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X\fi\fi X \if T\isinbookJPE In ``\unhbox\bookboxJPE\unskip''\if T\isbybookJPE, X\else\period \fi\fi X \if T\isbybookJPE (\unhbox\bybookboxJPE\unskip)\period \fi X \if T\ispublisherJPE \unhbox\publisherboxJPE\unskip \if T\isjourJPE, \else\if XT\isyrJPE \ \else\period \fi\fi\fi X \if T\istoappearJPE (To appear)\period \fi X \if T\ispreprintJPE Preprint\period \fi X \if T\isjourJPE \unhbox\jourboxJPE\unskip\ \fi X \if T\isvolJPE \unhbox\volboxJPE\unskip, \fi X \if T\ispagesJPE \unhbox\pagesboxJPE\unskip\ \fi X \if T\isyrJPE (\unhbox\yrboxJPE\unskip)\period \fi X X} X X%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X%%% SOME SMALL TRICKS%%%%%%%%%% X\def \breakline{\vskip 0em} X\def \script{\bf} X\def \norm{\vert \vert} X\def \endnorm{\vert \vert} X \def\cite#1{{\rm [#1]}} X \def\bref#1{{\rm [~\enspace~]}} % blank ref cite X \def\degree{\mathop{\rm degree}\nolimits} X \def\mod{\mathop{\rm mod}\nolimits} X \def\bT{{\bf T}} X \def\frac#1#2{{#1\over#2}} X \def\Norm{{\|~\enspace~\|}} X\TITLE COMPUTATION OF DOMAINS OF ANALYTICITY XFOR SOME PERTURBATIVE EXPANSIONS OF MECHANICS X\footnote{$^1$} X{\baselineskip=12pt\rm Preprint available through the mathematical physics electronic Xpreprint archive. Send mail to {\tt mp\_arc@math.utexas.edu} for details.} X\ENDTITLE X\AUTHOR XRafael de la Llave X\footnote{$^2$}{Supported by NSF grants.} X\footnote{$^3$}{e-mail: {\tt llave@math.utexas.edu}} X\FROM XDepartment of Mathematics XThe University of Texas at Austin XAustin, TX 78712 X\AUTHOR XStathis Tompaidis\footnote{$^4$}{e-mail: {\tt stathis@math.utexas.edu}} X\FROM XDepartment of Physics XThe University of Texas at Austin XAustin, TX 78712-1081 X\ENDTITLE X\ABSTRACT XWe compute the domain of analyticity Xof some perturbative expansions for invariant circles Xappearing in mechanics. We use Pad\'e approximants for the Xperturbative Xexpansions and Xintroduce methods to ascertain their domain of convergence. XWe also use non-perturbative methods based on direct Xcomputation of the invariant circles Xand, in analogy with Greene's criterion, approximation by Xcircles with rational rotation. XWe find that the domains computed by all the methods agree within Xthe limits of accuracy. The analytic structure of the expansions Xfor rational frequencies is clarified rigorously. X\ENDABSTRACT X X X\SECTION Introduction X XThe long term behavior of deterministic systems is very often studied Xby finding landmarks that organize the dynamics. X XAmong them, ones with very drastic effect are invariant circles. For Xexample, if the phase space of the system is two dimensional, existence Xof an invariant circle implies long term stability. X XIt is, therefore, not surprising that there has been a lot of Xeffort devoted to the computation of such objects. X XOne of the first and still widely used is the Poincar\'e--Lindstedt Xmethod (see, e.g., \cite{Po}, \S123 ff. or \cite{RA} for a computer Ximplementation) which consists on finding recursively the coefficients Xof an expansion of a parametrization of the invariant circle in powers Xof a small parameter. X XOne should notice that the study of the sense in which these series are Xvalid is rather subtle X(see e.g., \cite{Po}, \S146 for examples of Xdivergence, \S122 for different concepts of ``convergence''). X XGiven the practical importance of those invariant circles, it is Xinteresting to investigate the actual domains of analyticity of the functions Xdefined by those series. This can give a measure of confidence on Xapproximate calculations and, one could hope, also shed some light Xon the scenarios Xfor the breaking up of the validity Xof the conclusions drawn from perturbation theory. X XSimilar problems appear frequently Xin theoretical physics, for example in statistical mechanics Xwhen the boundaries of the domain of analyticity of Xfunctions defined perturbatively -- e.g. by low temperature Xexpansions -- correspond to phase transitions. XTherefore, besides their intrinsic interest, Lindstedt series can Xbe considered as a useful model for other series in theoretical physics Xsuch as low temperature perturbative expansions. In this context, one Xshould remark that there is considerable evidence that the Xphenomena happening at breakdown of Lindstedt series are very Xsimilar to those happening at breakdown of other series, both of them Xcan be interpreted as phase transitions whose main analytical features Xcan be described and predicted with the help of a renormalization group Xpicture. (See e.g., \cite{McK1}.) X XWe point out that in the case that the invariant circles X are ``normally hyperbolic'' X--- in particular attractive --- Xthere is a well developed theory to prove convergence of the expansions X\cite{Fe}. In other cases, convergence can be proved using the much Xsubtler K.A.M. theory (see \cite{Ze1}, \cite{Ze2}, \cite{Bo} for surveys). XThe latter is the usual case in the dynamical systems arising in classical Xmechanics as hamiltonian perturbations of integrable systems. XUnfortunately, in both cases the practical domains provided by these Xtheories are very conservative (much more so in the case of K.A.M. Xtheory) and throw little light on the behavior to be expected at Xbreakdown. X XRecently, in \cite{BC}, \cite{BCCF} it was proposed that for some of Xthese series arising in classical mechanics, one could use the so called XPad\'e method that is to compute the zeros of the Pad\'e approximant Xas a reasonable approximation to the domain of analyticity. This procedure, Xwhich we will call henceforth {\sl the Pad\'e method} Xhas been widely used in theoretical physics \cite{BGM}. X XUnfortunately, the Pad\'e method, even if very successful in practice, Xdoes not have a complete mathematical justification. Moreover, for the series Xin celestial mechanics very frequently the size of the coefficients of the Xperturbative expansion, vary widely in an erratic fashion. This makes Xthe numerical computation of Pad\'e approximants more difficult and throws Xsome doubt about the validity of the final result. One verification Xof the Pad\'e method by comparing its results with those of Xother methods was undertaken in \cite{FL1}. In that paper, Xthe Pad\'e method was applied to models similar to those considered in X\cite{BCCF} and the results were compared with those obtained by using X a complex Xversion of Greene's method (a partial Xjustification of this criterion can be found Xin \cite{FL2}, \cite{McK2}). The agreement obtained using the two methods was Xquite encouraging. X XIn this paper, we have extended the Xcomparison of the Pad\'e method with other Xindependent methods, some of them based in perturbative Xexpansions and others completely non-perturbative. X XWe have considered models which are not conservative for which we Xhave computed the perturbation expansions and applied the Pad\'e method. XBesides looking for the zeros of Xthe denominator of Pad\'e approximants, Xwe have explored other methods of ascertaining the Xdomain of analyticity of the Pad\'e approximants. XWe have also used non-perturbative methods to compare to the Xmethods based on the study of the perturbative expansions. XSome of the non-perturbative methods we have used are based on the Xfact that the map is dissipative, which makes it easy to Xcompute invariant sets. XWe have also proved an analogue of a partial justification of XGreene's criterion that makes Xit reasonable that the domains of analyticity for an invariant circle can be Xapproximated by those computed using rational frequencies. In that case, Xthe nature of the singularities Xcan be described in detail. Assuming a well known conjecture, this Xsingularity structure explains the behavior of the Pad\'e approximants Xthat we found. X XThe main reason to use dissipative systems in our study is that for them it Xis possible to locate the invariant tori by direct iteration and Xfollow them by non perturbative methods. XBesides the intrinsic interest of these dissipative systems -- they Xare reasonable models of some physical systems -- they can be used Xto provide some insight on the Poincar\'e--Lindstedt series for XHamiltonian systems. A first, heuristic, argument is that the structure of Xthe series is very similar -- in the dissipative case, however, they Xare more stable to compute numerically. Also Xone can argue heuristically that at the point at which Xthe invariant circle breaks down, the invariant circle has lost Xits hyperbolicity so that -- using the Xfact that our system can only have one non-zero Lyapunov Xexponent -- in an infinitesimal neighborhood, the system Xlooks like an area preserving system, so one could hope that Xsome of the results obtained about the behavior at Xbreakdown of circles would also apply to Xhamiltonian systems (of course the argument carries no weight for Xdynamical behaviors that in the neutral case are controlled by Xsubdominant terms). Indeed it has been argued \cite{R} that the Xrenormalization group picture developed for the breakdown of Xinvariant circles in hamiltonian systems can be extended to dissipative systems, Xif one scales the dissipation appropriately. X XThe results of our exploration are summarized in figures 1, 2, 3. XWe find it quite encouraging that so different methods give Xsimilar results, which seem to be within reasonable estimates Xof the margin of error for each of them. The methods Xthat seem to be the easiest to use and produce the more reliable Xresults for our system seem to be the non-perturbative methods Xdescribed in section 8. X X\SECTION Notation and Preliminaries. X XWe have considered the convergence of Lindstedt series for one particular Xmodel that we will, henceforth refer to as the ``rotating logistic'' map. X$$F_{\epsilon,\lambda,\omega} (r,\theta) X= (f_{\epsilon,\lambda}(r, \theta),\ \theta + \omega \bmod 1) X= (r^2 +\lambda +\epsilon \cos (2\pi \theta),\ \theta + \omega \bmod 1) X\EQ(logistic)$$ Xwhere $r$ is taken to be a complex number. X XNotice that for $\epsilon =0$ \equ(logistic) reduces to the well known Xlogistic map, which is well known to exhibit a very rich behavior. XA fixed point $r_0$ for the Xlogistic map becomes X an invariant circle $\{r_0\} \times \bT^1$ for X$F_{0,\lambda,\omega}$, filled by dense orbits if $\omega$ is irrational, Xor consisting of a Xfamily of periodic orbits for X$\omega$ rational. X XNotice that \equ(logistic) with $\omega$ irrational can Xbe considered as a quasi-periodic Xexcitation of the usual logistic map. Hence, it can appear as a physically Xreasonable model in all of the Xsituations where the logistic map appears, if we assume that they are modified Xby an external quasi-periodic force. For example in population Xdynamics or as a phenomenological model of processes with very strong Xdissipation that reduces the system to one-dimensional. X XAs the parameters $\lambda,\epsilon$ vary, this map exhibits a large variety Xof behaviors and bifurcations (suffice it to mention that for X$\epsilon =0$ it exhibits a Feigenbaum cascade of period doublings). XA study of the bifurcation diagrams for invariant tori was undertaken Xin \cite{K} and in \cite{AKL1} and there one can find detailed Xdescriptions of breakdown behavior for certain regions of parameters. XOf course, in \cite{AKL1} only real values of the parameters are considered. XIf we consider complex values of the parameters, some bifurcations Xthat do not appear in the real case, such as period $n$-tupling, $n> 2$, Xbecome possible. Indeed, they happen in the quadratic family Xfor certain complex values of $\lambda$. A K.A.M. argument Xsimilar to those in \cite{AKL1}, \cite{CI} can Xshow that such behaviors persist for sufficiently small values of $\epsilon$. X X X XIn this paper we will consider $\lambda$, X$\omega$ as fixed and explore the dependence Xon $\epsilon$ of the invariant circle. XHence, when it is not needed, Xwe will suppress $\lambda$, $\omega$ from Xthe notation for the map. X XWe will consider only $\lambda$'s real and somewhat smaller than the value Xfor which the first period doubling Xbifurcation occurs, Xwhich the work of \cite{AKL} suggests as not having any other Xbifurcation as $\epsilon$ changes Xtill breakdown. This hypothesis is also verified by our calculations, Xsince we compute the invariant circle for all the values of X$\epsilon$ for which it exists, very close to the value for Xwhich it breaks down, and we verify that the mechanism of destruction Xis very different from the simple $n$-tupling bifurcations. X X X X\SECTION Lindstedt expansions X XFollowing standard practice in XLindstedt methods, we observe that the graph of a map X$u_\epsilon : \torus \to \real$ Xis invariant under the map \equ(logistic) if and only if Xit satisfies X$$u_\epsilon (\theta +\omega) = \bigl[ u_\epsilon (\theta)\bigr]^2 X+\lambda + \epsilon \cos 2\pi \theta . X\EQ(Lindstedt)$$ X XIf we now assume an expansion in powers of $\epsilon$, X$u_\epsilon (\theta) = \sum_{n=0}^\infty \epsilon^n u^n (\theta)$ Xand substitute it in \equ(Lindstedt) we obtain: X$$\eqalign{ Xu^0 (\theta +\omega) & = u^0 (\theta)^2 +\lambda\cr Xu^n (\theta +\omega) & = 2u^0 (\theta) u^n (\theta) X+ \sum_{m=1}^{n-1} u^{n-m} (\theta) u^m (\theta) + \delta_{n,1} X\cos (2\pi\theta),\quad n>0\cr} X\EQ(recursion)$$ Xwhere $\delta_{n, 1}$ is the usual