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<html><head><title>Unstable periodic orbits</title></head><body bgcolor="#ffffff"><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>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>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>verbosity level</a> (0 = only fatal errors) <br> <font color=blue><tt> -h </tt></font>show this message<p> <a href=../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 theirstabilitiy (the most unstable eigenvalue). The dynamics is approximated using aGaussian kernel estimator of bandwidth <font color=blue><tt>-r | -v</tt></font>, where either <font color=blue><tt>-r</tt></font> or <fontcolor=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 <fontcolor=blue><tt>-n</tt></font> embedded points are tried as initialconditions. False minima can be cut off by limiting <fontcolor=blue><tt>-a</tt></font>, the error at the minimum. Very close trialpoints can be omitted by giving <font color=blue><tt>-w</tt></font>. Orbitswhich are less than <font color=blue><tt>-W</tt></font> apart are consideredidentical.<p>The stability is computed by iterating forward a small (set by <fontcolor=blue><tt> -s </tt></font>) initial perturbation to the orbit.<p>Orbits are written to <fontcolor=blue><em>file</em>_<tt>upo</tt>_<em>pp</em></font> where <em>pp</em> isthe desired period. If an orbit is found to have a sub-period, all results aregiven with respect to that. Orbits can be plotted in delay coordinates using<a href="upoembed.html">upoembed</a>.<p><font color=red><b>Note:</b></font>As you noticed, the UPOs are defined here ina rather loose sense, similarly in spirit to the use by <ahref="citation.html#so">So et al.</a> and other authors. Thus, the meredetection of such an orbit does not constitute evidence for low dimensionaldynamics or anything the like. <p><font color=red><b>Note:</b></font>The period is passed to the program insamples. This is different from what you may expect, since a "period 2" orbitof 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 theorbits seems to be quite reliable, the stabilities pose surprisingproblems. The chief reason is that they use information at a single point inphase space and no averaging over the whole attractor is involved. Values should be fine for comparisons, like in <ahref="test.html">surrogate data testing</a>. If absolute quantities are needed(like in <a href="http://www.nbi.dk/ChaosBook/">cycle expansions</a>), extracare 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><IMG SRC="upo.gif" alt="upo results"><p><a href="../contents.html">Table of Contents</a> * <a href="../../index.html" target="_top">TISEAN home</a></body></html>⌨️ 快捷键说明
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