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\title{Breakup of Invariant Tori for the Four Dimensional Semi-Standard
Map} \author{Erik M. Bollt \and James D. Meiss \\ \and
Program in Applied Mathematics\\ Box 526, University of
Colorado\\ Boulder CO 80309, USA} \date{October 1, 1992}
\begin{document} \setlength{\baselineskip}{20pt} \maketitle
\begin{abstract}
We compute the domain of existence of two-dimensional invariant tori
with fixed frequency vectors for a four-dimensional, complex,
symplectic map. The map is a generalization of the semi-standard map
studied by Greene and Percival; it has three parameters, $a_1$ and
$a_2$ representing the strength of the kicks in each degree of freedom,
and $\epsilon$, the coupling. The domain of existence of a torus in
$(a_1,a_2)$ is shown to be complete and log-convex for fixed $k =
\epsilon/a_1 a_2$. Explicit bounds on the domain for fixed $k$ are
obtained. Numerical results show that quadratic irrationals can be more
robust than the cubic irrational, ``the spiral mean."
\end{abstract}
\setlength{\baselineskip}{20pt}
\newpage
%\input{inc} %\input{psfig}
%\include{intro} %\include{coupling} %\include{incommensurate}
%\include{recursion} %\include{holomorphy} %\include{numerics}
%\include{conclusions} %\include {fig} %\bibliography{JDM}
%\end{document} \section{Introduction}
Stability of motion in Hamiltonian systems and symplectic mappings is
of great interest in many physical situations such as plasma and
accelerator confinement and stellar and planetary dynamics; an
understanding of stability is also of intrinsic theoretical interest.
The primary stability result is the KAM theorem which asserts that most
of the invariant tori of a nonlinear integrable Hamiltonian survive
upon a {\it small}, smooth perturbation \cite{Arnold}. The robust tori,
according to the theorem, are those which have sufficiently
incommensurate frequency vectors (they satisfy a Diophantine condition,
see \S{sec:incommensurate}). As a practical result, however, the KAM
theorem has several drawbacks. The first is that estimates of the
perturbation size for the destruction of tori are typically extremely
small: much smaller than the size indicated by numerical computations
for specific perturbations on specific tori (of course the theorem
guarantees the survival of any Diophantine torus for any small enough
perturbation). The second is that the theorem guarantees stability only
for systems of two degrees of freedom since the invariant tori have
half the dimension of phase space. None-the-less, computations indicate
that while a system of three degrees of freedom may not be rigorously
stable, it exhibits a ``practical stability" since orbits appear to
remain trapped near invariant tori for extremely long periods. To some
extent this is addressed by the Nekhoroshev theorem \cite{Nekhoroshev},
though this theorem requires extremely small perturbation sizes as
well.
For the case of two degrees of freedom, or equivalently area preserving
mappings, much progress has been made in determining the existence of
invariant tori. Three basic techniques have been used. The first is to
examine the stability of a sequence of periodic orbits whose
frequencies limit on the irrational frequency of interest---this gives
rise to the {\it residue criterion} \cite{Greene,MacKay82}. It yields
extremely accurate values for the parameters at which invariant circles
are destroyed and can be made rigorous \cite{MacKay91}. The second
method is a nonexistence criterion for twist maps, called converse KAM
theory \cite{Mather,MacKay-Percival}. The final technique is numerical
computation (in some cases using interval arithmetic) of the conjugacy
to pure rotation \cite{Percival,DeLaLave,Berretti-Marmi,Marmi-Stark}.
These methods can give accurate nonrigorous values for the critical
parameter for essentially arbitrary Diophantine frequencies, and can
also give reasonable rigorous values.
Though many have attempted to generalize these techniques to
Hamiltonian systems with more than two degrees of freedom, or
equivalently, symplectic maps of four or more dimensions, there has
been limited success in determining the existence of invariant tori.
Periodic orbit approximations to invariant tori have been obtained
\cite{Kook-Meiss,Muldoon}, and computations reveal that the stability
domains of periodic orbits limiting on an incommensurate frequency
vector may be converging for a volume preserving example
\cite{Mao-Helleman1}. However the existence of the limit is difficult
to prove \cite{Bernstein-Katok}, primarily because the ordering
property of periodic orbits on the circle no longer applies on the
torus. Converse KAM theory can be generalized to higher dimensions
\cite{MacKay-Meiss-Stark}, though in this case one must assume that the
tori are Lagrangian graphs.
One of the fundamental problems in these studies is number theoretic:
there is no satisfactory generalization of the continued fraction
theory to simultaneous approximation of several irrationals (perhaps
the most promising is that of Brentjes \cite{Brentjes}). In the case of
the residue criterion, it is the best approximants (convergents of a
continued fraction) whose properties converge to those of the invariant
circle. Furthermore, quadratic irrationals play a large role in these
studies because their continued fraction expansions are eventually
periodic (these give rise to self-similar structures). Finally, the
most robust tori appear to correspond to the class of quadratic
irrationals known as the {\it noble} numbers; these have a continued
fraction expansion with a tail of all one's. Roughly speaking, the
explanation for this is that the noble numbers are the most difficult
to approximate in the sense of Diophantine. The generalization of this
class to higher dimensions is unknown.
There has been some speculation that for four-dimensional mappings,
cubic irrationals will replace the quadratics. One reason for this is
that a periodic approximation scheme based on a Farey tree construction
necessarily leads to a frequency which is the eigenvalue of a
$3\times3$ matrix, and is therefore cubic
\cite{Guckenheimer,Kim-Ostlund}. However, even in this case it has been
difficult to determine if there is self-similar behavior near break-up
\cite{Mao-Helleman1}, and there is no evidence that cubic irrationals
are more robust than others.
In this paper we study the four-dimensional, complex, symplectic map
corresponding to the coupling of two semi-standard maps, as introduced
in \cite{Greene-Percival}. This map is the complex version of a
mapping introduced by Froeshl\'{e} \cite{Froeshle,Kook-Meiss}---we call
it the \semiF map. We generalize the method of Percival and Greene
\cite{Greene-Percival} to this case and find recursion formulae for the
Fourier coefficients of an invariant two torus with a fixed frequency
vector in \S{sec:recursion}. Existence of such a torus for small enough
parameter values is guaranteed providing the frequency vector satisfies
a Diophantine condition; we discuss this in \S{sec:incommensurate}.
Because of the simple structure of the Fourier series for the \semiF
map, we are able to apply some results from the theory of holomorphic
functions of several complex variables in \S{sec:holomorphy}, and show
that the domain of convergence of the Fourier series has a particular
form; it is complete and log-convex. Finally in \S{sec:numerics} we
compute these convergence domains for several example frequency
vectors, including quadratic and cubic irrationals.