Kronecker symbol. X XWe claim that the first equation in \equ(recursion) admits the two Xsolutions $u^0(\theta) = u^0 = \frac12 \pm \frac12 \sqrt{1-4\lambda}$ and no Xother continuous solution if $\lambda \in (-\frac34, \frac14)$. X XMoreover, once we choose one of the two solutions for the Xfirst equation, the second hierarchy of equations allows the Xrecursive determination of all the $u^n$'s. X XIn effect, if \equ(recursion) is to hold, X$$u^0 (\theta +n\omega) = \ell_\lambda^n \bigl( u^0 (\theta)\bigr)$$ Xwhere $\ell_\lambda$ denotes the logistic map $\ell_\lambda (x) = x^2+\lambda$. XFor the values of the parameter selected, it is well known --- see e.g. X\cite {G} --- that all bounded orbits of $\ell_\lambda$, except those Xstarting in $\frac12 +\frac12 \sqrt{1-4\lambda}$ and its preimages, Xconverge to $\frac12-\frac12 \sqrt{1-4\lambda}$. Since $\theta$ is an Xaccumulation point of $\theta +n\omega$, it follows that $u^0(\theta)$ Xshould be either one of the two fixed points. XIf the function $u^0$ is to be continuous, then it should be constant. X XTo prove the second assertion, we observe that if we recursively Xassume that $u^0,\ldots,u^{n-1}$ are known we can determine $u^n$ Xby solving an equation of the form X$$u^n (\theta +\omega) - 2u^0 u^n(\theta) = R^n (\theta) X\EQ(cohomology)$$ X XSuch equations can be conveniently analyzed using Fourier series Xsetting: X$$ X u^n (\theta) = \sum \hat u^{n,k} e^{2\pi ik\theta} X$$ Xand similarly for $R$, and other periodic functions. XWe have that the unique solution of X\equ(cohomology) is X$$\hat u^{n,k} = \hat R^{n,k} \big\slash (e^{2\pi ik\omega} - 2u^0). X\EQ(solution)$$ X XWe note that for the rotating logistic map it is easy to prove by Xinduction that the solutions $u^n$ of \equ(recursion) are Xtrigonometric polynomials with degree$(u^n) = n$. XHence, the $R^n$'s are also trigonometric polynomials. X XA similar argument would show that provided that the forcing term is Xa trigonometric polynomial in $\theta$, then the $u^n$'s are also Xtrigonometric polynomials and the degree of $u^n$ is a linear function Xof $n$. X XTo discuss convergence, it is convenient to adopt a more general point Xof view that will also turn out to apply to the case when the forcing Xterm is a periodic function of $\theta$ rather than just a trigonometric Xpolynomial. X XWe note that, when $\lambda \in (-\frac34, \frac14)$, $|2u^0|\ne 1$ Xhence $(e^{2\pi ik\omega} - 2u^0)$ is bounded away from zero Xuniformly in $k \in \integer$ so that X$$ X\sup_{k \in \integer} | e^{2\pi ik\omega} - 2u^0 |^{-1} \le K X\EQ(Kdefined) X$$ Xwhere $K \ge 1$ (for $\lambda \in (-\frac34, \frac14)$). XHence, X$ |\hat u^{n,k} | \le K |\hat R^{n,k}| $. XSo that, if we assume $|\hat R^{n,k} | \le Ae^{-\delta |k|}$ we Xwould obtain $|\hat u^{n,k} | \le KAe^{-\delta |k|}$ so that, for example, Xif $R$ is analytic in a certain domain, $\{\theta \mid |\Im \theta| < \delta\}$ Xso will be $u$. X XMoreover, it is very easy to estimate the recursion to obtain convergence. X X\CLAIM {Lemma}(convergence) XIf $|\epsilon| < \frac12 K^{-2} e ^{-2\pi \delta}$ X(where $K$ is as in \equ(Kdefined)), the series obtained by Xsumming the $u^n$'s obtained by \equ(recursion) converges uniformly Xon $\{\theta \mid |\Im \theta| < \delta\}$. X X\PROOF XFor $\delta >0$, Xif $f$ is an analytic function on the unit circle, Xwe can expand it in Fourier series, X$f(\theta) = \sum \hat f^k e^{2\pi ik\theta}.$ XWe denote by $\|f\|_\delta = \sup e^{\delta |k|} |\hat f^k|$. XIt is well known that this defines a norm on a Banach space of analytic Xfunctions. We denote such space by $C^{\omega,\delta}$. XWe observe that $\|f\cdot g\|_\delta \le \|f\|_\delta \|g\|_\delta$ Xand that if $u^n$ and $R^n$ are related as in \equ(solution), then X$\|u^n\|_\delta \le K\|R^n\|_\delta$. X XWe also observe that $R^1 (\theta) = \cos 2\pi \theta$ so that X$\|u^1 \|_\delta \le \frac12 K e^{2\pi\delta}$. X XThe recursion part of \equ(recursion) implies for $n>1$ X$$\|u^n \|_\delta \le K\|R^n\|_\delta X\le K\sum_{m=1}^{n-1} \|u^{n-m}\|_\delta \|u^m\|_\delta. X\EQ(estimates)$$ XBy induction, it is easy to show that X$$\| u^i\|_{\delta} \le \frac{\sigma_i}{K} X\bigl(\frac{K^2 e^{2 \pi \delta}}{2} \bigr)^i , X\qquad i \ge 1 $$ Xwhere $\sigma_1 = 1$, $\sigma_k = \sum_{j=1}^{k-1} \sigma_j \sigma_{k-j}$. XThis is certainly true for $i=1$ and the recursive bound \equ(estimates) Xshows that if it is true for $i \le n-1$ it is true for $i=n$. XWe see that if we denote X$$ \sigma(z) = \sum_{i=1}^{\infty} \sigma_i z^n $$ Xthen $\sigma(z)$ is a solution of $\sigma(z) = \sigma(z)^2 + z$. XSince $\sigma(z)$ also satisfies that $\sigma(0) = 0$, X$\sigma_0 = 0$, we conclude that $\sigma(z) = \frac12 - \frac12 \sqrt{1- 4z}$ and X$$ \sigma_1 = 1,\quad X \sigma_n = \frac1n {{2n-2} \choose {n-1} } \le \frac{4^{n-1}}{n}, X\qquad n \ge 2 .$$ XThis shows that $\sum_0^{\infty} u^n(\theta) \epsilon^n $ converges in Xthe $\Norm_{\delta}$ sense Xand proves the statement of \clm(convergence). X\QED X X\REMARK XWe point out that results similar to \clm(convergence) can Xbe proved as corollaries of the general theory of stability Xfor normally hyperbolic manifolds. We also point out that the Xdomain of applicability of these results converges to zero as X$\lambda$ converges to $-\frac34$, the value at which the logistic map Xexperiences the first period doubling bifurcation. By using the Xmuch more subtle K.A.M. theory it is possible to show that if X$\omega$ is Diophantine, one can get domains of convergence which are Xuniform in $\lambda \in [\lambda_-,\lambda_+]$ where $\lambda_-$, X$\lambda_+$ are some numbers that contain the bifurcation point. XWe refer to \cite{AKL2} or \cite{CI} for details of this theory. X X\REMARK XThe use of the norms $\Norm_\delta$ in the above proof is natural Xin view of the fact that, for fixed $\epsilon$, the maximal domains Xof analyticity in $\theta$ for $u_\epsilon(\theta)$ are of the form X$\{\theta \mid |\Im \theta| < \delta\}$. XThis can be seen by observing that if $u_\epsilon$ is defined for some X$\theta,$ then it is also defined using \equ(Lindstedt) for X$\theta +\omega$. So that the domains of analyticity in $\theta$ of X$u_{\epsilon}(\theta)$ have to be invariant Xunder irrational translation. X XTo study numerically the domain of analyticity of the map X$\epsilon \mapsto u_\epsilon$, it is easy to study the Xdomain of analyticity of maps $\epsilon \mapsto \Gamma[ u_\epsilon]$, Xwhere $\Gamma$ is an entire map from the space of analytic functions Xto the complex numbers. Clearly the domain of analyticity of X$\epsilon \mapsto \Gamma[ u_\epsilon]$ is bigger than the Xdomain of analyticity of the map $\epsilon \mapsto u_\epsilon$. XOne expects also, that many observables will lead to the same Xdomain of analyticity. Some observables that immediately come Xto mind are the evaluation of the function at certain values and Xthe Fourier coefficients. XWe conjecture that indeed, these simple observables give the optimal domain Xof analyticity. X X\CLAIM {Conjecture}(analdom) XFor a fixed $\lambda \in (-3/4, 1/4)$, $\omega$ irrational, $\theta \in \real$, Xthe function X$\epsilon\to u_\epsilon (\theta)$ is defined in a domain $D$ Xindependent of $\theta$. This domain agrees with the Xdomain of analyticity of $\epsilon \mapsto \hat u_{\epsilon}^k$. X XTo justify \clm(analdom) we see that if, Xfor a fixed $\theta$, the function $\epsilon\to u_\epsilon (\theta)$ Xcan be defined in a certain domain $D(\theta)$, \equ(Lindstedt) shows Xthat $\epsilon\to u_\epsilon (\theta +\omega)$ can be defined in a Xdomain that is at least as big. Therefore $D(\theta) \subset D(\theta +\omega)$. XIn our case, however, we expect that $D(\theta) = D(\theta +\omega)$ since Xthe only way that the domain of analyticity $D(\theta +\omega)$ could Xactually be bigger than $D(\theta)$ is by using that the function X$u\to u^2+\lambda$ is not invertible and can transform certain Xsingularities into analytic functions. Such behavior seems unlikely, Xespecially in view of our numerical computations. X XNotice that the argument for \clm(analdom) uses heavily Xthat $\omega$ is irrational and, if X$\omega$ were rational, the domain does depend on X$\theta$. (See section 9 for a discussion of analyticity domains in the Xcase that the frequency is rational.) XAlso, if the system we consider is not \equ(logistic) but Xhad special Xsymmetries that force all the coefficients in the expansion to Xhave odd parity, the function $u_\epsilon(\theta)$ Xvanishes identically at certain values of $\theta$, Xhence is trivially entire in $\epsilon$ for those values. X XOur computations are also evidence that the analyticity domain of Xall the observables mentioned in X\clm(analdom) are the same, so that it is a reasonable conjecture that Xthey agree with the domain of analyticity of $\epsilon \mapsto u_\epsilon$. X X\SECTION Pad\'e Approximations X XWe recall that a Pad\'e approximant $[M/N]$ to an analytic function Xis a rational function with numerator $P$ of degree $M$ and denominator $Q$ Xof degree $N$ whose Taylor expansion up to order $M+N$ agrees with that Xof $S$. X XWe can assume without loss of generality that $D(0)=1$. XIf we impose this normalization, under mild non-degeneracy conditions, Xthe Pad\'e approximant of order $[M/N]$ exists and is unique. X XWe refer to \cite{B}, \cite{BGM}, \cite{M}, \cite{Gi} Xfor a survey of mathematical results about Pad\'e approximants and Xtheir applications in problems of theoretical physics. XThey have indeed been widely used in almost all fields in Physics Xin which perturbative expansions and their breakdown play a role. X XIf the Taylor expansions of $S(\epsilon)$ and of $P(\epsilon)/D(\epsilon)$ Xare to match up to order $\epsilon^{N+M+1}$ we can write X$S(\epsilon) D(\epsilon) = P(\epsilon) +O(\epsilon^{N+M+1})$ which, Xtogether with the normalization $D(0)=1$, leads to the equations X$$\eqalign{ XP_i & = S_i + \sum_{j=1}^{\min (N,i)} S_{i-j} D_j\qquad 0\le i\le M\cr X0 & = S_i + \sum_{j=1}^{\min (N,i)} S_{i-j} D_j\qquad M< i\le M+N\cr} X\EQ(match)$$ X XNotice that the second set the equations involves only the $D$'s Xand that once we know the $D$'s, by substitution in the first set Xof equations, it is possible to compute the $P$'s. X XThere are other computationally more efficient methods to compute XPad\'e approximants based on recursions (see e.g., \cite{BGM} vol. 1, pg.66). XNevertheless, the algorithm sketched above has the advantage that, by Xusing careful standard numerical analysis routines we can obtain condition Xnumbers that give a measure of the reliability of the calculations. We Xwill give more details in the section devoted to numerical implementations. X XThe standard method to compute the domain of analyticity of $S(\epsilon)$ Xbased on Pad\'e approximants consists in computing $D$ and $P$ as before. XOne then expects that the boundary Xof the domain of analyticity for $S$ will be approximated by Xthe poles of $P/D$. That is, the zeros of $D$ which are not zeros of $P$ X(or at least zeros of a smaller order). XThis is expressed in the following quote (\cite{Gi}, pg. 310): X X{\sl ``Tous les theor\`emes de convergence des approximants de Pad\'e, Xainsi que les resultats num\'eriques indiquent que ces Xapproximants ont une tendence visible \'a reproduire les propriet\'es Xd' analycit\'e d' une fonction.''} X X\SECTION \vbox{ X\vbox{A new method to compute domains of analyticity from }\breakline X\vbox{Pad\'e approximants.