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\begin{document}

\begin{titlepage}
\begin{center}
{\large\bf Arnold Flames}\\
{\large and}\\
{\large\bf Resonance Surface Folds*}
\vskip .2in
Richard P. McGehee\\
School of Mathematics\\
University of Minnesota\\
Minneapolis, MN 55455\\
mcgehee@geom.umn.edu
\vskip .1in
Bruce B. Peckham\\
Department of Mathematics and Statistics\\
University of Minnesota at Duluth\\
Duluth, Minnesota  55812\\
bpeckham@arnold.d.umn.edu
\vskip .1in
%DRAFT:
 \today\\
%\vskip .2in
\end{center}

\begin{abstract}
Periodically forced planar oscillators are often studied by varying the
two parameters of forcing amplitude and forcing frequency.
For low forcing amplitudes, the study of the essential oscillator dynamics 
can be reduced to the study of families of circle maps.
The primary features of the resulting parameter plane bifurcation
diagrams are ``(Arnold) resonance horns'' emanating from zero forcing amplitude.
Each horn is characterized by the existence of a periodic orbit with a certain 
period and
rotation number.  In this paper we investigate divisions of these horns
into subregions -- different subregions corresponding to maps having
different numbers
of periodic orbits.  The existence of subregions having more than the
``usual'' one pair of attracting and repelling periodic
orbits implies the existence of ``extra folds'' in the 
corresponding surface of periodic points in the cartesian product of the
phase and parameter spaces.
The existence of more than one attracting and one repelling periodic orbit
is shown to be generic.
For some of the families we create, the resulting parameter plane bifurcation 
pictures appear in shapes we call ``Arnold flames.''
Results apply both to circle maps and forced oscillator maps.
\end{abstract}

\noindent *A similar version of this paper will appear in the {\it International
Journal of Bifurcation and Chaos}.

\end{titlepage}

\newpage

\section{Introduction}

Periodically forced planar oscillators arise as models of phenomena in many
settings in science, engineering, and mathematics.
By varying the forcing amplitude and 
frequency, a two-parameter family of differential equations is obtained.
Such families are often studied by considering the two-parameter family of
maps of the plane
generated by sampling the flow of the forced oscillator differential
equations at the time period of forcing.  For small forcing amplitudes, the
study can be further reduced to a two-parameter family of circle maps which
are perturbations of rigid rotations of the circle. An exposition of
this reduction is in a previous paper [McGehee \& Peckham, 1994].
% [GH, Bushard, Loud, MP1??]
We study such families in this paper;  see Eq. (\ref{mapF})
in Sec.~2.

Studies of such circle map families by Arnold [1983] and Hall [1984]
indicate the
presence of {\it (Arnold) resonance horns} emanating from each rational
point on the zero forcing amplitude axis in the parameter plane.
Each resonance horn is characterized by the existence of a periodic orbit
with a certain (rational) rotation number.
In the best-known example, the {\it standard circle map family} (Example 1,
Sec. 2.1),
at least for small enough values of the forcing amplitude,
all maps corresponding to parameter values
inside any one $p/q$ resonance horn have exactly two period-$q$ orbits and are 
all topologically conjugate.  The two orbits come together in a
period-$q$ saddle-node
bifurcation at the boundaries of each resonance region.
With the standard map,
the resonance horns, along with the curves ``in between'' them corresponding
to maps conjugate to a rigid rotation with an irrational rotation number,
are the complete bifurcation story.

It is quite possible, however, to have families of circle maps whose
resonance regions must be subdivided in order to respect topological equivalence. 
(We use resonance ``region'' and 
resonance ``horn'' interchangeably, often preferring resonance horn when 
restricting to parameter values near the tip of the full resonance region.)
Such a subdivision is necessary when 
maps in the same period-$q$ resonance region do not all have the same number of period-$q$ orbits.
The bifurcation phenomena associated with the birth and death of these extra 
orbits are
explored in this paper.

Perhaps the most familiar parameter plane
feature (but not the simplest, as we shall see)
we study is the {\it swallowtail}.
This feature is illustrated, for example, in two-parameter families of 
maps of the plane
studied by Schreiber, Dolnik, Choc, and Marek [1988] and in a 
two-parameter family
of maps of $\R^3$ studied by Frouzakis, Adomitas and Kevrekidis
[1991].
Parameter plane bifurcation diagrams from those two papers are reproduced
in our Figs. \ref{sdcm} and \ref{fak}, respectively.
The Schrieber, et. al. maps are intended to be a caricature of maps of
the plane generated
by a periodically forced planar oscillator family; the two parameters are
analogous to forcing frequency and forcing amplitude.  We focus on their $3/4$ 
resonance region.  It has tips at either ``end'' of the resonance
region: one at the analogue of zero forcing amplitude
($A=0$) and the other at a
Hopf bifurcation curve (the dash-dot line in Fig. \ref{sdcm}).
The complete resonance region provides a connection
between the two types of resonance horns; these two types are described in
more detail in a previous paper [Peckham, 1990].
The $1/5$ resonance region we reproduced from the Frouzakis, et. al. paper
emanates from a Hopf bifurcation curve: the straight dashed line in Fig. 
\ref{fak}; no forced oscillator interpretation
is intended.
In both bifurcation diagrams, the continuation
of the saddle-node curves which determine the boundary of the respective 
resonance regions results in swallowtail regions inside those resonance
regions.
The two cusps on the tail of each swallowtail are saddle-nodes with a higher
order degeneracy; this cusp is different from both the forced oscillator tip
and from the Hopf bifurcation tip.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{ A schematic of the $3/4$ resonance region from Fig. 3 of Schreiber, 
et. al. [1988].  The resonance region is in the $(T,A)$ parameter plane for a
family of maps of the plane
generated by a planar flow with ``impulse forcing.''
The $T$ parameter can be thought of as controlling
forcing frequency, the $A$ parameter as controlling forcing amplitude.
Note especially the swallowtail in the $3/4$ resonance region. \label{sdcm}}
\end{figure}

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{(Reproduced from Fig. 6a of Frouzakis, et. al. [1991]
with permission from the authors.) The $1/5$ resonance region
in the $(k,b)$ parameter plane for a family of maps
of $\R^3$.  Note especially the swallowtail in the resonance region.
\label{fak}}
\end{figure}

Although not emphasized in either the Schrieber, et. al. paper or the 
Frouzakis, et. al. paper, the maps
corresponding to
parameter values inside the swallowtail regions have four period-$q$
orbits instead of just a single attracting-repelling
pair as in the rest of each period-$q$
resonance region.  Because similar scenarios are present in 
the simpler setting of families of circle
maps, we discuss mainly circle maps in this paper.
As suggested in Aronson, McGehee, Kevrekidis, and Aris [1986], we 
study these scenarios by focusing on the fixed- and periodic-point
surfaces  in the phase $\times$ parameter space instead of just the
resonance regions in the parameter space.
Implications for forced oscillators are discussed briefly toward the
end of the paper in Sec.~5.

We encourage the reader to look ahead to Fig. \ref{swallowtail}
where we present our
two-parameter family of circle maps whose $0/1$ fixed-point resonance 
phenomena include a swallowtail.
Illustrated are the ``fixed-point resonance surface'' and its projection to 
the fixed-point ``resonance horn.''
The curves on the fixed-point surface are along its folds (i.e., its 
singularities with respect to projection to the parameter plane);
dynamically, these folds
are saddle-node bifurcation curves.  Note that when projecting to the
parameter plane, the saddle-node curves either mark the boundary of the 
fixed-point resonance horn or the swallowtail region inside the horn.  The 
fixed-point surface projects 2:1 outside the swallowtail region, but 4:1 inside. In other words,
there is an ``extra pair'' of fixed points for maps corresponding to parameter 
values inside the swallowtail.
This example will be explained 
further toward the end of Sec. 2, after presenting some simpler examples
which serve as ``building blocks'' for the swallowtail example.

Most of the examples we present can be thought of as falling into one of two 
broad classes:
those with extra pairs of orbits ``at'' the zero forcing amplitude horn tip,
and those
with extra pairs of orbits which are born only away from the horn tip.
The extra orbits in the swallowtail are away from the horn tip and thus 
belong to the latter class.  
The existence of extra orbits
``at'' the zero forcing 
amplitude tip of the 
a resonance region may be surprising, especially to persons 
who are familiar with the ``other'' type of Arnold resonance horns: those
with resonance horns
originating from a Hopf bifurcation curve in a two-parameter plane of a family 
of maps of the plane.  Arnold [1982] shows using local normal forms
that in such cases (at least for periods greater than four)
that maps corresponding to parameter values inside the 
resonance horn and near the (tip generically
have only one pair of period-$q$ orbits.  Thus, bifurcation
features are generically possible only away from the Hopf tip of resonance
regions.


In part, this paper discusses how to interpret features of a bifurcation
diagram for a given family
of maps. More, however, the emphasis is on generating families with
desired bifurcation scenarios.  In the process we discuss what bifurcation 
scenarios are possible for families of circle maps and forced oscillator maps.
In Sec. 2 we provide building blocks for bifurcation scenarios and use them
to generate families having bifurcation features such as the swallowtail.
In Sec. 3, we extend some of our constructions
from the fixed-point setting to the period-$q$ setting for $q \ge 2$.
We  show in Sec. 4
that the existence of subdivisions
in resonance regions is generic.
Implications for forced oscillators are discussed in Sec. 5.

The data for all surfaces and bifurcation curves shown in the figures in this
paper were computed using the authors' continuation
software [Peckham, 1988-94].  Numerical
data was then viewed using the three dimensional viewing software, {\it
Geomview} [Phillips et. al., 1993].

\section{Families of Circle Maps}

Consider the following family of maps of $\R^1$:

\begin{equation}
x \rightarrow F_{(\phi,\alpha)}(x) := x + \phi + \alpha 
g_{(\phi,\alpha)}(x)
\label{mapF}
\end{equation}
%
where $x, \phi$, and $\alpha$ are real, and
\begin {enumerate}
\item $g_{(\phi,\alpha)}(x) \in C^1$ as a function of $\phi, \alpha$ and
$\theta$, and $|{dg_{(\phi,\alpha)} \over
dx}| \le 1$
\item $g_{(\phi,\alpha)}(x + 2\pi) = g_{(\phi,\alpha)}(x)$
\item $\int_0^{2\pi} g_{(\phi,\alpha)}(x)dx = 0$ for all 
$\phi$ and $\alpha$.
\end {enumerate}
If we let $\theta = x$ (mod $2\pi$), and
$f_{(\phi,\alpha)} = F_{(\phi,\alpha)}$ (mod $2\pi$), then
$F_{(\phi,\alpha)}$
is a lift of a two-parameter family of degree one circle maps.
The maps are homeomorphisms when $0\le\alpha \le 1$.
Such maps arise from periodically forced planar oscillators as alluded to in
the introduction.
%[GH, Bushard, Loud or MP1, 1994 ???].
Because of the relationship to periodically forced oscillators, we
call the parameter $\alpha$ the forcing amplitude; the parameter $\phi$ 
corresponds to the forcing frequency; $g_{(\phi,\alpha)}$ is the forcing 
function.
When $g_{(\phi,\alpha)} (x) = \sin(x)$, the resulting map of Eq. 
(\ref{mapF}) is 
called
the standard (circle map) family; see Example 1 in Sec. 2.1 below.
We are primarily interested in periodic orbits of $f_{(\phi,\alpha)}$,
which are usually defined in terms of the lift $F_{(\phi,\alpha)}$.
\vskip .2in
\noindent
{\bf Definition:}
The point  $(\theta, \phi, \alpha)$ is a ``{\bf $p/q$ point}'' (that is,
$\theta$ is a $p/q$ point for $f_{(\phi,\alpha)}$) if $\theta := x$ (mod $2\pi$)
is a 
least-period-$q$ point for $f_{(\phi,\alpha)}$ and  
\begin{equation}
F_{(\phi,\alpha)}^q(x) = x + 2\pi p
\label{pqpt}
\end{equation}

\vskip .1in
\noindent
{\bf Definition:} The set of $p/q$ points $\in {\bf S}^1 \times
\R \times [0,1)$ is ``{\bf the $p/q$ resonance surface}.''
The projection of the $p/q$ resonance surface to the parameter plane is
``{\bf the
$p/q$ resonance horn}'' or ``{\bf the $p/q$ resonance region}.''
\vskip .2in

%\noindent
If $\theta$ is a $p/q$ point for $f_{(\phi,\alpha)}$ with eigenvalue not
equal to one 
(${\partial f^q \over \partial \theta} (\theta, \phi, \alpha) \ne 1$, where
$f_{(\phi, \alpha)}$ is abbreviated by $f$), then
the implicit function theorem guarantees local continuation of 
the $p/q$ point in the $(\theta, \phi, \alpha)$ space
as the parameters $\phi$ and $\alpha$ are varied.
So we make another definition:
\vskip .2in
\noindent
{\bf Definition:} A $p/q$ point $(\theta, \phi, \alpha)$ with  
${\partial f^q \over \partial \theta} (\theta, \phi, \alpha) = 1$
is a ``{\bf singular $p/q$ point}.''
\vskip .1in
\noindent The most common singularities we encounter are the saddle-node
bifurcation (eigenvalue 1) and the cusp bifurcation (eigenvalue 1 and a higher
order degeneracy).

