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\begin{document}
\title { Nontwist area preserving maps with reversing symmetry group }
\author{\large Emilia Petrisor}
\date{}
\maketitle
\bce Department of Mathematics\\
"Politehnica" University of Timisoara\\
Str. M. Viteazul, nr. 1\\
1900 Timisoara, Romania
\ece
\bigskip
\begin{abstract}
The aim of this paper is to give a theoretical explanation of the
rich phenomenology exhibited by nontwist mappings of the cylinder,
in numerical experiments reported in {[del-Castillo {\it et al},
1996]}, {[Howard \& Humpherys, 1995]},
and to give new insights in the dynamics of nontwist
standard--like maps. Our approach is based on the reversing
symmetric properties of the nontwist standard--like systems. We
relate the bifurcation of periodic orbits of such maps to their
position with respect to a group--invariant circle, and to a
critical circle, at whose points the twist property is violated.
Moreover, we show that appearance of the meanders, i.e. homotopically nontrivial
invariant circles of the cylinder,
that are not graphs of real functions of angular variable,
is a global bifurcation of a curve on the cylinder, with nodal--type singularities,
and invariant with respect to the action of a reversing symmetry group.
\end{abstract}
\section { Introduction }
Area preserving diffeomorphisms of the annulus $\S\times [a,b]$
or infinite cylinder $\cil$
were extensively studied since they represent the dynamics of Poincar\'{e}
maps associated to two degree of freedom Hamiltonian systems.
An area preserving map (a.p.m) $F:\cil\to\cil$ is a positive (negative) monotone twist map
if one of its lifts ${\tilde F}:{\R}^2\to{\R}^2$ is such that the function
$y\to p_1{\tilde F}(x,y)$ is strictly increasing (decreasing),
for any $x\in \S$
($p_1:{\R}^2\to\R$ is the projection $p_1(x,y)=x$).
The basic properties of the monotone twist area preserving maps can be found
in {[Katok, A. \& Hassenblatt B, 1995]}. One of them is
that a homotopically nontrivial $F$--invariant circle
is the graph of a continuous function $\varphi:\S\to\R$.
%The monotone twist property is one of the basic assumptions in KAM
%theory,
% dealing with the persistence under perturbations
%of the homotopically nontrivial invariant circles of an integrable
%diffeomorphism of the cylinder.\par
Last years physical phenomena whose evolution is described by the
dy\-na\-mics of nontwist area preserving diffeomorphisms were
studied. Namely, nontwist maps are appropriate models for some
processes in hydrodynamics, plasma physics, accelerator physics,
celestial mechanics (see ~{[del-Castillo {\it et al}, 1996],
[Howard \& Humpherys, 1995]} and references therein).\par
The prototype for such a map is a two parameter family
of area preserving nontwist diffeomorphisms $F_{\omega,k}$ of an
annulus $\S\times [a,b]$:
\beq\label{eq:nontwmg}
F(x,y)=\left (x+f_\omega\left(y-\frac{k}{2\pi}g(2\pi x)\right)
\,\, (\mbox{mod}\,\, 1),\,\, y-\frac{k}{2\pi}g(2\pi x)\right ),
\eeq
where $f_\omega:\R\to\R$, called {\it twist function}, has a unique
critical point in $(a,b)$, and it is an extremum,
while $g$ is a $2\pi$--periodic function of zero average.\par
Numerical investigations, that revealed a wealth of phenomena
in the dynamics of such maps, were done on the particular systems
corresponding to $g(x)=\sin x$, and $f_\omega(y)=y-\omega y^2$
~{[Howard \& Humpherys, 1995]}, repectively $f_\omega(y)=\omega(1-y^2)$
~{[del-Castillo {\it et al}, 1996],
[del-Castillo {\it et al}, 1997]}. Mainly, one has detected collision
and annihilation of
periodic points, global bifurcations of some invariant circles,
scaling properties.\par
A nontwist map corresponding to $g(x)=\sin x$, is nongeneric.
In the generic case $g$ has infinitely many harmonics.
As an example we will take
$g(x)=\sin(x-\arcsin(\epsilon \sin x))$, $\epsilon\in (0,1)$.\par
As far as we know, there exist only few theoretical results concerning
nontwist a.p.m. ~{[Sim\'{o}, 1998], [Delshams \& de la Llave, 1998],
[Franks \& Le Calvez, 1999]}. C. Sim\'{o} [1998]
proves the persistence of invariant circles
in a generic perturbation of a nontwist area preserving map, and
the existence of invariant circles that exhibit foldings in such a way that they
are not graphs of functions defined on $\S$. These circles are called
{\it meanders}. On the other hand, in {[Delshams \& de la Llave, 1998]} it is proved
the persistence of the invariant circles of a two-parameter family of nontwist
a.p.m. having rotation number
on the boundary of the range of the frequency, and it is given a partial
justification of Greene's criterion {[Greene, 1979]} for nontwist maps (numerical results
in {[del-Castillo {\it et al}, 1997]} were obtained on the basis of this criterion).
The paper {[Franks \& Le Calvez, 1999]} studies regions of instability for nontwist area
preserving maps.\par
In the present work we give a theoretical explanation of the rich
phenomenology displayed by the nontwist maps in numerical
experiments reported in {[del-Castillo {\it et al}, 1996], [Howard
\& Humpherys, 1995]}.
Our approach is based on the reversing symmetric properties of the
analysed nontwist area preserving systems. A nontwist
standard--like a.p.m. $F$ has a reversing symmetry group $\cG$,
generated by a classical symmetry (equivariance), and two reversing
symmetries: one orientation reversing, and another orientation
preserving. The fixed points of the orientation preserving
reversors record all global bifurcations underwent by the
$\cG$--invariant curve on the cylinder.
We prove that appearance of the meanders
is a global bifurcation of a curve with nodal singularities,
invariant with respect to the action of the reversing symmetry
group on the cylinder.\par
\section {Properties of Nontwist Standard-like Maps}
In the following, consider the circle $\S$ identified with the
interval $[0,2\pi)$.
Let $F:\cil\to\cil$ be a nontwist a.p.m., $\Pi:{\R}^2\to\cil$,
the covering projection defined by
$\Pi(X,Y)=(X\,\,(\mbox{mod}\,\,2\pi), Y)$,
and ${\tilde F}:{\R}^2\to{\R}^2$ a lift of $F$
to the covering space ${\R}^2$, i.e. $F \Pi=\Pi\tilde{F}$.
Associate to a point $(x,y)\in\cil$, the sequence
$(X_n,Y_n)=\tilde{F}^n(X_0,Y_0)$, where $n\in\N$, and $(X_0,Y_0)$
is a point of the preimage $\Pi^{-1}(x,y)\subset{\R}^2$. If the
limit: \beq \rho= \ds\lim_{n\to\infty}\ds\frac{X_n-X_0}{2n\pi} \eeq
exists, then it does not depend on the lift $\tilde{F}$, and it is called
the {\it rotation number} of the orbit generated by
$(x,y)\in\cil$.\par
An orbit ${\cal O}(x,y)=\{(x_n,y_n)=F^n(x,y), n\in\Z\}$ of a
nontwist a.p.m of the cylinder is called {\it rotational orbit} if
there is a homotopically nontrivial circle on the cylinder, that
interpolates the points of the orbit (here a circle is meant in the
topological sense).\par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $(x,y)=\Pi (X,Y)$ be a $q$--periodic point of the map $F$,
i.e. $F^q(x,y)=(x,y)$. Denoting by $(X_q,Y_q)=\tilde{F}^q(X,Y)$, we
have $\Pi(X_q,Y_q)=\Pi(X,Y)$. Hence $X_q-X\in 2\pi{\Z}$, $Y_q=Y$,
and thus a q-periodic point of $F$ is the projection of a point
$(X,Y)\in{\R}^2$ such that
\beq\begin{array}{ll} X_q=X+2p\,\pi,\,\, & p\in {\Z}\\
Y_q=Y &
\end{array}\eeq
\noindent The periodic point $(x,y)=\Pi(X,Y)$ is called
$(p,q)$-periodic point or periodic point of type $(p,q)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The behaviour of the nontwist maps, investigated in {[del-Castillo
{\it et al}, 1996], [del-Castillo {\it et al}, 1997]} and {[Howard
\& Humpherys, 1995]}, is captured by the {\it model map} defined on
the infinite cylinder $\cil$,
$F_{\omega,k}(x,y)=(\ovl{x},\ovl{y})$: \beq\label{eq:modelmap}
F_{\omega,k}(x,y)=(x+f_\omega(\ovl{y})\,\, (\mbox{mod}\,\,
2\pi),y+k\sin{ x}), \eeq where $f_\omega(y)=2\pi\omega+y^2$. This
choice for the form of the map, instead of that of type Eq. (\ref
{eq:nontwmg}) is subjective (it allows faster guesses for
particular initial conditions in different simulations). All
figures in this paper, excepting Fig. 4, display different features
of the system defined by Eq. (\ref {eq:modelmap}).
