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\TITLE
Numerical study of invariant sets of a volume-preserving map.
\ENDTITLE
\AUTHOR
Stathis Tompaidis
\footnote{$^1$}{Partial support from NSERC grant OGP-0121848}
\footnote{$^2$}{e-mail: {\tt stathis@lie.univ-rennes1.fr}}
%\footnote{$^3$}{Current address: IRMAR, Univ. de Rennes I, 35042 Rennes Cedex,
%France}
\FROM
%Department of Mathematics
%University of Toronto
%100 St. George Street
%Toronto, ON M5S 1A1
%Canada
IRMAR
Universit\'e de Rennes I
35042 Rennes Cedex
France
\ENDTITLE
\ABSTRACT
We study the behavior of invariant sets of a volume-preserving map,
that is a quasi-periodic perturbation of a symplectic map,
using approximation by periodic orbits. We present numerical
results for analyticity domains of invariant surfaces, behavior after
breakdown and a critical function describing breakdown of invariant surfaces
as a function of their rotation vectors. We discuss implications
of our results to
the existence of a renormalization group operator describing breakdown
of invariant surfaces.
\ENDABSTRACT
\runninghead{Rotating Standard Map. A Numerical Study}{S. Tompaidis}
\SECTION Introduction
Existence and persistence of invariant sets of dynamical
systems on which motion is, up to a smooth change of
variables, quasi-periodic, is a problem that has attracted considerable
attention for at least 100 years (see \cite{Po93}). Such sets have many
important, practical, applications as landmarks that organize the
long-term behavior. For Hamiltonian systems that are close to integrable,
the Kolmogorov-Arnol'd-Moser theorem guarantees that a majority
of such invariant sets persists for small enough perturbation (see
\cite{Kol54, Ar63, Mo62} or \cite{Ll93} for a self-contained introduction
to KAM theory and a proof). Results similar to the KAM theorem
can also be shown in the case of quasi-periodic perturbations
of symplectic maps.
Unfortunately, the analytical estimates are very conservative compared to
numerical indications, especially for high-dimensional systems.
In this paper we investigate numerically the domains of
existence of two-dimensional tori in a particular
two-parameter family of volume-preserving
maps and the behavior that occurs at breakdown. The maps we study are
quasi-periodic perturbations of a family of symplectic maps.
Our numerical algorithms are based on analytical results described in
\cite{T95, FL92-1}. The main idea is to use another landmark
of long-term behavior, periodic orbits, to determine the existence
and breakdown of tori.
For the case of two-dimensional systems, it was originally
observed in \cite{Gr79} that existence of invariant circles
has a strong influence on periodic orbits close to the
circle. In \cite{T95} it was proven that in symplectic maps of
any dimension, as well as in quasi-periodic perturbations
of them, existence of an invariant torus implies that
the behavior of the map in a neighborhood of the torus is
close to that of an integrable map. We use this result as an
indication of breakdown.
The system we will study is a three-dimensional model of a family of
volume-preserving maps. Motion in one of the coordinates
is rigid rotation with rotation number given by an appropriate diophantine
number. The other two coordinates
of the map are described by a perturbation of the
standard map. Properties of the map and existence of tori
have also been investigated in \cite{ACS91}. Our rationale for
studying this map (henceforth called the {\it
rotating standard map}) is similar to that used in experimental
physics, where one carefully prepares a sample in order to
observe certain phenomena. In our case, the rotating standard map
serves as a paradigm for phenomena that appear in higher-dimensional
maps.
An important problem is to describe behavior at breakdown of invariant
tori. In the two-dimensional case careful numerical experiments and
analytical arguments suggest that the breakdown of invariant
circles can be described by a fixed point, with a co-dimension one
stable manifold, of a renormalization-group operator (see \cite{McK82}).
We investigate whether such an approach generalizes in the case of the
rotating standard map. To consider such a generalization we use an
algorithm (called the Jacobi-Perron algorithm) to
approximate irrational points by rational ones. The algorithm reduces to
the continued-fraction method for the case of invariant circles.
Although it has not been widely used in the dynamical systems
literature, it has important measure-theoretic and convergence properties
that are useful in introducing a renormalization-group operator.
In \cite{Kos91} such an operator was constructed and it was shown
that convergence, under repeated application of the operator,
to a (trivial) fixed point implies existence of an invariant torus.
We investigate whether breakdown can be understood in terms of
a different fixed point.
The paper is organized as follows: in section 2 we present the
rotating standard map. In section 3 we show existence of periodic
orbits with any rotation vector and discuss efficient ways to
compute them. In section 4 we present the Jacobi-Perron algorithm and
describe its connection to an extension of modular transformations.
In section 5 we compute
domains of existence for a particular invariant torus and discuss whether
breakdown can be understood in terms of a fixed point of a
renormalization-group operator in certain regions of parameter space.
Finally, in section 6 we generalize the notion of critical function.
%and make a connection between the
%Jacobi-Perron algorithm and an extension of modular transformations.
\SECTION Notation and preliminaries
We will study the two-parameter family of three-dimensional volume-preserving
maps,
%hence\-forth called
the {\it rotating standard map}
$$
F_{\epsilon, k} : \real \times \torus \times \torus \to
\real \times \torus \times \torus$$
$$ \pmatrix{A_{n+1}\cr \theta_{n+1}\cr \phi_{n+1}\cr } =
F_{\epsilon, k} \pmatrix{A_n \cr \theta_n \cr \phi_n \cr} =
\pmatrix{
A_n - \frac{1}{2\pi}(k+\frac{\epsilon}{2\pi}\cos(2\pi\phi_n))
\sin(2\pi\theta_n) \cr
\theta_n + A_{n+1} \quad \mod 1\cr
\phi_n + \omega_2 \quad \mod 1 \cr}
\EQ(rotstd)$$
The parameters $\epsilon, k$ will be considered complex.
The value of the variable $\omega_2$ determines
if the standard map is perturbed periodically (for $\omega_2$ rational)
or quasi-periodically (for $\omega_2$ irrational).
