Find unstable period

-membedding dimension

-rabsolute kernel bandwidth

-vsame as fraction of standard deviation

-pperiod of orbit (1)

-wminimal separation of trial points (e)

-Wminimal separation of distinct orbits (e)

-amaximal error of orbit to be plotted (all plotted)

-sinitial separation for stability (e)

-nnumber of trials (all points)

-lnumber of values to be read (all)

-xnumber of values to be skipped (0)

-ccolumn to be read (1 orfile,#)

-ooutput file name, just-omeansfile_upo_pp

-Vverbosity level (0 = only fatal errors)

-hshow this messageverbosity level (add what you want):

1 = input/output

2 = print orbits found

4 = status after 1000 points

8 = status after 100 points

16 = status after 10 points

p --- \ / \ 2 | | x - f(x ,...) | / \ n+1 n / --- n=1is sought by a Levenberg-Marquardt scheme. The first

The stability is computed by iterating forward a small (set by ` -s `) initial perturbation to the orbit.

Orbits are written to *file*_`upo`_*pp* where *pp* 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
upoembed.

**Note:**As you noticed, the UPOs are defined here in
a rather loose sense, similarly in spirit to the use by So et al. and other authors. Thus, the mere
detection of such an orbit does not constitute evidence for low dimensional
dynamics or anything the like.

**Note:**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.

**Note:** 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 surrogate data testing. If absolute quantities are needed
(like in cycle expansions), 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.

>henon -l1000 | addnoise -v0.1 > data>upo -p6 -m2 -v0.1 -n70 data -ognuplot>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"