article published yes Tamal K. Dey author Andrew Haas author Michał Lipiński author dfffb474-4317-11ee-8f5c-fe3fc95a425e0000-0001-9789-9750 HeEd department IST-BRIDGE: International postdoctoral program project 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. Society for Industrial & Applied Mathematics2026 eng 1536-0040 2502.1936910.1137/25m1739406 251108-130 Dey, Tamal K., et al. “Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition.” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 25, no. 1, Society for Industrial &#38; Applied Mathematics, 2026, pp. 108–30, doi:<a href="https://doi.org/10.1137/25m1739406">10.1137/25m1739406</a>. Dey, Tamal K., Andrew Haas, and Michał Lipiński. “Computing a Connection Matrix and Persistence Efficiently from a Morse Decomposition.” <i>SIAM Journal on Applied Dynamical Systems</i>. Society for Industrial &#38; Applied Mathematics, 2026. <a href="https://doi.org/10.1137/25m1739406">https://doi.org/10.1137/25m1739406</a>. T. K. Dey, A. Haas, and M. Lipiński, “Computing a connection matrix and persistence efficiently from a morse decomposition,” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 25, no. 1. Society for Industrial &#38; Applied Mathematics, pp. 108–130, 2026. Dey TK, Haas A, Lipiński M. 2026. Computing a connection matrix and persistence efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems. 25(1), 108–130. T.K. Dey, A. Haas, M. Lipiński, SIAM Journal on Applied Dynamical Systems 25 (2026) 108–130. Dey TK, Haas A, Lipiński M. Computing a connection matrix and persistence efficiently from a morse decomposition. <i>SIAM Journal on Applied Dynamical Systems</i>. 2026;25(1):108-130. doi:<a href="https://doi.org/10.1137/25m1739406">10.1137/25m1739406</a> Dey, T. K., Haas, A., &#38; Lipiński, M. (2026). Computing a connection matrix and persistence efficiently from a morse decomposition. <i>SIAM Journal on Applied Dynamical Systems</i>. Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/25m1739406">https://doi.org/10.1137/25m1739406</a> 209802026-01-12T11:17:06Z2026-01-20T07:40:39Z