Content-Type: multipart/mixed; boundary="-------------0607121300269" This is a multi-part message in MIME format. ---------------0607121300269 Content-Type: text/plain; name="06-200.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-200.keywords" area-preserving nontwist maps, breakup of invariant tori, renormalization ---------------0607121300269 Content-Type: application/x-tex; name="NewFixedPoint2.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="NewFixedPoint2.tex" \documentclass[twocolumn,prl,amsfonts,floatfix,superscriptaddress]{revtex4} %% For submission, take off ``twocolumn'': %\documentclass[prb,amsfonts,floatfix,superscriptaddress]{revtex4} \ifx\pdfoutput\undefined \usepackage[dvips]{graphicx} \else \usepackage[pdftex]{graphicx} \usepackage[pdftex]{hyperref} \fi \usepackage{amsmath,amsfonts,amssymb,amscd,amsthm,epsf} \usepackage{subfigure} \usepackage{color} \newcommand{\enquote}[1]{``#1''} \graphicspath{{figs/}} \theoremstyle{remark} \newtheorem{remark}{Remark} \newcommand{\ie}{{i.e., }} \newcommand{\eg}{{e.g., }} \newcommand{\Rset}{{\mathbb R}} \newcommand{\Tset}{{\mathbb T}} \newcommand{\nc}{\newcommand} \newcommand{\xyvect}[2]{\left(\begin{matrix} #1 \\ #2 \end{matrix}\right)} \nc{\figref}[1]{Fig.~\ref{fig:#1}} \nc{\figsref}[2]{Figs.~\ref{fig:#1}-\ref{fig:#2}} \nc{\tabref}[1]{Table~\ref{tab:#1}} \nc{\tabsref}[2]{Tables~\ref{tab:#1}-\ref{tab:#2}} \nc{\secref}[1]{Sec.~\ref{sec:#1}} \nc{\secsref}[2]{Sec.~\ref{sec:#1}-Sec.~\ref{sec:#2}} \nc{\ssecref}[1]{Sec.~\ref{ssec:#1}} \nc{\ssecsref}[2]{Sec.~\ref{ssec:#1}-Sec.~\ref{ssec:#2}} \nc{\eqeqref}[1]{Eq.~\eqref{eq:#1}} \nc{\eqseqref}[2]{Eqs.~\eqref{eq:#1}-\eqref{eq:#2}} \nc{\thmref}[1]{Theorem~\ref{theo:#1}} \nc{\thmsref}[2]{Theorem~\ref{theo:#1}-\ref{theo:#2}} \nc{\rcite}[1]{Ref.~\onlinecite{#1}} \nc{\rcites}[1]{Refs.~\onlinecite{#1}} \nc{\qtq}[1]{{\qquad\text{#1}\qquad}} \nc{\vect}[1]{\boldsymbol{#1}} \definecolor{changed}{rgb}{0.3,0.3,0.3} \nc{\changed}[1]{{\bf previous version: }{\color {changed} #1}} \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