Discretization strategies for computing Conley indices and Morse decompositions of flows

Konstantin  Mischaikow; Marian  Mrozek; Frank  Weilandt

doi:10.3934/jcd.2016001

Article Contents

2016, Volume 3, Issue 1: 1-16. Doi: 10.3934/jcd.2016001

This issue Previous Article Next Article Towards a formal tie between combinatorial and classical vector field dynamics

Discretization strategies for computing Conley indices and Morse decompositions of flows

1.
Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019
2.
Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków

Received: April 2015

Revised: July 2016

Published online: August 2016

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Abstract

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:

Mathematics Subject Classification: Primary: 37B30, 37B35, 65G20; Secondary: 37-04, 37C10.

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