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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>Lyapunov exponents</TITLE><META NAME="description" CONTENT="Lyapunov exponents"><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="tex2html334" HREF="node27.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html332" HREF="TiseanHTML.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html326" HREF="node25.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html335" HREF="node27.html">The maximal exponent</A><B>Up:</B> <A NAME="tex2html333" HREF="TiseanHTML.html">Practical implementation of nonlinear </A><B> Previous:</B> <A NAME="tex2html327" HREF="node25.html">Nonlinear noise reduction in </A><BR> <P><H1><A NAME="SECTION00070000000000000000">Lyapunov exponents</A></H1><A NAME="seclyap"> </A>Chaos arises from the exponential growth of infinitesimal perturbations,together with global folding mechanisms to guarantee boundedness of thesolutions. This exponential instability is characterized by the spectrum ofLyapunov exponents [<A HREF="citation.html#EckRuelle">67</A>]. If one assumes a local decomposition ofthe phase space into directions with different stretching or contraction rates,then the spectrum of exponents is the proper average of these local rates overthe whole invariant set, and thus consists of as many exponents as there arespace directions. The most prominent problem in time series analysis is thatthe physical phase space is unknown, and that instead the spectrum is computedin some embedding space. Thus the number of exponents depends on thereconstruction, and might be larger than in the physical phase space. Suchadditional exponents are called <I>spurious</I>, and there are severalsuggestions to either avoid them [<A HREF="citation.html#spurious">68</A>] or to identify them. Moreover,it is plausible that only as many exponents can be determined from a timeseries as are entering the Kaplan Yorke formula (see below). To give a simpleexample: Consider motion of a high-dimensional system on a stable limitcycle. The data cannot contain any information about the stability of thisorbit against perturbations, as long as they are exactly on the limitcycle. For transients, the situation can be different, but then data are notdistributed according to an invariant measure and the numerical values are thusdifficult to interpret. Apart from these difficulties, there is one relevantpositive feature: Lyapunov exponents are invariant under smooth transformationsand are thus independent of the measurement function or the embeddingprocedure. They carry a dimension of an inverse time and have to be normalizedto the sampling interval.<P><BR> <HR><UL><A NAME="CHILD_LINKS"> </A><LI> <A NAME="tex2html336" HREF="node27.html#SECTION00071000000000000000">The maximal exponent</A><LI> <A NAME="tex2html337" HREF="node28.html#SECTION00072000000000000000">The Lyapunov spectrum</A></UL><HR><A NAME="tex2html334" HREF="node27.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html332" HREF="TiseanHTML.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html326" HREF="node25.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html335" HREF="node27.html">The maximal exponent</A><B>Up:</B> <A NAME="tex2html333" HREF="TiseanHTML.html">Practical implementation of nonlinear </A><B> Previous:</B> <A NAME="tex2html327" HREF="node25.html">Nonlinear noise reduction in </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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