}} X XThe usual method of computing domains of analyticity of functions Xis just to compute the zeros of the denominator of the XPad\'e approximation. XUnfortunately, the calculation of zeros of a polynomial frequently Xhas large condition numbers (see \cite{Wi} for some examples, X\cite{He} for a discussion of algorithms). This is particularly Xunfortunate since the calculation of Pad\'e approximants out of the Xcoefficients in the expansion is also very ill conditioned. X XThe previous remarks are especially true for the zeros which Xare not close to the origin. The elementary example -- which we learned Xfrom G. Baker -- X$$S(\epsilon ) = {1\over \epsilon -a_1} + {1\over \epsilon -a_2} X= \sum_{n=0} \epsilon^n X\left( \left( { 1\over a_1}\right)^{n+1} + \left( { 1\over a_2}\right)^{n+1} X\right)$$ Xshows that the information about the outermost pole is hidden in the Xhigh precision part of the coefficients of the expansion. For many of the Xperturbation expansions in classical mechanics, whose terms alternate widely Xin sizes, this seems particularly dangerous. X XSince the number of zeros of the denominator is roughly half the degree Xof $S(\epsilon)$ (in practice considerably less since sometimes the zeros Xof the numerator are also zeros of the denominator) it is clear that it is Xnot very easy to obtain a very detailed picture of the boundary since the Xcondition number for the computation of zeros worsens very fast with the Xnumber of zeros. X XIn the expansions in celestial mechanics such as the Lindstedt method, we Xcan take advantage of the presumed fact that Xwhen the frequency is irrational the domain of analyticity Xshould be independent of $\theta$, to do several calculations and obtain Xsignificantly more points in the boundary. Such enhancement is not Xavailable for most of the situations to which Pad\'e approximants are Xapplied. X XWe state the following conjecture: X X\CLAIM {Conjecture}(Padeapprox) XLet $f$ be one of the Xfucntions appearing in the XLindstedt-Poincar\'e Xexpansions. Then, $f$ Xis analytic in a topological disc ${\cal D} \subset \complex$ Xwith a natural boundary for $f$ and the Xsequence of $[N/N]$ Pad\'e approximants converges to X$f$ in measure, as $N \to \infty$, on any compact subset of ${\cal D}$. X XThe method we propose is based on \clm(Padeapprox). XIf, according to \clm(Padeapprox), $\frac{P_N(z)}{D_N(z)}$ converges as X$N \to \infty$, $\left|\frac{P_N(z)}{D_N(z)} - \frac{P_M(z)}{D_M(z)}\right|$ Xshould converge to Xzero as $N,M\to\infty$. XHence, a reasonable approximation to the boundary of convergence of the Xdiagonal Pad\'e series --- and hence, according to \clm(Padeapprox), Xto the domain of analyticity of the function --- could Xbe the level curve X$$\left| {P_N(z) \over D_N(z)} - {P_M(z) \over D_M(z)}\right| X= \delta$$ Xfor a reasonably small $\delta$ and sufficiently large $N,M$, Xand $z$ not in the neighborhood of a spurious pole of the $[M/M]$, $[N/N]$ Xapproximants. X XUnfortunately, the practical Ximplementation of the criterion Xcannot take the limit as $N,M$ tend to infinity Xbut rather just take some reasonably high value. XThen, the criterion involves a free parameter X$\delta$ as a function of the degree of the Xapproximation and the results could depend on its choice. XIn the examples we have considered, we have found that any Xchoice between $10^{-5}$ and $10^{-1}$ leads to results not more uncertain Xthan those obtained by finding the zeros of $D_M$. X XThe dependence on the parameter $\delta$ can be further reduced Xby plotting the level surface X$$ X\left| {P_{N_1}(z) \over D_{N_1}(z)} X- {P_{N_2}(z) \over D_{N_2}(z)} \right| + \cdots + X\left| {P_{N_{j-1}} (z) \over D_{N_{j-1}} (z)} X- {P_{N_j}(z) \over D_{N_j}(z)}\right| = \delta X$$ Xwhich, according to \clm(Padeapprox), will also provide an approximation to the Xdomain of analyticity. X X XUnfortunately, little is known about the Xcovergence of Pad\'e approximants in the case that the Xfunction has natural boundaries. XFor a class of functions Xwith natural boundaries -- but which remain Xquasianalytic accross the boundary -- X(see \cite{Gi}, pg. 306--309) \cite{GN} have shown that Xthe $[(N+J)/N]$ Pad\'e approximants converge in measure to the Xfunction as $N \to \infty$, in any closed, bounded region of the Xcomplex plane. Specifically: X X\CLAIM Theorem (Nuttal) XFor the function $f(z) = \sum_{n=1}^{\infty} X\frac{A_n}{(1-z\alpha_n)}$, where the $\alpha_n$ lie densely on the Xunit circle and $|A_n| < Ce^{-n^{1+\gamma}}, \gamma > 0$, the sequence Xof $[(N+J)/N]$ Pad\'e approximants to $f(z)$ converges in measure to $f$ Xas $N \to \infty$ in any closed, bounded region of the complex plane. X X XThe method of proof, as remarked in \cite{GN} can Xbe extended to other cases and that the condition X$|\alpha_n| = 1$ can be modified to X$|\alpha_n | < a$. They also discuss how Xslightly faster rates of growth in the coefficients, Xcould lead to divergence. X XResults such as this one make it reasonable to Xbe hopeful that convergence of the Pad\'e approximants is Xa reasonably general phenomenon and, hence, lend Xindirect support to \clm(Padeapprox) X XOur numerical results, also support X\clm(Padeapprox) XOn the other had we note that, in contrast with the X\clm(Nuttal), we to observe that the XPad\'e approximants of our Xfunctions seem to diverge outside of the Xdomain analyticity of the function. X XIt is not clear to us what could be a reasonably general Xcondition that implies convergence of Xthe Pad\'e approximants for the cases we are interested in. X X\SECTION Newton Method X XFor a fixed $\epsilon_0$, it is possible to solve equation X\equ(Lindstedt) using a Newton method. X XWe see that if $u_{\epsilon_0}$ fails to satisfy \equ(Lindstedt) by a small Xamount $R_{\epsilon_0} (\theta)$ i.e., X$$u_{\epsilon_0} (\theta +\omega) - u_{\epsilon_0} (\theta)^2 - \lambda - X\epsilon_0 \cos 2\pi \theta = R_{\epsilon_0}(\theta) X\EQ(error)$$ Xwe can try to improve the solution by setting it to X$u_{\epsilon_0}(\theta) + \Delta_{\epsilon_0} (\theta)$, Xwhere $\Delta_{\epsilon_0} (\theta)$ will be conveniently chosen. X XIf $\Delta_{\epsilon_0}(\theta)$ satisfies X$$\Delta_{\epsilon_0} (\theta +\omega) - 2u_{\epsilon_0} (\theta) X\Delta_{\epsilon_0} (\theta) = R_{\epsilon_0} (\theta) X\EQ(Newton)$$ Xthen, $u_{\epsilon_0}(\theta) +\Delta_{\epsilon_0} (\theta)$ Xwill satisfy \equ(Lindstedt) up to error terms which are much Xsmaller than $R_{\epsilon_0}(\theta)$. X XWe will postpone for the moment a discussion of the numerical Xdiscretizations used to solve \equ(Newton). X XThis method, as it is well known from even finite dimensional examples, Xhas the shortcoming that it Xonly converges when sufficiently good guesses are taken as starting points. X XIn our case one could perform a Xcontinuation method starting from very small values of $\epsilon$. That is, Xwhen one exact solution is found, we take it as an approximate solution Xfor the equation with a slightly bigger parameter. We note that the Xsolution when X$\epsilon = 0$ is known exactly. XWe could also take for some values of $\epsilon$ the sum of the Xseries \equ(recursion) as an initial guess. This would provide us with Xan independent verification that the series is converging to solutions of the Xequation. X XWe emphasize that the validity of the solution of the Newton method Xis independent of the method used to obtain the initial guess since, Xin the process of running the Newton method one checks that the equation Xis indeed satisfied. X XFor practical calculations in concrete problems the Newton method seems Xto have advantages over the method of expansion in powers of $\epsilon$. XFor example, notice that the expansions in powers of $\epsilon$ can only Xconverge on a disk of radius equal to the distance from the origin to the Xclosest singularity. If the shape of the analyticity domain is very Xdifferent from a disk, this implies that there will be several Xvalues of $\epsilon$ for which the invariant circles exist and for which Xthe $\epsilon$ expansion does not converge. For a continuation method, it Xis quite possible to obtain the solution in all the connected domain where Xthe solution exists. XNotice also that the method of expansion in powers requires the storage Xof many functions. The Newton method requires only the storage of one. XOn the other hand, one should also mention that if one requires solutions Xfor many values of the parameter, the $\epsilon$ expansion method could Xbe faster and gives more global information. X X\SECTION Multipoint Pad\'e Approximation X XThe multipoint Pad\'e approximation is an interesting compromise Xbetween rational interpolation and the Pad\'e approximation (which we Xcould consider as a degenerate case of interpolation in which all Xthe interpolation points are infinitesimally close). X XWe recall, see e.g. \cite{BGM} vol.2, pg.5, that given a set of points X$\{z_i\}_{i=1,k}$ in the complex plane and a function $S(z)$ with XTaylor expansions of order $\ell_i$ around those points, we define Xthe $[N/M]$ multipoint Pad\'e (or rational) approximant as: X$${P(z)\over D(z)}\ ,\quad \degree P=N\ ,\quad \degree D=M\ ,\quad XD(0)=1$$ Xsuch that $M+N+1 = \sum_{i=1}^k (\ell_i +1)$ and X$${d^s\over dz^s} \bigl( P(z) - D(z) S(z)\bigr) \big|_{z=z_i} =0 X\ ,\qquad s \le \ell_i\ ,\ i= 1,k.$$ XSetting X$$\eqalign{ X&P(z) = \sum_{j=0}^N P_j z^j\cr X&D(z) = \sum_{j=0}^M D_j z^j\ ,\quad D(0) = D_0 = 1\cr X&S(z) = \sum_{j=0}^{\ell_i} S_{i;j} (z-z_i)^j\ , i=1,k X}$$ Xwe require X$$\displaylines{ X\hfill \sum_{j=s}^N P_j {j!\over (j-s)!} \, z_i^{j-s} X= \left[ \sum_{n=0}^s {s\choose n} S^{(s-n)} D^{(n)} \right] (z_i)\hfill\cr}$$ Xwhere X$$S^{(s)} (z_i) \equiv {d^s \over dz^s} S(z)\big|_{z=z_i}.$$ XSince X$$\displaylines{\hfill XD^{(n)} (z_i) = \sum_{m=n}^M {m!\over (m-n)!} \, D_m z_i^{m-n}\ ,\qquad 0\le n\le M\hfill \cr X\hfill S^{(s-n)} (z_i) = (s-n)! \, S_{i;s-n}\ , \qquad 0\le s-n < M+N+1 \hfill \cr}$$ Xor X$$\sum_{j=s}^N P_j {j!\over (j-s)!} \, z_i^{j-s} X= s! \sum_{n=0}^s \sum_{m=n}^M {m\choose n} S_{i;s-n} D_m z_i^{m-n}$$ Xwhere, to simplify the notation, we extend the usual combinatorial Xnumbers by: X$${m\choose n} = X\cases{ X\displaystyle {m!\over n!(m-n)!}\ , &$m\ge n$\cr X0\ ,&$m1 \hfill \ \cr} X\EQ(recursionnewton)$$ X XIf we assume that we know $\Delta^1,\ldots,\Delta^{n-1}$ then $\Delta^n$ Xcan be found by solving an equation of the form X$$\Delta^n (\theta + \omega) - 2u_{\epsilon_i}(\theta) \Delta^n(\theta) = XR^n(\theta) \EQ(deltarecursion)$$ XTo solve \equ(deltarecursion) we prove the following lemma : X\CLAIM{Lemma}(deltabound) XIf $u_{\epsilon_i}(\theta)$ is an analytic function satisfying X\vskip1pt X\item{(i)} $\|2u_{\epsilon_i}\|_{\delta} \le A$ X\vskip1pt X\item{(ii)} For some $m \in \natural, 0<\gamma<1, \quad \| 2u_{\epsilon_i}(\theta) \cdots 2u_{\epsilon_i}(\theta+m\omega)\|_{\delta}\le\gamma$ X\vskip1pt Xthen, X\vskip1pt X Given any $R^n$ analytic in X$\{\theta \big| |\Im(\theta)| \le \delta\}$, Xwe can find a unique $\Delta^n$ solving X\equ(deltarecursion). Moreover, X\vskip1pt X$\| \Delta^n\|_{\delta} \le K \|R^n\|_{\delta}$, where $K$ Xcan be taken to be $K = A^m/(1 - \gamma^{1/m})$. X X X\PROOF XApplying \equ(deltarecursion) repeatedly we have X$$\eqalign{\Delta^n (\theta + \omega) & = \sum_{k=0}^N \left(\prod_{j=1}^k X\bigl(2u_{\epsilon_i}(\theta-j\omega)\bigr)\right) R^n(\theta -k\omega)\cr X&\qquad + \prod_{k=0}^N \Bigl( 2u_{\epsilon_i}\bigl(\theta - k\omega\bigr) X\Bigr) \Delta^n \bigl( \theta -N\omega\bigr).} X\EQ(iterated)$$ X XIf there is a bounded solution, then the last term in X\equ(iterated) should tend to zero, so that the only Xsolution of \equ(deltarecursion) should be: X$$ \Delta^n (\theta) = \sum_{k=0}^{\infty} \left(\prod_{j=1}^k( 2u_{\epsilon_i}(\theta X-j\omega))\right)R^n(\theta-k\omega). X\EQ(deltaR)$$ XConsidering blocks of length m we can bound the products appearing in X\equ(deltaR) by: X$$ \bigl\|\prod_{j=1}^k(2u_{\epsilon_i}(\theta-j\omega))\bigr\|_{\delta} X\le A^m \gamma^{[k/m]} = A^m [\gamma^{1/m}]^k$$ Xso that X$$\| \Delta \|_\delta \le \| R\|_\delta \frac{A^m}{1-\gamma^{1/m}} = K \|R\|_{\delta}.$$ X XThe above estimates show also that the series \equ(deltaR) converge absolutely, Xso that it is possible to rearrange terms and show that Xindeed it solves \equ(deltarecursion). X\QED XNow we can show convergence for the series \equ(recursionnewton). X X\CLAIM{Lemma}(deltaconv) XIf $ | \hat \epsilon | < \frac12 K ^{-2} e^{-2\pi \delta} $ (where $K$ as in X\clm(deltabound)) the series obtained by summing the $\Delta^n$'s computed Xfrom \equ(recursionnewton) converges uniformly on X$\{\theta \big| \ |\Im \theta | < \delta \}$. X X\PROOF XWe notice that $R^1(\theta) = \cos 2\pi \theta$ so that X$$\| \Delta^1 \|_{\delta} \le K\|R^1\|_{\delta} \le \frac12 K e^{2 \pi \delta}$$ XFrom \equ(recursionnewton) we have that for $n>1$ X$$ \|\Delta^n \|_{\delta} \le K\|R^n\|_{\delta} \le K \sum_{m=1}^{n-1} X\| \Delta^{n-m}\|_{\delta} \| \Delta^m\|_{\delta} X\EQ(deltabound)$$ XThis recursion is the same Xas \equ(estimates) and to obtain the Xestimates claimed, it suffices to use the same argument that Xwe used in \clm(convergence) X\QED X\REMARK X\clm(deltaconv), together with \clm(convergence) show that the mapping X$\epsilon \to u_\epsilon$ is analytic in $|\epsilon| \le \frac12 K^{-2}$ Xwhen we give the $u_\epsilon$ the topology induced by $\Norm_\delta$. XThis is much stronger than saying that for a fixed $\theta$ the series X$\sum_{n=0}^\infty u^n (\theta) \epsilon^n$ or $\sum_{n=1}^\infty \Delta^n(\theta) \hat \epsilon^n $ converge. X X\REMARK XNotice that the equations appearing in the recursion for $n>1$ are very Xsimilar to the equations we encountered in the study of the Newton method X(this is not a coincidence since the procedure we carried out is just a very Xexplicit form of the implicit function theorem --- the equation X\equ(Newton) is just inverting the derivative, which also plays a role Xin the implicit function theorem). X XAgain, once we have computed the expansions, in terms of X$\epsilon$, of the function by evaluating at Xdifferent values of $\theta$ we obtain several Xnumerators and denominators and several possible candidates for the domain Xof convergence. They should agree. X X X XWe point out that the hypotheses of \clm(deltabound) Xon the existence of a uniformly contractive analytic Xinvariant circle, for this model, are implied by the X\`a priori much weaker conditions that there exists Xa continuous invariant circle with negative Lyapunov Xexponent. Hence, the boundary of the domain of analyticity Xis given by the boundary of the domain of Xexistence of continuous circles with negative XLyapunov exponent. X X XWe recall -- see e.g. \cite{W} Thm 6.20 -- that rotations by an Xirrational number are uniquely ergodic. That is, they admit Xonly one invariant measure. Hence, it makes sense to speak of Xthe Lyapunov exponent of the invariant circle without specifying Xexplicitly the measure. X X\CLAIM Lemma(Lyap) XLet $u$ be a continuous function solving \equ(Lindstedt) Xwith $\omega$ irrational. XIf $\int \ln | 2 u( \theta) | d \theta < 0$, Xthen, we can find an $m$ such that X$$\bigl|2u(\theta) \cdot 2u(\theta + \omega) \cdots X2 u( \theta + Xm \omega)\bigr| \le \gamma < 1.