@article{22648,
  abstract     = {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 the 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.},
  author       = {Dey, Tamal K. and Lipiński, Michał and Soriano Trigueros, Manuel},
  issn         = {1615-3383},
  journal      = {Foundations of Computational Mathematics},
  keywords     = {Multivector field, Conley index, Morse decomposition, Bifurcation, Continuation, Zigzag persistence, Persistence barcode, Gentle algebra},
  publisher    = {Springer},
  title        = {{Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations}},
  doi          = {10.1007/s10208-026-09766-6},
  year         = {2026},
}