| 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 |