We say that $\bf x$ is a periodic orbit of type $(P/N)$,
$P\in \integer^2$, $N \in \natural^*(\equiv \natural - \{0\})$, if
$F_{\epsilon, k}^N ({\bf x}) = {\bf x}$ and
$\tilde F_{\epsilon, k}^N ({\bf \tilde x}) = {\bf \tilde x} + (P, 0)$,
where
$\tilde F_{\epsilon,k}, \bf \tilde x$ a (fixed) lift of $F_{\epsilon, k}, \bf x$
to the universal
cover of $\real \times \torus \times \torus$. We will
call $N$ the period of the orbit.
For $N$-vectors we will use the norm
$\|{\bf v}\|_N = \sum_{i=1}^N |v_i| $.
We define the {\it rotation vector} of an orbit of $\tilde F_{\epsilon, k}$
as the $2$-dimensional vector
$$ \omega = \lim_{i\to \infty}
\frac{\pi_1(\tilde F_{\epsilon, k}^i (A,\theta,\phi)) - (\theta,\phi)}{i}$$
if the limit exists, where $\pi_1$ the projection on the angle coordinates
$\pi_1 (A, \theta, \phi) = (\theta, \phi)$.
For a periodic orbit of type $(P/N)$
the rotation vector is $\omega = P/N$.
We are interested in the behavior of invariant sets with diophantine
rotation vector. A diophantine rotation vector of type $(K, \tau)$
is a $2$-vector $\omega$, such that
$$ | P\cdot\omega | \ge \frac{K}{\|P\|^\tau}, \quad P\in \integer^2,
P\ne 0, K> 0$$
It is well known (see \cite{Ar88}) that, for some $K_0$ and
fixed $K> K_0, \tau> 1$ the set
of vectors of type $(K, \tau)$ has positive Lebesgue measure in the unit
square.
In \cite{T95} it was shown that existence of invariant sets on
which motion is conjugate to rigid rotation with diophantine rotation
vector has certain implications for the properties of periodic
orbits in the neighborhood of the invariant set. As in \cite{T95}
we define the residue of a periodic orbit with period $N$
$$R({\bf x}) = \frac{1}{6} \left [ 3 - {\rm Tr}(D f^N ({\bf x})) \right ]
\EQ(residue)$$
Considering the lift of map \equ(rotstd) to the universal cover of
$\real \times \torus \times \torus $ we
write \equ(rotstd) in Lagrangian formulation
$$
\eqalign{
\theta_{n+1} - 2 \theta_n + \theta_{n-1} = &
-\frac{1}{2\pi} \left ( k + \frac{\epsilon}{2\pi}
\cos\left ( 2\pi \phi_n \right ) \right )
\sin ( 2\pi \theta_n) \cr
\phi_{n+1} - 2 \phi_n + \phi_{n-1} = & 0 \cr
}
\EQ(lagrotstd)
$$
We will use equations
\equ(lagrotstd) to numerically compute periodic orbits for
$F_{\epsilon, k}$.
\SECTION Existence and computation of periodic orbits
John Mather (see \cite{Ma91}) extended several of the properties of twist
maps of the annulus to finite compositions of twist maps.
Consider a periodic perturbation of the standard map
%(i.e. $\omega_2 = M/N \in \rational$),
and the twist maps of the annulus
$$f_n \pmatrix{p\cr q\cr} = \pmatrix{p -
\frac{1}{2\pi} \left( k + \frac{\epsilon}{2\pi}
\cos(2\pi ( \phi_0 + n \omega_2))\right)
\sin(2\pi q)\cr
q + p -
\frac{1}{2\pi} \left( k + \frac{\epsilon}{2\pi}
\cos(2\pi ( \phi_0 + n \omega_2))\right)
\sin(2\pi q) \cr}$$
with generating functions
$$h_n(q, q') = \frac12 (q-q')^2 - V_n(q)$$
where
$$V_n(q) = \frac{1}{2\pi^2} \left(k + \frac{\epsilon}{2\pi}
\cos(2\pi ( \phi_0 + n \omega_2))\right)
\cos(2\pi q).$$
The (finite) composition
$$f = f_1 \circ \cdots \circ f_N \EQ(fdef)$$
is the rotating standard map with $B_n \equiv \omega_2 = M/N$ and
with generating function $h$
$$ h = h_1 \ast \cdots \ast h_N \EQ(hdef)$$
where
$$ h_1 \ast h_2 (x_1, x_2) = \min_{\xi}
\left( h_1 (x, \xi) + h_2 (x', \xi) \right) $$
The operation $\ast$ (named {\it conjunction} in \cite{Ma91}) was used to show
that many of the results for twist maps are preserved.
In particular, for the rotating standard map
as in \equ(fdef)
(i.e. a finite composition of twist maps), there exist configurations
$\bf x$ for any $\omega \in \real$ such that $\bf x$ is an orbit of
$f$ with rotation number $\omega$ (see proposition 2.4 in \cite{Ma91}).
Moreover, if $\Gamma$ is a curve invariant under $f$, on which motion is
conjugate to rigid rotation with an irrational rotation number,
then $\Gamma$ consists of minimal configurations of $h$
(for a discussion about minimal configurations see \cite{Ma91}. For
a proof see \cite{Ma91} proposition 2.8).
The above theorems guarantee the existence of minimal periodic orbits for
our maps, with any period and any (rational) rotation vector.
To compute such periodic orbits for the map \equ(rotstd)
we use Newton's method in the space of finite length sequences.
Given a periodic orbit with rotation vector $(P_1/N, P_2/N)$ we can
eliminate the dependence on the $\phi$ variable in
\equ(lagrotstd) and transform the equations to
$$\theta_{n+1} - 2\theta_n + \theta_{n-1} =
-\frac{1}{2\pi} \left ( k + \frac{\epsilon}{2\pi}
\cos (2\pi (\phi_0 + \frac{n P_2}{N} )) \right ) \sin(2\pi \theta_n)
\EQ(intermediate)
$$
Equations \equ(intermediate) can be interpreted as the
Euler-Lagrange equations for a certain Lagrangian
(see \cite{Ma91}, \cite{KM89}).
