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\TITLE Numerical calculation of domains of analyticity
for perturbation theories in the presence of small divisors
\footnote{${}^{\rm 1}$ } {\rm This preprint is available from the
math-physics electronic preprints archive.
Send e-mail to {\tt mp\_arc@math.utexas.edu} for instructions}
\AUTHOR Corrado Falcolini
\footnote{${}^2$}{ Permanent address: Dipt. di Matematica,
II Universit\'a degli Studi di Roma ``Tor Vergata'',
Via del Fontanile di Carcaricola, 00133 Roma, Italia}
\footnote{${}^3$}{ e-mail address: {\tt FALCOLINI\%40085.decnet.cern.ch}},
Rafael de la Llave
\footnote{${}^4$}{Supported in part by National Science
Foundation Grants}
\footnote{${}^5$}{ e-mail address: {\tt llave@math.utexas.edu}}
\FROM Dept. of Mathematics
University of Texas at Austin
Austin TX 78712
\ENDTITLE
\ABSTRACT
We study numerically the complex domains of validity for K.A.M.\ theory
in generalized standard mappings. We compare methods based on Pad\'e
approximants and methods based on the study of periodic orbits.
\ENDABSTRACT
\SECTION Introduction
The goal of this paper is to study numerically the complex domains
of values of $\epsilon$ for which an standard like map from the
$\real^1 \times T^1$ to itself
$$T_\epsilon (p,q) = \bigl(p+\epsilon s(q)\ ,\ q+p+\epsilon s(q) \bigr)
\EQ(standard)$$
leaves invariant topologically non-trivial circles on which the motion is,
up to a smooth change of variables, a rigid rotation by an angle $\omega$
The well known K.A.M.\ theorem guarantees that provided $\omega$ is
Diophantine, there is a ball of positive radius on which there is
such a curve.
Unfortunately, the estimates that come out of the analytical proofs
are very conservative and, given the practical importance of constants, it
is of considerable interest to devise proofs without this failing or, at
least devise reliable methods for numerical computation. Problems such
as \equ(standard) provide convenient models of the general situation.
The reason to study domains of analyticity is that, in applications one
frequently uses perturbation expansions whose behaviour and usefulness
is affected by complex singularities even if the behaviour of the true
answer is perfectly well behaved for real values of $\epsilon$.
The case when $s(q) = {1\over2\pi} \sin (2\pi q)$ and $\omega = {\sqrt5-1
\over 2}$ was studied in [BC] by deriving a perturbation expansion in
powers of $\epsilon$ and using a Pad\'e approximant
of the series. They obtained
the surprising result that the domain of analyticity was a circle.
Several other families were studied in [BCCF].
In this paper, we consider the same problem for other values of $\omega$
and for $s$ of the form of a trigonometric polynomial of low degree.
We reimplemented from scratch the Pad\'e method of [BC],
[BCCF]
using an extended precision
library --- the reasons for doing so, are discussed in the
section about the Pa\'e method
-- and we used an independent method: a complex extension of
Greene's method, which seems to afford higher precision and for which some
theoretical justification is recently available [FL] [McK3].
(We point out the the justification used in [FL] works
enven for the case when the parameters are complex.)
At the end, we propose some tentative discussion of these results
for the renormalization group point of view about the phenomena of
disappearence of circles.
\SECTION Greene's residue criterion
\SUBSECTION The basis of Greene's criterion.
In a remarkable paper [Gr] observed that, for the standard map, it is
possible to ascertain the existence of an invariant circle with rotation
number $\omega$ by computing the ``residue'' of periodic orbits
of the type $N_n/D_n$ --- that is, orbits which satisfy
$$T_\epsilon^{N_n} (p,q) = (p,q+D_N)\ N_N/D_n\to \omega$$
If we define $R= (p,q;N_n;D_n) = Tr(DT_\epsilon^{N_n} (p,q)-2)$ then,
[Gr] asserts that there is an invariant circle of rotation number $\omega$
if and only if $R(p,q;N_n;D_n)\to 0$.
The importance of this criterion arises from the fact that it is very
easy to compute periodic orbits even of relatively high periods. [Gr] used
an algorithm first described in [DeV] that exploits the fact that maps
of the form \equ(standard) are ``reversible.'' There are other algorithms
(see eg. [KM] [M]).
We will see later that the method generalizes to the complex case.
Unfortunately, Greene's residue criterion is difficult to justify
rigorously and there are strong indications that, as stated, it is false
even for standard-like mappings which are not the standard map.
Nevertheless, it is possible to prove rigorously statements that serve
as justification of practical computations.
For example, it is possible to show that:
\CLAIM Theorem(justification)
Let $T_\epsilon$ be an analytic map as before, $\omega$
is Diophantine. Assume that
$\sup_{|\Im q |\le\delta} |T_\epsilon(p,q)|\le\Gamma\le\infty$,
$\sup_{|\Im q |\le\delta} |T_\epsilon^{-1} (p,q)|\le\Gamma\le\infty$,
and that there is a mapping $K:\torus^1\to\real\times\torus^1$
with $f(K(\varphi)) = K(\varphi+\omega)$ and that $\sup_{|\Im\varphi| }
\le\delta|K(\varphi) |\le\Gamma $.
