node27.html

an analysis software with souce code for the time series with methods based on the theory of nonline

Sec.<A HREF="node29.html#secdimension" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node29.html#secdimension"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A>.
<P>
<P><blockquote><A NAME="5542">&#160;</A><IMG WIDTH=345 HEIGHT=458 ALIGN=BOTTOM ALT="figure1368" SRC="img106.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img106.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figlyap1">&#160;</A>
  Estimating the maximal Lyapunov exponent of the CO<IMG WIDTH=6 HEIGHT=11 ALIGN=MIDDLE ALT="tex2html_wrap_inline6701" SRC="img39.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img39.gif"> laser data. The
  top panel shows results for the Poincar&#233; map data, where the average time
  interval <IMG WIDTH=21 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7371" SRC="img101.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img102.gif">. For comparison: The iteration of the
  radial basis function model of Fig.&nbsp;<A HREF="node21.html#figpredictINOrbf" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node21.html#figpredictINOrbf"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A> yields
  <IMG WIDTH=8 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7375" SRC="img103.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img104.gif">. In good approximation, <IMG WIDTH=108 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7379" SRC="img105.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img105.gif">. Here, the
  time window <I>w</I> to suppress correlated neighbors has been set to 1000, and
  the delay time was 6&nbsp;units.<BR>
</blockquote><P>
Let us discuss a few typical outcomes. The data underlying the top panel of
Fig.&nbsp;<A HREF="node27.html#figlyap1" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node27.html#figlyap1"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img39.gif"> laser
data. Since this laser exhibits low dimensional chaos with a reasonable noise
level, 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img115.gif"> per iteration (map data!), or, when introducing the
average time interval, 0.007 per <IMG WIDTH=9 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline7481" SRC="img116.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img116.gif">s. In the bottom panel we show the result
for the same system, but now computed on the original flow-like data with a
sampling rate of 1&nbsp;MHz. As additional structure, an initial steep increase and
regular oscillations are visible. The initial increase is due to non-normality
and effects of alignment of distances towards the locally most unstable
direction, and the oscillations are an effect of the locally different
velocities and thus different densities. Both effects can be much more dramatic
in less favorable cases, but as long as the regular oscillations possess a
linearly increasing average, this can be taken as the estimate of the Lyapunov
exponent. Normalizing by the sampling rate, we again find <IMG WIDTH=25 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap_inline7483" SRC="img117.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img117.gif">
0.007 per <IMG WIDTH=9 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline7481" SRC="img116.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img116.gif">s, but it is obvious that the linearity is less pronounced then
for the map-like data.  Finally, we show in Fig.&nbsp;<A HREF="node27.html#figlyap2" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node27.html#figlyap2"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/cross_ref_motif.gif"></A> an example
of a negative result: We study the human breath rate data used before. No
linear part exists, and one cannot draw any reasonable conclusion. It is worth
considering the figure on a doubly logarithmic scale in order to detect a power
law behavior, which, with power 1/2, could be present for a diffusive growth
of distances. In this particular example, there is no convincing power law
either.
<P>
<P><blockquote><A NAME="5715">&#160;</A><IMG WIDTH=341 HEIGHT=215 ALIGN=BOTTOM ALT="figure1470" SRC="img118.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img118.gif"><BR>
<STRONG>Figure:</STRONG> <A NAME="figlyap2">&#160;</A>
   The breath rate data (c.f. Fig.&nbsp;<A HREF="node23.html#figb" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node23.html#figb"><IMG  ALIGN=BOTTOM ALT="gif" SRC="icons/cross_ref_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node28.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="icons/next_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/next_motif.gif"></A> <A NAME="tex2html344" HREF="node26.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node26.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="icons/up_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/up_motif.gif"></A> <A NAME="tex2html338" HREF="node26.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node26.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="icons/previous_motif.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/icons/previous_motif.gif"></A>   <BR>
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<P><ADDRESS>
<I>Thomas Schreiber <BR>
Wed Jan  6 15:38:27 CET 1999</I>
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