We introduce (formally) the action $W$ for an orbit
$\{\theta_i\}_{i=-\infty}^{\infty}$
$$ W(\dots , y_i, \dots) =
\sum_{n=-\infty}^{\infty} g(y_n, y_{n+1}, n)
$$
where $y$ is the lift of $\theta$ and
$$g(x, y, i) = \frac{1}{2} (x-y)^2 - \frac{1}{4\pi^2} \left (
k + \frac{\epsilon}{2\pi} \cos (2\pi (\phi_0 + \frac{iP_2}{N})) \right )
\cos (2\pi x).$$
Even though $W$ is only formally defined, its gradient is well
defined and a sequence of points is an orbit
of \equ(rotstd) if and only if it is a critical point
of $W$ (see \cite{KM89}, \cite{Gol94}).
$$\frac{\partial}{\partial y_l} W = 0 \iff
\frac{\partial}{\partial y_l} [ g(y_{l-1}, y_l, l-1) + g(y_l, y_{l+1}, l)] =0,
\quad {\rm for\ all\ } l.$$
A minimal configuration for the generating function \equ(hdef)
corresponds to a critical point for
$W$, but not necessarily vice versa, since minimax configurations may exist
for certain values of the parameters.
% This observation can serve as a basis
% for a converse KAM theory, in the spirit of \cite{MP85, St88} to
% determine rigorous upper bounds for the breakdown of invariant curves.
For a periodic orbit of type $(P_1/N, P_2/N)$ we have
$y_{N+1} = y_1 + P_1/N, y_{0} = y_N - P_1/N$.
We redefine the action $W$ as
$$ W_N(y_1, \dots, y_N) = \sum_{n=1}^{N} g(y_n, y_{n+1}, n) $$
which leads to the system
$$\frac{\partial}{\partial y_l} [ g(y_{l-1}, y_l, l-1) + g(y_l, y_{l+1}, l)] =0,
\quad l=1\dots N
\EQ(periodsys)$$
To solve \equ(periodsys),
consider the operator
$${\cal T} : \real^N \to \real^N,\quad {\cal T} [{\bf y}] =
\frac{\partial}{\partial y_l} W_N , \quad l = 1, \dots, N.$$
Given an initial guess for the coordinates of the periodic orbit
${\bf y} = \allowbreak (y_1,\allowbreak \dots, y_N)$
we can improve it, setting ${\bf y'} = {\bf y} + {\bf \delta y}$.
Ignoring terms of order ${\bf \delta y \cdot \delta y}$ we have
$$D{\cal T}[{\bf y}] {\bf \delta y} = -{\cal T} [{\bf y}]$$
or,
$$A{\bf \delta y} =
\pmatrix{2-a_1&-1&\ldots&0&-1\cr
-1&2-a_2&\ldots&0&0\cr
\vdots&\vdots&\ddots&\vdots&\vdots\cr
0&0&\ldots&2-a_{N-1}&-1\cr
-1&0&\ldots&-1&2-a_N\cr}
\pmatrix{\delta y_1\cr
\delta y_2\cr
\vdots\cr
\delta y_{N-1}\cr
\delta y_N\cr}
= -{\cal T}[{\bf y}]
\EQ(newtonrotstd)$$
where
$a_i =(k + \frac{\epsilon}{2\pi}\cos(2\pi(\phi_0 + iP_2/N))) \cos (2\pi y_i)$.
This method of finding periodic orbits is very similar
to the one used in \cite{KM89} for the case of high-dimensional
symplectic maps. The only difference is that the coefficients
$a_i$ in our case depend not only on the coordinates of the periodic orbit,
but on the iteration number itself. Another method for numerically computing
certain periodic orbits of maps with symmetries uses
properties of periodic orbits with respect to symmetry lines of the map.
We have not made use of symmetry lines in our numerical algorithm.
%Since the orbits at $k=\epsilon=0$ are explicitly known,
%a continuation algorithm can be used, as long as certain, easy to check,
%non-degeneracy assumptions are satisfied.
Our numerical implementation of the method for a particular
$(P_1/N, P_2/N)$ periodic orbit used the following continuation algorithm:
\hfil \break
$\bullet$ Choose a precision-cutoff value $\delta$ (in our computations
we chose $\delta = 10^{-7}$).\hfil \break
$\bullet$ Choose a family of paths passing through
$k=\epsilon=0$. \hfil \break
$\bullet$ For a point along a path iterate Newton's algorithm, using as
initial guess the periodic orbit computed at the previous point on the path.
\hfil \break
$\bullet$ Proceed to the next point when the error between the computed
periodic orbit and its first iterate is less than $\delta$.
\SUBSECTION Iterative methods
The problem of solving the linear system \equ(newtonrotstd)
is greatly simplified by the fact that
the matrix $A$ is sparse,
with only $3N$ nonzero entries.
To solve systems of linear equations involving large, sparse matrices,
iterative methods have been developed
(similar problems appear frequently in
finite element and finite difference discretizations of
partial differential equations).
Iterative methods have the advantage that, for sparse matrices,
only order $N$ computation steps and storage space are required.
Given a linear system
$$Au=b$$
an iterative method successively approximates the true solution
$\bar u$
from an initial guess $u^{(0)}$. The iteration scheme is
$$u^{(n+1)} = G u^{(n)} + k$$
where $G = I -Q^{-1}A, k=Q^{-1}b$, for a suitable matrix $Q$. The
$Q$'s are chosen to be easily invertible (for example diagonal,
tridiagonal, or upper (lower) triangular matrices) so that
their inversion can be done in order $N$ computation steps.
An iterative method can be further
speeded up by the use of an acceleration procedure.
Such procedures are based on properties of the matrix $A$. For example,
Chebyshev acceleration uses information about the -- estimated -- range
of the eigenvalues of $A$, whereas the conjugate gradient method
minimizes a certain function of $A$. For a more complete description,
with many examples see \cite{YG88, YH81}.
Convergence of an iterative method is checked by monitoring
the norm of the error.
Convergence of some iterative methods to the true solution has been rigorously
demonstrated for the case that $A$ is a symmetric, positive-definite matrix.
Unfortunately we are
not aware of any general convergence result for matrices that are not symmetric.
In our computations we have used the package {\tt ITPACK 2C},
developed in the Center for Numerical Analysis in the University of Texas
at Austin (for a description of {\tt ITPACK 2C} see \cite{KRYG82}).
We used the Jacobi iterative method (where $Q $ is the diagonal
part of $A$) with either Chebyshev acceleration or conjugate gradient
acceleration. We verified that the time to converge to a solution,
within a specified precision, increased linearly
with $N$ for a type $(P_1/N, P_2/N)$ periodic orbit.