\section{Incommensurate Frequencies} \label{sec:incommensurate}
The convergence of the Fourier series for the semi-standard map has
been studied extensively in
\cite{Greene-Percival,Percival,Marmi-Stark}. In particular, rather
sophisticated techniques for computing convergence of this series were
developed in \cite{Percival}; these give accurate results for quite
general frequencies. In general one determines a parameter interval $
|a| < a^{ss}(\omega)$ for which there is an analytic invariant circle
with frequency $\omega$. Here $a^{ss}$, the critical function, is zero
for every rational value and exhibits a maximum for \begin{equation}
\label{golden}
\omega = \gamma \equiv \frac{1+\sqrt{5}}{2} \;. \end{equation}
The critical function appears to have a local maximum at each of the
{\it noble frequencies}: those equivalent to $\gamma$ under a modular
transformation, or equivalently which have a continued fraction
expansion whose elements are all $1$ beyond some level.
These results also apply to the \semiF map when $\epsilon =0$. Thus an
invariant torus of frequency $\wvec = (\omega_1,\omega_2)$ exists
within the rectangle $\{(a_1,a_2,\epsilon) : |a_1| < a^{ss}(\omega_1),
|a_2| < a^{ss}(\omega_2), \epsilon = 0 \}$. Furthermore, since
the \semiF map is an analytic perturbation of a twist map, KAM theory
implies that for sufficiently small values of the three parameters
$(a_1,a_2,\epsilon)$ there exists an invariant torus analytically
conjugate to the rotation $\tvec \mapsto \tvec + 2\pi \wvec$ providing
the frequency vector satisfies a Diophantine condition.
For $d$-dimensions, the set of Diophantine vectors ${\cal D}_\mu$
consists of those $\wvec \in \Real^d$ for which there exists a $C > 0$
such that for all $({\bf p},q) \in \Integer^{d+1}$ \begin{equation}
\label{Diophantine}
|{\bf p} \cdot \wvec - q| \ge \frac{C}{||{\bf p}||^{\mu}} \;,
\end{equation} where $||{\bf p}|| = \max (|p_1|,...,|p_d|)$. It is easy
to see that if $\mu > d$, the measure of ${\cal D}_\mu$ approaches one
as $C \rightarrow 0$; however, the measure of ${\cal D}_d$ is zero.
Certainly if $\wvec \in {\cal D}_\mu$ then it is {\it incommensurate},
that is $1$ and $\omega_1, ...,\omega_d$ are linearly independent over
the rationals. For $d>1$ one must distinguish between commensurate
vectors and {\it resonant} vectors. While the former satisfy {\it some}
rational relation ${\bf p} \cdot \wvec = q$, the latter have {\it all}
components rational and correspond to periodic orbits. A
straightforward generalization of Greene's method \cite{Greene} to
higher dimensions would use resonant vectors, e.g.
\cite{Guckenheimer}, instead of commensurate vectors. However, in KAM
theory it is commensurabilities which cause the problems, not just
resonances.
Though there exist many Diophantine vectors, a result of Minkowski
implies that every $\wvec$ can be closely approximated in a certain
sense \cite{Cassels}:
\begin{theorem} \label{minkowski}
For any $\wvec \in \Real^d$ there are infinitely many
integer
vectors
$({\bf p},q)$ such that when $K=1$ \begin {equation}
|\bf p\rm \cdot \wvec -q|<
{{K} \over {||\bf p\rm ||}^{d}}
\;.
\end{equation} If $d=1$ then $K$ can be replaced by $
1/ \surd 5$ but
nothing smaller.
\end{theorem} To our knowledge, the minimal value of $K$ for
$d>1$ is not known.
One class of frequency vectors which are Diophantine are those
constructed from algebraic irrationals \cite{Cassels}:
\begin{theorem} \label{algebraic}
If the components of $\wvec$ are incommensurate and
elements
of a real
algebraic field of degree $d+1$, then $\wvec \in {\cal
D}_d$. \end{theorem} Recall that an algebraic field
generated by $\xi \in \Real$ of degree $n$ is defined as the set of
numbers of the form $$R(\xi) = {P(\xi) \over Q(\xi)}$$ where P and Q
are polynomials of degree $n$ with integer coefficients.
One would expect that a frequency vector $\wvec$ which is more
incommensurate, in the sense of having a larger Diophantine constant
$C$ and smaller exponent $\mu$ would tend to persist for higher
perturbations. This is numerically verified for the standard and
semi-standard maps where the noble numbers give local maxima of
$a^{ss}$, and are also the ``most" irrational in the sense of
Diophantine. Unfortunately, to our knowledge, there are no results in
the theory of simultaneous approximations which determine a class of
frequency vectors analogous to the noble numbers. Indeed one of the
main reasons for our numerical investigation is to attempt to develop a
technique for determining this class.
We will choose several simple frequency vectors as examples for our
study. In addition to the golden mean, we will use the quadratic
irrationals \begin{equation} \label{silver}
\sigma \equiv \sqrt{2} = [2,2,2,2....] \equiv [2^{\infty}] \;;
\ \ \zeta \equiv \frac{1+\sqrt{2}}{5+4\sqrt{2}} = [
0,4,2^{\infty}]\;. \end{equation} The expressions on the right
above give the continued fraction expansions. Setting $\wvec =
(\gamma,\sigma)$ or $(\gamma, \zeta)$ yields two incommensurate
frequency vectors since $\sqrt{{5 \over 2}}$ is irrational. Furthermore
by Theorem \ref{algebraic}, both of these vectors are in ${\cal D}_2$,
since they are elements of the algebraic field of degree three
generated by $\xi = \sqrt{2} + \sqrt{5}$. This is easy to see, since
any cubic polynomial in $\xi$ has the form $P(\xi)= a + b\sqrt{2} +
c\sqrt{5} + d\sqrt{10}$ for $a,b,c,d \in \Integer$. Thus $\gamma$,
$\sigma$, and $\zeta$ are all in $R(\xi)$.
Finally we consider a cubic irrational, the real solution of
\begin{equation} \label{eq:spiral} \begin{array}{lcl}
\tau^3 &=& \tau+1 \\ \tau &\simeq&
1.32471795724474602596090885447809734 \\
&\simeq&
[1,3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,2,1,1,3,1,...] \;.
\end{array} \end{equation} This so called ``spiral mean'' frequency,
was introduced in \cite{Kim-Ostlund} as a possible analogue of the
golden mean since in the Ostlund-Kim version of the Farey tree, $\tau$
has a simple periodic construction. The number $\tau$ is Diophantine
since according to a theorem of Roth, every algebraic irrational is in
${\cal D}_{1+\delta} \; \forall \delta > 0$. Thus the critical
function $a^{ss}(\tau) \ne 0$; however, determining its value is
difficult because the continued fraction elements appear unbounded
\cite{Mao-Helleman2}. None-the-less, for the four-dimensional case we
will study the vector $( \tau, \tau^2)$ which is in the cubic field
generated by $\tau$, and so an element of ${\cal D}_2$. Furthermore
$\tau$ is the smallest of the ``{\it PV} numbers," which implies that
the rational vectors on the Farey sequence approaching $(\tau, \tau^2)$
converge more slowly than any other algebraic pair \cite{Kim-Ostlund}.