In this Section, we use forcing functions $g_{(\phi,\alpha)}$ which are all
independent of $\phi$.  Consequently, we will
usually abbreviate $g_{(\phi,\alpha)}$
with $g_\alpha$ for the rest of Sec. 2.
We will return to the more general form in Sec. 3.
In this case,  changing $\phi$ merely translates the
circle map $f_{(\phi, \alpha)}$ and its 
lift $F_{(\phi, \alpha)}$.
This makes the fixed-point bifurcations especially easy to understand.
As $\phi$ is increased, fixed points are born at
$\theta$ values corresponding to local maxima of 
$g_\alpha$; fixed points
die at $\theta$ values of local minima.
Usually (if $g_\alpha^{'} = 0$ and $g_\alpha^{''}\ne0$)
the births and deaths of fixed points occur in saddle-node 
bifurcations.

\begin{figure}
%\centerline{ \input{plot_fpbygraph}}
\caption{Fixed points by graph.
$F_{(-0.4, 0.9)}(x) = x + (-0.4) + 0.9((0.7 {\sin(2x) \over 2} + 0.3 \sin(2\pi-h(.9) {\pi \over 2})$,
where $h(.9) = 0.972$.  The form of $F$ is chosen to 
match the form used later in the paper in Example 12. \label{fpbygraph}}
\end{figure}

This is illustrated in Fig. \ref{fpbygraph},
where we show the graphs of $y=x$
and of $F_{(\phi,\alpha)}(x) = x + \phi
+ \alpha g_\alpha(x)$ for
$g_\alpha=0.4\sin(x)+0.6{\sin(2x) \over 2}$,
with $\phi= -0.5, \alpha = 0.9$. 
By the definition of a $0/1$ point, such points correspond
to the intersections of the two curves.  Increasing $\phi$ now translates up
the graph of  $x + \phi + \alpha g_\alpha(x)$. New pairs of $0/1$
fixed points are obviously ``born'' as $\phi$ increases when the values of
local maxima of 
$\alpha g_\alpha(x) + \phi$
push through height zero.  Similarly, pairs of orbits 
coalesce and ``die'' as values of local minima increase thru zero.
In Fig. \ref{fpbygraph}, one attracting-repelling pair of fixed points has
already been born;  as $\phi$ continues to increase, a second pair will 
be born, then a pair will die, and finally another pair will die, leaving
no $0/1$ fixed points for any larger values of $\phi$.

Restricted to
$\alpha =$ constant, the $0/1$ resonance region occupies a line segment from
$\phi = -\alpha M {\rm \ to\ } \phi = -\alpha m$ where
$M$ and $m$ are the global maximum and 
global minimum, respectively, of $g_\alpha(x)$.
This follows directly from Eq. (\ref{pqpt}).
When $g_\alpha$ is independent of $\alpha$, 
the $0/1$ resonance region is bounded by the straight lines 
$\phi = -\alpha M {\rm \ and\ } \phi = -\alpha m$.
More generally, considering $p/q$ resonance regions for $q$'s other than
one, Hall [1984] has shown that if $g_{(\phi,\alpha)}$ is independent
of $\phi$ and $\alpha$
then the $p/q$ resonance horns generically open out of each
point $(2\pi p/q, 0)$ on the $\phi$ axis as ``wedges,''
that is, the two sides of the resonance regions have different slopes.
He uses the Fourier series expansion
of $g(x)$ to obtain his results.
The issue of how many period-$q$ orbits exist inside the period-$q$ horn is not
addressed.
Hall's results apply to our more general family because his results
are local -- the rate at which resonance horns open near the zero forcing 
amplitude tip of the $p/q$ resonance horn depends only on the Fourier series
for the single map $g_{(p/q,0)}(x)$.

In the following subsections, we present several example functions $g_\alpha$ which determine two-parameter families of circle maps via Eq. (\ref{mapF}).
The various $g_\alpha$'s are chosen to illustrate the variety of features
that the corresponding $0/1$ resonance surfaces and associated bifurcation 
diagrams can exhibit.  We start in Sec. 2.1
with several $g_\alpha$'s that are independent of $\alpha$. 
We focus on the number of $p/q$ orbits existing for parameter
values in a $p/q$ resonance region, not just whether or not a 
$p/q$ orbit exists.
In Secs. 2.2 through 2.6, we use the features of the Sec. 2.1
examples to construct  $g_\alpha$'s which do depend on $\alpha$ and
whose associated bifurcation diagrams consequently are more ``interesting.''
See also Koslovsky [1991] for two similar examples.
In Sec. 3 we extend these constructions to
create similar features in resonance horns
for periodic orbits with periods greater than one.

%\item Solve for $g(x) = x$.  Key points at $g'(x) = 0$.

\subsection{Preliminary examples}

All the $g_\alpha$'s in this subsection will be independent of $\alpha$.
The resulting circle maps are defined Eq. (\ref{mapF}).
The independence of $g_\alpha$ on $\alpha$ implies
the fixed-point (possibly degenerate) saddle-node curves are all of the form
$\{(\theta, \phi, \alpha) | \theta = \theta_{c_i}, \phi = -V_{c_i}\alpha\}$, where $\theta_{c_i}$
are the critical points of $g$, and $V_{c_i} = g(\theta_{c_i})$
are the corresponding critical values.  This is because Eq. (\ref{mapF}) 
implies
that the curves all satisfy $F_{(-V_{c_i} \alpha, \alpha)}(x) = x$, and
$F_{(-V_{c_i} \alpha, \alpha)}^{'}(x) = 0$.
The projection to the parameter plane of the two curves corresponding to the 
global maximum and global minimum of $g$ form the
boundary of the $0/1$ resonance region.

\vskip .2in
\noindent{\bf Example 1: The standard family.}
By setting
$$ g_\alpha(x) = g(x) = G_1(x) := \sin(x)$$

\noindent
in Eq. (\ref{mapF}),
we obtain the much studied ``standard family of circle maps.''
We restrict $\alpha$ to $0\le \alpha < 1$ to ensure our circle maps are
homeomorphisms.
From Eq. (2), we see that the $0/1$ resonance surface (of fixed points)
is given by $\phi = -\alpha \sin x$; this projects to the $0/1$ resonance
horn $|\phi| \le \alpha$,
$0\le\alpha<1$.
The singular $0/1$ points are
the $0/1$ points with $x = \pi/2$ or $3\pi/2$.  These (saddle-node) curves
project to $\phi = \mp \alpha$ in the parameter plane.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 1, the Standard Family: $g(x) = \sin(x)$.
The $0/1$ resonance 
surface and its projection to the parameter plane.  \label{stdmap}}
\end{figure}

Figure \ref{stdmap} shows the
$0/1$ fixed-point surface and its projection to the parameter plane.  The
forcing amplitude $\alpha$ is restricted to $[0,1]$; $\theta$ is restricted to
$[0,2\pi]$; the parameter plane projection is shown at $\theta = 10$.
%1
%COLOR Version
"%The color
%
% BLACK and White version
The shading 
%
of the saddle-node curves is the same on both the surface and on
the curves' parameter plane projection.
%2
%COLOR Version
%The straight red line
%
%BLACK and White version
The dark straight line
%
parallel to the $\theta$ axis is a representative phase line.
It is extended to
the parameter plane projection to assist in identification of
its parameter value.  The 
phase portrait for this line has two fixed points at the intersection 
points
with the $0/1$ surface.  Successive iterates march in the positive $\theta$
direction for starting $\theta$ values ``in front'' of the surface ; iterates
march in the negative $\theta$
direction for starting points behind the surface.
%3
%COLOR Version
%The yellow lines
%
% BLACK and White version
The two thin straight lines parallel to the $\theta$ axis
%
are included merely to aid in visualizing the projection to
the parameter plane.
%
%Figure \ref{stdmap}b is a redrawn version of the
%boundaries of the $0/1$ resonance region.  Figure \ref{stdmap}c will be 
%explained later in Sec. 2.5.
This same figure format is used throughout the
paper for almost all our subsequent examples.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 1, the Standard Family: $g(x) = \sin(x)$.  Several resonace 
surfaces and their projections to the parameter plane.  \label{stdmapall}}
\end{figure}

To further place the context of our study we have included a more complete
picture of resonance surfaces and the resonance regions which are
their projections to the parameter space.
In Fig. \ref{stdmapall} are 
the $0/1, 1/3, 1/2, 2/3$, and $1/1$ resonance surfaces
and corresponding projections.  The parameter plane
bifurcation picture for this 
example is well known [Devaney, 1989].  It
has a resonance horn corresponding to each rational number.  Each resonance 
region, of course, has an associated resonance surface in the phase $\times$
parameter space.  Since we are
restricting $\alpha$ to $[0,1]$, the circle maps are all homeomorphisms
and uniqueness of rotation numbers implies that none of the resonance regions
may overlap.  The following examples also have resonance regions corresponding
to each rational number, but we are focusing only on the $0/1$ resonance 
surfaces and regions.  As we shall see in Sec. 3.1,
all the features we create for a $0/1$
resonance can be created for any $p/q$ resonance, as well.

\vskip .2in
\noindent
{\bf Example 2: One pair of extra folds -- coincident projections.}
\nobreak
Let $$ g_\alpha(x) = g(x) =  G_{2,0}(x) :={\sin(2x)\over 2}.$$
Analogous to the standard family, the fixed point surface is given by
$\phi = -\alpha {\sin (2x) \over 2}$; this projects to the $0/1$ resonance
horn $|\phi| \le {\alpha \over 2}$,
$0\le\alpha<1$.
The singular $0/1$ points are
the $0/1$ points with $x = \pi/4, 3\pi/4, 5\pi/4$, or $7\pi/4$. 
%4
%COLOR Version
%The two green
%
% BLACK and White version
The two light
%
(saddle-node) curves correspond to the maxima of $g(x)$,
$x = \pi/4$ and $5\pi/4$, and both project
to $\phi = {-\alpha \over 2}$ in the parameter plane;
%5
%COLOR Version
%The two blue
%
% BLACK and White version
The two dark
%

(saddle-node) curves correspond
to the minima, $x = \pi/4$ and $5\pi/4$, and both project to 
$\phi = {+\alpha \over 2}$ in the parameter plane.
See Fig. \ref{copro}.
As with the standard map in Fig. \ref{stdmap}, the 
%5
%COLOR Version
%red line
% BLACK and White version
dark straight line parallel to the $\theta$ axis 
%
is a
representative phase line.  For this example, however, we have four instead
of two fixed points; two extra fixed points exist. 