\vskip 0 em
Then, there exists a constant $D>0$ -- depending on the
Diophantine properties of the
number $\omega$ -- such that
for every periodic orbit $x$ of
type $M/N$ with $|\omega -M/N|\le 1/N $
$$|R (x)|^{1/N}\le D e^{-D\Gamma\delta^{1+\nu} |\omega - M/N| }$$
We observe that the proof of \clm(justification) in [FL] consists in
computing normal forms up to high enough order around the invariant
circle and observe that as a corollary of the proof,
one obtains also upper bounds for the residues of orbits
whose rotation number is similar to $\omega$,
%upper bounds for the residues of all
%periodic orbits close to the rotation number.
This justification, as
stated in [FL], carries over without any change to the case that the
parameters and, hence the circles, take complex values.
We also emphasize that the justification implies that all periodic orbits
with a certain rotation number close to $\omega$ will have a small residue.
In general, we expect to have many periodic orbits with the same rotation
number.
We refer to [PJ] for a discussion of the combinatorics of the calculation
of periodic orbits based on critical lines.
If, for any of them, the residue is large, we can take it as an indication
that the invariant circle has disappeared.
Notice that if the residue of an orbit is not zero, the eigenvalues of
the derivative are not 1. Hence, applying the implicit function theorem we
conclude that if for some $\epsilon_0$ we can find $x_{\epsilon_0}$ such that
$T_{\epsilon_0}^N x_{\epsilon_0} = x_{\epsilon_0} + (0,D)$ and
$|R(x_{\epsilon_0} ;N,D)| = \alpha \ne 0$,
we can find a neighborhood $|\epsilon -\epsilon_0| \le \rho$
on which we can find a unique $x_\epsilon$ satisfying
$T_\epsilon^N (x_\epsilon) = x_\epsilon + (0,D)$.
Moreover, $R(x_\epsilon; N,D)$ will be an analytic function
of $\epsilon$.
One, then, expects that the equation $|R(x_\epsilon ;N,D)| =\alpha$ will define
a smooth curve in the $\epsilon$ plane in the neighborhood where it is
defined. (It seems that $R(x_\epsilon ;N,D)$ has very large derivatives
in the cases of interest, hence, the ciurve has reasonably small derivatives.)
This remark and the partial justification of the residue criterion suggest
the following algorithms.
\CLAIM Algorithm(continuation)
Fix $\alpha >0$.
\vskip1pt
\item{1)} Find a real number $\epsilon_0$ and a periodic orbit for
$T_{\epsilon_0}$ satisfying
$$|R(x_{\epsilon_0} ;N;D) | = \alpha$$
\vskip1pt
\item{2)} Given that we know $\epsilon_n = r_n e^{i\theta_n}$ satisfying
$|(x_{\epsilon_n};N;D)| =\alpha$, fix $\theta_{n+1} >\theta_n$
$|\theta_{n+1}-\theta_n||$ small, and then find $r_{n+1}$ in such a
way that $\epsilon_{n+1} = r_{n+1} e^{i\theta_{n+1}}$ satisfies
$R(x_{\epsilon_{n+1}};N;D)|=\alpha$.
\CLAIM Algorithm(paths)
Fix $\alpha>0$ and a family of paths $\epsilon = P_\theta (t)$ with
$t\in\real$ being the parameter along the path and $\theta$ being an
index for the paths (e.g., $P_\theta (t) = te^{2\pi i\theta}
or P_\theta(t) = \theta + \imath t $
\vskip1pt
\item{1)} If we fix $\theta,t$, we find one periodic orbit $x_{\theta,t}$
of type $N/D$ performing of a certain topologically type. Then,
$R(x_{\theta,t};N,D)$ is a function of $\theta,t$.
\vskip1pt
\item{2)} For fixed $\theta$, we can consider $R(x_{\theta,t};N,D)-\alpha$
as a function of $t$ and feed it to a reliable zero finder. This gives us
a critical value $t_\theta^*$.
\vskip1pt
\item{3)} Cycling other different $\theta$'s we can obtain a curve
$\epsilon = P_\theta (t^*(\theta))$, of critical points.
Notice that \clm(continuation) is just a version of the well known
continuation methods and that it is quite well understood how to write
safeguards that examine that the conditions of the implicit function
theorem apply so that if the program runs without detecting an error,
we may be confident that indeed there is a curve in $\gamma(x_{\epsilon_0};
N;D)$ in the complex plane for which the residue of the orbit of the
topological type takes the value
The method of \clm(paths) is somewhat more delicate to implement, and is
slower to run.
The main reason is that, for fairly high values of the parameter there are
different orbits of type $N/D$ and it is necessary to take special
precautions to ensure that all the evaluations of $R(x_{\theta,t};N,D)$
required by the zero finder are on the same family, specially if the
evaluations are on fairly separate values.
We avoided this problem by ensuring that the intervals for $t$ were
not very large, (they were centered in the last successful value) and we
kept the last found periodic orbit as a guess.
We also point out that \clm(paths) is somewhat slower. Nevertheless, we
found it useful to have several algorithms so that comparing their results
could give us confidence on their reliability. As discussed later, we also
compared the results obtained by using different algorithms for finding
zeros, and computing orbits.
