Conley-Morse persistence barcode: a homological signature of a combinatorial bifurcation

Abstract: Bifurcation is one of the major topics in the theory of dynamical systems. It characterizes the nature of qualitative changes in parametrized dynamical systems. In this work, we study combinatorial bifurcations within the framework of combinatorial multivector field theory--a young but already well-established theory providing a combinatorial model for continuous-time dynamical systems (or simply, flows). We introduce Conley-Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. The 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 data analysis (TDA). Specifically, we consider a persistence module obtained from a zigzag filtration of topological pairs (formed by index pairs defining the Conley index) over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them with algorithms from TDA known for processing zigzag filtrations.