% main.tex
%
% main of paper:
% AREA PRESERVING NONTWIST MAPS:
% RENORMALIZATION STUDY OF THE TRANSITION TO CHAOS
%
%
%\documentstyle[12pt, pf]{article}
%\pagestyle{empty}
\documentstyle[12pt]{article}
% updated.....February, 1996 %
%%*******************************************************************
%% MARGIN PARAMETERS (for basically all papers with exception of camera ready)
%
\def\margins{\textwidth 6.5in
\evensidemargin 0.0in
\oddsidemargin 0.0in
\marginparwidth -5in
\textheight 8.5in
\topmargin -.5in
\topskip 0.0in}
%
%% JOURNAL ABBREVIATIONS
\def\PR{Phys.~Rev.}
\def\PRep{Phys.~Reports}
\def\APJ{Ap.~J.}
\def\JPP{J.~Plasma Phys.}
\def\JFM{J.~Fluid Mech.}
\def\JGR{J.~Geophys. Res.}
\def\GRL{Geophys. Res. Lett.}
\def\PL{Phys.~Lett.}
\def\PF{Phys.~Fluids}
\def\PRL{Phys.~Rev.~Lett.}
\def\PoP{Phys.~Plasmas}
\def\PPCF{Plasma Phys. Contr. Fusion}
\def\NF{Nucl.~Fusion}
\def\APL{Appl.~Phys.~Lett.}
\def\PSS{Phys.~Solid State}
\def\RMP{Rev.~Mod.~Phys.}
\def\JCP{J.~Comput.~Phys.}
%%********************************************************************
%% DEFINITIONS WITHIN THE BODY OF PAPER
\def\ackline{\subsection*{Acknowledgments}}
\def\partsupport{\subsection*{Acknowledgments}
This work was supported in part by the U.S. Dept.~of Energy
contract No.~DE-FG03-96ER-54346}
\def\doeline{This work was supported by the U.S. Dept.~of Energy
contract No.~DE-FG03-96ER-54346.}
\def\support{\subsection*{Acknowledgments}
This work was supported by the U.S. Dept.~of Energy
contract No.~DE-FG03-96ER-54346.}
\def\ifsnf{\small Institute for Fusion Studies, The
University of Texas at Austin, Austin, Texas~~78712~~USA}
\def\ifsem{\small\em Institute for Fusion Studies, The
University of Texas at Austin, Austin, Texas~~78712~~USA}
\def\phys{Department of Physics\\ The University of Texas at Austin\\ Austin,
Texas~~78712~~USA}
\def\ifs{Institute for Fusion Studies\\ The
University of Texas at Austin\\ Austin, Texas~~78712~~USA}
\def\physifs{Department of Physics and Institute for Fusion Studies\\
The University of Texas at Austin\\ Austin, Texas~~78712~~USA}
\def\ifsphys{Institute for Fusion Studies and Department of Physics\\
The University of Texas at Austin\\ Austin, Texas~~78712~~USA}
\def\ifsfrc{Institute for Fusion Studies and Fusion Research Center\\
The University of Texas at Austin\\ Austin, Texas~~78712~~USA}
\def\frc{Fusion Research Center\\ The
University of Texas at Austin\\ Austin, Texas~~78712~~USA}
%%********************************************************************
%% PHYSICS OF PLASMAS STYLES %%
%
\def\ifspop{\small\em Institute for Fusion Studies, The
University of Texas at Austin\\ \small\em Austin, Texas~~78712~~USA}
\def\physifspop{\small\em Department of Physics and Institute for Fusion
Studies\\
\small\em The University of Texas at Austin, Austin, Texas~~78712~~USA}
\def\ifsphyspop{\small\em Institute for Fusion Studies and Department of
Physics\\
\small\em The University of Texas at Austin, Austin, Texas~~78712~~USA}
\def\ifsfrcpop{\small\em Institute for Fusion Studies and Fusion Research
Center,
The University of Texas at Austin, Austin, Texas~~78712~~USA}
\def\frcpop{\small\em Fusion Research Center, The
University of Texas at Austin, Austin, Texas~~78712~~USA}
%
%%*******************************************************************
%%APPENDIX ABBREVIATIONS
\newcounter{abc}
\def\Apa{\clearpage\section*{Appendix A}
\renewcommand{\theequation}{A\arabic{equation}}
\setcounter{equation}{0}}
\def\Apb{\clearpage\section*{Appendix B}
\renewcommand{\theequation}{B\arabic{equation}}
\setcounter{equation}{0}}
\def\Apc{\clearpage\section*{Appendix C}
\renewcommand{\theequation}{B\arabic{equation}}
\setcounter{equation}{0}}
\def\levels#1{\renewcommand{\theequation}{\arabic{equation}\alph{abc}}
\setcounter{abc}{1}}
\def\add#1{\addtocounter{equation}{-1}\newline\addtocounter{abc}{1}}
\def\wholenum{\renewcommand{\theequation}{\arabic{equation}}}
%%*********************************************************************
%% MATH ABBREVIATIONS
%
\def\tdots{\begin{array}{c}
\large\,\,\cdot\\[-7pt]
\large\cdot\cdot{{}}\end{array}}
\def\intup{\dsp\int^{-\infty}_\infty}
\def\angle#1{{\left<#1\right>}}
\def\det{{\rm det}}
\def\emode{{$\et_e$~mode}}
\def\imode{{$\et_i$~mode}}
\def\smax{{\mathop{\scriptstyle\max}}}
\def\smin{{\mathop{\scriptstyle\min}}}
\def\lambar{{\mathchar'26\mskip-9mu\lambda}}
\def\ibar{{\mbox{$\;\iota\!\!$-}\ }}
\def\rhobar{{\mbox{$\;\rho\!\!\!\!\raisebox{-4pt}{--}$}}}
\def\doublesum{\mathop{\sum\sum}}
\def\dspdoublesum{\mathop{\dsp\sum\dsp\sum}}
\def\onehalf{{\textstyle\frac{1}{2}}}
\def\onethree{{\textstyle\frac{1}{3}}}
\def\onefour{{\textstyle\frac{1}{4}}}
\def\onefive{{\textstyle\frac{1}{5}}}
\def\onesix{{\textstyle\frac{1}{6}}}
\def\oneeight{{\textstyle\frac{1}{8}}}
\def\twothree{{\textstyle\frac{2}{3}}}
\def\threetwo{{\textstyle\frac{3}{2}}}
\def\threefour{{\textstyle\frac{3}{4}}}
\def\threeeight{{\textstyle\frac{3}{8}}}
\def\fourthree{{\textstyle\frac{4}{3}}}
\def\fivefour{{\textstyle\frac{5}{4}}}
\def\fivetwo{{\textstyle\frac{5}{2}}}
\def\fivesix{{\textstyle\frac{5}{6}}}
\def\fivetwelve{{\textstyle\frac{5}{12}}}
\def\ninefour{{\textstyle\frac{9}{4}}}
\def\EB{{\bfE\bf\times\bfB}}
\newcommand\trule{\rule{0pt}{2.6ex}} %top strut
\newcommand\brule{\rule[-1.2ex]{0pt}{0pt}} %bottom strut
\newcommand\bottomrule{\rule[1.2ex]{0pt}{0pt}}
\def\rec{\rule{.8ex}{.9ex}}
\def\ha{\mbox{\huge$a$}}
\def\Delsimsim{\begin{array}{c}{\bfDe}\\[-8pt]\approx\end{array}}
\def\subnos#1{\Delsimsim_{\mbox{$\raisebox{3em}{\scriptsize #1}$}}}
% $$\subnos{875}=(\nu-\bfq^{\dagger}\cdot\ha^{-1}\cdot\bfb)^{-1}%
% \bfDe_{51}\,\bfq^{\dagger}$$%
\def\m@th{\mathsurround=0pt}
\def\n@space{\nulldelimiterspace=0pt \m@th}%1
\def\biggg#1{{\mbox{$\left#1\vbox to 20.5pt{}\right.\n@space$}}}%2
\def\Biggg#1{{\mbox{$\left#1\vbox to 23.5pt{}\right.\n@space$}}}%3
\def\Bigggg#1{{\mbox{$\left#1\vbox to 40pt{}\right.\n@space$}}}%4
\def\Biggggg#1{{\mbox{$\left#1\vbox to 50pt{}\right.\n@space$}}}%5
\def\Bigggggg#1{{\mbox{$\left#1\vbox to 60pt{}\right.\n@space$}}}%6
\def\Biggggggg#1{{\mbox{$\left#1\vbox to 80pt{}\right.\n@space$}}}%7
\def\noal#1{\noalign{\noindent\rm#1}}
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%\lines\raisebox{-6pt}{$\circ$}\lines
\def\linecirc{$\qquad\quad\underline{\phantom{suzymitchellsuzymitchell
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\def\lege{{\mathop{\stackrel{<}{\scriptstyle >}}\nolimits}}
\def\dotequal{{\mathop{~=~}\limits^.}}
\def\dotminus{{\mathop{-}\limits^.}}
\def\dotedot{\mathop{=}\limits^{\bf\cdot}_{^{\scriptstyle{\bf\cdot}}}}
%\def\ded{\mathop{\stackrel{\doteq}{\ .\ }}\nolimits}
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\def\overapprox#1#2{\lower 2pt\vbox{\baselineskip 0pt\lineskip - 1pt
\ialign{$\nms#1\hfil##\hfil$\crcr#2\crcr\approx\crcr}}}
\def\gtapprox{\mathrel{\mathpalette\overapprox>}} % greater than or approx.
\def\ltapprox{\mathrel{\mathpalette\overapprox<}} % less than or approx.
\def\gtsim{\mathrel{\mathpalette\oversim>}} % greater than or sim.
