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<H2><A NAME="SECTION00071000000000000000">The maximal exponent</A></H2>
<A NAME="seclyapmax"> </A>
The maximal Lyapunov exponent can be determined without the explicit
construction of a model for the time series. A reliable characterization
requires that the independence of embedding parameters and the exponential law
for the growth of distances are checked [<A HREF="citation.html#Holger" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#Holger">69</A>, <A HREF="citation.html#rose" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#rose">70</A>] explicitly.
Consider the representation of the time series data as a trajectory in the
embedding space, and assume that you observe a very close return <IMG WIDTH=18 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7443" SRC="img107.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img107.gif"> to
a previously visited point <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline6691" SRC="img38.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img38.gif">. Then one can consider the distance
<IMG WIDTH=94 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline7447" SRC="img108.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img108.gif"> as a small perturbation, which should grow
exponentially 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img109.gif">. If one finds that <IMG WIDTH=85 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline7451" SRC="img110.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img110.gif"> then <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"> is (with probability one) the maximal
Lyapunov exponent. In practice, there will be fluctuations because of many
effects, which are discussed in detail in [<A HREF="citation.html#Holger" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#Holger">69</A>]. Based on this
understanding, one can derive a robust consistent and unbiased estimator for
the maximal Lyapunov exponent. One computes
<BR><A NAME="eqS"> </A><IMG WIDTH=500 HEIGHT=58 ALIGN=BOTTOM ALT="equation5543" SRC="img111.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img111.gif"><BR>
If <IMG WIDTH=61 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline7455" SRC="img112.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img24.gif"> and for a reasonable range of <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif">, then this
slope can be taken as an estimate of the maximal exponent <IMG WIDTH=13 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline7463" SRC="img113.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img113.gif">.
<P>
The formula is implemented in the routine <a href="../../../../../tppmsgs/msgs0.htm#39" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/lyapunov.html">lyapunov</a> in a
straightforward way. (The program <a href="../dresden/lyap_r.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/dresden/lyap_r.html">lyap_r</a> implements the very similar
algorithm of Ref. [<A HREF="citation.html#rose" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#rose">70</A>] where only the closest neighbor is followed for
each reference point. Also, the Euclidean norm is used.) Apart from parameters
characterizing the embedding, the initial neighborhood size <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif"> is of
relevance: The smaller <IMG WIDTH=6 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline6495" SRC="img3.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif">, the large the linear range of <I>S</I>, if there
is 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" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img3.gif">
from below. It is not always necessary to extend the average in Eq.(<A HREF="node27.html#eqS" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node27.html#eqS"><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>) over
the whole available data, reasonable averages can be obtained already with a
few hundred reference points <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline7473" SRC="img114.gif" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/img114.gif">. If some of the reference points have very
few neighbors, the corresponding inner sum in Eq.(<A HREF="node27.html#eqS" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/node27.html#eqS"><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>) is dominated by
fluctuations. Therefore one may choose to exclude those reference points which
have less than, say, ten neighbors. However, discretion has to be applied with
this parameter since it may introduce a bias against sparsely populated
regions. This could in theory affect the estimated exponents due to
multifractality. Like other quantities, Lyapunov estimates may be affected by
serial correlations between reference points and neighbors. Therefore, a
minimum time for |<I>n</I>-<I>n</I>'| can and should be specified here as well. See also
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