eqno(6)
$$
Notice that for $\tau> m-1$, the union over $\gamma$ of the above sets has
full mesure.
For positive $\rho$ we define the
extension $\Gamma_\rho$ of $\Gamma$ as
$$
\Gamma_\rho(\gamma,\tau)=\left(\bigcup_{I\in\Gamma(\gamma,\tau)}
B_\rho(I)\right)\cap\G \ , \eqno(7)
$$
where $B_\rho(I)$ is the closed sphere of radius $\rho$ and center $I$. Then,
following the lines of the proof of lemma 4.3 of ref.~[6] it is easy to
proof the following

\noindent{\bf Lemma.}
\quad {\it Consider the system of differential equations~(1); assume
that $f$, $g$ and $\omega$ are $C^2$ on the closure of
$\G\times\toro^m\times[0,\eta]$. Moreover assume that there exist a
positive $r$ and a constant $\C$ such that, for each fixed
$\eps\in[0,\eta]$ the $C^2$ norm of $\fp (I,\phi,\eps)$ is bounded
by $\C K^{-r}$ on $\G\times\toro^m$. Then, provided $\eps$ is small
enough, there exists a differentiable coordinate transformation defined
on $\Gamma_\rho(\gamma,\tau)$, with
$$
\rho=O(\eps^{(\tau +1)/\alpha} )
$$
($\alpha = 2 \tau + r +1$) such that in the new variables the system (1)
takes the averaged form (2) with $p=r/\alpha$. Moreover, for each fixed
$\eps\in[0,\tau]$, the $C^1$ norms of $\R$ and of $\tilde g$ are
bounded on $\Gamma_\rho(\gamma,\tau)\times\toro^m$. }

We point out that the assumption on $\fp$ is implied for example by the
assumption that the Fourier
series of the differential of order $(r+2)$ of $f$ with respect to the
angles is absolutely summable.
This, in turn, is implied e.g. by the property that $f$ admits more than
$(r+m+2)$ derivatives with respect
to the angles.

Assume now that there exists a
$J_0\in\Gamma_\rho$ which is an hyperbolic zero of $\Fm$. Then
$T_0:=J_0\times\toro^m$ is a normally hyperbolic invariant torus of
the truncated system. So, using the theory of normally
hyperbolic manifolds, one can conclude that the system (4) has a true invariant
normally hyperbolic torus which is $\eps^{r / \alpha}$ close to the
torus $T_0$. However, this is true only if the
deformed invariant torus is contained in $\Gamma_\rho$. In order to ensure
this property uniformly
in $\eps$, we first assume that $\omega(J_0)$ is diophantine (otherwise,
since $\rho \to 0$ as
$\eps \to 0$, there would exist $\eps_*$ such that for $\eps < \eps_*$ one has
$J_0\not \in \Gamma_\rho$). Secondly, the size $\rho$ of the set
$\Gamma_\rho(\gamma,\tau)$ must be larger than the distance
between the deformed torus and the
unperturbed one $T_0$. Looking at the proof of the theorem on the
persistence of invariant iperbolic manifolds (see [4]) it is easy to see
that the distance between the torus $T_0$ and the invariant torus of
system (2) is $O(\eps^{r/\alpha})$. So, provided
$r > \tau + 1$, the invariant torus of the system (2) is also an invariant
torus of the system (1), and we have the following

\noindent {\bf Proposition\quad}{\it In the same hypotheses and
notation as in the lemma above, assume also that $\Fm$ has a hyperbolic
zero $J_0$, which for some positive $\gamma,\tau$ is contained in
$\Gamma(\gamma,\tau)$, and that the constant $r$ is larger than
$\tau+1$. Then, for $\eps$ small enough, the system (1) has an
invariant torus $O(\eps^{r/\alpha})$ close, in the $C^1$ topology, to
the torus $I=J_0$, $\phi\in\toro^{m}$.}


We remark that if $\tau>m-1$ then, generically, a zero $J_0\in\G$ satifyies
the hypotheses of the proposition above. Therefore our assumptions are
generically satisfied, for example in the class of the functions $f$
admitting more than $(2m+2)$ derivatives
with respect to the angles.

We would also like to point out that the regularity we need is less than the
one required when using KAM techniques, where more than $3m-1$
derivatives are required (see e.g. [7]).

\bigskip\bigskip

{\bf Acknowledgements}

This work was performed during a visit of D.B. in Loughborough, and
the final version of the paper completed during a visit of G.G. in
Milano. We acknowledge the support of a L.U.T.
PAS Research Committee grant, which made possible the visit of D.B. in
Loughborough.

\

{\bf References}

[1] V.I.~Arnold, V.V.~Kozlov, A.I.~Neishtadt: Mathematical aspects of
classical and celestial mechanics. In V.I.~Arnold (ed.): {\it Dynamical
Systems III}, Encyclopaedia of mathematical sciences, vol.~3,
Springer-Verlag (Berlin 1988)

[2] N.N.~Bogoljubov, Ju.A.~Mitropoliskii, A.M.~Samolienko: {\it Methods of
Accelerated Convergence
in Nonlinear Mechanics}; Springer, Berlin 1976

[3] K.~Yagasaki: ``{Chaotic motions near homoclinic manifolds and resonant
tori in quasi-periodic
perturbations of planar Hamiltonian-systems}''; {\it Physica D} {\bf 69},
232-269 (1993)

[4] N.~Fenichel: ``Persistence and smoothness of invariant manifolds for
flows''; {\it Ind. Univ. Math. J} {\bf 21}, 193--226 (1971)

[5] M.W.~Hirsch, C.C.~Pugh, M.~Shub: {\it Invariant Manifolds.} Lecture
Notes Mathematics {\bf
583}, Springer, Berlin 1977.

[6] M.~Andreolli, D.~Bambusi, A.~Giorgilli: ``{On a weakened form of the
averaging principle in
multifrequency systems}''. {\it Nonlinearity} {\bf 8}, 283--293 (1995).

[7] J. P\"oschel: ``{Integrability of hamiltonian systems on Cantor
sets}''. {\it Comm. Pure and
Appl. Math.} {\bf 35}, 653--695 (1982).



\bye