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\TITLE
NATURE OF SINGULARITIES FOR ANALYTICITY DOMAINS OF INVARIANT CURVES
\footnote{$^1$}
{\baselineskip=12pt\rm Preprint available
through the mathematical physics electronic
preprint archive. Send mail to {\tt mp\_arc@math.utexas.edu} for details.}
\ENDTITLE
\AUTHOR
Rafael de la Llave
\footnote{$^2$}{Supported by NSF grants.}
\footnote{$^3$}{e-mail: {\tt llave@math.utexas.edu}}
and Stathis Tompaidis
$^2$
\footnote{$^4$}{e-mail: {\tt stathis@math.utexas.edu}}
\FROM
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712-1082
\ENDTITLE
\ABSTRACT
We present theoretical arguments (based on infinite dimensional bifurcation
theory) and numerical evidence (based on non-perturbative methods)
that the boundaries of analyticity of invariant curves
can be described as an accumulation of branch points,
which are typically of order 2.
We show how this fact would explain previous numerical results of
several authors and how it suggests more efficient numerical
algorithms, which
we implement.
\ENDABSTRACT
Invariant curves have been
very important in the development of dynamical systems because they
organize the long term behavior.
A considerable amount of
numerical work has been performed recently
trying to understand the phenomena that happen when they disappear.
In previous numerical investigations the breakdown of invariant curves
has been attributed to the presence of a natural boundary in their analyticity
domain (see \cite{BC}, \cite{BCCF}, \cite{FL}, \cite{LT}).
In \cite{BM} complex rotation numbers were introduced and an unexpected
structure of the analyticity domain was numerically observed.
This paper sheds light on the nature of the singularities
of invariant curves for standard-like maps seen as analytic functions of
a perturbation parameter.
We present a mechanism, supported by theoretical arguments and numerical
evidence, that can
explain the numerical
results from the previous studies.
We expect that the results would be
typical for many systems in which breakdown of K.A.M curves occur.
Standard-like maps are area-preserving twist maps given by:
$$p_{n+1} = p_n + \epsilon S(q_n), \qquad
q_{n+1} = q_n + p_{n+1} \quad (\mod 2\pi)
\EQ(stdmap)$$
where $S(q)$ is an odd trigonometric polynomial in $q$. For $S(q) = \sin(q)$
this reduces to the standard map, for which numerous studies exist (\cite{Ch},
\cite{Au}, \cite{Gr}). Standard-like maps are not only canonical examples of
twist maps but have also
appeared as models of several phenomena in different areas
of physics, ranging from plasma to solid state physics.
To study invariant
curves we will find it more convenient to use the second order ``Lagrangian''
recurrence:
$$q_{n+1} - 2q_n + q_{n-1} = \epsilon S(q_n). \EQ(lagrangian)$$
We will study an invariant curve of rotation number $\omega$ that can
be parametrized in terms of a
parameter $\theta$ :
$$q_n = \theta_n + u(\theta_n ; \epsilon, \omega) \EQ(param)$$
requiring that
$\theta_{n+1} = \theta_n + \omega, \mod 2\pi$. The function
$u$, called the hull function in \cite{Au}, conjugates the dynamics
of the standard-like map to a rigid rotation with rotation number $\omega$.
If an invariant curve for \equ(lagrangian) parameterized by $u$, exists :
$$\Delta_{\omega}[u](\theta) - \epsilon S(\theta + u(\theta ; \epsilon, \omega))
= 0 \EQ(homology)$$
where
$\Delta_{\omega}[u](\theta) = u(\theta + \omega; \epsilon, \omega) -2u(\theta; \epsilon, \omega) + u(\theta - \omega; \epsilon, \omega).$
The Poincar\'e-Lindstedt perturbation method consists in expanding $u$ in
series in $\epsilon, \theta$ :
$$u(\theta; \epsilon, \omega) = \sum_{n=1}^\infty \sum_{k = -\infty}^\infty
\hat u_{n,k} \epsilon^n e^{ik\theta} \EQ(lindstedt)$$
and matching coefficients by order in $\epsilon$ and the Fourier mode.
For $\omega$ real, Diophantine (i.e. $\omega$ that satisfies
$| q\omega - p| \ge C | q|^{-r}$, for $C>0, r>2, p,q \in \integer$),
using methods from KAM theory \cite{CC} showed that there exists a
solution to \equ(homology) for $\epsilon$ sufficiently small.
The solution is unique under the condition $\int u d\theta = 0$.