We emphasize that, given the justification outlined in
\clm(justification) for reasonably high $N,D$ we should consider
the critical lines produced by the algorithm as lying
outside the domain of analyticity, so that the most sensible
approximation of the true domain of analyticity we can form is the
intersection of the regions bounded by the computed curves.
Even if each of the curves is smooth, the final domains obtained taking
intersections could very well have cusps. Taking the limit
of a finite number of intersections allows that the final result
is very complicated.
The algorithms \clm(continuation), \clm(paths) depend on having
reliable methods for computing periodic orbits.
We have used two. First, we observe that Greene-DeVogelere method
can be generalized to complex maps. We quickly review appendix A of
[Gr] to check that the method described there only involves algebraic
manipulations that are valid for complex numbers.
This method is based on the observation that $T_\epsilon$ in
\equ(standard) can be written $T_\epsilon = I_{1,\epsilon}\circ
I_{2,\epsilon}$ with $I_{1,\epsilon}^2 = I_{2,\epsilon}^2 = Id$.
$$\eqalign{
I_1 (p,q) & = \bigl( p- \epsilon s(q),-q\bigr)\cr
I_2 (p,q) & = (p,-q+p)\cr}$$
We denote by $\Omega_1$ the set of points left fixed by $I_1$. If one
point $(p_0,q_0) \in\Omega_1$ and $T_\epsilon^N (p_0,q_0)\in\Omega_1$
then, $T_\epsilon^{2N} (p_0,q_0) = (p_0,q_0)$. In effect
$$\eqalign{ T_\epsilon^{2N} (p_0,q_0)
&= T_\epsilon^{N-1}\circ I_{2,\epsilon} \circ I_{1,\epsilon} T_\epsilon^{N-1}
I_{2,\epsilon} I_{1,\epsilon} (p_0,q_0) =\cr
&= T_\epsilon^{N-1} \circ I_{2,\epsilon} \circ T_\epsilon^{N-1}
\circ I_{2,\epsilon} (p_0,q_0)\cr}$$
Since $I_{2,\epsilon}\circ T_\epsilon = I_{2,\epsilon}^2 \circ I_{1,\epsilon}
= I_{1,\epsilon}$
and
$T_\epsilon \circ I_{1,\epsilon} = I_{2,\epsilon} I_{1,\epsilon}^2 =
I_{2,\epsilon}$ we obtain that $T_\epsilon^{2N} (p_0,q_0)$.
The set $\Omega_1$ can be calculated explicitly since, using the form of
$I_1$ we obtain $(p_0,q_0) \in\Omega$ if and only if
$$\eqalign{ & p_0-\epsilon s(q_0)= p_0\cr
&q_0 = -q_0 +k\cr}$$
when $s(0) = s(1/2) =0$, which is the case for the trigonometric
polynomials we have considered, the above equations are equivalent to
$q_0=0$, $q_0=1/2$.
We will refer to these two vertical lines as the ``critical lines.''
The algorithm for searching for periodic orbits consists in searching
along a critical line for the points that, after a certain number of
iterations, come back to another critical line. This amounts to
finding the zeros of a function of one variable.
In the case that the variable is real there are excellent zero finders
that exploit the mean value property.
In the complex case, however, one has to use other methods such as the
secant method.
We however found that one obtains better results if one tries to
minimize the target function using a Powell method (see [PFTV])
considering the complex variable as two real variables.
Notice that the secant method requires a reasonably good guess to converge
to the solution, specially in the case that the functions whose zeros we
are computing is very rapidly oscillating. For our purposes this can
be achieved by taking as a guess for the location of the periodic orbit
for one value of the parameter, the location of the periodic orbit for
a previously computed similar value and then increasing. Nevertheless, the
method becomes delicate to use and we preferred to use Powell's method
for most of the calculation.
Another algorithm we used was based on the variational principle for periodic
orbits that plays an important role in Aubry-Mather theory.
\CLAIM Lemma(variational)
A sequence of angles $\{q_1,\ldots,q_N\}$ is the projection of a periodic
orbit $\{(p_1,q_1),\ldots,(p_N,q_N)\}$ of type $M/N$ for \equ(standard)
if and only if $(q_1,\ldots,q_N)$ is a critical point of the function
$${\cal S} (q_1,\ldots,q_N) =
\sum_{i=1}^{N-1} {1\over2} (q_i-q_{i+1})^2 + \epsilon \cS (q_i)
+ {1\over2} (q_N -q_1+M)^2 +\epsilon s(q_i)$$
where $S' (q) = s(q)$. (Notice that the condition
$ \int s(q) dq = 0 $ ensures that we can find such an $S$
defined on the circle.