\SUBSECTION Direct methods
In \cite{KM89} a direct algorithm was proposed to solve
an equation similar to \equ(newtonrotstd) in the case of high-dimensional
symplectic maps.
The algorithm had the advantages of an iterative method in that it required
only order $N$ steps and order $N$ storage space.
We implemented a similar method, taking advantage of the
structure of the matrix $A$.
The method is based on the fact that $A$ is very close to a tridiagonal
matrix, namely it is {\it cyclic tridiagonal} (i.e. tridiagonal with
two additional entries at the corners).
We first solve the tridiagonal linear system in order $N$ steps
using the Thomas algorithm (see \cite{YG88 vol.2 pg 587}).
The algorithm takes order $N$ computational steps to
perform Gaussian elimination, due to the special form of
a tridiagonal matrix.
After finding the solution to the tridiagonal problem we
can add corrections, due to the terms at the two corners,
either using the Sherman-Morrison, or the
Woodbury methods (see \cite{FPTV92 pg.73-77} for a description of the
methods and an implementation).
The algorithm can be easily extended to higher-dimensional cases.
Our implementation is designed to be flexible,
so that periodic orbits in different maps (either volume-preserving
or symplectic) and even higher-dimensional
systems can be computed by simply changing some map-dependent
definitions in a file.
\SECTION Jacobi-Perron approximation schemes
Approximation of irrational numbers by rationals has been important in
the study of breakdown of invariant curves in twist maps of the annulus since
the work of Greene (see \cite{Gr79}). Greene conjectured that
the behavior of periodic orbits with rotation numbers that
are continued-fraction convergents of a diophantine,
irrational number, determines the existence of
an invariant curve with rotation number equal to the irrational. MacKay
(see \cite{McK82}) constructed a renormalization-group operator in
spaces of analytic maps, that
changes the rotation number of an invariant curve by eliminating
the first continued fraction coefficient. For irrationals with periodic
continued fraction expansions (and in particular the golden mean
$\gamma = (\sqrt{5} - 1)/2$ ) he provided evidence, based on careful
numerical work and analytical arguments,
that the breakdown of invariant curves is described by
a fixed point of the renormalization operator with a co-dimension one
stable manifold.
\SUBSECTION Description of the algorithm
The Jacobi-Perron algorithm is one -- of many -- generalization of the
continued fraction algorithm in higher dimensions.
Detailed description of
the algorithm and proofs of the results we present here can be found
in \cite{Ber71, Sch73, Kos91, Lag93}.
Given a point $(\omega_1, \omega_2) \in (0,1)\times (0,1)$ the
Jacobi-Perron convergents
$P_n / N_n$,$ P_n = (P_{1_n}, P_{2_n}) \in \natural^2$,
$N_n \in \natural$ are recursively defined by
$$\eqalign{
P_{n+1} & = k_{n+1} P_n + l_{n+1} P_{n-1} + P_{n-2} \cr
N_{n+1} & = k_{n+1} N_n + l_{n+1} N_{n-1} + N_{n-2} \cr
}
\EQ(jacper1)$$
where the integer coefficients $k_{n+1}, l_{n+1}$ are determined by
the Jacobi-Perron map
$$\eqalign{
(\omega_1^{n+1}, \omega_2^{n+1}) & =
(\{ \frac{1}{\omega_2^n}\}, \{ \frac{\omega_1^n}{\omega_2^n}\} )\cr
(k_{n+1}, l_{n+1}) & =
([\frac{1}{\omega_2^n}], [\frac{\omega_1^n}{\omega_2^n}] )\cr}
\EQ(jacper2)$$
with initial values
$$\eqalign{
& (\omega_1^0, \omega_2^0) = (\omega_1, \omega_2) \cr
& P_0 = (0, 0), \quad P_{-1} = (1, 0), \quad P_{-2} = (0,1)\cr
& N_0 = 1, \quad N_{-1} = N_{-2} = 0\cr}$$
The Jacobi-Perron algorithm is
a linear simplex-splitting algorithm (see \cite{Lag93}).
For the case of points in the unit square a consequence is that if
three successive Jacobi-Perron approximants define a triangle, all approximants
of higher order (and the point being approximated) will lie inside that
triangle. This property is shared with another commonly used algorithm,
the Farey-tree approximation scheme (see \cite{KO86}).
Lagarias studied the rate of convergence of the Jacobi-Perron
algorithm for a set
of points of (Lebesgue) measure one in the unit square.
Consider a point in the unit square $w$ and a close-by point with rational
coordinates $r = (p_1/q, p_2/q)$. The Roth exponent of $r$ is defined as
$$ \eta (r, w) = -\frac{\log\| w - r \|_2}{\log q}. $$
Let $r_i$ be the $i^{\rm th}$ Jacobi-Perron approximant to a point $w$.
The {\it best approximation exponent} for $w$ using the Jacobi-Perron
scheme is defined as
$$ \eta_{b} \equiv \limsup_{i\to \infty} \{ \eta (r_i(w)) \}$$
and the {\it uniform approximation exponent} as
$$ \eta_{u} \equiv \liminf_{i\to \infty} \{
\min (\eta (r_i(w)), \eta(r_{i+1}(w)), \eta(r_{i+2})) \}.$$
The exponent $\eta_b$ gives the rate of convergence
for the best possible approximant towards
a point, while the exponent $\eta_u$ is an --asymptotic -- estimate for
the rate at which all the vertices of the triangle enclosing $w$ approach $w$.
Lagarias showed, using methods from ergodic theory (see \cite{Lag93}, also
\cite{Kos91}) that for the Jacobi-Perron algorithm $\eta_b$ and
$\eta_u$ are constant in a set of measure one in the unit square.
He conjectured that the constant values are in fact equal. In \cite{Bald92}
(see also \cite{Kos91})
numerical methods were used to estimate $\eta_b = 1.374 \pm 0.002$,
an estimate that, coupled with Lagarias' conjecture, suggests that the
triangles formed from successive Jacobi-Perron approximants become, in the
limit, needle-shaped.
