% Test patterns of special characters
%
% exclamation mark ! double quote " number #
% tilde ~ at @ period .
% asterisk * plus + comma ,
% minus - backslash \ forward slash /
% colon : semicolon ; less than <
% equals = greater than > question mark ?
% left parenthesis ) right parenthesis ) circumflex (caret) ^
% left bracket [ right bracket ] underscore _
% left brace { right brace } vertical bar |
% left single quote ` percent % ampersand &
% right single quote ' dollar $
\documentstyle[12pt]{article}
\textwidth=16.2cm \textheight=23cm
\oddsidemargin=0mm \evensidemargin=0mm \topmargin=-5mm
\newcommand{\mR}{{\bf R}} \newcommand{\mT}{{\bf T}}
\newcommand{\mZ}{{\bf Z}}
\newcommand{\Iall}{I^{\rm all}} \newcommand{\kall}{k^{\rm all}}
\newcommand{\oall}{\omega^{\rm all}} \newcommand{\tall}{\theta^{\rm all}}
\newcommand{\zpart}{z^{\rm part}} \newcommand{\Bpart}{B^{\rm part}}
\newcommand{\half}{{\textstyle \frac{1}{2}}}
\newcommand{\dFdI}{{\displaystyle \frac{\partial F}{\partial\Iall}}}
\newcommand{\const}{{\rm const}}
\begin{document}
\baselineskip=15.5pt
\parindent=0mm
{\large\bf The 1991 AMS-MSC: 58F27, 58F30, 58F36, 70H05}
\vspace{7mm}
{\large\bf M.~B.~Sevryuk}
\vspace{5mm}
{\small
Institute of Energy Problems of Chemical Physics
The Russia Academy of Sciences
Lenin prospect~38, Bldg.~2, Moscow~117829, Russia
E-mail: rusin@chph.ras.ru
}
\vspace{5mm}
\baselineskip=19.4pt
{\LARGE\bf Invariant tori \\ of intermediate dimensions \\
in Hamiltonian systems: \\ A brief survey}
\vspace{7mm}
\baselineskip=15.5pt
\hrule
\vspace{3mm}
{\small
In the present paper, we survey recent results on the existence and the
structure of Cantor families of invariant tori of dimensions $p>n$
in a neighborhood of families of invariant $n$-tori in Hamiltonian systems
with $d\geq p$ degrees of freedom.
}
\vspace{3mm}
\hrule
\baselineskip=19.4pt
\parindent=5mm
\vspace{7mm}{\large\bf
1. Whitney-smooth families of invariant tori
}\vspace{5mm}
Invariant submanifolds of the phase space are one of the main subjects of
research
in Hamiltonian dynamics as well as in the general theory of dynamical systems.
Of all the invariant manifolds of Hamiltonian systems, invariant tori (and
asymptotic surfaces to tori) are best studied and occur most frequently.
This is due, in the long run, to the fact that any connected compact Abelian
finite-dimensional Lie group is a torus. Invariant tori of Hamiltonian systems
possess a number of remarkable properties under examination in the KAM
(Kolmogorov--Arnold--Moser) theory. The aim of the present survey is to discuss
the recent achievements pertaining to one of these properties, namely,
the so called excitation of elliptic normal modes. For simplicity, in the
sequel all the systems will be autonomous and real-analytic (many of the
results presented below can be generalized {\sl mutatis mutandis\/} to
infinitely differentiable and even finitely smooth systems).
To start with, we will list some main properties of invariant tori in
Hamiltonian flows. The relevant bibliography and/or proofs are given,
e.g., in recent works \cite{Laz}--\cite{Vill} which contain surveys for
various branches of Hamiltonian dynamics.
\vspace{2mm}
{\bf A.} A ``typical'' (in the sense to be explained below) invariant
$n$-torus of a ``typical'' analytic Hamiltonian system with $n+m$ degrees
of freedom ($n\geq 0$, $m\geq 0$) is analytic itself and {\em isotropic\/}
(i.e., the restriction of the symplectic structure to this torus vanishes),
while the dynamics in a neighborhood of the torus is extremely regular
{\em in the approximation linear with respect to the distance from the
torus}. To be more precise, in a neighborhood of the torus, one can introduce
the coordinates $\varphi\in\mT^n=(\mR/2\pi\mZ)^n$, $I\in\mR^n$, $z\in\mR^{2m}$
($I$ and $z$ range near the origins) in which the torus itself is given by
the equations $\{I=0,z=0\}$, the symplectic structure takes the form
\begin{equation}
{\textstyle \sum_{i=1}^ndI_i\wedge d\varphi_i+
\sum_{j=1}^mdz_j\wedge dz_{j+m}, }
\label{eqn1}
\end{equation}
and the Hamilton function takes the form
\[
H=c+\langle\omega,I\rangle+
\half\langle z,Bz\rangle+O(|I|^2+|z|^3).
\]
Here $\langle\cdot,\cdot\rangle$ denotes the inner product of two vectors,
$c\in\mR$, $\omega\in\mR^n$, and $B$ is a symmetric $2m\times 2m$ matrix
(independent of $\varphi$), $\det B\neq 0$, the frequencies
$\omega_1,\ldots,\omega_n$ satisfying the standard strong incommensurability
condition: there exist positive constants $\tau_0$ and $\gamma_0$ such that
\begin{equation}
|\langle k,\omega\rangle|\geq\gamma_0|k|^{-\tau_0}
\label{eqn2}
\end{equation}
for all $k\in\mZ^n\setminus\{0\}$. The equations of motion in the coordinates
$(\varphi,I,z)$ have the form
\begin{equation}
\dot{\varphi}=\omega+O(|I|+|z|^2), \quad
\dot{I}=O(|I|^2+|z|^3), \quad
\dot{z}=\Omega z+O(|I|^2+|z|^2).
