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\draft
\title{An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems}
\author{C. Chandre$^1$, H. R. Jauslin$^1$, and G. Benfatto$^2$}
\address{$^1$Laboratoire de Physique, CNRS, Universit\'e de Bourgogne,
BP 400, F-21011 Dijon, France}
\address{$^2$Dipartimento di Matematica,
Universit\`a di Roma ``Tor Vergata'',
Via della Ricerca Scientifica, I-00133 Roma, Italy}
\maketitle
\begin{abstract}
We construct an approximate renormalization scheme for Hamiltonian
systems with two degrees of freedom. This scheme is a combination of
Kolmogorov-Arnold-Moser (KAM) theory
and renormalization-group techniques. It makes the connection between the
approximate renormalization procedure derived by Escande and Doveil, and
a systematic expansion of the transformation. In particular, we show that
the two main approximations, consisting in keeping only the quadratic
terms in the actions and the two main resonances, keep the essential
information on the threshold of the breakup of invariant tori.
\end{abstract}
\pacs{PACS numbers: 05.45.+b, 64.60.Ak}
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\section{Introduction}
In 1981, Escande and Doveil~\cite{escandedoveil} set up an approximate
renormalization scheme for Hamiltonian systems with two degrees of
freedom, in order to study the breakup of invariant tori, and especially
to compute the threshold of stochasticity. Their scheme
was motivated by Chirikov's resonance overlap criterion~\cite{chirikov}, and by
Greene's results~\cite{greene} about the link between the existence of a
torus with the stability of neighboring periodic orbits. They established
the relevance of a sequence of these periodic orbits for the breakup of
invariant tori, by setting an approximate transformation which focuses
successively on smaller scales, i.e. acting like a microscope in phase space.\\
Due to the complexity of the phase space of a non-integrable Hamiltonian,
their method requires strong approximations to obtain explicit expressions.
Basically, two approximations were involved:\\
(1) a quadratic approximation in the actions: their transformation
produces terms that are higher than quadratic in the actions;
in order to remain in the same class of Hamiltonians, they neglect these
higher order terms,\\
(2) a two-resonance approximation: they only keep the two main resonances
at each iteration of the transformation.\\
The idea was to keep only the most relevant features of the mechanism of the
breakup of a given torus.\\
In this article, we construct an approximate scheme using the same two
approximations. We establish the
connection between Escande's scheme~\cite{escandedoveil,escande}
and the KAM-RG transformation derived in Refs.\ \cite{gallben,govin,chandre}.
The aim is to show that an exact renormalization transformation
can be approximated by a
simple transformation: It can be
useful to derive approximate explicit expressions of universal parameters, and
to see what are the most relevant terms responsible for the breakup of
invariant tori.
The results we obtain support the general idea that the irrelevant terms
of the renormalization
transformation can be eliminated with little loss of accuracy in the
parameters associated to the breakup of
invariant tori.\\
The transformation $\mathcal{R}$ we define has two main parts: a KAM
transformation which
is a canonical change of coordinates that reduces the size of the perturbation
from $\epsilon$ to $\epsilon^2$,
and a renormalization transformation which is a combination of a shift of
the resonances and a rescaling of momentum and energy.\\
It acts on the following class of Hamiltonians with two degrees of freedom,
quadratic in the action variables $\u{A}=(A_1,A_2)$, and described by three
even scalar functions of the angles $\u{\phi}=(\phi_1,\phi_2)$:
\begin{eqnarray}
H(\u{A},\u{\phi})=&&\frac{1}{2}\left(1+m(\u{\phi})\right)
(\u{\Omega}\cdot\u{A})^2 \nonumber \\
&& +\left[\u{\omega}_0+g(\u{\phi})\u{\Omega}
\right]\cdot\u{A}+f(\u{\phi}),\label{hamiltonian}
\end{eqnarray}
where $m$, $g$, and $f$ are of zero average.
The vector $\u{\omega}_0$ is the frequency vector of the considered torus and
$\u{\Omega}=(1,\alpha)$ is some other constant vector not parallel to
$\u{\omega}_0$.
The perturbation $(m,g,f)$ is of order $O(\varepsilon)$. \\
The renormalization-group approach is based on the following general picture:
The idea is to construct the transformation $\mathcal{R}$
as a generalized canonical
change of coordinates acting on some space of Hamiltonians such that the
iteration of $\mathcal{R}$ converges to a fixed point. If the
perturbation is smaller than critical,
$\mathcal{R}$ should converge to a Hamiltonian of type~(\ref{hamiltonian})
with $(m,g,f)=0$, which is integrable, and the equations
of motion show that the torus with frequency vector
$\u{\omega}_0$ is located at $\u{A}=0$.
All Hamiltonians attracted by this trivial fixed point have an
invariant torus
of that frequency (this can be considered as an
alternative version of the KAM theorem~\cite{koch}).
If the perturbation is larger than critical, the system does not have a KAM
torus of the considered frequency and the iteration of $\mathcal{R}$ diverges.
The domain of
convergence to the trivial fixed point and the domain of divergence
are separated by a {\it critical surface}
invariant under the action of $\mathcal{R}$. The main hypothesis of the
renormalization-group approach is that there should be another nontrivial
fixed point (or more generally, a fixed set) on this critical surface,
that is attractive for Hamiltonians
on that surface. From its existence, one can expect to deduce universal
properties in the mechanism of the breakup of invariant tori.\\
Many aspects of this general picture are still at the stage of conjecture,
supported by some results in the perturbative regime \cite{koch}, by
numerical works \cite{govin,chandre,cgjk,abad}, and by analogies with
the related problem for area-preserving maps \cite{mackay,mackayL}.
