Bifurcation is one of the major topics in the theory of dynamical systems. It
characterizes the nature of qualitative changes in parametrized dynamical
sistemi. In questo lavoro, 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). Nello specifico, Noi
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.

Questo articolo esplora i giri e le loro implicazioni.

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