We observe that this variational principle generalizes to the complex case
if we consider
$$\widetilde{\cal S} (q_1,\ldots,q_N) =
{\cal S} (q_1,\ldots,q_N) \overline{{\cal S} (q_1,\ldots,q_N)}$$
because then, we obtain, considering $\varphi_i,\overline{\varphi}_i$ as
independent variables
$$\eqalign{
{\partial \widetilde{\cal S} (q_1,\ldots,q_N) \over
\partial q_i} & = \overline{ {\cal S} (q_1,\ldots,q_N)} \
{\partial {\cal S} (q_1,\ldots,q_N) \over \partial \varphi_i} \cr
{\partial \widetilde{\cal S} (q_1,\ldots,q_N) \over
\partial \overline{q}_i} & = {\cal S} (q_1,\ldots,q_N) \
{\partial \overline{ {\cal S} (q_1,\ldots,q_N)} \over
\partial \overline{q}_i} \cr
&= {\cal S} (q_1,\ldots,q_N)
\overline{ \left( {\partial {\cal S} (q_1,\ldots,q_N) \over \partial q_i}
\right) } \cr}$$
If we add a large enough constant to $\cal S$ that ensures that
${\cal S} (\varphi_1,\ldots, \varphi_n) \ne 0$,
we see that the variational equations for $\widetilde{\cal S}$ with
$\varphi_i,\overline{\varphi}_i$ as independent variables.
Inspired by the paper [Go], which uses the gradient flow
of the action as a functional in the space of orbits
to prove
the existence of critical points we implemented the system of differential
equations
$$ {dq_i \over dt} = - {\partial\over\partial\varphi_i}
\widetilde{\cal S} (q_1,\ldots,q_N) =
= \overline{{\cal S} (q_1,\ldots,q_N)} \
{\partial {\cal S} (q_1,\ldots,q_N) \over \partial q_i}
\EQ(flow)$$
The solutions of these equations --- implemented using e.g., an standard
Runge-Kutta solver --- converge rather quickly to approximate
solutions, specially if we take as initial conditions points which are
close to being a solution (e.g., the solutions for similar parameter
values). Notice that the R.H.S.\ of the equations is very easy to
evaluate since most of the terms that are obtained taking the derivative
of $\cal S$ vanish.
In this case, the periodic orbits can be compared with those
obtained using the secant method and verified directly to $x$ orbits.
\SECTION The Pad\'e method
In [BC] Berretti and Chierchia suggested the use of Pad\'e approximants
to study the analyticity domain of the expansion in powers of $\epsilon$
of the solutions of
$$\Delta_\omega u_\epsilon (x) + \epsilon s \bigl( x+u_\epsilon (x)\bigr)
= 0
\EQ(conjugating)$$
where $\Delta_\omega$ is the operator defined by
$$\Delta_\omega u_\epsilon (x) = u_\epsilon (x+\omega) - 2u_\epsilon
(x) + u_\epsilon (x-\omega)$$
where $u_\epsilon :T^1 \to \real$ satisfies
$$u_\epsilon (x+1) = u_\epsilon (x)
\EQ(normalization)$$
The function $u_\epsilon$ is called the ``hull function'' by Aubry and if
we define
$$K_\epsilon (x) = \left( {u_\epsilon (x) - u_\epsilon (x-\omega)-\omega \atop
u_\epsilon (x) +x}\right)
\EQ(K)$$
then:
$$T_\epsilon \circ K_\epsilon (x) = K_\epsilon (x+\omega)
: K_\epsilon (x+1) = K_\epsilon (x)
\EQ(intertwining)$$
so that $K_\epsilon$ conjugates the motion on an invariant circle to a
rotation $\omega$. $u_\epsilon (x)+x$ gives the conjugacy of a
rotation of the motion of the first component.
Hence, finding solutions of \equ(conjugating) for a fixed $\epsilon$
implies that there is indeed a circle with the motion on it being
conjugated to a rotation.
Conversely, finding a solution of \equ(intertwining) implies that
there is a solution of \equ(conjugating) as can be verified by
direct calculation.
Hence, we define the domain of validity of K.A.M.\ theorem as the domain
of $\epsilon$ for which it is possible to find a solution of
\equ(intertwining) or \equ(conjugating).
Using Birkhoff theorem [Ma], and Herman's theorem on conjugacy to
rotations of smooth diffeomorphisms of the circle for Diophantine
rotation numbers, [He],
one can show that for real $\epsilon$ and Diophantine $\omega$, the
existence of analytic solutions of \equ(conjugating) is equivalent
to the existence of topologically nontrivial invariant circles for
\equ(standard) with rotation number $\omega$.
Nevertheless, for complex $\epsilon$, we do not know any version of
Aubry Mather theorem that could show that invariant circles are graphs.
Nevertheless, Herman's theorem still applies, so that, in the complex
case, can guarantee that \equ(intertwining) is equivalent to existence
of topologically invariant circles but, nevertheless it seems reasonable
to assume that they agree.
It is not excluded that \equ(conjugating) does not admit a solution
but that, nevertheless \equ(intertwining) does. That will happen,
for example, if the invariant circle is not a graph.
We point out that if $u_\epsilon (x)$ is a solution of \equ(conjugating)
so is $u_\epsilon (x+\eta_\epsilon) - \eta_\epsilon$ but, except for
this, the solution is unique. We can and will choose always a solution
satisfying a normalization condition
$$\int u_\epsilon (x)\,dx = 0
\EQ(gauge)$$
A power series expansion for
$u_\epsilon (x) = \sum_{i=1}^\infty \epsilon^i, u^{[i]} (x)$
can be computed by matching powers of $\epsilon$ if we have a way of
computing the expansion in powers of
$s(x+u_\epsilon (x)) = \sum_{i=0} \epsilon^i s^{i} (x)$
in terms of the expansion in powers of $\epsilon$ for $u$.
