Discretization strategies for computing Conley indices and Morse decompositions of flows
Konstantin Mischaikow Marian Mrozek Frank Weilandt
Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameter to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow.
keywords: Morse decomposition rigorous numerical algorithm. Conley index interval arithmetic
Towards a formal tie between combinatorial and classical vector field dynamics
Tomasz Kaczynski Marian Mrozek Thomas Wanner
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: multivalued maps simplicial complex discrete dynamical system Combinatorial vector field discrete Morse theory isolated invariant set Conley theory isolating block.

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