As we will see in \S{sec:numerics}, the frequencies enter the Fourier
expansion for $\xvec (\tvec )$ solely in terms of the small
denominators \begin{equation} \label{denom} D_\nvec=4\sin^2( \pi \nvec
\cdot \wvec) \;; \end{equation} that is, the ${\bf n}^{th}$ Fourier
coefficient of $\xvec(\tvec)$ is divided by $D_\nvec$. For Diophantine
frequency vectors, $D_\nvec$ is bounded from below; in fact
\eq{Diophantine} implies that if $\wvec \in{ \cal D}_{2}$, then
$1/D_\nvec < {\cal O}(||\nvec||^4)$. Unfortunately, there is no theory
analogous to the continued fraction theory which provides the values of
$\nvec$ for which there are large peaks in $1/D_\nvec$. In fig.
\figdenominator we show a plot of the values of $\nvec$ for which
$(D_\nvec(\gamma,\sigma) ||\nvec||^4)^{-1} > 1.0\times10^{-2}$ and
$5.0\times10^{-5}$. As this figure shows, these peaks are quite
isolated and rare. Thus, following the results for the semi-standard
map, one would expect the Fourier coefficients to have similar isolated
peaks, and for the convergence determination of the Fourier series to
be quite delicate. However, as we will see in \S{sec:numerics}, this
is fortuitously not the case. \section{Coupling of Two Semi-standard
Maps} \label{sec:coupling}
The {\it semi-standard}, area preserving map was introduced by Greene
and Percival \cite{Greene-Percival} as a numerically simpler model than
the standard map for the investigation of the analytic properties of
invariant circles. In Lagrangian form, the semi-standard map takes
$\{x_{t-1},x_{t}\} \mapsto \{x_t,x_{t+1}\}$ and is defined by
\begin{equation} \label {semistd}
\delta^2 x_t \equiv x_{t+1}-2x_t+x_{t-1}=i a e^{ix_t} \;;
\end{equation} this is a map on $\Complex^2$. The notation $\delta^2$
is reminiscent of the second derivative operator.
In this paper we study a four dimensional generalization, analogous to
the map introduced by Froeshl\'{e} \cite{Froeshle,Kook-Meiss}. Letting
$\xvec_t \in \Complex ^2$, the \semiF map is \begin{equation}
\delta^2\xvec_t \equiv \xvec_{t+1}-2\xvec_t+\xvec_{t-1}= {\bf
F}(\xvec_t) \;, \label{semiF} \end{equation} where
\begin{equation}\label{force} {\bf F}(\xvec) \equiv i
\left( \begin{array}{ccc}
a_1e^{ix^{(1)}}&+& \epsilon e^{ix^{(1)}+ix^{(2)}} \\
a_2e^{ix^{(2)}}&+& \epsilon e^{ix^{(1)}+ix^{(2)}}
\end{array} \right) \;. \end{equation} There are three
parameters, the strength of the kicks for each component semi-standard
map ($a_1, a_2$) and $\epsilon$, the strength of the coupling of the
two maps. \Eq{semiF} is symplectic since ${\bf F}$ is the gradient of a
scalar potential (see for example \cite{Kook-Meiss}).
We are looking for solutions $\xvec_t$ of \eq{semiF} which lie on an
invariant two-torus homotopic to the trivial torus defined by the
momentum ${\bf y}_t \equiv \xvec_t -\xvec_{t-1}$ being constant. In
fact we demand that this torus be analytically conjugate to a uniform
rotation on the angle variable $\tvec$ with a given frequency vector
$\wvec$. These tori include those found by KAM theory. The conjugacy
is represented by the following commuting diagram
\begin{equation}\label{eq:3}
\begin{array}{clcr}
\xvec_t & \longrightarrow & \xvec_{t+1} \\ \downarrow &
\ & \downarrow \\ \xvec(\tvec) & \longrightarrow &
\xvec(\tvec+2 \pi\wvec) \; . \end{array}
\end{equation}
Thus, for a given $\wvec$, an invariant torus for \eq{semiF} is given
by
\begin{equation} \label{eq:4}
\xvec_t = \xvec(\tvec+2\pi \wvec t) \;, \end{equation} for
$\tvec \in \Torus^2$. The homotopy condition implies that
\begin{equation} \label{eq:5}
\xvec(\tvec+2\pi\mvec)=\xvec(\tvec)+2\pi\mvec \ \ \forall \mvec
\in \Integer ^2 \;, \end{equation} thus $\xvec(\tvec)$ is
coperiodic with $\tvec$; \begin{equation} \label{coperiodic}
\xvec(\tvec)=\tvec+\chivec(\tvec) \end{equation} where
$\chivec(\tvec)$ is doubly $2\pi$ periodic. If we suppose that $\xvec$
is analytic, it can be expanded in a Fourier series
\begin{equation}
\label{Fourier}
\xvec(\tvec)=\tvec+\sum_{\nvec \in \Integer^{2}}
\chivec_\nvec e^{i \nvec \cdot \tvec}
\end{equation}
Inserting \eq{eq:4} into \eq{semiF} yields the Percival form of the
mapping \begin{equation}\label{Froeshle}
\delta^2\xvec(\tvec) \equiv \xvec(\tvec+2 \pi \wvec)-
2\xvec(\tvec)+\xvec(\tvec- 2 \pi \wvec)= {\bf
F}(\xvec(\tvec)) \; .
\end{equation}
Inserting the series (\ref{Fourier}) into \eq{Froeshle} will yield
equations determining the Fourier coefficients $\chivec_\nvec$; these
will be obtained in \S{sec:recursion}. \section{Recursion Relation}
\label{sec:recursion}
In this section we will derive recursion relations for the Fourier
coefficients of ${\bf x}(\tvec)$, the solution to \eq{Froeshle}. For
the semi-standard map it was possible to find a solution analytic in
the upper half $\theta$ plane. In this case only the positive Fourier
coefficients are nonzero. This is one advantage over the series for
real mappings where all the Fourier coefficients must be considered. In
the case at hand, since the force, \eq{force}, has only positive
imaginary exponentials, we can also find solutions analytic in the
domain $\{(\theta_1,\theta_2) : \rm{Im}{(\theta_1)} \geq 0 ,
\rm{Im}{(\theta_2)}\geq 0 \}$ , so that only positive Fourier
coefficients are needed
It is convenient to define $\uvec \in \Complex^2$ as \begin{equation}
\label{complex} \uvec= \left( \begin{array}{c}
a_1 e^{i\theta_1} \\ a_2 e^{i \theta_2} \end{array}
\right) = \left(
\begin{array}{cc}
u_1 \\ u_2 \end{array} \right) \;. \end{equation} The
advantage of this definition, is that the parameters $a_1$ and $a_2$
will not appear in any of our expansions. Further, using
\eq{coperiodic} we define \begin{equation} \label{eq:12}
{\bf g}(\uvec)=i({\bf x}(\tvec)-\tvec)=i \chivec (\tvec) \;.