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\ialign{$\nms#1\hfil##\hfil$\crcr#2\crcr\sim\crcr}}}
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\ialign{$\nms#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}
\def\leftrightarrowfill{$\nms\mathord\leftarrow\mkern-6mu
\cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}
\def\overdoublearrow#1{\vbox{\ialign{##\crcr
\leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
$\hfil\displaystyle{#1}\hfil$\crcr}}}
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\upequal#1{\overline{\overline#1}}
\def\mapleftsuper#1{\smash{\mathop{\longleftarrow}\limits^{#1}}}
\def\maprightsub#1{\smash{\mathop{\longrightarrow}\limits_{#1}}}
\def\mapleftsub#1{\smash{\mathop{\longleftarrow}\limits_{#1}}}
\def\spmb#1{\setbox0=\hbox{$\scriptstyle#1$}%for script\pmb
\kern-.015em\copy0\kern-\wd0
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\kern.05em\copy0\kern-\wd0
\kern-.025em\raise.0433em\box0 }
\def\proof{{\parskip7pt\noindent{\sl Proof}.\ \ }}
\def\drop#1{\noindent\smash{\lower\baselineskip\hbox{\caps #1}}}
%%*******************************************************
%% ABBREVIATIONS USED IN MATH MODE
\def\Re{\mathop{\rm Re}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\mhd{{\rm MHD}}
\def\mfp{{\rm mfp}}
\def\move{\hspace*{20pt}}
\def\rot{{\rm rot}}
\def\const{{\rm const}}
\def\kev{{\rm keV}}
\def\mev{{\rm MeV}}
\def\cm{{\rm cm}}
\def\nc{{\rm nc}}
\def\etal{{\it et~al.}}
\def\crit{{\rm crit}}
\def\ext{{\rm ext}}
\def\sgn{{\rm sgn}\,}
\def\eff{{\rm eff}}
\def\cosh{{\rm cosh}}
\def\sech{{\rm sech}}
\def\Ampere{{\rm Amp\'ere}}
\def\Alfven{{\rm Alfv\'en}}
\def\Alfvenic{{\rm Alfv\'enic}}
\def\bs{{\scriptstyle\pmb\ast}}
\def\beginab{\begin{abstract}}
\def\endab{\end{abstract}}
\def\beginenum{\begin{enumerate}}
\def\endenum{\end{enumerate}}
\def\begindoc{\begin{document}}
\def\enddoc{\end{document}}
\def\bq{\begin{equation}}
\def\eq{\end{equation}}
\def\bqy{\begin{eqnarray}}
\def\eqy{\end{eqnarray}}
\def\bqyn{\begin{eqnarray*}}
\def\eqyn{\end{eqnarray*}}
\def\bc{\begin{center}}
\def\ec{\end{center}}
\def\bfll{\begin{flushleft}}
\def\efll{\end{flushleft}}
\def\bflr{\begin{flushright}}
\def\eflr{\end{flushright}}
\def\bigskip{\vspace{\bigskipamount}}
\def\medskip{\vspace{\medskipamount}}
\def\smallskip{\vspace{\smallskipamount}}
\def\dfr{\displaystyle\frac}
\def\dsp{\displaystyle}
\def\bskip{\baselineskip}
\def\smallin{\hbox{{\small\rm I}\kern-.2em\hbox{{\small\rm N}}}}
\def\ZZ{\hbox{{\rm Z}\kern-.4em\hbox{\rm Z}}}
\def\IN{\hbox{{\rm I}\kern-.2em\hbox{{\rm N}}}}
\def\IR{\hbox{{\rm I}\kern-.2em\hbox{{\rm R}}}}
\def\IP{\hbox{{\rm I}\kern-.2em\hbox{{\rm P}}}}
\def\IC{\hbox{{\rm I}\kern-.2em\hbox{{\rm C}}}}
\def\II{\hbox{{\rm I}\kern-.2em\hbox{{\rm I}}}}
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\def\ID{\hbox{{\rm I}\kern-.2em\hbox{{\rm D}}}}
\newcommand{\IH}{\mbox{\hspace{3.0pt}\rule{.9pt}{8.2pt}
\hspace{-7.0pt}$\rm{H}$}}
\def\endbox{$\rule{2mm}{2mm}$}
\def\wh#1{{\widehat{#1}}}
\def\wt#1{{\widetilde{#1}}}
\def\pp#1{\frac{\p}{\p #1}}
\def\dd#1{\frac{d}{d #1}}
%%****************************************************
\arraycolsep 1.5pt
\tabcolsep 1.5pt
\def\jot{12pt}
\hfuzz 5pt
\vfuzz 5pt
\def\citenum#1{{\def\@cite##1##2{##1}\cite{#1}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ABBREVIATED GREEK
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\al{\alpha}
\def\be{\beta}
\def\Be{\Beta}
\def\de{\delta}
\def\De{\Delta}
\def\ep{\epsilon}
\def\et{\eta}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\io{\iota}
\def\ka{\kappa}
\def\la{\lambda}
\def\La{\Lambda}
\def\na{\nabla}
\def\om{\omega}
\def\Om{\Omega}
\def\p{\partial}
\def\ph{\phi}
\def\Ph{\Phi}
\def\ps{\psi}
\def\Ps{\Psi}
\def\rh{\rho}
\def\si{\sigma}
\def\Si{\Sigma}
\def\ta{\tau}
\def\th{\theta}
\def\Th{\Theta}
\def\ti{\tilde}
\def\Up{\Upsilon}
\def\varep{\varepsilon}
\def\ze{\zeta}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CALIGRAPHY LETTERS (SCRIPT!) (upper case only)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\cala{{\cal A}}
\def\calb{{\cal B}}
\def\calc{{\cal C}}
\def\cald{{\cal D}}
\def\cale{{\cal E}}
\def\calf{{\cal F}}
\def\calg{{\cal G}}
\def\calh{{\cal H}}
\def\cali{{\cal I}}
\def\calj{{\cal J}}
\def\calk{{\cal K}}
\def\call{{\cal L}}
\def\calm{{\cal M}}
\def\caln{{\cal N}}
\def\calo{{\cal O}}
\def\calp{{\cal P}}
\def\calq{{\cal Q}}
\def\calr{{\cal R}}
\def\cals{{\cal S}}
\def\calt{{\cal T}}
\def\calu{{\cal U}}
\def\calv{{\cal V}}
\def\calw{{\cal W}}
\def\calx{{\cal X}}
\def\caly{{\cal Y}}
\def\calz{{\cal Z}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%THIS NEXT MACRO REPLACES THE TEX \bar COMMAND
%\def\baroverletter#1{\setbox1=\hbox{$#1$}
% \dimen1=\ht1
% \advance\dimen1 by 1.25pt % was 1pt
% \dimen2=\wd1
% \advance\dimen2 by -2pt % was -1pt
% \rlap{\hspace{1pt}\rule[\dimen1] % was .5pt
% {\dimen2}{.35pt}}\box1} % was .25pt
%
\def\baroverletter#1{\setbox1=\hbox{$#1$}
\dimen1=\ht1
\advance\dimen1 by 1pt
\dimen2=\wd1
\advance\dimen2 by -1pt
\rlap{\hspace{.5pt}\rule[\dimen1]
{\dimen2}{.35pt}}\box1} % was .25pt chgd. 2/11/89 whm
%
\def\overA{{\baroverletter A}}
\def\overB{{\baroverletter B}}
\def\overC{{\baroverletter C}}
\def\overD{{\baroverletter D}}
\def\overE{{\baroverletter E}}
\def\overF{{\baroverletter F}}
\def\overG{{\baroverletter G}}
\def\overH{{\baroverletter H}}
\def\overI{{\baroverletter I}}
\def\overJ{{\baroverletter J}}
\def\overK{{\baroverletter K}}
\def\overL{{\baroverletter L}}
\def\overM{{\baroverletter M}}
\def\overN{{\baroverletter N}}
\def\overO{{\baroverletter O}}
\def\overP{{\baroverletter P}}
\def\overQ{{\baroverletter Q}}
\def\overR{{\baroverletter R}}
\def\overS{{\baroverletter S}}
\def\overT{{\baroverletter T}}
\def\overU{{\baroverletter U}}
\def\overV{{\baroverletter V}}
\def\overW{{\baroverletter W}}
\def\overX{{\baroverletter X}}
\def\overY{{\baroverletter Y}}
\def\overZ{{\baroverletter Z}}
%%%%% THE FOLLOWING ARE BARS OVER GREEK!
\def\overxi{{\baroverletter\xi}}
\def\overka{{\baroverletter\kappa}}
\def\overal{{\baroverletter\alpha}}
\def\overbe{{\baroverletter\beta}}
\def\overGa{{\baroverletter\Gamma}}
\def\overga{{\baroverletter\gamma}}
\def\overcale{{\baroverletter{\cale}}}
\def\overvartheta{{\baroverletter\vartheta}}
\def\overth{{\baroverletter\theta}}
\def\overTh{{\baroverletter\Theta}}
\def\overvarphi{{\baroverletter\varphi}}
\def\overla{{\baroverletter\lambda}}
\def\overna{{\baroverletter\nabla}}
\def\overchix{{\baroverletter\chix}}
\def\overnu{{\baroverletter\nu}}
\def\overpsi{{\baroverletter\psi}}
\def\overPsi{{\baroverletter\Psi}}
\def\overphi{{\baroverletter\phi}}
\def\overrho{{\baroverletter\rh}}
\def\oversi{{\baroverletter\si}}
\def\overmu{{\baroverletter\mu}}
\def\overPhi{{\baroverletter\Phi}}
\def\overom{{\baroverletter\omega}}
\def\overOm{{\baroverletter\Omega}}
\def\overpar{{\baroverletter\partial}}
\def\overOm{{\baroverletter\Omega}}
\def\overDe{{\baroverletter\Delta}}
\def\overde{{\baroverletter\delta}}
\def\overtau{{\baroverletter\tau}}
\def\overze{{\baroverletter\zeta}}
\def\overep{{\baroverletter\epsilon}}
\def\overell{{\baroverletter\ell}}
\def\overeta{{\baroverletter\eta}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LOWER CASE LETTERS
\def\overa{{\baroverletter a}}
\def\overb{{\baroverletter b}}
\def\overc{{\baroverletter c}}
\def\overd{{\baroverletter d}}
\def\overe{{\baroverletter e}}
\def\overf{{\baroverletter f}}
\def\overg{{\baroverletter g}}
\def\overh{{\baroverletter h}}
\def\overi{{\baroverletter i}}
\def\overj{{\baroverletter j}}
\def\overk{{\baroverletter k}}
\def\overl{{\baroverletter l}}
\def\overm{{\baroverletter m}}
\def\overn{{\baroverletter n}}
\def\overo{{\baroverletter o}}
\def\overp{{\baroverletter p}}
\def\overq{{\baroverletter q}}
\def\overr{{\baroverletter r}}
\def\overs{{\baroverletter s}}
\def\overt{{\baroverletter t}}
\def\overu{{\baroverletter u}}
\def\overv{{\baroverletter v}}
\def\overw{{\baroverletter w}}
\def\overx{{\baroverletter x}}
\def\overy{{\baroverletter y}}
\def\overz{{\baroverletter z}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%THIS NEXT MACRO IS FOR SUB AND SUPER CHARACTERS WITH OVERBARS
%\def\sbaroverletter#1{\setbox1=\hbox{\scriptsize$#1$}
% \dimen1=\ht1
% \advance\dimen1 by 1.25pt % was 1pt
% \dimen2=\wd1
% \advance\dimen2 by -2pt % was -1pt
% \rlap{\hspace{1pt}\rule[\dimen1] % was .5pt
% {\dimen2}{.35pt}}\box1} % was .25pt
%
\def\subbaroverletter#1{\setbox1=\hbox{\scriptsize$#1$}
\dimen1=\ht1
\advance\dimen1 by 1pt
\dimen2=\wd1
\advance\dimen2 by -1pt
\rlap{\hspace{.5pt}\rule[\dimen1]
{\dimen2}{.35pt}}\box1} % was .25pt chgd. 2/11/89 whm
%
\def\soverA{{\subbaroverletter A}}
\def\soverB{{\subbaroverletter B}}
\def\soverC{{\subbaroverletter C}}
\def\soverD{{\subbaroverletter D}}
\def\soverE{{\subbaroverletter E}}
\def\soverF{{\subbaroverletter F}}
\def\soverG{{\subbaroverletter G}}
\def\soverH{{\subbaroverletter H}}
\def\soverI{{\subbaroverletter I}}
\def\soverJ{{\subbaroverletter J}}
\def\soverK{{\subbaroverletter K}}
\def\soverL{{\subbaroverletter L}}
\def\soverM{{\subbaroverletter M}}
\def\soverN{{\subbaroverletter N}}
\def\soverO{{\subbaroverletter O}}
\def\soverP{{\subbaroverletter P}}
\def\soverQ{{\subbaroverletter Q}}
\def\soverR{{\subbaroverletter R}}
\def\soverS{{\subbaroverletter S}}
\def\soverT{{\subbaroverletter T}}
\def\soverU{{\subbaroverletter U}}
\def\soverV{{\subbaroverletter V}}
\def\soverW{{\subbaroverletter W}}
\def\soverX{{\subbaroverletter X}}
\def\soverY{{\subbaroverletter Y}}
\def\soverZ{{\subbaroverletter Z}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% LOWER CASE LETTERS
\def\sovera{{\subbaroverletter a}}
\def\soverb{{\subbaroverletter b}}
\def\soverc{{\subbaroverletter c}}
\def\soverd{{\subbaroverletter d}}
\def\sovere{{\subbaroverletter e}}
\def\soverf{{\subbaroverletter f}}
\def\soverg{{\subbaroverletter g}}
\def\soverh{{\subbaroverletter h}}
\def\soveri{{\subbaroverletter i}}
\def\soverj{{\subbaroverletter j}}
\def\soverk{{\subbaroverletter k}}
\def\soverl{{\subbaroverletter l}}
\def\soverm{{\subbaroverletter m}}
\def\sovern{{\subbaroverletter n}}
\def\sovero{{\subbaroverletter o}}
\def\soverp{{\subbaroverletter p}}
\def\soverq{{\subbaroverletter q}}
\def\soverr{{\subbaroverletter r}}
\def\sovers{{\subbaroverletter s}}
\def\sovert{{\subbaroverletter t}}
\def\soveru{{\subbaroverletter u}}
\def\soverv{{\subbaroverletter v}}
\def\soverw{{\subbaroverletter w}}
\def\soverx{{\subbaroverletter x}}
\def\sovery{{\subbaroverletter y}}
\def\soverz{{\subbaroverletter z}}
%%%%
\def\sovercale{{\subbaroverletter{\cale}}}