Other (non measure-theoretic) properties of the Jacobi-Perron algorithm
that are of interest are the following %(for a proof see \cite{XXXXXX} )
\hfil \break
\noindent $\bullet$ For all points in the unit square
$k_n \ge 1, k_n \ge l_n \ge 0$. \hfil \break
\noindent $\bullet$ The triangle formed by three successive Jacobi-Perron
approximants contains no rational point with denominator smaller than the
largest denominator of the vertices. \hfil \break
%\noindent $\bulllet$
In analogy with periodic continued-fractions with period 1, we introduce
golden means of the Jacobi-Perron algorithm $\omega$ for which
$k_n =k, l_n=l$. The polynomial
$P_\omega(t) = t^3 - kt^2 -lt -1$ is called the characteristic polynomial
of $\omega$. Golden means have the following properties: \hfil \break
\noindent $\ast$ $k < \tau < k+1, 0 < |\tau_1 |, |\tau_2| < 1$ where
$\tau$ is the root of $P_\omega$ of maximal absolute value and $\tau_1, \tau_2$
the remaining roots. \hfil \break
\noindent $\ast$ $(\omega_1, \omega_2) = (\tau - k, 1/\tau)$. \hfil \break
\noindent $\ast$ $\|q_n \omega - (p_{1_n}, p_{2_n}) \|_2\le C(\omega) \kappa^n$
where $\kappa = \max (|\tau_1|, |\tau_2|) < 1$.
In \cite{Kos91}, Kosygin constructed a renormalization-group operator
in the space of symplectic
maps of $\real\times\real \times \torus\times\torus $.
The action of the operator on a map that has an invariant surface with
rotation vector $(\omega_1, \omega_2) = ((k_1, l_1), (k_2, l_2), \dots )$
produces a new map with an invariant surface with rotation vector
$(\omega_1', \omega_2') = ((k_2, l_2), \dots)$. For
golden means of the Jacobi-Perron algorithm it was shown that,
if the original map, under repeated action of the renormalization-group
operation converges to a (trivial) map (which is a fixed point of the
renormalization-group operator), then the original map admits
an invariant surface, on which motion is conjugate to rigid rotation
with rotation vector $(\omega_1, \omega_2)$. Moreover, the trivial fixed point
is attractive, i.e. maps in its neighborhood admit an invariant surface
(this result can also be viewed as
a renormalization-group proof of the KAM theorem for
invariant surfaces with rotation vectors golden means of the Jacobi-Perron
algorithm).
\SUBSECTION Connection between Jacobi-Perron algorithm and ${\bf PSL}_3(\integer)$
The Jacobi-Perron algorithm has a natural connection with $3\times 3$
matrix transformations.
We introduce an extension of modular transformations
of one complex variable to two complex variables
$$(z_1, z_2) \to (\frac{az_1 + bz_2 + c}{gz_1 + hz_2 + i},
\frac{dz_1 + ez_2 + f}{gz_1 + hz_2 + i})\EQ(extmodul)$$
Successive transformations obey the rules of matrix multiplication
of $3\times 3$ matrices
$$ M = \pmatrix{a&b&c\cr d&e&f\cr g&h&i \cr}$$
where $M$ and $-M$ are identified and (because of invariance of the
transformation under scaling) $|\det M | = 1$. The group of transformations
defined by \equ(extmodul) is isomorphic to the projective group
${\bf PSL}_3(\integer)$. It is known that ${\bf PSL}_3(\integer)$ is
finitely generated by the elementary matrices
$T_{ij} = I + e_{ij}, i\ne j, i,j = 1,2,3$
where $e_{ij}$ the $3\times 3$ matrix with the only nonzero entry the element
$i,j$ which is equal to 1. On the other hand the Jacobi-Perron operator
can be viewed as a subgroup of ${\bf PSL}_3(\integer)$ generated by the
matrices
$$ T_1 = \pmatrix{1&0&1\cr 0&1&0\cr 0&0&1\cr},
T_2 = \pmatrix{1&0&0\cr 0&1&1\cr 0&0&1\cr},
U = \pmatrix{0&1&0\cr 0&0&1\cr 1&0&0\cr}$$
$T_1$ represents translations in the first coordinate, $T_2$ translations
in the second coordinate, and $U$ generalized inversion.
Given a point $(\omega_1, \omega_2)$ in the unit square,
the Jacobi-Perron operator can be written as \hfil \break
\centerline{$(\omega_1^{(0)}, \omega_2^{(0)}) = (\omega_1, \omega_2)$}
\centerline{$(\omega_1^{(n+1)}, \omega_2^{(n+1)}) = T_1^{-k_n}T_2^{-l_n}U(
\omega_1^{(n)}, \omega_2^{(n)})\quad n\ge 1$}
%the Jacobi-Perron can be written as
%$$\eqalign{(\omega_1^{(0)}, \omega_2^{(0)}) & = (\omega_1, \omega_2)\cr
%(\omega_1^{(n+1)}, \omega_2^{(n+1)}) & = T_1^{-k_n}T_2^{-l_n}U(
%\omega_1^{(n)}, \omega_2^{(n)})\quad n\ge 1}$$
where $k_n, l_n$ the unique non-negative integers such that
($\omega_1^{(n+1)}, \omega_2^{(n+1)})$ is inside the unit square. The
coefficients $k_n, l_n$ form the Jacobi-Perron expansion of
$(\omega_1, \omega_2)$.
We can use the Jacobi-Perron operator to move between points with the
same tail coefficients,
since if $\omega^{(0)} = ((k_0, l_0), (k_1, l_1), \dots )$,
then $\omega^{(n)} = ((k_n, l_n), (k_{n+1}, l_{n+1}), \dots)$. The difference
with the lower dimensional case studied in \cite{BPV90} is that the
Jacobi-Perron transformation corresponds to
a subgroup of ${\bf PSL}_3(\integer)$,
whereas the continued fraction algorithm corresponds to
${\bf PSL}_2(\integer)$.
%The connection between the Jacobi-Perron algorithm and ${\bf PSL}_3(\integer)$
%may be useful in understanding the behavior at breakdown.
\SECTION Domains of existence of invariant surfaces
Existence of invariant surfaces on which motion is, up to a change of
variables, rigid rotation, plays a significant role in determining
long term dynamics in many physical applications.
In the case of the rotating standard map the existence of
two-dimensional invariant tori presents a complete barrier to
phase-space diffusion. As was shown in \cite{PW94}
invariant tori in high-dimensional Hamiltonian systems
also guarantee long-term stability for orbits in their neighborhood.
