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Tool to draw the templates of chaotic attractors.

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CATE stands for Chaotic Attractor TEmplate.

cate is a libre software tool (licensed under GNU Lesser General Public License v3.0 only) to draw the templates of chaotic attractors.

Chaotic attractors are solutions of deterministic processes, of which the topology can be described by templates. We consider templates of chaotic attractors bounded by a genus-1 torus described by a linking matrix.

This tool first validates a linking matrix by checking continuity and determinism constraints. The tool then draws the template corresponding to the linking matrix, and optimizes the compactness of the representation. The representation is saved as a Scalable Vector Graphics (SVG) file.


cate is developed as a research project.

The motivation for a tool such as cate, an introduction to linking matrices, and a description of the optimization logic are described, inter alia, in a publication presented at the conference Graph Drawing 2018.

This publication is available in the conference proceedings (doi:10.1007/978-3-030-04414-5_8, paywalled), or on arXiv (arxiv:1807.11853).

If you use cate for a publication, please cite as follows:

Maya Olszewski, Jeff Meder, Emmanuel Kieffer, Raphaël Bleuse, Martin Rosalie, Grégoire Danoy, and Pascal Bouvry. Visualizing the Template of a Chaotic Attractor. In Graph Drawing, volume 11282 of Lecture Notes in Computer Science, 106–119. Springer, 2018. doi:10.1007/978-3-030-04414-5_8.

Or you may use, at your convenience, the following BibTeX entry:

  author    = {Olszewski, Maya and
               Meder, Jeff and
               Kieffer, Emmanuel and
               Bleuse, Rapha{\"{e}}l and
               Rosalie, Martin and
               Danoy, Gr{\'{e}}goire and
               Bouvry, Pascal},
  title     = {{V}isualizing the {T}emplate of a {C}haotic {A}ttractor},
  booktitle = {Graph Drawing},
  series    = {Lecture Notes in Computer Science},
  volume    = 11282,
  pages     = {106--119},
  publisher = {Springer},
  year      = 2018,
  month     = sep,
  doi       = {10.1007/978-3-030-04414-5_8},
  isbn      = {978-3-030-04413-8},


cate is packaged as a regular Python package, and is published on PyPI. It hence can easily be installed with pip.

For more details on how to install a Python package, one can refer to

The latest stable (recommended) version can be installed with the following command (assuming pip is installed):

$ pip install cate

It is recommended to use a virtual environment to install cate. Again, one can refer to to get a more comprehensive overview.

On a typical Linux environment, the typical commands to use would be:

$ python3 -m venv cate_venv
$ source cate_venv/bin/activate
$ pip install cate

This will create a new virtual environment in the cate_venv subdirectory, and configure the current shell to use it as the default python environment. This will then install cate in this new environment without interfering with the already installed packages.

One would then exit this environment either by exiting the current shell, or by typing the command deactivate.

Further uses of cate only require to activate the virtual environment with the following command:

$ source cate_venv/bin/activate


Given a linking matrix, we can draw its template with cate by following these steps. For instance, let’s consider the following linking matrix:

5x5 linking matrix

This linking matrix describes a template made of five strips. cate uses JSON files as an input. A linking matrix has to be described as an array made of arrays of integers, with a row-major order. For the above example linking matrix, one can encode it as follows. Note that whitespaces, new lines, … are insignificant, but do improve the readability.

[[2, 1, 0, 0, 0],
 [1, 1, 0, 0, 0],
 [0, 0, 0, 0, 0],
 [0, 0, 0, 1, 1],
 [0, 0, 0, 1, 2]]

To draw the compact template of this example linking matrix, it is as simple as calling cate with the name of the input file as the first argument.

$ cate 5x5_001.json
[  INFO  ] Input matrix
[  INFO  ]   [2, 1, 0, 0, 0]
[  INFO  ]   [1, 1, 0, 0, 0]
[  INFO  ]   [0, 0, 0, 0, 0]
[  INFO  ]   [0, 0, 0, 1, 1]
[  INFO  ]   [0, 0, 0, 1, 2]
[  INFO  ] Starting constructing the tree
[  INFO  ] Maximum possible template length: 2
[  INFO  ] Finished constructing the tree
[  INFO  ] Starting creation of the SVG template
[  INFO  ] Shortest template
[  INFO  ]   Level 1: (0, 1), (3, 4)
[  INFO  ] Finished creation of the SVG template

cate has created a SVG whose default file is template.svg.

template of the 5x5 linking matrix

The comprehensive set of elementary matrices of size 5x5 and 6x6 is available in the official repository of cate (see The depicted example corresponds to the 5x5_001.json linking matrix.

The comprehensive list of the supported options and their usage is available by typing cate -h.

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