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 2, Coincident projections: $g(x) = \sin(2x)$.  \label{copro}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 3: One pair of extra folds -- noncoincident projections.}
We now construct our first example having a resonance region which must be 
further subdivided in order to respect topological equivalence.  The subregions
within the resonance region are inequivalent because they have different numbers
of fixed points for the corresponding maps.
In the previous example, the fact that the two maxima of
$g(x) = {\sin(2x) \over 2}$ are equal-valued
causes the 
two corresponding saddle-node curves to project to the same parameter space 
line. This line marks the left-hand side of the fixed point resonance horn.
The two equal-valued minima cause the other two saddle-node curves to 
project to the right-hand side of the resonance region.
To separate these coincident projections, we perturb the previous example
by letting
\begin{equation}
 g_\alpha(x) = g(x)= G_{2,\epsilon}(x) := 
(1 - \epsilon){\sin(2x) \over 2} +\epsilon \sin(x).
\label{g2e}
\end{equation}
%
where $0 \le \epsilon < .5$ in order to ensure that $G_{2,\epsilon}(x)$ still
has two local maxima and two local minima.
The surface of $0/1$ points is given
by $\phi = -\alpha g_\alpha(x)$, as in the previous two examples,
but it is no longer trivial 
to solve analytically for the saddle-node curves in the phase $\times$
parameter space, or for their projection to the parameter plane.  These 
bifurcation curves are, however, relatively easy to trace out numerically.
The result for $\epsilon = 0.1$
is Fig. \ref{noncopro}; this should be compared to Fig. \ref{copro}, which 
is the unperturbed
$\epsilon = 0$ case.  The two outside saddle-node curves in the parameter
plane denote the $0/1$ resonance horn boundaries, while the two inside curves
separate the resonance horn into the center region, in which there exist
two attracting and two repelling fixed points, and the two outside regions, 
where there exists
only a single attracting-repelling  pair of fixed points.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{ Example 3, Noncoincident projections: $g(x) = 0.9{\sin(2x) \over 2}
+ 0.1\sin(x)$.  \label{noncopro}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 4: Multiple pairs of extra folds.}
The previous example can easily be extended to obtain a resonance horn with 
$2n$ saddle-node curves by
choosing $g$ to be a periodic function with
$n$ local maxima and $n$ local minima.
The $n$ maxima ensure the saddle-node
births of $n$ pairs of fixed points as $\phi$ increases
(for $\alpha \in (0,1)$ fixed); the $n$ minima ensure $n$ saddle-node deaths.
If the $2n$ maximum and minimum values 
are distinct, the $2n$ saddle-node curves will project to distinct lines in
the parameter plane.  For example, to obtain $n-1$ extra pairs of folds,
we could use

\begin{equation}
 g_\alpha(x) = g(x) = G_{n,\epsilon}(x) :=
(1-\epsilon){ \sin(nx) \over n} + \epsilon \sin(x + \tilde{B})
\label{gne}
\end{equation}
%
where where $0<\epsilon < .5$ in order to ensure $G_{n,\epsilon}(x)$ has
$n$ local maxima and $n$ local minima and $\tilde{B}$
is chosen to ensure all the local extremum values are distinct.
This prevents curves which are distinct in the phase $\times$ parameter 
space from having coincident projections to the parameter space.
%For $n$ even we choose $\tilde{B}$ to be zero; for $n$ odd we choose 
%$\tilde{B}$ to be ${\pi \over 2n}$.

\vskip .2in
\noindent
{\bf Example 5: An infinity of folds.}
\nobreak
By choosing $g$ to be a periodic function with an infinite number of local
maxima and minima, we obtain an example with an infinite number of
saddle-node curves inside the $0/1$ resonance region.  Such an example can
have an
infinite number of topological equivalence classes inside the $0/1$ resonance
region, as well as parameter values with an infinite number of fixed points
without having any interval of fixed points.
We use the $C^\infty$ function

$$ g_\alpha(x) = g(x)= G_\infty(x) :=
{1 \over K} \sin({1 \over x} + \tilde{B})H(x/2\pi)$$
where $H(x)$ is a $C^\infty$ 
function which is $C^\infty$ flat at $x = 0$ and $x = 1$ and
positive on $(0,1)$,
$\tilde{B}$ is chosen so that the average value of $\sin({1 \over x} + \tilde{B})H(x/2\pi)$
is zero on $[0,2\pi]$,
and $K$ is the maximum derivative of 
$\sin(1/x + \tilde{B})H(x/2\pi)$ on $[0,2\pi]$.
Then $G_\infty(x)$ satisfies the three forcing function conditions
listed at the beginning of Sec. 2.
We choose $H(x)$ to have the form 
$$ H(x) = 4 h(x) h(1-x)$$
where $h(x)$ is a $C^\infty$ function which is zero for $x \le
0$ and $1$ for $x \ge1$, $C^\infty$ flat at zero and one, and strictly
monotonic on $[0,1]$.
(For computation throughout this paper we use the $C^1$ function
$h(x) = 3x^2 - 2x^3, \ \  {\rm for\ } x \in [0,1].$)
This makes $H(x)$ symmetric about $x=.5$ where it obtains its maximum value of $1$.
Sketches of $h(x)$ and $H(x)$, which are also used later in the paper,
are shown in Fig. \ref{hH}.

\begin{figure}
%\centerline{ \input{ plot_bumps} }
\caption{Smooth bump functions $h(x)$ and $H(x)$.  \label{hH}}
\end{figure}

No such analytic example exists since $g$ having an infinite number of local
maxima or minima would force $g'$ to be identically zero.  Since $g$ is
required to have average value zero (condition (3) at the beginning of
Sec. 2), it must therefore be identically zero as well.  This would
result in a $0/1$ resonance horn with ``width'' zero: Eq. (\ref{mapF}) 
indicates
that if $\phi = 0$ every point would be a $0/1$ fixed point,
but if $\phi \ne 0$, no point
would be a $0/1$ fixed point.

\subsection{Flames}

We now move to examples using functions $g_\alpha$ which actually depend on
$\alpha$.  We will use $\alpha \in [0,1]$ as a
``homotopy'' parameter
to trade off between, for instance, the different $g$'s in the 
preliminary examples of the previous section.
More specifically, the examples in this subsection
have the form
$g_\alpha(x) =  (1 - \alpha) \tilde{g}(x) + \alpha \sin(x + \tilde{B})$
where $\tilde{g}$ has multiple maxima and minima as do Examples 2 through 5
from the previous subsection, and $\tilde{B}$
is chosen to make sure the extremum
values of $g_\alpha(x)$ are distinct.
As $\alpha$ increases from $0$ to $1$, all but a single max-min pair of 
singularities must disappear.  The disappearance generically occurs at cusps --
the tips of the ``flames'' in the parameter space bifurcation
diagrams.
This is fairly immediate from Eq. (\ref{mapF}) since the disappearance of
a max-min pair for $g_\alpha$ implies that there is an $x$ value, say $p$,
and an $\alpha$ value, say $a$, 
at which $g^{'}_a(p)=0$, and $g^{''}_a(p) = 0$;
we expect that generically $g^{'''}_a(p) \ne 0$.  These conditions and
Eq. (\ref{mapF}) imply that there is a $\phi$ value, say $\phi_0$ such that
$F_{(\phi_0,a)}(p) = p, F^{'}_{(\phi_0,a)}(p) = 1, 
F^{''}_{(\phi_0,a)}(p) = 0$, but $F^{'''}_{(\phi_0,a)}(p) \ne 0$,
the generic conditions for a cusp bifurcation [Guckenheimer \& Holmes, 1982].
See Koslovsky [1991] for a fomal proof of the existence of the cusp.
\vskip .2in
\noindent
{\bf Example 6: Single Flame.}
\nobreak
To obtain a single flame inside the $0/1$ resonance horn, we define 

%$$  g_\alpha(x) := (1-h(\alpha))G_{2,\epsilon}(x) + h(\alpha)\sin(x)$$
$$  g_\alpha(x) := (1-\alpha)G_{2,\epsilon}(x) + \alpha\sin(x)$$
$$= (1 - A(\alpha)){\sin(2x) \over 2} + A(\alpha) \sin(x + B(\alpha))$$

\noindent
where
$A(\alpha) = \epsilon (1-\alpha) + \alpha = \epsilon + (1 - \epsilon)
\alpha, B(\alpha)=0$,
$G_{2,\epsilon}(x)$ has one extra pair of folds,
as defined in Eq. (\ref{g2e}) above,
and $\epsilon \in [0, 0.5)$.  The second form of 
$g_\alpha$ is given for comparison
with other examples of the same form in Secs. 2.3 through 2.6; any
family of that form can be viewed as a ``path through the 
$(A,B)$ parameter space'' as $\alpha$ changes.  The $(A,B)$ parameter space
is more fully described in Sec. 2.5 below.

Three snapshots of $g_\alpha(x)$ are shown in Fig. \ref{flame}a for $\alpha \in
[0,1]$. Note the four local extrema for $\alpha = 0.1$ and only two local
extrema for $\alpha = 0.9$.
The resulting $0/1$ resonance surfaces and bifurcation diagram
are shown in Fig. \ref{flame}b
for $\epsilon = .1$, but
with $\alpha$ replaced by $h(\alpha^2)$, both to smooth the bifurcation pictures
and to increase the size of the flame; recall the graph of $h(x)$ in Fig.
\ref{hH}.
The $\alpha$ value of the cusp (the top tip of the flame) is the value
which separates the $g_\alpha$'s with two local maxima and two local minima
from those with a single maximum and a single minumum. 

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{ Example 6, Single Flame:
$g_\alpha(x) =  (1 - A(\alpha)){\sin(2x) \over 2} + A(\alpha) \sin(x)$,
where $A(\alpha) = 0.1 + 0.9 \alpha$. (a) The evolution of 
$g_\alpha(x): \alpha = 0.1, 0.5, 0.9$.  
(b) The $0/1$ resonance surface and its projection to the parameter plane
(with $\alpha$ replaced by $h(\alpha^2)$).
\label{flame}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 7: Multiple Flames.}
More generally, to have $n-1$ flames:
%$$ g_\alpha(x) :=
%h(\alpha) \sin(x + \tilde{B})) + (1 - h(\alpha)){\sin(nx) \over n}$$

%$$  g_\alpha(x) := (1-h(\alpha))G_{n,\epsilon}(x) +  
$$  g_\alpha(x) := (1-\alpha)G_{n,\epsilon}(x) +  
\alpha\sin(x+\tilde{B})$$
where $\tilde{B}$ is chosen as in the definition of
$G_{n,\epsilon}(x)$ in Eq. (\ref{gne}) above to avoid coincident projections
to the phase plane fo distinct saddle-node curves.

See Fig. \ref{twoflame} for the $0/1$ resonance surface and bifurcation 
diagram for a double flame created by letting
$n=3, \epsilon = 0$, and $\tilde{B} = {\pi \over 4}$; we also replaced
$\alpha$ with $h(\alpha)$ for visual enhancement.
The inset at the lower right
shows the double flame parameter space projection more clearly.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 7, Double Flame:
$g_\alpha(x) = (1-h(\alpha))\sin(3x) +  h(\alpha)\sin(x+{\pi \over 4})$.
The inset shows more clearly the double flame in the
projection to the parameter plane.  \label{twoflame}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 8: An infinity of Flames.}
Define
$$ g_\alpha(x) :=
%(1 - h(\alpha))G_\infty(x) + h(\alpha) \sin(x))$$
(1 - \alpha)G_\infty(x) + \alpha \sin(x))$$
where $G_\infty$ is the $C^\infty$ function with an infinite number of maxima
and minima described for the infinity of folds in Example 5 above.