Similar methods have been used in [P], [GP], [BC].
Then, equation \equ(conjugating) becomes
$$\Delta_\omega u^i (x) + s^{i-1} (x) = 0 \ ;\qquad u_0 (x) =0
\EQ(recursive)$$
Notice that $s^{i-1}$ can be computed in terms of $u^0,\ldots,u^{i-1}$
so that \equ(recursive) should be considered as an equation that allows
the computation of $u^i$ given that we know $u^0,\ldots,u^{i-1}$.
The theory of inversion of the operator $\Delta_\omega$ is worked out
in detail in severl places, for example [SM], \S32. We just recall that,
if $\omega$ is Diophantine
and $s^{i-1}$ is analytic and satisfies
$$\int s^{i-1} (x)\, dx = 0
\EQ(compatibility)$$
it is possible to find a unique $u^i$ solving \equ(recursive) and
\equ(gauge).
To complete the proof of the claim that \equ(recursive) can be solved
by matching orders, we only have to show that if $u^0,\ldots,u^{i-1}$
solve \equ(recursive), then $s^{i-1}$ satisfies \equ(compatibility).
To prove that, we observe that if $u$ is a periodic function and
$\Delta_\omega u(x) + s\left(u(x)+ x\right) = R(x)$, then
$$\eqalign{ \int R(x) \bigl( 1+u'(x)\bigr)
& = \int u(x+\omega) +\int u(x-\omega) - 2\int u(x) + \cr
&\qquad \int u(x+\omega) u' (x) +\int u(x-\omega) u' (x)
- 2\int u(x) u' (x)\cr
&\qquad + \int s(x+u(x)) (1+u' (x))\cr}$$
The first three terms cancel out and, realizing that,
integrating by parts we have,
$\int u(x+\omega) u' (x) = \int u (x) u' (x-\omega)$
we can write the last four terms terms as
$$\int {d\over dx} u(x) u(x-\omega)
- \int {d\over dx} (u(x))^2 + \int {d\over dx} s(x+u(x))$$
Hence, the integral vanishes as claimed.
If we denote by $u_\epsilon^{[\le i-1]} (x)
= \sum_{j=0}^{i-1} u^j (x)\epsilon^j$ and, analogously,
$s_\epsilon^{[\le i-1]} (x) = \sum_{j=0}^{i-1} s^j (x)\epsilon^j $
we observe that:
$$s(x+u_\epsilon^{[\le i-1]} (x))
= s_\epsilon^{[\le i-1]} (x) + O (\epsilon^i)$$
Hence
$$\eqalign{& \Delta_\omega u_\epsilon^{[\le i-1]} (x)
+ \epsilon s(x+u_\epsilon^{[\le i-1]} (x)) \cr
&\qquad = \Delta_\omega u_\epsilon^{[\le i-1]} x
+ \epsilon s_\epsilon^{[\le i-2]} (x) + \epsilon^i s^{i-1} (x)
+ O(\epsilon^{i+1})\cr}$$
If the $u^i$ are determined solving \equ(recursive)
$\Delta_\omega u^{[\le i-1]} (x) +\epsilon s_\epsilon^{[\le i-2]} (x)=0$.
Hence, applying the previous argument, we obtain
$$\int (\epsilon^i s^{i-1} (x) + O(\epsilon^{i+1}))
(1+u_\epsilon^{[\le i-1]} (x)')=0$$
Hence
$$\epsilon^i \int s^{i-1} (x)\, dx = O(\epsilon^{i+1})$$
and therefore
$\int s^{i-1} (x)\, dx =0$
as claimed.
We furthermore claim that it is possible to implement very efficient
algorithms that implement the recursion described above.
We observe that $\Delta_\omega \exp (2\pi ikx) =
2(\cos (2\pi k\omega)-1) \exp (2\pi ikx)$
so that if we discretize the $u^i$'s in terms of Fourier series the linear
equations \equ(recursive) to be solved are diagonal.
The $s^i$ can be computed in terms of the $u^i$ very efficiently if we use
a trick that is described in [Kn], Vol.2, p.507.
We observe that if we denote by
$$\eqalign{ E(\epsilon,x) & = \exp (2\pi ik u_\epsilon (x)) \cr
{\partial\over\partial\epsilon} E(\epsilon,x) &
= 2\pi ik\exp (2\pi ik u_\epsilon (x)) {\partial\over\partial\epsilon}
u_\epsilon (x) \cr
& = 2\pi ik E(\epsilon,k) {\partial\over\partial \epsilon}
u_\epsilon (x)\cr}
\EQ(diffeq)$$
If we expand $E(\epsilon,x) = \sum_{j=0} E^i (x) \epsilon^i$, we obtain
matching the coefficients of $\epsilon^N$ in \equ(diffeq)
$$(N+1) E^{N+1} (x) =
2\pi ik\sum_{j=0}^N E^{N-j} (x) (j+1) u^{j+1} (x)
\EQ(recursionexp)$$
Together with the initial condition $E^0(x)=1$, \equ(recursionexp) allows
one to recursively compute $E^{N+1}$ once one knows $E^0(x) ,\ldots,
E^N(x)$.