For $k=0$, the model map is an integrable system: \bea
\ovl{x}&=&x+2\pi\omega+y^2\,\, (\mbox{mod}\,\, 2\pi)\nonumber\\
\ovl{y}&=&y \eea The twist condition is violated along the circle
$\S\times \{0\}$, i.e. $\ds\frac{\partial \tilde{F}_1}{\partial
y}(x,0)=0$, $\all x\in\R$. Each point on the invariant circle
$\S\times \{y\}$ has the rotation number equal to
$\omega+y^2/{2\pi}$. Hence the critical circle is the circle of
minimum rotation number among all circles that foliate the annulus
(if we chose $f_\omega(y)=2\pi\omega-y^2$, then this would be the
circle of maximum rotation number).\par
Observe that the map $f_\omega$ in all families of nontwist a.p.m.
mentioned here, is symmetric with respect to a line $y=y^*$ (i.e.
$f(2y^*-y)=f(y),$ $\all y\in\R$), and it is either a strictly
convex or a strictly concave function.
In the following, consider the family of diffeomorphisms $F_{\omega,k}$ of
the cylinder defined by a strictly convex
twist function $f_\omega$, which is
symmetric with respect to the line
$y=y^*$, ($y^*$ is the point of extremum for $f_\omega$), and $g(x)=\sin \, x$.
Call such a map {\it nontwist standard-like map}.
The systems corresponding to strictly concave twist functions
exhibit similar behaviour.\par
%In the case of the
% model map, $f_\omega(y)=2\pi\omega+y^2$ is symmetric with
% respect
% to the line $y=0$.\par
Next, we drop the indices
$\omega,k$ and $(\mbox{mod}\,\, 2\pi)$ for the first component of
each mapping of the cylinder. If ${\tilde F}$ is a lift of the nontwist
standard--like map $F$, then
$\ds\frac{\partial{\tilde F}_1}{\partial y}=0$
along the curve $C: y+k\sin{ x}=y^*$. This curve is called
the {\it nonmonotone curve} or
{\it circle} of the nontwist map $F$. It is not a
$F$--invariant circle.\par
\bdf An orbit $\{(x_n,y_n), n\in\Z\}$, of a nontwist
standard--like map $F$ is a {\it pseudomonotone orbit} if there
exist $i,j\in \Z$, such that $f'(y_i+k\sin {x_i})>0$, and
$f'(y_j+k\sin {x_j})<0$. A $q$--periodic orbit of $F$,
$\{z_0,z_1,\ldots,z_{q-1}\}$,
$z_i=(x_i,y_i)$, for which
there exists $j=\ovl{0,q-1}$, such that $z_j\in C$, is called {\it
nonmonotone periodic orbit}.\edf
A nontwist standard-like map is symmetric with respect to the
transformation
$S$ of the cylinder, defined by
\beq
S(x,y)=\left (x+\pi, 2y^*-y \right ),
\eeq
\noi i.e. $S F=F S$. Observe that $S^2=\mbox{id}$, that
is $S$ is an involution of the cylinder.
On the other hand, $F$ is reversible with respect to the
involution $R$ defined by: \beq R(x,y)=\left (-x, y+k\sin{ x}
\right ), \eeq i.e. $F^{-1}=R F R$. $R$ is called {\it involutive
reversor} of the map $F$. \noi $F$ decomposes as a product of
involutions $F=I R$, where \beq I(x,y)=\left ( -x+f(y),y\right ).
\eeq
Associate to these involutions the basic symmetry lines
denoted ${\L}_0,{\L}_1$, ${\L}_0:=\mbox{Fix}(R)$, and ${\L}_1:=\mbox{Fix}(I)$.
Each symetry line has two components:
\beq
\begin{array}{l}
{\L}_0\left \{\begin{array}{l} \Gamma_{0}: x=0\\
\Gamma'_{0}: x=\pi
\end{array}\right . \\
\\
{\L}_1\left \{\begin{array}{l} \Gamma_{1}: x=f(y)/2\,\,
(\mbox{mod}\,\, 2\pi)\\
\Gamma'_{1}: x=f(y)/2+\pi\,\, (\mbox{mod}\,\, 2\pi)
\end{array}\right .
\end{array}
\eeq
Observe that \beq \label{eq:efectS} S(\Gamma_i)= \Gamma'_i,\,\,
i=0,1 \eeq Next we recall some properties of periodic orbits of a
reversible system.
\bdf A $R$--invariant orbit ${\cal O}(x,y)=\{ f^j(x,y), j\in {\Z}
\}$ of
the $R$-reversible system $F$, i.e. $R({\cal O}(x,y))={\cal O}(x,y)$, is
called $R$-symmetric orbit or orbit symmetric with
respect to $R$. \edf
Denote by ${\L}_{2n}=F^n({\L}_0)$ and ${\L}_{2n+1}=F^n({\L}_1)$,
and call them symmetry lines of order $2n$, respectively, $2n+1$.
It is known that periodic orbits of a reversible system, which are
$R$--symmetric, can be found
as the intersections of the symmetry lines ${\L}_i,{\L}_j$.
Namely, a point belonging to ${\L}_i\cap {\L}_j$ is periodic
and its period $q$ divides $\vert i-j\vert$.
$R$--symmetric periodic orbits have a special position on the basic
symmetry lines {[de Vogelaere, 1958]}:
\bpr\label{pozper} A R--symmetric $q$-periodic orbit ${\cal
O}(x,y)$ of the revesible diffeomorphism $F$, $F=I R$, has the
property:\par
i) if $\,q$ is odd, the orbit has a unique point on the symmetry
line ${\L}_0$,
and a unique point on ${\L}_1$. If $\,(x,y)\in{\L}_0$ then
$F^{(q+1)/2}(x,y)\in{\L}_1$.\par
ii) if $\,q$ is even, the orbit does not intesect both symmetry
lines ${\L}_0$, ${\L}_1$,
but it intersects twice one of them. If $\,(x,y)\in{\L}_j$,
then $ F^{{q}/{2}}(x,y)\in{\L}_j, j=0,1.$
\epr
Because the symmetry $S$ is an involution that commutes with $R$
and $I$, it is straightforward to see that $S R$ and $S I$ are
involutions, too, and $F$ decomposes as $F=(S I) (S R)$. Denote by
$\cal G$ the group generated by $S, R, I$ and call it the {\it
reversing symmetry group} of the nontwist standard-like map $F$.
For a more general reversing symmetry group of a map see {[Lamb,
1992]}. The name of reversing symmetry group is due to the fact
that it contains symmetries of the map $F$, and reversing
symmetries i.e. involutive reversors of $F$. The composition of two
reversors is a symmetry ($F$ is a symmetry for itself), and the
composition of a symmetry with an involutive reversor is an
involutive reversor (in the case of a nontwist standard--like map,
not in general).\par
While the reversors $R$ and $I$ are orientation reversing, $S R$
and $S I$ are orientation preserving. Hence their fixed point set
is not a curve, but two points: $\mbox{Fix}(R
S)=\{{\p}_0,{\p}_0'\}$, where \beq
{\p}_0\left(\frac{\pi}{2},y^*-\frac{k}{2}\right ),\,\,\,
{\p}'_0\left(\frac{3\pi}{2},y^*+\frac{k}{2}\right), \eeq and
$\mbox{Fix}(I S)=\{{\p}_1,{\p}'_1 \}$, \beq
{\p}_1\left(\left(\frac{1}{2}f(y^*)+\frac{\pi}{2}\right )\,\,
(\mbox{mod}\,\, 2\pi),y^*\right),\,\,\,
{\p}'_1\left(\left(\frac{1}{2}f(y^*)+\frac{3\pi}{2}\right )\,\,
(\mbox{mod}\,\, 2\pi),y^*\right)
\eeq
Observe that \beq\label{relpoints1} S({\p}_0)={\p}'_0, \,\,\,
R({\p}_0)={\p}'_0 \eeq \noi and \beq\label{relpoints2}
S({\p}_1)={\p}'_1, \,\,\,I({\p}_1)={\p}'_1 \eeq
Our purpose is to analyse how the symmetry $S$ and the reversors
$R$ and $S R$ manifest themselves in the dynamics of the map
$F$.\par
The hypothesis that the twist function $f$ of nontwist
standard--like map $F$ is a strictly convex function (i.e.
