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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>Dimensions and entropies</TITLE><META NAME="description" CONTENT="Dimensions and entropies"><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="tex2html364" HREF="node30.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html362" HREF="TiseanHTML.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html356" HREF="node28.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html365" HREF="node30.html">Correlation dimension</A><B>Up:</B> <A NAME="tex2html363" HREF="TiseanHTML.html">Practical implementation of nonlinear </A><B> Previous:</B> <A NAME="tex2html357" HREF="node28.html">The Lyapunov spectrum</A><BR> <P><H1><A NAME="SECTION00080000000000000000">Dimensions and entropies</A></H1><A NAME="secdimension"> </A>Solutions of dissipative dynamical systems cannot fill a volume of the phasespace, since dissipation is synonymous with a contraction of volume elementsunder the action of the equations of motion. Instead, trajectories are confinedto lower dimensional subsets which have measure zero in the phase space. Thesesubsets can be extremely complicated, and frequently they possess a fractalstructure, which means that they are in a nontrivial wayself-similar. Generalized dimensions are one class of quantities tocharacterize this fractality. The <EM>Hausdorff dimension</EM> is, from themathematical point of view, the most natural concept to characterize fractalsets [<A HREF="citation.html#EckRuelle">67</A>], whereas the <EM>information dimension</EM> takes intoaccount the relative visitation frequencies and is therefore more attractive forphysical systems. Finally, for the characterization of measured data, othersimilar concepts, like the <EM>correlation dimension</EM>, are more useful. Onegeneral remark is highly relevant in order to understand the limitations of anynumerical approach: dimensions characterize a set or an invariant measure whosesupport is the set, whereas any data set contains only a finite number ofpoints representing the set or the measure. By definition, the dimension of afinite set of points is zero. When we determine the dimension of an attractornumerically, we extrapolate from finite length scales, where the statistics weapply is insensitive to the finiteness of the number of data, to theinfinitesimal scales, where the concept of dimensions is defined. Thisextrapolation can fail for many reasons which will be partly discussedbelow. Dimensions are invariant under smooth transformations and thus againcomputable in time delay embedding spaces.<P>Entropies are an information theoretical concept to characterize the amount ofinformation needed to predict the next measurement with a certainprecision. The most popular one is the Kolmogorov-Sinai entropy. We willdiscuss here only the correlation entropy, which can be computed in a much morerobust way. The occurrence of entropies in a section on dimensions has to dowith the fact that they can be determined both by the same statistical tool.<P><BR> <HR><UL><A NAME="CHILD_LINKS"> </A><LI> <A NAME="tex2html366" HREF="node30.html#SECTION00081000000000000000">Correlation dimension</A><UL><LI> <A NAME="tex2html367" HREF="node31.html#SECTION00081100000000000000">Takens-Theiler estimator</A><LI> <A NAME="tex2html368" HREF="node32.html#SECTION00081200000000000000">Gaussian kernel correlation integral</A></UL> <LI> <A NAME="tex2html369" HREF="node33.html#SECTION00082000000000000000">Information dimension</A><UL><LI> <A NAME="tex2html370" HREF="node34.html#SECTION00082100000000000000">Entropy estimates</A></UL></UL><HR><A NAME="tex2html364" HREF="node30.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif"></A> <A NAME="tex2html362" HREF="TiseanHTML.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif"></A> <A NAME="tex2html356" HREF="node28.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html365" HREF="node30.html">Correlation dimension</A><B>Up:</B> <A NAME="tex2html363" HREF="TiseanHTML.html">Practical implementation of nonlinear </A><B> Previous:</B> <A NAME="tex2html357" HREF="node28.html">The Lyapunov spectrum</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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