\label{eqn3}
\end{equation}
Here $\Omega=J_mB$, whereas $J_m$ is the matrix of the canonical skew-symmetric
product in $\mR^{2m}$:
\begin{equation}
J_m=\left( \begin{array}{cc} 0 & -E_m \\ E_m & 0 \end{array} \right) ,
\label{eqn4}
\end{equation}
$E_m$ being the identity $m\times m$ matrix. Without loss of generality one
can set $c=0$.
\vspace{2mm}
{\sc Remark~1.}
An analytic function of $\varphi$, $I$, $z$ equal to $O(|I|^2+|z|^3)$ can
be represented as a sum of terms of the form $I_uI_v\chi$, $I_uz_vz_w\chi$,
and $z_uz_vz_w\chi$ where $\chi$ are analytic functions of $\varphi$, $I$, $z$.
\vspace{2mm}
{\sc Remark~2.}
The flow on the initial torus is {\em parallel\/} (the equations of motion
can be reduced to the form $\dot{\varphi}=\omega$ with a constant vector
$\omega$), {\em ergodic\/} (the frequencies $\omega_1,\ldots,\omega_n$ are
independent over rationals) and, moreover, {\em Diophantine\/} (the infinite
system of inequalities (\ref{eqn2}) is met). Ergodic parallel flows on tori are
otherwise said to be {\em quasi-periodic}. Invariant tori carrying
quasi-periodic (or Diophantine) flows are sometimes themselves said to be
quasi-periodic (respectively Diophantine).
\vspace{2mm}
{\sc Remark~3.}
In coordinates $(I,z)$, the matrix of the variational equation (of order
$2m+n$) along the torus under consideration has the form
\[
\left( \begin{array}{cc} 0 & 0 \\ 0 & \Omega \end{array} \right) .
\]
In particular, it is independent of $\varphi$. This property is called
{\em reducibility\/} of the torus. The term ``Floquet torus'' is also used
\cite{BHS1,BHS2}. The torus is said to be {\em elliptic\/} if all the
eigenvalues of matrix $\Omega$ are purely imaginary, and is said to be
{\em hyperbolic\/} if all the eigenvalues of $\Omega$ lie outside the
imaginary axis.
\vspace{2mm}
{\sc Remark~4.}
In fact, any quasi-periodic invariant torus of a Hamiltonian system is
isotropic automatically provided that the symplectic structure is exact
(determines the zero cohomology class) \cite{BHS2,Herman}.
\vspace{2mm}
{\sc Remark~5.}
Hamiltonian systems on manifolds with a non-exact symplectic structure
admit also Diophantine invariant tori of dimension {\em greater\/} than the
number of degrees of freedom. As a rule, those tori are {\em coisotropic\/}
(their tangent space at any point contains its skew-orthogonal complement).
We will not consider such tori in the present paper.
\vspace{2mm}
{\bf B.} Reducible Diophantine isotropic invariant $n$-tori of ``typical''
Hamiltonian systems with $n+m$ degrees of freedom are not isolated in the
phase space (for $n>0$) but are organized into {\em Whitney-smooth\/}
$n$-parameter families. The parameter $\xi$ labeling the tori ranges in
some Cantor set $\Xi\subset\mR^n$ of positive Lebesgue measure, the
$2n$-dimensional Hausdorff measure of the union of all the tori $T_{\xi}$
of a given family being positive. The coordinate functions
$\varphi_{\xi}:M\to\mT^n$, $I_{\xi}:M\to\mR^n$, $z_{\xi}:M\to\mR^{2m}$
on the phase space $M$,\footnote{to be more precise, on some neighborhood
of the union of the tori} that normalize (in the first approximation) the
flow near the torus $T_{\xi}$, can be treated as functions defined on
$M\times\Xi$. Whitney-smoothness means that these functions are extendible
to functions defined on the whole space $M\times\mR^n$, analytic in the
first argument $\zeta\in M$ and infinitely differentiable in the second
argument $\xi\in\mR^n$. In particular, the tori $T_{\xi}$ themselves, the
frequency vectors $\omega=\omega(\xi)$, and the matrices $\Omega=\Omega(\xi)$
depend on $\xi$ in a Whitney-smooth fashion.
\vspace{2mm}
{\sc Remark.}
For $n\leq 1$, the $n$-parameter families of invariant $n$-tori in question
are analytic rather than merely Whitney-smooth. For $n=0$, these families are
isolated equilibrium points, whereas for $n=1$, they are one-parameter
families of close trajectories (one can take, e.g., the value of the
Hamilton function as a parameter).
\vspace{2mm}
Hamiltonian systems with $n+m$ degrees of freedom that possess Whitney-smooth
$n$-parameter families of reducible Diophantine isotropic invariant $n$-tori
constitute a set with non-empty interior in the functional space of all the
Hamiltonian systems with $n+m$ degrees of freedom \cite{BHS1,BHS2}.
It is this property of Hamiltonian dynamics that is called the
``typicality'' of such systems in the present paper. It is the {\em families
of tori\/} of this form rather than individual tori that are the elementary
``bricks'' the whole complex of quasi-periodic motions in Hamiltonian systems
consists of.
\vspace{2mm}
{\bf C.} In spite of quasi-periodicity and reducibility of each invariant
$n$-torus $T_{\xi}$ belonging to a Whitney-smooth $n$-parameter family
($\xi\in\Xi\subset\mR^n$), the dynamics in a neighborhood of the family
as a whole is extremely complicated.