$f''(y)>0$, $\all\,\, y\in\R$), ensures the existence of
nonmonotone $\cal G$--invariant annuli (a nonmonotone annulus is
one having as the complement, two $S$--symmetric unbounded
components, $U^+$, $U^-$ $\subset\cil$, and the restriction
$F_{\vert U^+}$, respectively $F_{\vert U^-}$, is a positive,
respectively,
a negative monotone twist map). We will show that interesting
behaviour, and bifurcations occurs within such an annulus. Denote
by $A$, the largest annulus, whose all rotational
orbits are pseudomonotone or monotone.\par
The dynamics in monotone twist regions is well known (see {[Katok
\& Hassenblatt, 1995]}). One of the basic results, for an exact
area preserving monotone twist map of the infinite cylinder,
asserts that for any $p\in \Z$, $q\in \N\setminus \{0\}$ relative
prime numbers, there exist two periodic orbits of the same period
$q$, and rotation number $\frac{p}{q}$, one elliptic and another
hyperbolic. If the system is close to an integrable one, the two
orbits form a chain of islands, called Poincar\'{e}--Birkhoff
chain. Elliptic points are surrounded by invariant circles, and
hyperbolic points are connected by heteroclinic orbits.\par
Because of the $S$--symmetry of a nontwist standard--like map,
Poincar\'{e}--Birkhoff chains exist in pairs. $S$ maps the points
$(x_n,y_n)$ of an orbit to $(x_n+\pi,2y^*-y_n)$, $\forall n\in \Z$,
hence two $S$--symmetric orbits have the same rotation number.
Denote by $C^q_u$, $C^q_l$ the upper, respectively, the lower
Poincar\'{e}--Birkhoff chain of period $q$, $C^q_l=S(C^q_u)$. Next
we investigate the relative position on the basic symmetry lines of
elliptic and hyperbolic points of the $S$--symmetric pair
$(C^q_u,C^q_l)$.
\bpr Let $C^q_u$, $C^q_l$ be two $S$-- symmetric
Poincar\'{e}--Birkhoff chains of odd period $q$. The two chains
have points of opposite stability type on the same component of a
basic symmetry line.\epr
\begin{dem}
Let $(x_0,y_0)$ be an elliptic point of the chain $C^q_u$ and
$(x'_0,y'_0)$ a hyperbolic one. If, for instance, ${\cal
O}(x_0,y_0)$ intersects $\Gamma_i$, and ${\cal O}(x'_0,y'_0)$
intersects $\Gamma'_i$, $i\in\{0,1\}$, then because an odd periodic
orbit intersects both basic symmetry lines ${\L}_0$,
${\L}_1$, we can suppose that ${\cO}(x_0,y_0)$ intersects
$\Gamma'_j$ too, while ${\cO}(x'_0,y'_0)$ has a point on $\Gamma_j$,
$j\in\{0,1\}$, $j\neq i$. But $S(\Gamma_i)=\Gamma'_i$,
hence $S({\cO}(x_0,y_0))$ intersects $\Gamma'_i$
and $\Gamma_j$, while $S({\cO}(x'_0,y'_0))$ intersects
$\Gamma_i$ and $\Gamma'_j$.
\end{dem}
Because on the same component of a basic symmetry line there exists
two points of opposite stability type, belonging to
different chains that are $S$--symmetric (see Fig.1), when the
parameter $k$ is fixed and $\omega$ increases, these points
approach each other, and can collide.
\bpr \label{pr:opar}
Let ${\cal O}(x_0,y_0)$ be a periodic orbit of period $2q$ and type $t$
(elliptic or hyperbolic), intersecting one
of the two components of the
symmetry line ${\L}_i$, $i=0,1$.
The orbit $S({\cal O}(x_0,y_0))$
has the same type $t$, and intersects
the same component of the line ${\L}_i$.
\epr
\begin{dem}
To make a choice suppose that $(x_m,y_m)=F^m(x_0,y_0)\in\Gamma_i$, $i\in\{0,1 \}$.
Then, by Eq. (\ref {eq:efectS}), $S(x_m,y_m)$ lies on $\Gamma'_i$.
Because an orbit of even period
intersects twice the same symmetry line ${\L}_i$ we have that there
is an
$n\in \{ 0,1,\ldots, 2q-1\}\setminus \{m \}$
such that $(x_n,y_n)\in\Gamma_i$. Because equivariances preserve
the stability type of periodic orbits, ${\cal O}(x_0,y_0)$ and
$S({\cal O}(x_0,y_0))$ are both, in the nondegenerate case, either
hyperbolic or elliptic.
\end{dem}
\begin{rmk} This result shows that even periodic orbits exist in pairs of
the same
stability type on a component of a basic symmetry line (Fig. 1).
In this case a
hyperbolic--hyperbolic
and elliptic--elliptic collision is possible.
\end{rmk}
\begin{figure}[h]
\bce
%\includegraphics{fig1p.eps}
\caption{ Two pairs of Poincar\'{e}-Birkhoff chains of period 3,
respectively 4, outside the nonmonotone annulus of the model map
Eq. (\ref {eq:modelmap}), corresponding to $k=0.3$, $\omega=0.231$;
The figure also displays the components of the basic symmetry
lines, the nonmonotone circle $C$ and the $\cal G$--invariant curve
$\gamma_s$ (the curve containing four marked points.}
\ece
\end{figure}
Besides Poincar\'{e}--Birkhoff chains, invariant circles of the
same rotation number exist in $S$--symmetric pairs, too.\par
The amount of (reversing) symmetry of a subset $\Lambda$ of the
cylinder is measured
by its isotropy subgroup
$\Sigma_\Lambda\subseteq{\cal G}$, \beq
\Sigma_\Lambda=\{\sigma\in{\cal G}\vert\,\, \sigma
\Lambda=\Lambda\} \eeq We are interested in determining the
isotropy group of an orbit of the system, or of a $F$--invariant
homotopically nontrivial circle on the cylinder. For example, the
union of two $S$--symmetric invariant circles has ${\cal G}$ as
isotropy group. This set has two connected components. Next, we
are interested whether there exists a connected $\cal G$-invariant
subset $\Lambda$ of the cylinder or not.
\bpr\label{pr:fshear} A nontwist standard--like map has at most
one $\cal G$-invariant homotopically nontrivial circle.
\epr
\begin{dem}
Let $SR, F, S$ be the generators of the group $\cal G$, and
${\P}_0(\frac{\pi}{2},y^*-\frac{k}{2})$,
${\P}_1(\frac{1}{2}f(y^*)+\frac{\pi}{2},y^*)\in{\R}^2$. Consider
the lift $U$ of $SR$ such that
$U({\P}_0)={\P}_0$,
and the lift $V$ of $SI$, with $V({\P}_1)={\P}_1$.
$U$ is a nonlinear symmetry with respect to the point ${\P}_0$,
i.e. the differential $DU({\P}_0)$ has the matrix \beq \left [
\begin{array}{rr} -1& 0\\ 0& -1 \end{array}\right]. \eeq
Hence there exists an infinity of
$U$--invariant curves containing the point ${\P}_0$. But at most
one of these curves has a $F$-invariant projection onto $\cil$,
i.e. for all points $(X,Y)$ lying on this curve, and for any $n\in
\Z$, there exist $m$ and $l \in \Z$ such that $U{\tilde
F}^n(X,Y)={\tilde F}^m(X,Y)+(2\pi\,l,0)$ ($\tilde F=U V$).
If such a curve exists, it contains the point
${\P}'_0(\frac{3\pi}{2},y^*+\frac{k}{2})$. Indeed, let $n$ be an
arbitrary integer. Then, $U{\tilde F}^n({\P}'_0)={\tilde
F}^{-n}U({\P}'_0)+(2\pi\,l,0)$ for some $l\in\Z$ (here for chosen
$U,V$, $l=0$). Hence $U{\tilde F}^n({\P}'_0)={\tilde
F}^{-n}({\P}'_0+(-2\pi,0))=$ ${\tilde F}^{-n}({\P}'_0)+(-2\pi,0)$.
The projection
onto the cylinder of this curve is a homotopically nontrivial circle,
because it
contains the arc of ends $U({\P}'_0)$, ${\P}'_0$.
Denote this circle by $\gamma_s$. Obviously it is a connected set.
Finally we have to prove that $\gamma_s$ is $S$-invariant.
$S\gamma_s=S(F(\gamma_s))=F(S\gamma_s)$. Hence $S\gamma_s$ is
$F$--invariant. On the other hand,
$S\gamma_s=S(SR\gamma_s)=SR(S\gamma_s)$, i.e. $S\gamma_s$ is also
$SR$--invariant. Because there is at most one $F$, and
$SR$--invariant circle, we can conclude that $S\gamma_s=\gamma_s$.
\end{dem}
The $\cal G$--invariant circle is in fact the {\it shearless
curve} defined in {[del-Castillo {\it et al}, 1996]} using
approximation by periodic orbits. The figures in this paper
display the $\cG$--invariant circle of the model map, as the
circle containing four marked points, as in:
\bpr
The fixed points ${\p}_0,{\p}'_0$, ${\p}_1,{\p}'_1$ of the reversors
$S R$, $I R$, belong to the $\cal G$--invariant curve. \epr
\begin{dem}
It is clear that ${\p}_0$ and ${\p}'_0$ belong to $\gamma_s$.
Let us show that the orbit of ${\p}_1$ is $SR$--invariant.
We prove that for any $n\in \Z$ there exists an $m\in \Z$
such that $SR\,F^n({\p}_1)=F^m({\p}_1)$.
$SR\,F^n({\p}_1)$=$F^{-n}(SR({\p}_1))=$ $F^{-n}SR\,IR({\p}_1)=F^{-n}F^{-1}{\p}_1$
$=F^{-n-1}({\p}_1)$.
