Persistent Homology of Morse Decompositions in Combinatorial Dynamics
Tamal K. Dey, Mateusz Juda, Tomasz Kapela, Jacek Kubica, Michał Lipiński, Marian Mrożek · SIAM Journal on Applied Dynamical Systems · 2019
We investigate combinatorial dynamical systems on simplicial complexes considered as finite topological spaces. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of Morse decompositions of such systems as a tool for validating the reconstruction. Our approach may be viewed as a step toward applying the classical persistence theory to data collected from a dynamical system. We present experimental results on two numerical examples.