\vspace{2mm}
1) In the space between the tori, there are other families of invariant tori
of smaller dimensions $p$ ($1\leq p\leq n-1$), so called {\em cantori\/}
(invariant sets of Cantor structure), and chaotic motion zones which surround
$p$-tori and cantori (if $n>1$). Roughly speaking, the tori of dimensions
$p0$ then in an arbitrarily small neighborhood of the $n$-torus $T$,
there are invariant tori of dimension $n+\nu$ as well as those of all the
intermediate dimensions $n+1,\ldots,n+\nu-1$. It is the existence of such
tori of dimensions greater than $n$ that is called the {\em excitation of
the elliptic normal modes\/} of torus $T$. In the sequel, we will consider
two approaches to studying the families of ``excited'' tori of dimensions
$p>n$. These approaches differ mainly in the genericity conditions imposed
on the initial family of $n$-tori.
\vspace{5mm}{\large\bf
2. The formal normal form of the Hamilton function in a neighborhood
of an invariant torus
}\vspace{5mm}
For simplicity, suppose first that the initial $n$-torus $T$ is elliptic,
and $\pm i\omega^N_1,\ldots,\pm i\omega^N_m$ are the eigenvalues of matrix
$\Omega$ in (\ref{eqn3}). The numbers $\omega_1,\ldots,\omega_n$ are called
the {\em intrinsic\/} frequencies (or just frequencies) of torus $T$, while
the numbers $\omega^N_1,\ldots,\omega^N_m$ are called the {\em normal\/}
frequencies of torus $T$ \cite{BHS1,BHS2} (the superscript $N$ is from the
word ``normal''). If the complete collection of frequencies
$(\omega,\omega^N)$ is Diophantine:
\[
|\langle k,\omega\rangle+\langle k^N,\omega^N\rangle|\geq
\gamma_1(|k|+|k^N|)^{-\tau_1}
\]
for all $k\in\mZ^n$, $k^N\in\mZ^m$, $|k|+|k^N|>0$, then under an appropriate
choice of the signs of numbers $\omega^N_1,\ldots,\omega^N_m$, the Hamilton
function $H$ can be reduced in a neighborhood of torus $T$ to a
{\em Birkhoff-like normal form}
\begin{equation}
H=\langle \omega,I\rangle+\langle \omega^N,I^N\rangle+F(I,I^N)
\label{eqn5}
\end{equation}
by a formal canonical transformation. Here $I^N\in\mR_+^m$,
$I^N_j=\frac{1}{2}(z_j^2+z_{j+m}^2)$ for $1\leq j\leq m$, and $F$ is a formal
power series without constant and linear terms \cite{Vill,Jorba1} ($\mR_+^m$
being the space of vectors of length $m$ with non-negative components). In
the notation $z_j=\sqrt{2I^N_j}\cos\varphi^N_j$,
$z_{j+m}=\sqrt{2I^N_j}\sin\varphi^N_j$ ($\varphi^N\in\mT^m$), the symplectic
structure (\ref{eqn1}) takes the form
\[
{\textstyle \sum_{i=1}^ndI_i\wedge d\varphi_i+
\sum_{j=1}^mdI^N_j\wedge d\varphi^N_j, }
\]
so that the equations of motion afforded by Hamilton function (\ref{eqn5})
have the form
\[
\dot{I}=0, \quad
\dot{I}^N=0, \quad
\dot{\varphi}=\partial H/\partial I, \quad
\dot{\varphi}^N=\partial H/\partial I^N.
\]
Each manifold
\begin{equation}
\left\{ I=C=\const\in\mR^n, \;\; I^N=C^N=\const\in\mR_+^m\right\}
\label{eqn6}
\end{equation}
is a formal invariant torus in our system of dimension $n+r$ where $r$ is
the number of {\em positive\/} components of vector $C^N$ ($0\leq r\leq m$).
The flow on torus (\ref{eqn6}) is parallel, and the torus itself is reducible,
elliptic and isotropic. Assume for definiteness that the first $r$ components
of vector $C^N$ are positive while the remaining components are equal to zero.
One easily sees that the intrinsic $n+r$ frequencies of torus (\ref{eqn6})
are
\[
\omega_i+\partial F/\partial I_i, \quad 1\leq i\leq n, \qquad
\omega^N_j+\partial F/\partial I^N_j, \quad 1\leq j\leq r,
\]
whereas the normal $m-r$ frequencies are
\[
\omega^N_j+\partial F/\partial I^N_j, \quad r+1\leq j\leq m
\]
(of course, the derivatives are taken at the point $(C,C^N)$). Torus $T$
corresponds to the values $C=0$, $C^N=0$. Its neighborhood is a formal union
of invariant tori (\ref{eqn6}) of dimensions $n,n+1,\ldots,n+m$.
We wonder what will persist in this picture if one takes into account the
divergence of the transformation that reduces the Hamilton function to form
(\ref{eqn5}).
\vspace{5mm}{\large\bf
3. Excitation of elliptic normal modes of an invariant torus
}\vspace{5mm}
Consider the general case of torus $T$ with an arbitrary number $\nu$
($0\leq\nu\leq m$) of pairs of purely imaginary eigenvalues of matrix $\Omega$
in (\ref{eqn3}). Fix an arbitrary integer $r$ in the interval $0\leq r\leq\nu$
and $r$ pairs $\pm i\omega^N_1,\ldots,\pm i\omega^N_r$ of purely imaginary
eigenvalues of matrix $\Omega$. Our problem is to construct reducible
Diophantine isotropic invariant $(n+r)$-tori with intrinsic frequencies close
to $\omega_1,\ldots,\omega_n,\omega^N_1,\ldots,\omega^N_r$ in a neighborhood
of $n$-torus $T$. Introduce the notation
\[
(\omega_1,\ldots,\omega_n,\omega^N_1,\ldots,\omega^N_r)=\oall\in\mR^{n+r},
\]
so that
\[
\oall_i=\omega_i, \quad 1\leq i\leq n, \qquad
\oall_{n+j}=\omega^N_j, \quad 1\leq j\leq r.