\def\soverpartial{{\subbaroverletter\partial}}
\def\soverDelta{{\subbaroverletter\Delta}}
\def\soverzeta{{\subbaroverletter\zeta}}
\def\soverth{{\subbaroverletter\theta}}
\def\sovermu{{\subbaroverletter\mu}}
\def\soverom{{\subbaroverletter\omega}}
\def\soverOm{{\subbaroverletter\Omega}}
%%%%
\def\soverlambda{{\subbaroverletter\lambda}} %added 5-27-89
\def\soverf{{\subbaroverletter f}} %edited 5-31-89
%%****************************************************
%\catcode`@=11
%\def\n@me#1{\csname #1\endcsname}
%\def\n@medef#1{\expandafter\edef\csname #1\endcsname}
%
% macros for defining another family of mathematica characters
%
% newmathfam#1#2 defines a new font family called #1fam
% which use font #2
% sets textfont to #2 at 10pt
% sets scriptfont to #2 at 7pt and
% sets scriptscriptfont to #2 at 5pt
% defines \#1 to switch to the new family
%
%\def\newf@m{\alloc@ 8\fam \chardef \sixt@@n} % a non "\outer" version of
%\newfam
%\def\newmathfam#1#2{
% \edef\famname{\n@me{#1fam}}
% \expandafter\newf@m\famname
% \expandafter\expandafter\expandafter\font\n@me{#1text} = #2 at 10pt
% \expandafter\expandafter\expandafter\font\n@me{#1script} = #2 at 7pt
% \expandafter\expandafter\expandafter\font\n@me{#1scriptscript} = #2 at 5pt
% \expandafter\expandafter\expandafter\textfont\n@me{#1fam} = \n@me{#1text}
% \expandafter\expandafter\expandafter\scriptfont\n@me{#1fam} = \n@me{#1script}
% \expandafter\expandafter\expandafter\scriptscriptfont\n@me{#1fam} =
% \n@me{#1scriptscript}
% \n@medef{#1}{\fam=\n@me{#1fam}}}
%
%
%\catcode`\@=12
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\begin{document}
%\bskip24pt
\bskip18pt
% title.tex
%
% version Jan 08 96
%
% Title and Abstract of paper:
% AREA PRESERVING NONTWIST MAPS:
% RENORMALIZATION AND TRANSITION TO CHAOS%
%
\title{Renormalization and Transition to Chaos\\
in Area Preserving Nontwist Maps}
\author{D.~del-Castillo-Negrete,\thanks{Present address:
Scripps Institution of Oceanography, University
of California at San Diego, La Jolla, California
92093-0230, email: diego@fawlty.ucsd.edu}~~J.~M.~Greene,\thanks{General Atomics
Inc., San Diego, CA 92186--9784.}~~
{\rm and} P.~J.~Morrison\\ \physifs}
\maketitle
\begin{abstract}
\bskip12pt
The problem of transition to chaos, i.e.\ the destruction of
invariant circles or KAM (Kolmogorov-Arnold-Moser) curves, in area
preserving {\em nontwist} maps is studied within the renormalization group
framework. Nontwist maps are maps for which the twist condition is violated
along a curve known as the shearless curve. In renormalization language this
problem is that of finding and studying the fixed points of the
renormalization group operator ${\cal R}$ that acts on the space of maps. A
simple period-two fixed point of ${\cal R}$, whose basin of attraction
contains the nontwist maps for which the shearless curve exists, is found.
Also, a critical period-twelve fixed point of ${\cal R}$, with two unstable
eigenvalues, is found. The basin of attraction of this critical fixed point
contains the nontwist maps for which the shearless curve is at the threshold
of destruction. This basin defines a new universality class for the
transition to chaos in area preserving maps.
\end{abstract}
% introduction.tex
%
% version Jan 1996
%
% Section 1 of paper:
% AREA PRESERVING NONTWIST MAPS:
% RENORMALIZATION STUDY OF THE TRANSITION TO CHAOS
\section{Introduction}
A fundamental problem of Hamiltonian dynamics is to understand
the behavior of an integrable Hamiltonian system when subject to perturbation.
In terms of the action-angle variables $(J, \theta)$ of the integrable system,
the Hamiltonian for the perturbed system in the case of
one degree-of-freedom can be written as
\bq
\label{ham}
H = H_0(J) + H_1(J, \theta, t) \, ,
\eq
where $H_0$ is the Hamiltonian of the integrable system and the perturbation is
represented by $H_1$. Since the pioneering work of Poincar\'{e}, it has been
known that the dynamics for Hamiltonians of this form is far from trivial.
Typically, the phase space consists of a complicated mixture of integrable
(confined to invariant tori) and nonintegrable (chaotic) trajectories. Thus,
the problem of the transition to chaos is to determine which trajectories of
$H_0$ remain integrable and which become chaotic under the effect of $H_1$.
When the perturbation is periodic in time, i.e.\
$H(J, \theta, t+T) = H(J, \theta, t)$,
the essential aspects of the dynamics are captured by the so-called
Poincar\'{e} map, which is obtained by plotting the phase space coordinates of
the trajectories at times $t=T, 2T, 3T, \ldots, nT, \ldots$. Since, in
general, Hamilton's equations preserve the volume of phase space, the
Poincar\'{e} map is an area preserving map. Accordingly, the behavior of
Hamiltonian systems can be understood by studying area preserving maps, which
are relatively simpler mathematical objects than
differential equations (see for example
\cite{MacKay-Meiss-1987,Meiss-1992,Reichl-1992} and references therein). In
particular, the transition to chaos for Hamiltonians of the form of
Eq.~(\ref{ham}) can be studied with area preserving maps of the form
\bqy
x_{i+1}&=&x_i+\Omega(y_{i+1})+ f(x_i,y_{i+1})\nonumber \\
y_{i+1}&=&y_i+g(x_i,y_{i+1})\, ,
\label{gen-map}
\eqy
where the area preservation condition requires $\partial f/\partial x_{i} +
\partial g/\partial y_{i+1}=0$. The map variables $(x,y)$ correspond to the
action-angle coordinates $(J, \theta)$, the function $\Omega$ corresponds to
the unperturbed frequency $\partial H_0 / \partial J$, and the functions $f$
and $g$ correspond to the
perturbation $H_1$.
When $f$ and $g$ are zero the map is integrable: successive iterations of
initial conditions lie on straight horizontal lines that wrap around the
periodic $x$-domain. The {\em rotation number} of an orbit is defined, when it
exists, by $\omega := \lim_{i \rightarrow \infty} x_i/i$, where in this
definition the $x$-coordinate is lifted to the real line (i.e. $x$ is not taken
to be periodic). Orbits with irrational rotation numbers fill one-dimensional
dense sets called invariant tori (circles) or KAM
(Kolmogorov-Arnold-Moser) curves. On the other hand, periodic orbits have
rational
rotation numbers. Under the effect of the perturbation some KAM curves are
broken whereas others are merely deformed---they remain topologically
equivalent to straight lines. The problem of the transition to chaos in area
preserving maps is to determine which KAM curves persist and which are destroyed
by a nonintegrable perturbation of the map.
In the present paper we study the transition to chaos in the following
area preserving map:
\bqy
x_{i+1}&=&x_i+a(1-y^2_{i+1})
\label{nt-gen1}\\
y_{i+1}&=&y_i-b\,\sin\, (2\pi x_i) \,,
\label{nt-gen2}
\eqy
where, $a$ and $b$ are real numbers, and the domain
of interest is $D := \{(x,y)\,|\, y\in (-\infty,\infty) \ {\rm and}\ x\in
(-1/2, 1/2) \ {\rm mod}\ 1\}$. Following the terminology of
\cite{del-Castillo-Morrison-1993}, we call this map the {\em standard nontwist
map} because it violates the {\em twist condition},
\bq
\frac{\partial x_{i+1}}{\partial y_{i}} \neq 0 \,,
\label {twist-cond}
\eq
which is the map analogue of the {\em nondegeneracy condition}
for Hamiltonian systems,
\bq
\frac{\partial^2 H_0}{\partial J^2} \neq 0\, .
\label{deg-cond}
\eq
The point where the twist condition fails can be an
extremum (if $\partial x_{i+1}/\partial y_i$ changes sign), as in the
case of the standard nontwist map, or an
inflection point (if $\partial x_{i+1}/\partial y_i$ does not change sign).
The most interesting and challenging case, the one addressed here, is that of
an extremum.
The study of the transition to chaos in area preserving nontwist maps,
and equivalently in degenerate
Hamiltonian systems, is a problem of both theoretical and
practical relevance. Mathematically the problem is of interest because many
results in the theory of area preserving maps, including the KAM theorem
\cite{Moser-1962}, depend upon the twist condition. Only recently
there have been attempts to extend KAM
theory to nontwist maps
\cite{Llave-1995}. From a physics perspective, degenerate Hamiltonian
systems and nontwist maps are important because such systems naturally occur in
a variety of problems of fluid dynamics, plasma physics, celestial mechanics,
accelerator physics, condensed matter physics, and ray optics in wave guides,
among others. (For a discussion of some of these applications
see \cite{del-Castillo-1994,del-Castillo-etal-1995}.)
For $b=0$ the standard nontwist map is integrable. In this case, the twist
condition is violated along the line $y=0$, which we call the {\em shearless
curve} because along it the shear, $\partial x_{i+1}/\partial y_{i}$,
vanishes. A
precise and general (for $a$ and $b$ nonzero) definition of the shearless curve
is given in Sec.~2. As $a$ and $b$ deviate form zero, the
shearless curve bends and eventually breaks. The problem of transition to chaos
in nontwist maps is to understand when and how this shearless curve breaks.
Here we study this problem, restricting attention to the case in which the
rotation number of the shearless curve is equal to the
inverse golden-mean, $1/\gamma := (\sqrt{5}-1)/2$.
The transition to chaos in area preserving
maps exhibits {\em critical scaling} behavior
\cite{Kadanoff-1981,Shenker-Kadanoff-1982}. This means that
at the threshold of its destruction (i.e.\ at criticality), a KAM curve
possesses nontrivial scaling properties. In particular, critical KAM curves
are fractals, well-known geometrical objects that remain invariant
under appropriate successive spatial rescalings. These scaling properties are
believed to be universal in the sense that they depend only on very general
features of the map. In a way akin to what is done in the theory of phase
transitions, one can introduce {\em universality classes} for classifying the
fundamentally different ways in which the transition to chaos can take place.
These universality classes group together all the maps that share the same
scaling properties at criticality, even though the maps might ``look''
different.
Thus, a fundamental problem is to determine the possible universality classes of
the transition to chaos in area preserving maps. In
\cite{MacKay-1982,MacKay-1983,Greene-1993} it was shown that this problem
can be studied
using renormalization group techniques.
The goal of renormalization in area preserving maps is to
provide a framework for the study of a KAM curve with a given rotation number.