To regularize the convergence for the series \equ(lindstedt)
Berretti and Marmi (see \cite{BM}) introduced
rotation numbers with nonzero imaginary part -- an
effect somewhat similar to adding dissipation to the system.
When $\Im \omega \ne 0$
a simple argument based on the majorant method
shows that
$u$ is analytic in $\theta$ in a neighborhood
of zero in the $\epsilon$ plane. The existence and the critical behavior
of the invariant curve when $\omega$ is real are determined by the behavior
of the singularities of the solution to \equ(homology) as
$\Im \omega \to 0$.
The singularity structure of the analyticity domain in $\epsilon$ of
$u(\theta; \epsilon, \omega)$
for
$\Im \omega \ne 0$, can be determined using
bifurcation theory.
We will view \equ(homology) as an equation in the space of continuous
functions (of the variable $\theta$)
with zero average (denoted henceforth as $C^0_0$) and introduce
the operator ${\cal T}_\omega : C_0^0 \to C_0^0$, where :
$${\cal T}_\omega [u](\theta) = \epsilon \Delta_\omega^{-1}
S(\theta + u(\theta; \epsilon, \omega)). \EQ(operator)$$
Then \equ(homology) is reduced to a fixed point problem for
${\cal T}_\omega$ in $C_0^0$.
It is easy to see that the operator
$D{\cal T}_\omega [u]$ is compact
in this infinite dimensional space (it maps bounded sets into pre-compact ones).
A well known theorem about the spectrum of compact operators states that
all the eigenvalues of $D{\cal T}_\omega$ are isolated with
finite multiplicity apart
from an eigenvalue at zero (see \cite{Ru}). A simple bifurcation
occurs for values $\epsilon_0$ of $\epsilon$
such that $D{\cal T}_\omega [u]$ has a simple
eigenvalue $1$ and :
$$\int
v (\theta) \Delta_\omega^{-1}
\bigl [ S(\theta+ u(\theta; \epsilon_0, \omega)) \bigr ] d\theta \ne 0
\EQ(bif1)$$
$$ \int
v (\theta) \Delta_\omega^{-1}
\bigl [ S''(\theta+ u(\theta; \epsilon_0, \omega)) v^2(\theta) \bigr ]
d\theta \ne 0 \EQ(bif2)$$
where $v(\theta)$ is a
null vector for $D{\cal T}_\omega [u(\theta; \epsilon_0, \omega)] - I$
at $\epsilon = \epsilon_0$ :
$$\bigl [
I - \epsilon_0 \Delta_\omega^{-1} S' (\theta + u(\theta; \epsilon_0, \omega))
\bigr ] v = 0, \quad v \ne 0 \EQ(bif3)$$
(see \cite{GS}, \cite{ChH}).
At such a value $\epsilon_0$, $u$ considered as a function of
$\epsilon$, exhibits branch points of order $2$. We note that, as usual
in bifurcation theory, this is the expected behavior for a typical function.
If $S$ were to depend on extra parameters, we expect that
bifurcation points of order 3 will appear in a set
of co-dimension 2 in parameter
space.
\picture{1a}{$S(q) = \sin q.\quad \omega = 0.2i.$
The poles and zeros of the Pad\'e
approximant [14/14] for $\theta = 0.23$
(Set 1) superimposed with the poles of the Pad\'e
approximant for the derivative of the logarithm of $u'$ (Set 2)
and the path used in the continuation method (Set 3).}
We anticipate that points
that satisfy the above conditions
are located throughout
the $\epsilon$ complex plane.
Some of the branch points of the analyticity domain seem to form, at
$\Im \omega = 0$, $\omega$ Diophantine, a natural boundary, as
reported in \cite{BC}, \cite{LF}.
To form this natural boundary, we expect some of the branch points
to move, as $\Im \omega \to 0$,
towards the origin with a ``speed'' depending on their position (the further
from the origin the faster they move).
The bifurcation theory analysis also predicts that if $\epsilon_0$ is a
bifurcation point for
${\cal T}_\omega$ then $-\epsilon_0$ is also a bifurcation point
since
$u(\theta; -\epsilon_0 , \omega) = u(\theta - \pi; \epsilon_0 , \omega)$
is a solution of \equ(homology) for $\epsilon = -\epsilon_0$ and
the conditions \equ(bif1), \equ(bif2) are -- verified to be --
satisfied. This prediction
can be used to test the accuracy of the numerical methods.
\picture{1b}{
The values of the solution to \equ(stdmap) along the path depicted on figure 1a.
Set 1 are the values through the first loop and Set 2 the values through
the second loop.}
To study numerically the behavior of the solution to \equ(homology),
we continued the solution between points along a
path encircling a branch point (how a branch point can be located will
be discussed later) using a non-perturbative (Newton) method.
