Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes

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 decomp...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 17; no. 6; pp. 1585 - 1633
Main Author Mrozek, Marian
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2017
Springer
Springer Nature B.V
Subjects
Algorithms
Analysis
Applications of Mathematics
Attractors (mathematics)
Combinatorial analysis
Complexes (Mathematics)
Computer Science
Differential equations
Economics
Fields (Mathematics)
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
United States
Repeller
65P99
Topology of finite sets
37B35
Attractor
Primary 54H20
18G35
Conley index theory
Morse decomposition
Discrete Morse theory
37B30
Secondary 57Q10
Morse inequalities
Combinatorial vector field
06A06
55U15
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Summary: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.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-016-9330-z