In particular, the relation between the properties of the nontrivial
renormalization fixed point and the geometric properties of the invariant
torus at the instability threshold are not well established. The
coincidence of the critical coupling of one-parameter families at
which a torus breaks up, with the boundary of attraction of the trivial
fixed point is supported by numerical studies \cite{govin,chandre,cgjk}.\\
In Sec.\ \ref{sec:kam}, we describe the KAM part of the transformation,
and we make explicit
the two approximations involved in this scheme: the quadratic approximation
and the two-resonance approximation.
In Sec.\ \ref{sec:ren}, we present the
renormalization transformation which is a combination
of the KAM part, a shift of the resonances,
and rescalings of actions and energy.
In Sec.\ \ref{sec:res}, we give
our numerical results, and in particular, we show that the approximate scheme
contains the essential features of the exact one.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{KAM transformation}
\label{sec:kam}
We perform a canonical transformation $\mathcal{U}_F:(\u{\phi},\u{A})
\mapsto (\u{\phi}',\u{A}') $
defined by a generating function
$F(\u{A}',\u{\phi})$~\cite{goldstein,gallavottiL}
characterized by a
scalar function $X$ of the action and angle variables, of the form
\begin{equation}
F(\u{A}',\u{\phi})=\u{A}'\cdot\u{\phi}+ X(\u{A}',\u{\phi}),
\end{equation}
leading to
\begin{eqnarray}
&& \label{eqn:tcan1}
\u{A}=\d \frac{\partial F}{\partial \u{\phi}}
=\u{A}'+\frac{\partial X}{\partial \u{\phi}},\\
&& \label{eqn:tcan2}
\u{\phi}'=\d \frac{\partial F}{\partial \u{A}'}
=\u{\phi}+\frac{\partial X}{\partial \u{A}'}.
\end{eqnarray}
The function $X$ is constructed such that in
$\mathcal{H}\circ\mathcal{U}_F$ the perturbation terms of first order
in $\epsilon$ are equal to zero.
Inserting Eq.\ (\ref{eqn:tcan1}) into Hamiltonian (\ref{hamiltonian}), one
obtains the expression of the Hamiltonian
in the mixed representation of new action variables and old angle variables:
\begin{eqnarray}
\tilde{H}({\u A}',{\u \varphi})=&&(\u{\Omega}\cdot\u{A}')^2/2
+ {\u \omega}_0 \cdot {\u A}'\nonumber \\
&& + \u{\omega}(\u{A}')\cdot\frac{\partial X}{\partial \u{\phi}}+
h(\u{A}',\u{\phi}) +O(\varepsilon^2),
\end{eqnarray}
where
\begin{eqnarray}
&& \u{\omega}(\u{A}')=
\u{\omega}_0+(\u{\Omega}\cdot\u{A}') \u{\Omega},\\
&& h(\u{A}',\u{\phi})=\frac{1}{2}m(\u{\phi})(\u{\Omega}\cdot\u{A}')^2+
g(\u{\phi})\u{\Omega}\cdot\u{A}'+f(\u{\phi}).
\end{eqnarray}
The equation that determines $X$ is thus:
\begin{equation}
\label{eqn:eqX}
\u{\omega}(\u{A}')\cdot\frac{\partial X}{\partial \u{\phi}}+
h(\u{A}',\u{\phi})=0.
\end{equation}
We recall that the functions $m$, $g$, and $f$ are of order
$O(\epsilon)$; as a consequence, $X$ is also of order
$O(\epsilon)$.
Equation (\ref{eqn:eqX}) has the solution
\begin{equation}
\label{eqn:X}
X(\u{A}',\u{\phi})=\sum_{\nu \in {\Bbb Z}^2}
X_{\nu}(\u{A}') \sin(\u{\nu}\cdot\u{\phi}),
\end{equation}
where, if we write $h(\u{A},\u{\phi})=\sum_{\nu} h_{\nu}(\u{A})
\cos(\u{\nu}\cdot\u{\phi})$,
\begin{equation}
\label{eqn:Xnu}
X_{\nu}(\u{A}')=-\d \frac{h_{\nu}(\u{A}')}{\u{\omega}(\u{A}')\cdot\u{\nu}}.
\end{equation}
The denominator of $X_{\nu}$ depends on the actions: thus, by power
expansion, it generates terms
that are
higher than quadratic in the actions.
In order to remain in the same space of Hamiltonians~(\ref{hamiltonian}),
we expand the Hamiltonian to
the second order in the actions and neglect the order $O(A^3)$.
The justification for such an approximation is that we are interested
in the torus with frequency vector $\u{\omega}_0$
which is located at $\u{A}=0$ for the trivial fixed point.\\
We consider Hamiltonians~(\ref{hamiltonian}) with only two Fourier modes
which are the two main resonances defined as follows:
For a frequency vector $\u{\omega}_0$, the resonances are given by
the vectors $\u{\nu}_n=(p_n,q_n)$ which are the sequence of the best
rational approximations. They are characterized precisely
by the following property:
$|\u{\omega}_0\cdot\u{\nu}_n| < |\u{\omega}_0\cdot\u{\nu}|$, for any
$\u{\nu}\equiv (p,q)\not=\u{\nu}_n$ such that $|q|