\]
For the sequel, we have to point out exactly what norms in $\mR^L$ and
$\mZ^L$ we are considering. Under $|\cdot|$, we will always understand the
$l_1$-norm:
\begin{equation}
|x|=|x_1|+\cdots+|x_L|.
\label{eqn7}
\end{equation}
Besides, we will also use the Euclidean norm $l_2$ to be denoted by
$\|\cdot\|$:
\begin{equation}
\|x\|^2=x_1^2+\cdots+x_L^2.
\label{eqn8}
\end{equation}
Assume all the eigenvalues
$\pm i\omega^N_1,\ldots,\pm i\omega^N_r,\pm\lambda_1,\ldots,\pm\lambda_{m-r}$
of matrix $\Omega$ to be pairwise distinct and to satisfy, together with the
frequencies $\omega_1,\ldots,\omega_n$, the following condition: there exist
positive constants $\tau$ and $\gamma$ such that
\begin{equation}
|i\langle\kall,\oall\rangle+\langle l,\lambda\rangle|\geq
\gamma|\kall|^{-\tau}
\label{eqn9}
\end{equation}
for all $\kall\in\mZ^{n+r}\setminus\{0\}$, $l\in\mZ^{m-r}$, $|l|\leq 2$.
Then, under an appropriate choice of the signs of numbers
$\omega^N_1,\ldots,\omega^N_r$, for any $R>0$ the Hamilton function $H$
can be reduced to the form
\begin{equation}
\renewcommand{\arraystretch}{1.4} \begin{array}{cl}
H \;\; = & \langle\oall,\Iall\rangle+
\half\langle\zpart,[\Bpart+f(\Iall)]\zpart\rangle+F(\Iall)+ \\
& O(|\zpart|^3)+U(\varphi,I,z)
\end{array} \renewcommand{\arraystretch}{1.}
\label{eqn10}
\end{equation}
by an ($R$-dependent) canonical transformation analytic in a neighborhood of
torus $T$ \cite{Vill,Jorba1}.\footnote{This transformation depends on $R$
discontinuously, but piecewise analytically.} Here $\Iall\in\mR^{n+r}$,
\[
\Iall_i=I_i, \quad 1\leq i\leq n, \qquad
\Iall_{n+j}=\half(z_j^2+z_{j+m}^2)\geq 0, \quad 1\leq j\leq r;
\]
$\zpart=(z_{r+1},\ldots,z_m,z_{m+r+1},\ldots,z_{2m})\in\mR^{2(m-r)}$;
$\Bpart$ is an $R$-independent symmetric matrix of order $2(m-r)$ such that
the eigenvalues of the matrix $J_{m-r}\Bpart$ are
$\pm\lambda_1,\ldots,\pm\lambda_{m-r}$ ($J_{m-r}$ is defined by formula
(\ref{eqn4}) with $m$ having been replaced by $m-r$); $f$ is an analytic
mapping of a neighborhood of the origin in $\mR^{n+r}$ to the space of
symmetric matrices of order $2(m-r)$, and $f(0)=0$; $F$ is a power series
without constant and linear terms convergent in a neighborhood of the
origin in $\mR^{n+r}$; the remainder $U$ satisfies the inequality
\[
|U(\varphi,I,z)|0$, one has
\begin{equation}
\mathop{\rm mes}(W(R)\setminus A)\leq
c_1\exp\left[ -c_2R^{-2/(\tau+1)}\right] ,
\label{eqn13}
\end{equation}
where $\tau$ is the exponent entering inequalities\/ {\rm (\ref{eqn9})},
positive constants $c_1$ and $c_2$ do not depend on $R$, and\/ {\rm mes}
denotes the Lebesgue measure in $\mR^{n+r}$.}
\vspace{2mm}
Condition (\ref{eqn11}) means that the mapping
\[
\Iall\mapsto\oall+\frac{\partial F}{\partial\Iall}(\Iall)
\]
is a local diffeomorphism at the origin. We can conclude that through torus
$T$, there passes a Cantor $(n+r)$-parameter family of reducible Diophantine
isotropic invariant $(n+r)$-tori with frequencies close to
$\oall_1,\ldots,\oall_{n+r}$, these tori ``condensing'' exponentially while
approaching the initial torus $T$. As a matter of fact, the family of
$(n+r)$-tori obtained is Whitney-smooth.
Inequalities (\ref{eqn9}) involve Diophantine conditions on the numbers
$\omega^N_1,\ldots,\omega^N_r$. For $n\geq 1$, these conditions are not
restrictive because the initial torus $T$ belongs to a Whitney-smooth
$n$-parameter family of invariant $n$-tori, and generically almost all the tori
in this family meet condition (\ref{eqn9}). On the other hand, for $n=0$
torus $T$ is just an isolated equilibrium point, and the Diophantine condition
on the normal frequencies $\omega^N_1,\ldots,\omega^N_r$ is no longer a
genericity condition: for $r>1$ one can make those frequencies, e.g.,
commensurable by an arbitrarily small perturbation of the Hamilton function.
