Towards a formal tie between combinatorial and classical vector field dynamics

59. Tomasz Kaczynski, Marian Mrozek, Thomas Wanner:
    Towards a formal tie between combinatorial and classical vector field dynamics
    Journal of Computational Dynamics 3(1), pp. 17-50, 2016.

Abstract

Forman’s combinatorial vector fields on simplicial complexes are a discrete analogue of classical flows generated by dynamical systems. Over the last decade, many notions from dynamical systems theory have found analogues in this combinatorial setting, such as for example discrete gradient flows and Forman’s discrete Morse theory. So far, however, there is no formal tie between the two theories, and it is not immediately clear what the precise relation between the combinatorial and the classical setting is. The goal of the present paper is to establish such a formal tie on the level of the induced dynamics. Following Forman’s paper from 1998, we work with possibly non-gradient combinatorial vector fields on finite simplicial complexes, and construct a flow-like upper semi-continuous acyclic-valued mapping on the underlying topological space whose dynamics is equivalent to the dynamics of Forman’s combinatorial vector field on the level of isolated invariant sets and isolating blocks.

Links

The published version of the paper can be found at https://doi.org/10.3934/jcd.2016002.

Bibtex

@article{kaczynski:etal:16a,
   Author = {Tomasz Kaczynski and Marian Mrozek and Thomas Wanner},
   title = {Towards a formal tie between combinatorial and classical
            vector field dynamics},
   journal = {Journal of Computational Dynamics},
   volume = {3},
   year = {2016},
   number = {1},
   pages = {17--50},
   doi = {10.3934/jcd.2016002}
   }