Several methods, both analytical and numerical, have been used in
two-dimensional systems to determine the domain of existence of invariant
curves (see \cite{CC88, Ran87, LR91, BC90, FL92-2}). Unfortunately,
estimates based on analytical methods are very conservative in the
case of higher-dimensional systems. The, widely believed, most accurate
among the numerical methods for two-dimensional systems is based on a
conjecture of Greene concerning the behavior of periodic orbits approaching
the invariant curve (see \cite{Gr79} and also \cite{FL92-1, McK92}
for a rigorous, partial justification).
Results in \cite{T95, FL92-1} provide justification for a similar criterion
in higher-dimensional models. We reproduce the relevant results
from \cite{T95}
\CLAIM{Theorem}(residue_anal)
(Theorem 2.4 in \cite{T95})\hfil \break
%Let
%$f:\torus^{d+e}\times \real^d\to \torus^{d+e}\times \real^d \in C^r, r>1$
%(analytic)
%satisfy:\hfill \break
%$\bullet
%f|_{\torus^{d}\times \real^d}$ is a non-degenerate symplectic map,\hfil \break
%$\bullet
%f|_{\torus_e}$ is rigid rotation with a diophantine rotation vector.
%\hfil \break
%Assume that $f$ admits a $C^r$ (analytic) invariant
%surface $\Gamma$, homotopic to $\torus^{d+e}$,
%on which the motion is
%$C^r$ (analytically) conjugate to rigid rotation with
%rotation vector ${\bf \omega}$ of type $(K, \tau)$.
%Moreover assume that in a neighborhood
%of $\Gamma$ there are periodic orbits $x_{(P/N)}$ of type ($P/N$) for
%of the derivative $Df^N (x_{(P/N)})$ satisfy
%$$|\lambda_i -1 | \le D_k \| N{\omega} - P\|_d^{k/2} N ,\ i = 1,\dots ,2d$$
%$$(|\lambda_i -1 | \le \tilde D_1 N
%\exp ( -\tilde D_2 \| N\omega - P\|_d^{\frac{-1}{2(1+\tau)}}),\ i = 1,\dots, 2d
%)$$
%The remaining $e$ eigenvalues are identically $1$.
Let
$f:\torus^{d+e}\times \real^d\to \torus^{d+e}\times \real^d \in C^r, r>1$
(analytic)
be a quasi-periodic skew-product of a symplectic map satisfying
%(i), (ii)
non-degeneracy conditions, over $\torus^e$ such that
$f|_{\torus_e}$ is rigid rotation with a diophantine rotation vector.
Assume that $f$ admits a $C^r$ (analytic) invariant
surface $\Gamma$, homotopic to $\torus^{d+e}$,
on which the motion is
$C^r$ (analytically) conjugate to rigid rotation with
rotation vector ${\bf \omega}$ of type $(K, \tau)$.
Moreover assume that in a neighborhood
of $\Gamma, f$ there are periodic orbits $x_{(P/N)}$ of type ($P/N$) for
$\|N \omega - P \|_{d+e}$ small enough.\hfil \breakline
Then, for $k \in \natural, k < \frac{r-1}{\tau}$
we can find $D_k > 0$, such that $2d$ of
the eigenvalues $\lambda_1, \dots, \lambda_{2d}$
of the derivative $Df^N (x_{(P/N)})$ satisfy
$$|\lambda_i -1 | \le D_k \| N{\omega} - P\|_d^{k/2} N ,\ i = 1,\dots ,2d$$
$$(|\lambda_i -1 | \le \tilde D_1 N
\exp ( -\tilde D_2 \| N\omega - P\|_d^{\frac{-1}{2(1+\tau)}}),\ i = 1,\dots, 2d )$$
The remaining $e$ eigenvalues are identically $1$.
\REMARK
For the rotating standard map the eigenvalues of the derivative are
completely determined by the trace (since one eigenvalue is identically
1 and the map is volume-preserving). Therefore, instead of monitoring the
eigenvalues we will study the behavior of the {\it residue}
$$R({\bf x}) = \frac16 (3-{\rm Tr}(Df^N({\bf x})))$$
along a periodic orbit ${\bf x}$ with period $N$.
\REMARK
In the case of hyperbolic invariant sets semi-conjugate to rigid rotation,
it was shown in \cite{FL92-1} that the derivative along
periodic orbits approaching the invariant set has
exponentially increasing eigenvalues (increasing with the period of the orbit).
Although the invariant sets of the rotating standard map
are never hyperbolic (due to rigid rotation in the second angle
coordinate) we have observed that in the absence of an invariant
surface, the eigenvalues of periodic orbits approaching a limit
set are exponentially increasing.
\clm(residue_anal) (which also holds for complex
maps and complex invariant sets) suggests the following algorithm: \hfil \break
\noindent $\bullet$ Fix a value $\alpha > 0$ (this value will be used as
the cutoff criterion for determining breakdown).\hfil \break
\noindent $\bullet$ Choose a family of paths in the parameter space.
\hfil \break
\noindent $\bullet$ Compute periodic orbits with rotation vectors close to
the rotation vector of an invariant set of interest, along the paths, and
determine the point along the path when the residue of
the periodic orbit satisfies
$|R(x)| > \alpha$.
\hfil \break
\REMARK The value of $\alpha$ plays only a minor role for periodic orbits
with large period. The reason is that if an invariant curve exists,
the eigenvalues of the derivative of the map approach 1 exponentially
as the period increases, whereas if the invariant surface has "disintegrated"
then at least one eigenvalue is expected to be exponentially
large. The reasoning is valid as long as the breakdown of an
invariant surface is well-defined, for
example one value along the path in the parameter space.
\REMARK The algorithm is based on continuation of periodic orbits using
Newton's method in a space of finite sequences, as presented in section 3.
Using numerically computed condition numbers we can determine the validity of
the continuation scheme. As long as no eigenvalue of the derivative of
the map along the periodic orbit is 1, the implicit function theorem
guarantees the success of the continuation method for small enough steps
along the path. Notice that one eigenvalue in the
case of the rotating standard map is identically 1, due to
rigid rotation in the second angle coordinate, but
does not influence Newton's
method, as described by \equ(newtonrotstd).