Given the solution of \equ(homology) at
$\epsilon_0, u(\theta; \epsilon_0, \omega) = u_{\epsilon_0}(\theta)$ we write
for $u(\theta; \epsilon, \omega) = u_{\epsilon_0} (\theta) + \eta (\theta)$ :
$$\Delta_\omega [u_{\epsilon_0}] (\theta) -
\epsilon S(\theta + u_{\epsilon_0}(\theta)) = R(\theta) \EQ(remainder)$$
or, ignoring terms of order $\eta^2$:
$$\Delta_\omega [\eta](\theta) -
(\epsilon -\epsilon_0) S_u(\theta + u_{\epsilon_0}(\theta))\eta(\theta)
= - R(\theta) \EQ(newton).$$
\vfill
\eject
\picture{2a}{
Same as figure 1a.
$S(q) = \sin q.\quad \omega = \frac{\sqrt{5}-1}{2} + 0.1i$. Pad\'e approximant [
48/48], $\theta = 0.23$.}
The existence of a solution for \equ(newton) for $|\epsilon - \epsilon_0|$
small, $\Im \omega \ne 0$ and $\epsilon$ not a bifurcation point
for ${\cal T}_\omega$ is guaranteed, based on an argument using the
implicit function theorem and the contraction mapping principle.
To monitor the solution along the path we chose to evaluate
$u$ at a fixed value of $\theta$. The results are largely independent
of the choice of the observable (another choice could be the k-th
Fourier coefficient of $u$). A
problem arises only when certain values of the observable impose an
additional symmetry
on the problem (for example $u(1/2; \epsilon, \omega) = 0$ for all
$\epsilon, \omega$). The fact that we can detect the same singular
behavior using different observables is very similar to detecting
a phase transition in physical phenomena with an underlying
renormalization group structure, through the behavior of different
physical properties,
e.g. thermal and electrical conductivity.
The results of our computations, following the solution of \equ(homology)
at fixed values of $\theta$ are shown in figures 1b, 2c, 3b, 4b.
\vfill
\eject
\picture{2b}{
Detail of figure 2a. Sets 1-3 the same.
The path depicted by set 3 was found to encircle 2 branch points. Set 4
depicts a second path used for the continuation method, encircling only one
branch point.}
They are consistent with the theoretical predictions (in figure 2
where two branch points are in close proximity a path that encloses
only one branch point was constructed, depicted by Set 4 in figure 2b).
The existence of branch points of order 2 in the analyticity domain of $u$
can be detected by the positions of the poles and
zeros of Pad\'e approximants (Pad\'e approximants have also been used in many
areas of Physics to identify singular behavior -- for
an overview and details see \cite{BGM}).
According to a
-- not yet proven in general but supported by
numerical experiments and proven in particular cases -- conjecture of
John Nuttall,
high order diagonal Pad\'e approximants to
functions with a finite number of branch points
have most of their poles
and zeros along arcs
emanating from the branch points.
Other poles and zeros occur in nearby pairs (see \cite{N1} for a proof
for a particular class of functions, and \cite{N2} for the conjecture).
Although $u$ could have an infinite number of branch points,
due to numerical roundoff and truncation errors,
only the branch points closest to the origin (i.e a finite number)
are numerically computable using the series \equ(lindstedt).
Our computations confirm that for several standard-like maps, for
$\Im \omega \ne 0$, both the zeros and the poles of diagonal Pad\'e
approximants, computed for a fixed $\theta$,
lie on arcs (for the case of the standard map \cite{BM}
reported a similar behavior only for the poles of Pad\'e approximants).
The results of our computations are shown in
figures 1a, 2a, 2b, 3a, 4a.
\picture{2c}{
The values of the solution to \equ(stdmap) along the paths
depicted on figure 2b. It takes three turns to come back to the
original solution for the path depicted by Set 3 in figures 2a,b.
Set 1 are the values through the first loop, Set 2
the values through the second loop and Set 3 the values through the third
loop along this path.
It takes two turns to come back to the original solution for the path
depicted by Set 4 in figure 2b.
Set 4 are the
values through the first loop and Set 5 through the second loop along this
second path.
}
The neighborhoods of the points where the lines of poles and zeros of
the diagonal Pad\'e approximants emanate from, were found to include one or
more branch points of order 2 (using the non-perturbative continuation method).
\picture{3a}{
Same as figure 1a.
$S(q) = \sin q + \sin 3q.\quad \omega = \frac23 + 0.01i$.
