Towards aformal tie between combinatorial and classical vector field dynamics

Tomasz  Kaczynski; Marian  Mrozek; Thomas Wanner

doi:10.3934/jcd.2016002

Article Contents

2016, Volume 3, Issue 1: 17-50. Doi: 10.3934/jcd.2016002

This issue Previous Article Discretization strategies for computing Conley indices and Morse decompositions of flows Next Article On the numerical approximation of the Perron-Frobenius and Koopman operator

Towards a formal tie between combinatorial and classical vector field dynamics

1.
Département de mathématiques, Université de Sherbrooke, 2500 boul. Université, Sherbrooke, Qc, J1K2R1
2.
Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. St. Łojasiewicza 6, 30-348 Kraków
3.
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030

Received: September 2015

Revised: June 2016

Published online: September 2016

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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 [6], 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.

Keywords:

Mathematics Subject Classification: Primary: 37B30; Secondary: 37B35, 37E15, 57M99, 57Q05, 57Q15.

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