\end{equation} Since by ansatz, only the positive coefficients will be
needed in the Fourier expansion of $\chivec (\tvec)$, ${\bf g}(\uvec)$
has a Taylor expansion \begin{equation}\label{taylor}
{\bf g}(\uvec)=\sum_{\nvec\in \Natural^2} \bvec_\nvec
\uvec^\nvec
\equiv \sum_{n_1=0}^{\infty} \sum_{n_2=0}^{\infty}
\left(
\begin{array}{c}
b_{(n_1,n_2)}^{(1)} \\ b_{(n_1,n_2)}^{(2)} \end{array}
\right)
u_1^{n_1} u_2^{n_2} \;. \end{equation} Here we use standard
multi-index notation for the vector exponentiation: while $\uvec \in
\Complex^2$ and $\nvec \in \Natural^2$, $\uvec^\nvec \equiv u_1^{n_1}
u_2^{n_2} \in \Complex$. In addition to the expansion of $\bf g$, we
will need the expansions of its exponential as well: \begin{equation}
\label{exponential}
e^{g_i(\uvec)}=\sum_{\nvec \in \Natural^2} c_\nvec^{(i)}
\uvec^\nvec \end{equation} where $i = 1,2$. In terms of the new
variables, the map \eq{Froeshle} takes the form \begin{equation}
\label{eq:16} \delta^2 {\bf g}(\uvec)=
- \left(
\begin{array}{c}
u_1 e^{g_1(\uvec)} \\ u_2 e^{g_2(\uvec)} \end{array}
\right) - k\left(
\begin{array}{c}
u_1u_2 e^{g_1(\uvec)+g_2(\uvec)}\\ u_1u_2
e^{g_1(\uvec)+g_2(\uvec)} \end{array} \right)
\end{equation} where \begin{equation}\label{k-define}
k=\frac{\epsilon}{a_1a_2} \end{equation} is the coupling
parameter. Note that these equations depend upon the three parameters
$a_1,a_2$ and $\epsilon$ solely through $k$.
Substituting \eqs{taylor}-(\ref{exponential}) into \eq{eq:16} and
noting that for a term in the Fourier series the operator $\delta^2$
becomes $-D_\nvec$, as defined by \eq{denom}, yields the recursion
relation for $\bvec_\nvec$ : \begin{equation} \label{b-recursion}
D_\nvec \bvec_\nvec= \left(
\begin{array}{c}
c_{\nvec-(1,0)}^{(1)} \\ c_{\nvec-(0,1)}^{(2)}
\end{array} \right) + k \sum_{\mvec=(0,0)}^{\nvec-(1,1)} \left(
\begin{array}{c}
c_\mvec^{(1)}c_{\nvec-\mvec-(1,1)}^{(2)}\\
c_\mvec^{(1)}c_{\nvec-\mvec-(1,1)}^{(2)} \end{array}
\right) \;. \end{equation} If $\wvec$ is incommensurate, then
$D_\nvec$ is nonzero, so that \eq{b-recursion} defines $\bvec_\nvec$.
In fact $\bvec_\nvec$ is a convolution sum of $\{\cvec_\mvec\}$ for
those $\mvec \prec \nvec$ where we define the partial order $\prec$ on
integer vectors by $\mvec \prec \nvec$ if $m_i \le n_i$, and $\mvec \ne
\nvec$.
A simple derivative identity allows us to find the $\cvec_\mvec$
coefficients. \begin{equation}
\frac{d}{du_j}e^{g_i(\uvec)}=[\frac{d}{du_j}g_i(\uvec)]e^{g_i(\uvec)},
\end{equation} which upon substitution of
\eqs{taylor}-(\ref{exponential}) yields \begin{equation}
\label{c-recursion}
n_jc_\nvec^{(i)}=\sum_{\mvec \ne{(0,0)}}^\nvec m_j
b_\mvec^{(i)}
c_{\nvec-\mvec}^{(i)} \;.
\end{equation} Note that \eq{c-recursion} allows the two forms, j=1 or
2, for $\nvec$ off the axis (these are equivalent), but for $\nvec$ on
the axis, only one is valid because of a required division by a zero
value of $n_j$.
Examining \eq{c-recursion} reveals that $\cvec_\nvec$ is a function of
strictly previous $\cvec_\mvec$, but up to current $\bvec_\nvec$,
therefore the process must be started by generating $\bvec_\nvec$.
Since the choice of initial phase $\tvec$ is arbitrary, we can set
\begin{equation}\label{b0} \bvec_{\bf 0} = {\bf 0} \;. \end{equation}
Examination of the mapping \eq{eq:16} yields in addition
\begin{equation} \label{boundaries}
b^{(2)}_{(n_1,0)}= 0 \;; \ \ b^{(1)}_{(0,n_2)} = 0 \;.
\end{equation} Similarly, \eq{b0} and \eq{exponential} imply that
$\cvec_{\bf 0} = {\bf 1}$, and \eq{c-recursion} yields \begin{equation}
c^{(2)}_{(n_1,0)}= 0 \; ; \ \ c^{(1)}_{(0,n_2)}= 0 \; .
\end{equation} Finally, the recursion (\ref{b-recursion}) implies that
the values $b^{(1)}_{(n_1,0)}$ and $b^{(2)}_{(0,n_2)}$ are identical to
those for the semi-standard map with frequencies $\omega_1$ and
$\omega_2$, respectively.
This completes the recursion algorithm which allows $\bvec_\nvec$ to be
built as an explicit function of previous $\bvec_\nvec$ and
$\cvec_\nvec$ coefficients. Note that if $k>0$ then $\bvec_\nvec$ is
positive and real, a big advantage in their computation. Since
equation \eq{taylor} actually represents two series, one in each
component of the vector ${\bf g}$, the domain of convergence of ${\bf
g}(\uvec)$ is the intersection of the domains of convergence of each
component's series. \section{The Domain of Convergence}
\label{sec:holomorphy}
In this section, we review some relevant results on the domain of
convergence for power series in several complex variables. Let $\zvec=
(z_1,...z_d) \in \Complex^d$, and for $\mvec \in \Natural^d$, define
${\bf z}^m = z_1^{m_1}z_2^{m_2}....z_d^{m_d} \in \Complex$. We consider
a power series, \begin{equation}
S=\sum_{\mvec \in \Natural^d}^{} {b}_{\mvec}\zvec^\mvec \;,
\end{equation} similar to the series obtained in the previous section.
We denote the radii by $r_j = |z_j|$. The projection onto the radius
space is denoted $\Pi: \Pi(\zvec) = (r_1, r_2,....r_d)$. The subset
$\Complex^{d*} = \{\zvec : z_j \ne 0\}$ excludes points for which any
component of $\zvec$ is zero.