\subsection{Center folds}

Flames in the previous section were created by ensuring that $g_\alpha$ had
``extra'' extrema for $\alpha$ near zero, but only a single maximum and
single minimum for $\alpha$
near one.  If instead, as $\alpha$ increases from zero to one, we start with
no extra extrema, ensure that extra max-min pairs are born by $\alpha = 0.5$,
and ensure that they die as $\alpha$ approaches one, we effectively ``slide''
the flame up out of the bottom tip of the $0/1$ resonance horn.  Since
such examples create extra folds in the $0/1$ resonance surface which project
to the middle of the $0/1$ resonance horn, we call them ``center folds.''
We create these examples simply by replacing 
$\alpha$ in the Basic Flames section with $1 - H(\alpha)$, or any
other $C^\infty$ function that travels from 1 to 0 to 1 as $\alpha$ increases
from 0 to 1.
This has the effect of weighting the $\sin(x + B)$ term heavily
for values of
$\alpha$ near both zero and one, but weighting the ``other'' term,
the one causing multiple folds, for $\alpha$ near .5.  Thus, as $\alpha$
increases from zero to one, all families 
start with no extra folds, acquire extra folds, then lose all extra folds.

\vskip .2in
\noindent
{\bf Example 9: Single Center Fold.}
Define
$$ g_\alpha(x) :=
(H(\alpha))G_{2,\epsilon}(x) + (1 - H(\alpha)) \sin(x)$$
$$= (1 - A(\alpha)){\sin(2x) \over 2} + A(\alpha) \sin(x + B(\alpha))$$
where $A(\alpha) = 1 -
(1-\epsilon) H(\alpha), B(\alpha)=0$.
As with the ``single flame'' example, the second form  of
$g_\alpha$ is for later comparison with other examples.
See Fig. \ref{centerfold} for the center fold with $\epsilon = 0.3$.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 9, Center Fold:
$g_\alpha(x) = (1 - A(\alpha)){\sin(2x) \over 2} + A(\alpha) \sin(x)$,
where $A(\alpha) = 1 -0.7 H(\alpha)$.  \label{centerfold}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 10: Multiple Center Folds}
For $n-1$ extra pairs of folds, define
%$$ g_\alpha(x) :=
%A(\alpha) \sin(x + B) + (1 - A(\alpha)){\sin(nx) \over n}$$
%or
$$ g_\alpha(x) :=
H(\alpha)G_{n,\epsilon}(x) + (1 - H(\alpha)) \sin(x + \tilde{B})$$
%where $A(\alpha) = .3 (1-h(\alpha)) + h(\alpha) = .3 + .7
%[2 h(\alpha)-1)]^2$.
where $\tilde{B}$ is chosen to have the same value as in the definition of
$G_{n,\epsilon}(x)$ in Eq. (\ref{gne}) above.
%
% Fig 8: n=3, $\epsilon = .3$.

\vskip .2in
\noindent
{\bf Example 11: An infinity of Center Folds.}
Define
%$$ g_\alpha(x) :=
%A(\alpha) \sin(x + B) + (1 - A(\alpha))G_\infty(x)$$
%where $A(\alpha) = .3 (1-h(\alpha)) + h(\alpha) = .3 + .7
%[2 h(\alpha)-1]^2$.
%or
$$ g_\alpha(x) :=
(H(\alpha))G_\infty(x) + (1 - H(\alpha)) \sin(x)$$
where $G_\infty(x)$ was defined in Example 5 above.

\subsection{Saddle-node crossings}

The emphasis on all the examples presented to this point has been controlling 
the births and deaths of extra pairs of fixed points.  We now present some
examples for which we control the crossing of saddle-node curves for families 
having extra pairs of orbits.
The examples are constructed
using the simple idea that saddle-node curves cross when values of any two
local extrema for $g_\alpha$ cross as $\alpha$ changes.

\vskip .2in
\noindent
{\bf Example 12: A single left-side crossing.}
Define
$$ g_\alpha(x) :=
(1 - .3){\sin(2x) \over 2} + .3\sin(x + B(\alpha))$$
where $B(\alpha) = 2\pi - {\pi \over 2} h(\alpha)$.
From the symmetry of the sine function, it is apparent that when
$\alpha = 0.5$, $g_\alpha(x)$
has two local maxima with equal values.  Thus, the ``crossing'' takes place
at $\alpha = 0.5$.  This is corroborated by the snapshots of $g_\alpha(x)$
for $\alpha= 0.1, 0.5, 0.9$ in Fig. \ref{leftcross}a.  Note that the
gobal maximum for $g_\alpha(x)$ switches from one local maximum to the
other as $\alpha$ passes
through $0.5$.
Figure \ref{leftcross}b shows the $0/1$ resonance surface and its projection
to the parameter plane.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 12, Left-side crossing: 
$ g_\alpha(x) =
0.7{\sin(2x) \over 2} + .3\sin(x +  2\pi - {\pi \over 2} h(\alpha))$.
(a) The evolution of 
$g_\alpha(x): \alpha = 0.1, 0.5, 0.9$.
(b) The $0/1$ resonance surface and its projection to the parameter plane.  \label{leftcross}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 13: A ``full twist'' with one extra fold.}
Define
$$ g_\alpha(x) :=
(1 - \epsilon){\sin(2x) \over 2} + \epsilon\sin(x + B(\alpha))$$
where $B(\alpha) = 2 \pi - 2\pi h(\alpha), \epsilon \in (0,0.5)$.

Since the two local maximum values are coincident at
$B(\alpha) = {\pi \over 4}$ and ${5\pi \over 4}$, and the two local minimum
values are coincident at $B(\alpha) = {3\pi \over 4}$ and ${7\pi \over 4}$,
letting $B(\alpha)$ vary from zero to $-2\pi$, causes two left side saddle-node
crossings alternating with two right side saddle-node crossings.
The result for $\epsilon = 0.3$ is in Fig. \ref{twist}.

\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 13, Full twist: $ g_\alpha(x) =
0.7{\sin(2x) \over 2} + .3\sin(x +  2\pi - {2 \pi} h(\alpha))$.  \label{twist}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 14: A ``full twist'' with n-1 extra folds.}
This is a slight generalization of the previous example. Define
$$ g_\alpha(x) :=
(1 - \epsilon){\sin(nx) \over n} + \epsilon \sin(x + B(\alpha))$$
where $B(\alpha) = 2\pi-2\pi h(\alpha), \epsilon \in (0,0.5)$.

%See Fig. 11: $n=3, \epsilon = 0.3$. Include???

\subsection{More complicated flames and folds}
We now consider bifurcation diagrams created by controlling both the
saddle-node 
crossings and the birth/death of an extra pair of fixed points.
We use the following form which has already been used for the standard map,
coincident projection, noncoincident projection,
single flame, single fold, left side crossing, and 
full twist with one extra fold examples:

\begin{equation}
g_{\alpha}(x) = {(1-A(\alpha)) \over 2}\sin(2x) + 
A(\alpha)\sin(x + B(\alpha))
\label{ABmodel}
\end{equation}

As $A(\alpha)$ and $B(\alpha)$ vary in Eq. (\ref{ABmodel}), 
two types of changes are effected:
\begin{enumerate}
\item The number of relative extrema of $g_\alpha$ changes from 
four (two maxima, two minima) to two (one maximum, one minimum)
as $A(\alpha)$ increases from zero to one.
\item For $A(\alpha)$ sufficiently small to ensure $g_\alpha$ has 
four relative extrema, the values of the relative extrema cross as $B(\alpha)$ 
varies:
\begin{enumerate}
\item The values of the two local maxima cross
 on the two line segments with $B = \pi/4$ or $5\pi/4$;
\item The values of the two local minima cross
on the two line segments with $B = 3\pi/4$ or $7\pi/4$.
\end{enumerate}
\end{enumerate}

Changes in features in the
bifurcation diagram for  the circle maps of Eq. (\ref{mapF})
are effected as the coefficients $(A(\alpha), B(\alpha))$ pass between the
five regions of $(A,B)$ space of $[0,1] \times S^1$ shown in the numerically
computed Fig. \ref{AB}.
\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{(a) The two-mode parameter space: $g_\alpha(x) = {1-A \over 2} \sin(2x)
+ A \sin(x)$.  (b) The path through this two-mode parameter space
which corresponds to the left edge swallowtail of Example 15.  \label{AB}}
\end{figure}

When $(A(\alpha), B(\alpha))$ passes from the four-extrema region to the
two-extrema region (avoiding the four cusp points of Fig. \ref{AB}a),
the corresponding
bifurcation diagram for Eq. (\ref{mapF}) has a cusp (a flame tip).
%look back to the corresponding paths for the single flame and single
% center fold in Figs. \ref{flame}c and \ref{centerfold}c, respectively.
When $(A(\alpha), B(\alpha))$ passes across an equal maximum value
line (on $B = \pi/4$ or $5\pi/4$) the bifurcation diagram for the circle
map family of Eq. (\ref{mapF})
has a crossing of saddle-node curves on the left side of its $0/1$ resonance
region; similarly,
a crossing of minima of $g_\alpha$ as $\alpha$ varies effects
a right-side crossing of saddle-nodes.
%recall the left-side crossing and full
%twist from Figs. \ref{leftcross}c and \ref{twist}c, respectively.

Allowing both  $A(\alpha)$ and $B(\alpha)$
to vary as $\alpha$ varies leads to bifurcation diagrams having both
saddle-node crossings and cusps.  The path through the
$(A(\alpha), B(\alpha))$ space illustrated
in Fig. \ref{AB}b corresponds to Example 15 below,
our left edge swallowtail example.
The reader may wish to return to Examples 1, 2, 3, 6, 9, 12 and 13
to identify each one's corresponding path through
the $(A,B)$ space of Fig. \ref{AB}a.


%\vskip .2in
%\noindent
%{\bf Example 15: Left outside single flame.}
%We use Eq. (\ref{ABmodel}) with
%$A(\alpha) = .3 + .7 [h(\alpha)]^2$,
%and 
%$B(\alpha) = {2\pi} - {3\pi \over 4} h(\alpha)$.
%See Fig. 12.

%\vskip .2in
%\noindent
%{\bf Example 16: Double outside single flame.}
%We use Eq. (\ref{ABmodel}) with
%$A(\alpha) = .3 + .7 [h(\alpha)]^2$,
%and 
%$B(\alpha) = {2\pi} - {5\pi \over 4} h(\alpha)$.
%See Fig. 13.

\vskip .2in
\noindent
{\bf Example 15: Left edge ``swallowtail.''}
%
This example illustrates the creation of the ``swallowtail'' feature mentioned
in the introduction.  This motivating example for our study has been constructed
by combining the simple building blocks of birth/death of extra pairs of 
fixed points with a crossing of left side saddle-node curves.  We have
used Eq. (\ref{ABmodel}), with 
$A(\alpha) = 1 - .7 H(\alpha)$,
and 
$B(\alpha) = {2\pi} - {\pi \over 2} h(\alpha)$.
See Fig. \ref{swallowtail}.
\begin{figure}
%\centerline{ \psfigp{figure=./Nplots/fig2.ps,height=20pc,width=330pt } }
\caption{Example 15, the left edge swallowtail:
$g_{\alpha}(x) = {(1-A(\alpha)) \over 2}\sin(2x) + 
A(\alpha)\sin(x + B(\alpha))$,
with $A(\alpha) = 1 - .7 H(\alpha)$, and 
$B(\alpha) = {2\pi} - {\pi \over 2} h(\alpha)$.  \label{swallowtail}}
\end{figure}

\vskip .2in
\noindent
{\bf Example 16: Make your own.}
Sketch a $0/1$ resonance region bounded by 
saddle-node curves which use the following building block
``moves'' as $\alpha$ is increased:
increasing from two to four saddle-node curves at a downward pointing cusp,
decrease from four to two saddle-node curves at an upward pointing cusp,
crossing two lefthand saddle node curves (only when four curves exist),
and crossing two righthand saddle node curves (only when four curves exist).
Parametrize a path through the AB parameter space which realizes these moves.
Numerically compute the saddle-node curves to see your bifurcation creation.
%\vskip .2in
%\noindent
%{\bf Example 18: Flame with a full twist.}
%We use Eq. (\ref{ABmodel}) with
%$A(\alpha) = .3 + .7 [h(\alpha)]^2$,
%and 
%$B(\alpha) = {2\pi} - {9\pi \over 4} h(\alpha)$.
%See
%Fig. 10$a_2$.