$S({\p}_1)$ $={\p}'_1$ implies that ${\cal O}({\p}'_1)\in\gamma_s$.
\end{dem}
The orbits of these particular points are hence related by:
${\cal O}({\p}_i)=$\\ $SR ({\cal O}({\p}'_i))$, $i=0,1$. Next we derive necessary
and sufficient conditions that the orbits of these points to be
S--invariant. First, recall that
as in the case of orientation reversing
reversors, if $\ell_0=\mbox{Fix}(S R)$,
and $\ell_1=\mbox{Fix}(S I)$, then, denoting by
$\ell_{2n}=F^n(\ell_0)$, $\ell_{2n+1}=F^n(\ell_1)$, we have the
following result:
If $\ell_{i}\cap \ell_0\neq\emptyset$, $i\neq 0$, or
$\ell_{i}\cap \ell_1\neq\emptyset$, $i\neq 1$,
then the points of intersection are
periodic points of period $i$, respectively $\vert i-1\vert$.
\bpr The orbit of a point in $\ell_0\cup\ell_1$ has full isotropy
group iff it is a periodic point of even period. \epr
\begin{dem} First we prove that a point in $\ell_0\cup\ell_1$, say
${\p}_0$, has a $S$--invariant orbit iff it is a periodic point of
even period. Indeed,
${\p}_0$ has a $S$--symmetric orbit
iff for any $n\in\Z$, there is a $m\in\Z$, such that
$SF^n({\p}_0)=F^m({\p}_0)$. By the $S$--symmetry of the map $F$,
and the fact that $S({\p}_{0})={\p}'_0$, this relation is
equivalent to $F^n({\p}'_0)=F^m({\p}_0)$ or
$F^{n-m}({\p}'_0)={\p}_0$. But this means that $\ell_{2(n-m)}\cap
\ell_0\ni{\p}_0$, i.e. ${\p}_0$ is a $2(n-m)$ periodic point.\par
If one of the points of $\ell_0$ ($\ell_1$) has an even periodic
orbit, then the second one belongs to its orbit. Namely, if
${\p}_0$ (${\p}_1 $) is $2q$--periodic then $F^q({\p}_0)={\p}'_0$
($F^q({\p}_1)={\p}'_1$). Hence, because of Eqs.
(\ref{relpoints1})-(\ref{relpoints2}) it is easy to see that even
periodic orbits of the points ${\p}_i$, $i=0,1$, have
$R$--symmetric orbits.
\end{dem}
\brmk Eq. (\ref{eq:efectS}) ensures that a periodic orbit of
period $2q$ of one of the points ${\p}_i$, $i=0,1$, contains a
point of tangential intersection between one of the lines of
${\L}_{2q}$ and one of ${\L}_0$ or one of the lines of
${\L}_{2q+1}$ and one of ${\L}_1$.\ermk
\section{Description of the Route to Collision \\ of Periodic
Points}
\subsection{Collision of odd periodic orbits}
Collision of $q$--periodic points of a nontwist standard--like
map $F$ was detected
numerically in {[del-Castillo {\it et al}, 1996]},
{[Howard \& Humpherys, 1995]}. The route to
collision/annihilation is described in {[Sim\'{o}, 1998]}
using the interpolating Hamiltonian of a small perturbation of
an integrable nontwist
a.p.m. The approximating
Hamiltonian captures the behaviour of the map $F^q$,
corresponding to a fixed small value of $k$
and to a range of the parameter $\omega$, for which the fixed points
of the map $F^q$
lie within the annulus $A$ of nonmonotone dynamics.\par
Next we extract more information
on the route to collision, derived from reversing symmetry properties
of the interpolating Hamiltonian of a nontwist standard--like
map.
Exploiting symmetry properties of the
nontwist standard--like map
$F$, we are also
able to characterize the position of the points of collision with
respect to the $\cal G$--invariant curve, and in some particular cases with
respect to the nonmonotone circle $C$.\par
%First, we point out that elliptic periodic points cannot lie on
the circle %$\gamma_s$
%because the surrounding invariant circles
%would intersect $\gamma_s$.\par
Let us discuss the scenario of elliptic--hyperbolic
collision--annihilation of odd periodic points.
In {[Sim\'{o}, 1998]} it is derived the interpolating Hamiltonian
for a generic perturbation of the integrable nontwist area
preserving map \beq F_0(x,y)=(x+\omega+y^2+O(y^3),y)\eeq
Incidentally, the chosen truncation:
\beq\label{eq:intham} H(x,y)=\mu
y+\frac{1}{3}y^3+\cos{x}\eeq of the Hamiltonian function defines a
vector field \beq X_H=\left(-\frac{\partial H}{\partial
y},\,\,\frac{\partial H}{\partial x}\right),\eeq having a
reversing symmetry group $\cal G$ generated by the reversor
$R(x,y)=(-x,y)$, and the symmetry $S(x,y)=(x+\pi,-y)$, i.e. $R
X_H=-X_H R$, and $DS\circ X_H=X_H\circ S$. It is known that
Hamiltonian normal forms are almost always reversible [Lamb \&
Roberts, 1998]. The symmetry $S$, however,
induces a feature of the
elliptic--hyperbolic collision that is not exhibited by the
generic perturbations of the integrable systems, as we shall see
next.\par
Starting with the integrable (unperturbed) system
$F_0(x,y)=(x+\omega+y^2+O(y^3),y))$, which is $S$--symmetric with
respect to the transformation $S(x,y)=(x+\pi ,-y)$ one can use the
same method as in {[Sim\'{o}, 1998]},
with some additional assumptions on the perturbation of the
generating function of $F_0$, that ensures $S$--symmetry for the
associated perturbed map $F_\varepsilon$, to
construct
the interpolating Hamiltonian $H$ for $F_\varepsilon$.
We are not interested here in the method
of construction of the symmetric interpolating Hamiltonian, but in the further
information given by its reversing symmetry properties.
Its truncation is now Eq. (\ref {eq:intham}), and it
describes the
behaviour of the map $F^q$, $q$--odd, where $F$ is a nontwist
standard--like map
corresponding to a fixed small value of $k$, and a range of $\omega$
for which $q$--periodic orbits are in the nonmonotone annulus
$A$.\par
For $\mu<0$ the Hamiltonian system has two pairs of equilibrium
points, one above the energy curve $H(x,y)=0$, and another bellow
it (Fig. 2a) :
\begin{figure}[h]\label{fig:evham}
\begin{center}
%\includegraphics{fig2a.eps}\hspace{-1.75cm}\includegraphics{fig2b.eps}\hspace{-1.75cm}
%\includegraphics{fig2c.eps}\hspace{-1.75cm}\includegraphics{fig2d.eps}
a)\hskip 2.5cm b)\hskip 2.5cm c)\hskip 2.5cm d)
\caption{The route
to collision of equilibrium points of the Hamiltonian system
defined by Eq. (\ref {eq:intham}) Discussion in text.}
\end{center}
\end{figure}
\noindent ${\bf u}_h(0,\sqrt{-\mu})$, ${\bf u}_e
(\pi,\sqrt{-\mu})$, respectively ${\bf d}_e (0,-\sqrt{-\mu})$,
${\bf d}_h(\pi,-\sqrt{-\mu})$, where ${\bf u}$ (${\bf d}$) stands
for up (down), and the indices $e, h$ for elliptic, respectively,
hyperbolic. As $\mu$ increases, the two pairs of equilibrium
points approach each other on the symmetry lines $x=0, x=\pi$ of
the reversor $R$, and at $\mu=\mu_c=-(9/4)^{1/3}$ the upper and
lower hyperbolic points cross the
${\cal G}$--invariant energy curve $H(x,y)=0$ (Fig. 2b and Fig. 3a).
Because $SR$ is a reversor, we have the following
relation between the stable and unstable manifolds $W^s$, $W^u$, of the
corresponding hyperbolic points:
\beq\label{eq:usmanif} SR (W^{s,u}({\bf u}_h))= W^{u,s}(SR({\bf u}_h))=W^{u,s}({\bf d}_h)\eeq
Hence along with hyperbolic points, the union of their $F$--invariant manifolds
is $SR$--invariant, too, and as a consequece the ${\cal G}$--invariant
curve $H(x,y)=0$
is the
union of saddles and their invariant manifolds.
This curve contains in fact
three rotational circles (Fig. 3a): one containing distinct saddles ${\bf u}_h,{\bf d}_h$
and heteroclinic arcs connecting them. These arcs contain the
fixed points $(\pi/2,0)$, $(3\pi/2,0)$ of
the reversor $SR$. The other two rotational circles are homoclinic manifolds
to ${\bf u}_h$, respectively, ${\bf d}_h$. These last two
rotational circles are not $S$--invariant, only their union is $S$--invariant.
In order to find out
the sense of flowing on the three circles,
evaluate the vector field $X_H$ at the fixed points $(\pi/2,0)$, $(3\pi/2,0)$ of
the reversor $SR$. We get that at $\mu=\mu_c$,
the sense of flowing on the heteroclinic arcs is anti--clockwise.
