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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN"><!--Converted with LaTeX2HTML 96.1-h (September 30, 1996) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><HTML><HEAD><TITLE>Finding unstable periodic orbits</TITLE><META NAME="description" CONTENT="Finding unstable periodic orbits"><META NAME="keywords" CONTENT="TiseanHTML"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="TiseanHTML.css"></HEAD><BODY bgcolor=ffffff LANG="EN" > <A NAME="tex2html265" HREF="node20.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html263" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html257" HREF="node18.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html266" HREF="node20.html">Locally linear prediction</A><B>Up:</B> <A NAME="tex2html264" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html258" HREF="node18.html">Simple nonlinear prediction</A><BR> <P><H2><A NAME="SECTION00053000000000000000">Finding unstable periodic orbits</A></H2><A NAME="secupo"> </A>As an application of simple nonlinear phase space prediction, let us discuss amethod to locate unstable periodic orbits embedded in a chaotic attractor. Thisis not the place to review the existing methods to solve this problem, somereferences include [<A HREF="citation.html#auerbach">47</A>, <A HREF="citation.html#biham">48</A>, <A HREF="citation.html#so">49</A>, <A HREF="citation.html#schmelcher">50</A>]. The TISEAN packagecontains a routine that implements the requirement that for a period <I>p</I> orbit<IMG WIDTH=119 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6947" SRC="img54.gif"> of a dynamical system like Eq.(<A HREF="node5.html#eqmap"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) acting ondelay vectors<BR><A NAME="equpo"> </A><IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation4912" SRC="img55.gif"><BR>With unit delay, the <I>p</I> delay vectors contain <I>p</I> different scalar entries,and Eq.(<A HREF="node19.html#equpo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) defines a root of a system of <I>p</I> nonlinear equations in<I>p</I> dimensions. Multidimensional root finding is not a simple problem. Thestandard Newton method has to be augmented by special tricks in order toconverge globally. Some such tricks, in particular means to select differentsolutions of Eq.(<A HREF="node19.html#equpo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) are implemented in [<A HREF="citation.html#schmelcher">50</A>]. Similarlyto the problems encountered in nonlinear noise reduction, solvingEq.(<A HREF="node19.html#equpo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) <EM>exactly</EM> is particularly problematic since <IMG WIDTH=23 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6957" SRC="img56.gif">is unknown and must be estimated from the data. In Ref. [<A HREF="citation.html#so">49</A>], approximatesolutions are found by performing just one iteration of the Newton method foreach available time series point. We prefer to look for a <EM>least squares</EM>solution by minimizing<BR><A NAME="equpo2"> </A><IMG WIDTH=500 HEIGHT=44 ALIGN=BOTTOM ALT="equation4916" SRC="img57.gif"><BR>instead. The routine <a href="../docs_f/upo.html">upo</a> uses a standard Levenberg-Marquardt algorithm tominimize (<A HREF="node19.html#equpo2"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>). For this it is necessary that <IMG WIDTH=23 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6957" SRC="img56.gif"> issmooth. Therefore we cannot use the simple nonlinear predictor based on locallyconstant approximations and we have to use a smooth kernel version,Eq.(<A HREF="node18.html#eqpredictkernel"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>), instead. With <IMG WIDTH=149 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline6961" SRC="img58.gif">, the kernelbandwidth <I>h</I> determines the degree of smoothness of <IMG WIDTH=23 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6957" SRC="img56.gif">. Trying tostart the minimization with all available time series segments will produce anumber of false minima, depending on the value of <I>h</I>. These have to bedistinguished from the true solutions by inspection. On the other hand, we canreach solutions of Eq.(<A HREF="node19.html#equpo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) which are not closely visited in the timeseries at all, an important advantage over close returnmethods [<A HREF="citation.html#auerbach">47</A>].<P><P><blockquote><A NAME="4908"> </A><IMG WIDTH=345 HEIGHT=315 ALIGN=BOTTOM ALT="figure727" SRC="img53.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figupo"> </A> Orbits of period six, or a sub-period thereof, of the Hénon map, determined from noisy data. The Hénon attractor does not have a period three orbit.<BR></blockquote><P>It should be noted that, depending on <I>h</I>, we may always find good minima of(<A HREF="node18.html#eqpredictkernel"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>), even if no solution of Eq.(<A HREF="node19.html#equpo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>), or noteven a truly deterministic dynamics exists. Thus the finding of unstableperiodic orbits in itself is not a strong indicator of determinism.We may however use the cycle locations or stabilities as a discriminatingstatistics in a test for nonlinearity, see Sec. <A HREF="node35.html#secsurro"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.While the orbits themselves are found quite easily, it is surprisinglydifficult to obtain reliable estimates of their stability in the presence ofnoise. In <a href="../docs_f/upo.html">upo</a>, a small perturbation is iterated along the orbit and theunstable eigenvalue is determined by the rate of its separation from theperiodic orbit.<P>The user of <a href="../docs_f/upo.html">upo</a> has to specify the embedding dimension, the period (whichmay also be smaller) and the kernel bandwidth. For efficiency, one may chooseto skip trials with very similar points. Orbits are counted as distinct onlywhen they differ by a specified amount. The routine finds the orbits, theirexpanding eigenvalue, and possible sub-periods. Figure <A HREF="node19.html#figupo"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> shows thedetermination of all period six orbits from 1000 iterates of the Hénon map,contaminated by 10% Gaussian white noise.<P><HR><A NAME="tex2html265" HREF="node20.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html263" HREF="node16.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html257" HREF="node18.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html266" HREF="node20.html">Locally linear prediction</A><B>Up:</B> <A NAME="tex2html264" HREF="node16.html">Nonlinear prediction</A><B> Previous:</B> <A NAME="tex2html258" HREF="node18.html">Simple nonlinear prediction</A><P><ADDRESS><I>Thomas Schreiber <BR>Wed Jan 6 15:38:27 CET 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
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