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Mathematics > Dynamical Systems

arXiv:1506.00018 (math)
[Submitted on 29 May 2015 (v1), last revised 21 May 2016 (this version, v3)]

Title:Conley-Morse-Forman theory for combinatorial multivector fields

Authors:Marian Mrozek
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Abstract:We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.
Subjects: Dynamical Systems (math.DS); Algebraic Topology (math.AT); Combinatorics (math.CO); Numerical Analysis (math.NA)
MSC classes: 54H20, 37B30, 37B35, 65P99, 57Q10, 18G35, 55U15, 06A06
Cite as: arXiv:1506.00018 [math.DS]
  (or arXiv:1506.00018v3 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.1506.00018

Submission history

From: Marian Mrozek [view email]
[v1] Fri, 29 May 2015 20:14:44 UTC (856 KB)
[v2] Mon, 13 Jul 2015 10:55:49 UTC (857 KB)
[v3] Sat, 21 May 2016 06:55:57 UTC (998 KB)
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