Hence on the homoclinic
arc to ${\bf u}_h$ the sense is
from the right to left, and the same sense for its $S$--symmetric arc,
homoclinic to ${\bf d}_h$.\par
$\mu=\mu_c$ is a point of global bifurcation of the ${\cal
G}$--invariant
curve $H(x,y)=0$. At $\mu=\mu_c$ it is a curve with singularities,
having a topological type different from that of a circle (Fig 3a).
For $\mu< \mu_c$, by implicit function theorem, it
is a graph of a differentiable function defined on ${\bf
S}^1\equiv[0,2\pi)$ (Fig 2a), while
for $\mu>\mu_c$ the ${\cal G}$--invariant
curve has foldings in such a way that it is no longer a graph
(Fig.2c).\par
On the other hand, when $\mu$ crosses $\mu_c$, the hyperbolic
points leave their elliptic partner in the Poincar\'{e}--Birkhoff
chain, and homoclinic loops to these hyperbolic points encircle
elliptic points, lying on
the same component of the symmetry line, of the $S$--symmetric
chain (Fig. 2c).
A global bifurcation from
heteroclinic to homoclinic connections occurs.\par
For $\mu\leq 0$ not only
the ${\cal G}$--invariant curve, but also its neighbors
$H(x,y)=h$, $h\in (H({\bf d}_h),H({\bf u}_h))$,
fold (Fig. 2c). That is why, in {[Sim\'{o}, 1998]}, they are
called {\it meanders}. Note that as $\mu$ increases from $\mu_c$ to $0$
the interval $(H({\bf d}_h),H({\bf u}_h))$ $=(\frac{2}{3}(\sqrt{-\mu})^3-1,
-\frac{2}{3}(\sqrt{-\mu})^3+1)$ gets larger, i.e. the strip of
meanders enlarges, because new rotational orbits are
born (see Fig. 3).
\begin{figure}[h]\label{fig:jurshear}
\begin{center}
%\includegraphics{fig3a.eps}\hspace{-1.75cm}\includegraphics{fig3b.eps}\hspace{-1.75cm}
%\includegraphics{fig3c.eps}\hspace{-1.75cm}\includegraphics{fig3d.eps}
a)\hskip 2.5cm b)\hskip 2.5cm c)\hskip 2.5cm d)
\caption{The birth
of orbits as $\mu$ increases in $[\mu_c,0)$. The first picture
shows the $\cal G$--invariant curve, $H(x,y)=0$, at $\mu=\mu_c$,
while the next, the new born $S$--symmetric meanders. }
\end{center}
\end{figure}
\brmk According to C. Sim\'{o} {[1998]} the period of the $\cal
G$--invariant curve decreases as $\mu$ increases from $\mu_c$ to
$0$. As a consequence, for each $\mu\in (\mu_c,0)$ the new--born orbit
is the
$\cG$--invariant circle, while the former has generated two $S$--symmetric
circles. Hence, we have a "central" source of orbits.\ermk
As $\mu$ approaches $0$, the homoclinic loops shrink to the
elliptic points, and at $\mu=0$ the hyperbollic--elliptic collision
occurs (Fig. 2d), and a rotational circle $H(x,y)=H({\bf u}_e)=
H({\bf d}_e)$, having cusps
at points of collision,
is born. For $\mu>0$ the
system has no equilibrium points and no meanders. The energy
curves $H(x,y)=h$ are again graphs.\par
After the above analysis, we can conclude that a nontwist
standard--like mapping $F$ exhibits a rich phenomenology in the
ranges of $\omega$ for which odd periodic points lie within the
nonmonotone annulus $A$ of the cylinder. For each fixed small
parameter $k$ in the definion of the map $F$, denote by
$\omega_1(p/q)$ the value of the parameter $\omega$ for which the
$S$--symmetric hyperbolic odd--periodic orbits of rotation number
$p/q$, enter the $\cal G$--invariant curve, and by $\omega_2(p/q)$
the collision threshold of the odd periodic orbits having the same
rotation number $p/q$. Note that hyperbolic--elliptic collision
occurs in a strip including the $\cal G$--invariant circle, but
not exactly on this circle.
\par
Because of the $S$--symmetry of the system, collision of the two
pairs of $SR$--symmetric odd periodic points occurs simultaneously.
In the generic case, when such a symmetry
is not present
this collision
occurs in stages, as we can see in Fig. 4d, where upper three periodic
orbits have already annihilated, while the lower still persist
within
a homoclinic chain.
\begin{figure}[h]\label{fig:nontw}
\bce
%\includegraphics{fig4a.eps}\includegraphics{fig4b.eps}
\hspace{1cm} a) \hspace{6cm} b)
%\includegraphics{fig4c.eps}\includegraphics{fig4d.eps}
\hspace{1cm} c) \hspace{6cm} d)
\caption{
The phase space of the
nontwist a.p.m. $F(x,y)=(\ovl{x},\ovl{y})$,
$\ovl{x}=x+f_\omega(\ovl{y})$, $\ovl{y}=y+k\,\sin (
x-\arcsin(\varepsilon\sin(x))$, $k=0.2$, $\varepsilon=0.38$,
displaying a pair of three
periodic chains in a nonmonotone annulus, whose periodic points collide in stages;
$\omega$ is respectively, a) 0.3285 b)
$0.3297655$, c) $0.3318$, d) $0.332$.}
\ece
\end{figure}
% The same delay in colllison of a pair of five--periodic points
%against the other
%was identified
%for the nontwist map $F(x,y)=(\ovl{x},\ovl{y})$,
%$\ovl{x}=x+f_\omega(\ovl{y})$, $\ovl{y}=y+k\sin\, x$,
%$f_\omega(y)=2\pi\omega+y^2+y^3/6$, $k=0.3, \omega=0.39800061$.
%The term $y^3/6$ has broken the symmetry $S$ of the system.
In the generic case, the corresponding nontwist map has merely
reversing symmetries, and no symmetries (equivariances). More
precisely, the system is double weakly reversible {[Roberts \&
Quispel, 1992]}. Exploiting this property we will explain, in a
forthcoming paper, the collision in stages of periodic points of
opposite stability type and common rotation number,
observed in the dynamics of H\'{enon} map, too {[Van der Weele \& Valkering, 1990]}.\par
The birth of meanders of nontwist standard-like mappings is a
consequence of a global bifurcation of a $\cal G$--invariant
homotopically nontrivial circle on the cylinder, where $\cal G$ is
a group acting on $\cil$:
\bpr Let ${\cal G}$ be the group acting on
the cylinder $\cil$, generated by the transformations $R,S$,
$R(x,y)=(-x,y)$, $S(x,y)=(x+\pi,-y)$, ${\bf S}^1\equiv [0,2\pi)$.
A family $c_\mu$: $\Phi(x,y,\mu)=0$ of
$\cal G$-- invariant analytic curves on a domain $\{(x,y)\vert\,\,
\mbox{Im}(x)\mu_0$ they are not graphs any more.\epr
\begin{dem}
Let $\Phi$ be an analytic function on a domain, as above, and its
level curves $L_h:\,\Phi(x,y)=h$. Such a curve is $\cal
G$--invariant if along with a point $(x,y)\in L_h$, $R(x,y)$ and
$S(x,y)$ belong to $L_h$, too. Zero level set of such a function
satisfying: \bea \Phi(-x,y)=&\Phi(x,y)\\
\Phi(x+\pi,-y)
=&-\Phi(x,y)\eea is the only ${\cal G}$--invariant curve on the
cylinder. In this case $\Phi$ is periodic and even in $x$. Its
Fourier expansion is: \beq
\Phi(x,y)=a_0(y)+\ds\sum_{n=0}^{\infty}a_n(y)\cos\, x\eeq
The
$S$--invariance of the zero level set implies that $a_0$ is an
odd function, the sum is taken over odd natural numbers, $n=2k+1$,
and $a_{2k+1}$ are even analytical functions, for all $k\geq 1$.
Hence: \beq \Phi(x,y)=a_0(y)+\sum_{k=0}^\infty
a_{2k+1}(y)\cos(2k+1)x\eeq Truncating $a_0(y)$ at $ay+b\,y^3$, and
the sum at $\cos\, x$, let us investigate when the curve
$ay+b\,y^3+\cos\, x=0$
can have
singular points. A simple computation gives that for $ab<0$ the
curve has critical points. Taking $b>0$ and rescaling, we get
$\mu\,y+\frac{1}{3}y^3+\cos x=0$, where $\mu=a<0$. For
$\mu_0=-(9/4)^{1/3}$ our curve has nodal--type singular points.
Implicit function theorem ensures that for $\mu<\mu_0$ the curve
is a graph, while for $\mu>\mu_0$ it has foldings in such a way
that is not a graph.
\end{dem}
This result explains the appearance of invariant circles that are
meanders, in the dynamics of nontwist standard--like maps.
