Conley-Morse-Forman theory for combinatorial multivector fields

2015/05/29 by Marian Mrożek, Mrozek, Marian · 1 citation
Computer Science · Mathematics · #06A06 #18G35 #37B30 #37B35 #54H20 #55U15 #57Q10 #65P99 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Digital Image Processing Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Numerical Analysis (math.NA) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1506.00018

openalex publication_date 2015/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Fundamental Group Algorithm for low dimensional tessellated CW complexes

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