Pad\'e approximant [28/28], $\theta = 0.23$.}
\picture{3b}{
Same as figure 1b.}
Knowledge of the structure of the singularities of the analyticity domain
of the series \equ(lindstedt) allows for more efficient numerical algorithms
for locating the singularities.
One possibility is based on the following observation
(see also \cite{BGM}):
If f has an isolated branch point of order 2 at $x_0$,
then for x close to $x_0$,
$f'(x) \approx A(x-x_0)^{-1/2}$ and
$ \frac{d}{dx} \ln f'(x) = f'' / f' \approx - 0.5 / (x-x_0) $.
So if a function f exhibits an isolated branch point of order 2, then the
derivative of the logarithm of $f'$ exhibits a pole
in the same position.
Similarly, if for some $n \in \integer, n \ge 0$,
$\frac{d^n}{dx^n} f(x) \approx A(x-x_0)^\gamma, \gamma < 0$
the Pad\'e approximant to
$\frac{d}{dx} \ln f^{(n)} (x) = f^{(n+1)}(x) / f^{(n)}(x)$
exhibits a pole at $x=x_0$.
Pad\'e approximants are much better suited to approximate
functions with simple poles than with isolated branch points.
\picture{4a}{
Same as figure 1a.
$S(q) = \sin q + \sin 3q.\quad \omega = \frac{\sqrt{5}-1}{2} + 0.1i$.
Pad\'e approximant [28/28], $\theta = 0.23$.}
An $[N/M]$ Pad\'e approximant for $f''/f'$ can be computed
as follows. Let [$N/M$]($x$) = $P(x)/Q(x)$ where $P$, $Q$
are polynomials of order N, M respectively and $Q(0) = 1$.
Then
$$ \frac{f''(x)}{f'(x)} = \frac{P(x)}{Q(x)} + O (x^{N+M+1})$$
which is equivalent to, for $Q(0) = 1, f'(0) \ne 0$
$$ f''(x)Q(x) = f'(x) P(x) + O(x^{N+M+1}).$$
The coefficients of $P, Q$ are determined by
matching coefficients up to order $x^{N+M}$. Since this involves
solving a linear system, condition numbers can be used to determine the
accuracy of the solution.
We found that this algorithm
gives much smaller condition numbers
than the use of
a straightforward diagonal Pad\'e approximant.
Moreover, according to our conjecture, all the poles of the
Pad\'e approximant are singularities of the function rather than
being artifacts of the method, as in the case of straightforward Pad\'e
approximants.
Our results are
shown in figures 1a, 2a, 2b, 3a, 4a.
We also implemented algorithms to compute Pad\'e approximants for
ratios of higher derivatives, but we did not find a significant
difference.
\picture{4b}{
Same as figure 1b.}
Finally, we have observed striking geometric properties for
the analyticity domain of the solution of \equ(homology) for rotation
numbers with rational real part. This suggests an underlying
renormalization group explanation of the phenomenon.
We are currently carrying out investigations in that direction.
PACS numbers: 03.20.+i, 05.45.+b
\SECTION References
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\endref
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\jour{Phys. Rev. Lett.}
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\pages{1443--1446}
\yr{1992}
\endref
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\pages{119--161}
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\by{B.~V.~Chirikov}
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\pages{263--379}
\yr{1979}
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\by{S.~N.~Chow, J.~K.~Hale}
\book{Methods of Bifurcation Theory}
\publisher{Springer--Verlag, New York}
\yr{1982}
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\paper{Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors}
\jour{Jour. Stat. Phys.}
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\yr{1992}
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\by{M.~Golubitsky, D.~G.~Schaeffer}
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\publisher{Springer--Verlag, New York}
\yr{1985}
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\by{J.~M.~Greene}
\paper{A method for determining a stochastic transition}
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\pages{1183--1201}
\yr{1979}
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\no{LT}
\by{R. de la Llave, S. Tompaidis}
\paper{Computation of domains of analyticity for some perturbative expansions from mechanics}
\jour{Physica D}
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\pages{55--81}
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\endref
\ref
\no{N1}
\by{J.N. Nutall}
\paper{The convergence of Pad\'e approximants to functions with branch points}
\inbook{Pad\'e and rational approximation, E.B.~Saff, R.H.~Varga (eds.)}
\pages{101--109}
\publisher{Academic Press, New York}
\yr{1977}
\endref
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\by{J.N. Nutall}
\paper{Letter to Stathis Tompaidis, dated January 8, 1993}
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\by{W. Rudin}
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\publisher{McGraw Hill}
\yr{1973}
\endref
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