Because of this global bifurcation, and the
continuity of the system, the rotational circles, near the $\cal
G$--invariant one, are meanders, too, for $\mu\in (\mu_0,0)$
(Figs. 3b, 3c, 3d).\par
The above described route to elliptic/hyperbolic collision
of odd periodic orbits of nontwist standard--like mapings does not give any
information on the position of periodic points with respect to the
nonmonotone circle. Next we analyse this aspect.\par
Consider a nontwist standard--like map $F$, whose twist function
$f$ has $y^*$ as the extremum point. Hence the nonmonotone circle
$C$ has the equation: $y+k\sin x =y^*$.
Denote by ${\bf n}_0=(0,y^*)$, ${\bf n}'_0=(\pi,y^*)$
the points of the intersection $C\cap {\cal L}_0$, and ${\bf
n}_1,{\bf n}'_1\in$ $C\cap {\cal L}_1$. Their orbits are
$R$--symmetric nonmonotone orbits. Hence if these particular
points have rotational orbits, then the $\cal G$--invariant curve
lies "between" the orbits ${\cal O}({\bf n}_0)$ ${\cal O}({\bf
n}'_0)$, respectively ${\cal O}({\bf n}_1)$, ${\cal O}({\bf
n}'_1)$. As a consequence, for $k$ small (i.e. the system is close
to an integrable one) if $\gamma_s$ is a meander, its height,
defined as the difference, along the meander, of the maximal and
minimal values of $y$, is less then the minimum of the Hausdorff
distances between the pairs of invariant circles or $S$ and
$R$--symmetric nonmonotone chains containing these special
points.\par
For each fixed small parameter $k$ of a nontwist standard--like
map, the meander $\gamma_s$, corresponding to $\omega\in
(\omega_1(p/1), \omega_2(p/1)]$, $p\in \Z$ (i.e. the meander
associated to fixed points),
seems to have the largest height among all $\cal G$--invariant
meanders associated to any other odd periodic orbit. This property
is due to the fact that in this range of the parameter $\omega$,
the
$R$--symmetric nonmonotone points have non--rotational orbits,
allowing the meanders to exceed the amplitude of the nonmonotone
circle $C$. To illustrate this property, let us analyse the route
to collision of the fixed points of the model map Eq. (\ref
{eq:modelmap}) corresponding to $\omega>0$ (the case $\omega\leq
0$ implies some sign changes). For $n-1<\omega \leq n$, $F$ has
the fixed points $(0, \pm \sqrt{2n\pi-2\pi\omega})$,
$(\pi, \pm \sqrt{2n\pi-2\pi\omega})$, of rotation number $p/q=n/1$.
% the $S$--symmetric pairs are above, respectively bellow the
%nonmonotone circle, and their collision occurs on $C$. Fixed
points are points of intersection of the basic symmetry lines
${\L}_0,{\L}_1$. Hence their collision takes place when one of the
lines of ${\L}_0$ has tangential contact to one of ${\L}_1$, i.e.
when $\omega=n\in \N$.
The route to collision of fixed points corresponds to the
variation of the parameter $\omega$ in intervals of the form
$(n-1,n]$, $n\in \N$. The points of collision are $(0,0)$ on
$\Gamma_0$ and $(\pi,0)$ on $\Gamma'_0$, i.e. collision occurs on
the nonmonotone circle $C$. % Characteristic equation of the
differential $DF$, at a %fixed point $(x_0,y_0)$ is \beq
%\lambda^2-(2+f'(y_1)k\cos(x_0))+1=0\eeq Hence nonmonotone fixed
%points ($f'(y_1)=0 \Leftrightarrow y_0+k\sin\, x_0=0$) are
%parabolic points. %At the %collision threshold, $p_0, p_1'$ and
$p_0',p_1$ have the same %abscissa, hence as we already know the
shearless curve is not a %graph, but it will bifurcate to a
graph.
Next we "follow" the fixed points in their way towards
collision. In this way three topological changes in the
phase space occur:
\begin{enumerate}
\item For $\omega$ close to $n-1$, the points $(0,0)$, $(\pi,0)$
have rotational orbits (see Fig. 5a). As $\omega$ increases
the fixed points approach each other, hence the homoclinic
manifolds to hyperbolic points, that are nearer to $C$, get closer and
closer. At a
threshold $\omega_0(n/1)$, the point
$(0,0)$ (hence $(\pi,0)$) crosses the homoclinic
manifold (Fig. 5b) and a global bifurcation
occurs:
the rotational
orbits of points $(0,0)$ , $(\pi,0)$ become
orbits on invariant circles that are contractible to elliptic fixed points
(Fig. 5c).
\begin{figure}[h]\label{fig:fixe}
\bce
%\includegraphics{Fig5a.eps}\includegraphics{Fig5b.eps}
\hspace{1cm} a) \hspace{5cm} b)
%\includegraphics{Fig5c.eps}\includegraphics{Fig5d.eps}
\hspace{1cm} c) \hspace{5cm} d)
\caption{ $k=0.2$; Orbits of the points $(0,0)$, $(\pi,0)$ (these points are
marked with $\diamond$,
while the points $p_i, p'_i\in \gamma_s$, $i=0,1$,
with crosses) for different values of $\omega$; a) $\omega=0.88$: the points
$(0,0)$, $(\pi,0)$
have rotational orbits; b) $\omega=0.888345$: orbits homoclinic
to hyperbolic fixed points; c) $\omega=0.9$: orbits surrounding
elliptic fixed points; d) $\omega=0.955$: the meander $\gamma_s$
having larger height than the amplitude of the nonmonotone circle
$C$. }
\ece
\end{figure}
\item The second global bifurcation was already discussed above: at
$\omega=\omega_1(n/1)$,
$\gamma_s$ bifurcates from a graph to a meander.
\item At the collision threshold $\omega=\omega_2(n/1)$, the points
$(0,0)$, $(\pi,0)$ have again rotational orbits. Hence a
reconnection of a homotopically nontrivial circle occured. At the
reconnection threshold, these points are cusps. \end{enumerate}
Next we investigate whether the orbits of the special points
${\bf n}_0$, ${\bf n}'_0$, and ${\bf n}_1,{\bf n}'_1$
can be
included in $\gamma_s$ or not. Note that the nonmonotone circle
$C$ is $S$--invariant, but not $R$ or $F$--invariant, and
along with ${\bf
n}_0,{\bf n}'_0$, the points $F^{-1}({\bf n}_0)$, $F^{-1}({\bf
n}'_0)$ belong to $C$, too. \bpr Let $F$ be a nontwist
standard--like map, having the twist function $f_\omega$. For each
small fixed $k$, the orbits of the $R$--symmetric nonmonotone
points ${\bf n}_0(0,y^*)$, ${\bf n}'_0(\pi,y^*)$, are included in
the graph $\gamma_s$, but ${\cal O}({\bf n}_1)\not\subset
\gamma_s$, iff $\omega$ is a solution of the equation
$f_\omega(y^*)=(2m+1)\pi$, $m\in\Z$. \epr
\begin{dem}
Suppose that
$\gamma_s$ is the
projection of the $\mbox{graph}(\varphi)$, where $\varphi:{\bf R}\to{\bf R}$
is a continuous $2\pi$--periodic function. As a consequence of $S$,
and $R$--invariance of $\gamma_s$,
$\varphi$ satisfies the conditions: \beq\begin{array}{l}\varphi(x)+\varphi(x+\pi)=2y^*\\
\varphi(x)-\varphi(-x)=-k\sin x\end{array}\eeq
The additional information that $\gamma_s$ contains the points ${\bf n}_0(0,y^*)$ and
${\bf p}_0(\frac{\pi}{2},y^*-\frac{k}{2})$ leads to $\varphi(x)=-\frac{k}{2}\sin
x+y^*$.
Taking into account that \beq F^{-1}(0,y^*)= (-f(y^*),y^*+k\sin
(f(y^*)))\in C\eeq we get that $F^{-1}(0,y^*)\in\gamma_s$, too.
This implies $f_\omega(y^*)=n\pi$. For $n$ even, ${\cal L}_1$ has
tangential contact to ${\cal L}_0$, i.e. the fixed points collide.
>From the analysis of the approximating Hamiltonian, we know that
at the collision threshold the circle $\gamma_s$ is still a
meander. For $\omega$ that makes $f_\omega(y^*)=(2m+1)\pi$, $m\in
\Z$, ${\cal L}_2$ has tangential contact to ${\cal L}_0$, thus in
this case collision of two $2$--periodic orbits of the same
stability type occurs. Hence for such a value of the parameter
$\omega$ the orbit of the $R$--symmetric nonmonotone point
$(0,y^*)$ and its $S$--symmetric point $(\pi,y^*)$, is included in
$\gamma_s$ which is a graph (see below the collision of even
periodic orbits) .\par
The proof for the
converse statement is straightforward.