$$ X X\PROOF XDenote by $\phi(\theta)$ a continuous function X$\phi(\theta) \ge \ln | 2 u(\theta) |$, X$ \int \phi(\theta) d \theta = \gamma_1 < 0$. X(Such function can be obtained Xby setting $\phi( \theta) = \ln( \max( |2 u(\theta)| , \rho))$ Xfor sufficiently small $\rho > 0$.) XWe recall that (see e.g \cite{W} Thm 6.19) that for Xa uniquely ergodic map, the Birkhoff sums of a continuous Xfunction converge uniformly. XIn our case, they should converge uniformly to $\gamma_1$. XHence, we can find $m$ such that X$\phi(\theta) + \phi(\theta + \omega) + \cdots X + \phi(\theta + m \omega) \le \ln \gamma < 0.$ X Since $\phi(\theta) \ge \ln( | 2 u(\theta)|)$, Xthe lemma is established. X\QED X\CLAIM{Lemma}(analyticu) XLet $u$ be a continuous map solving \equ(Lindstedt) and Xsuch that, for some $m \in \natural, 0< \gamma < 1$, X$ | 2 u( \theta) \cdot 2 u(\theta + \omega) \cdots 2 u(\theta + m \omega) | \le \gamma < 1$, for $\theta \in \torus^1.$ XThen $u$ is analytic. X XNotice that, even if \clm(Lyap) requires that $\omega$ is irrational, X\clm(analyticu) works even for rational $\omega$. X X\PROOF XConsider the graph transform operator X$$ X\Gamma[u](\theta) = X u(\theta - \omega)^2+ \lambda + \epsilon\cos 2 \pi (\theta-\omega). X\EQ(graphtransform) X$$ XIt is equivalent that $u$ is a fixed point of X$\Gamma$ and that it satisfies \equ(Lindstedt). X XWe will show that, under the hypotheses Xof \clm(analyticu), we can find a Xball in $C^0$ and in $C^{\omega,\delta}$, for $\delta$ Xsufficiently small, with non-empty Xintersection, on which $\Gamma^{m+1}$ is a contraction with the Xcorresponding norms and that the $C^0$ ball contains the Xgiven fixed point $u$. Then, if we take a point belonging to Xthe intersection of the two balls, by the uniqueness part Xof the contraction mapping theorem, by iterating it, it Xhas to converge to $u$. On the other hand, by the contraction Xon $C^{\omega,\delta}$, it converges to an analytic fixed point. X XTo prove the claim, we observe that Xthe derivative of $\Gamma^{m+1}$ at the fixed point $u$ Xcan be computed both in X$C^0$ and in $C^{\omega, \delta}$ as: X$$ XD\Gamma^{m+1}(u)[\eta](\theta) = 2 u( \theta - \omega) X\cdot 2 u( \theta - 2 \omega)\cdots 2 u(\theta - (m+1) \omega) X\eta( \theta - (m+1) \omega) X\EQ(derivativeformula) X$$ XWe note that for other functions in place of the logistic, Xthe derivative of $\Gamma^{m+1}$ Xis still obtained by a multiplication operator and a shift. X XDenote by $K_\alpha$ the heat kernel and Xobserve that $ \| u - K_\alpha u\|_{C^0}$ converges uniformly to $0$ Xas $\alpha \to 0$. Also X$\|K_\alpha u\|_{\delta} \to_{\scriptstyle \delta \to 0} X\|K_\alpha u\|_{C^0} \to_{\scriptstyle \alpha \to 0} \|u\|_{C^0}$ Xwhere the convergence, Xdue to the periodicity of u, is uniform in $\alpha, \delta$. X XUsing the previous observations Xby choosing $\alpha$,$\delta$ sufficiently small, Xwe can get that X$\| \Gamma^{m+1} [K_\alpha u ] - K_\alpha u\|_\delta$ Xis arbitrarily small and X$\| D \Gamma^{m+1}( K_\alpha u) \|_\delta$ Xis as close to $\gamma$ as desired, Xin particular less than $1$. X XMoreover $D^2 \Gamma^{m+1}(u) ( \eta_1 ,\eta_2) = \sum_{i,j} \left(\prod_{i' \ne i \atop i' \ne j} 2u(\theta - {i'}\omega)\right) \eta_1( \theta - i \omega) \eta_2( \theta - j \omega).$ X Hence if we pick a neighborhood of Xradius $\rho$ in the $C^{\omega,\delta}$ space, Xwe can bound the size of the second derivative uniformly in X$\alpha, \delta$ as they become arbitrarily small. X XHence for all $\alpha$, $\delta$ sufficiently small, Xit is possible to get a $C^{\omega,\delta}$ ball around $K_\alpha u$ Xso that $\Gamma^{m+1}$ is a contraction on it. XWe can also arrange that $K_\alpha u$ is Xin the $C^0$ ball around $u$ for which $\Gamma^{m+1}$ is Xa $C^0$ contraction. X\QED X X\SECTION Non-perturbative methods X XIf the invariant curve is contractive, it can be approximated numerically Xby iterating the map \equ(logistic) forward. XBy changing $\epsilon$ in small steps X we can compute the invariant curve for Xall the domain in which it is Xcontractive. Because of \clm(deltaconv), X\clm(convergence), which guarantee analyticity Xin $\epsilon$ for contractive invariant circles, Xif we succeed in finding parameter values along a path that goes Xthrough the origin, for which the orbits Xsettle onto an invariant curve Xwe can ensure that these points belong to the domain of analyticity of Xthe invariant curve. X XOn the other hand, if we find real values of $\epsilon$ Xfor which some orbits escape to infinity X-- by the quadratic behavior of the map it suffices to check that X$r$ is larger than a certain number -- Xwe are sure that there are no invariant circles separating the Xbeginning of the orbit and infinity. X XWhen $\epsilon$ is complex, the existence of orbits escaping to infinity Xdoes not preclude the existence of an invariant circle in X$\torus^1 \times \complex$. Nevertheless, the fact that the basin of attraction Xof the invariant circle shrinks to nothing is indication of a sudden change in Xbehavior. If it was just that the invariant circle lost stability, Xthis could be discovered by iterating backwards. For all the values of the Xparameters we are reporting after breakdown, we found no evidence of Xinvariant circles. X XSo, if we increment $\epsilon$ along a path, and find values Xfor which the orbit settles in an invariant circle and Xvery nearby values for which all orbits seem to escape, Xwe conclude that these values belong to the boundary of the Xdomain of analyticity. By taking several paths of a Xfamily, e.g radii, we can obtain a reasonable estimate of the boundary. XSince the boundary can bend back with respect to a family of paths, Xwe should use several families to obtain a better approximation. X XOne side effect of this Xalgorithm is that it allows to compute the invariant curves just Xbefore breakdown. For real values of $\epsilon$ the invariant curve Xremains smooth until breakdown. At breakdown it undergoes a saddle node Xbifurcation of invariant circles and disappears (the theory of this phenomenon Xis worked out in \cite{AKL1}). For complex values of $\epsilon$ the invariant Xcurve disappears by becoming very oscillating at small scales. XThis phenomenon requires further study but we will Xnot concern with it here. X XWe emphasize that just finding the values of $\epsilon$ Xfor which there are points that do not escape Xwhich are close to other for which escape takes place, Xdoes not Xprovide X always with a reliable approximation for the Xdomain of analyticity of the invariant circle. XOne has to check that the invariant set in which the Xorbit settles is an invariant circle. XIndeed, for some values of $\lambda$, the invariant circles Xexperience period doubling bifurcations. These periodic circles are Xbarriers for the escape but nevertheless are not direct continuations Xof the original invariant circle. X XIn the practical implementation, we have just checked visually Xfor some of the values that indeed the attracting set was a circle. XFor the values of $\lambda$ we report, the evidence that the Xsets remain one dimensional up to extremely close to breakdown Xis very clear. XEven if we have not been able to obtain a mathematical Xargument that shows that there are no other bifurcations of the circle Xfor the values of $\lambda$ that we have considered, the numerical Xresults show that these bifurcations, if they happen, Xcan only occupy a very small region in the parameters space. X X\def\omegab{{\bar \omega}} X\def\deltab{{\bar \delta}} X\SECTION Behavior at rational frequencies X XIn this section we study the analyticity domains for the case that the Xfrequency is rational. A motivation for this is that it is possible to show X-- see Theorem 9.1 below for a precise statement -- that the analyticity Xdomains at irrational $\omega$ are approximated by those at rational Xfrequencies $p/q$ when $p/q \approx \omega$. Nevertheless, when the frequency Xis rational, many of the analytical problems become algebraic and we can Xperform a detailed analysis. In particular, we can discuss in detail what is Xthe nature of the singularities when the perturbation expansions break down. X XWe also point out that, when $\omega$ is rational, several of the domains Xthat we conjectured to agree for $\omega$ irrational, differ. Nevertheless, Xwe observe in our numerical experiments that Xas the rational numbers approach their irrational Xlimiting values these domains seem to approach each other. This may serve Xas additional support for these conjectures. X XWe start by discussing some justification for the Xapproximation of the analyticity domains for circles with Xfrequency $\omega$ by those of Xapproximate frequencies. X X\CLAIM{Theorem}(Greenanal) XIf, for fixed $\epsilon$, $\lambda$, and $\omega$ irrational there exists an Xanalytic $u_{ \omega}(\theta)$ that satisfies \equ(Lindstedt) and X\item {$(i)$} X$\|2 u_\omega(\theta -\omega)2 u_\omega(\theta - 2 \omega) \cdots 2u_\omega( \theta - m \omega)\|_\delta \le \gamma < 1$ X\vskip1pt Xthen, X\vskip1pt Xfor $\omegab$ in a neighborhood of $\omega$, there exists an analytic X$u_{\omegab}(\theta)$ satisfying \equ(Lindstedt). X X\PROOF XWe will use a version of the Xcontractive mapping theorem similar to those we used in Xsections 6 and 7. XWe introduce the operator $\Gamma_\omega$ Xas in \equ(graphtransform). Since the dependence in $\omega$ Xwill play an important role, we make it explicit in the notation. X XSince $u_{\omega}$ satisfies X$\Gamma_\omega[ u_\omega] - u_\omega = 0$ Xwe see that we also have for $\deltab \le \delta$, X$$\|\Gamma_\omegab[ u_\omega] - u_\omega \|_\deltab \le 2 \| u_\omega\|_\deltab \| u_\omega'\|_\deltab| \omega - \omegab| \le K |\omega - \omegab|$$ Xwhere $K$ depends on $u_\omega$, $\delta$, $\deltab$ Xbut not on $|\omega - \omegab|$. X XFrom that, it is very easy to show that X$\Gamma_\omegab^m$ has a fixed point. XUsing condition $(i)$, it is Xpossible to show that $D \Gamma^m_\omegab [u_\omega]$, Xwhich can be expressed as multiplication by Xshifted versions of $u_\omega$, is Xa contraction in $\| \quad\|_\deltab$ Xwith a factor as close as desired to $\gamma$, if Xwe assume that $\omega - \omegab$ is small enough. XWe can also bound the $D^2 \Gamma^m_\omegab$ in a neighborhood Xof $u_\omega$. X XThe argument to prove that $\Gamma_\omegab$ has a fixed point Xis only slightly more complicated. XWe observe that, proceeding as in \clm(deltabound), Xgiven $R \in C^{\omega,\deltab}$ Xwe can solve the equation Xfor $\eta$, $D \Gamma_\omegab[ u_\omega] \eta = R$ Xand we have $\| \eta\|_\deltab \le K \|R\|_\deltab$ Xwhere $K$ can be chosen uniformly for $|\omega - \omegab|$ Xsufficiently small. XIf we consider the auxiliary operator X$\Phi(v) = -\left( D\Gamma_\omegab(u_\omega) - {\rm Id} \right)^{-1} \left( \Gamma_\omegab(v) - v \right) + v,$ Xwe Xsee that $\Phi$ is a contraction Xin $\|\quad\|_\deltab$ of Xa factor $1/2$ in a neighborhood Xof $u_\omega$ that can be chosen uniformly as X$|\omega - \omegab| $ is small. XSince $\| \Phi(u_\omega) - u_\omega\|_\deltab$ Xcan be made as small as desired by choosing X$\omega - \omegab$ to be small, we conclude that X$\Phi$ has a fixed point for all $\omegab$ Xin a neighborhood of $\omega$. But a fixed point Xof $\Phi$ is a fixed point of X$\Gamma_\omegab$. X\QED X\REMARK X\clm(Greenanal) is an analog of the justification Xof Greene's criterion for the approximation of Xinvariant curves by periodic orbits for the case of Hamiltonian systems, Xin the spirit of \cite{FL2}, in the case of Xhyperbolic circles. (See also \cite{McK2}.) X XThe main consequence of X\clm(Greenanal) is that Xall the non-perturbative methods X based on the Xcontinuation of attractive invariant circles Xwill provide a reasonable approximation Xto the domain of parameters for which there is an attractive Xinvariant circle. According to X\clm(deltaconv) for the values of $\lambda$ Xwe considered, Xthis agrees with the domain of analyticity. X XNow, we start to discuss the perturbative methods. XWe first observe that Xthe Lindstedt series for the case of rational frequencies do not exhibit Xsmall denominators, as can be easily established from \equ(Kdefined), and Xthe calculation of the XPad\'e approximants goes through. X X XTo study the nature of the boundary Xof the domain of analyticity Xfor the invariant circles for $\omega = p/q$ it is more convenient to Xinvestigate the behavior of the $q^{th}$ iterate of \equ(logistic). We have: X$$ F^q_{\lambda,\epsilon,\omega}(r, \theta) =(P_{\lambda, \epsilon, \theta}(r), \theta) X\EQ(iterate)$$ Xwhere $P_{\lambda,\epsilon,\theta}$ is a polynomial in $r$ Xof degree $2^q$, with Xonly even orders of $r$. XHence, $\theta$ enters only as a parameter in the Xdynamics of the $q^{\rm th}$ iterate of the map. X XPeriodic orbits of period $q$ are solutions of: X$$r=P_{\lambda,\epsilon,\theta}(r). X\EQ(fixedpoint)$$ X XSince \equ(fixedpoint) is a polynomial equation in $r$ Xof degree $2^q$, in general, Xit has $2^q$ Xdistinct solutions. The only way for the solution to a polynomial equation Xto lose analyticity on its dependence to the coefficients is if two or more Xsolutions coincide (there we may get a branch point). XIf we fix $\lambda$, $\theta$ and choose one solution for $\epsilon = 0$ and Xthen vary $\epsilon$ over the complex Xplane and follow that solution, the possible Xbranch points for the solution in terms Xof $\epsilon$, are the values of $\epsilon$ Xsuch that our solution is a root of \equ(fixedpoint) Xof multiplicity at least two. XOf course, there could be values of the parameter for Xwhich the root becomes double but which do not cause Xa loss of analyticity of the branches of the solutions. XThis, nevertheless, can only happen in degenerate situations X(e.g transcritical saddle node). In practice, once we have Xobtained a finite number of candidates, it is easy to verify that the Xnon-degeneracy conditions that imply that there is a branch point Xtake place. X X\REMARK XNotice that the position of the branch points depends on $\theta$, and that Xalthough generically there are branch points, there can be values of $\theta$ Xsuch that some branch points disappear. For example for $\omega = 1/1$ the Xposition of the branch point is determined by X $\lambda + \epsilon \cos(2 \pi \theta) = 1/4$, Xand for $\theta = 1/4$ the branch point is at $\infty$. X X XNotice that one important consequence Xof the previous discussion is that in the Xcase that $\omega$ is rational, the only possible Xways of breaking analyticity is branch points and that, Xtypically, we expect that these branch points are of Xorder $2$. This has important consequences for the Xbehavior of Pad\'e approximants. XAccording to a long standing conjecture, Pad\'e approximants of functions Xwith branch points tend to arrange their poles and zeros along lines Xthat are uniquely determined from the position of the poles ( see X\cite{BGM} vol.1, pg. 51, \cite{Gi} pg. 288, \cite{N}). XThe results we obtained Xfrom the Pad\'e approximants are remarkably close to the behavior Xoutlined above, since the poles and zeros of the Pad\'e approximants lie Xalong lines that emanate from the branch point and go radially to Xinfinity (forming a Mittag-Leffler star, see \cite{BGM} vol.1, pg. 50). XWe have also noticed that the poles (and the Xzeros) of the Pad\'e approximant tend to accumulate to the branch point. XAs the denominator increases the Xnumber of the branch points gets bigger and for large values of the denominator Xthe branch points tend to accumulate to Xthe natural boundary investigated in previous sections. X XSimilar behavior for the poles of the Pad\'e approximants for the Lindstedt Xseries, for invariant circles of the standard map Xwith complex $\omega$ with rational real part, Xwas reported in \cite{BM}. XThe poles (the zeros were not investigated in \cite{BM}) lied Xalong lines that emanate from points that tend to the origin as X$\Im \omega \to 0$, and go radially to infinity. It can be argued that the Xresemblance is due to the absence of small denominators for complex frequencies. XIt can also be argued that the effect of the imaginary part of the frequency Xis very similar to introducing dissipation in the system. XBased on this analogy, we conjecture X\CLAIM Conjecture(BM) XThe behavior observed in X\cite{BM} corresponds to branch points in the complex domain. XIn particular, the zeros of the numerator of the Pad\'e approximant Xshould also be in the same line in which the poles were found. X XSince this paper is mainly concerned with the rotating logistic Xmap, we will not discuss the standard map any further. X XTo compute the position of the branch points for the case of the rational Xfrequencies we fixed $\lambda$, $\theta$. XWe observe that the polynomial in $r$ given by X$P_{\epsilon, \lambda, \theta}(r) - r$ Xis a polynomial whose coefficients are Xpolynomials in $\epsilon$. XWe recall that the discriminant of a polynomial X$P(x)$ of degree $N$ is defined as disc($P(x)$) $= \prod_{i>j} (x_i - x_j)^2$ Xwhere $x_i$ are the roots. Hence, a polynomial has double Xroots if and only if the discriminant is zero. The importance of this remark is Xthat the discriminant of a polynomial is an algebraic function of the Xcoefficients. In particular, if the coefficients are polynomials in Xanother variable, the discriminant will be a Xpolynomial in the auxiliary variable. Reasonably efficient algorithms Xexist to compute discriminants of polynomials. XIn particular, the resultant algorithm X-- see e.g \cite{Kn} vol. 2 -- works in the case when the Xcoefficients are polynomials. X XNotice that to Xcompute analyticity Xdomains we are only interested in the Xcollisions of a particular root with the others, however the discriminant Xwill vanish whenever any two roots collide. XSo that finding all the values of $\epsilon$ for Xwhich the discriminant vanishes will provide us Xwith a discrete subset that includes, but which could be bigger than, Xthe set of branch points of the periodic solution that we are tracing. XUnfortunately, it is not so easy to decide whether a place where Xthe discriminant vanishes corresponds to a branch point of Xthe root we are tracing. This requires a global continuation method Xand one should follow all the Riemann sheets, since the roots Xchange identity by going into different sheets. XThis is a question that merits a more detailed study, Xbut we have not pursued it in this paper. We just observe that, Xin the cases that we studied, many of the values of X$\epsilon$ where the Xdiscriminant vanishes are indeed at the tip of Xthe accumulation of zeros and poles of the XPad\'e approximations. X XThe study of the domain of no escape becomes more Xcomplicated in the case of rational frequencies than what it was for Xirrational frequencies. XThe case $\omega = 1/1$ is relatively simple, X$P_{\lambda, \epsilon, \theta}(r) = r^2 + \lambda + \epsilon \cos (2\pi\theta)$ X and for any $\theta$ such that $\cos(2\pi\theta)\not= 0$ we get Xa distorted quadratic family $F_q = z^2 + \epsilon$. There exist only two Xfixed points, with only one of them attractive for some domain in the X$\epsilon$ plane. The domain of existence of the attractive fixed point Xis the main cardioid of the Mandelbrot set and can be computed by : X$$ z_0 = F_q(z_0),\quad | F'_q (z)|_{z=z_0} < 1$$ Xor X$$ |1 \pm \sqrt{1-4\epsilon}| < 1. X\EQ(stablefixed)$$ XThe cusp of the cardioid is located at the branch point of the domain of Xanalyticity of the roots and corresponds to a collision of the Xattractive and repelling fixed points. X XFor $\omega = p/q, \quad q>1$ the situation is no longer simple. XTo understand the full dynamics of the problem one has to consider Xiterations of polynomial maps of degree $2^q$ in the Xcomplex domain (for an overview see \cite{Bl1}). XSuch maps have $2^q$ fixed points, and although our solution follows one of Xthem, that is attractive for some domain in the parameter space, it can Xundergo a bifurcation and either disappear Xor become unstable. On the same time other stable fixed points, to which Xforward iteration of the map will converge, may still exist. XThe behavior of the map under forward iteration, is characterized Xby the behavior of the critical points of the map ($r_{cr}$ such that, X$P_{\lambda,\epsilon,\theta}'(r_{cr}) = 0$) under forward iteration. XIf an attractive periodic point exists, then at least one critical Xpoint belongs to its' basin of attraction (see \cite{Br}). XOne can recover the behavior of all the Xcritical points at a fixed $\theta$ by looking at the behavior of the Xcritical point at zero of X$P_{\lambda, \epsilon, \theta + n \frac{p}{q}}(r), \quad n=0,\ldots, q-1$, Xsince : X$$\eqalign{P_{\lambda, \epsilon, \theta}'(r) &= \left( X\underbrace{F_{\lambda, \epsilon, p/q}\circ \cdots X\circ F_{\lambda, \epsilon, p/q}}_{q \quad \rm times}(r, \theta)\right)' \hfill \cr X&= \underbrace{F_{\lambda, \epsilon, p/q}' \circ \cdots X\circ F_{\lambda, \epsilon, p/q}}_{q\quad \rm times}(r, \theta) X\underbrace{F_{\lambda, \epsilon, p/q}' \circ \cdots X\circ F_{\lambda, \epsilon, p/q}}_{q-1\quad \rm times}(r, \theta) \cdots XF_{\lambda, \epsilon, p/q}'(r, \theta).}$$ XThis feature is characteristic of our problem. X XDepending on the initial conditions and the numerical Ximplementation, the domain of no escape for a particular choice will Xbe a subset of the domain where at least one critical point remains bounded. XCaution should be taken at the interpretation of these results, since, as Xwas pointed out, the solution we are following may have changed stability Xas we varied $\epsilon$ in the complex plane and the forward iteration Xmay have followed a different solution (not necessarily a fixed point Xeither). Studies for Xcubic polynomials have been performed in \cite{BH},\cite{Bl2},\cite{Mil}. X XThe cusps of the domain of existence of at least one attractive fixed point Xcorrespond to saddle-node bifurcations between attractive and Xrepelling fixed points. The branch points in the domain of analyticity of Xthe root we follow form a subset of the set of points where Xthe cusps occur. X XTo interpret the dependence on $\theta$ we note that Xdifferent $\theta$ corresponds to a different slice of the parameter space. XFor the cases $\omega = 1/1$ and $\omega = 1/2$, the behavior for different X$\theta$'s can be recovered from the behavior for $\theta=0$ after Xthe scaling transformation $\epsilon \to \epsilon\cos(2\pi\theta)$. For Xother $\omega$'s such a simple scaling no longer exists. X X\SECTION Numerical Implementation X XWe have written a package in {\tt C} to manipulate one-dimensional XFourier series. This package has the feature that the arithmetic Xis done through function calls so that by changing a definitions file, Xwe can switch the arithmetic from single to extended precision. X XThe use of extended precision is a convenient way of handling Xthe severe numerical instabilities of the recursion, the Pad\'e approximation Xand the search for zeros. XWe have used the arithmetic library of the public domain program X{\tt PARI/GP}. X XTo increase the accuracy of the computation of the terms of the Lindstedt Xexpansion, we used a technique also used in \cite{FL1}. XWe considered the expansion in powers not of $\epsilon$ but of X$\epsilon/\rho$ where $\rho$ is chosen so as to make the series Xhave radius of convergence~1. XThe value of $\rho$ is determined from a preliminary run of the program. X XFrom this series expansion, we solved \equ(match) using Gaussian Xelimination verifying that the condition was always much smaller Xthan the accuracy of the previous results. The actual algorithm Xwas a translation into {\tt C} of the well known X{\tt DECOMP} and {\tt SOLVE} from \cite{FMM}. X XTo find the zeros of the denominator we used the routines ``{\tt xzroot}'', X``{\tt zroot}'' from ``Numerical Recipes'' translated to Xuse the {\tt PARI/GP} arithmetic and then checked if the answer Xwas indeed correct by evaluating the Xpolynomial and requiring that Xthe result was smaller than a tolerance. We eliminated zeros of the denominator Xthat are also zeros of the numerator. For some values of $N, M, \theta$ Xwe encountered spurious poles, that disappeared as $N, M, \theta$ Xchanged, but we did not Xeliminate them from the figures. X XTo implement the Newton method we discretized \equ(Newton) by Xtruncating the Fourier series representation of the function X$$u_{\epsilon_0} (\theta) = \sum_{k=-n}^n \hat u_{\epsilon_0,k} Xe^{2\pi ik\theta}\ ,\quad X\Delta(\theta) = \sum_{\ell=-n}^n \hat\Delta_\ell e^{2\pi i\ell\theta}\ ,\quad XR(\theta) =\sum_{m=-n}^n \hat R_m e^{2\pi im\theta}$$ Xwhere $n$ is a large number ($n\sim 100$). X XEquation \equ(Newton) reduces to X$$\mathop{{M}}_\approx \cdot \mathop{{\Delta}}_\sim X= - \mathop{{R}}_\sim$$ Xwhere X$$\displaylines{\hfill X\mathop{{\Delta}}_\sim X= (\hat\Delta_{-n},\ldots,\hat\Delta_n)^T\quad ,\quad X\mathop{{R}}_\sim = (\hat R_{-n},\ldots, \hat R_n)^T \hfill\cr X\hfill M_{k\ell} = \cases{ X2u_{\epsilon_0,k-\ell} - \delta_{k\ell} e^{2\pi i\omega k} X&,\quad $|k-\ell| \le n$\cr X0&,$\quad |k-\ell| >n$\cr}\hfill \cr}$$ XWe verified that the Xprocedure was converging in a quadratic fashion Xfor the cases that we studied. XThis gives us confidence that the Ximplementation was correct and that Xsources of error that make the truncated derivative Xdifferent from the derivative of the Xtruncation are small. XWhen $R(\theta)$ stopped decreasing we stopped the procedure. X XAs for the multipoint Pad\'e method we considered sequences of points Xwhich were distributed according to a sequence of powers of a fixed Xcomplex number. We used roots of unity, which leads to points evenly Xdistributed in a circle or a number of modulus slightly smaller than one, Xwhich leads to a spiral converging to the origin. XAgain the equations for the interpolation equations were solved by XGaussian elimination so as to have an estimate of the condition. XWe observed that, for the same number of points, distributing the points Xon a spiral seemed to lead to smaller condition numbers than distributing Xthem on a circle. X XNotice that the equations \equ(logistic) are invariant under the Xchange $\epsilon\to -\epsilon$, $\theta\to \theta+\pi$, Xtherefore the final domains of analyticity should be invariant Xunder the change $\epsilon\to-\epsilon$. XSince the coefficients of the series at zero are real, Xthe domains of analyticity should be invariant also under Xthe change $\epsilon\to \epsilon^*$. XSince the numerical methods we used were not built