Of course, once we have computed the complex exponential, it is quite easy
to compute trigonometric functions. Notice that the recursion relations
\equ(recursionexp) only require multiplication, and addition of Fourier
series and multiplication of Fourier series by a scalar.
By examining the recursion relations \equ(recursionexp) and \equ(recursive)
it is possible to show
\CLAIM Proposition(degrees)
If $s(x)$ in \equ(standard) is a trigonometric polynomial of degree $d$,
then
$$\eqalign{
&\hbox{\rm degree }(u^i) = 2di-d\qquad i\ge 1\cr
&\hbox{\rm degree }(s^i) = 2di+d\qquad i\ge 0\cr}$$
\PROOF
Proceeding inductively, we assume that the conclusions are true for
$i\le N$.
Then, \equ(recursionexp) shows $\hbox{deg}(E^N) = 2Nd$.
Using the usual addition formulas for trigonometric functions, we see that
$\hbox{deg }s^N = 2Nd+d$ and observing that \equ(recursive) is
diagonal on Fourier series $\hbox{deg }U^{N+1} = 2Nd+d$
which is the inductive hypothesis for $i=N+1$. A small calculation
shows that the hypothesis is satisfied for $i=1$.
\QED
\REMARK
We emphasize that the existence of a K.A.M.\ torus for a value of
$\epsilon$ is equivalent to the existence of a function satisfying
\equ(conjugating) so that we should consider the expansion
$u_\epsilon = \sum \epsilon^j u^j (x)$ as an expansion taking place
in a certain space of functions.
This is obviously related but, in principle not equivalent to considering
the domains of analyticity of the functions obtained specializing the
expansion for values of $x$.
For example, for the standard mapping $u^i (1/2) =0$ so that for
$x=1/2$, the series expansion vanishes identically and, in particular
converges.
If we denote by $\Omega_{\epsilon,\theta}$ the domain of convergence of
$K_\epsilon (\theta)$ solving \equ(intertwining) the domain of validity
of the K.A.M.\ theorem is
$$\Omega_\epsilon = \bigl\{ \epsilon \mid \exists \ \delta >0\ ,\
\{\epsilon \} \times \{ |\Im \theta| <\delta\} \subset
\Omega_{\epsilon,\theta}\bigr\}$$
In practice, however, it is more practical --- following [BC] --- to
compute $\Omega_\epsilon (\theta)$ which is the domain of convergence
of $u_\epsilon (\theta)$ for a fixed $\theta$.
Notice that \equ(intertwining) implies that if $x$ is in the domain
of $K_\epsilon$, so is $x+\omega$. It follows that the domain of $K_\epsilon$
in the $\theta$ variable should be a strip.
On the other hand, from \equ(K)
it follows that for a fixed $\epsilon$, $\Omega_\theta^u$, the domain
of $u_\epsilon$ in the $\theta$ variable is related to $\Omega_\theta^K$
the domain of $K$ in the $\theta$ variable by
$$\Omega_\theta^K = \Omega_\theta^j \cap \Omega_\theta^u -\omega$$
In the examples we have considered the numerical results suggest that
$\Omega_\theta^K = \Omega_\theta^u$ and, hence, both are a strip around
the real axis.
Notice that the fact that the domain of convergence in the $\theta$
variable can be computed for fixed $\epsilon$ also using the Pad\'e method.
This could be considered a check of the computation of the perturbation
expansion and of the Pad\'e method, specially because there are
renormalization group predictions about the scaling behaviour of the domain
of convergence of the function $K$.
Given a series $S(\epsilon) = \sum \epsilon^i S_i$ we define
the Pad\'e approximant of type $N,M$ as the polynomials $P(\epsilon), D
(\epsilon)$ such that:
$$\eqalign{&\degree (P)=N\ ,\quad
\degree (D) = M\ ;\quad D(0)=1\cr
&P(\epsilon) /D(\epsilon) = S(\epsilon) + \theta (\epsilon^{N+M+1}) \cr}
\EQ(quotient)$$
The fact that $P,D$ exist can be obtained by observing that
\equ(quotient) is equivalent to
$$S(\epsilon) D(\epsilon) = P(\epsilon) + O (\epsilon^{N+M+1})$$
and, hence, expressing it in terms of the coefficients of the polynomials
$$\eqalign{ & S_i + \sum_{j=1}^M S_{i-j} D_j = P_j\qquad i\le N\cr
&S_i + \sum_{j=1}^M S_{i-j} D_j = 0\qquad N< i< N+M\cr}
\EQ(linearsystem)$$
The second equations determine $D_j$ and, substituting them on the first,
we can obtain the $P_i$.
Even if it is quite possible to solve \equ(linearsystem) directly, it is also
possible to derive recursion relations for the Pad\'e approximants for
different $M,N$ that allow a rapid calculation.
We have, in general, preferred to use a direct solution of the linear
system \equ(linearsystem) because it allows us to obtain an estimate
of the condition of the problem.
It so seems that $u^i(1/2-\sigma)$ converges in a finite domain so that
$K'_\epsilon (1/2)$ has a finite domain.
