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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 maximal exponent</TITLE><META NAME="description" CONTENT="The maximal exponent"><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="tex2html346" HREF="node28.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html344" HREF="node26.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html338" HREF="node26.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html347" HREF="node28.html">The Lyapunov spectrum</A><B>Up:</B> <A NAME="tex2html345" HREF="node26.html">Lyapunov exponents</A><B> Previous:</B> <A NAME="tex2html339" HREF="node26.html">Lyapunov exponents</A><BR> <P><H2><A NAME="SECTION00071000000000000000">The maximal exponent</A></H2><A NAME="seclyapmax"> </A>The maximal Lyapunov exponent can be determined without the explicitconstruction of a model for the time series. A reliable characterizationrequires that the independence of embedding parameters and the exponential lawfor the growth of distances are checked [<A HREF="citation.html#Holger">69</A>, <A HREF="citation.html#rose">70</A>] explicitly.Consider the representation of the time series data as a trajectory in theembedding space, and assume that you observe a very close return <IMG WIDTH=18 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7443" SRC="img107.gif"> toa previously visited point <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif">. Then one can consider the distance<IMG WIDTH=94 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7447" SRC="img108.gif"> as a small perturbation, which should growexponentially in time. Its future can be read from the time series:<IMG WIDTH=120 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7449" SRC="img109.gif">. If one finds that <IMG WIDTH=85 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline7451" SRC="img110.gif"> then <IMG WIDTH=8 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7375" SRC="img103.gif"> is (with probability one) the maximalLyapunov exponent. In practice, there will be fluctuations because of manyeffects, which are discussed in detail in [<A HREF="citation.html#Holger">69</A>]. Based on thisunderstanding, one can derive a robust consistent and unbiased estimator forthe maximal Lyapunov exponent. One computes<BR><A NAME="eqS"> </A><IMG WIDTH=500 HEIGHT=58 ALIGN=BOTTOM ALT="equation5543" SRC="img111.gif"><BR>If <IMG WIDTH=61 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7455" SRC="img112.gif"> exhibits a linear increase with identical slope for all<I>m</I> larger than some <IMG WIDTH=20 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6617" SRC="img24.gif"> and for a reasonable range of <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">, then thisslope can be taken as an estimate of the maximal exponent <IMG WIDTH=13 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7463" SRC="img113.gif">.<P>The formula is implemented in the routine <a href="../docs_f/lyapunov.html">lyapunov</a> in astraightforward way. (The program <a href="../docs_c/lyap_r.html">lyap_r</a> implements the very similaralgorithm of Ref. [<A HREF="citation.html#rose">70</A>] where only the closest neighbor is followed foreach reference point. Also, the Euclidean norm is used.) Apart from parameterscharacterizing the embedding, the initial neighborhood size <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif"> is ofrelevance: The smaller <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">, the large the linear range of <I>S</I>, if thereis one. Obviously, noise and the finite number of data points limit <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif">from below. It is not always necessary to extend the average in Eq.(<A HREF="node27.html#eqS"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) overthe whole available data, reasonable averages can be obtained already with afew hundred reference points <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7473" SRC="img114.gif">. If some of the reference points have veryfew neighbors, the corresponding inner sum in Eq.(<A HREF="node27.html#eqS"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) is dominated byfluctuations. Therefore one may choose to exclude those reference points whichhave less than, say, ten neighbors. However, discretion has to be applied withthis parameter since it may introduce a bias against sparsely populatedregions. This could in theory affect the estimated exponents due tomultifractality. Like other quantities, Lyapunov estimates may be affected byserial correlations between reference points and neighbors. Therefore, aminimum time for |<I>n</I>-<I>n</I>'| can and should be specified here as well. See alsoSec.<A HREF="node29.html#secdimension"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>.<P><P><blockquote><A NAME="5542"> </A><IMG WIDTH=345 HEIGHT=458 ALIGN=BOTTOM ALT="figure1368" SRC="img106.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figlyap1"> </A> Estimating the maximal Lyapunov exponent of the CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif"> laser data. The top panel shows results for the Poincaré map data, where the average time interval <IMG WIDTH=21 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7371" SRC="img101.gif"> is 52.2 samples of the flow, and the straight line indicates <IMG WIDTH=58 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline7373" SRC="img102.gif">. For comparison: The iteration of the radial basis function model of Fig. <A HREF="node21.html#figpredictINOrbf"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> yields <IMG WIDTH=8 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7375" SRC="img103.gif">=0.35. Bottom panel: Lyapunov exponents determined directly from the flow data. The straight line has slope <IMG WIDTH=66 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline7377" SRC="img104.gif">. In good approximation, <IMG WIDTH=108 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7379" SRC="img105.gif">. Here, the time window <I>w</I> to suppress correlated neighbors has been set to 1000, and the delay time was 6 units.<BR></blockquote><P>Let us discuss a few typical outcomes. The data underlying the top panel ofFig. <A HREF="node27.html#figlyap1"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> are the values of the maxima of the CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif"> laserdata. Since this laser exhibits low dimensional chaos with a reasonable noiselevel, we observe a clear linear increase in this semi-logarithmic plot,reflecting the exponential divergence of nearby trajectories. The exponent is<IMG WIDTH=58 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline7479" SRC="img115.gif"> per iteration (map data!), or, when introducing theaverage time interval, 0.007 per <IMG WIDTH=9 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline7481" SRC="img116.gif">s. In the bottom panel we show the resultfor the same system, but now computed on the original flow-like data with asampling rate of 1 MHz. As additional structure, an initial steep increase andregular oscillations are visible. The initial increase is due to non-normalityand effects of alignment of distances towards the locally most unstabledirection, and the oscillations are an effect of the locally differentvelocities and thus different densities. Both effects can be much more dramaticin less favorable cases, but as long as the regular oscillations possess alinearly increasing average, this can be taken as the estimate of the Lyapunovexponent. Normalizing by the sampling rate, we again find <IMG WIDTH=25 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7483" SRC="img117.gif">0.007 per <IMG WIDTH=9 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline7481" SRC="img116.gif">s, but it is obvious that the linearity is less pronounced thenfor the map-like data. Finally, we show in Fig. <A HREF="node27.html#figlyap2"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A> an exampleof a negative result: We study the human breath rate data used before. Nolinear part exists, and one cannot draw any reasonable conclusion. It is worthconsidering the figure on a doubly logarithmic scale in order to detect a powerlaw behavior, which, with power 1/2, could be present for a diffusive growthof distances. In this particular example, there is no convincing power laweither.<P><P><blockquote><A NAME="5715"> </A><IMG WIDTH=341 HEIGHT=215 ALIGN=BOTTOM ALT="figure1470" SRC="img118.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figlyap2"> </A> The breath rate data (c.f. Fig. <A HREF="node23.html#figb"><IMG ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif"></A>) exhibit no linear increase, reflecting the lack of exponential divergence of nearby trajectories.<BR></blockquote><P><HR><A NAME="tex2html346" HREF="node28.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html344" HREF="node26.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html338" HREF="node26.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html347" HREF="node28.html">The Lyapunov spectrum</A><B>Up:</B> <A NAME="tex2html345" HREF="node26.html">Lyapunov exponents</A><B> Previous:</B> <A NAME="tex2html339" HREF="node26.html">Lyapunov exponents</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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