Several types of subsets of $\Complex^d$ are of interest. The {\it
domain of convergence} of a series is the interior of the set of points
for which it converges absolutely. A {\it polydisk} is the appropriate
generalization of a disk: $P(a) = \{\zvec : |z_j| < |a_j|,\; j =
1,...,d\}$. A {\it Reinhardt domain} is a domain $R$ such that $R =
\Pi^{-1}(\Pi(R))$; that is, if it contains a point with radii $r_j$,
then it must contain every point with those same radii, regardless of
phases. Reinhardt domains are conveniently pictured in the radius space
$\Pi(\Complex^d) = \Real^d$. A Reinhardt domain is {\it complete} if
for every $\zvec \in R$, the polydisk $P(\zvec) \subset R$; thus a
complete domain contains all points with smaller radii. Finally a
domain $D$ is {\it log-convex} if the set \begin{equation}
\log(\Pi(D)) \equiv \{(\log(r_1), \log(r_2),....\log(r_d)): z
\in
\Complex^{d*} \cap D\}
\end{equation} is a convex subset of $\Real^d$.
We will use the following theorem \cite{Range,Kaup}: \begin{theorem}
\label{th:convergence}
If $S$ converges for all orderings of its terms at a point
$\zvec$ then it converges absolutely to a holomorphic function.
The domain of convergence, $D$, of $S$ is the interior of the
set for which $|b_\mvec \zvec^\mvec|$ is bounded. Furthermore
$D$ is a log-convex, complete Reinhardt domain. Conversely, if
$|b_\mvec \zvec^\mvec|$ is unbounded then there is an ordering
of the terms in $S$ for which it diverges. \end{theorem}
The proof of this theorem is straightforward. Its most unusual aspect
is that the domain of convergence is log-convex, which we will discuss
in more detail. Suppose $\zvec, \xvec \in \Complex^{d*} \cap D$. Then
for $\alpha + \beta = 1$ let $\uvec$ be any point in $\Complex^{d*}$
such that \begin{equation}
\Pi(\uvec) = (r_1^\alpha s_1^\beta, ...,r_d^\alpha s_d^\beta)
\;, \end{equation} where $r_j$ and $s_j$ are the radii of
$\zvec$ and $\xvec$, respectively. Then, since $S$ converges at both
$\zvec$ and $\xvec$, $B = \sup (|b_\mvec \zvec^\mvec|, |b_\mvec
\xvec^\mvec|)$ exists, and \begin{equation}
|b_\mvec \uvec ^\mvec| = |b_\mvec| \prod_{i=1}^{d}
r_i^{m_i \alpha} s_i^{m_i \beta} \le
B^{\alpha+\beta} = B
\end{equation} is bounded as well. Thus $S$ converges at $\uvec$. Now
since \begin{equation}
\log(\Pi(\uvec)) = \alpha \log(\Pi(\zvec)) + \beta
\log(\Pi(\xvec)) \;, \end{equation} we have shown that $D$ is
log-convex.
The application of Theorem \ref{th:convergence} to our system is
straightforward since the series \eq{taylor} has the desired form; it
yields the interesting result \begin{corollary}\label{convergence2}
For fixed $k$ defined by \eq{k-define}, an analytic invariant
torus with Diophantine frequency $\wvec$ of the \semiF map
exists in a parameter domain in $(a_1,a_2)$ which is complete
and log-convex. \end{corollary} In particular for fixed $k$,
completeness implies that the domain of convergence is simply
connected, and its boundary projected onto the radius space can be
expressed as a graph of a function $r_1(r_2)$ or $r_2(r_1)$.
As we will see in the next section, the calculation of these domains is
possible with reasonable accuracy using the requirement that the terms
in the series must be bounded. \section{Numerical Results}
\label{sec:numerics}
Determination of the sequence of $\{\bvec_\mvec\}$ of Fourier
coefficients of $\chivec(\tvec)$ using the recursion algorithm of
\S{sec:recursion} is straightforward, since they are real and positive
for $k \geq 0$. The next issue is to numerically find the domain of
absolute convergence, which from \S{sec:holomorphy}, is the set of
$\uvec \in \Complex^2$ for which $|\bvec_\mvec \uvec^\mvec|$ is
bounded. We begin by noting that the series (\ref{taylor})
\begin{equation} \label{g-series}
\gvec(\uvec)=\sum_{n=0}^{\infty} \sum_{m=0}^{\infty}
\bvec_{m,n} u_1^m u_2^n
\end{equation} converges absolutely in the polydisk $P(\uvec)$ if the
reordered series \begin{equation} \label{eq:slopesum}
\sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \bvec_{m,n} r_{1}^{m}
r_{2}^{n}
=\sum_{n=0}^{\infty} r_1^n \Bvec_n (s) \;. \end{equation} converges.
Here we define the slope $s \equiv r_2/r_1$, and the diagonal
coefficient, \begin{equation} \label{diagonal}
\Bvec_n(s) \equiv \sum_{m=0}^{n} \bvec_{n-m,m}s^m
\end{equation} which, most importantly, is expressed as a finite sum.
It follows that the radius of convergence for the $i^{th}$ component is
\begin{equation} \label{log-radius}
\log(r_1^{(i)}(s)) = - \lim_{n \rightarrow \infty} \frac{ \log{
B_n^{(i)}(s)}}{n} \end{equation} for each fixed $s$. Since the domain
of convergence is complete according to Theorem \ref{th:convergence},
$r_1(s)$ is a single valued function.
To avoid numerical overflow for large $s^m$, we use \eq{diagonal} when
$s \leq 1$, and use a corresponding formula with the slope defined as
$r_1/r_2$ otherwise. Furthermore, we can take advantage of definition
(\ref{diagonal}) by computing the coefficients $\bvec_\mvec$ in a
triangular domain $m+n \le N$. This saves about a factor of ten in
computing time over using the square domain. Computer memory
constraints lead us to chose $N=255$ as our matrix dimension.
The efficacy of this method depends upon estimating \eq{log-radius},
the asymptotic growth rate of the Fourier coefficients. We first
consider $\wvec=(\gamma,\sigma)$ where the components were defined by
\eqs{golden} and (\ref{silver}). Figure \figcontour is a logarithmic
contour plot of $b_\mvec^{(1)}$. It can be seen that $\bvec_\mvec$
grows rapidly as $\mvec$ grows, indeed the maximal values of $b_\mvec$
in the figure are ${\cal O}(10^{150})$. This is a result of the
recursion algorithm, which shows that $\bvec_\mvec$ is a combination of
all the previous $\cvec_\mvec$. When $k \ge 0$, the $\cvec_\mvec$
coefficients are positive and $\cvec_\mvec > \cvec_\nvec$ for $\mvec
\succ \nvec$. Whenever there is a near commensurability, $D_\mvec$ is
small, and $\bvec_\mvec$ takes a sudden jump. This can be seen in the
contour plot as a serrating of the contour lines. The neighboring
coefficients for greater $\mvec$ are influenced by this jump, but the
recursion algorithm serves to spread and dissipate the extra height.
In other words, the coupling serves to dampen the commensurabilities.
This partly accounts for the stepping up nature of the contour plot.
Fortunately, the limit (\ref{log-radius}) is not as difficult to
evaluate for this problem as it could be in general. It turns out that
$\log{B_n}$ behaves quite linearly as a function of n; this can be seen
clearly in fig. \figslopes. The small spikes visible on a given ``line"
of $(\log [B_n(s)],n)$ are due to near commensurabilities.