%\vskip .2in
%\noindent
%{\bf Example 19: Center fold with a full twist.}
%We use Eq. (\ref{ABmodel}) with
%$A(\alpha) = 1 - .7 [H(\alpha)]^2$,
%and 
%$B(\alpha) = {2\pi} - {\pi \over 2} h(\alpha)$.
%See
%Fig. 10$b_2$.
%
%\vskip .2in
%\noindent
%{\bf Example 20: Overlapping double folds.}
%Fig. 15. Include???

\subsection{Discussion}

%{\bf Condensed from Sec. 6.1.}
The form of the forcing function in Eq. (\ref{ABmodel}) really includes
all bifurcation diagrams attainable for any two-mode truncation of the 
Fourier series for $g_\alpha$: $A_0(\alpha) + A_1(\alpha) \sin(x + B_1(\alpha)) 
+ A_2(\alpha) \sin(2x + B_2(\alpha))$.
This is because $A_0(\alpha)$ is 
zero due to the zero average value condition following Eq. (\ref{mapF}),
$B_2(\alpha)$ can be eliminated by a translation in $\theta$, and 
$A_2(\alpha)$ can be replaced with $(1 - A_1(\alpha)) \over 2$ 
because it is only the relative 
size of $A_1(\alpha)$ and $A_2(\alpha)$ that determines the number of
relative extrema for 
$g_\alpha$.  In general, bifurcations attainable from an
$N$-mode truncation can be thought of as paths through a $2N-2$ dimensional
parameter space. In this sense, the 
standard map family is the only possibility for
a one-mode truncation.  Systematic generalizations to determine all possible bifurcation diagrams with truncations at higher higher modes
are of course possible, but we choose only to include
only a few examples with higher modes.

%\subsection{Handles (Include?)}

\section{Period-$q$ Flames}

\subsection{Equivariant examples}

All the bifurcation features described so far in the paper have been
generated for the $0/1$ fixed-point resonance.  It turns out that
a slight modification allows any of our fixed-point features to
appear instead for any $p/q$ resonance.  Recall that all our examples
have lifts of the form of Eq. (\ref{mapF}):
$F_{(\phi,\alpha)}(x) := x + \phi + \alpha  g_{(\phi,\alpha)}(x)$.
If we replace $g_{(\phi,\alpha)}(x)$ with ${g_{(\phi,\alpha)}(qx) \over q}$,
we obtain a circle  map that
is equivariant with respect to rotations by $2\pi p/q$. (The division by $q$ 
is to preserve the condition that our forcing function have derivative
bounded by one in absolute value; recall the conditions following Eq. (\ref{mapF}).)
The lift becomes equivariant with respect to translations by $2\pi p/q$;
that is, if 
$\tilde{F}_{(\phi,\alpha)}(x) := x + \phi + {\alpha  \over q} 
g_{(\phi,\alpha)}(qx)$,
then $\tilde{F}_{(\phi,\alpha)}(x + {2\pi p \over q}) = 
\tilde{F}_{(\phi,\alpha)}(x) + {2\pi p \over q}$.

From this it follows fairly easily that if $\theta$ is a
fixed point for $F_{(\phi,\alpha)}$  then
$\{{x \over q}, {x \over q} + {2 \pi p \over q},
..., {x \over q} + {2 \pi p(q-1) \over q}, {x \over q} + {2 \pi p} \}$
is an orbit for
$\tilde{F}_{({\phi \over q} + {2\pi p \over q},\alpha)}$.  That is, the 
orbit of $\theta = {x \over q}$ (mod  $2\pi$) under the circle map corresponding
to lift $\tilde{F}_{({\phi \over q} + {2\pi p \over q},\alpha)}$ is a $p/q$ 
orbit.
This correspondence between $0/1$ points of the circle map with lift
$F_{(\phi,\alpha)}$ and $p/q$ orbits with lift
$\tilde{F}_{({\phi \over q} + {2\pi p \over q},\alpha)}$
turns out to be one-to-one whenever the equivariant
lift $F_{(\phi,\alpha)}$ is a homeomorphism; we can ensure
$F_{(\phi,\alpha)}$ is a homeomorphism by restricting
$\alpha$ to be in $[0,1]$.
% {\bf Need more to justify 1-1 ????????}

Consequently, all the least-period-$q$ resonance regions for $\tilde{F}$
will have exactly
the same bifurcation features as the $0/1$ resonance region for
$F$.  The only difference is that the $p/q$ resonance regions for $\tilde{F}$
will be contracted by a factor of $1 \over q$ in the horizontal 
($\phi$) direction.
Thus any bifurcation feature from Sec. 2 can be made to appear
in any $p/q$ resonance region.

\subsection{General examples}

Analytic studies of period-$q$ bifurcation phenomena for $q>1$ are much
more difficult for general lifts of circle maps of the form
$F_{(\phi,\alpha)}(x) := x + \phi + \alpha  g_{(\phi,\alpha)}(x)$ where
$g_{(\phi,\alpha)}(x)$ 
lacks the special structure of the above subsection which led to $F$ (called
$\tilde{F}$ in the above subsection) being
equivariant.  We can, however, show that when the
forcing amplitude $\alpha$ is small,
only the Fourier modes of $g_{(\phi,\alpha)}$
which are multiples of $q$ affect whether or not ``extra''
folds exist on a  $p/q$ resonance surface as it emanates from 
the circle $({p \over q}, 0,\theta)$ for $\theta \in [0,2\pi) = S^1$ in the 
three-dimensional phase $\times$ parameter space.  Unfortunately, the 
analysis is to lowest order terms in $\alpha$, so it guarantees nothing 
about the number of extra folds for values of $\alpha$ away from zero.

The low forcing amplitude analysis is still of use to us in at least the
following two contexts.  First, we can combine it with
some results from complex dynamics to create examples with the
guarantee that any extra folds that exist on any $p/q$ resonance surface
will disappear as $\alpha$ increases
sufficiently close to $\alpha = 1$; this will in turn
guarantee the existence of flames in the corresponding
$p/q$ resonance regions.  We present this construction following the low forcing
amplitude analysis.
Second, the low forcing amplitude analysis helps
us to show that the existence of extra folds on resonance surfaces is a 
generic property of circle maps whose lifts are given by Eq. (\ref{mapF})
and therefore a generic property for periodically forced oscillator families.
This program is detailed in Sec. 4.

\subsubsection{Low forcing amplitude analysis}

We assume the starting point is
the lifted circle map of Eq. (\ref{mapF}).
We use subscripts on
the $x$ variable to indicate iterations:

$$x_{n+1} = F_{(\phi,\alpha)}(x_n)
= x_n + \phi + \alpha g_{(\phi,\alpha)}(x_n)$$
Straightforward computation leads to:
$$x_n = x_0 + n\phi + \alpha \sum_{k=0}^{n-1} g_{(\phi,\alpha)}(x_k)$$

Since the existence of a $p/q$ point requires $x_q = x_0 + 2\pi p$, 
as in Eq. (2), we require
\begin{equation}
q \phi + \alpha \sum_{k=0}^{q-1} g_{(\phi,\alpha)}(x_k) = 2 \pi p
\label{qfa}
\end{equation}

To obtain a power series expansion around the parameter point $(\phi,\alpha)=
({2\pi p \over q},0)$, we replace the parameter $\phi$ with the small parameter
$\epsilon$ via
$\phi = {2 \pi p \over q} + \epsilon$.

So Eq. (\ref{qfa}) becomes
\begin{equation}
q \epsilon + \alpha \sum_{k=0}^{q-1} g_{({2\pi p \over q} + 
\epsilon,\alpha)}(x_k) = 0
\label{qea}
\end{equation}

Using the Fourier expansion $g_{({2\pi p \over q} + \epsilon,\alpha)}(x) = 
\sum_{n=-\infty}^\infty
c_n(\epsilon,\alpha)e^{inx}$, switching the order of summation, noting that
$\sum_{k=0}^{q-1} e^{ink2\pi p \over q}$ is $q$ for $n=mq$ and $0$ otherwise, alters  Eq. (\ref{qea}) to 

\begin{equation}
q \epsilon + q\alpha \sum_{m=-\infty}^{\infty} 
c_{mq}(0,0)e^{iqmx_0}+O(\alpha^2,\alpha \epsilon) = 0
\label{qeq}
\end{equation}

By the implicit function theorem,  Eq. (\ref{qeq}) can be solved for $\epsilon$
as a function of $\alpha$ and $x_0$:

\begin{equation}
\epsilon = -\alpha \sum_{m=-\infty}^{\infty} 
c_{mq}(0,0)e^{iqmx_0}+O(\alpha^2)
\label{perqsurf}
\end{equation}

Hall [1984] used Eq. (\ref{perqsurf}) to conclude that period-$q$ resonance
horns generically
open as wedges, not cusps.  That is, the sides of the resonance horns
are not tangent at the $(\phi, \alpha) = (p/q, 0)$ tip.
This follows from the generic assumption
that not all of the coefficients of form $c_{\pm nq}$ are zero.  As pointed
out to us by Hall, the standard map, and all the examples
of this paper for which $g_{(p/q,0)} (x)$ has only a finite number of 
terms in
its Fourier expansion, are nongeneric in this sense: all coefficients of form
$c_{\pm nq}$ {\it are} zero for {\it all} $q$'s larger than
the largest 
mode with nonzero coefficient in its Fourier expansion.
For the standard map, this nongenericity is 
manifested in every non-fixed-point resonance horn: they all
open as cusps, not wedges.  This is consistent with Fig. \ref{stdmapall}.

\subsubsection {Flames in $p/q$ resonance horns}

We use Eq.  (\ref{perqsurf}) to interpret how many
period-$q$ orbits there are for various parameter values.  More specifically,
if we fix $\alpha$ and vary $x_0$ from zero to $2\pi$, the projection to the $(\epsilon,\alpha)$ parameter
space travels back and forth in the $\epsilon$ direction 
along a line segment.
The endpoints of the line segment are on the boundary of the
$p/q$ resonance horn in the parameter space.
The number of period-$q$ points for any parameter value
on this line segment is equal to the number of times we pass through that
parameter point as $x_0$ varies from zero to $2\pi$.  The number 
of period-$q$ orbits is the number of period-$q$ points divided by $q$.
Except at $\epsilon$ parameter values ($\alpha$ is still fixed)
which correspond to
critical values for $\epsilon$ as a function of $x_0$ via
Eq. (\ref{perqsurf}), such as at saddle-node orbits, 
the number of period-$q$ orbits will be divisible by $2$.
One attracting-repelling
pair of orbits is the minimum possible number.  Any further pairs
correspond to extra pairs of periodic orbits. (Note: There isn't 
a natural fixed pairing of attracting with repelling periodic orbits except
for orbits near a saddle-node bifurcation; the term
{\it pair} is therefore being loosely used.)

With this in mind, we note that, for $\alpha$ and $\epsilon$
small enough, there will be 
$n-1$ extra pairs of $p/q$ orbits if $g_{(\phi,\alpha)}$ has a
Fourier series at $(\phi,\alpha)=(p/q, 0)$
for which coefficients $c_{\pm nq}$ sufficiently dominate the
coefficients $c_{\pm mq},
m \ne n$.
If $g_{(\phi,\alpha)}$ is independent of $\phi$, then the small $\alpha$
part of all $p/q$ resonance
regions may be studied at once since the Fourier series for
$g_{(p/q,0)}$ is the same for all $p/q$.
For example, if $g_{(\phi,\alpha)}(x) = \sin(6x)$, the fixed point surface
will have $5$ extra fixed point pairs ($6$ total pairs)
resulting in $10$ extra folds ($12$ total folds) on the 
$p/1$ surfaces;
the period-two surface will have $2$ extra period-$2$ orbits resulting in 
$8$ extra folds on the $p/2$ surfaces;
the period-three surface will have one extra pair of orbits resulting in 
$6$ extra folds on the $p/3$ surfaces; 
and the period-$6$ surfaces will have no extra fold pairs.