In figures 1a,1b,1c we follow
periodic orbits along paths in the parameter plane.
We start the paths at $\epsilon=k=0$, where we know the coordinates of
the periodic orbits. The initial value of $\phi_0$ was taken to be $1/2$.
In figure 1a we investigated the domain of existence of the invariant
curve with rotation vector $\omega$ with Jacobi-Perron expansion
$(1,1)^\infty$ for real values of $k, \epsilon$.
We observe that for the boundaries for successive approximants
crossed each other for many sets of parameter values (a similar
phenomenon was observed in \cite{FL92-2} for standard-like
maps in the complex domain). The phenomenon is more pronounced for large values
of $k$ (see figure 1b).
In \cite{ACS91} a similar
phenomenon was observed but for a different invariant surface
(with rotation vector the spiral mean)
using a different approximation scheme (an extension
of the Farey-tree expansion).
We observed however that for certain paths in the
parameter plane the boundaries for successive approximants are well-ordered
and their successive positions follow
a power law. Such paths are the
$\epsilon=0$ path (corresponding to the well-understood case of the
standard map) and the $k=0$ path (see figure 2).
In figure 1c we present the domain of analyticity of the
same invariant surface for a small, fixed value of $k$ ($k=0.2$)
and for complex values of $\epsilon$. The absence of
crossings and the scaling observed in the figure suggests that
behavior at breakdown could be understood in terms of a fixed point
of a renormalization operator in a space of maps with complex variables.
For invariant surfaces that break down to invariant sets semi-conjugate
to rotation, we
can study the transition by monitoring the behavior of close-by
periodic orbits. The natural question is whether an analog of
Aubry-Mather theory applies to higher-dimensional systems and
what are the invariant sets that an invariant surface breaks down to.
The continuation algorithm allows to compute periodic orbits of
%-- relatively --
high period close to breakdown. Based
on the results in figure 1a we identified
breakdown of the invariant surface with rotation vector
$(1,1)^\infty$ at $\epsilon = 1.75, k =0.2$ and $\epsilon = 3.55, k= 0$.
We remark that although crossings between boundaries where different
periodic orbits had residue 1 were common, the residue of each
periodic orbit along a path seemed to behave regularly. %(i.e.
%we did not observe residues and then becoming smaller,
%something that would indicated breakdown and re-generation of
%an invariant surface as Wilbrink observed in
The transition
occurring at breakdown was investigated in figures 3a, 3b, 3c, 3d, 3e.
We observe that the invariant set develops gaps, analogous to
the ones that appear as an invariant curve breaks down to an Aubry-Mather set
in two-dimensional systems. Such gaps have been used to
rigorously show (using a computer assisted proof) that
breakdown has occurred (see \cite{Mu89} for the case of a four-dimensional
symplectic map). Notice that the largest gap occurs at $\theta = \phi = 0$
where the potential attains its maximum.
\REMARK For the case of the
rotating standard map we have shown in section 3
that periodic orbits for all rational
rotation vectors exist. However we have not been able to rigorously
determine whether
they converge to an invariant set as their rotation vector tends to a
diophantine rotation vector.
In figures 4a, 4b, 4c we investigated the behavior of the residue and the
stability of periodic orbits along two paths in parameter space.
The stability is quantified by the stability exponent of the
periodic orbit defined, for an orbit $x$ of period $N$ as
$$\lambda = \frac{1}{N} \log |Df^N(x)| $$
where, by $|Df^N(x)|$, we mean the biggest eigenvalue of the matrix.
The stability exponent tends, in the limit $N\to \infty$, to the
Lyapounov exponent of the invariant set, as long as the orbit tends to
the invariant set and motion on the invariant set is semi-conjugate to
rigid rotation with diophantine rotation vector (in which case there
is only one invariant measure and it makes sense to talk of Lyapounov
exponents without specifying it). Notice that, due to nature of the
map, one eigenvalue of the derivative
is always one and the other two are reciprocal.
\REMARK It is important to monitor the algorithm that computes the
periodic orbits to guarantee
that the type of periodic orbit we are following is the same. As
was remarked in \cite{OV92} periodic orbits of higher-dimensional
symplectic maps
may undergo bifurcations and follow a different (low-dimensional)
torus from the one we are interested in. To avoid this problem we
have monitored the size of the condition numbers for the solution of
the linear system \equ(newtonrotstd). When the condition numbers
were large we decreased the stepsize of the algorithm.
Moreover we have tried to choose periodic orbits with rotation vectors
whose components are
irreducible, to avoid bifurcations similar to the ones observed in \cite{OV92}.
As can be observed from the behavior of the stability exponent in figures
4b,4c there were indeed bifurcations for some of
the periodic orbits with reducible components.
\SECTION Critical function
In \cite{Pe82} Percival introduced a critical function for two-dimensional
twist maps. It is a function of the rotation number
and its value corresponds to the perturbation strength
at which the invariant curve breaks down (in the
case of the standard map there are indications from
renormalization-group theory that breakdown occurs at a
specific value for each rotation number).
Based on the renormalization-group description, Buric et al.
\cite{BPV90} observed that the
critical function for the semi-standard map
has transformation properties under the modular group that allow
its rapid computation. These transformation properties have also been
considered as additional evidence of an underlying renormalization-group
transformation.
In analogy to \cite{Pe82}
we introduce a critical function $K(\omega)$ for the rotating standard map,
from the space of rotation vectors $\omega$ to the parameter space
$(\epsilon, k)$. Given a rotation vector $\omega$, the values of the critical
function are the values of the parameters at which breakdown occurs.
Similar to the two-dimensional case,
this function is discontinuous in a set of large measure (since it vanishes
at all rationally dependent rotation vectors but is non-zero
at a set of large measure).
%Difficulties that arise in this
%generalization are the many parameters
%involved.
We will use the value of the residue of a nearby periodic orbit as an indication
of breakdown of an invariant surface.
In figure 5
we present the logarithm of a slice of
the critical function for the rotating standard map,
at a fixed value of $k=0.2$. The figure was generated using periodic orbits
and taking into account the symmetries of
the critical function $\omega_1 \to 1-\omega_1$, $\omega_2 \to 1-\omega_2$.
