# Periodic Orbits in the Rössler Equations

This Web page contains the data referred to in the paper
*Computer Assisted Method for Proving Existence
of Periodic Orbits* by Pawel Pilarczyk,
published in *Topological Methods in Nonlinear Analysis*,
1999 Vol 13 No. 2, 365-377.
Please, consult this paper for a more precise explanation
of the data files available below.

All the data is in Unix-type text files.
Since these files are very large, I have compressed them
with the popular program "gzip" for your convenience.

The cubes of which the isolating neighbourhood
constructed in the proof of Theorem 1 in Section 7 is built
are listed in the file neighbhd.
The result of reduction of this neighbourhood,
i.e., the set *N'*, consists of cubes
listed in the file domain.

The file cubicmap
contains the cubical approximation of the time-2 map on *N'*
computed with the Rössler method applied to the cubes
which form the neighbourhood.
The format of this file is compatible with the input format
of the program *chmap* written by M. Mazur and J. Szybowski
for the computation of a chain map corresponding
to this almost perfect multivalued cubical map.
The cubes forming the range of this map, i.e., the set *N''*,
are listed in the file range.

The program for the computation of a chain selector of an almost perfect
multivalued cubical map produces the files
chmapcub
and chmap_cy.
The first one contains the values of the chain map
on each generator of the domain, referred further to as *Cx*,
and the other contains a list of all the 3-dimensional generators
of the range, referred further to as *Cy*.

Finally, the files chains_x,
chains_y
and chainmap
contain the definition of the chain complex *Cx*,
the chain complex *Cy* and the chain map
in the format of my program
for homology computation.
In this format, each generator in each dimension
gets its own number, starting with 1.
In the formulas for the boundaries of the generators,
these numbers are referred to,
preceded with the sign "+" or "-" as needed.

If you have any questions or remarks,
you are welcome to contact the author of the paper by e-mail.

Note [March 17, 2003]:
The new and significantly expanded program for the homology computation
(by Pawel Pilarczyk) currently available with the sources
at http://chomp.rutgers.edu/software/ already contains the reduction
of cubical sets (and much more) in itself and is capable of computing the map
induced in homology by the above-mentioned map in reasonably
short time without the use of any other programs, including *chmap*.