The basis of the Pad\'e method lies on the observation that the Pad\'e
approximants frequently converge on much larger domains than the Taylor
expansion since the approximating class can accommodate bad behaviours
such as poles that cannot be accommodated by a Taylor expansion. Hence,
frequently, the domains of analyticity of a function can be as
ascertained by looking for the zeros of the denominator of the Pad\'e
approximant.
This can be justified in several cases, (see [BG] vol. I chapter 6
for a review of the many results about convergence of Pad\'e series
on the full domain that are known).
Unfortunately,
there are cases (see e.g., [BO], p.400) where the Pad\'e approximants
of type [M:1] converge in a domain strictly smaller than the ball
of convergence of the Taylor expansion or, the more elaborate Gammel's
counterexample (see e.g. [BG] vol I. p 284 ff. )
Besides the theoretical considerations of existence of counterexamples,
there are several reasons that make the Pad\'e approximants delicate
to compute. We refer to [BG] for a very complete discussion but let
us point some and the precautions we have taken to avoid them:
$\bullet$\quad The system \equ(linearsystem) tends to be very
ill-conditioned so that small numerical errors in the coefficients of the
series could be enormously amplified.
$\bullet$\quad The computation of zeros which have larger magnitude
then other zeros requires a large precision. This can be understood if
we consider the function $S(\epsilon) = {1\over 1-\epsilon} +
{1\over 1-a\epsilon}$, the $i^{th}$ Taylor coefficient is $S_i = 1+a^i$
so that if $a\ll 1$, the information to compute the zero $1/a$, is
contained in the last decimal figures.
To the difficulties, we may add the fact that polynomials with random
coefficients tend to have their zeros on circles. Hence, it is difficult
to assess the validity of a circle using the Pad\'e method.
We implemented the Pad\'e method using the high precision library of the
public domain program {\tt PARI}.
A first implementation used macros to generate function calls. Significant
parts of the code were tested on previous programs [DL]. Later, we
produced another implementation using the {\tt GP} programming language
that is part of the {\tt PARI} system.
One advantage of doing so was that the language has primitives to
manipulate polynomials so that the algorithms to compute the recursion
relation could be coded very tersely. (Unfortunately, taking care of
starting and final values made the programs difficult to read.)
In the {\tt C} language implementation we could obtain estimates of the
condition number and made sure that the precision of the numbers was
enough to obtain meaningful results.
In the case that $s(q)$ in \equ(standard) is
a trigonometric polynomial that contains
only odd powers of $\sin$, we have
$s(q+1/2) = - s(q)$. Hence $T_\epsilon (p,q) = T_{-\epsilon} (p,q+1/2)
- 0,1/2)$. Using this symmetry we can conclude that the domain
in the $\epsilon$ plane for which circles exist
should be invariant under reflections accross the origin.
It is possible to
use this symmetry to decrease the number of coefficients in the expansion to be
handled and, hence, increase the speed and accuracy.
We decided not to do that so that we can study $s$'s which contain
even frequencies with the same programs.
Also the consequences of the symmetry can be
used as an verification of the accuracy achieved.
A simple trick that seems to increase the precision of the calculation
is to use instead of power series in the variable $\epsilon$ to use
power series in the variable $\epsilon/\rho$ where $\rho$ is a number
chosen phenomenologically so that the series has a radius of convergence
of about 1. The choice of $\rho$ was done by performing the calculation
once, estimating the radius and restarting the calculation with $\rho$
chosen to be the value suggested by the first calculation.
We also point out that the zeros of the denominator need not be a
singularity of the approximant if they are a zero of the numerator also.
Hence, we implemented a second pass which eliminated the zeros that
also correspond to an approximate zero of the numerator.
In many practical applications of Pad\'e methods, it is also customary to
consider as spurious the zeros that are far away from those of other
approximants. We have not implemented these methods.
\SECTION Results
Our results can be summarized in the accompanying pictures.
In Figure 1, we represented the zeros of the Pad\'e approximants of type
$[95:95]$ for the angles $\theta = {i\over 10} + 0.03456$ as well as the
curve of critical residue for orbits of type $18/13$ evaluated for the
standard map. Figure~2 represents the same calculation with the zeros of the
denominator corresponding to a very small numerator removed.
The map used, the degree of the Pad\'e approximant and the type of the
periodic orbit are indicated on the line above the circle.
In Figure 3 we represented several critical curves corresponding to
different orbits for the standard map.
Figures 4, 5, 6 present the results of similar calculations
performed for the standard like map with
$$S(x) = {1\over2} \sin (2\pi x)
+ {1\over 20\cdot 2\pi} \sin (2\cdot 2\pi x)$$
while Figures~7, 8, 9 present the calculations for
$$S(x) = \sin (2\pi x) + {1\over 30\cdot 2\pi} \sin (3\cdot 2\pi x)\ .$$
We draw attention to the fact explained in the previous discussion that
the zeros of largest modulus violate the symmetry requirements.
Nevertheless, those of small modulus satisfy it to a very high accuracy.
As expected, the zeros of high modulus do not seem to be reliable.
Let us highlight some important points
A)\quad The results seems to suggest that the domain of analyticity of
the $\epsilon$ expansion of $K_\epsilon$ for fixed $\theta$ is
independent of $\theta$. The analyticity domain for $u_\epsilon$ seems
to be constant except in $\theta = 1/2$.