The excitation of elliptic normal modes of an equilibrium point of a
Hamiltonian system has been considered in a much larger number of works
(see, e.g., \cite{Eliasson,Poeschel}) and has been studied much thoroughly
than the excitation of normal modes of an invariant torus of arbitrary
dimension. It turns out that if among the eigenvalues of the linearization
of a Hamiltonian system at an equilibrium point $0$, there are $\nu\geq 1$
pairs of purely imaginary numbers, then generically this system possesses
reducible Diophantine isotropic invariant tori of all the dimensions
$r=1,2,\ldots,\nu$ in any neighborhood of $0$ (for $r=1$ this assertion is
the classical Lyapunov center theorem which does not require the KAM theory
methods). To prove this statement, one does not need the corresponding
collections of eigenfrequencies to be Diophantine, there suffice some
nondegeneracy and nonresonance conditions insensitive to small perturbations
of the Hamilton function. However, to obtain exponential ``condensation''
of the tori while approaching $0$ for $r\geq 2$, it is necessary to impose
certain Diophantine hypotheses. In paper \cite{Delshams}, such ``condensation''
was proven for the first time for a particular case of $m$-tori in a
neighborhood of an elliptic equilibrium point of a Hamiltonian system
with $m$ degrees of freedom. In that paper, the theorem on an exponentially
small estimate for the measure of the complement of the union of the
invariant tori was also generalized to the case where the collection of
the eigenfrequencies is not Diophantine but only ``quasi-Diophantine''
(Diophantine ``up to precision $\delta$''). The property of being
``quasi-Diophantine'' is structurally stable, i.e., it persists under small
perturbations. To obtain an exponentially small estimate in the
``quasi-Diophantine'' case, one should neglect a neighborhood of the
equilibrium point of radius $O(\delta)$.
\vspace{5mm}{\large\bf
4. Excitation of elliptic normal modes of an analytic family of
invariant tori
}\vspace{5mm}
The proof and even the statement of theorem~1 exploit the advanced
technique of normal forms for a Hamiltonian system in a neighborhood
of an invariant torus. The nondegeneracy condition (\ref{eqn11}) and the
nonresonance condition (\ref{eqn12}) involve terms of the normal form
(\ref{eqn10}) for the Hamilton function that are of degree 4 in the ``normal
coordinates'' $z$ (since $2\Iall_j=z_{j-n}^2+z_{j-n+m}^2$ for
$n+1\leq j\leq n+r$). The left-hand side of the exponential estimate
(\ref{eqn13}) involves (via $W(R)$) terms of the normal form for the Hamilton
function that are of arbitrarily high degree in the ``normal coordinates''.
It turns out that for $n\geq 1$, the excitation of normal modes of invariant
tori can be established without the normal form technique and without
considering the terms of the Hamilton function of degrees higher than 2
in the ``normal coordinates'' \cite{BHS2,SevUMN,SevAMS}. For this purpose,
one has to change somewhat the statement of the problem and to
formulate {\em new\/} nondegeneracy and nonresonance conditions for a
Hamiltonian system $X_0$ possessing an {\em analytic\/} (not merely
Whitney-smooth) $n$-parameter family of reducible isotropic invariant
$n$-tori with parallel flows. Since for $n\geq 2$ the space of such
{\em partially integrable\/} systems is of infinite codimension, the existence
of invariant tori of dimensions larger than $n$ should be verified in this
context not for the system $X_0$ only but {\em for all its sufficiently
small perturbations\/} as well (perturbed systems are already generic).
Consider a Hamiltonian system governed by a Hamilton function of the form
\begin{equation}
H=H(\varphi,I,z)=P(I)+\half\langle z,B(I)z\rangle+O(|z|^3)
\label{eqn14}
\end{equation}
with respect to symplectic structure (\ref{eqn1}), where $\varphi\in\mT^n$
($n\geq 1$), $I\in G\subset\mR^n$ ($G$ being a bounded connected domain in
$\mR^n$), $z$ ranges in a neighborhood of the origin in $\mR^{2m}$, and
$2m\times 2m$ matrix $B(I)$ is symmetric for each value of $I$. The Hamilton
function (\ref{eqn14}) affords the equations of motion
\begin{equation}
\dot{\varphi}=\omega(I)+O(|z|^2), \quad
\dot{I}=O(|z|^3), \quad
\dot{z}=\Omega(I)z+O(|z|^2),
\label{eqn15}
\end{equation}
where $\omega(I)=\partial P(I)/\partial I$ and $\Omega(I)=J_mB(I)$ while
$J_m$ is matrix (\ref{eqn4}). The system in question thus possesses the
$n$-parameter family of reducible isotropic invariant $n$-tori
$\{I=\const,z=0\}$, the flow on each torus being parallel with frequency
vector $\omega(I)$.
Now suppose that for each $I\in G$, among the eigenvalues of matrix
$\Omega(I)$, there are $\nu$ pairs ($0\leq\nu\leq m$) of purely imaginary
numbers. We wonder whether system (\ref{eqn15}) has invariant tori of
dimensions $n+1,\ldots,n+\nu$ near the $2n$-dimensional surface $\{z=0\}$
and whether sufficiently small perturbations of this system possess
invariant tori of dimensions $n,n+1,\ldots,n+\nu$.
Fix an arbitrary integer $r$ in the interval $0\leq r\leq\nu$. Assume that
for each $I\in G$, all the eigenvalues of matrix $\Omega(I)$ are pairwise
distinct, and fix $r$ pairs $\pm i\omega^N_1(I),\ldots,\pm i\omega^N_r(I)$
of purely imaginary eigenvalues of matrix $\Omega(I)$ that depend on $I$
analytically. Denote the remaining eigenvalues of this matrix by
\[
\renewcommand{\arraystretch}{1.4} \begin{array}{ll}
\pm i\theta_j(I), & 1\leq j\leq\nu-r, \\
\pm\eta_j(I), & 1\leq j\leq m-\nu-2s, \\
\pm\alpha_j(I)\pm i\beta_j(I), & 1\leq j\leq s,
\end{array} \renewcommand{\arraystretch}{1.}
\]
where $0\leq s\leq\frac{1}{2}(m-\nu)$ while $\theta_j(I)$, $\eta_j(I)$,
$\alpha_j(I)$, $\beta_j(I)$ are analytic functions. Introduce the notation
\[
\renewcommand{\arraystretch}{1.4} \begin{array}{l}
\oall=\oall(I)=(\omega_1,\ldots,\omega_n,\omega^N_1,\ldots,\omega^N_r), \\
\tall=\tall(I)=(\theta_1,\ldots,\theta_{\nu-r},\beta_1,\ldots,\beta_s).