Xtaking into account such symmetries, the accuracy with Xwhich they are reflected in the final result Xcan give an estimate of the accuracy of the whole procedure. X XFor the non-perturbative methods we considered paths which Xwere radial, horizontal and vertical and found no discrepancies Xexcept in the places where the boundary shadows some of the Xpoints in the boundary. X XSome symbolic manipulation packages such as X{\tt MAXIMA} and {\tt MAPLE} implement Xthe resultant algorithm, for the calculation of the discriminant, Xin such a way that it is possible to compute with polynomial coefficients. XWe indeed implemented such calculations. XNevertheless, we found it more efficient to Xcompute the discriminant by evaluating the discriminant Xfor a discrete set of values of $\epsilon$ and then, Xusing the knowledge that the discriminant is a Xpolynomial in $\epsilon$ whose degree Xwe know, interpolate. XTo find the interpolating polynomial we adapted the routine ``{\tt toeplz}'' Xfrom \cite{FPTV} to Xbe able to use {\tt PARI/GP}. XWe found the results to agree with those obtained using the Xsymbolic algebra packages. X X\SECTION References X X X\ref\no{AKL1}\by{R.A. Adomaitis, I.G. Kevrekidis, R. de la Llave}\paper{Predicting the complexity of disconnected basins of attraction for a noninvertible system}. Preprint\endref X\ref\no{AKL2}\by{R.A. Adomaitis, I.G. Kevrekidis, R. de la Llave}\paper{Global bifurcations for the rotating logistic map. Some rigorous and numerical results}. Preprint\endref X\ref\no{B}\by{G.Baker}\book{Essentials of Pad\'e approximants}\publisher{Academic Press}\yr{1975}\endref X\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 X\ref\no{BCCF}\by{A. Berretti, A. Celletti, L. Chierchia, C. Falcolini}\paper{Natural boundaries for area preserving twist maps}\jour{Preprint}\endref X\ref\no{BH}\by{B. Branner, J. H. Hubbard}\paper{The iteration of cubic polynomials. Part I: The global topology of parameter space}\jour{Acta Mathematica}\vol{160}\pages{143--206}\yr{1988}\endref X\ref\no{Bl1}\by{P. Blanchard}\paper{Complex analytic dynamics on the Riemann sphere}\jour{Bull. Am. Math. Soc.}\vol{11}\pages{85--141}\yr{1984}\endref X\ref\no{Bl2}\by{P. Blanchard}\paper{Disconnected Julia sets}\inbook{Chaotic Dynamics and Fractals, M. F. Barnsley, S. G. Demko ed.}\publisher{Academic Press}\yr{1986}\endref X\ref\no{BGM}\by{G. Baker, M. Graves--Morris}\book{Pad\'e Approximants}\publisher{Addison Wesley}\yr{1981}\endref X%\ref\no{BGW}\by{G. Baker, J. L. Gammel, J. G. Wills}\paper{An investigation of the applicability of the Pad\'e approximant method}\jour{Jour. Math. Anal. App.}\vol{2}\pages{405--418}\yr{1961}\endref X\ref\no{BM}\by{A. Berretti, S. Marmi}\paper{Standard map at complex rotation numbers : Creation of natural boundaries}\jour{Phys. Rev. Lett.}\vol{68}\pages{1443--1446}\yr{1992}\endref X\ref\no{Bo}\by{J.-B. Bost}\paper{Tores invariants des syst\`emes dynamiques hamiltoniens}\jour{ Seminaire Bourbaki no. 639, Ast\'erisque}\vol{133--134} \pages{113}\yr{1986}\endref X\ref\no{Br}\by{H. Brolin}\paper{Invariant sets under iterations of rational functions}\jour{Arkiv for Matematik}\vol{6}\pages{103--144}\yr{1965}\endref X\ref\no{CE}\by{P. Collet, J.-P. Eckmann}\book{Iterated maps of the interval as dynamical systems}\publisher{Birkhauser, Boston}\yr{1980}\endref X\ref\no{CI}\by{A. Cenciner, G. Ioos}\paper{Bifurcations de tores invariants}\jour{Arch. Rational Mech. Anal.}\vol{69}\pages{109--198}\yr{1979}\endref X\ref\no{Fe}\by{N. Fenichel}\paper{Persistence and smoothness of invariant manifolds for flows}\jour{Int. Math. Jour.}\vol{21}\pages{193--226}\yr{1971}\endref X\ref\no{FL1}\by{C. Falcolini, R. de la Llave}\paper{Numerical Calculation of domains of analyticity for perturbation theories in the presence of small divisors}\jour{Jour. Stat. Phys.} \vol{67}\pages{645--666}\yr{1992}\endref X\ref\no{FL2}\by{C. Falcolini, R. de la Llave}\paper{A rigorous partial justification of Greene's criterion}\jour{Jour. Stat. Phys.} \vol{67}\pages{609--643}\yr{1992}\endref X\ref\no{FMM}\by{G.E. Forsythe, M.A. Malcolm, C. E. Moler}\book{Computer methods for mathematical computations}\publisher{Prentice Hall, Englewood Cliffs}\yr{1977}\endref X\ref\no{FPTV}\by{W.H. Press, B.P. Flannery, S. Teukolski, W. T. Vetterling}\book{Numerical Recipes}\publisher{Cambridge Univ. Press, Cambridge}\yr{1986}\endref X\ref\no{G}\by{J. Guckenheimer}\paper{Sensitive dependence on initial conditions for one dimensional maps}\jour{Comm. Math. Phys.}\vol{70}\pages{133--160}\yr{1979}\endref X\ref\no{Gi}\by{J. Gilewicz}\book{Approximants de Pad\'e}\publisher{Springer--Verlag}\yr{1978}\endref X%\ref\no{GN}\by{J. L. Gammel, J. Nuttall}\paper{Convergence of Pad\'e approximants to quasi-analytic functions beyond natural boundaries}\jour{Jour. Math. Anal. App.}\vol{43}\pages{694--696}\yr{1973}\endref X\ref\no{He}\by{P. Henrici}\book{Applied and computational complex analysis I} \publisher{J. Wiley, New York}\yr{1974}\endref X\ref\no{Ka}\by{K. Kaneko}\book{Collapse of tori and genesis of chaos in dissipative systems}\publisher{Book Scientific, Singapore}\yr{1986}\endref X\ref\no{Kn}\by{D.E. Knuth}\book{The art of computer programming, vol. II 2$^{nd}$ edition}\publisher{Addison Wesley}\yr{1981}\endref X\ref\no{M}\by{J.C. Mason}\paper{Some applications and drawbacks of Pad\'e approximants}\inbook{Approximation theory and applications} \pages{207} \publisher{Academic Press, New York}\yr{1981}\endref X\ref\no{McK1}\by{R.S. McKay}\paper{Renormalization in area preserving maps}\jour{Princeton thesis}\yr{1982}\endref X\ref\no{McK2}\by{R.S. McKay}\paper{On Greene's residue criterion}\jour{Nonlinearity}\vol{5}\pages{161--187}\yr{1992}\endref X\ref\no{N} \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 X\ref\no{Po}\by{H. Poincar\'e}\book{Les m\'ethodes nouvelles de la m\'ecanique c\'eleste}\publisher{Gauthier Villars}\yr{1899}\endref X%\ref\no{Pom}\by{Ch. Pommerenke}\paper{Pad\'e approximants and convergence in capacity}\jour{J. Math. Anal. Appl.}\vol{41}\pages{775--780}\yr{1973}\endref X\ref\no{R} \by{D. Rand} \paper{Universality for the breakdown of dissipative golden invariant tori} \inbook{ Proceedings of the 1986 I.A.M.P. meeting} \pages{537--547} \publisher{World Scientific, Singapore} \yr{1987} \endref X\ref\no{RA}\by{R. H. Rand, D. Armbruster}\book{Perturbation methods, bifurcation theory and computer algebra}\publisher{Springer--Verlag, New York}\yr{1987}\endref X\ref\no{SB}\by{J. Stoer, R. Burlisch}\book{Introduction to numerical analysis}\publisher{Springer--Verlag, N.Y.}\yr{1980}\endref X\ref\no{Si}\by{D. Singer}\paper{Stable orbits and bifurcations of maps of the interval}\jour{SIAM Jour. App. Math.}\vol{35}\pages{260}\yr{1978}\endref X\ref\no{W}\by{P. Walters}\book{An introduction to ergodic theory}\publisher{Springer--Verlag, N.Y.}\yr{1982}\endref X\ref\no{Wi}\by{H. Wilkinson} \paper{The perfidious polynomial}\inbook{Studies in numerical analysis, H. Golub ed.}\publisher{M.A.A., Washington}\yr{1985}\endref X\ref\no{Ze1}\by{E. Zehnder}\paper{Generalized implicit function theorems with applications to some small divisor problems I}\jour{Comm. Pure and Appl. Math.}\vol{28}\pages{91--140}\yr{1975}\endref X\ref\no{Ze2}\by{E. Zehnder}\paper{Generalized implicit function theorems with applications to some small divisor problems II}\jour{Comm. Pure and Appl. Math.}\vol{29}\pages{49--111}\yr{1976}\endref X\SECTION Captions X XFigure 1. XThe boundary of no escape (Set 1), X computed by iterating the map forward and using a continuation method until Xescape occurs (three families of paths (radial, vertical, horizontal) Xare used and the results are superimposed) with Xthe poles of the Pad\'e approximants $[50/50]$ for several angles (Set 2) Xand with the poles of the Pad\'e approximants for several Fourier coefficients Xof the $\epsilon$ expansion (Set 3) for X$\lambda = 0.2$. X XFigure 2. XThe boundary of no escape (Set 1) Xwith Xthe poles of the Pad\'e approximants $[50/50]$ for several angles (Set 2) Xfor $\lambda = 0.1$. X XFigure 3. Boundaries where the Pad\'e approximants $[24/24]$ and $[20/20]$ Xfor $\lambda = 0.2, \theta = 0.72$ Xdiffer more than $\delta$, superimposed with the boundary of no escape under Xforward iteration (Set 0). For Set 2 $\delta = 3 \times 10^{-5}$, Xfor Set 3 $\delta = 3 \times 10^{-4}$, for Set 4 $\delta = 2 \times 10^{-3}$, Xfor Set 5 $\delta = 2 \times 10^{-2}$. X XFigure 4. The poles of the multi-point Pad\'e approximant $[90/90]$ for X$\lambda = 0.2 $ and several angles Xsuperimposed with the boundary of no escape under forward iteration (Set 1). XExpansion Xin $\epsilon$ around 21 points with order of the expansion around zero 20 Xand around the other points 7. Set 2 : Expansion around $z_k = 0.1 Xe^{2 \pi i k /20} , k= 1,\ldots , 20$. Set 3 : Expansion around X$z_k = ( 0.19 e^{2 \pi i/20})^k , k=1, \ldots ,20$. X XFigure 5. Invariant curve close to breakdown. An example Xof a saddle node bifurcation. X$\lambda = 0.2$, $\epsilon= 0.589$. X XFigure 6. Invariant curve close to breakdown. XAn example of an Xinvariant curve becoming discontinuous. XReal part of the invariant circle for X$\lambda = 0.2$, $\epsilon = 0.22 + 0.677i$. X XFigure 7. Invariant curve close to breakdown. XAn example of an Xinvariant curve becoming discontinuous. XImaginary part of the invariant circle for X$\lambda = 0.2$, $\epsilon = 0.22 + 0.677i$. X X XFigure 8. Branch points (Set 2) and branch cuts (Set 3) Xfor rational frequency superimposed Xwith the domain of existence of an Xattractive fixed point (Set 1). The zeros and the Xpoles of the $[40/40]$ Pad\'e approximant seem to converge to the Xbranch cut on a straight line. X$\omega = 1/2$, $\lambda=0.2$, $\theta= 0$. X XFigure 9. As in figure 6 with $\omega = 2/3$, $\lambda=0.2$, $\theta= 0$. 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X1828 1988 D X1881 1966 D X1925 1943 D X1962 1918 D X1974 1886 D X1933 1833 D X1854 1764 D X1855 1723 D X1849 1678 D X1838 1629 D X1890 1608 D X1935 1585 D X1984 1564 D X2037 1547 D X2087 1529 D X2126 1505 D X2180 1491 D X2248 1486 D X2288 1465 D X2305 1429 D X2310 1383 D X2359 1368 D X2405 1352 D X2446 1331 D X2478 1304 D X2519 1284 D X2558 1263 D X2606 1250 D X2654 1238 D X2708 1232 D X2758 1225 D X2809 1218 D X2859 1212 D X2909 1208 D X2955 1198 D X3005 1195 D X3051 1186 D X3100 1185 D X3148 1181 D X3195 1178 D X3242 1176 D X3287 1171 D X3331 1165 D X3376 1161 D X3419 1153 D X3462 1145 D X3503 1131 D X3544 1118 D X3584 1098 D X3626 1080 D X3668 1062 D X3712 1051 D X3754 1021 D X3799 998 D X3844 973 D X3890 942 D X3940 947 D X3988 921 D X4037 944 D X4087 942 D X4133 970 D X4179 995 D X4222 1028 D X4265 1051 D X4309 1062 D X4351 1080 D X4393 1098 D X4433 1118 D X4474 1131 D X4515 1145 D X4558 1153 D X4601 1161 D X4645 1168 D X4690 1171 D X4737 1173 D X4782 1178 D X4829 1181 D X4877 1185 D X4924 1189 D X4974 1192 D X5020 1201 D X5070 1205 D X5118 1212 D X5168 1218 D X5219 1225 D X5269 1232 D X5323 1238 D X5371 1250 D X5419 1263 D X5461 1282 D X5494 1309 D X5531 1331 D X5578 1347 D X5618 1368 D X5667 1383 D X5672 1429 D X5686 1467 D X5736 1482 D X5797 1491 D X5851 1505 D X5894 1527 D X5940 1547 D X5993 1564 D X6042 1585 D X6095 1604 D X6131 1632 D X6128 1678 D X6122 1723 D X6123 1764 D X6044 1833 D X6012 1883 D X6015 1918 D X6048 1944 D X6096 1966 D X6145 1989 D X6184 2015 D X6209 2045 D X6243 2074 D X6267 2106 D X6281 2139 D X6303 2172 D X6414 2189 D X6161 2265 D X6189 2294 D X6183 2329 D X6223 2358 D X6243 2389 D X6077 2435 D X6040 2469 D X6006 2501 D X5966 2532 D X5874 2562 D X5744 2590 D X5747 2590 D X5866 2611 D X5926 2632 D X6015 2653 D X6015 2674 D X6015 2695 D X6045 2716 D X6075 2737 D X6105 2758 D X6254 2779 D X6254 2800 D X6254 2821 D X6224 2842 D X6164 2863 D X6194 2884 D X6194 2905 D X6164 2926 D X6403 2947 D X6403 2968 D X6433 2989 D X6313 3010 D X6313 3031 D X6283 3052 D X6283 3073 D X6283 3094 D X6254 3115 D X6224 3136 D X6194 3157 D X6194 3178 D X6134 3199 D X6105 3220 D X6075 3241 D X6015 3262 D X6015 3283 D X6015 3304 D X6045 3325 D X6045 3346 D X6075 3367 D X6105 3388 D X6134 3409 D X6134 3430 D X6134 3451 D X6134 3472 D X6134 3493 D X6134 3514 D X6134 3535 D X6134 3556 D X6075 3577 D X6045 3598 D X5985 3619 D X5956 3640 D X5896 3660 D X5836 3681 D X5717 3702 D X5687 3723 D X5687 3744 D X5687 3765 D X5687 3786 D X5628 3807 D X5568 3828 D X5538 3849 D X5509 3870 D X5479 3891 D X5449 3912 D X5360 3933 D X5211 3954 D X5061 3975 D X4853 3996 D X4614 4017 D X4525 4038 D X4465 4059 D X4406 4080 D X4376 4101 D X4287 4122 D X4257 4143 D X4227 4164 D X4197 4185 D X4138 4206 D X4108 4227 D X4048 4248 D X4018 4269 D X2230 2590 D X2111 2611 D X2051 2632 D X1962 2653 D X1962 2674 D X1932 2695 D X1932 2716 D X1902 2737 D X1872 2758 D X1843 2779 D X1723 2800 D X1723 2821 D X1753 2842 D X1813 2863 D X1783 2884 D X1813 2905 D X1813 2926 D X1574 2947 D X1544 2968 D X1544 2989 D X1664 3010 D X1664 3031 D X1694 3052 D X1694 3073 D X1694 3094 D X1723 3115 D X1753 3136 D X1783 3157 D X1783 3178 D X1843 3199 D X1872 3220 D X1932 3241 D X1962 3262 D X1962 3283 D X1962 3304 D X1932 3325 D X1932 3346 D X1902 3367 D X1872 3388 D X1843 3409 D X1843 3430 D X1843 3451 D X1843 3472 D X1843 3493 D X1843 3514 D X1813 3535 D X1843 3556 D X1872 3577 D X1932 3598 D X1992 3619 D X2051 3640 D X2081 3660 D X2141 3681 D X2260 3702 D X2260 3723 D X2290 3744 D X2290 3765 D X2290 3786 