It is important to note that while KAM theory deals with the persistence
of dense sets of invariant curves, the renormalization approach deals
only with individual KAM curves of prescribed rotation numbers. This loss of
generality is compensated for by a gain in precision: renormalization group
estimates for the persistence of a KAM curve are considerably better than
estimates provided by KAM theory, which are generally too conservative.
The basic idea of renormalization is embodied in the {\em renormalization group
operator} ${\cal R}$, which maps the function space of area preserving maps
into itself. Iteration of this operator, which takes an area preserving map
into another such map, enables one to study a KAM curve on
successively smaller spatial scales and successively longer time scales.
{}From this ``space-time zooming" the fate of the KAM curve can be determined.
More formally, the destruction or persistence of the KAM curve of a map
$M$ is
determined by the asymptotic behavior of ${\cal R}$ acting repeatedly on $M$.
As will be explained in Sec.~3, the asymptotic behavior of the operator
${\cal R}$ is largely determined by its fixed points, which are maps
invariant
under renormalization. Thus, in renormalization language, the problem of
transition to chaos corresponds to the problem of finding and studying critical
fixed points of ${\cal R}$. Since the discovery of the critical period-one
fixed point for twist maps \cite{MacKay-1982,MacKay-1983}, other fixed
points have been found in standard {\em twist} maps
\cite{Greene-etal-1987,Wilbrink-1987,Wilbrink-1988,Ketoja-MacKay-1989,
Wilbrink-1990,Greene-Mao-1990}. In the present paper, we demonstrate the
existence of a new, higher order, critical fixed point of the renormalization
operator: the one associated with the transition to chaos of the $1/\gamma$
KAM shearless curve that occurs in nontwist maps. This fixed point has two
unstable eigenvalues. We also show that the simple fixed point for nontwist
maps is a period-two fixed point of ${\cal R}$.
In the next section we review previous results on the
transition to chaos in the standard nontwist map
\cite{del-Castillo-1994,del-Castillo-etal-1995}
and study the spatial scaling properties of the shearless curve at criticality.
In Sec.~3, after reviewing the renormalization group formalism, we discuss
the simple and critical fixed points corresponding to nontwist maps, and
compute the two unstable eigenvalues of the critical fixed point. Section~4
contains the conclusions. The present work is based in part on
\cite{del-Castillo-1994}.
% chaos.tex
%
% version Jan 08 1995
%
% Section 2 of paper:
% AREA PRESERVING NONTWIST MAPS:
%RENORMALIZATION STUDY OF THE TRANSITION TO CHAOS
\section{Transition to Chaos}
In this section we consider the destruction of the shearless curve. The first
subsection contains the summary of previous results, while the second
subsection contains the discussion of the spatial scaling properties of the
shearless curve at criticality.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Summary of previous results}
The analysis of the transition to chaos requires
the use of periodic orbits. Therefore, we review here our previous results on
periodic orbits in the standard nontwist map
\cite{del-Castillo-1994,del-Castillo-etal-1995}.
A point ${\bf x}:=(x,y)$ generates a {\it periodic orbit} of {\it order $n$}
if after $n$ iterations it returns to itself; i.e.\ $M^n{\bf x}={\bf x}$.
The rotation number associated with a periodic orbit
is the rational number $m/n$, where $n$ is the order of the periodic
orbit and $m$ is the integer number of times the orbit cycles through the
$x$-domain before returning to its initial position.
The standard nontwist map is reversible and accordingly can be decomposed into
a product of involutions:
$M=I_1I_0$, where
$I_0{\bf x} := $$\left( -x,y - b \sin (2 \pi x)\right)$ and
$ I_1{\bf x} := $$\left(-x +a(1-y^2),y\right)$.
The invariant sets of the involution maps,
${\cal I}_{0,1}:=\{{\bf x} | I_{0,1}{\bf
x}={\bf x} \}$, form the {\em symmetry lines} of the map.
For the standard nontwist map, ${\cal I}_0$ is the union of the following
symmetry lines
\bq
s_1=\{(x,y) \mid x=0\}\,,\quad\quad s_2= \{(x,y) \mid x=1/2\}\,,
\eq
while the invariant set ${\cal I}_1$ is the union of
\bq
s_3=\{(x,y) \mid x=a(1-y^2)/2\}\,, \quad \:
s_4= \{(x,y) \mid x=a(1-y^2)/2+1/2\} \,.
\eq
Symmetry lines reduce the search for periodic orbits
to a one-dimensional root finding problem, which is described
further in \cite{del-Castillo-1994,del-Castillo-etal-1995}.
Because of the violation of the twist condition, periodic orbits in the
standard nontwist map come in pairs; that is, contrary to what
happens typically in twist maps, there are {\em two} periodic orbits with
the {\em same} rotation number on each symmetry line. This is evident when
$b=0$, in which case periodic orbits with rotation number $m/n$ on
$s_1$, for example, are located at $(0,\pm \sqrt{1-(m/n)/a})$. We call the
periodic orbit with the larger
$y$-coordinate the {\em up} orbit and that with the smaller $y$-coordinate the
{\em down} orbit. As the map parameters are varied the up and down periodic
orbits
on a symmetry line can collide giving rise to a rich variety of
bifurcations including separatrix reconnection, a global bifurcation
that changes the phase space topology in the vicinity of the shearless curve.
At the collision point, the residue, (cf.\ \cite{Greene-1979}),
and the Poincar\'{e} index of the up and down periodic orbits vanish.
Given a rational number $r/s$, the {\em $r/s$ bifurcation curve},
$b:=\Phi_{r/s}(a)$, is defined as the locus of points $(a,b)$ for which
the $r/s$ periodic orbits are at the point of collision.
Given an irrational number $\sigma$, and a sequence of rational numbers
$\{r_i/s_i\}$ such that $\lim_{i \rightarrow \infty} r_i/s_i = \sigma$,
the {\em $\sigma$ bifurcation curve} is defined by
$b=\Phi_{\sigma}(a):=\lim_{i \rightarrow \infty} \Phi_{r_i/s_i}(a)$.
By construction, for $(a,b)$ values below $\Phi_{\sigma}$
all the periodic orbits with rotation numbers $r/s < \sigma$ are below
their collision point and thus exist and can be found by using the
symmetry line formalism.
Given two integer numbers $r$ and $s$, an
{\em $r/s$ nontwist map} is defined as
a map satisfying the following two conditions: (i) The map has either no
periodic orbits with rotation number greater than $r/s$, or it has no
periodic orbits with rotation number less than $r/s$. (ii) The map does have
periodic orbits with rotation number equal to $r/s$, and these orbits have
zero residue and zero Poincar\'{e} index. The condition on the Poincar\'{e}
index is imposed to ensure that the zero residue periodic orbits are at
the bifurcation point where the up and down periodic orbits have collided.
Bifurcation curves can be used to construct $\sigma$ nontwist maps,
where $\sigma$ is either a rational or an irrational number. In particular,
the standard nontwist map with $(a,b)$ values restricted to the
$\sigma$ bifurcation curve is a one-parameter $\sigma$ nontwist
map.
For a $\sigma$ nontwist map, a {\em shearless curve} is defined as a
curve with rotation number equal to $\sigma$.
Throughout this (and the previous \cite{del-Castillo-etal-1995})
paper we concentrate on the study of the
$1/\gamma$ shearless curve. This curve, when it exists, can be found
approximately as follows: First, construct an approximation to the
$\Phi_{1/\gamma}$ bifurcation curve; this was done by computing
$\Phi_{F_i/F_{i+1}}$ for Fibonacci ratios $F_i/F_{i+1}$ up to
$75,025/121,393$, and by using the scaling relation of
Eq.~(\ref{b-perp-sca}). Using this, the $1/\gamma$ shearless curve
can be approximated for an $(a,b)$ value on $\Phi_{1/\gamma}$ by the
set of up and down periodic orbits with rotation numbers
$\{F_{2i-1}/F_{2i}\}$; we considered rational approximants up to
$46,368/75,025$.
To compute the critical $(a,b)$ values for the destruction of the
shearless curve we used the residue criterion \cite{Greene-1979}, according
to which a KAM curve exists (does not exist) if the residues of the periodic
orbits approximating it converge to zero (infinity). The parameter
value(s) at which the residues exhibit a nontrivial convergence
(i.e.\ $\neq 0, \: \infty$) defines the critical point.
Employing the residue criterion, we found
in \cite{del-Castillo-1994,del-Castillo-etal-1995}
that the critical point for the destruction
of the $1/\gamma$ shearless curve in the standard nontwist map is:
\bq
\label{ab_critical}
(a_c,b_c) = (0.686049,\ 0.742493131039) \, .
\eq
At this critical value, the residues of the down periodic orbits on $s_1$ and
$s_4$, as well as the residues of the up periodic orbits on $s_2$ and $s_3$
converge to the {\em six-cycle} $\{H_1, H_2, H_3, H_4, H_5, H_6\}$ where:
\bqy
H_1&=2.325 \pm 0.002 \ \qquad H_2&=2.575 \pm 0.020 \\
H_3&=-0.599 \pm 0.010 \qquad H_4&=-1.283 \pm 0.001 \\
H_5&=2.575 \pm 0.020 \ \qquad H_6&=1.548 \pm 0.037 \, .
\eqy
On the other hand, the residues of the up periodic orbits on $s_1$ and $s_4$, as
well as the residues of the down periodic orbits on $s_2$ and $s_3$ converge
to the six-cycle $\{H_1, -H_2, H_6, H_4, -H_5, H_3\}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spatial scaling at criticality}
Now consider the spatial scaling of the shearless curve at criticality. To
this
end it is convenient to introduce {\em symmetry line coordinates},
\bqy
\label{sym-co}
\hat{x} := x - a(1-y^2)/2 \ , \qquad \hat{y} := y-y_{s} \ ,
\eqy
where $y_{s}$ is the $y$-coordinate of the point where the shearless
curve intersects the $s_3$ symmetry line. In these coordinates,
the $s_3$ symmetry line becomes a straight line that intersects the shearless
curve at the origin.
Figure~\ref{shearless}(a) shows, in symmetry line coordinates, a portion
of the $1/\gamma$ shearless curve at criticality. Figure~\ref{shearless}(b)
shows a magnification, centered at the origin, of Fig.~\ref{shearless}(a); the
$x$-coordinate is scaled by a factor of $321.92$ and the $y$-coordinate is
scaled by a factor of $463.82$. The spatial self-similar (fractal)
structure of the
$1/\gamma$ shearless curve at criticality is displayed by the remarkable
similarity between Figs.~\ref{shearless}(a) and (b). To understand the
origin of this self-similarity, up periodic orbits with rotation numbers
$55/89$ and
$17711/28657$ are shown in Fig.~\ref{spatial-renorm}(a), again using symmetry
line coordinates. (Equivalent results are obtained for down periodic
orbits.) In this figure, the circles denote the coordinates of the $55/89$ up
periodic orbit, while the crosses denote the coordinates of
the $17711/28657$ up periodic orbit after the spatial rescaling
$(\hat{x},\hat{y}) \rightarrow (321.92 \, \hat{x}\,,\, 463.82 \, \hat{y})$.
A similar plot is presented in Fig.~\ref{spatial-renorm}(b), where the
$144/233$ up periodic orbit (circles) is plotted along with the
spatially rescaled $46368/75025$ up periodic orbit (crosses).
{}From Fig.~\ref{spatial-renorm}(a) (Fig.~\ref{spatial-renorm}(b)) it is
seen that if the spatial coordinates are rescaled then, close to
$(\hat{x},\hat{y})=(0,0)$, a periodic orbit with rotation number
$F_{21}/F_{22}=17711/28657$ ($F_{23}/F_{24}=46368/75025$) is transformed into a
periodic orbit with rotation number $F_9/F_{10}=55/89$
($F_{11}/F_{12}=144/233$).
