Journal
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
Volume 17, Issue 6, Pages 1585-1633Publisher
SPRINGER
DOI: 10.1007/s10208-016-9330-z
Keywords
Combinatorial vector field; Conley index theory; Discrete Morse theory; Attractor; Repeller; Morse decomposition; Morse inequalities; Topology of finite sets
Funding
- Polish National Science Center under Maestro Grant [2014/14/A/ST1/00453]
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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.
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