D X2349 3807 D X2379 3828 D X2439 3849 D X2468 3870 D X2498 3891 D X2528 3912 D X2617 3933 D X2766 3954 D X2916 3975 D X3124 3996 D X3363 4017 D X3452 4038 D X3512 4059 D X3571 4080 D X3601 4101 D X3690 4122 D X3720 4143 D X3750 4164 D X3780 4185 D X3839 4206 D X3869 4227 D X3929 4248 D X3959 4269 D X2230 2590 D X2111 2569 D X2051 2548 D X1962 2527 D X1962 2506 D X1932 2485 D X1932 2464 D X1902 2443 D X1872 2422 D X1843 2401 D X1723 2380 D X1723 2359 D X1753 2338 D X1813 2317 D X1783 2296 D X1813 2275 D X1813 2254 D X1574 2233 D X1544 2212 D X1544 2191 D X1664 2170 D X1664 2149 D X1694 2128 D X1694 2107 D X1694 2086 D X1723 2065 D X1753 2044 D X1783 2023 D X1783 2002 D X1843 1981 D X1872 1960 D X1932 1939 D X1962 1918 D X1962 1897 D X1962 1876 D X1932 1855 D X1932 1834 D X1902 1813 D X1872 1792 D X1843 1771 D X1843 1750 D X1843 1729 D X1843 1708 D X1843 1687 D X1843 1666 D X1813 1645 D X1843 1624 D X1872 1603 D X1932 1582 D X1992 1561 D X2051 1541 D X2081 1520 D X2141 1499 D X2260 1478 D X2260 1457 D X2290 1436 D X2290 1415 D X2290 1394 D X2349 1373 D X2379 1352 D X2439 1331 D X2468 1310 D X2498 1289 D X2528 1268 D X2617 1247 D X2766 1226 D X2916 1205 D X3124 1184 D X3363 1163 D X3452 1142 D X3512 1121 D X3571 1100 D X3601 1079 D X3690 1058 D X3720 1037 D X3750 1016 D X3780 995 D X3839 974 D X3869 953 D X3929 932 D X3959 911 D X5747 2590 D X5866 2569 D X5926 2548 D X6015 2527 D X6015 2506 D X6015 2485 D X6045 2464 D X6075 2443 D X6105 2422 D X6254 2401 D X6254 2380 D X6254 2359 D X6224 2338 D X6164 2317 D X6194 2296 D X6194 2275 D X6164 2254 D X6403 2233 D X6403 2212 D X6433 2191 D X6313 2170 D X6313 2149 D X6283 2128 D X6283 2107 D X6283 2086 D X6254 2065 D X6224 2044 D X6194 2023 D X6194 2002 D X6134 1981 D X6105 1960 D X6075 1939 D X6015 1918 D X6015 1897 D X6015 1876 D X6045 1855 D X6045 1834 D X6075 1813 D X6105 1792 D X6134 1771 D X6134 1750 D X6134 1729 D X6134 1708 D X6134 1687 D X6134 1666 D X6134 1645 D X6134 1624 D X6075 1603 D X6045 1582 D X5985 1561 D X5956 1541 D X5896 1520 D X5836 1499 D X5717 1478 D X5687 1457 D X5687 1436 D X5687 1415 D X5687 1394 D X5628 1373 D X5568 1352 D X5538 1331 D X5509 1310 D X5479 1289 D X5449 1268 D X5360 1247 D X5211 1226 D X5061 1205 D X4853 1184 D X4614 1163 D X4525 1142 D X4465 1121 D X4406 1100 D X4376 1079 D X4287 1058 D X4257 1037 D X4227 1016 D X4197 995 D X4138 974 D X4108 953 D X4048 932 D X4018 911 D XLT0 X6486 4346 M X(Set 2) Rshow X6654 4346 C2 X3687 4134 C2 X3687 4134 C2 X3687 1046 C2 X3567 1135 C2 X3567 1135 C2 X3567 4045 C2 X3567 4045 C2 X4690 4060 C2 X4690 4060 C2 X4923 1200 C2 X4923 1200 C2 X5122 998 C2 X5122 998 C2 X5122 4182 C2 X5122 4182 C2 X2796 3983 C2 X2771 987 C2 X2771 987 C2 X2771 987 C2 X2771 4193 C2 X2771 4193 C2 X5669 1293 C2 X5669 1293 C2 X5669 3887 C2 X5669 3887 C2 X2283 1502 C2 X2283 3678 C2 X2110 2590 C2 X5882 1842 C2 X6007 1300 C2 X6007 3880 C2 X1957 2590 C2 X1900 3262 C2 X1900 1918 C2 X1900 1918 C2 X6168 3369 C2 X6168 3369 C2 X6168 1811 C2 X6168 1811 C2 X1807 1402 C2 X1807 3778 C2 X6193 2590 C2 X6193 2590 C2 X1728 2196 C2 X1728 2984 C2 X1728 2984 C2 X6258 2175 C2 X6258 3005 C2 X4170 4062 C2 X4170 1118 C2 X3638 1044 C2 X3638 4136 C2 X4506 4060 C2 X4506 1120 C2 X3419 1196 C2 X3419 3984 C2 X3287 1093 C2 X3287 4087 C2 X4722 1156 C2 X4722 4024 C2 X5004 4027 C2 X5004 1153 C2 X2967 1039 C2 X2967 4141 C2 X2810 3960 C2 X2810 1220 C2 X5473 1208 C2 X5473 3972 C2 X5721 1560 C2 X5721 3620 C2 X5739 1362 C2 X5739 3818 C2 X2233 657 C2 X2233 4523 C2 X2212 3407 C2 X2212 1773 C2 X2142 1439 C2 X2142 3741 C2 X5870 2590 C2 X2094 2590 C2 X6003 2590 C2 X1931 2590 C2 X1905 2857 C2 X1905 2323 C2 X6080 1857 C2 X6080 3323 C2 X1887 3278 C2 X1887 1902 C2 X6153 3030 C2 X6153 2150 C2 X6680 2590 C2 X3718 4144 C2 X3718 1036 C2 X4303 4232 C2 X4303 948 C2 X4430 1102 C2 X4430 4078 C2 X3518 4019 C2 X3518 1161 C2 X3364 4116 C2 X3364 1064 C2 X4641 4035 C2 X4641 1145 C2 X3182 1112 C2 X3182 4068 C2 X4878 4009 C2 X4878 1171 C2 X2911 1178 C2 X2911 4002 C2 X5301 1120 C2 X5301 4060 C2 X5534 3626 C2 X5534 1554 C2 X2358 1426 C2 X2358 3754 C2 X5869 2590 C2 X5949 3730 C2 X5949 1450 C2 X6024 2590 C2 X1935 2590 C2 X1925 1902 C2 X1925 3278 C2 X1924 3789 C2 X1924 1391 C2 X6087 3020 C2 X6087 2160 C2 X6243 1915 C2 X6243 3265 C2 X1722 2893 C2 X1722 2287 C2 X1397 3155 C2 X1397 2025 C2 X6584 1567 C2 X6584 3613 C2 X4031 4253 C2 X4031 927 C2 X3703 1103 C2 X3703 4077 C2 X4399 4134 C2 X4399 1046 C2 X3437 4135 C2 X3437 1045 C2 X4602 4066 C2 X4602 1114 C2 X3247 4063 C2 X3247 1117 C2 X4778 4024 C2 X4778 1156 C2 X4808 4377 C2 X4808 803 C2 X3027 1135 C2 X3027 4045 C2 X5138 3979 C2 X5138 1201 C2 X2499 1263 C2 X2499 3917 C2 X5623 1436 C2 X5623 3744 C2 X2123 1311 C2 X2123 3869 C2 X5870 2590 C2 X1958 2590 C2 X1956 1848 C2 X1956 3332 C2 X6034 2590 C2 X1904 2997 C2 X1904 2183 C2 X6086 1923 C2 X6086 3257 C2 X6147 2300 C2 X6147 2880 C2 X1606 2590 C2 X1575 1979 C2 X1575 3201 C2 X6545 3534 C2 X6545 1646 C2 X6927 2590 C2 X4241 4230 C2 X4241 950 C2 X3598 1157 C2 X4412 4114 C2 X4412 1066 C2 X3423 1114 C2 X3423 4066 C2 X3423 4066 C2 X4593 1183 C2 X4659 1137 C2 X4659 4043 C2 X4659 4043 C2 X3159 1166 C2 X3159 4014 C2 X5001 1165 C2 X5001 4015 C2 X2839 1134 C2 X2839 4046 C2 X5540 1276 C2 X5540 3904 C2 X2424 4021 C2 X2424 1159 C2 X2303 3759 C2 X2303 1421 C2 X5693 3742 C2 X5693 1438 C2 X5962 2590 C2 X6023 1526 C2 X6023 3654 C2 X1949 1904 C2 X1949 3276 C2 X6101 1919 C2 X6101 3261 C2 X1853 2267 C2 X1853 2913 C2 X1793 2590 C2 X1714 3427 C2 X1714 1753 C2 X6299 2273 C2 X6299 2907 C2 X6420 2590 C2 XLT0 X6486 4206 M X(Set 3) Rshow X6654 4206 T2 X3889 844 T2 X4088 4336 T2 X4476 602 T2 X3501 4578 T2 X3172 4133 T2 X4805 1047 T2 X5806 2590 T2 X2171 2590 T2 X5980 1274 T2 X1997 3906 T2 X1985 1381 T2 X5992 3799 T2 X1916 2593 T2 X6061 2587 T2 X4240 4226 T2 X3737 954 T2 X3722 4221 T2 X4255 959 T2 X3603 943 T2 X4374 4237 T2 X3576 4224 T2 X4401 956 T2 X4561 4147 T2 X3416 1033 T2 X3381 4174 T2 X4596 1006 T2 X4982 4291 T2 X2995 889 T2 X2632 4328 T2 X5345 852 T2 X5685 1255 T2 X2292 3925 T2 X5726 4002 T2 X2251 1178 T2 X2251 3752 T2 X5726 1428 T2 X5737 3726 T2 X2240 1454 T2 X5964 3988 T2 X2013 1192 T2 X1878 3289 T2 X6099 1891 T2 X1868 2576 T2 X6109 2604 T2 X6109 3295 T2 X1868 1885 T2 X6458 2736 T2 X1519 2444 T2 X1349 1900 T2 X6628 3280 T2 X6716 1549 T2 X1261 3631 T2 X3881 879 T2 X4096 4301 T2 X4604 829 T2 X3373 4351 T2 X3159 822 T2 X4818 4358 T2 X5815 573 T2 X2162 4607 T2 X5817 2594 T2 X2160 2586 T2 X2109 1422 T2 X5868 3758 T2 X5870 1314 T2 X2107 3866 T2 X1971 1832 T2 X6006 3348 T2 X1794 2550 T2 X6183 2630 T2 X6189 1864 T2 X1788 3316 T2 X4084 4221 T2 X3893 959 T2 X3434 4214 T2 X4543 966 T2 X2933 724 T2 X5044 4456 T2 X2931 4270 T2 X5046 910 T2 X2323 3933 T2 X5654 1247 T2 X5869 3702 T2 X2108 1478 T2 X6025 2627 T2 X1952 2553 T2 X6149 2967 T2 X1828 2213 T2 X6155 1892 T2 X1822 3288 T2 X3737 850 T2 X4240 4330 T2 X4462 974 T2 X3515 4206 T2 X3031 4174 T2 X4946 1006 T2 X5501 1236 T2 X2476 3944 T2 X2397 1485 T2 X5580 3695 T2 X2278 1600 T2 X5699 3580 T2 X5877 2765 T2 X2100 2415 T2 X1973 2649 T2 X6004 2531 T2 X1896 3303 T2 X6081 1877 T2 X3815 970 T2 X4162 4210 T2 X4287 1002 T2 X3690 4178 T2 X3605 1016 T2 X4372 4164 T2 X3471 1028 T2 X4506 4152 T2 X3433 4174 T2 X4544 1006 T2 X3424 4201 T2 X4553 979 T2 X2988 4279 T2 X4989 901 T2 X5127 4381 T2 X2850 799 T2 X5312 1019 T2 X2665 4161 T2 X5520 3965 T2 X2457 1215 T2 X5606 1283 T2 X2371 3897 T2 X2266 1449 T2 X5711 3731 T2 X5754 1432 T2 X2223 3748 T2 X2115 1469 T2 X5862 3711 T2 X5872 4494 T2 X2105 686 T2 X1952 2705 T2 X6025 2475 T2 X1906 3283 T2 X6071 1897 T2 X1849 1824 T2 X6128 3356 T2 X1810 2240 T2 X6167 2940 T2 X6223 2544 T2 X1754 2636 T2 X1737 3325 T2 X6240 1855 T2 X6547 2084 T2 X1430 3096 T2 Xstroke Xgrestore Xend Xshowpage END_OF_FILE if test 20399 -ne `wc -c <'figure1.ps'`; then echo shar: \"'figure1.ps'\" unpacked with wrong size! fi # end of 'figure1.ps' fi if test -f 'figure2.ps' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'figure2.ps'\" else echo shar: Extracting \"'figure2.ps'\" \(19614 characters\) sed "s/^X//" >'figure2.ps' <<'END_OF_FILE' X%!PS-Adobe-2.0 X%%Creator: gnuplot X%%DocumentFonts: Courier X%%BoundingBox: 50 50 554 770 X%%Pages: (atend) X%%EndComments X/gnudict 40 dict def Xgnudict begin X/Color false def X/gnulinewidth 5.000 def X/vshift -46 def X/dl {10 mul} def X/hpt 15.5 def X/vpt 15.5 def X/vpt2 vpt 2 mul def X/hpt2 hpt 2 mul def X/vpt4 vpt 4 mul def X/hpt4 hpt 4 mul def X/Lshow { currentpoint stroke moveto X 0 vshift rmoveto show } def X/Rshow { currentpoint stroke moveto X dup stringwidth pop neg vshift rmoveto show } def X/Cshow { currentpoint stroke moveto X dup stringwidth pop -2 div vshift rmoveto show } def X/DL { Color {setrgbcolor [] 0 setdash pop} X {pop pop pop 0 setdash} ifelse } def X/BL { stroke gnulinewidth 2 mul setlinewidth } def X/AL { stroke gnulinewidth 2 div setlinewidth } def X/PL { stroke gnulinewidth setlinewidth } def X/LTb { BL [] 0 0 0 DL } def X/LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def X/LT0 { PL [] 0 1 0 DL } def X/LT1 { PL [4 dl 2 dl] 0 0 1 DL } def X/LT2 { PL [2 dl 3 dl] 1 0 0 DL } def X/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def X/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def X/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def X/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def X/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def X/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def X/M {moveto} def X/L {lineto} def X/P { stroke [] 0 setdash X currentlinewidth 2 div sub moveto X 0 currentlinewidth rlineto stroke } def X/D { stroke [] 0 setdash 2 copy vpt add moveto X hpt neg vpt neg rlineto hpt vpt neg rlineto X hpt vpt rlineto hpt neg vpt rlineto closepath stroke X P } def X/B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add moveto X 0 vpt2 neg rlineto hpt2 0 rlineto 0 vpt2 rlineto X hpt2 neg 0 rlineto closepath stroke X P } def X/C { stroke [] 0 setdash exch hpt sub exch vpt add moveto X hpt2 vpt2 neg rlineto currentpoint stroke moveto X hpt2 neg 0 rmoveto hpt2 vpt2 rlineto stroke } def X/T { stroke [] 0 setdash 2 copy vpt 1.12 mul add moveto X hpt neg vpt -1.62 mul rlineto X hpt 2 mul 0 rlineto X hpt neg vpt 1.62 mul rlineto closepath stroke X P } def X/D2 { stroke [] 0 setdash 2 copy vpt2 add moveto X hpt2 neg vpt2 neg rlineto hpt2 vpt2 neg rlineto X hpt2 vpt2 rlineto hpt2 neg vpt2 rlineto closepath stroke X P } def X/A2 { stroke [] 0 setdash vpt2 sub moveto 0 vpt4 rlineto X currentpoint stroke moveto X hpt2 neg vpt2 neg rmoveto hpt4 0 rlineto stroke X } def X/B2 { stroke [] 0 setdash 2 copy exch hpt2 sub exch vpt2 add moveto X 0 vpt4 neg rlineto hpt4 0 rlineto 0 vpt4 rlineto X hpt4 neg 0 rlineto closepath stroke X P } def X/C2 { stroke [] 0 setdash exch hpt2 sub exch vpt2 add moveto X hpt4 vpt4 neg rlineto currentpoint stroke moveto X hpt4 neg 0 rmoveto hpt4 vpt4 rlineto stroke } def X/T2 { stroke [] 0 setdash 2 copy vpt2 1.12 mul add moveto X hpt2 neg vpt2 -1.62 mul rlineto X hpt2 2 mul 0 rlineto X hpt2 neg vpt2 1.62 mul rlineto closepath stroke X P } def Xend X%%EndProlog X%%Page: 1 1 Xgnudict begin Xgsave X50 50 translate X0.100 0.100 scale X90 rotate X0 -5040 translate X0 setgray X/Courier findfont 140 scalefont setfont Xnewpath XLTa X1008 2590 M X6969 2590 L X3989 491 M X3989 4689 L XLTb X1008 491 M X1071 491 L X6969 491 M X6906 491 L X924 491 M X(-1) Rshow X1008 911 M X1071 911 L X6969 911 M X6906 911 L X924 911 M X(-0.8) Rshow X1008 1331 M X1071 1331 L X6969 1331 M X6906 1331 L X924 1331 M X(-0.6) Rshow X1008 1750 M X1071 1750 L X6969 1750 M X6906 1750 L X924 1750 M X(-0.4) Rshow X1008 2170 M X1071 2170 L X6969 2170 M X6906 2170 L X924 2170 M X(-0.2) Rshow X1008 2590 M X1071 2590 L X6969 2590 M X6906 2590 L X924 2590 M X(0) Rshow X1008 3010 M X1071 3010 L X6969 3010 M X6906 3010 L X924 3010 M X(0.2) Rshow X1008 3430 M X1071 3430 L X6969 3430 M X6906 3430 L X924 3430 M X(0.4) Rshow X1008 3849 M X1071 3849 L X6969 3849 M X6906 3849 L X924 3849 M X(0.6) Rshow X1008 4269 M X1071 4269 L X6969 4269 M X6906 4269 L X924 4269 M X(0.8) Rshow X1008 4689 M X1071 4689 L X6969 4689 M X6906 4689 L X924 4689 M X(1) Rshow X1008 491 M X1008 554 L X1008 4689 M X1008 4626 L X1008 351 M X(-1) Cshow X1604 491 M X1604 554 L X1604 4689 M X1604 4626 L X1604 351 M X(-0.8) Cshow X2200 491 M X2200 554 L X2200 4689 M X2200 4626 L X2200 351 M X(-0.6) Cshow X2796 491 M X2796 554 L X2796 4689 M X2796 4626 L X2796 351 M X(-0.4) Cshow X3392 491 M X3392 554 L X3392 4689 M X3392 4626 L X3392 351 M X(-0.2) Cshow X3989 491 M X3989 554 L X3989 4689 M X3989 4626 L X3989 351 M X(0) Cshow X4585 491 M X4585 554 L X4585 4689 M X4585 4626 L X4585 351 M X(0.2) Cshow X5181 491 M X5181 554 L X5181 4689 M X5181 4626 L X5181 351 M X(0.4) Cshow X5777 491 M X5777 554 L X5777 4689 M X5777 4626 L X5777 351 M X(0.6) Cshow X6373 491 M X6373 554 L X6373 4689 M X6373 4626 L X6373 351 M X(0.8) Cshow X6969 491 M X6969 554 L X6969 4689 M X6969 4626 L X6969 351 M X(1) Cshow XLTb X1008 491 M X6969 491 L X6969 4689 L X1008 4689 L X1008 491 L X140 2590 M Xcurrentpoint gsave translate 90 rotate 0 0 moveto X(Im epsilon) Cshow Xgrestore X3988 211 M X(Real epsilon) Cshow X4088 1 M X(Figure 2) Cshow XLT0 XLT0 X6486 4486 M X(Set 1) Rshow X6654 4486 D X3989 4080 D X4018 4059 D X4048 4038 D X4078 4038 D X4108 4038 D X4138 4017 D X4167 3996 D X4197 3996 D X4227 3996 D X4257 3954 D X4287 3933 D X4316 3933 D X4346 3933 D X4376 3912 D X4406 3891 D X4436 