Thus, at criticality, in the vicinity of $(\hat{x},\hat{y})=(0,0)$ the map is
invariant under a simultaneous rescaling of spatial coordinates and
rotation numbers. Since the rescaling of rotation
numbers amounts to shifting the Fibonacci sequence by twelve,
$F_{2k+12-1}/F_{2k+12} \rightarrow F_{2k-1}/F_{2k}$, it is convenient to
write the spatial scaling using factors that are powers of twelve;
namely, $(\hat{x},\hat{y}) \rightarrow (\alpha^{12} \,
\hat{x}, \beta^{12} \, \hat{y})$, where
$\alpha=\sqrt[12]{321.92}$, $\beta=\sqrt[12]{463.82}$.
In this way, the map at criticality remains invariant after
twelve successive shiftings of rotation numbers
$F_{i}/F_{i+1} \rightarrow F_{i-1}/F_{i}$ and twelve successive spatial
rescalings $(\hat{x},\hat{y}) \rightarrow (\alpha \,\hat{x}, \beta\,
\hat{y})$. In the next section it will be shown that this is
equivalent to saying that the map at criticality is invariant
under the twelfth iterate of the renormalization group operator.
To formalize the previous ideas, let $(\tilde{x}_i,\tilde{y}_i)$ denote the
symmetry line coordinates of the up $F_{i-1}/F_{i}$ periodic orbit
closest to $(0,0)$ (similar results are obtained using the down-periodic
orbit), i.e.\ closest to the point where the shearless curve intersects
the $s_3$ symmetry line. Then, in the limit $n\rightarrow \infty$, it
is observed numerically that periodic orbits approach the shearless curve
(i.e. $\tilde{x}$ and $\tilde{y}$ converge to zero) according
to the power laws
\bqy
\label{xy-sca}
\tilde{x}_{2n} = {\cal X}(n)\, \alpha^{-2n} \ \qquad
\tilde{y}_{2n} = {\cal Y} (n) \, \beta^{-2n} \ ,
\eqy
where ${\cal X}$, ${\cal Y}$ are the
period-six functions (${\cal X}(n+6)={\cal X}(n)$ and
${\cal Y}(n+6)={\cal Y}(n)$) shown in Table~\ref{xy-scaling}.
Note that only coordinates $(\tilde{x}_i,\tilde{y}_i)$ with $i=2n$
have been considered. This is because only periodic orbits with
rotation numbers $\{F_{i-1}/F_{i}\}$ where $i=2n$ exist.
For this same reason, the exponent of $\alpha$ and
$\beta$ is $-2n$, and functions ${\cal X}$ and ${\cal Y}$ are
period six, rather than period twelve.
Equations~(\ref{sym-co}), (\ref{xy-sca}), and the periodicity
condition on ${\cal Y}$ imply
\bq
y_{s}= \lim_{n\rightarrow \infty} \frac{y_{n+1} \: y_{n+6}
- y_{n} \: y_{n+7}}
{\left ( y_{n+1} - y_{n} \right ) -
\left ( y_{n+7} - y_{n+6}\right )} \approx 0.2225230
\label{ysh}
\eq
and
\bqy
\alpha = \lim_{n \rightarrow \infty} \left |\;
\frac{{\tilde x}_{2n}}{{\tilde x}_{2n+12}} \; \right | ^{1/12}
\!\!\approx 1.618 \,, \qquad
\beta = \lim_{n \rightarrow \infty} \left | \;
\frac{{\tilde y}_{2n}}{{\tilde y}_{2n+12}} \; \right | ^{1/12}
\!\!\approx 1.668 \, ,
\label{alpha-beta}
\eqy
where in the calculation we have used periodic orbits
with rotation numbers up to $46368/75025$. Note that, as expected,
$\alpha^{12} = 321.92$, and $\beta^{12} = 463.82$
are the scaling factors used in Fig.~\ref{shearless}.
% renorm.tex
%
% version Jan 08 1996
%
% Section 3 of paper:
% AREA PRESERVING NONTWIST MAPS:
% RENORMALIZATION STUDY OF THE TRANSITION TO CHAOS
%
%
\section{Renormalization}
In the previous section it was shown that after a spatial rescaling,
at criticality, orbits with high period are mapped into orbits of
lower period (as shown e.g.\ in Fig.~(\ref{spatial-renorm})).
In this section the renormalization group formalism is used to explore this
invariance in greater detail.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Renormalization group operator}
Following \cite{MacKay-1982,MacKay-1983,Greene-1993} we define the
renormalization
operator for the $1/\gamma$ KAM curve in terms of pairs of commuting maps.
A {\em pair of commuting maps} is an ordered pair of maps, $(U,T)$, such that
$UT=TU$. The orbit of a point ${\bf x}$
generated by $(U,T)$ is the set of points $\{U^qT^p {\bf x} \}$,
where $q$ and $p$ are integers.
Given the set of periodic orbits $\{{\bf x}_i\}$ of
$M$ with rotation numbers $\{m_i/n_i\}$,
the commuting map pair $(U,T)$ associated with $M$ is defined by the condition
$U^{m_i}T^{n_i} {\bf x}_i={\bf x}_i$. Commuting map pairs are useful
because they provide a simple way for defining the renormalization group
operator.
For the $1/\gamma$ KAM curve, the renormalization group operator
is defined as follows \cite{MacKay-1982,MacKay-1983}:
\bq
\label{ren-def}
{\cal R} \left( \begin{array}{c} U \\T \end{array} \right) :=
B \left( \begin{array}{c} T \\TU \end{array}\right) B^{-1} \,.
\eq
This operator contains both time and space renormalization.
The {\em space renormalization} is represented by the operator $B$,
which rescales the $(x,y)$ coordinates, i.e.\ $(x,y) \rightarrow B(x,y)$
where
\bq
B=\left( \begin{array}{cc} \alpha & 0 \\0 & \beta \end{array} \right).
\eq
The values of $\alpha$ and $\beta$ are chosen to give the
appropriate magnification of the phase space in the vicinity of the
$1/\gamma$ KAM curve.
The idea of {\em time renormalization} is to transform periodic orbits
with large periods into periodic orbits with smaller periods, which amounts to
a rescaling of time. To understand how Eq.~(\ref{ren-def}) accomplishes
this, note
that:
\begin{quote}
If ${\bf x}$ is a periodic orbit of $(U,T)$ with rotation number
$F_{j-1}/\,F_{j}$, then ${\bf x}$ is a periodic orbit of
$(\tilde{U},\tilde{T})= (T,TU)$ with
rotation number $F_{j-2}/F_{j-1}$.
\end{quote}
The proof of this result is straightforward:
$\tilde{U}^{F_{j-2}} \tilde{T}^{F_{j-1}} = T^{F_{j-2}}
(TU)^{F_{j-1}} =$
\linebreak
$T^{F_{j-2}+ F_{j-1}} U^{F_{j-1}}
=U^{F_{j-1}} T^{F_j}$,
where the commutation relation $TU=UT$ and the definition of the Fibonacci
sequence, $F_{j}=F_{j-1}+F_{j-2}$,
have been used.
Hence, if ${\bf x}$ is a periodic orbit of
$(U,T)$ with rotation number $F_{j-1}/\,F_j$, then $U^{F_{j-1}}
T^{F_{j}}\,
{\bf x}\, = {\bf x}$, and since $U^{F_{j-1}} T^{F_{j}}\,
= \tilde{U}^{F_{j-2}} \tilde{T}^{F_{j-1}}, $ it is concluded that
${\bf x}$ is a periodic orbit of
$(\tilde{U},\tilde{T})$ with rotation number ${F_{j-2}}/\,F_{j-1}$.
By induction it is apparent that an orbit with rotation number
$F_{j-1}/\,F_{j}$
under $(U,T)$ is transformed into an orbit with rotation
number $F_{j-n-1}/\,F_{j-n}$ under ${\cal R}^n (U,T)$. Evidently ${\cal R}$
shifts the rotation number of the periodic
orbits, an operation that is equivalent to rescaling time.
To better understand the action of ${\cal R}$ on the space of
maps it is convenient to introduce coordinates for this domain. This is done
by using the residues. A map $M$ will be assigned coordinates $(R_{[1]},
R_{[2]},
\ldots,
R_{[i]},\ldots)$, where $R_{[i]}$ is the residue of the $F_i/F_{i+1}$
periodic orbit
of $M$ that approximates the $1/\gamma$ KAM curve.
Since the residues of a map are independent of the coordinates
used, maps related by coordinate changes of the $(x,y)$ space
will have the same coordinates in the space of maps.
In fact, using the residues as coordinates amounts
to dividing the space of maps into equivalence classes that contain maps
with the same values of the residues for the periodic orbits approximating
the KAM
curve under consideration. This is advantageous because the destruction
of a given KAM curve only depends upon the values of the residues.
Let $(\tilde {R_{[1]}}, \tilde {R_{[2]}}, \ldots \tilde {R_{[i]}}, \dots )$
denote the
coordinates of ${\cal R}(M)$. Since a periodic orbit of $M$ with rotation
number
$F_{i-1}/F_{i}$ is transformed into a periodic orbit of ${\cal R}(M)$ with
rotation number $F_{i-2}/F_{i-1}$, $\tilde{R_{[i]}}=R_{[i+1]}$ for
$i=1,2, \ldots$.
Hence in residue coordinates, the renormalization operator acts simply as a
shift (or translation of coordinates).
One can view the operator ${\cal R}$ as defining a dynamical system in the
space of maps. The existence of the $1/\gamma$ KAM curve in a map $M$ is then
determined by the asymptotic behavior of ${\cal R}$ acting repeatedly on $M$.
For example, if the coordinates of a map $M$ have a tail of zeroes, i.e.
if $M = (R_{[1]},\ldots, R_{[i]}, 0, 0,0,\ldots )$, then the
sequence $\{M, {\cal R}M, {\cal R}^2M, {\cal R}^3 M, \ldots \}$ will
converge to
the map $T=(0, 0, \ldots 0, \ldots)$. Since the residues of $T$ are all zero,
the $1/\gamma$ KAM curve exists in $T$ and therefore it exists in $M$,
which is,
from a renormalization point of view, equivalent to $T$. This is the
renormalization group interpretation of the residue criterion.
The {\em fixed points} of ${\cal R}$, which are maps invariant under
the renormalization, play a crucial role in the asymptotic behavior of
${\cal R}$. In particular, if a map $M$ is in the basin of attraction of
a fixed point $P$, then ${\cal R}^n M \rightarrow P$ as $n \rightarrow
\infty$.
{}From the renormalization point of view, all of the maps located in the
basin
of attraction of a fixed point are equivalent to the fixed point.
There are two kinds of fixed points: {\em simple} fixed points and
{\em critical} fixed points. A simple fixed point is an integrable map,
and its basin of attraction contains all the maps for which the KAM
curve under study exists. The problem of KAM theory, namely the study of the
persistence of invariant circles under perturbation, is translated in
renormalization language as the problem of showing that the simple fixed
point is
an attractor of all maps in its vicinity. The critical fixed point is the
map for
which the KAM curve under consideration is at the threshold of its destruction,
i.e.\ at criticality. All the maps in the basin of attraction of the
critical fixed
point are, from the renormalization point of view, equivalent to the fixed
point and
thus exhibit the same universal transition to chaos. This is the
renormalizationinterpretation of universality.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simple fixed point}
For twist maps, the simple fixed point is a period-one orbit of the
renormalization operator \cite{MacKay-1983}. For nontwist maps,
no period-one fixed points exist. However, a
period-two orbit of the renormalization operator that corresponds to the
simple fixed point does exist. This period-two fixed point is given
explicitly by
\bqy
\label{ut-plus}
U_+ \left( \begin{array}{c} x \\y \end{array} \right)=
\left( \begin{array}{c} x-\gamma+y^2/\gamma \\y \end{array} \right) \
\qquad
T_+ \left( \begin{array}{c} x \\y \end{array} \right)=
\left( \begin{array}{c} x+1+y^2 \\y \end{array} \right)
\eqy
and
\bqy
\label{ut-minus}
U_- \left( \begin{array}{c} x \\y \end{array} \right)=
\left( \begin{array}{c} x-\gamma-y^2/\gamma \\y \end{array} \right)
\ \qquad
T_- \left( \begin{array}{c} x \\y \end{array} \right)=
\left( \begin{array}{c} x+1-y^2 \\y \end{array} \right) \ .