\end{array} \renewcommand{\arraystretch}{1.}
\]
The conditions guaranteeing the existence of invariant tori of dimension
$n+r$ in system (\ref{eqn15}) and any of its sufficiently small perturbations
can be expressed in terms of the intrinsic frequencies $\omega_i$ of the
unperturbed $n$-tori $\{I=\const,z=0\}$ and the eigenvalues $\pm i\omega^N_j$,
$\pm i\theta_j$, $\pm\alpha_j\pm i\beta_j$ of matrices $\Omega(I)$, but
those conditions are rather complicated and involve derivatives with respect
to variables $I$ of an arbitrarily high order. Let $q\in\mZ_+^n$, $x\in\mR^n$,
$e\in\mR^{n+r}$, $\mu\in\mZ$, $Q\in\mZ$, $Q>0$, $l\in\mZ^{\nu-r+s}$, where
$\mZ_+^n=\mZ^n\cap\mR_+^n$ denotes the set of integer vectors of length $n$
with non-negative components. Introduce the notation
\[
D^q\oall=
\frac{\partial^{|q|}\oall}{\partial I_1^{q_1}\cdots\partial I_n^{q_n}},
\quad D^q\tall=
\frac{\partial^{|q|}\tall}{\partial I_1^{q_1}\cdots\partial I_n^{q_n}},
\]
\[
x^q=x_1^{q_1}\cdots x_n^{q_n},
\]
\[
\rho(I,Q)=\min_{\|e\|=1}\max_{\mu=0}^Q\max_{\|x\|=1}
\left| \sum_{|q|=\mu}\langle e,D^q\oall(I)\rangle x^q\right| ,
\]
\[
\psi(I,Q,l)=\max_{\mu=0}^Q\max_{\|x\|=1}
\left| \sum_{|q|=\mu}\langle l,D^q\tall(I)\rangle x^q\right| .
\]
Recall that the norms $|\cdot|$ and $\|\cdot\|$ are respectively defined by
equalities (\ref{eqn7}) and (\ref{eqn8}).
\vspace{2mm}
{\sc Theorem~2 {\rm \cite{BHS2,SevAMS}}.}
{\sl Suppose that for each $I\in G$, all the eigenvalues of matrix $\Omega(I)$
are pairwise distinct and the following nondegeneracy and nonresonance
conditions on the couple $(\omega(I),\Omega(I))$ are satisfied:
1) Nondegeneracy condition: there exists an integer $Q>0$ such that
$\rho(I,Q)>0$ for each $I\in G$.
2) Nonresonance condition: for any $I\in G$, $l\in\mZ^{\nu-r+s}$,
$k\in\mZ^{n+r}$ such that
\[
1\leq|l|\leq 2, \quad
1\leq\|k\|\leq\frac{\psi(I,Q,l)}{\rho(I,Q)},
\]
the inequality $\langle k,\oall(I)\rangle\neq\langle l,\tall(I)\rangle$ holds.
Then for any $\sigma>0$, any sufficiently small Hamiltonian perturbation of
system\/ {\rm (\ref{eqn15})} possesses, in the $\sigma$-neighborhood of the
surface $\{z=0\}$, a Whitney-smooth $(n+r)$-parameter family of reducible
Diophantine isotropic invariant $(n+r)$-tori, the $2(n+r)$-dimensional
Hausdorff measure of the union of all these tori being positive (the
perturbation smallness needed depends on $\sigma$). The frequencies of
an $(n+r)$-torus located near the $n$-torus $\{I=I_0,z=0\}$ are close to
$\oall_1(I_0),\ldots,\oall_{n+r}(I_0)$.}
\vspace{2mm}
Note that the nondegeneracy condition in this theorem is equivalent to the
following geometric condition $\Psi$: the image of the frequency map
$\oall:G\to\mR^{n+r}$ does not lie in any linear hyperplane passing through
the origin \cite{BHS1,BHS2}. This condition is very weak (for example, among
the maps $\oall$ satisfying condition $\Psi$, there are maps whose image
is a one-dimensional curve in the frequency space $\mR^{n+r}$).
Although no one of theorems 1 or 2 is a corollary of the other due to
different scenarios of the existence of tori and entirely different
nondegeneracy and nonresonance conditions, theorem~2, as a whole, seems to be
a weaker statement than theorem~1. Theorem~2 contains no exponential
estimates, it is not applicable for $n=0$, whereas for $n\geq 2$ it describes
only systems close to partially integrable ones (integrable on the surface
$\{z=0\}$). On the other hand, theorem~2 can be easily carried over to
{\em reversible\/} systems (the excitation of elliptic normal modes of
invariant tori in reversible systems is considered in, e.g.,
\cite{BHS2,Chaos}), in particular, for the case where the manifold
$\Sigma$ of the fixed points of the reversing involution is subject to
inequality $\dim\Sigma<\mathop{\rm codim}\Sigma-1$ (cf.~\cite{Chaos}). The
proof and even the formulation of the analogue of theorem~1 for reversible
systems in the case $\dim\Sigma<\mathop{\rm codim}\Sigma-1$ seem to be a
very difficult problem.
