From Finite Vector Field Data to Combinatorial Dynamical Systems in the Sense of Forman

TL;DR

This paper presents a novel method to construct combinatorial dynamical systems from finite vector field data using a linear minimization approach, effectively capturing complex behaviors like cycles and chaos.

Contribution

The paper introduces a new algorithm combining simplicial complexes, linear minimization, and matchings to derive combinatorial dynamical systems from vector data, with extensions for improved resolution and gradient matchings.

Findings

Successfully retrieved cyclic behavior in Lotka-Volterra model

Captured chaotic behavior of Lorenz attractor

Enhanced resolution with barycentric subdivision

Abstract

The main goal is to construct a combinatorial dynamical system in the sense of Forman from finite vector field data. We use a linear minimization problem with binary variables and linear equality constraints. The solution of the minimization problem induces an admissible matching for the combinatorial dynamical system. They are three main steps for the method: Construct a simplicial complex, compute a vector for each simplex, solve the minimization problem, and apply the induced matching. We argue the effectiveness of the method by testing it on the Lotka-Volterra model and the Lorenz attractor model. We are able to retrieve the cyclic behaviour in the Lotka-Volterra model and the chaotic behaviour of the Lorenz attractor. Two extensions to the algorithm are shown. We use the barycentric subdivision to obtain a better resolution. We add conditions on the minimization problem to obtain a solution which induce a gradient matching.