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<H1><A NAME="SECTION00030000000000000000">Phase space representation</A></H1>
<A NAME="secdelay">&#160;</A>
Deterministic dynamical systems describe the time evolution of a system in some
phase space <IMG WIDTH=52 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline6541" SRC="img8.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img8.gif">.  They can be expressed for example by
ordinary differential equations
<BR><A NAME="eqode">&#160;</A><IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation4408" SRC="img9.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img9.gif"><BR>
or in discrete time <IMG WIDTH=55 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline6543" SRC="img10.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img10.gif"> by maps of the form
<BR><A NAME="eqmap">&#160;</A><IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation4413" SRC="img11.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img11.gif"><BR>
A time series can then be thought of as a sequence of observations
<IMG WIDTH=89 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6545" SRC="img12.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img12.gif"> performed with some measurement function <IMG WIDTH=22 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6547" SRC="img13.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img13.gif">.  Since
the (usually scalar) sequence <IMG WIDTH=30 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline6549" SRC="img14.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img14.gif"> in itself does not properly represent
the (multidimensional) phase space of the dynamical system, one has to employ some
technique to unfold the multidimensional structure using the available data.
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan  6 15:38:27 CET 1999</I>
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