The consequence of the above discussion is that we can control the number
of extra orbits near the $(p/q, 0)$ tip of the $p/q$ resonance horn.
Unfortunately, since the period-$q$ analysis was ``local,'' it says nothing 
about controlling the number of pairs of orbits which exist for higher
values of $\alpha$.  We can, however, obtain some control by utilizing
some results from complex dynamics.  The following lemma was
communicated to us by R. L. Devaney, N. Fagella and G. R. Hall; the first published version we are aware of (with essentially 
the same proof) appears in Yakobson [1985].

\vskip .2in
\noindent
{\bf Lemma :}
Let  $f_{(\phi,\alpha)}$ be the standard family:
$f_{(\phi,\alpha)}(\theta) = \theta + \phi + \alpha \sin(\theta)$. 
Then
for any fixed values of $\phi$ and $\alpha$ with $\alpha \in [0,1)$, the 
corresponding circle map $f_{(\phi,\alpha)}$ can have at most one attracting
periodic orbit.

\vskip .1in
\noindent
{\bf Sketch of proof:}  Consider the complex family 
$S_{(\phi,\alpha)}(z)=ze^{i\alpha}e^{{\beta \over 2}(z-{1 \over z}})$.
The family $S_{(\phi,\alpha)}$ restricted
to the unit circle is the real standard map
$f_{(\phi,\alpha)}$ [Fagella, 1994].
Each  $S_{(\phi,\alpha)}$ is also a complex analytic map on
$\C \backslash \{0\}$ and therefore any attracting orbit must attract a critical
point [Devaney, 1989].
Each  $S_{(\phi,\alpha)}$ has critical points satisfying
$\alpha z^2 +2z + \alpha^2 = 0$. For $\alpha<1$, these critical points, $c_1$
and $c_2$, are on the negative real axis and satisfy
$c_2 = {1 \over \overline{c_1}}$.
Since $S(1/\overline{z}) = 1 / \overline{S(z)}$, the two critical orbits are ``symmetric with respect to the unit circle.''  Thus, any attracting
periodic orbit must attract not just one, but both critical points.  So
there can exist only one attracting periodic orbit. 
%
\hfill $\Box$

\vskip .1in

This enables us to construct a single example with flames in both a
$p_1/q_1$ and a $p_2/q_2$ resonance horn.

\vskip .2in
\noindent
{\bf Example 19: Flames for two resonances.} Define

$$ g_\alpha(x) = (1-\alpha)( {\sin(n_1 q_1 x) \over q_1} + {\sin(n_2 q_2 x)
\over q_2}) + \alpha \sin(x) $$
%$$ g_\alpha(x) = (1-h(\alpha)) g_0(x) + h(\alpha) \sin(x) $$

If $q_1$ and $q_2$ are relatively prime, this example will have $n_1-1$
extra pairs of $p_1/q_1$ orbits, causing $2(n_1-1)q$ extra folds
in the $p_1/q_1$ resonance surface for small $\alpha$; similarly, there
will be $n_2-1$
extra pairs of $p_2/q_2$ orbits for small $\alpha$.
Because any extra orbits must disappear as $\alpha$ increases toward one
(see the above Lemma),
both the  $p_1/q_1$ and $p_2/q_2$ resonance regions will have flames.
\vskip .2in
Example 19 could be easily generalized to create flames in any finite number
of resonance regions whose periods are all relatively prime.  The extreme case
is a the following single example having flames in an infinity of resonance regions.  The price , however, is that only a finite number of flames will
necessarily persist under perturbation.

\vskip .2in
\noindent
{\bf Example 20: Flames in an infinity of horns.}
We now present an analytic example having an infinite number of resonance
horns with flames.  Define
$$ g_\alpha(x) = (1-\alpha) g_0(x) + \alpha \sin(x)$$
where $g_0(x)$ is a $C^\omega$ map causing the associated family of circle
maps, $F_{(\phi,\alpha)} = x + \phi + \alpha g_0(x)$ to have
infinitely many horns with extra
folds.  Thus, as $\alpha$ increases from zero to one, the circle
maps corresponding 
to the form of $g_\alpha$ for Example $20$ will have ``extra'' folds in the 
infinity of horns which must all
die out, forming the infinitely many flame tips.

To define $g_0(x)$, we first let $r$ be any real number.  We will construct
$g_0$ so
that the corresponding family of circle maps has a $p/q$ horn with a flame
for an infinite sequence of $p/q$'s which accumulate on $r$. 
First choose a sequence $\{ {p_n \over q_n} \} \rightarrow r$.
Require also that the $q_n$'s are all relatively prime.
Then define $g_0$
by prescribing its Fourier series coeffients to be $c_{\pm 2 q_n}
= e^{-2q_n}$ for all $q_n$'s in the chosen sequence of ${p_n \over q_n}$'s,
and zero for all other coefficients.
Such a function is $C^\omega$
since the Fourier series coefficents decay
exponentially [Wheeden \& Zygmund, 1977].
(This follows from the fact that, after complexifying
$g_0(x)$ to $g_0(z)$, we have that $g_0(z) = F \circ G(z)$, where $G(z)= e^{iz}$
is entire and $F(w) = \sum_{k=-\infty}^\infty c_k w^k$ converges, by our choice
of $c_k$'s, for $|w|<e$.
In particular, $g_0(z)$ is analytic for $z$ real, since $|e^{ix}|=1<e$.)
As in the previous example, each period-$q_n$
surface for $x_{n+1} = f(x_n) = x_n + \phi + \alpha g_\alpha(x)$ has an ``extra'' pair of period-$q_n$
orbits associated with it if $0 < \alpha << 1$, but no extra orbits for
$0 << \alpha < 1$.  Consequently,
every ${p_n \over q_n}$ horn will have a flame in it.
\vskip .1in
%{\bf Not stable in analytic or $C^\infty$ or any $C^k$ topology. 
%Do we care????  (Comment added before this example.)}

\section{Flames are Generic}

Let $\Upsilon$ be the function space of lifts of two-parameter families
of circle homeomorphisms of the form of Eq. (\ref{mapF}):
$F_{(\phi,\alpha)}(x) = x + \phi + \alpha g_{(\phi,\alpha)}(x)$.
Assume the dependence of the homeomorphisms is $C^\infty$ as a function
defined on the cartesian product of the phase space $\R$ with the
$(\phi, \alpha)$ parameter
plane.  Endow $\Upsilon$ with the $C^\infty$ topology [Ruelle, 1989]. 
We now show that the property of having only one
attracting and one repelling orbit in each $p/q$ resonance region (as with
the standard family) is 
nongeneric for two-parameter families in $\Upsilon$.
Equivalently, the property of having 
resonance regions with
internal bifurcation features is shown to be generic.
The rest of this section is devoted to this task.
\vskip .1in
\noindent
{\bf Theorem:}
There is an open dense set of families in $\Upsilon$
having the property that the family has at least two hyperbolic
attracting and two hyperbolic
repelling period-$q$ orbits in at least part of one $p/q$ resonance region.
\vskip .2in
\noindent
We will need the following two lemmas for the proof of the theorem.

\vskip .2in
\noindent
{\bf Lemma 1:} 
Let $F_{(\phi,\alpha)}(x) \in \Upsilon$.  Let $p/q$ be any reduced rational
number.  Since $g$ is at least $C^1$, $g_{(p/q,0)}(x)$ equals
its Fourier series expansion [Wheeden \& Zygmund, 1977] which we write
in the following form
(the average value zero condition on $g$
forces the constant term to be zero):
%
\begin{equation}
g_{(p/q,0)}(x) = \sum_{k=1}^\infty A_k \sin(kx + B_k)
\label{fser}
\end{equation}
If $2q|A_{2q}| > q|A_q| + \sum_{k=3}^\infty kq|A_{kq}|$ then for $\alpha$ a
fixed
positive but sufficiently small number, the $p/q$ 
resonance region for $F_{(\phi,\alpha)}$ will include an open interval of $\phi$
values for which the corresponding circle map has
at least four
$p/q$ orbits.
\vskip .1in
\noindent
{\bf Proof:}
Define $\tilde{g}(x) = \sum_{k=1}^\infty A_{kq} \sin(kqx + B_k)$.
%
%We will first show that the hypotheses of Lemma 1 ensure that
%$\tilde{g}(x)$ 
%has $2q$ local maxima and $2q$ local minima for $x \in [0, 2\pi)$.
%Equation  (\ref{perqsurf}) will then give us the desired result for small
%positive values of $\alpha$.
%
Recall that $\epsilon = 0$ in Eq. (\ref{perqsurf}) corresponds to 
$\phi = p/q$.
That equation also shows that only the Fourier
modes of $g_{(p/q,0)}(x)$ which are multiples of $q$ need to be considered
to determine the existence of extra orbits near $\alpha=0$.
This is why we defined $\tilde{g}(x)$ as we did.

Note that ${d \over dx}(A_{2q}\sin(2qx + B_{2q})) =
2 q A_{2q}\cos(2qx + B_{2q})$ which equals $2qA_{2q}$ for $x={2\pi k - B_{2q}
 \over 2q}, k=0,...,2q-1$, and equals
$-2qA_{2q}$ for $x={2\pi k + \pi -B_{2q} \over 2q}, k=0,...,2q-1$.  The inequality in
the hypotheses of the lemma guarantees that the magnitude of the contribution
to $\tilde{g}'(x)$ due to the sum of the rest of the Fourier modes in $\tilde{g}(x)$ is dominated
at the $4q$ points just listed by the contribution to  $\tilde{g}'(x)$ due to
the $2q^{th}$ mode.  Thus, for $x \in [0,2\pi), \tilde{g}(x)$ must have at
least $2q$ points with
positive slopes alternating with $2q$ points having negative slopes.
%This forces $\tilde{g}(x)$ to have at least 
%$2q$ local maxima and $2q$ local minima for $x \in [0, 2\pi)$.

To show that this implies the existence of at least $4q$ distinct $p/q$
orbits for positive values of $\alpha$, note first that
Eq. (\ref{perqsurf}) can be rewritten (dropping the subscript on $x_0$) as

\begin{equation}
\epsilon_\alpha(x) = -\alpha \tilde{g}(x)+O(\alpha^2).
\label{perqsurf2}
\end{equation}

\noindent
By fixing $\alpha$ sufficiently small, Eq. (\ref{perqsurf2}) implies
that $\epsilon_\alpha$ has $2q$ points with positive slopes alternating with 
$2q$ points having
negative slopes for $x \in [0,2\pi)$.

Since all points on a period-$q$ orbit must exist at the same
parameter value, we must have
$\epsilon(x) = \epsilon((F_{(\epsilon(x), \alpha)}(x))$.  Thus it is sufficient
to consider the graph of $\epsilon_\alpha(x)$ only on a ``fundamental interval,"
say for $x \in [0, F_{(\epsilon(0), \alpha)}(0))$.  On this fundamental interval
there must exist two points with positive slopes alternating with
two points with negative
slopes.  Straightforward calculus arguments lead to the existence of two
local maxima and two local minima on the fundamental interval, with the two
local maxima both strictly larger that both local mimima.  That is, there exist
$\epsilon$ values $M_1 \ge M_2 > m_2 \ge m_1$, where the $M_i$'s are local
maxima and the $m_i$'s are local mimima.  For any $\epsilon_0 \in (m_2,M_2)$,
this forces at least four $x$ values in the fundamental interval
$[0, F_{(\epsilon(0), \alpha)}(0))$ to satisfy $\epsilon(x) = \epsilon_0$.
These four $x$ values represent four distinct $p/q$ orbits, so
$(m_2, M_2)$ is the open interval of $\phi$ values claimed to be in the $p/q$ 
resonance region for our fixed small value of $\alpha$.


