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 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.