\end{dem}
\brmk Depending on the twist function
$f_\omega$, the equation $f_\omega(y^*)=(2m+1)\pi$, can have or
not a solution $\omega$. For the model map, and the nontwist
standard like--mapings investigated in {[del-Castillo {\it et al},
1996]}, {[Howard \& Humpherys, 1995]}, this equation has
solutions.\ermk
\bpr
For a nontwist standard--like map, the orbit of the point ${\bf
n}_0(0,y^*)\in C$ is periodic of period three, and included in the
${\cal G}$--invariant circle iff it contains one of the points
${\bf n}_1, {\bf n}'_1\in C\cap {\cal L}_1$.\epr
\begin{dem}
Let $F(x,y)=(\ovl{x},\ovl{y})$ be a nontwist standard--like map
of the cylinder, i.e. $F(x,y)=(x+f_\omega(\ovl{y}),y+k\sin x)$.
$F^{-1}(0,y^*)= (-f(y^*),y^*+k\sin (f(y^*)))\in C$, and $F(0,y^*)=
(f(y^*),y^*)$. $(0,y^*)$ is a periodic point of period three iff
$F^2(0,y^*)=F^{-1}(0,y^*)$, i.e. iff $f(y^*)+f(y^*+k\sin
(f(y^*))=-f(y^*)+2m\pi$, $m\in\Z$. But the last relation means
$F^{-1}(0,y^*)\in{\cal L}_1$.
\end{dem}
\brmk Because of the $S$--symmetry of the nonmonotone circle and
of the $\cal G$--invariant circle $\gamma_s$, the point ${\bf
n}'_0(\pi,y^*)\in C$ is also a three periodic point, and its
orbit lies on $\gamma_s$ iff $F^{-1}(\pi,y^*)\in{\cal L}_1$.
Obviously in this case the two $S$--symmetric three-periodic
orbits are orbits of the hyperbolic points (Fig.6a).\ermk
The above discussion points out that $R$--symmetric nonmomonotone
points of a nontwist standard--like map can have periodic orbits
on the $\cal G$--invariant circle.
\begin{figure}[h]
\bce
%\includegraphics{fig6a.eps}\includegraphics{fig6b.eps}
\hspace{0.9cm} a) \hspace{5.75cm} b)
\caption{ $k=0.4$; a) $\omega=0.660446$: hyperbolic points
of period three on the $\cal G$--invariant curve; the fixed points
of the reversors $SR$, $SI$ have heteroclinic orbits to hyperbolic
points.
b) $\omega=0.66066$: the meandering curve $\gamma_s$ that bifurcated from
$\cal G$--invariant curve displayed in a). The position of the
points ${\bf p}_i$,
${\bf p}'_i$, $i=0,1$ on $\gamma_s$, is also shown.} \ece
\end{figure}
In the most cases $R$--symmetric nonmomonotone points have orbits
outside the ${\cal G}$--invariant circle. Computer experiments
shows that these orbits can densely fill an invariant circle, or
can have orbits within a Poincar\'{e}--Birkhoff chain. In Fig. 7
the points ${\bf n}_0, {\bf n}'_0$ have six-periodic elliptic
orbits on the two sides of the circle $\gamma_s$.
\begin{figure}[h]
\bce
%\includegraphics{fig7.eps}
\caption{ $k=0.4$, $\omega=0.1370511$: $R$--symmetric nonmonotone points
${\bf n}_0, {\bf n}'_0$ have six--periodic elliptic orbits. Between
their Poincar\'{e}--Birkhoff chains lies the $\cal G$--invariant
circle.}
\ece
\end{figure}
We chose this example to show that unlike the elliptic fixed
points, elliptic periodic orbits can be nonmonotone. More
precisely, if $F=(F_1,F_2)$ is a nontwist area preserving map, and
$(x_0,y_0)$ a fixed point, then the differential $DF(x_0,y_0)$ has
the matrix: \beq J(x_0,y_0)=\left [ \begin{array}{cc}
\frac{\partial F_1}{\partial x }& 0\\ & \\ \frac{\partial
F_2}{\partial x}& \frac{\partial F_2}{\partial y}\end{array}\right
](x_0,y_0)\eeq Area preserving property implies $(\frac{\partial
F_1}{\partial x }\frac{\partial F_2}{\partial y})(x_0,y_0)=1$, and
as the consequence $DF(x_0,y_0)$ cannot have complex eigenvalues.
\subsection{Collision of even periodic orbits}
The even periodic orbits of a nontwist standard--like map $F$ have
a different route to collision from the odd ones. First, by
Proposition \ref{pozper} ii) an even periodic orbit intersects
both components of a basic symmetry line, and none of the second.
Moreover, by Proposition \ref{pr:opar} two $S$--symmetric even
periodic orbits might undergo a hyperbolic/hyperbolic,
elliptic/elliptic collision. The behaviour of the map $F^{2q}$
restricted to a nonmonotone annulus of the cylinder is captured by
$F^2$. Because $F$ is $R$--reversible, and factorizes as $F=IR$,
$F^2$ is doubly reversible, i.e. it has two reversors as we can
see from the decompositions: $F^2=(IRI)R$, $F^2=I(RIR)$. Moreover
$F^2$ is $S$--symmetric, too. Hence collision of even periodic
orbits of a nontwist standard like map (or conversely the birth of
even periodic orbits) is a bifurcation of the fixed points of a
nontwist area preserving map having a reversing symmetry group
generated by two involutive reversors $R_1, R_2$ and an involutive
symmetry $S$. Namely, it is the bifurcation of two fixed points,
one $R_1$--symmetric and another $R_2$--symmetric. The
approximating Hamiltonian for the dynamics of such a map on a
nonmonotone annulus, was derived in {[Sim\'{o}, 1998]}:
\beq\label{parham} H(x,y)=\mu y+\frac{1}{3}y^3-y\cos\, 2x\eeq Next
we point out the symmetry breaking or symmetry increasing
bifurcations occuring at the collision of equilibrium points of
the Hamiltonian vector field $X_H$, and the global bifurcations
underwent by the ${\cal G}$--invariant curve.\par
The vector field $X_H$ is reversible with respect to the reversors
$R_1,R_2$, defined by $R_1(x,y)=(-x,y)$, $R_2(x,y)=(\pi-x,y)$,
and $S$--symmetric, where $S(x,y)=(-x,-y)$. Because of the
symmetry $T=R_2R_1$, which is a rotation along each circle
$\S\times\{y\}$, $T(x,y)=(x+\pi,y)$, the dynamics of the vector
field $X_H$ can be recovered from that corresponding to the
Hamiltonian $K=H\varphi$, where $\varphi(x,y)=(x/2,y)$:
\beq\label{newham}
K(x,y)=\mu y+\frac{1}{3}y^3-y\cos {x}\eeq The new Hamiltonian
vector field $X_K$ defined on the cylinder has the reversing
symmetry group ${\cal G}$, generated by the reversor
$R(x,y)=(-x,y)$, and the symmetry $S(x,y)=(x,-y)$. The circle
$\S\times \{0\}$ is $\cal G$--invariant. For $\mu<-1$, on the
symmetry line $x=\pi$ there exists a pair of saddle equilibrium
points $(\pi,\pm\sqrt{-1-\mu})$, while for $\mu<1$ on the symmetry
line $x=0$ are located the centers $(0,\pm\sqrt{1-\mu})$ (Fig 8a).
At $\mu=-1$, the two saddles collide, and the resulting point
$(\pi,0)$, which is one of the two fixed points of the reversor
$SR$, has ${\cal G}$ as the isotropy group (Fig.8b). For $\mu<-1$
the $\cal G$--invariant curve is the rotational circle
$\S\times\{0\}$. At $\mu=-1$, besides this circle, the energy
curve $K(x,y)=0$ has two additional branches, whose union is $\cal
G$--invariant, namely, the upper ($y>0$) and the lower ($y<0$) arc
of the closed curve $c:\,\,-1+\frac{1}{3}y^2-\cos{x}=0$ (Fig. 8c).
