Computing Connection Matrix and Persistence Efficiently from a Morse Decomposition

Bibliographic Details
Title: Computing Connection Matrix and Persistence Efficiently from a Morse Decomposition
Authors: Dey, Tamal K., Lipiński, Michał, Haas, Andrew
Publication Year: 2025
Collection: Computer Science
Mathematics
Subject Terms: Mathematics - Dynamical Systems, Computer Science - Computational Geometry, Mathematics - Algebraic Topology
More Details: Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix.These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures - such as attractors, repellers, and orbits - in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in~\cite{DLMS24}. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of~\cite{DLMS24} and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.
Document Type: Working Paper
Access URL: http://arxiv.org/abs/2502.19369
Accession Number: edsarx.2502.19369
Database: arXiv
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