\eqy
It is easy to check that ${\cal R} (U_{\pm},T_{\pm}) = (U_{\mp},T_{\mp})$,
and thus ${\cal R}^2 (U_{\pm},T_{\pm})= (U_{\pm},T_{\pm})$ with
\bq
B=\left ( \begin{array}{c c} -\gamma & 0 \\ 0 & -\gamma \end{array}\right)\ .
\eq
If ${\bf x}$ is a periodic orbit of $(U_{\pm},T_{\pm})$ with
rotation number $w_{\pm}=m/n$, then by definition
$U^m_{\pm}T^n_{\pm} {\bf x} = {\bf x}$
which, upon using Eqs.~(\ref{ut-plus}) and (\ref{ut-minus}), implies
$n (1 \pm y^2) + m (-\gamma \pm y^2/\gamma)=0$. This last equation
gives the following expression for the rotation number as a function
of $y$:
\bq
\label{ut-winding}
w_{\pm}(y) = \frac{\gamma \left (1 \pm y^2 \right )}{\gamma^2 \mp y^2} \,.
\eq
Accordingly, the map pairs of the period-two fixed point correspond to
the following integrable nontwist maps:
\bqy
(U_{\pm},T_{\pm}) \Longleftrightarrow \left \{
\begin{array}{l}
x_{i+1} = x_i + w_{\pm} (y_{i+1})
\\ y_{i+1} = y_i \,.
\end{array}
\right .
\eqy
Observe that at $y=0$, as expected, $\partial w_{\pm} /\partial y_{i}=0$
and $w_{\pm}=1/\gamma$. Therefore, $(U_{\pm},T_{\pm})$ are nontwist maps
with a shearless curve of rotation number equal to $1/\gamma$ at y=0.
Upon a change of coordinates the standard nontwist map of
Eqs.~(\ref{nt-gen1}) and (\ref{nt-gen2}) with $b=0$ is equivalent to the
map $(U_-,T_-)$. On the other hand, for $y$ close to zero, the map
$(U_+,T_+)$ is equivalent, up to a coordinate change, to a standard
nontwist map with an ``inverted shear"; i.e., to the map:
$x_{i+1} = x_i + a(1+y_{i+1}^2)$, $y_{i+1}=y_{i}$. Because of
this, the map $(U_+,T_+)$ ($(U_-,T_-)$) only possesses periodic
orbits with rotation numbers greater (less) than $1/\gamma$.
As said before, the simple fixed point is important because its basin of
attraction contains all the nontwist maps for which the $1/\gamma$ shearless
curve exists. In particular, the standard nontwist map, with $(a,b)$
values on the $b=\Phi_{1/\gamma}(a)$ bifurcation curve and with $a1$, $\mu_i^n\to \infty$ as $n \to \infty$, and
the eigenvalue is called unstable. On the other hand, if
$|\mu_i|<1$, $\mu_i^n\to 0$ as $n \to \infty$ and the eigenvalue
is called stable. Following the terminology used in
the theory of critical phenomena we refer to the eigenvectors
with unstable eigenvalues as {\em relevant eigenvectors},
and to the eigenvectors with
stable eigenvalues as {\em irrelevant eigenvectors}.
Thus, only perturbations $\delta \Lambda$ with components along
the relevant eigenvectors lead to departures from criticality.
Since the relevant eigenvectors span the tangent space of ${\cal U}$
at $\Lambda$, the dimension of ${\cal U}$ equals the number of
relevant eigenvectors. This number is closely related to the
number of independent control parameters necessary to put the system
at criticality. In particular, if criticality is observed at an isolated point
in parameter space, then the number of relevant eigenvectors equals the number
of control parameters. In the simplest case, e.g. the standard map, there
is only one control parameter. However, as we have seen before,
for the standard nontwist map we have to adjust two parameters
in order to achieve criticality, which occurs at the isolated point
$(a_c,b_c)$.
Thus, in this case, there are two relevant eigenvectors, and
the unstable manifold ${\cal U}$ is two-dimensional as shown in
Fig.\ref{cartoon}. All maps on ${\cal S}$ are, upon renormalization, equivalent
to $\Lambda$ and therefore share all the scaling properties of $\Lambda$. For
example, the spatial scaling properties of the $1/\gamma$ shearless curve in the
standard nontwist map at criticality are shared by $\Lambda$, and
all the maps on ${\cal S}$. Departures from criticality also
exhibit universal scaling behavior because {\em all} the departures
are governed by the {\em same} relevant eigenvectors.
The main difficulty in computing the relevant (i.e.\ unstable) eigenvalues
of $\wh{\cal R}$ is that
the space of maps is
infinite dimensional whereas the $(a,b)$-parameter space has only two
dimensions.
Obviously, with only two parameters the space of maps can not completely
explored.
However, the two dimensional unstable manifold ${\cal U}$ in the
vicinity of ${\Lambda}$ can be
explored using $(a,b)$ values close to $(a_c,b_c)$.
In order to do this, we define a renormalization operator in $(a,b)$ space,
$\rho(a_n,b_n) = (a_{n+1},b_{n+1})$, such that for
$(a,b)$ close to $(a_c,b_c)$:
\bq
\lim_{n\rightarrow \infty} \rho^n (a, b) = (a_c, b_c) \ ,
\label{rho_def1}
\eq
\bq
\rho\,(a_c,b_c) =(a_c,b_c) \ ,
\label{rho_def2}
\eq
and
\bq
\wh{\cal R} \left (M (\rho (a, b)) \right )
\label{rho_def3}
= M(a, b) \ .
\eq
Conditions (\ref{rho_def1}) and (\ref{rho_def2}) mean that $(a_c,b_c)$ is an
attracting fixed point of $\rho$, and (\ref{rho_def3}) means that
$\rho$ acting on the space of parameters is the
inverse of $\wh{\cal R}$ acting on the space of maps.
This last condition is the key property that will be used to
compute the eigenvalues of $\wh{\cal R}$ from the eigenvalues of $\rho$.
Near $(a_c,b_c)$, the operator $\rho$ can be linearized
\bq
\rho(a_c+\delta a, b_c+\delta b) \approx
(a_c,b_c) + D\rho(a_c,b_c)\cdot(\delta a, \delta b) \ ,
\label{rho_lin}
\eq
and if $\phi_i$, $i=1, 2$, are the eigenvectors of $D \rho$ with
eigenvalues $\nu_i$, then
\bq
\rho \left( (a_c, b_c)+\phi_i \right) \approx (a_c,b_c) + \nu_i \phi_i \ .
\label{rho_eigen}
\eq
In the limit $n \to \infty$, $M(\rho^n(a,b))$ is on the unstable
manifold ${\cal U}$ and, because of (\ref{rho_def3}), there is a one-to-one
correspondence between paths in $(a,b)$ space generated by $\rho$, and
paths in ${\cal U}$ generated by $\wh{\cal R}^{-1}$.
This implies that the unstable eigenvalues of $\wh{\cal R}$ are the
inverses of the stable eigenvalues of $\rho$, that is $\mu_i=1/\nu_i$.
Denoting the unstable eigenvalues of ${\cal R}$ by $\delta_1$ and $\delta_2$,
it is concluded from (\ref{def}) that
\bq
\delta_i = (1/\nu_i)^{1/12} \ .
\label{delta}
\eq
To compute the eigenvalues $\nu_i$, we will study the
linear behavior of $\rho$ along two direction in $(a,b)$ space: one
transverse and the other tangent to the $1/\gamma$ bifurcation curve
at $(a_c,b_c)$ as shown in Fig.~\ref{a_b_paths}.
\subsubsection {Computation of $\delta_1$}
To compute the first eigenvalue consider the following sequence of parameter
values:
\bq
\label{perp-seq}
\Sigma_{\perp} =
\{(a_c,b_{[2]}) ,\; (a_c,b_{[3]}), \ldots (a_c,b_{[n]}), \ldots \} \ ,
\eq
where
$b_{[n]}:=\Phi_{[n]}(a_c)$. (Recall $[n]=F_n/F_{n+1}$.)
This sequence approaches the critical point $(a_c,b_c)$ in a direction
``perpendicular" to the $\Phi_{1/\gamma}$ bifurcation curve as shown in
Fig.~\ref{a_b_paths}.
The action of the renormalization operator $\rho$
on this sequence is defined
as
\bq
\rho \, (a_c, b_{[n]}) := (a_c, b_{[n+12]}) \ .
\label{rho_perp}
\eq
By construction, this definition satisfies conditions (\ref{rho_def1}) and
(\ref{rho_def2}). Condition (\ref{rho_def3}) will be satisfied provided
\bq
\wh{\cal R}\left( M (a_c, b_{[2n+12]}) \right) = M\left(a_c, b_{[2n]}
\right) \ .
\label{Rrho_perp}
\eq
Denoting by
$R_{[m]}(a,b)$ the residue of the $F_m/F_{m+1}$ down periodic
orbit of the standard nontwist map at $(a,b)$, and
remembering that $\wh{\cal R}$ shifts the rotation number by twelve,
(\ref{Rrho_perp}) becomes
\bq
\label{res-inv-perp}
R_{[2m-1]}(a_c,b_{[2n]}) = R_{[2m-1+12]}(a_c,b_{[2n+12]}) \ .
\eq
Table~\ref{delta1-renorm}, shows the values of $R_{[2m-1]}(a_c,b_{[12]})$
and $R_{[2m-1+12]}(a_c,b_{[24]})$ for down periodic orbits
for $m=1,\, 2,\, \dots 6$. These values show that in the limit
$n \to \infty$, $n-m$ finite, (\ref{res-inv-perp}) is satisfied and
thus $\rho$, as defined in (\ref{rho_perp}), satisfies condition
(\ref{rho_def3}).
Having defined $\rho$ on the sequence $\Sigma_{\perp}$, let us study the
linear behavior of $\rho$ along
this sequence.
It is observed numerically that in the limit $n \to \infty$,
\bq
\label{b-perp-sca}
b_{[n+1]}=b_c+B_{\perp}(n) \, \nu_1^{n/12} \ .
\eq
a result that is accurate, for $n>20$, to eleven significant figures.
In Eq.~(\ref{b-perp-sca}) $B_{\perp}(n)$ is the
period-twelve function, $B_{\perp}(n+12)=B_{\perp} (n)$, given in
Table~\ref{B-function}, $b_c$ is the critical
$b$-coordinate of (\ref{ab_critical}) and
\bq
\label{b-perp-eigen}
\nu_1=\lim_{n\rightarrow \infty}
\left( \frac{b_{[n+12]}-b_c}{b_{[n]}-b_c} \right) \ .
\eq
>From Eq.~(\ref{b-perp-eigen}) we have that for large $n$,
\bq
b_{[n+12]} \approx b_c + \nu_1\, \left( b_{[n]} - b_c \right) \ .
\eq
Using this expression in (\ref{rho_perp}) yields
\bq
\rho\left(a_c,b_{[n]}\right) \approx
(a_c,b_c) +\nu_1 \, \left(0, b_{[n]}-b_c \right) \ ,
\eq
thus, according to (\ref{rho_eigen}), $(0,b_{[n]}-b_c)$ is an eigenvector
of $\rho$
with eigenvalue $\nu_1$.
>From (\ref{b-perp-eigen}) with $n=12$, and (\ref{delta}) the following
estimate for the first unstable eigenvalue of ${\cal R}$
is obtained:
\bq
\label{delta1}
\delta_1 \approx 2.683 \ .