Because the sense of flowing on these arcs is from the right to
left, while on the the arc $\S\times \{0\}\setminus \{(\pi,0)\}$
is opposite, each of these arcs form a "vortex" along with the
circle $\S\times\{0\}$. Hence we have a pair of $S$--symmetric
vortices. When $\mu$ crosses $-1$ a pair of $R$--asymmetric
saddles are born on $y=0$: $(\arccos{\mu},0)$,
$(2\pi-\arccos{\mu},0)$ (Fig.8d). With these saddles are also
associated two $S$--symmetric vortices
(Fig. 8e). As $\mu$
increases towards $1$, the closed curve $c':\,\,
\mu+\frac{1}{3}y^2-\cos{x}=0$ shrinks, forcing the two encircled
centers to approach each other on the symmetry line $x=0$, and
the two $R$--asymmetric saddles to get closer on $y=0$. At
$\mu=1$, these four points collide at $(0,0)$, which is the second
fixed point of the reversor $SR$, and now it is a point of maximal
isotropy group. This collision leads to the annihilation of
equilibrium points, because for $\mu>1$, the vector field has no
equilibrium points (Fig. 8f).
\begin{figure}[h]
\begin{center}
%\includegraphics{fig8a.eps}\hspace{-1.75cm}\includegraphics{fig8b.eps}\hspace{-1.75cm}
%\includegraphics{fig8bp.eps}
\hspace{1cm} a)\hspace{2cm} b)\hspace{2cm} c)
\noindent%\includegraphics{fig8c.eps}\hspace{-1.75cm}\includegraphics{fig8cp.eps}\hspace{-1.75cm}\includegraphics{fig8d.eps}
\hspace{3cm} d) \hspace{2cm} e) \hspace{2cm} f) \hspace{2cm}
\caption{The route to collision of equilibrium points of the
system defined by the Hamiltonian Eq. (\ref {newham}). In a) and
b) $\S\equiv [0,2\pi)$, while in the next figures $\S\equiv
[-\pi,\pi)$. a) Two
equilibrium points of the same stability type on each symmetry line;
b) Collision of the
saddles on $y=0$; c) The $\cal G$--invariant curve at the
threshold of collision of saddles; d) $R$--asymmetric saddles, and
associated vortices; e) $\cal G$--invariant curve for $\mu\in
(-1,1)$; f) The phase space after annihilation of equilibria.}
\end{center}
\end{figure}
\bigskip
Note that besides the collision of equilibrium points at $\mu=-1$,
respectively at $\mu=1$, these values of the parameter $\mu$ are
also thresholds of global bifurcation of the $\cal G$--invariant
energy curve of the Hamiltonian Eq. (\ref {newham}).\par
In Figs. 9a, 9b we
display the ${\cal G}$--invariant curve of the model map
corresponding to $k=0.4$ and $\omega$ at the point of colision of
period six hyperbolic points , respectively, after the birth of
$R$--asymmetric hyperbolic points.
\begin{figure}[h]
\bce %\includegraphics{fig9d.eps}\includegraphics{fig9c.eps}
\hspace{0.75cm} a) \hspace{7cm} b)
\caption{$k=0.4$. The six periodic chains shown in Fig. 7,
entered the nonmonotone annulus and merged; a) At
$\omega=0.153285015$, the parabolic orbit has ${\cal G}$ as
isotropy group, and the arcs of invariant manifolds form pairs of
vortices encircling elliptic points; b) For $\omega=0.1533$ it is
shown the special topological type of the $\cal G$--invariant
curve.}
\end{center}
\end{figure}
\vskip 0.75cm
We can conclude that the global bifurcations of the
$\cal G$--invariat curve of a nontwist standard like map are
recorded by the fixed points of the orientation preserving
involutive reversors $SR$, and $SI$. At the threshold of
bifurcation to a meander, these special points have heteroclinic
orbits to odd periodic hyperbolic points. Thus this threshold is
also one of bifurcation of the limit sets of all the points ${\bf
p}_i,{\bf p}'_i$, $i=0,1$. In the nongeneric case, at the
thresholds of global bifurcations of the $\cal G$--invariat curve,
one of the pair of points $({\bf p}_0,{\bf p}'_0)$, $({\bf
p}_1,{\bf p}'_1)$, and only one, lies on an even periodic orbit of
maximal isotropy group. Hyperbolic points collide first, and then
elliptic ones. Before collision, each orbit of hyperbolic points
has as the isotropy group, the group generated by $R$. At the
collision threshold the isotropy group is $\cG$ and the new born
$R$--asymmetric hyperbolic orbits lie on the $\cG$--invariant
curve but they have trivial isotropy group. At the threshold of
collision of elliptic orbits and hyperbolic points coming from
transverse direction we have again a full isotropy group for the
parabolic orbit.
\subsection{Higher order meanders}
>From the analysis of the approximating Hamiltonian Eq. (\ref
{eq:intham}) we found out that, for $\mu\in (\mu_c,0)$, new orbits
of the associated vector field $X_H$ are born. As a consequence,
in the case of a nontwist standard--like map corresponding to a
fixed small value of the parameter $k$, if $p_1,q_1$ are relative
prime integers, $q_1$, an odd number, and $\omega_1(p_1/q_1)$ is
the point of bifurcation of the $\cG$--invariant circle from a
graph to a meander, then for $\omega\in (\omega_1(p_1/q_1),
\omega_2(p_1/q_1))$, $\gamma_s$ is a meander. In this case we call
it the first order meander associated to the periodic orbits of
type $(p_1,q_1)$. In the case of the model map corresponding to
$k=0.3$, $p_1=1, q_1=1$, we have $\omega_1(1/1)=0.9065375$ and
$\omega_2(1/1)=1$. As $\omega$ increases from $\omega_1$ to
$\omega_2$, new orbits are born. By Remark 3.1, there is a
"central" source of orbits. Namely, orbits of irrational rotation
number, and even periodic orbits appear on the $\cG$--invariant
circle, while odd periodic orbits of type $(p_2,q_2)$ appear
within a narrow strip containing $\gamma_s$, but not on
$\gamma_s$. For $p_2/q_2$ sufficiently close to the rotation
number $\rho$ of the $\cG$--invariant circle, i.e. at
$\omega_2(p_2/q_2)$ the $(p_2,q_2)$--type orbits are born as cusps
on two $S$--symmetric invariant circles, and each cusp bifurcates
to a pair of points belonging to $R$--symmetric $(p_2,q_2)$--type
orbits, one elliptic,
and another hyperbolic, lying each within a homoclinic
chain, not a heteroclinic one. Moreover at the threshold
$\omega_2(p_2/q_2)$ the $\cG$--invariant curve bifurcates from a
first order meander to a second order meander, i.e. $\gamma_s$
folds further, new $q_2$ folds being generated. If
$\omega_1(p_2/q_2)$ is the point at which the $S$--symmetric
hyperbolic orbits enter the $\cG$--invariant curve, then for
$\omega\in (\omega_2(p_2/q_2),\omega_1(p_2/q_2))$ the circle
$\gamma_s$ is a second order meander. Moreover, as $\omega$
increases in this interval the reverse process occurs, namely
orbits are destroyed.\par
Fig. 10 displays for the model map, the
$(p=2, q=45)$ orbits as cusps of two $S$--symmetric invariant
circles (Fig. 10a), a part of the second order meander with $45$
folds, associated to the periodic orbits of type $(2,45)$ (Fig.
10b), and the $S$--symmetric hyperbolic orbits of period $45$ on
the $\cal G$ invariant curve (Fig. 10c).
\begin{figure}[h]
\begin{center}
%\includegraphics{p45creer.eps}\includegraphics{p45shear.eps}
\hspace{0.75cm} a)\hspace{7.5cm} b)
%\includegraphics{p45gams.eps}
\hspace{1cm} c)
\caption{Fig. $k=0.3$; magnification of the
rectangle $[1.5,2.5]\times [-1.5.0]$ a) $\omega=0.9299775$:
$S$--symmetric circles with $45$ cusps each; this is the
threshold of creation of periodic orbits of rotation number
$\frac{2}{45}$; b) $\omega=0.93$: second level meanders on
$\gamma_s$ associated to the periodic orbits of type $(p=2, q=45)$
c) $\omega=0.9300026$: $2/45$--periodic hyperbolic points lie on
the $\cal G$--invariant curve.}
\end{center}
\end{figure}
Because destruction of
periodic orbits of nontwist standard--like maps take place by
collision, we {\bf conjecture} that the above described process
repeats at every scale. More precisely, if $\gamma_s$ is a first
order meander associated to odd periodic orbits of type
$(p_1,q_1)$, then there is a number of subintervals of type
$[\omega_2,\omega_1]\subset$
$[\omega_1(p_1/q_1),\omega_2(p_1/q_1)]$ associated to odd periodic
orbits that are created for $\omega\in
[\omega_1(p_1/q_1),\omega_2(p_1/q_1)]$. For each such subinterval
$[\omega_2(p_2/q_2),\omega_1(p_2,q_2)]$ there exists another
number of subintervals of type $[\omega_1, \omega_2]$ associated
to periodic orbits that are destroyed and so on. In this context,
if $(p_n/q_n)$ is
a sequence of rational numbers, with $(q_n)$ an increasing sequence
of odd numbers, such that:
\beq\begin{array}{l} [\omega_1(p_1/q_1), \omega_2(p_1/q_1)]
\supset [\omega_2(p_2/q_2), \omega_1(p_2/q_2)]\supset\cdots\\
\supset [\omega_1(p_{2k-1}/q_{2k-1}),\omega_2(p_{2k-1}/q_{2k-1})]
\supset[\omega_2(p_{2k}/q_{2k}),\omega_1(p_{2k}/q_{2k})]\supset\cdots\end{array}\eeq
and $\{\omega_*\}$ is the intersection of the decreasing sequence
of compact intervals, we wonder whether for $\omega=\omega_*$ the
$\cG$--invariant curve is a fractal curve or not.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
This work has been partially supported by the CNCSIS grant no.
35094/1999. The author is very indebted to professor Carles
Sim\'{o}, from University of Barcelona, for providing the paper
{[Sim\'{o}, 1998]} while it was in a preprint form.
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\end{document}