By contrast, the Fourier coefficients for the semi-standard map,
$b_n$, depend only on the small denominator and the single previous
coefficient $c_{n-1}$, (which is in turn implicitly a function of the
coefficients $b_1 ,..., b_{n-1}$). Resonances are extremely important,
and primary, secondary, and even tertiary prominences can be observed,
so that the Fourier coefficients have an extremely spiked profile. In
the four-dimensional case the coupling between the frequencies $\wvec$
appears to play the dominant role. Resonances gain and lose prominence
in a delicate balancing of the coupling between frequencies, which can
be seen as shadows of vertical lines in \figslopes.
To determine the radius in \eq{log-radius} we performed a least squares
fit of variable data sets. The top and bottom ends of the fit were
allowed to float by $n=10$ points each, and the fit with the lowest
residual was automatically chosen. This eliminates the problem of a
given fit falling just above or below a resonance spike. RMS errors in
the slope fit are typically $\sigma_c =0.003 \pm 0.001$, which leads us
to expect at least 2 decimal accuracy in the $r_i$ values.
Using the three frequency pairs, $\wvec=(\gamma,\sigma)$,
$(\gamma,\zeta)$, and $(\tau,\tau^2)$, we generate the respective
$(r_1,r_2) = (a_1,a_2)$ curves for various coupling constants k, and
for each of the $B_n^{(i)}$ components. These domains of convergence
are displayed in figs. \figkplota-\figkplotc. $B_n^{(1)}$ is
represented as solid curves, and $B_n^{(2)}$ as dashed curves.
Figure \figlogplot displays the $(a_1,a_2)$ curves for
$\wvec=(\gamma,\sigma)$ on a log- log scale. This example shows that D
is log-convex in accord with Theorem \ref{th:convergence}. The sharp
bends seen in some of the curves are due to the regular spacing of
angles on a grid, which the log scale makes especially prevalent near
the axes.
Here we will discuss the behavior of the curves for the first
component, $B_n^{(1)}$ (solid curves). When $a_2\rightarrow 0$, $r_1$
must approach $a^{ss}(\omega_1)$ since the map \eq{eq:16} becomes
uncoupled in this limit, and $B_n^{(1)}$ becomes the coefficients of
the semi-standard map with frequency $\omega_1$. We call the $r_1$ axis
the ``dominant axis" for $B_n^{(1)}$; similarly, the $r_2$ axis will be
the dominant axis for $B_n^{(2)}$. This behavior can be seen in figs.
\figkplota-\figkplotc as all the various curves intersect the dominant
axis at $a^{ss}(\omega_1)$. For reference, Table 1 gives the critical
values of the semi-standard map for the various frequencies. Note that
the curves in fig. \figkplota and \figkplotc actually overestimate the
correct values on the axis; for example in fig. \figkplota, the
intersection with the $r_2$ axis occurs near 0.985, while Table 1
implies that the correct value is 0.966. This overestimate is due to
the fact that we compute the coefficients only out to the $255^{th}$
Fourier coefficient, and that near the axes the spikes in the $B_n$
curves become more prominent (see further discussion of this below).
For the semi-standard mapping more sophisticated fitting techniques
(e.g. \cite{Percival}) are required for an accurate evaluation of the
critical function. For our mapping we believe that, away from the axes,
the radius curves are actually more accurate than this indicates. In
fig. \figkplotb, the intersection with the $r_1$ axis appears to be
much lower than the value $r_1 = 0.979$ given in Table 1; however these
curves actually rise rapidly to the correct (actually overestimated)
value as $r_2 \rightarrow 0$. It is interesting that in this case even
though the values on axis are quite different, the convergence boundary
has adjusted itself to be nearly square for small $k$. Finally, this
rapid rise---approaching $a^{ss}(\omega_1)$ at a sharp angle, does not
violate log-convexity, as required by Theorem \ref{th:convergence}.
\begin{table} \label{table} \[\begin{array}{cl}\omega &{\rm a}^{\rm
ss}\\ \gamma &\rm 0.979661\\ \sigma &\rm 0.966165\\ \zeta &\rm
0.833726\\ \tau &\rm 0.657\\ {\tau }^{2}&0.660\\ \gamma \rm +\sigma
&\rm 0.09\\ \gamma \rm +\zeta &\rm 0.66\\ \tau \rm +{\tau
}^{2}&0.33\end{array}\]
\caption{Critical values for the semi-standard map.}
\end{table}
The figures also show that the solid curves limit to $a^{ss}(\omega_2)$
on the $r_2$ axis, which we call the ``subdominant" axis for
$B_n^{(1)}$. This phenomena requires some explanation. When $\epsilon
\equiv 0$, the boundary of domain of convergence for $B_n^{(1)}$ is
$r_1^{(1)} = a^{ss}(\omega_1)$, independent of $r_2$; the numerical
results for nonzero $\epsilon$, however, imply that $r_2$ limits to
$a^{ss}(\omega_2)$ on the $r_2$ axis. This also occurs for the domain
of convergence of the second component $B_n^{(2)}$: $r_1 \rightarrow
a^{ss}(\omega_1)$ as $r_2 \rightarrow 0$. To explain this phenomena,
consider for example the small slope limit of $B_n^{(2)}(s)$.
\Eq{diagonal} implies that \begin{equation} {B}_{n}^{(2)}(s) =
{b}_{n,1}^{(2)}\ s+\ {\cal O}({s}^{2}) \end{equation} where the $s^0$
term vanishes according to \eq{boundaries}. Using the recursion
relation (\ref{b-recursion}) implies \begin{equation} \label{small-s}
B_n^{(2)}(s) \simeq sk{\frac{D_{(n,0)}}{D_{(n,1)}}} b _{(n,0)}^{(1)}
\end{equation} Thus using \eq{log-radius} the radius of convergence is
\begin{equation} \log(r_1^{(2)}) \simeq \log
\left[a^{ss}(\omega_1)\right]\ -\ \lim\limits_{n\ \rightarrow \ \infty}
{\frac{1}{n}}\log\left({\frac{D_{(n,0)}}{D_{(n,1)}}}\right)
\end{equation} The last limit in fact is zero, since by the Diophantine
condition \eq{Diophantine}, the ratio of the denominators is bounded by
${\cal O}(n^4)$. A similar result holds for the first component of the
mapping along the $r_2$ axis, so we have shown that \begin{equation}
\label{subdom} \lim\limits_{s\ \rightarrow \rm
\ 0}^{}\ {r}_{1}^{(2)}(s)\ =\ {a}^{ss}(\omega _1),\ \
\lim\limits_{s\ \rightarrow \rm \ \infty
}^{}\ {r}_{2}^{(1)}(s)\ =\ {a}^{ss}(\omega _2) \end{equation}
Furthermore \eq{subdom}, together with completeness implies that the
the domain of convergence is bounded by the rectangle \begin{equation}
a_1 \leq a^{ss}(\omega_1),\ \ \ a_2 \leq a^{ss}(\omega_2)
\end{equation} In fact the figures show that as $k \rightarrow 0$ the
domain of convergence approaches this rectangle. Our interpretation of
this is that for small but nonzero k, the singularity corresponding to
$r_2=a^{ss}(\omega_2)$ is still present, though weakened (the
``residue" of this singularity, limits to zero as $k \rightarrow 0$,
but it is still present for any nonzero k). This causes a difficulty
with our numerical scheme for finding $r_1(r_2)$ when $k$ is small; we
we discuss this further below.