The proofs of theorems 1 and 2 are given in original works \cite{Vill,Jorba1}
and \cite{BHS2,SevAMS}, respectively. Both the theorems can be carried over to
``Hamiltonian systems with discrete time'', i.e., exact symplectic
diffeomorphisms.
\vspace{5mm}{\large\bf
5. Bibliographical remarks
}\vspace{5mm}
In this concluding section, we list, without claiming to be complete,
some basic works pertaining to two aspects of the theory under consideration:
the {\em existence\/} of invariant tori of dimensions $p>n$ in a neighborhood
of invariant tori of dimension $n\geq 2$ and {\em exponential
``condensation''\/} of invariant tori of dimensions $p\geq n$ while
approaching a given invariant torus of dimension $n\geq 0$.
Diophantine Lagrangian invariant $(n+m)$-tori in a neighborhood of invariant
$n$-tori in Hamiltonian systems with $n+m$ degrees of freedom were first
considered by V.~I.~Arnold \cite{Arn1,Arn2}. A.~D.~Bruno \cite{Bruno1,Bruno2}
studied {\em analytic\/} families of Diophantine isotropic invariant tori
of dimensions $p\geq n$ (not necessarily reducible) passing through a given
invariant $n$-torus in systems with $n+m$ degrees of freedom. Cantor
$p$-parameter families of invariant $p$-tori near invariant $n$-tori in
systems with $n+m$ degrees of freedom for any $n$, $m$ and $p$
($2\leq n\leq p\leq n+m$) were constructed in works by \`A.~Jorba,
J.~Villanueva \cite{Vill,Jorba1}, H.~W.~Broer, G.~B.~Huitema, and the author
\cite{BHS2,SevAMS}.
An exponentially small estimate for the measure of the complement of the
union of invariant tori was first obtained by A.~I.~Ne\u{\i}shtadt
\cite{Neish}. In paper \cite{Neish}, a nearly-integrable system with two
degrees of freedom was considered in the presence of the so called proper
degeneracy, the measure of the complement of the union of invariant tori
turning out to be exponentially small with respect to the perturbation
magnitude. In the present survey, we have described the phenomenon of
exponential ``condensation'' of a family of invariant $p$-tori while
approaching an invariant $n$-torus in a Hamiltonian system with $n+m$ degrees
of freedom. Here exponential smallness of the ``slits'' between the $p$-tori
is understood with respect to the distance from the given $n$-torus. The
exponential ``condensation'' effect was discovered by A.~Morbidelli and
A.~Giorgilli \cite{MG2} who considered the case $p=n$, $m=0$. The results of
\cite{MG2} were confirmed numerically in \cite{Lega}. The case $p=m$, $n=0$ was
examined by A.~Delshams and P.~Guti\'errez \cite{Delshams} while the case where
$p=n$ and $m$ is arbitrary, by \`A.~Jorba and J.~Villanueva \cite{Jorba2}.
Finally, \`A.~Jorba and J.~Villanueva have proven exponential ``condensation''
for any $n$, $m$ and $p$ \cite{Vill,Jorba1}.
Recently, \`A.~Jorba, J.~Villanueva and co-authors obtained exponential
estimates for the measure of ``exceptional sets'' or the magnitude of
``remainders'' of normal forms in some other problems of the KAM theory as
well (see, e.g., \cite{Vill,Jorba2,Jorba3,Jorba4}). Exponentially small
effects occur frequently in analytic dynamical systems in rather diverse
situations.
The applications of the results discussed in the present survey to some
problems of celestial mechanics are considered in, e.g.,
\cite{Vill,Arn1,Arn2,Jorba2}.
\vspace{3mm}
I am indebted to V.~I.~Arnold who introduced me to the whimsical world
of Cantor families of invariant tori. I am also very grateful to
H.~W.~Broer and G.~B.~Huitema for extremely fruitful cooperation during
the last several years and to \`A.~Jorba who gave me the preprints of his
papers \cite{Jorba1,Jorba2,Jorba3,Jorba4} prior to publication and the
manuscript of the thesis work of his student J.~Villanueva \cite{Vill}.
\newpage
\baselineskip=15.5pt
\parindent=0mm
{\small
\begin{thebibliography}{10}
\bibitem{Laz}
{\em Lazutkin~V.~F.}
KAM Theory and Semiclassical Approximations to Eigenfunctions.
Berlin, Springer, 1993.
\bibitem{ChG}
{\em Chierchia~L., Gallavotti~G.}
Drift and diffusion in phase space //
Ann.\ Institut Henri Poincar\'e, Physique Th\'eorique,
1994, V.~60, No.~1, p.~1--144.
\bibitem{MF}
{\em Mather~J.~N., Forni~G.}
Action minimizing orbits in Hamiltonian systems //
In: Transition to Chaos in Classical and Quantum Mechanics,
ed.\ {\em Graffi~S.}
(Lect.\ Notes Math., V.~1589).
Berlin, Springer, 1994, p.~92--186.
\bibitem{MacKay}
{\em MacKay~R.~S.}
Recent progress and outstanding problems in Hamiltonian dynamics //
Physica~D, 1995, V.~86, No.~1--2, p.~122--133.
\bibitem{Kozlov}
{\em Kozlov~V.~V.}
Symmetries, Topology, and Resonances in Hamiltonian Mechanics.
Berlin, Springer, 1996.
\bibitem{BHS1}
{\em Broer~H.~W., Huitema~G.~B., Sevryuk~M.~B.}
Families of quasi-periodic motions in dynamical systems depending on
parameters //
In: Nonlinear Dynamical Systems and Chaos,
eds.\ {\em Broer~H.~W., van Gils~S.~A., Hoveijn~I., Takens~F.}
(Progress Nonlinear Differ.\ Equations and Their Appl., V.~19).
