# Excision-Preserving Cubical Approach to the Algorithmic
Computation of the Discrete Conley Index

# Software and Examples

This page contains additional information and data referred to
in the paper *Excision-preserving cubical approach
to the algorithmic computation of the discrete Conley index*
by Paweł Pilarczyk and Kinga Stolot.

A preprint of this paper can be downloaded
here: [

PDF ]
[

PS ]

## Software

The programs `homcubes`, `homcub2l`
and `liftcubes.pl` mentioned in Section 4 of the paper
are included in the Computational Homology Project Software; see the
CHomP Website
for the source code, compilation instructions, description of the software,
and precompiled versions for some operating systems.

## Example Index Pairs

The example index pairs are defined in 5 files each, for the purpose
of homology computation by the programs `homcubes`
and `homcub2l`, although the latter requires only the first 3 files
as listed below.
For clarity, the names of the files end with letters that indicate
the contents of the file as follows:

`f` – a definition of the combinatorial cubical
multivalued map on X
`x` – a list of cubes in the set X\A
`a` – a list of cubes in the set A
`y` – a list of cubes in Y\B = X\A
as part of the codomain of the map
`b` – a list of cubes in B = A + F(A)

The files that contain the definitions
of the cubical sets and the combinatorial cubical multivalued map
are prepared in the text format accepted by the CHomP software
(see a

description of these formats), which is also human-readable
and can be easily processed for the use by other software.

The file excision.zip contains the definition
of the index pair and index map illustrted in Figure 1 of the paper.
This index pair was constructed manually and does not come from any
real computations. Its purpose is to illustrate the excision problem
in an extremely simple setting.

For testing the speed of the new approach to the computation
of the index map, each of the examples listed below was processed
in the three ways described in the paper and indicated below.
A computer with the Intel® Xeon® 5030 2.66 GHz processor
was used for the tests.

First, the old program
`homcubes` alone was used to compute the endomorphism
induced in homology by the combinatorial cubical multivalued map
with the following command (where `ex` was replaced
with the corresponding part of the file names in each example):

`homcubes -i ex_f.map ex_x.cub ex_a.cub ex_y.cub ex_b.cub`

Note that this approach may sometimes fail, as illustrated in Table 1
in the paper.

Second, the new program `homcub2l` was used to compute
the same endomorphism. Note that this program uses the double-layer
construction described in our paper. The command used
to run the computations was like the following one:

`homcub2l ex_f.map ex_x.cub ex_a.cub`

Finally, the alternative construction suggested in the paper was tested
using the following series of commands:

`cnvmvmap
ex_f.map ex_f.mp`

`liftcubes.pl
ex_f.mp ex_x.cub ex_a.cub ex_y.cub ex_b.cub`

`homcubes
-i _ex_f.mp _ex_x.cub _ex_a.cub _ex_y.cub _ex_b.cub`

The first command was necessary to convert the text format
in which the combinatorial cubical multivalued map was stored
to the format accepted by the Perl script `liftcubes`.
Note that the time used by the conversion was very short
in comparison with the time used for the homology computation.

The data files and execution logs are available here in the
following files:

- ex1.zip – Example 1 from [4]
cited in the paper, based on files available at Oliver Junge's
web page
- ex2.zip - Example 2 from [4]
cited in the paper, based on files available at Oliver Junge's
web page
- ex3.zip - Example 3 from [4]
cited in the paper, based on files available at Oliver Junge's
web page
- v2u.zip - an index pair constructed
for an unstable periodic trajectory observed in the time-
*t*
discretization of the Vanderpol equations in the plane
with reversed time
- les1.zip - a sample index pair
obtained in the actual computations for a nonlinear 2-dimensional
Leslie population model, mentioned in Table 1 in the paper
- les2.zip - a sample index pair
obtained in the actual computations for a nonlinear 2-dimensional
Leslie population model, plotted in Figure 5 in the paper

## Comparison of Index Pair Definitions

The three index pairs discussed in the paper as a comparison
of two different index pair definitions (see Table 2 in the paper)
were constructed in an algorithmic way using a program in C++ written
especially for this purpose.
Since this program uses both the
CHomP and
CAPD libraries
and its compilation might be complicated, a compiled version for Windows
is available below, in addition to its source code:

- indpdemo.cpp – the source code
of the program that constructs various index pairs compared in the paper
- indpdemo-win.zip –
the above-mentioned program compiled for Windows
- indpdemo.zip –
the scripts for running the computations, as well as the results
of these computations collected together

## Embedding the double-layer structure into full cubes

In the paper we mention an alternative approach to forcing the excision
property. Namely, instead of creating the double-layer structure,
we propose to increase the dimension of the full cubes by 1
and to put the cubes corresponding to various layers at specific
"height" in the embedding space. Below is a note which describes this
approach and explains it in details (with illustrations):

- liftcubes.pdf – a note
on representing double-layer cubical sets with (ordinary) cubical sets