Figure \figbns displays the coefficients $B_n^{(1)}$ and $B_n^{(2)}$
for $s= 10^{- 25}$ and $\wvec=(\gamma,\sigma)$. In the limit of small
slope $B_n^{(1)} \approx b_{n,0}^{(1)}$ which are the Fourier
coefficients for the semi-standard map \cite{Greene-Percival}. Thus the
upper plot is indistinguishable from that for the semi-standard map.
The lower half of fig. \figbns shows $B_n^{(2)}$ for small slope. The
profile exhibits spikes and valleys corresponding to a complicated
coupling between $\sigma$ resonances and the still important $\gamma$
resonances, as shown in \eq{small-s}. Furthermore, \eq{small-s} implies
that the profile approaches a limiting form as $s \rightarrow 0$, even
though the magnitude of $B_n^{(2)}$ approaches zero. Likewise,
$B_n^{(2)}$ near the $r_2$, axis yields the semi-standard coefficients
for $\omega_2 = \sigma$, while $B_n^{(1)}$ goes to zero, while
similarly converging to a fixed profile.
As the domain of convergence plots show, the rectangular domain for
small $k$ contains the domain for any finite $k$. This follows from the
completeness of the domain of convergence, and the fact that the curves
limit to the semi-standard values on the axes. This fact can be used as
an upper-bound when discussing the question of which torus is ``last."
We also computed $r_1(r_2)$ curves for negative values of k. By the
same argument as above, the negative k curves intersect the axis at
$a^{ss}(\omega_1)$ and $a^{ss}(\omega_2)$. Otherwise the curves are
qualitatively similar to those shown in figs. \figkplota-\figkplotc,
so we omit the plots. Since the domain of convergence depends only on
$k$, these curves provide the boundary of existence in four of the
octants in $(a_1,a_2, \epsilon)$ space, the other four being determined
by the positive $k$ results.
As $k$ increases all of the boundaries in figs. \figkplota-\figkplotc
become hyperbolic in shape. This can be seen most clearly in fig.
\figkplotb, for $\wvec=(\gamma,\zeta)$. The large $k$ limit
corresponds to \begin{equation} \epsilon \gg (a_1 , a_2).
\end{equation} Taking this to the extreme, we set $a_1=a_2=0$, then
\eqs{semiF}-(\ref{force}) have the form \begin{equation} \label{eq:hyp}
\delta^2 \xvec=i\epsilon \left( \begin{array}{c}
e^{ix_1+ix_2}\\ e^{ix_1+ix_2} \end{array} \right).
\end{equation} Defining the new variables \begin{equation}
\xi_1=x_1+x_2 \end{equation} \begin{equation} \xi_2=x_1-x_1,
\end{equation} and adding and subtracting the components of \eq{eq:hyp}
yields a new map. \begin{equation} \label{eq:hypmap}
\begin{array}{l} \delta^2 \xi_1= 2i \epsilon e^{i \xi_1}\\
\delta^2 \xi_2=0. \end{array} \end{equation}
Thus, there exists an invariant torus for $(\xi_1,\xi_2)$ up to some
critical value \begin{equation} \label{eq:maxeps} 2
\epsilon=a^{ss}(\omega_1+\omega_2). \end{equation} Now \eq{eq:hyp} is
approximately valid for small $a_1$ and $a_2$, so we expect that as $k
\rightarrow \infty$, using $\epsilon = k a_1 a_2$, the fixed $k$
boundary will limit to \begin{equation} \label{hyperbola} r_1r_2 =
\frac{a^{ss}(\omega_1+\omega_2)}{2k} \end{equation} which defines a
hyperbola.
Thus we have three analytic bounds on the domain of existence of a
torus: \begin{equation} \label{bounds} a_1 < a^{ss}(\omega_1), \ \ \
a_2 < a^{ss}(\omega_2), \ \ \ \epsilon < 0.5 a^{ss}(\omega_1+\omega_2)
~~, \end{equation} though the last equation is not rigorously derived.
As a confirmation, Table 1 shows that $a^{ss}(\gamma+\sigma)$ is much
smaller than $a^{ss}(\gamma+\zeta)$ and $a^{ss}(\tau+\tau^2)$. Thus,
\eq{hyperbola} predicts that the curves for $\wvec=(\gamma,\sigma)$
should become hyperbolae more quickly than for other $\wvec$ curves, as
we do in fact observe.
As mentioned earlier, the scheme (\ref{eq:slopesum})-(\ref{log-radius})
for finding $r_1(s)$ has numerical problems when $k \ll 1$. For such
small k, the singularity on one axis is dominant over the singularity
on the other axis. To illustrate the problem, consider a simple
example which has a similar imbalance in the prominence of its
singularities. Let \begin{equation} \label{exampleseries}
S(r_1,r_2)=\frac{\alpha}{\alpha-r_1} + \frac{\delta \beta}{\beta-r_2} =
\sum_{m,n} b_{m,n}r_1^mr_2^n~~~. \end{equation} Here small values of
$\delta$ simulate small values of $k$; however, for any nonzero
$\delta$, the domain of convergence of this series is the rectangle
$\{(r_1,r_2): r_1 < \alpha, r_2 < \beta\}$.
We examine the behavior of equations
(\ref{eq:slopesum})-(\ref{log-radius}) when applied to
\eq{exampleseries} by a perturbation analysis near $s=0$. For a finite
$n$, the algorithm gives an error in $r_1$ of
\begin{equation} \label{error} \Delta r_1 \ \sim \ -\ {\frac{\delta
\alpha }{\rm n}}\ {\left({{\frac{\alpha s}{\beta }}}\right)}^{n}~~~.
\end{equation} Thus the method works well provided $s < \beta /
\alpha$, but fails drastically for larger $s$. In our computations, the
slope is never larger than one; we switch to the inverse of the slope
when $s=1$. Thus, supposing $\beta < \alpha$ the method fails in a cone
$\beta / \alpha < s < 1$. So for the Froeshl\'e mapping, we also
expect that slopes within a similar cone will give bad results if k is
too small. That this is true can be seen as a slight loss of convexity
for the smallest values of k along the subdominant axis in figs.
\figkplota- \figkplotc. In practice we are unable to lower $k$ below
$10^{-5}$ in the computations.
Finally, our $r_1(r_2)$ data can be displayed in terms of the coupling
parameter $\epsilon$, instead of k. Figures \figkplotb and \figkplotc
are converted via \eq{k-define} to the three dimensional graphs seen in
figs. \figtda and \figtdb. Here we see in a new way the importance of
the sum frequency $(\omega_1+\omega_2)$ through \eq{bounds}. Numerical
overflow for large k prevents us from calculating the curves for
$\epsilon$ too close to its maximum value.
