Computing a connection matrix and persistence efficiently from a morse decomposition

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 Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of Dey et al. 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.