About non-uniqueness when removing closed orbits in Morse-Smale vector fields

2024/10/03 by Clemens Bannwart, Bannwart, Clemens · 1 citation
Mathematics · #37D15 #57R25 #Advanced Differential Equations and Dynamical Systems #Algebraic Topology (math.AT) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2410.02363

openalex publication_date 2024/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a Morse-Smale vector field on a smooth manifold, Franks described how one can replace a closed orbit of index k by two rest points of index k+1 and k, using a local perturbation. Combined with classical results about gradient-like vector fields, this gives a method of assigning different topological or algebraic structures to Morse-Smale vector fields. We show that there are multiple non-equivalent ways of following this procedure and illustrate this non-uniqueness in various examples. We describe the consequences of this non-uniqueness to the endeavour of assigning CW complexes or chain complexes to Morse-Smale vector fields in a canonical way.

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