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### Open Access Journals

JCD

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.

JCD

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.

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