3870 D X4465 3870 D X4495 3849 D X4525 3828 D X4555 3828 D X4585 3807 D X4614 3807 D X4644 3807 D X4674 3786 D X4704 3786 D X4734 3786 D X4763 3786 D X4793 3786 D X4823 3765 D X4853 3765 D X4883 3765 D X4912 3765 D X4942 3744 D X4972 3744 D X5002 3744 D X5032 3723 D X5061 3723 D X5091 3723 D X5121 3702 D X5151 3702 D X5181 3702 D X5211 3681 D X5240 3681 D X5270 3681 D X5300 3660 D X5330 3660 D X5360 3660 D X5389 3640 D X5419 3640 D X5449 3640 D X5479 3619 D X5509 3619 D X5538 3598 D X5568 3577 D X5598 3577 D X5628 3577 D X5658 3556 D X5687 3556 D X5717 3556 D X5747 3535 D X5777 3535 D X5807 3514 D X5836 3451 D X5866 3451 D X5896 3430 D X5926 3409 D X5956 3367 D X5985 3346 D X6015 3325 D X6045 3325 D X6075 3304 D X6105 3304 D X6134 3304 D X6164 3283 D X6194 3262 D X6224 3241 D X6254 3220 D X6283 3199 D X6313 3178 D X6343 2947 D X6373 2926 D X6403 2905 D X6433 2905 D X6462 2884 D X6492 2884 D X6522 2863 D X6552 2863 D X6582 2863 D X6611 2842 D X6641 2821 D X6671 2716 D X6701 2716 D X6731 2716 D X6760 2695 D X6790 2632 D X6820 2632 D X6850 2611 D X6880 2611 D X3989 4080 D X3959 4059 D X3929 4059 D X3899 4059 D X3869 4038 D X3839 4017 D X3810 3996 D X3780 3996 D X3750 3996 D X3720 3954 D X3690 3933 D X3661 3933 D X3631 3933 D X3601 3912 D X3571 3891 D X3541 3870 D X3512 3870 D X3482 3849 D X3452 3828 D X3422 3828 D X3392 3807 D X3363 3807 D X3333 3807 D X3303 3807 D X3273 3786 D X3243 3786 D X3214 3786 D X3184 3786 D X3154 3765 D X3124 3765 D X3094 3765 D X3065 3765 D X3035 3765 D X3005 3744 D X2975 3744 D X2945 3744 D X2916 3723 D X2886 3723 D X2856 3702 D X2826 3702 D X2796 3702 D X2766 3681 D X2737 3681 D X2707 3681 D X2677 3681 D X2647 3660 D X2617 3660 D X2588 3640 D X2558 3640 D X2528 3640 D X2498 3619 D X2468 3598 D X2439 3598 D X2409 3577 D X2379 3577 D X2349 3577 D X2319 3556 D X2290 3556 D X2260 3556 D X2230 3535 D X2200 3535 D X2170 3514 D X2141 3451 D X2111 3451 D X2081 3430 D X2051 3409 D X2021 3367 D X1992 3346 D X1962 3325 D X1932 3325 D X1902 3304 D X1872 3304 D X1843 3304 D X1813 3283 D X1783 3262 D X1753 3220 D X1723 3220 D X1694 3199 D X1664 3199 D X1634 2947 D X1604 2926 D X1574 2905 D X1544 2905 D X1515 2884 D X1485 2884 D X1455 2884 D X1425 2863 D X1395 2842 D X1366 2842 D X1336 2821 D X1306 2716 D X1276 2716 D X1246 2716 D X1217 2695 D X1187 2632 D X1157 2632 D X1127 2611 D X1097 2611 D X3989 1100 D X3959 1121 D X3929 1121 D X3899 1121 D X3869 1142 D X3839 1163 D X3810 1184 D X3780 1184 D X3750 1184 D X3720 1226 D X3690 1247 D X3661 1247 D X3631 1247 D X3601 1268 D X3571 1289 D X3541 1310 D X3512 1310 D X3482 1331 D X3452 1352 D X3422 1352 D X3392 1373 D X3363 1373 D X3333 1373 D X3303 1373 D X3273 1394 D X3243 1394 D X3214 1394 D X3184 1394 D X3154 1415 D X3124 1415 D X3094 1415 D X3065 1415 D X3035 1415 D X3005 1436 D X2975 1436 D X2945 1436 D X2916 1457 D X2886 1457 D X2856 1478 D X2826 1478 D X2796 1478 D X2766 1499 D X2737 1499 D X2707 1499 D X2677 1499 D X2647 1520 D X2617 1520 D X2588 1541 D X2558 1541 D X2528 1541 D X2498 1561 D X2468 1582 D X2439 1582 D X2409 1603 D X2379 1603 D X2349 1603 D X2319 1624 D X2290 1624 D X2260 1624 D X2230 1645 D X2200 1645 D X2170 1666 D X2141 1729 D X2111 1729 D X2081 1750 D X2051 1771 D X2021 1813 D X1992 1834 D X1962 1855 D X1932 1855 D X1902 1876 D X1872 1876 D X1843 1876 D X1813 1897 D X1783 1918 D X1753 1960 D X1723 1960 D X1694 1981 D X1664 1981 D X1634 2233 D X1604 2254 D X1574 2275 D X1544 2275 D X1515 2296 D X1485 2296 D X1455 2296 D X1425 2317 D X1395 2338 D X1366 2338 D X1336 2359 D X1306 2464 D X1276 2464 D X1246 2464 D X1217 2485 D X1187 2548 D X1157 2548 D X1127 2569 D X1097 2569 D X3989 1100 D X4018 1121 D X4048 1142 D X4078 1142 D X4108 1142 D X4138 1163 D X4167 1184 D X4197 1184 D X4227 1184 D X4257 1226 D X4287 1247 D X4316 1247 D X4346 1247 D X4376 1268 D X4406 1289 D X4436 1310 D X4465 1310 D X4495 1331 D X4525 1352 D X4555 1352 D X4585 1373 D X4614 1373 D X4644 1373 D X4674 1394 D X4704 1394 D X4734 1394 D X4763 1394 D X4793 1394 D X4823 1415 D X4853 1415 D X4883 1415 D X4912 1415 D X4942 1436 D X4972 1436 D X5002 1436 D X5032 1457 D X5061 1457 D X5091 1457 D X5121 1478 D X5151 1478 D X5181 1478 D X5211 1499 D X5240 1499 D X5270 1499 D X5300 1520 D X5330 1520 D X5360 1520 D X5389 1541 D X5419 1541 D X5449 1541 D X5479 1561 D X5509 1561 D X5538 1582 D X5568 1603 D X5598 1603 D X5628 1603 D X5658 1624 D X5687 1624 D X5717 1624 D X5747 1645 D X5777 1645 D X5807 1666 D X5836 1729 D X5866 1729 D X5896 1750 D X5926 1771 D X5956 1813 D X5985 1834 D X6015 1855 D X6045 1855 D X6075 1876 D X6105 1876 D X6134 1876 D X6164 1897 D X6194 1918 D X6224 1939 D X6254 1960 D X6283 1981 D X6313 2002 D X6343 2233 D X6373 2254 D X6403 2275 D X6433 2275 D X6462 2296 D X6492 2296 D X6522 2317 D X6552 2317 D X6582 2317 D X6611 2338 D X6641 2359 D X6671 2464 D X6701 2464 D X6731 2464 D X6760 2485 D X6790 2548 D X6820 2548 D X6850 2569 D X6880 2569 D X6876 2590 D X6796 2631 D X6771 2672 D X6680 2709 D X6666 2748 D X6670 2788 D X6622 2824 D X6545 2856 D X6454 2884 D X6381 2911 D X6339 2942 D X6324 2976 D X6321 3012 D X6325 3050 D X6334 3089 D X6336 3127 D X6337 3166 D X6280 3190 D X6231 3215 D X6199 3245 D X6163 3272 D X6112 3294 D X6041 3307 D X5990 3327 D X5956 3352 D X5936 3382 D X5916 3412 D X5880 3435 D X5823 3448 D X5836 3494 D X5793 3513 D X5750 3532 D X5700 3545 D X5650 3558 D X5600 3569 D X5556 3584 D X5513 3598 D X5472 3613 D X5429 3626 D X5388 3640 D X5350 3655 D X5306 3665 D X5259 3672 D X5215 3680 D X5177 3692 D X5133 3699 D X5094 3710 D X5055 3721 D X5017 3731 D X4975 3737 D X4939 3749 D X4897 3754 D X4858 3761 D X4819 3768 D X4778 3771 D X4740 3779 D X4701 3784 D X4664 3791 D X4626 3798 D X4589 3804 D X4554 3816 D X4520 3830 D X4486 3844 D X4452 3861 D X4419 3883 D X4385 3905 D X4348 3917 D X4309 3926 D X4273 3946 D X4237 3973 D X4196 3983 D X4156 3992 D X4117 4030 D X4075 4042 D X4032 4049 D X3988 4063 D X3945 4049 D X3902 4042 D X3860 4030 D X3821 3995 D X3781 3979 D X3740 3976 D X3705 3943 D X3666 3932 D X3629 3917 D X3592 3902 D X3558 3883 D X3525 3861 D X3491 3844 D X3457 3830 D X3423 3816 D X3387 3807 D X3351 3798 D X3315 3788 D X3276 3784 D X3237 3779 D X3199 3771 D X3158 3768 D X3117 3764 D X3078 3757 D X3036 3752 D X2999 3740 D X2960 3731 D X2922 3721 D X2883 3710 D X2844 3699 D X2803 3690 D X2759 3682 D X2715 3674 D X2671 3665 D X2627 3655 D X2589 3640 D X2548 3626 D X2505 3613 D X2464 3598 D X2421 3584 D X2377 3569 D X2320 3562 D X2277 3545 D X2223 3534 D X2184 3513 D X2141 3494 D X2154 3448 D X2101 3434 D X2064 3411 D X2041 3382 D X2021 3352 D X1987 3327 D X1932 3309 D X1869 3292 D X1814 3272 D X1778 3245 D X1746 3215 D X1697 3190 D X1640 3166 D X1641 3127 D X1643 3089 D X1647 3050 D X1652 3012 D X1644 2977 D X1638 2942 D X1596 2911 D X1523 2884 D X1427 2856 D X1360 2824 D X1298 2789 D X1311 2748 D X1297 2709 D X1211 2672 D X1186 2631 D X1101 2590 D X1186 2549 D X1211 2508 D X1297 2471 D X1311 2432 D X1298 2391 D X1360 2356 D X1427 2324 D X1523 2296 D X1596 2269 D X1638 2238 D X1644 2203 D X1652 2168 D X1647 2130 D X1643 2091 D X1641 2053 D X1640 2014 D X1697 1990 D X1746 1965 D X1778 1935 D X1814 1908 D X1869 1888 D X1932 1871 D X1987 1853 D X2021 1828 D X2041 1798 D X2064 1769 D X2101 1746 D X2154 1732 D X2141 1686 D X2184 1667 D X2223 1646 D X2277 1635 D X2320 1618 D X2377 1611 D X2421 1596 D X2464 1582 D X2505 1567 D X2548 1554 D X2589 1540 D X2627 1525 D X2671 1515 D X2715 1506 D X2759 1498 D X2803 1490 D X2844 1481 D X2883 1470 D X2922 1459 D X2960 1449 D X2999 1440 D X3036 1428 D X3078 1423 D X3117 1416 D X3158 1412 D X3199 1409 D X3237 1401 D X3276 1396 D X3315 1392 D X3351 1382 D X3387 1373 D X3423 1364 D X3457 1350 D X3491 1336 D X3525 1319 D X3558 1297 D X3592 1278 D X3629 1263 D X3666 1248 D X3705 1237 D X3740 1204 D X3781 1201 D X3821 1185 D X3860 1150 D X3902 1138 D X3945 1131 D X3988 1117 D X4032 1131 D X4075 1138 D X4117 1150 D X4156 1188 D X4196 1197 D X4237 1207 D X4273 1234 D X4309 1254 D X4348 1263 D X4385 1275 D X4419 1297 D X4452 1319 D X4486 1336 D X4520 1350 D X4554 1364 D X4589 1376 D X4626 1382 D X4664 1389 D X4701 1396 D X4740 1401 D X4778 1409 D X4819 1412 D X4858 1419 D X4897 1426 D X4939 1431 D X4975 1443 D X5017 1449 D X5055 1459 D X5094 1470 D X5133 1481 D X5177 1488 D X5215 1500 D X5259 1508 D X5306 1515 D X5350 1525 D X5388 1540 D X5429 1554 D X5472 1567 D X5513 1582 D X5556 1596 D X5600 1611 D X5650 1622 D X5700 1635 D X5750 1648 D X5793 1667 D X5836 1686 D X5823 1732 D X5880 1745 D X5916 1768 D X5936 1798 D X5956 1828 D X5990 1853 D X6041 1873 D X6112 1886 D X6163 1908 D X6199 1935 D X6231 1965 D X6280 1990 D X6337 2014 D X6336 2053 D X6334 2091 D X6325 2130 D X6321 2168 D X6324 2204 D X6339 2238 D X6381 2269 D X6454 2296 D X6545 2324 D X6622 2356 D X6670 2392 D X6666 2432 D X6680 2471 D X6771 2508 D X6796 2549 D X6876 2590 D X6880 2590 D X6850 2611 D X6790 2632 D X6790 2653 D X6790 2674 D X6760 2695 D X6671 2716 D X6671 2737 D X6671 2758 D X6671 2779 D X6671 2800 D X6641 2821 D X6582 2842 D X6552 2863 D X6462 2884 D X6403 2905 D X6373 2926 D X6343 2947 D X6343 2968 D X6343 2989 D X6343 3010 D X6343 3031 D X6343 3052 D X6343 3073 D X6343 3094 D X6343 3115 D X6343 3136 D X6343 3157 D X6313 3178 D X6283 3199 D X6254 3220 D X6224 3241 D X6194 3262 D X6164 3283 D X6075 3304 D X6015 3325 D X5985 3346 D X5956 3367 D X5956 3388 D X5926 3409 D X5896 3430 D X5836 3451 D X5836 3472 D X5866 3493 D X5807 3514 D X5777 3535 D X5687 3556 D X5568 3577 D X5538 3598 D X5479 3619 D X5389 3640 D X5330 3660 D X5211 3681 D X5121 3702 D X5061 3723 D X4942 3744 D X4853 3765 D X4674 3786 D X4585 3807 D X4525 3828 D X4495 3849 D X4436 3870 D X4406 3891 D X4376 3912 D X4287 3933 D X4257 3954 D X4257 3975 D X4167 3996 D X4138 4017 D X4048 4038 D X4018 4059 D X1097 2590 D X1127 2611 D X1187 2632 D X1187 2653 D X1187 2674 D X1217 2695 D X1306 2716 D X1306 2737 D X1306 2758 D X1306 2779 D X1276 2800 D X1336 2821 D X1395 2842 D X1455 2863 D X1515 2884 D X1574 2905 D X1604 2926 D X1634 2947 D X1634 2968 D X1634 2989 D X1634 3010 D X1634 3031 D X1634 3052 D X1634 3073 D X1634 3094 D X1634 3115 D X1634 3136 D X1634 3157 D X1664 3178 D X1694 3199 D X1723 3220 D X1753 3241 D X1783 3262 D X1813 3283 D X1902 3304 D X1962 3325 D X1992 3346 D X2021 3367 D X2021 3388 D X2051 3409 D X2081 3430 D X2141 3451 D X2141 3472 D X2141 3493 D X2170 3514 D X2230 3535 D X2319 3556 D X2379 3577 D X2439 3598 D X2498 3619 D X2588 3640 D X2647 3660 D X2766 3681 D X2856 3702 D X2916 3723 D X3035 3744 D X3154 3765 D X3273 3786 D X3392 3807 D X3452 3828 D X3482 3849 D X3541 3870 D X3571 3891 D X3601 3912 D X3690 3933 D X3720 3954 D X3720 3975 D X3810 3996 D X3839 4017 D X3929 4038 D X3959 4059 D X1097 2590 D X1127 2569 D X1187 2548 D X1187 2527 D X1187 2506 D X1217 2485 D X1306 2464 D X1306 2443 D X1306 2422 D X1306 2401 D X1276 2380 D X1336 2359 D X1395 2338 D X1455 2317 D X1515 2296 D X1574 2275 D X1604 2254 D X1634 2233 D X1634 2212 D X1634 2191 D X1634 2170 D X1634 2149 D X1634 2128 D X1634 2107 D X1634 2086 D X1634 2065 D X1634 2044 D X1634 2023 D X1664 2002 D X1694 1981 D X1723 1960 D X1753 1939 D X1783 1918 D X1813 1897 D X1902 1876 D X1962 1855 D X1992 1834 D X2021 1813 D X2021 1792 D X2051 1771 D X2081 1750 D X2141 1729 D X2141 1708 D X2141 1687 D X2170 1666 D X2230 1645 D X2319 1624 D X2379 1603 D X2439 1582 D X2498 1561 D X2588 1541 D X2647 1520 D X2766 1499 D X2856 1478 D X2916 1457 D X3035 1436 D X3154 1415 D X3273 1394 D X3392 1373 D X3452 1352 D X3482 1331 D X3541 1310 D X3571 1289 D X3601 1268 D X3690 1247 D X3720 1226 D X3720 1205 D X3810 1184 D X3839 1163 D X3929 1142 D X3959 1121 D X6880 2590 D X6850 2569 D X6790 2548 D X6790 2527 D X6790 2506 D X6760 2485 D X6671 2464 D X6671 2443 D X6671 2422 D X6671 2401 D X6671 2380 D X6641 2359 D X6582 2338 D X6552 2317 D X6462 2296 D X6403 2275 D X6373 2254 D X6343 2233 D X6343 2212 D X6343 2191 D X6343 2170 D X6343 2149 D X6343 2128 D X6343 2107 D X6343 2086 D X6343 2065 D X6343 2044 D X6343 2023 D X6313 2002 D X6283 1981 D X6254 1960 D X6224 1939 D X6194 1918 D X6164 1897 D X6075 1876 D X6015 1855 D X5985 1834 D X5956 1813 D X5956 1792 D X5926 1771 D X5896 1750 D X5836 1729 D X5836 1708 D X5866 1687 D X5807 1666 D X5777 1645 D X5687 1624 D X5568 1603 D X5538 1582 D X5479 1561 D X5389 1541 D X5330 1520 D X5211 1499 D X5121 1478 D X5061 1457 D X4942 1436 D X4853 1415 D X4674 1394 D X4585 1373 D X4525 1352 D X4495 1331 D X4436 1310 D X4406 1289 D X4376 1268 D X4287 1247 D X4257 1226 D X4257 1205 D X4167 1184 D X4138 1163 D X4048 1142 D X4018 1121 D XLT0 X6486 4346 M X(Set 2) Rshow X6654 4346 C2 X3819 1229 C2 X3819 3951 C2 X4272 3977 C2 X4272 1203 C2 X4453 3862 C2 X4453 1318 C2 X3507 4174 C2 X3507 1006 C2 X3464 1322 C2 X3464 3858 C2 X4573 1350 C2 X4573 3830 C2 X3298 1364 C2 X3298 3816 C2 X4816 1349 C2 X4816 3831 C2 X3070 3777 C2 X3070 1403 C2 X5086 1456 C2 X5086 3724 C2 X2780 3684 C2 X2780 1496 C2 X2745 1387 C2 X2745 3793 C2 X5394 3750 C2 X5394 1430 C2 X5468 1634 C2 X5468 3546 C2 X2310 3526 C2 X2310 1654 C2 X5929 3714 C2 X5929 1466 C2 X2007 3388 C2 X2007 1792 C2 X6076 1765 C2 X6076 3415 C2 X6317 2196 C2 X6317 2984 C2 X1581 3007 C2 X1581 2173 C2 X1541 2590 C2 X6437 2590 C2 X6456 3101 C2 X6456 2079 C2 X1439 3715 C2 X1439 1465 C2 X3776 3933 C2 X3776 1247 C2 X4245 1245 C2 X4245 3935 C2 X3554 1025 C2 X3554 4155 C2 X4468 1329 C2 X4468 3851 C2 X3435 1313 C2 X3435 3867 C2 X4593 3817 C2 X4593 1363 C2 X3279 3801 C2 X3279 1379 C2 X4