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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>The Lyapunov spectrum</TITLE><META NAME="description" CONTENT="The Lyapunov spectrum"><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="tex2html354" HREF="node29.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html352" HREF="node26.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html348" HREF="node27.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html355" HREF="node29.html">Dimensions and entropies</A><B>Up:</B> <A NAME="tex2html353" HREF="node26.html">Lyapunov exponents</A><B> Previous:</B> <A NAME="tex2html349" HREF="node27.html">The maximal exponent</A><BR> <P><H2><A NAME="SECTION00072000000000000000">The Lyapunov spectrum</A></H2><P>The computation of the full Lyapunov spectrum requires considerably more effortthan just the maximal exponent. An essential ingredient is some estimate of thelocal Jacobians, i.e. of the linearized dynamics, which rules the growth ofinfinitesimal perturbations. One either finds it from direct fits of locallinear models of the type <IMG WIDTH=120 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7515" SRC="img119.gif">, such that the first rowof the Jacobian is the vector <IMG WIDTH=17 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6979" SRC="img61.gif">, and <IMG WIDTH=90 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline7519" SRC="img120.gif"> for<IMG WIDTH=83 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7521" SRC="img121.gif">, where <I>m</I> is the embedding dimension. The <IMG WIDTH=17 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6979" SRC="img61.gif"> is given bythe least squares minimization <IMG WIDTH=190 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline7527" SRC="img122.gif">where <IMG WIDTH=26 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7529" SRC="img123.gif"> is the set of neighbors of <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif"> [<A HREF="citation.html#Eckmann">45</A>, <A HREF="citation.html#sano">71</A>]. Orone constructs a global nonlinear model and computes its local Jacobians bytaking derivatives. In both cases, one multiplies the Jacobians one by one,following the trajectory, to as many different vectors <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7533" SRC="img124.gif"> in tangent spaceas one wants to compute Lyapunov exponents. Every few steps, one applies aGram-Schmidt orthonormalization procedure to the set of <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7533" SRC="img124.gif">, andaccumulates the logarithms of their rescaling factors. Their average, in theorder of the Gram-Schmidt procedure, give the Lyapunov exponents in descendingorder. The routine <a href="../docs_c/lyap_spec.html">lyap_spec</a> uses this method, which goes back to [<A HREF="citation.html#sano">71</A>]and [<A HREF="citation.html#Eckmann">45</A>], employing local linear fits. Apart from the problem ofspurious exponents, this method contains some other pitfalls: It <I>assumes</I>that there exist well defined Jacobians, and does not test for theirrelevance. In particular, when attractors are thin in the embedding space, some(or all) of the local Jacobians might be estimated very badly. Then the wholeproduct can suffer from these bad estimates and the exponents arecorrespondingly wrong. Thus the global nonlinear approach can be superior, if amodeling has been successful, see Sec. <A HREF="node16.html#secpredict"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<P>In Table <A HREF="node28.html#tabLyap1"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> we show the exponents of the stroboscopic NMRlaser data in a three dimensional embedding as a function of the neighborhoodsize. Using global nonlinear models, we find the numbers given in the lasttwo rows. More material is discussed in [<A HREF="citation.html#KantzSchreiber">2</A>]. The spread ofvalues in the table for this rather clean data set reflects the difficulty ofestimating Lyapunov spectra from time series, which has to be done with greatcare. In particular, when the algorithm is blindly applied to data from arandom process, it cannot internally check for the consistency of theassumption of an underlying dynamical system. Therefore a Lyapunov spectrum iscomputed which now is completely meaningless.<P><P><blockquote><A NAME="5717"> </A> <BR><TABLE COLS=6 BORDER RULES=GROUPS cellpadding=5 width=80%><COL ALIGN=LEFT><COL ALIGN=CENTER><COL ALIGN=RIGHT><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=CENTER><TBODY><TR><TD colspan=3 VALIGN=BASELINE ALIGN=LEFT NOWRAP>method </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=13 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7463" SRC="img113.gif"> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7539" SRC="img126.gif"> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=14 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7541" SRC="img127.gif"> </TD></TR></TBODY><TBODY><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP colspan=2>local linear </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <I>k</I>=20 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 0.32 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -0.40 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -1.13 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP COLSPAN=2> ``</TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <I>k</I>=40 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 0.30 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -0.51 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -1.21 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP COLSPAN=2> ``</TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> <I>k</I>=160 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 0.28 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -0.68 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -1.31 </TD></TR></TBODY><TBODY><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP COLSPAN=3>radial basis functions</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 0.27 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -0.64 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -1.31 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP COLSPAN=3> polynomial</TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 0.27 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -0.64 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> -1.15 </TD></TR></TBODY></TABLE><p><STRONG>Table:</STRONG> <A NAME="tabLyap1"> </A> Lyapunov exponents of the NMR laser data, determined with a three-dimensional embedding. The algorithms described in Sec. <A HREF="node27.html#seclyapmax"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> give <IMG WIDTH=104 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7549" SRC="img125.gif"> for the largest exponent. </blockquote><P><P>The computation of the first part of the Lyapunov spectrum allows for someinteresting cross-checks. It was conjectured [<A HREF="citation.html#KaplanYorke">72</A>], and is foundto be correct in most physical situations, that the Lyapunov spectrum and thefractal dimension of an attractor are closely related. If the expanding andleast contracting directions in space are continuously filled and only onepartial dimension is fractal, then one can ask for the dimensionality of a(fractal) volume such that it is invariant, i.e. such that the sum of thecorresponding Lyapunov exponents vanishes, where the last one is weighted withthe non-integer part of the dimension:<BR><A NAME="eqlyapKY"> </A><IMG WIDTH=500 HEIGHT=42 ALIGN=BOTTOM ALT="equation5736" SRC="img128.gif"><BR> where <I>k</I> is the maximum integer such that the sum of the <I>k</I> largest exponentsis still non-negative. <IMG WIDTH=34 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7555" SRC="img129.gif"> is conjectured to coincide with the informationdimension.<P>The Pesin identity is valid under the same assumptions and allows to computethe KS-entropy:<BR><A NAME="eqlyapPesin"> </A><IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="equation5738" SRC="img130.gif"><BR><HR><A NAME="tex2html354" HREF="node29.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html352" HREF="node26.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html348" HREF="node27.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html355" HREF="node29.html">Dimensions and entropies</A><B>Up:</B> <A NAME="tex2html353" HREF="node26.html">Lyapunov exponents</A><B> Previous:</B> <A NAME="tex2html349" HREF="node27.html">The maximal exponent</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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