Structure of optimal gradient flows bifurcations on closed surfaces

2024/08/20 by Ovtsynov, Illia, Prishlyak, Alexandr
#37c10 #37c15 #37c20 #Dynamical Systems (math.DS) #FOS: Mathematics #G.1.7

paper · doi:10.48550/arxiv.2408.10687

Abstract

We consider structure of typical gradient flows bifurcations on closed surfaces with minimal number of singular points. There are two type of such bifurcations: saddle-node (SN) and saddle connections (SC). The structure of a bifurcation is determinated by codimension one flow in the moment of bifurcation. We use the chord diagrams to specify the flows up to topological trajectory equivivalence. A chord diagram with a marked arc is complete topological invariant of a SN-bifurcations and a chord diagram with T-insert -- of SC-bifurcations. We list all such diagrams for flows on norientable surfaces of genus at most 2 and nonorientable surfaces of genus at most 3. For each of diagram we found inverse one that correspond the inverse flow.

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