\eq
This eigenvalue gives the rate of departure from $\Lambda$ for $(a,b)$
values off the $1/\gamma$ bifurcation curve. Next we compute the
second eigenvalue giving the rate of departure from $(a_c,b_c)$ for
$(a,b)$ values along the $1/\gamma$ curve.
%%%%%%%%%%%%%%%%%%%%%
% parallel eigenvalue
%%%%%%%%%%%%%%%%%%%%%
\subsubsection {Computation of $\delta_2$}
To compute the second eigenvalue we have to approach $(a_c,b_c)$
following a sequence of $(a,b)$ parameter values
such that the corresponding path, $M(a,b)$, in the space of
maps, leaves $\Lambda$ along the direction of the second
eigenvector as, for example, the trajectory that starts at $P_2$
in Fig.~\ref{cartoon}. It is natural to expect that parameter values
approaching $(a_c,b_c)$ along the $1/\gamma$ bifurcation curve,
$b=\Phi_{1/\gamma}(a)$, satisfy this condition.
That is, as $a\to a_c$, $M(a,\Phi_{1/\gamma}(a))$
leaves $\Lambda$ along the direction of the second eigenvector.
In terms of the operator $\rho$ this means that the tangent
of the $1/\gamma$ bifurcation curve at $(a_c,b_c)$ is an eigenvector
of $\rho$, and that there is a correspondence between
$(a,b)$ points on $\Phi_{1/\gamma}$ close to $(a_c,b_c)$, and points on the
unstable manifold generated by the second eigenvector.
Thus, following a similar approach to the one used before, we compute
the second eigenvalue $\delta_2$ by studying the standard nontwist map
for $(a,b)$ values on $\Phi_{1/\gamma}$ close to $(a_c,b_c)$.
However, in this case the problem is much harder because in practice it is
necessary to compute the $1/\gamma$ bifurcation curve to extreme
accuracy, otherwise the first eigenvalue dominates the result.
That is, if the $(a,b)$ values
are not close enough to the $1/\gamma$ bifurcation curve, the
departure of $M(a,b)$ from $\Lambda$ under the renormalization
operator will have a dominant component along the direction of the
first eigenvector.
The strong effect of the first eigenvalue
is one of the reasons why it is so difficult to compute the
critical parameter value $(a_c,b_c)$.
In fact for $(a_c,b)$ with $b=\Phi_{[24]}(a_c)$
($[24]=75,025/121,393$, $|[24]-1/\gamma|\sim 10^{-11}$)
the evolution of $M(a_c,b)$ under
${\cal R}$ is dominated by the first eigenvalue, even though
$(a_c,b)$ is very close to the $1/\gamma$ bifurcation curve.
This is the reason why,
as explained in \cite{del-Castillo-etal-1995}, to compute $(a_c,b_c)$,
that is to have $M(a_c,b_c)$ on the stable manifold ${\cal S}$,
it is necessary to compute $b_c$ to twelve digit accuracy.
Since finding a sequence of $(a,b)$ parameter values that
are exactly on $\Phi_{1/\gamma}$ is difficult, we will
use an approximate sequence, namely we will use
$(a,b)$ values on
$F_{2n}/F_{2n+1}$ bifurcation curves for $n=1, 2, \ldots$.
In the limit $n\to \infty$, this sequence approach
$(a_c,b_c)$ along $\Phi_{1/\gamma}$.
Note that for these parameter
values, there are two $1/\gamma$ KAM curves, the so-called up and
down curves. Thus we will consider the sequence
\bq
\label{par-seq}
\Sigma_{\parallel}=\{ (a_c^{[2]},b_c^{[2]}), (a_c^{[4]},b_c^{[4]}), \ldots,
(a_c^{[2n]},b_c^{[2n]}), \ldots \} \ ,
\eq
where for n=1, 2, \dots, $(a_c^{[2n]},b_c^{[2n]})$ is the point
on the $F_{2n}/F_{2n+1}$ bifurcation curve at which the up (and down)
$1/\gamma$ KAM curve is critical.
The renormalization operator $\rho$ acting on this sequence
is defined as
\bq
\rho(a_c^{[2n]},b_c^{[2n]}) := (a_c^{[2n+12]},b_c^{[2n+12]}) \ .
\eq
This definition clearly satisfies (\ref{rho_def1}) and (\ref{rho_def2}),
and condition
(\ref{rho_def3}) will be fulfilled provided
\bq
\label{res-inv-par}
R_{[m]}(a_c^{[2n]},b_c^{[2n]}) = R_{[m+12]}(a_c^{[2n+12]},b_c^{[2n+12]}) \ .
\eq
Table~(\ref{delta2-renorm}) shows numerical evidence that support
the validity of this
relation.
Having defined $\rho$ we turn now
to the problem of studying the behavior of $\rho$ near $(a_c,b_c)$.
In the limit
$n \rightarrow \infty$, the $\{a_{[2n]}\}$
sequence satisfies the
scaling relation
\bq
\label{a-par-sca}
a_c^{[2n]}=a_c+A_{\parallel}(n) \, \nu_2^{n/6} \ ,
\eq
where $A_{\parallel}(n)$ is a period-six function,
$A_{\parallel}(n+6)=A_{\parallel}(n)$,
$a_c$ is the critical $a$-coordinate of (\ref{ab_critical}),
and
\bq
\label{a-par-eigen}
\nu_2=\lim_{n\rightarrow \infty}
\left( \frac{a_c^{[2n+12]}-a_c}{a_c^{[2n]}-a_c} \right) \ .
\eq
Thus, for large $n$
\bq
\label{a-par-lin}
a^{[2n+12]}_c \approx a_c + \nu_2 \, \left(a^{[2n]}_c - a_c\right) \ .
\eq
On the other hand, the $b$ values scale as
\bq
\label{b-par-sca}
b_c^{[2n]} = \Phi_{1/\gamma} \left(a_c^{[2n]}\right) +
B_{\parallel}(n) \, \tilde{\nu_1}^{n/6} \, ,
\eq
where $B_{\parallel}$ is a period-six function, and
\bq
\label{b-par-eigen}
\tilde{\nu_1} = \lim_{n\rightarrow \infty}
\left( \frac
{b_c^{[2n+12]}-\Phi_{1/\gamma}\Big(a_c^{[2n+12]}\Big)}
{b_c^{[2n]}-\Phi_{1/\gamma}\Big(a_c^{[2n]}\Big)}
\right) \ .
\eq
Accordingly, for large $n$
\bq
\label{b-par-lin}
b_c^{[2n+12]} \approx b_c +
\tilde{\nu}_1 \left ( b_c - \Phi_{1/\gamma}\left(a_c^{[2n]}\right)\right) \ .
\eq
>From (\ref{a-par-lin}) and (\ref{b-par-lin}), we then conclude that for
large $n$
\bq
\label{a-b-par-lin}
\rho(a_c^{[2n]},b_c^{[2n]}) \approx (a_c, b_c) +
\tilde{\nu}_1 \left (0, b_c - \Phi_{1/\gamma}\left(a_c^{[2n]}\right)\right) +
\nu_2 \left(a^{[2n]}_c - a_c, 0\right) \ .
\eq
As discussed before, the eigenvalues of ${\cal R}$ are related
to those of $\rho$ by (\ref{delta}). The approximate value of $\tilde{\nu}_1$
as determined from (\ref{b-par-eigen}) gives, using to (\ref{delta}),
$\tilde{\delta}\approx2.748$, which within numerical error equals $\delta_1$.
Thus, the second term on the right hand side of (\ref{a-b-par-lin}) is the
component along the previously found eigenvector. This component is not zero in
this case because the sequence
$\Sigma_{\parallel}$ is not exactly on $\Phi_{1/\gamma}$. On the other
hand, the third term of (\ref{a-b-par-lin}) gives the projection of
the departure from $\Lambda$ along the second eigenvector, whose eigenvalue
according to (\ref{a-par-eigen}) has the approximate value
\bq
\delta_2 \approx 1.511 \ .
\eq
% conclusions.tex
%
% Section 4 paper:
%
% version jan 08 1996
%
% AREA PRESERVING NONTWIST MAPS:
% RENORMALIZATION STUDY OF THE TRANSITION TO CHAOS
\section{Conclusions}
In this paper we have presented a renormalization group study
of the transition to chaos in area preserving nontwist maps, maps that violate
the twist condition. These maps represent a type of degenerate
Hamiltonian system. They occur in many applications: chaotic transport in
fluid dynamics, reversed shear discharges in tokamak plasmas, and
trajectories about oblate planets in celestial mechanics, to name
a few. From a mathematical point of view, nontwist maps are interesting
because, as mentioned before, most of the theorems (including the KAM
theorem) assume the twist condition; many well-known and powerful results for
area preserving maps remain to be proved for nontwist maps.
The present work was based on the study of the standard nontwist map,
a prototype nontwist map that violates the twist condition along a
curve called the shearless curve. In
\cite{del-Castillo-1994,del-Castillo-etal-1995}
the critical parameter values for the destruction of the shearless
curve with rotation number equal to the inverse golden mean $1/\gamma=
(1-\sqrt{5})/2$ were computed, and it was shown that, at criticality,
the residues converge to a period-six cycle. The objective
of the present paper was to analyze these results in the renormalization
group framework.
Following a review of our previous results,
the spatial scaling properties of the $1/\gamma$ shearless curve
were studied. It was shown that, at criticality, the shearless curve
is invariant under the spatial rescaling $(\hat{x},\hat{y}) \rightarrow
(\alpha^{12} \hat{x}, \beta^{12}\hat{y})$ where $\alpha \approx 1.618$, and
$\beta \approx 1.668$. It was also shown that periodic orbits (in the vicinity
of the symmetry line) remain invariant under the simultaneous spatial
rescaling, $(\hat{x},\hat{y}) \rightarrow (\alpha^{12} \, \hat{x}, \beta^{12}
\, \hat{y})$, and shifting of rotation numbers,
$F_{2k+12-1}/F_{2k+12} \rightarrow F_{2k-1}/F_{2k}$.
Two fixed points of the renormalization group operator,
associated with the transition
to chaos in nontwist maps, were obtained: the simple period-two
fixed point and the critical period-twelve fixed point.
An explicit expression for the simple fixed point was presented.
This fixed point corresponds to integrable nontwist maps and its
basin of attraction contains all those maps for which the
shearless curve exists. The critical fixed point corresponds to
a nontwist map at criticality, and its basin of attraction defines
a new universality class for the transition to chaos. The standard
nontwist map at the critical parameter values is in the basin
of attraction of this critical fixed point and is thus representative of the
new universality class. Finally, the unstable eigenvalues of the period-twelve
fixed point were computed. Since the parameter space is two-dimensional, there
are two different directions available for departure from criticality, and
therefore, there are the two unstable eigenvalues: $\delta_1 \approx 2.683$ and
$\delta_2\approx 1.511$.
\subsection*{Acknowledgement}
This work was funded by the US Dept.~of
Energy under No.~DE-FG05-80ET-53088. DdCN
acknowledges partial support by the Universidad Nacional Autonoma
de M\'{e}xico, and the University Corporation for Atmospheric Research
Postdoctoral Program in Ocean Modeling.
% Fig-cap.tex
%
% Figure captions for:
% AREA PRESERVING NONTWIST MAPS: RENORMALIZATION AND
% AND TRANSITION TO CHAOS
% by D. del-Castillo-Negrete, J.M. Greene, and P.J. Morrison
%
% verison submitted to Physica D
% Jan 10 1996
\newcounter{figlist}
\subsection*{FIGURE CAPTIONS}
\begin{list}
{FIG.~\arabic{figlist}.}{\usecounter{figlist}
\setlength{\labelwidth}{.55in}
\setlength{\leftmargin}{.55in}}
\item Self-similar structure of the $1/\gamma$ shearless curve
at criticality. In case~(a) the shearless curve has been plotted in
symmetry-line coordinates. Case~(b) is a magnification of (a)
by a factor of $321.92$ in the $x$-direction and $463.82$ in the $y$-direction.
