Abstract
We introduce a three-step method to construct on a simplicial complex a combinatorial dynamical system in the sense of Forman from vector field data by means of a linear minimization problem, where the solution to this problem induces an admissible matching for the dynamical system. We show that the matrix of the minimization problem is unimodular, allowing us to relax the problem from binary-valued to real-valued and permitting its resolution in polynomial time. We demonstrate the effectiveness of the method on the Lotka–Volterra model and on the Lorenz attractor model. We also describe three potential extensions to our method: how barycentric subdivision can be applied to the simplicial complex to increase the resolution and obtain a solution that better fits the underlying dynamics of the data, how to add constraints to the minimization problem to fix the number of critical simplices, and how to add constraints to obtain a solution that induces a gradient matching.
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Acknowledgements
I would like to thank the Institut des Sciences Mathématiques (ISM) for their financial support. I will also like to thank Tomasz Kaczynski for discussions beneficial for this work and Marc Ethier for editing this paper.
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Institut des Sciences Mathématiques, Université du Québec à Montréal (14540).
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Partial financial support was received from Institut des Sciences Mathématiques (ISM). The author has no conflict of interest to declare that are relevant to the content of this article.
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Desjardins Côté, D. From finite vector field data to combinatorial dynamical systems in the sense of Forman. J Appl. and Comput. Topology 8, 669–694 (2024). https://doi.org/10.1007/s41468-024-00181-w
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DOI: https://doi.org/10.1007/s41468-024-00181-w
Keywords
- Computational topology
- Simplicial complex
- Combinatorial dynamical system
- Finite vector field data
- Matching algorithm