B)\quad For the standard map, the domains of critical residue are nested
onto each other in the same way that they intersect on the real line.
That is, the domains of critical residue do not intersect.
C)\quad The results seems to indicate that the domains of analyticity computed
by the Pad\'e method and by taking the intersections of the domains bounded
by the critical residue curves seem to agree.
Point C) is interesting for practical applications since the complex
extension of Greene's method seems to be much easier to implement and
has the capability of dealing with very asymetrical domains.
We point out that, according to the complex Greene method, the domain
of analyticity for the $\epsilon$ expansion of the standard map
is {\it not\/} a circle. The vertical and horizontal diameters differ by
about 1\% which internal consistency checks suggests is between one and
two orders of magnitude bigger than the reliability of the complex
Greene method.
Perhaps the most interesting feature uncovered by our calculations is that
for some families the critical residue lines cross so that for certain
values of the parameters orbits of high period have not become hyperbolic
whereas some of the low period ones have already experienced the
transition.
This can be interpreted as indicating that the dynamics induced by the
renormalization group in the space of maps has more complicated
features than just the McKay fixed point so that successive iterates
of the R.G.\ transformation are on different sides of the co-dimension
one surfaces obtained by requiring the residue of a particular orbit to
have a critical value.
Notice also that, according to the simple renormalization group scenario,
the codimension one surfaces in the space of maps
$\Sigma_n$ defined by the condition
that the periodic orbit in the critical line of period
$F_n$ becomes hyperbolic are
images of each other under the renormalization group and
accumulate towards the stable manifold
in a well defined order. Hence, the order in which the
orbits become critical should be a universal property.
The numerical evidence preesented in this paper seems
to corroborate these predictions for the
standard map family and for families close to it.
Nevertheless, it seems to
contradict both predictions for families of the form
\equ(standard) but with noticeably different functions.
We consider this to be evidence that the
standard renormalization group scenario has
only a local validity and that the renormalization group operator has
a dynamics with more complicated features than just a saddle point.
More evidence along these lines and theoretical discussion
from the point of view of the R.G. method
reaching similar conclusions
can be found in [W1], [W2] and [KMcK].
We also point out that the fact that the domain of
analyticity appears to be smooth for families such as the standard map
can be explained by assuming that the domain
of analyticity of the invariant circle
in the unstable manifold of the renormalization group --
which in the complex extension is
a manifold of complex dimension 1 -- has an smooth boundary.
Then, the fact that the boundary becomes non-smooth is,
again an indication that
by changing the family,
we change from a universality class to another.
Given the fact that the order of the crossings seems to change in
a quite complicated way,
It is natural to conjecture that this change of
universality class is also given
by the fact that the renormalization group
has a complicate dynamics.
If this conjecture were true, it would have profound implications
for the dynamics on the unstable manifold of the fixed point.
We recall that restricted to the unstable manifold,
the renormalization group transformation is
an analytic map of one complex variable and that the
domain of existence of the invariant circle is the
domain of attraction of the fixed point.
A theorem of Brolin
(see [Br] Thm. 9.1) says that for rational transformations
on the Riemman sphere, the only possible boundaries of
domains of attraction for first order fixed points that have tangents are
circles or straight lines.
We think it could be very useful to study in detail the global
dynamicss of the renormalizaation group
restricted to the global stable manifold of McKay's
fixed point.
It is also worth remarking that, independently of the
conjecture about the smoothness of the
the boundary, this boundary of
of attraction of the trivial fixed point
is an invariant set.
The points on it, neither converge to the
trivial fixed point under succesive renormalizations nor
escape to infinity. This suggest that the invarinat
circles associated with them will
have rather peculiar scaling properties.
\SECTION Acknowledgements
We thank E.~Cheney, G.~Baker,
A.~Berretti, L.~Chiercia, A.~Celletti, R.~McKay and M.~Muldoon for
discussions. We have used the public domain software packages
{\tt PARI} and {\tt XGRAPH} and thank the authors for making available
such fine tools. The work of R.~de la Llave has been supported by
National Science Foundation grants.
\SECTION References
\ref
\no{Br}
\by{H. Brolin}
\paper{Invarinat sets under iterations of rational functions}
\jour{Arkiv for Matematik}
\vol{6}
\pages{103-144}
\yr{1965}
\endref
\ref
\no{BaG}
\by{ G. Baker,M. Graves--Morris }
\book{Pad\'e Approximants}
\publisher{Addison Wesley}
\yr{1981}
\endref
\ref
\no{BO}
\by{ C.M. Bender, S.A. Orszag}
\book{ Advanced Mathematical Methods for Scientists and Engineers}
\publisher{ McGraw Hill, N.Y.}
\yr{ 1978}
\endref
\ref
\no{BC}
\by{A. Berretti, L. Chierchia}
\paper{On the complex analytic structure of the golden invariant curve for the standard map}
\jour{Nonlinearity}
\vol{3}
\pages{39-44}
\yr{1990}
\endref
\ref
\no{DeV}
\by{R. De Vogelaere}
\paper{On the structure of symmetric periodic solutions of conservative systems, with applications}
\inbook{Contributions to the theory of nonlinear oscillations vol. IV}
\publisher{Princeton Univ. Press}
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