\hfill $\Box$

\vskip .2in
\noindent
{\bf Lemma 2:}
Let $g(x) = \sum_{k=1}^\infty A_k \sin(kx + B_k)$ be a $C^\infty$ function on
${\bf S}^1$.
Let ${\bf {\it N}}$ be an open neighborhood of $g$ in the 
$C^\infty$ topology.  Then there exists a $\tilde{g} \in {\bf {\it N}}$ and
a $q \in {\bf Z}$ such that if
$\tilde{g}(x) = \sum_{k=1}^\infty \tilde{A}_k \sin(kx + \tilde{B}_k)$, then
$2q|\tilde{A}_{2q}| > q|\tilde{A}_q| + \sum_{k=3}^\infty kq|\tilde{A}_{kq}|$. 

\vskip .1in
\noindent
{\bf Proof:}
By the definition of the $C^\infty$ topology ([Ruelle, 1989, Appendix B], for
example), any $C^\infty$ neighborhood contains a $C^r$ neighborhood for some
positive integer
$r$.  That is, there exists a $r_1 \in {\bf Z^+}$ and an $\epsilon_1 >0$ 
such that $\{ \tilde{g} : ||\tilde{g} - g||_{r_1} < \epsilon_1 \} \subset 
{\bf {\it N}}$, where $||\tilde{g} - g||_{r_1} = \max_{0\le r\le r_1}
||\tilde{g}^{(r)} - g^{(r)}||$ and $||\tilde{g}^{(r)} - g^{(r)}|| =
\max_{x \in [0,2\pi]}|\tilde{g}^{(r)}(x) - g^{(r)}(x)|$.
Recall also that $g \in C^r$ implies
$|A_k| = o({1 \over k^r})$ [Zygmund, 1968]
and $g \in C^\infty$ implies $g\in C^r$ for all nonnegative integers $r$.
Therefore, $g \in C^{r_1}$ and so
\begin{equation}
|A_k| k^{r_1} \rightarrow 0 {\rm \ as\ } k \rightarrow \infty.
\label{Akzero}
\end{equation}

Now choose $q \in {\bf Z}^+$ large enough so that
$|A_k| k^{r_1} < {1 \over 2} {\epsilon_1 \over 2^{r_1}}$ for all $k > q$.
Let $A =  {1 \over 2} {\epsilon_1 \over 2^{r_1} q^{r_1}}$.
This choice of $A$ implies that for all for all $k > q$:

\begin{equation}
{ |A_k| k^{r_1} \over q^{r_1} } < A <  {\epsilon_1 \over 2^{r_1}q^{r_1}}
\label{chooseA}
\end{equation}

Define $\tilde{A} = {\rm sgn}(A_{2q}) A$. ($A_{2q}$ is defined from the Fourier 
series for $g$ in the statement of Lemma 2.)  Define $\tilde{B} = B_{2q}$. Finally define
$\tilde{g}(x) = g(x) + \tilde{A} \sin(2qx + \tilde{B})$.

To show $\tilde{g} \in  {\bf {\it N}}$: $||\tilde{g} - g||_{r_1}
= ||\tilde{A} \sin(2qx + \tilde{B})||_{r_1} = (2q)^{r_1}A < \epsilon_1$ by
the second inequality in Eq. (\ref{chooseA}).
(In the case of the analytic topology instead of the $C^\infty$ topology,
the $C^0$ norm
can be used, but $x$ must be allowed to be complex. Then
 $||\tilde{g} - g||_0
= ||\tilde{A} \sin(2qx + \tilde{B})||_0 < 2A < \epsilon_1$ since
$|\sin(2qz + \tilde{B})| < 2$ for $z$ in a sufficiently small neighborhood
of the real line.  See the notes at the end of this Section.)

To show the required dominance of the $\tilde{A}_{2q}$ term over the rest of
the $\tilde{A}_{kq}$ terms:
$$q |\tilde{A}_q| + \sum_{k=3}^\infty kq|\tilde{A}_{kq}| =
q|A_q| + \sum_{k=3}^\infty kq|A_{kq}|
\le \sum_{k=1}^\infty kq|A_{kq}| $$
$$ <  \sum_{k=1}^\infty kq {A q^{r_1} \over (kq)^{r_1} }
{\rm \ by\ the\ first\ inequality\ in\ Eq.\ (\ref{chooseA})}$$
$$= A q \sum_{k=1}^\infty {1 \over k^{r_1-1}} < 2Aq $$
$$ \le  2q|\tilde{A}_{2q}| {\rm \ ,\ \ since\ \ } 
|\tilde{A}_{2q}| = A + |\tilde{A}|$$
\hfill $\Box$
\vskip .2in
Now we are in a position to prove the genericity of flames:

\vskip .1in
\noindent
{\bf Proof of the theorem:}
The openness in $\Upsilon$ of the families with a resonance region having
at least two hyperbolic attracting and
two hyperbolic repelling orbits follows from the structural stability of
hyperbolic 
orbits.  So we need only verify the denseness of such families in $\Upsilon$.
Since families of the form of Eq. (\ref{mapF}) differ only via their
respective forcing functions, $g_{(\phi, \alpha)}$,
the function space $\Upsilon$, defined at the beginning of this section,
is equivalent to the function space $\Upsilon_1$, defined as
the function space of 
smooth two-parameter families of smooth forcing functions.
Thus, for a given forcing function, $g_{(\phi, \alpha)}$, and a neighborhood
$N_1$ of  $g_{(\phi, \alpha)}$ in $\Upsilon_1$, we must produce 
a $\tilde{g}_{(\phi, \alpha)}$ in $N_1$ whose {\it corresponding} family of circle homeomorphisms will have a $p/q$ resonance region having at
least two attracting and two repelling period-$q$ orbits.

The strategy is to choose some resonance $p_1/q_1$ and
determine a perturbation as in Lemma 2, with $g_{(p_1/q_1,0)}(x)$
playing the role of $g$ in Lemma 2, which guarantees via Lemma 1
that the $p_1/q_1$ 
resonance region has the desired extra orbits.  This perturbation can
either be applied to $g_{(\phi, \alpha)}$ for all values of $\phi$ and $\alpha$, or just to maps having parameter values in a neighborhood of $(p_1/q_1,0)$.
But there is a
technical glitch: $q_1$ must be greater than
the value of $q$ that is chosen by Lemma 2, but $q$ is determined
only after the Fourier series for $g_{(p_1/q_1,0)}$ is known.
So we need a uniform choice of $q$ that will enable us to proceed with Lemma
2 for any forcing function $g_{(\phi,0)}$ with $\phi$ in a nontrivial
interval.
Then we can choose a rational number $p_1/q_1$ in this $\phi$ interval with
$q_1 > q$ and perturb to obtain the desired extra $p_1/q_1$ orbits.
An additional arbitrarily small perturbation can be made (by Sard's Theorem),
if necessary, to
ensure two orbits are hyperbolically attracting and two are hyperbolically 
repelling.
So the following lemma completes the proof.

\hfill $\Box$
\vskip .2in
%
\noindent
{\bf Lemma 3:}
Let $g_\mu(x)$, $\mu \in \R$ be a $C^0$ family of $C^{r+1}, r\in {\bf Z}^+$,
periodic functions
having Fourier series expansions:
$$g_\mu(x) =  \sum_{k=1}^\infty A_k(\mu) \sin(kx + B_k(\mu)).$$
Let $K$ be a compact set in $\R$.
Then $|A_k(\mu)| k^{r} \rightarrow 0$ as $k \rightarrow \infty$
uniformly for $\mu \in K$.
\vskip .1in
\noindent
{\bf Proof:}
Note that convergence for each fixed $\mu$ is already established for 
$C^r$ periodic functions by
Eq.~(\ref{Akzero}).
From Wheeden \& Zygmund [1977],
we have that $f \in C^1$ implies

$$|A_k| \le {1 \over 2 \pi} \int_0^{2 \pi} |f(x) - f(x + {\pi \over k})|dx
\le \{\max_{x\in \S} f'(x) \} {\pi \over k}.$$
%
By applying this inequality to $g_\mu^{(r)}(x)$ whose Fourier series 
expansion is
$$\sum_{k=1}^\infty k^r A_k(\mu) \sin(kx + B_k(\mu)),$$ 
we obtain
$$|A_k(\mu)| k^r \le \{\max_{x\in \S, \mu \in K} g_\mu^{(r+1)}(x) \} {\pi \over k}.$$
Since $g_\mu \in C^{r+1}$,
the maximum in this expression exists and is finite, so the uniform convergence 
of $|A_k(\mu)| k^r$ to zero is established.
\hfill $\Box$
\vskip .1in
Notes about replacing the $C^\infty$ topology in the Theorem:
\begin{enumerate}
\item Parameter dependence need not be $C^\infty$, only $C^2$.
$C^2$ dependence is necessary to use the implicit function theorem in
going from Eq. (\ref{qeq}) to Eq. (\ref{perqsurf}).
\item The theorem's result is also true if the family of homeomorphisms is 
assumed to be analytic and $\Upsilon$ is endowed with the analytic
topology instead of the $C^\infty$ topology.  Lemmas 1 and 3 need no change;
The 
adjustment needed for the analytic topology in Lemma 2 was pointed out in 
the proof of that lemma.
Parameter dependence still only needs to be $C^2$.
\item  The theorem's result does not appear to hold if the family of 
homeomorphisms is assumed to be $C^r$ and is endowed with the $C^r$ topology
because our proof of Lemma 3 requires the family to be  $C^{r+1}$.
If, however, we assume the family
$F_{(\phi, \alpha)}(x)$ is $C^{r+1}$ as a function of 
$x$ but endow $\Upsilon$ with the $C^r$ topology, the theorem will hold.
Parameter dependence still only needs to be $C^2$.
\end{enumerate}

%\section{Folds on resonance surfaces near Hopf bifurcations}

%\subsection{Example of a center fold or at least a cusp in the horn. INCLUDE?}

%\subsection{Example of a family with a sequence of fold cusps approaching a 
%resonant point on the Hopf curve. INCLUDE?}

\section{Comments}

\subsection{Implications for forced oscillators}

All the features we produced in this paper were using circle maps. There is 
a fairly standard construction in order to generate families of lifts of 
circle maps of the form of Eq. (\ref{mapF}) from a periodically forced
planar oscillator of form

\begin{equation}
{{d{\bf x}}\over {{dt}}}={\bf V}({\bf x})+\alpha {\bf W}({\bf x},\omega t)
\label{pfo}
\end{equation}
where ${\bf x}\in \R^2;\omega,\alpha, t\in \R$; {\bf W}
is periodic with period one in its second variable; for
$\alpha =0$ the nonautonomous flow has
a repelling equilibrium point inside a
normally hyperbolic attracting limit cycle.
%
We have reviewed this construction in previous papers
([McGehee \& Peckham, 1994], [Peckham, 1990]).

The reverse process of generating a forced oscillator family from
a family of circle map lifts of the form of Eq. (\ref{mapF}) is
also possible, but the process is not unique and not quite so straightforward.
A sketch of one possible procedure is as follows.  Put the original circle
map family on the unit circle in $\R^2$.  Extend to a map of the plane
by making the unit circle globally attracting, except for the origin, which
is a repelling fixed point.  Compose this family of maps of $\R^2$ with
a linear contraction with a globally contracting fixed point.  Make the 
``contraction'' the identity at $\alpha=0$ and of increasing contractivity
as the forcing amplitude $\alpha$ is increased. (This and similar
types of ``impulse
forcing'' have been used by us as well as others in previous studies
([Schreiber, {\it et. al.}, 1988], [Peckham, 1990])
to build caricatures of maps which are generated by periodically forced
oscillators.)  The composition results in a family of diffeomorphisms of $\R^2$
which can then be suspended to produce a flow in $\R^2 \times \S$.  This flow
can be thought of as the original periodically forced planar oscillator.