\label{shearless}
\item Self-similar structure of periodic orbits in the standard nontwist map at
criticality.
In (a) the circles denote the $F_{9}/F_{10}=55/89$ down periodic orbit
in symmetry line
coordinates, and
the crosses denote the $F_{21}/F_{22}=17711/28657$ down periodic orbits in
rescaled,
$(\hat{x},\hat{y}) \rightarrow (\alpha^{12}\hat{x}, \beta^{12}\hat{y})$,
symmetry line coordinates. In (b) the $F_{11}/F_{12}=144/233$ down
periodic orbits are
plotted
with circles along with the spatially rescaled down
$F_{23}/F_{24}=46368/75025$ periodic orbit denoted
by the crosses. These two plots show that, at criticality, the
periodic orbits near the symmetry line remain invariant
under a simultaneous spatial rescaling and shifting of rotation numbers.
\label{spatial-renorm}
\item Trivial self-similar structure of the integrable nontwist map.
The circles denote the $F_{5}/F_{6}=8/13$ up and down periodic orbits, and
the crosses the spatialy rescaled, $(x,y) \rightarrow
(\gamma^{2} x, \gamma^{2} y)$, $F_{7}/F_{8}=21/34$ up and down periodic
orbits. The plot shows that the integrable map is invariant
under a simultaneous spatial rescaling and shifting of rotation
numbers.
\label{trivial}
\item Iterations of the sixth power of the renormalization
group operator, ${\cal R}^6$, projected onto the $(R_{[1]},R_{[7]})$
plane in the space of maps. For visualization purposes the iterations
are shown in two separate panels. The stars denote the orbit of ${\cal R}^6$
acting on $M_c$, the standard nontwist map at criticality. Since $M_c$
is in the basin of attraction of the critical period-twelve fixed point
of ${\cal R}$, the stars converge to the period-two cycle
$\{(H_{1}, H_{4}), (H_{4}, H_{1})\}$. The circles and crosses denote
orbits of ${\cal R}^6$ acting on the standard nontwist map
below and above criticality, respectively.
\label{fixed-point}
\item Paths in $(a,b)$ parameter space used to compute the
relevant (i.e. unstable) eigenvalues.
$(a_c,b_c)$ is the critical value, and
$\Phi_{1/\gamma}$ denotes the $1/\gamma$ bifurcation curve, i.e.\ the
set of $(a,b)$ values for which the rotation number of the
shearless curve is $1/\gamma$.
$\Sigma_{\perp}$, which approaches
$(a_c,b_c)$ in a direction tranverse to $\Phi_{1/\gamma}$, yields
the dominant unstable eigenvalue $\delta_1$.
In the limit $(a,b)\to (a_c,b_c)$, $\Sigma_{\parallel}$ approaches
$(a_c,b_c)$ along $\Phi_{1/\gamma}$, and yields the second unstable
eigenvalue $\delta_2$.
Maps $M(a,b)$ for
$(a,b)$ values on $\Sigma_{\perp}$ and
$\Sigma_{\parallel}$, follow paths in the space of maps like
the ones in Fig.~6 starting on $P_1$ and $P_2$, respectively.
\label{a_b_paths}
\item Cartoon illustrating
the dynamics of the twelfth iterate of the renormalization
group operator, $\wh{\cal R}$, in the neighborhood of
the critical fixed point, $\Lambda$,
in the infinite dimensional space of maps. The axes label the
residue coordinates. ${\cal U}$ is the two dimensional unstable
manifold, whose tangent
space at $\Lambda$ is spanned by the two relevant eigenvectors with unstable
eigenvalues $\delta_1$, and $\delta_2$. ${\cal S}$ is a
codimension-two manifold containing all the nontwist
maps for which the $1/\gamma$ shearless curve is critical. This
infinite dimensional manifold defines a universality class for
the transition to chaos in nontwist maps. Maps
near ${\cal S}$, like $P_1$ and $P_2$, initially follow
${\cal S}$ and then depart from $\Lambda$ along
${\cal U}$.
\label{cartoon}
\end{list}
% version jan 08 1996
%
\newcounter{list}
\subsection*{TABLE CAPTIONS}
\begin{list}%
{TABLE~\arabic{figlist}.}{\usecounter{figlist}
\setlength{\labelwidth}{.55in}
\setlength{\leftmargin}{.55in}}
\item Spatial scaling functions ${\cal X}$ and ${\cal Y}$ for
up-periodic orbits.
\label{xy-scaling}
\item Scaling function $B_{\perp}(n)$.
\label {B-function}
\item Residue invariance for $\delta_1$.
\label {delta1-renorm}
\item Residue invariance for $\delta_2$.
\label {delta2-renorm}
\end{list}
% version Jan 08 1996
%\newcommand\trule{\rule{0pt}{2.7ex}} %top strut
%\documentstyle[12pt]{article}
%\begin{document}
%\pagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Table with values of the X and Y UP and scaling functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabular}{ccc}
\makebox[0.5 in]{$n$}
& \makebox[0.8 in]{${\cal X}(n)$}
& \makebox[0.8 in]{${\cal Y}(n)$}
\end{tabular}
\nopagebreak
\begin{tabular}{|c||c|c||} \hline\hline\trule
\makebox[0.5 in]{$1$}
& \makebox[0.8 in]{$1.303$}
& \makebox[0.8 in]{$1.387$}
\\[0.2 in]
\makebox[0.5 in]{$2$}
& \makebox[0.8 in]{$1.363$}
& \makebox[0.8 in]{$2.925$}
\\[0.2 in]
\makebox[0.5 in]{$3$}
& \makebox[0.8 in]{$1.306$}
& \makebox[0.8 in]{$1.073$}
\\[0.2 in]
\makebox[0.5 in]{$4$}
& \makebox[0.8 in]{$1.262$}
& \makebox[0.8 in]{$1.052$}
\\[0.2 in]
\makebox[0.5 in]{$5$}
& \makebox[0.8 in]{$1.516$}
& \makebox[0.8 in]{$1.105$}
\\[0.2 in]
\makebox[0.5 in]{$6$}
& \makebox[0.8 in]{$2.109$}
& \makebox[0.8 in]{$0.783$}
\\[0.2 in] \hline
\end{tabular}
\hspace {4 in} {\bf TABLE 1}
\vspace {1.5 in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% B function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabular}{cccc}
\makebox[0.5 in]{$n$}
& \makebox[0.8 in]{$B_{\perp}(n)$}
& \makebox[0.5 in]{$n$}
& \makebox[0.8 in]{$B_{\perp}(n)$}
\end{tabular}
\nopagebreak
\begin{tabular}{|c|c||c|c|} \hline\hline\trule
\makebox[0.5 in]{$1$}
& \makebox[0.8 in]{$1.2337$}
& \makebox[0.5 in]{$7$}
& \makebox[0.8 in]{$1.1295$}
\\[0.2 in]
\makebox[0.5 in]{$2$}
& \makebox[0.8 in]{$-2.0434$}
& \makebox[0.5 in]{$8$}
& \makebox[0.8 in]{$-1.6287$}
\\[0.2 in]
\makebox[0.5 in]{$3$}
& \makebox[0.8 in]{$1.8613$}
& \makebox[0.5 in]{$9$}
& \makebox[0.8 in]{$1.8445$}
\\[0.2 in]
\makebox[0.5 in]{$4$}
& \makebox[0.8 in]{$-1.3599$}
& \makebox[0.5 in]{$10$}
& \makebox[0.8 in]{$-2.3512$}
\\[0.2 in]
\makebox[0.5 in]{$5$}
& \makebox[0.8 in]{$1.6866$}
& \makebox[0.5 in]{$11$}
& \makebox[0.8 in]{$1.6863$}
\\[0.2 in]
\makebox[0.5 in]{$6$}
& \makebox[0.8 in]{$-1.3433$}
& \makebox[0.5 in]{$12$}
& \makebox[0.8 in]{$0.3038$}
\\[0.2 in] \hline
\end{tabular}
\hspace {4 in} {\bf TABLE 2}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Residue and b renormalisation for delta_1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabular}{ccc}
\makebox[0.5 in]{$m$}
& \makebox[1.4 in]{$R_{[2m-1]}(a_c,b_{[12]})$}
& \makebox[1.4 in]{$R_{[2m-1+12]}(a_c,b_{[24]})$}
\end{tabular}
\nopagebreak
\begin{tabular}{|c||c|c|} \hline\hline\trule
\makebox[0.5 in]{$1$}
& \makebox[1.4 in]{$2.778$}
& \makebox[1.4 in]{$2.328$}
\\[0.2 in]
\makebox[0.5 in]{$2$}
& \makebox[1.4 in]{$2.652$}
& \makebox[1.4 in]{$2.596$}
\\[0.2 in]
\makebox[0.5 in]{$3$}
& \makebox[1.4 in]{$-0.759$}
& \makebox[1.4 in]{$-0.609$}
\\[0.2 in]
\makebox[0.5 in]{$4$}
& \makebox[1.4 in]{$-1.334$}
& \makebox[1.4 in]{$-1.292$}
\\[0.2 in]
\makebox[0.5 in]{$5$}
& \makebox[1.4 in]{$2.673$}
& \makebox[1.4 in]{$2.674$}
\\[0.2 in]
\makebox[0.5 in]{$6$}
& \makebox[1.4 in]{$1.572$}
& \makebox[1.4 in]{$1.510$}
\\[0.2 in] \hline
\end{tabular}
\hspace{4 in} {\bf TABLE 3}
\vspace {0.4 in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Residue and a, b renormalisation for delta_2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{tabular}{ccc}
\makebox[0.5 in]{$m$}
& \makebox[1.4 in]{$R_{[m]}(a_c^{[4]},b_c^{[4]})$}
& \makebox[1.4 in]{$R_{[m+12]}(a_c^{[16]},b_c^{[16]})$}
\end{tabular}
\nopagebreak
\begin{tabular}{|c||c|c|} \hline\hline\trule
\makebox[0.5 in]{$5$}
& \makebox[1.4 in]{$0.5435$}
& \makebox[1.4 in]{$0.5241$}
\\[0.2 in]
\makebox[0.5 in]{$6$}
& \makebox[1.4 in]{$0.1479$}
& \makebox[1.4 in]{$0.1474$}
\\[0.2 in]
\makebox[0.5 in]{$7$}
& \makebox[1.4 in]{$0.3397$}
& \makebox[1.4 in]{$0.3165$}
\\[0.2 in]
\makebox[0.5 in]{$8$}
& \makebox[1.4 in]{$0.2083$}
& \makebox[1.4 in]{$0.1938$}
\\[0.2 in]
\makebox[0.5 in]{$9$}
& \makebox[1.4 in]{$0.2799$}
& \makebox[1.4 in]{$0.2429$}
\\[0.2 in]
\makebox[0.5 in]{$10$}
& \makebox[1.4 in]{$0.2340$}
& \makebox[1.4 in]{$0.1887$}
\\[0.2 in]
\makebox[0.5 in]{$11$}
& \makebox[1.4 in]{$0.2610$}
& \makebox[1.4 in]{$0.1820$}
\\[0.2 in] \hline
\end{tabular}
\hspace {4 in} {\bf TABLE 4}
%\end{document}
% Bibliography.tex
%
% version Jan 08 1996
%
% Bibliography of paper:
% AREA PRESERVING NONTWIST MAPS:
% PERIODIC ORBITS AND TRANSITION TO CHAOS
%
% Version submitted to Physica D on February 24 1995
\begin{thebibliography}{99}
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%\listoffigures
%\listoftables
\end{document}