We chose the points at which to evaluate the critical function randomly,
and then approximated the chosen points with a nearby (eventually) golden mean
of the Jacobi-Perron algorithm (i.e. with Jacobi-Perron
expansion $(\dots, (1,1)^\infty )$).
We truncated the Jacobi-Perron expansion to obtain rational points close
to the (eventual) golden means.
All the periods used were between 300 and 800.
We used a continuation method to follow the periodic orbits up to
the perturbation value of $\epsilon$ at which the residue becomes $1$.
We assigned that perturbation strength as the value of the critical function
(the cases when breakdown has already occurred for $\epsilon =0$ have
been considered as if breakdown occurred at $\epsilon = 0$).
Our procedure is similar to the one used in \cite{BPV90} for the
computation of the critical function for the semi-standard map.
Figure 5 took 30 hours of CPU time on an IBM RS/6000 370.
It was plotted using {\tt gnuplot}, by computing the values at the grid
points using weighted interpolation (this process has the undesirable effect of
"smoothing" the function in certain regions).
We have not been able to identify transformation properties of the
critical function under the action of the Jacobi-Perron operator for the
rotating standard map. We face two problems, the first being a possible
problem for any multi-dimensional approximation algorithm, while the second
pertaining to the Jacobi-Perron algorithm in particular:
\noindent (a) We do not know how many parameters are
necessary to describe the critical fixed point of a renormalization-group
algorithm. In hopes of simple behavior (similar to the one
uncovered by MacKay in the case of the twist maps of the annulus) we
have used one-dimensional parameter paths in the parameter space.
Notice however that, as indicated in figure 1a, there are regions
in parameter space where the renormalization group behavior is more
complicated.
\noindent (b) The Jacobi-Perron algorithm is not symmetric with respect to
the two coordinates.
In the case of twist maps (when the rotation vector is a real number)
it is possible to produce a linear combination of $\log K(\omega)$,
$\log K(U\omega)$ to cancel the leading order
logarithmic singularities of the critical function.
It would be interesting to determine whether such cancelations are
possible in the case of the rotating standard map or high-dimensional
symplectic maps.
\SECTION Acknowledgments
Most of this work was completed at the University of Toronto.
I would like to thank Rafael de la Llave, Claudio Albanese, Luis Seco, John Im,
Arturo Olvera, Jim Meiss and Robert MacKay
for useful discussions and suggestions during this work.
Finally I want to express my thanks to the people that
developed and maintain {\tt gnuplot} for making such a fine program available.
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\eject
\SECTION Captions
{\noindent \bf Figure 1}.\hfil\break
a) Domain of existence for the
invariant surface with golden rotation vector $(1,1)^\infty$ for real
values of the parameters. The periodic orbits chosen are
Jacobi-Perron convergents to $(1,1)^\infty$.
The different sets plotted correspond
to the boundary where the residue of periodic orbits is 1. Set 1
has rotation vector $(44/81,68/81)$, Set 2 $(81/149,125/149 )$,
Set 3 $(149/274, 230/274 )$, Set 4 $(274/504,423/504 )$, Set 5
$(504/927, 778/927 )$, Set 6 $(927/1705, 1431/1705 )$ and Set 7
$(1705/3136, 2632/3136)$.
\hfil\break
b) Detail of figure 1a.
\hfil\break
c) Domain of existence for the
invariant surface with golden rotation vector $(1,1)^\infty$ for fixed $k=0.2$,
complex $\epsilon$. Sets 1-7 same as in figure 1a.
{\noindent \bf Figure 2}.\hfil\break
Scaling of the points where the
residue becomes 1, for $k=0$ for successive Jacobi-Perron approximants
to $(1,1)^\infty$. The approximant order is denoted in the horizontal
axis with the first approximant $(7/13, 11/13)$ and the last -- eleventh --
$(3136/5768, 4841/5768)$. The value of $\epsilon_{\rm cr}$ is
determined from the next approximant $(5768/10609, 8904/10609)$.
{\noindent \bf Figure 3}.\hfil\break
a) A periodic orbit
close to the invariant surface with rotation vector $(1,1)^\infty$.
The invariant surface is close to breakdown $(k = 0.2, \epsilon = 1.7)$.
Pe\-riodic orbit ro\-ta\-tion ve\-ctor
$(10609/19513, 16377/19513)$ and residue
$-0.831$.
\hfil\break
b) A periodic orbit
close to an invariant set with rotation vector $(1,1)^\infty$
after breakdown $(k = 0.2, \epsilon = 1.8)$.
Periodic orbit rotation vector $(10609/19513, 16377/19513)$ and residue
$-7.01\times 10^{12}$.
\hfil\break
c) A little before breakdown. Rotation vector $(1,1)^\infty$,
$k=0, \epsilon = 3.5$. Periodic orbit rotation vector
$(10609/19513, 16377/19513)$, residue $-5.41\times 10^{-3}$.
\hfil\break
d) A little after breakdown. Rotation vector $(1,1)^\infty$,
$k=0, \epsilon = 3.6$. Periodic orbit rotation vector
$(10609/19513, 16377/19513)$, residue $-2.05\times 10^3$.
\hfil\break
e) Continuation of the periodic orbit in figure 3d. $k=0, \epsilon =4.0$.
Residue $-1\times 10^{78}$.
{\noindent \bf Figure 4}.\hfil\break
a) Residue of periodic orbits approximating an invariant set with rotation
vector $(1,1)^\infty$ along the path $k=0$. The different sets correspond
to different periodic orbits. Set 1 $(149/274, 230/274 )$,
Set 2 $(274/504,423/504 )$, Set 3
$(504/927, 778/927 )$, Set 4 \break
$(927/1705, 1431/1705 )$, Set 5
$(1705/3136, 2632/3136)$, Set 6 $(3136/5768, 4841/5768)$ and
Set 7 $(5768/10609, 8904/10609)$.\hfil\break
b) Stability index $\lambda$ of the periodic orbits in figure 4a.
\hfil\break
c) Stability index $\lambda$ of periodic orbits approximating an invariant
set with rotation vector $(1,1)^\infty$ along the path $k=0.2$. Sets 1-7
same as in figure 4a.
{\noindent \bf Figure 5}.\hfil\break
Logarithm of the
critical function at $k=0.2$. The values of the function are computed using
the residue of 6576 periodic orbits and interpolating
on a $150\times 150$ grid.
\end