Because resonance regions for periodically forced oscillators typically
close in a Hopf bifurcation at higher forcing amplitudes
([Peckham, 1990], [McGehee \& Peckham, 1994]), 
any extra orbits must eventually be eliminated as the forcing amplitude
increases.  Because extra orbits are generic at zero forcing
amplitude, as established in the Theorem of Sec. 4, this establishes
the genericity of flames in bifurcation diagrams for periodically forced
oscillators.

\subsection{Flames in applications}
We have presented in this paper a variety of bifurcation scenarios which
might occur for periodically forced oscillators.  On the other hand, the
only feature which we have seen in numerical studies of periodically forced 
oscillators is the swallowtail. The question remains why flames, for example,
are not widely seen in past numerical
studies of periodically forced oscillators.  There seem to be several possible 
answers.  First, the features might actually be
present in periodically forced
oscillators that have been studied, even for low period resonances, but 
numerical work has just overlooked them.
Second, the
form of the periodically forced oscillators studied in the literature is
such that the corresponding circle maps, like the standard family,
are nongeneric and do not have features such as flames.
Third, these features may exist in examples studied, but only in
resonance regions with periods higher than those numerically studied.

Determining which of these answers applies in each case is not an easy 
task.  It would require being able to calculate the coefficients of at least
the lower mode coefficients in a Fourier series expansion for $g_\alpha$
in the circle map of Eq. (\ref{mapF}) directly from the differential equation
of the form of Eq. (\ref{pfo}).

\subsection{Topology of resonance surfaces}

It turns out that the bifurcations described in this paper all deal with creating extra folds in the various resonance surfaces.  While these folds
are singularities with respect to the projection to the parameter plane,
they are not topologically significant for the resonance surfaces. 
% {\bf Refer to SDCM and/or ACHM???}
Work in progress considers resonance surfaces which have various numbers of 
topological handles.  Such examples can be constructed via Eq.
(\ref{mapF}) by allowing 
$g_{(\phi,\alpha)}$ to actually depend on $\phi$.

\subsection{Acknowledgements}

The authors wish to thank G. R. Hall for enlightening and clarifying
discussions on topics related to this paper.  Thanks also to Alec Norton,
who first suggested the possibility of the theorem we presented in Sec. 4.

This research was partially supported by NSF Grants DMS-9020220 and
DMS-9206957.  Both authors acknowledge the use of Geometry Center
facilities and staff.

\vfill \eject

\section {References}

Arnold, V.I. [1982] 
{\it Geometrical Methods in the Theory of Ordinary Differential Equations},
(Springer Verlag, New York).\\

\noindent
Arnold, V.I. [1983] 
{\it Russian Mathematical Surveys}, {\bf 38} 215.\\

\noindent
Aronson, D.G., McGehee, R.P., Kevrekidis, I.G. \& Aris, R. [1986],
``Entrainment Regions for Periodically Forced Oscillators,''
{\it Phys. Rev. A} {\bf 33} (3), 2190-2192.\\


\noindent
Aronson D.G., Chory M.A., Hall G.R., McGehee R.P. [1983], 
``Bifurcations from an invariant circle for two-parameter families of maps
 of the plane: a computer assisted study,'' {\it Comm. Math. Phys.}, {\bf 83},
303-354.\\

\noindent
Devaney R. L. [1989], 
{\it An introduction to Chaotic Dynamical Systems},
(Addison-Wesley, New York).\\

\noindent
Fagella, Nuria,
``The Complex Standard Family,"
preprint.\\

\noindent
Frouzakis C.E., Adomaitis R.A. and Kevrekidis I.G. [1991],
``Resonance phenomena in an adaptivelly-controlled system",
{\it International Journal of Bifurcations and Chaos} {\bf 1}(1), 83-106.\\

\noindent
Guckenheimer J and Holmes P [1983] {\it Nonlinear Oscillations,
Dynamical Systems and Bifurcations of Vector Fields}, Applied Mathematical
Sciences, {\bf 42} (Springer Verlag, New York).\\

\noindent
Hall G.R. [1984], ``Resonance Zones in two-parameter families of
circle homeomorphisms,'' {\it SIAM J. Math. Anal.}{\bf 15} (6), 1075-1081.\\

%\noindent
%Herman M, personal communication.\\
%
\noindent
Kai T.\& Tomita K. [1979], ``Stroboscopic phase portrait of a forced
nonlinear oscillator,'' {\it Progr. Theor. Phys.}, {\bf 61}(1), 54-73.\\

\noindent
Koslovsky O.S. [1991], ``Resonance zone boundaries for families of 
circle diffeomorphisms,'' {\it Physica D} {\bf (54)}, 1--4.\\

\noindent
McGehee R.P. \& Peckham B.B. [1994], ``Resonance Surfaces for
Forced Oscillators,'' Geometry Center Research Report GCG70 and
{\it Experimental Mathematics} {\bf 3}(3), 221-244.\\

\noindent
Peckham, B.B. [1990],``The Necessity of the Hopf Bifurcation for
Periodically Forced oscillators with closed resonance regions,''
{\it Nonlinearity} {\bf (3)}, 261-280.\\

\noindent
Peckham, B.B. [1988-94], {\it To Be Continued ...}, a continuation software
package for discrete dynamical systems (continually under development).\\

\noindent
Phillips et. al. [1993], Phillips M., Levy S. and Munzner T., ``Geomview: An Interactive Geometry Viewer,'' {\it Notices of the Amer. Math. Soc.} {\bf 40} 
(1993), 985--988.  This software and the accompanying manual are available
by anonymous ftp from the host {\bf geom.umn.edu},
in the directory {\bf pub/software}.\\

\noindent
Ruelle D. [1989], {\it Elements of Differentiable Dynamics and
Bifurcation Theory}, (Academic Press, Inc., San Diego).\\

\noindent
Schreiber I., Dolnik M., Choc P. \& Marek M. [1988],
``Resonance Behaviour in Two-Parameter Families of Periodically Forced
Oscillators,'' {\it Physics Letters A}, {\bf 128, 2}, 66-70.\\

\noindent
Yakobson M.V. [1985],
``Number of periodic trajectories for analytic diffeomorphisms of the circle,'' 
{\it Functional Analysis and its Applications}, {\bf (19) 1}, 79-80.\\

\noindent
Wheeden R. L. \& Zygmund A. [1977], {\it Measure and Integral},
(Marcel Dekker, New York).\\

\noindent
Zygmund, A., [1988, c1959] {\it Trigonometric Series}
(Cambridge University Press, New York).

\pagebreak
\section {Complete Figure Captions}
\begin{enumerate}
\item [1.]  A schematic of the $3/4$ resonance region from Fig. 3 of Schreiber, 
et. al. [1988].  The resonance region is in the $(T,A)$ parameter plane for a
family of maps of the plane
generated by a planar flow with ``impulse forcing.''
The $T$ parameter can be thought of as controlling
forcing frequency, the $A$ parameter as controlling forcing amplitude.
Note especially the swallowtail in the $3/4$ resonance region. 
\item [2.] (Reproduced from Fig. 6a of Frouzakis, et. al. [1991]
with permission from the authors.) The $1/5$ resonance region
in the $(k,b)$ parameter plane for a family of maps
of $\R^3$.  Note especially the swallowtail in the resonance region.
\item [3.] Fixed points by graph.
$F_{(-0.4, 0.9)}(x) = x + (-0.4) + 0.9((0.7 {\sin(2x) \over 2} + 0.3 \sin(2\pi-h(.9) {\pi \over 2})$,
where $h(.9) = 0.972$.  The form of $F$ is chosen to 
match the form used later in the paper in Example 12.
\item [4.] Example 1, the Standard Family: $g(x) = \sin(x)$.
The $0/1$ resonance 
surface and its projection to the parameter plane.
\item [5.] Example 1, the Standard Family: $g(x) = \sin(x)$.  Several resonace 
surfaces and their projections to the parameter plane.
\item [6.] Example 2, Coincident projections: $g(x) = \sin(2x)$.
\item [7.] Example 3, Noncoincident projections: $g(x) = 0.9{\sin(2x) \over 2}
+ 0.1\sin(x)$.
\item [8.] Smooth bump functions $h(x)$ and $H(x)$.
\item [9.] Example 6, Single Flame:
$g_\alpha(x) =  (1 - A(\alpha)){\sin(2x) \over 2} + A(\alpha) \sin(x)$,
where $A(\alpha) = 0.1 + 0.9 \alpha$. (a) The evolution of 
$g_\alpha(x): \alpha = 0.1, 0.5, 0.9$.  
(b) The $0/1$ resonance surface and its projection to the parameter plane
(with $\alpha$ replaced by $h(\alpha^2)$).
\item [10.]  Example 7, Double Flame:
$g_\alpha(x) = (1-h(\alpha))\sin(3x) +  h(\alpha)\sin(x+{\pi \over 4})$.
The inset shows more clearly the double flame in the
projection to the parameter plane.
\item [11.] Example 9, Center Fold:
$g_\alpha(x) = (1 - A(\alpha)){\sin(2x) \over 2} + A(\alpha) \sin(x)$,
where $A(\alpha) = 1 -0.7 H(\alpha)$.
\item [12.] Example 12, Left-side crossing: 
$ g_\alpha(x) =
0.7{\sin(2x) \over 2} + .3\sin(x +  2\pi - {\pi \over 2} h(\alpha))$.
(a) The evolution of 
$g_\alpha(x): \alpha = 0.1, 0.5, 0.9$.
(b) The $0/1$ resonance surface and its projection to the parameter plane.
\item [13.] Example 13, Full twist: $ g_\alpha(x) =
0.7{\sin(2x) \over 2} + .3\sin(x +  2\pi - {2 \pi} h(\alpha))$.
\item [14.] (a) The two-mode parameter space: $g_\alpha(x) = {1-A \over 2} 
\sin(2x) + A \sin(x)$.  (b) The path through this two-mode parameter space
which corresponds to the left edge swallowtail of Example 15.
\item [15.] Example 15, the left edge swallowtail:
$g_{\alpha}(x) = {(1-A(\alpha)) \over 2}\sin(2x) + 
A(\alpha)\sin(x + B(\alpha))$,
with $A(\alpha) = 1 - .7 H(\alpha)$, and 
$B(\alpha) = {2\pi} - {\pi \over 2} h(\alpha)$.
\end{enumerate}

\end{document}

\begin{thebibliography}{xxxxxx}

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{\it Geometrical Methods in the Theory of Ordinary Differential Equations},
(Springer Verlag, New York), 1983.
%
\bibitem[AMKA]{AMKA}Aronson D.G., McGehee R.P., Kevrekidis I.G. \& Aris R.,
``Entrainment Regions for Periodically Forced Oscillators,''
{\it Phys. Rev. A} {\bf 33} (3) 2190-2192, 1986.
%
%\bibitem[ACHM]{ACHM}Aronson D.G., Chory M.A., Hall G.R., McGehee R.P., 
%``Bifurcations from an invariant circle for two-parameter families of maps
% of the plane: a computer assisted study,'' {\it Comm. Math. Phys.}, {\bf 83},
%303-354, 1983.
%
\bibitem[ACHM]{ACHM}Aronson D.G., Chory M.A., Hall G.R., McGehee R.P., 
``Bifurcations from an invariant circle for two-parameter families of maps
 of the plane: a computer assisted study,'' {\it Comm. Math. Phys.}, {\bf 83},
303-354, 1983.
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%
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{\it An introduction to Chaotic Dynamical Systems},
Addison-Wesley, New York, 1989.
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``The Complex Standard Family,"
preprint.
%
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``Resonance phenomena in an adaptivelly-controlled system",
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%
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%
\bibitem[Ha]{Ha}Hall G.R., ``Resonance Zones in two-parameter families of
circle homeomorphisms,'' {\it SIAM J. Math. Anal.}{\bf 15} (6), 1075-1081, 1984.
%
%\bibitem[Ha2]{Hall}Hall G.R., thesis paper??.
%
\bibitem[He]{He} Herman M, personal communication.
%
\bibitem[KT]{KT}Kai T., Tomita K., ``Stroboscopic phase portrait of a forced
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