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area-preserving nontwist maps, breakup of invariant tori, renormalization
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\begin{document}
\title{On a new fixed point of the renormalization group operator for area-preserving maps}
\author{K.~Fuchss} \affiliation{Department of Physics and Institute
for Fusion Studies, The University of Texas at Austin, Austin, TX
78712}
\author{A.~Wurm} \affiliation{Department of Physical \& Biological
Sciences, Western New England College, Springfield, MA 01119}
\author{P.J.~Morrison} \affiliation{Department of Physics and
Institute for Fusion Studies, The University of Texas at Austin,
Austin, TX 78712}
\date{\today}
\begin{abstract}
The breakup of the shearless invariant torus with winding number
$\omega=\sqrt{2}-1$ is studied numerically using Greene's residue
criterion in the standard nontwist map. The residue behavior and
parameter scaling at the breakup suggests the existence of a new
fixed point of the renormalization group operator (RGO) for
area-preserving maps. The unstable eigenvalues of the RGO at this
fixed point and the critical scaling exponents of the torus at
breakup are computed.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%
%\section{Introduction}
%\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%
Area-preserving nontwist maps are low-dimensional models of physical
systems whose Hamiltonians locally violate a nondegeneracy condition
(see below) as described, \eg in
\rcites{del_castillo96,apte03,wurm05}. Some applications are the
study of magnetic field lines in toroidal plasma
devices\cite{stix76,oda95,balescu98,horton98,morrison00,ullmann00,petrisor03}
and stellarators\cite{davidson95,hayashi95} (plasma physics), and
traveling waves,\cite{del_castillo93} coherent structures,
self-consistent transport\cite{del_castillo02} (fluid dynamics), and
particle accelerators.\cite{gerasimov86}
Nontwist regions have also been shown to appear generically in the
phase space of area-preserving maps that have a tripling bifurcation
of an elliptic fixed point.\cite{dullin00,vanderweele88} Additional
references can be found in \rcites{wurm05, del_castillo96}.
Of particular interest from a physics perspective is the breakup of
invariant tori, consisting of quasiperiodic orbits with irrational
winding number,\footnote{An {\it orbit} of an area-preserving map
$M$ is a sequence of points
$\left\{\left(x_i,y_i\right)\right\}_{i=-\infty}^{\infty}$ such that
$M\left(x_i,y_i\right) = \left(x_{i+1},y_{i+1}\right)$. The {\it
winding number} $\omega$ of an orbit is defined as the limit $\omega
= \lim_{i\to\infty} (x_i/i)$, when it exists. Here the
$x$-coordinate is ``lifted'' from $\Tset$ to $\Rset$. A {\it
periodic orbit} of period $n$ is an orbit $M^n \left( x_i,
y_i\right) = \left( x_i+m, y_i\right)$, $\forall \:i$, where $m$ is
an integer. Periodic orbits have rational winding numbers
$\omega=m/n$. An {\it invariant torus} is a one-dimensional set $C$,
a curve, that is invariant under the map, $C = M(C)$. Orbits
belonging to such a torus generically have irrational winding
number.} that often correspond to transport barriers in the physical
system, i.e., their existence determines the long-time stability of
the system. In nontwist maps, the invariant tori that appear to be
the most resilient to perturbations are the so-called {\it
shearless} tori, which correspond to local extrema in the winding
number profile of the map.
Invariant tori at breakup exhibit scale invariance under specific
phase space re-scalings, which are observed to be universal for
certain classes of area-preserving maps. To interpret these results,
a renormalization group framework has been developed (see, e.g.,
\rcites{mackay83, del_castillo97,apte03,apte05a}). For twist maps,
it is well understood which fixed point, cycle, or strange attractor
of the renormalization group operator (RGO) is encountered within a
given class of maps, depending on properties of the winding number
of the critical torus (see \rcite{chandre02} for a recent review).
For nontwist maps, however, only results for the single class of
shearless critical noble tori, i.e., shearless critical tori with
winding numbers that have a continued fraction expansion tail of
1's, are known. The result reported in this letter represents the
first new fixed point for nontwist maps.
A tool for studying the breakup of a torus with given winding number
is Greene's residue criterion, originally introduced in the context
of twist maps.\cite{greene79} This method is based on the numerical
observation that the breakup of an invariant torus with irrational
winding number $\omega$ is determined by the stability of nearby
periodic orbits. Some aspects of this criterion have been proved for
nontwist maps.\cite{delshams00}
To study the breakup, one considers a sequence of periodic orbits
with winding numbers $q_n/p_n$ converging to $\omega$,
$\lim_{n\rightarrow\infty} q_n/p_n=\omega$. The elements of the
sequence converging the fastest are the convergents of the continued
fraction expansion of $\omega$, i.e., $[n]:=q_n/p_n=[a_0, a_1,
\ldots, a_n]$, where
%
\begin{equation}
\label{eq:cf} \omega=[a_0, a_1, a_2, \ldots] =
a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \ldots}}.
\end{equation}
%
The stability of the corresponding orbits is determined by their
residues, $R_n= [2-{\rm Tr}(DM^{p_n})]/4$, where ${\rm Tr}$ is the
trace and $DM^{p_n}$ is the linearization of the $p_n$ times
iterated map $M$ about the periodic orbit: An orbit is elliptic for
$0