Conley-Morse barcodes - an algebraic signature of a combinatorial bifurcation
- Michał Lipiński (Institute of Science and Technology Austria)
Abstract
Bifurcation theory is one of the major topics in dynamical systems. Its goal is to understand the nature of qualitative changes in a parametrized dynamical system. In this project, 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.
In this talk, I will explain the idea behind Conley-Morse barcodes, a compact algebraic descriptor of (combinatorial) bifurcations. The barcodes capture structural changes in the dynamical system at the level of Morse decomposition and provide a characterization of the nature of observed transitions in terms of the Conley index.
The construction of Conley-Morse barcodes builds upon ideas from topological data analysis. Specifically, we construct a persistence module obtained from a zigzag filtration of topological pairs (formed by index pairs defining the Conley index) over a poset, which can be decomposed into simple intervals.