2025/04/23 by Tamal K. Dey, Dey, Tamal K., Michał Lipiński +3
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · #37B30 55N31 #Algebraic Topology (math.AT) #Cell Image Analysis Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Slime Mold and Myxomycetes Research #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2504.17105
openalex publication_date 2025/04/23 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems -- a young but already well-established theory. We introduce the Conley-Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of Conley-Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.