In many ways, it is these three dimensional plots which are most useful
when deciding a partial order to determine the ``last invariant torus."
One concept of ordering of the domains of convergence is to choose a
directed curve in $(a_1,a_2, \epsilon)$ beginning at the origin. One
could linearly order the domains of convergence in terms of the order
of intersection of the domain boundaries with this curve. This
motivates the following local definition of order.
\bf{Curve based Order}: \it{An $\wvec$ torus persists longer than a
$\muvec$ torus along a curve $\xivec(t)$ for which $\xivec(0) =
\bf{0}$, if $\xivec(t)$ intersects the boundary of the domain of
convergence of the $\muvec$ torus first.}\rm
The simplest example of a parameterized family is a line emanating from
the origin. Another example is a parabolic ray $a_1 = t, a_2 = st,
\epsilon = k s t^2$ for fixed $s$ and $k$. Figures
\figkplota-\figkplotc order domains in this sense.
In general, one wants to do more than compare two surfaces using a
single point from each surface, which is all a curve based order
allows. In some sense, one may want to incorporate the information of
the entire surface in a comparison. This motivates the definitions of
the following global comparisons.
\bf{Metric based Order}: \it{For a given metric, an $\wvec$-torus
persists longer than a $\muvec$-torus if the boundary of the domain for
$\wvec$ has a point farther from the origin than that for $\muvec$.}\rm
This definition for ordering is limited in that it requires the choice
of a metric.
If one surface is completely contained inside another, then that torus
is more persistent than the other according to any definition, since
containment is a topological notion. Thus we define the \it{partial
ordering}
\bf{Topological Order}: \it{An $\wvec$ torus persists longer than a
$\muvec$ torus if the domain for $\wvec$ contains that of $\muvec$.}\rm
Of course, the surfaces for two different frequencies will intersect in
general, and then the topological ordering does not apply. In our
examples, the surface for $(\tau,\tau^2)$ is completely contained
inside that of $(\gamma,\zeta)$, and therefore the $(\gamma,\zeta)$
torus is more persistent. The complete containment of the
$(\gamma,\zeta)$ surface is partly due to the fact that each of
$(a^{ss}(\gamma),a^{ss}(\zeta),a^{ss}(\gamma+\zeta))$ are greater than
their counterparts
$(a^{ss}(\tau),a^{ss}(\tau^2),a^{ss}(\tau+\tau^2))$. On the other hand
in order to compare the $(\gamma,\sigma)$ and $(\gamma,\zeta)$ tori,
note that though $a^{ss}(\sigma)) > a^{ss}(\zeta))$,
$a^{ss}(\gamma+\sigma)< a^{ss}(\gamma+\zeta)$. Thus the surfaces must
intersect, and therefore there can only be parameterized comparisons.
\section{Conclusions} \label{sec:conclusions}
We have determined the domain of existence of invariant two-tori
analytically conjugate to a rotation for the \semiF mapping by
expanding the conjugacy function in a Fourier series in the angle
variables. The \semiF mapping has the advantage that two of the
parameters can be eliminated in the Fourier series, so that the
boundary of existence of the tori in all three parameters can be
obtained with a single parameter sweep. We have studied the boundary of
the domain for several frequency vectors, all of which are elements of
a cubic algebraic field, and therefore satisfy Diophantine conditions.
The boundary of these domains appears to be smooth; rather
surprisingly, it appears smooth even when the parameters have opposite
signs (i.e. negative values of $k$). We have shown that when projected
on the parameters $(a_1,a_2)$ for fixed $k = \epsilon/a_1 a_2$, the
boundary is log-convex and complete, and that as $k \rightarrow 0$ the
domain limits to the rectangle corresponding to the domain for the
uncoupled mappings. Furthermore, numerical results imply that the
domain is bounded by the critical function for the sum frequency, as
shown by \eq{eq:maxeps}.
The methods and theorems of this paper are not restricted to the
four-dimensional version of \eq{semiF}. They also apply to the 2d
dimensional complex \semiF map, providing only that each occurrence of
$x^{(j)}$ in an exponential, $\exp{(im x^{(j)})}$, in the force has the
same sign. The main bottleneck is computing the Fourier coefficients
recursively which involves a (d-1) degree iterated convolution sum,
where d is the dimension. Computing the $m^d$ coefficients would take
${\cal O}(m^{2d})$ steps, making computer time a major problem in
practice. In the same vein, more complicated forcing terms in
\eq{force} could also be considered, but similar time constraints may
be a problem.
There are a number of open questions left by our study.
1) When the Fourier series does not converge, does there exist an
invariant Cantor set for the mapping (a cantorus)? Results for twist
mappings near the anti-integrable limit show the existence of cantori
for all frequencies \cite{MacKay-Meiss}. What is the nature of the
invariant set when the Fourier coefficients for $x^{(2)}$ converge,
but those for $x^{(1)}$ do not, as seen especially in Fig. \figkplotc?
One is tempted to think it is a Cantor set of circles.
3) Are all invariant tori for the \semiF mapping analytically conjugate
to a rotation? Perhaps all tori with Diophantine frequency vectors?
4) Is there an extension of the converse KAM theory of
\cite{MacKay-Meiss-Stark} to complex mappings?
2) Is there an extension of some of the results of Theorem
\ref{th:convergence} to real valued four-dimensional mappings of some
class? It is possible that such a map may also have a log-convex
domain in the proper coordinates.
5) Can one use similar techniques to study the existence of invariant
circles for a four-dimensional mapping? In \cite{MacKay-Meiss-Stark} it
was suggested that circles may last longer than any tori.
6) Which class of frequency vectors correspond to the most persistent
invariant tori? In this paper we compare several likely candidates,
but do not present evidence that there are not more persistent tori. In
searching for a particularly persistent torus, a first step might be to
maximize the values of $a^{ss}(\omega_1), a^{ss}(\omega_2)$, and
$a^{ss}(\omega_1+\omega_2)$. Which class of frequency vectors does
this? Of course since denominators containing all $\mvec \cdot \wvec$
occur, the most persistent class of frequencies may be that with
maximal Diophantine constant $C$ in \eq{Diophantine}. Since
incommensurate algebraic frequency vectors form a field, any elements
of such a field will have the same $C$. Moreover, since a degree three
algebraic field has the minimal exponent $\mu$ in \eq{Diophantine}, it
seems reasonable that it is such a field which will be most
persistent. Of course the definition of persistence will depend on the
choice of a partial ordering, and even then it is not clear how
dependent upon the specific model the results would be.
\section*{Acknowledgements} Partial support for this research was
obtained from the NSF through grant DMS-9001103 and by the ONR through
grant N00014-91-4037. We would aslo like to thank John Greene for
helpful discussions regarding the convergence of the semi-standard map
critical function.
For figures, send to
Erik Bollt
Program in Applied Mathematics
University of Colorado, Boulder 80309-0526
(303)492-4668 -4066FAX
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