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<head><title>Unstable periodic orbits</title></head>
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<h3>Unstable periodic orbits</h3>
<font color=blue><tt>upo <font color=red> -m# [-r# | -v#]</font> [-p# -w# -W# -a# -s# -n# -o </tt><em>outfile</em><tt> -l# -x# -c# -V# -h] </tt><em> file(s)</em>
</font>
<blockquote>
<br> <font color=red><tt> -m </tt></font>embedding dimension
<br> <font color=red><tt> -r </tt></font>absolute kernel bandwidth
<br> <font color=red><tt> -v </tt></font>same as fraction of standard deviation
<br> <font color=blue><tt> -p </tt></font>period of orbit (1)
<br> <font color=blue><tt> -w </tt></font>minimal separation of trial points (e)
<br> <font color=blue><tt> -W </tt></font>minimal separation of distinct orbits (e)
<br> <font color=blue><tt> -a </tt></font>maximal error of orbit to be plotted (all plotted)
<br> <font color=blue><tt> -s </tt></font>initial separation for stability (e)
<br> <font color=blue><tt> -n </tt></font>number of trials (all points)
<br> <font color=blue><tt> -l </tt></font>number of values to be read (all)
<br> <font color=blue><tt> -x </tt></font>number of values to be skipped (0)
<br> <font color=blue><tt> -c </tt></font><a href="../general.html#column" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/general.html#column">column to be read</a> (1 or <font color=blue><em>file</em></font>,#)
<br> <font color=blue><tt> -o </tt></font><a href="../general.html#outfile" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/general.html#outfile">output file name</a>, just <font color=blue><tt> -o </tt></font>means <font color=blue><em>file</em></font><font color=blue><tt>_upo</tt></font><font color=blue><tt>_pp</tt></font>
<br> <font color=blue><tt> -V </tt></font><a href="../general.html#verbosity" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/general.html#verbosity">verbosity level</a> (0 = only fatal errors)
<br> <font color=blue><tt> -h </tt></font>show this message
<p>
<a href="../general.html#verbosity" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/general.html#verbosity">verbosity level</a> (add what you want):
<p>
<font color=blue> 1</font> = input/output<br>
<font color=blue> 2</font> = print orbits found<br>
<font color=blue> 4</font> = status after 1000 points<br>
<font color=blue> 8</font> = status after 100 points<br>
<font color=blue> 16</font> = status after 10 points<br>
</blockquote>
Find unstable period <font color=blue><tt>-p</tt></font> orbits and their
stabilitiy (the most unstable eigenvalue). The dynamics is approximated using a
Gaussian kernel estimator of bandwidth <font color=blue><tt>-r | -v</tt></font>, where either <font color=blue><tt>-r</tt></font> or <font
color=blue><tt>-v</tt></font> must be given.
The minimum of
<pre>
p
---
\ / \ 2
| | x - f(x ,...) |
/ \ n+1 n /
---
n=1
</pre>
is sought by a Levenberg-Marquardt scheme. The first <font
color=blue><tt>-n</tt></font> embedded points are tried as initial
conditions. False minima can be cut off by limiting <font
color=blue><tt>-a</tt></font>, the error at the minimum. Very close trial
points can be omitted by giving <font color=blue><tt>-w</tt></font>. Orbits
which are less than <font color=blue><tt>-W</tt></font> apart are considered
identical.
<p>
The stability is computed by iterating forward a small (set by <font
color=blue><tt> -s </tt></font>) initial perturbation to the orbit.
<p>
Orbits are written to <font
color=blue><em>file</em>_<tt>upo</tt>_<em>pp</em></font> where <em>pp</em> is
the desired period. If an orbit is found to have a sub-period, all results are
given with respect to that. Orbits can be plotted in delay coordinates using
<a href="upoembed.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/upoembed.html">upoembed</a>.
<p>
<font color=red><b>Note:</b></font>As you noticed, the UPOs are defined here in
a rather loose sense, similarly in spirit to the use by <a
href="../chaospaper/citation.html#so" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/chaospaper/citation.html#so">So et al.</a> and other authors. Thus, the mere
detection of such an orbit does not constitute evidence for low dimensional
dynamics or anything the like.
<p>
<font color=red><b>Note:</b></font>The period is passed to the program in
samples. This is different from what you may expect, since a "period 2" orbit
of the Lorenz equations may turn out to have period 137 or whatever. In fact,
the program has been written for map like, or Poincaré section data.
It is also fair to say that it hasn't been tested extensively.
<p>
<font color=red><b>Note:</b></font> While the existence and locations of the
orbits seems to be quite reliable, the stabilities pose surprising
problems. The chief reason is that they use information at a single point in
phase space and no averaging over the whole attractor is involved. Values
should be fine for comparisons, like in <a
href="test.html" tppabs="http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.0/docs/wuppertal/test.html">surrogate data testing</a>. If absolute quantities are needed
(like in <a href="../../../../../tppmsgs/msgs0.htm#18" tppabs="http://www.nbi.dk/ChaosBook/">cycle expansions</a>), extra
care has to be taken. The user might consider using an alternative approach,
for example via the cycle Jacobians as obtained from a locally linear fit.
<p>
<hr>
<h3>Usage example</h3>
Try also just running: <font color=blue><tt>gnuplot upo.gnu</tt></font> in the
<font color=blue><tt>examples</tt></font> directory.
<p>
<pre>
> <font color=blue><tt>henon -l1000 | addnoise -v0.1 > data</tt></font>
> <font color=blue><tt>upo -p6 -m2 -v0.1 -n70 data -o</tt></font>
gnuplot> <font color=blue><tt>plot '< cat data | embed -d1' notitle w do,
\
'< cat data_upo_06 | upoembed -d1' index 3 title "fixed point", \
'< cat data_upo_06 | upoembed -d1' index 1 title "period 2", \
'< cat data_upo_06 | upoembed -d1' index 0 title "period 6", \
'< cat data_upo_06 | upoembed -d1' index 2 title "period 6"</tt></font>
</pre>
<p>
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<p>
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