Basel, Birkh\"auser, 1996, p.~171--211.
\bibitem{BHS2}
{\em Broer~H.~W., Huitema~G.~B., Sevryuk~M.~B.}
Quasi-Periodic Motions in Families of Dynamical Systems:\ Order
amidst Chaos
(Lect.\ Notes Math., V.~1645).
Berlin, Springer, 1996.
\bibitem{Vill}
{\em Villanueva~J.}
Normal Forms around Lower Dimensional Tori of Hamiltonian Systems.
PhD thesis, Universitat Polit\`ecnica de Catalunya, Barcelona (Spain), 1997.
\bibitem{Herman}
{\em Herman~M.~R.}
In\'egalit\'es ``a priori'' pour des tores lagrangiens invariants
par des diff\'eo\-morphis\-mes symplectiques //
Publ.\ Math.\ IHES, 1990, V.~70, p.~47--101.
\bibitem{MG1}
{\em Morbidelli~A., Giorgilli~A.}
On a connection between KAM and Nekhoroshev's theorems //
Physica~D, 1995, V.~86, No.~3, p.~514--516.
\bibitem{Jorba1}
{\em Jorba~\`A., Villanueva~J.}
On the normal behaviour of partially elliptic lower dimensional tori
of Hamiltonian systems //
Nonlinearity, 1997, V.~10, No.~4, p.~783--822.
\bibitem{Eliasson}
{\em Eliasson~L.~H.}
Perturbations of stable invariant tori for Hamiltonian systems //
Ann.\ Sc.\ Norm.\ Super.\ Pisa, Cl.\ Sci., IV~Ser.,
1988, V.~15, No.~1, p.~115--147.
\bibitem{Poeschel}
{\em P\"oschel~J.}
On elliptic lower dimensional tori in Hamiltonian systems //
Math.\ Zeitschr., 1989, V.~202, No.~4, p.~559--608.
\bibitem{Delshams}
{\em Delshams~A., Guti\'errez~P.}
Estimates on invariant tori near an elliptic equilibrium point of a
Hamiltonian system //
J. Differ.\ Equat., 1996, V.~131, No.~2, p.~277--303.
\bibitem{SevUMN}
{\em Sevryuk~M.~B.}
Excitation of normal modes of invariant tori in Hamiltonian systems //
Uspekhi Mat.\ Nauk, 1996, V.~51, No.~5, p.~171 (in Russian).
\bibitem{SevAMS}
{\em Sevryuk~M.~B.}
Excitation of elliptic normal modes of invariant tori
in Hamiltonian systems //
In: Topics in Singularity Theory. V.~I.~Arnold's 60th Anniversary Collection,
eds.\ {\em Khovanski\u{\i}~A.~G., Varchenko~A.~N., Vassiliev~V.~A.}
(Amer.\ Math.\ Soc.\ Transl., Ser.~2, V.~180)
(Adv.\ Math.\ Sci., V.~34).
Providence, AMS, 1997, p.~209--218.
\bibitem{Chaos}
{\em Sevryuk~M.~B.}
The iteration-approximation decoupling in the reversible KAM theory //
Chaos, 1995, V.~5, No.~3, p.~552--565.
\bibitem{Arn1}
{\em Arnold~V.~I.}
On the classical perturbation theory and the problem of stability
of planetary systems //
Sov.\ Math.\ Dokl., 1962, V.~3, No.~4, p.~1008--1012.
\bibitem{Arn2}
{\em Arnold~V.~I.}
Small denominators and problems of stability of motion in classical
and celestial mechanics //
Russian Math.\ Surveys, 1963, V.~18, No.~6, p.~85--191.
\bibitem{Bruno1}
{\em Bruno~A.~D.}
The sets of analyticity of a normalizing transformation //
The USSR Academy of Sciences Institute of Applied Mathematics, preprints
97 and 98, 1974 (in Russian).
\bibitem{Bruno2}
{\em Bruno~A.~D.}
Local Methods in Nonlinear Differential Equations.
Berlin, Springer, 1989.
\bibitem{Neish}
{\em Ne\u{\i}shtadt~A.~I.}
Estimates in the Kolmogorov theorem on the persistence
of quasi-periodic motions //
J. Appl.\ Math.\ Mech., 1981, V.~45, No.~6, p.~766--772.
\bibitem{MG2}
{\em Morbidelli~A., Giorgilli~A.}
Superexponential stability of KAM tori //
J. Stat.\ Phys., 1995, V.~78, No.~5--6, p.~1607--1617.
\bibitem{Lega}
{\em Lega~E., Froeschl\'e~C.}
Numerical investigations of the structure around an invariant KAM torus
using the frequency map analysis //
Physica~D, 1996, V.~95, No.~2, p.~97--106.
\bibitem{Jorba2}
{\em Jorba~\`A., Villanueva~J.}
On the persistence of lower dimensional invariant tori under
quasiperiodic perturbations //
J. Nonlinear Science, 1997, V.~7, No.~5, p.~427--473.
\bibitem{Jorba3}
{\em Jorba~\`A., Sim\'o~C.}
On quasiperiodic perturbations of elliptic equilibrium points //
SIAM J. Math.\ Anal., 1996, V.~27, No.~6, p.~1704--1737.
\bibitem{Jorba4}
{\em Jorba~\`A., Ram\'{\i}rez-Ros~R., Villanueva~J.}
Effective reducibility of quasiperiodic linear equations
close to constant coefficients //
SIAM J. Math.\ Anal., 1997, V.~28, No.~1, p.~178--188.
\end{thebibliography}
}
\end{document}