Steps to see the movie of the evolution of the non-escaping points in
the Michelson system
\[ x'=y, y'=z, z'=\lambda -y + x^2,\]
where $\lambda <0$.
We show the evolution with respect to $|\lambda|$ by displaying the
points in the Poincar\'e section $z=0, z'>0$. The Poincar\'e map is well
defined except in a tiny domain around the $z$ axis. Scaling $y$ by the
factor $2 |\lambda|^{1/6}$ the domain of interest is approximately the
half unit disk. It approaches the half disk when $|\lambda| \to 0$.
The marked points are the ones that do not escape after a long
transient. The step used both in $x$ and $2 |\lambda|^{1/6} y$ to scan
the domain of interest is 1/4000.
Initial points are taken on the domain
$(y,x)\in [-1,0] \times [-1.05,1.05]$.
Note that the Poincar\'e map preserves a measure whose density in $z=0$
is proportional to $z'$ and, hence, it is absolutely continuous with
respect to the Lebesgue measure on the $(y,x)$ plane.
1) Download the file moviehsn.tar.gz from http://www.maia.ub.es/dsg/moviehsn
The size is near 24 Mb.
2) Unpack the file using
tar xvfz moviehsn.tar.gz
which creates a directory moviehsn, with a size near 123 Mb.
The directory contains 595 data files, a file with gnuplot commands
named "movie.gnu" and a copy of this "readme.txt" file.
3) Move to the directory moviehsn
cd moviehsn
Open a window on that directory, start gnuplot and type
load 'movie.gnu'
and the movie starts. On top of the window open by gnuplot appears
the current value of $|\lambda|$, with step 0.0001.
4) The current default values are:
Size of the window open by gnuplot: 900 x 640 pixels
Location of the left upper corner of the window coincident with the
left upper corner of the screen
Delay after the plots: 0.0 seconds
These values can be changed by editing the movie.gnu file. For
instance, to change the size and the delay to 800 x 569 pixels and
0.5 seconds, just uncomment the second line and comment the third
one. That is, change
#set term x11 size 800,569 position 0,0;d=0.5
set term x11 size 900,640 position 0,0;d=0.0
to
set term x11 size 800,569 position 0,0;d=0.5
#set term x11 size 900,640 position 0,0;d=0.0
To repeat the movie with the same size and delay 0.3 second, change
set term x11 size 900,640 position 0,0;d=0.0
to
set term x11 size 900,640 position 0,0;d=0.3
5) At the end of the movie ($|\lambda|=0.0589$) there is a pause and the
window where movie.gnu has been started prompts to press intro.
Then it shows a magnified plot for $|\lambda|=0.01948$, shortly
before the last invariant curves surrounding the period 5 islands
break down.
Press intro again to see the plot for $|\lambda|=0.01949$, when these
curves have disappeared and the points in a chaotic zone related to
the tangle of the period 5 hyperbolic points are gone.
Press intro again to see the superposition of last two plots, the
first one in red and the second one in blue.
6) Pressing intro once and again similar plots appear for
$|\lambda|=0.02606, 0.02607, 0.03692$ and $0.03693$
just before and after destruction of invariant curves around period 4
and period 3 islands.
7) This has been tested using different linux versions and different
gnuplot versions, starting on Version 4.2 patchlevel 6
The time for the movie part (that is from $|\lambda|=0.0001$ to
$|\lambda|=0.0589$ with step $0.0001$) depends on the computer,
number of processors, graphical board, etc. In different tests
we have found values ranging from 45 to 25 seconds.
To these values one has to add 589*d when a delay of d seconds is
used.
It will be tested on windows.
Enjoy the presentation!
Carles Sim\'o