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Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018  pp. 93–115.

Global Bifurcations in Generic One-Parameter Families with a Separatrix Loop on S2

Authors:  Yu. Ilyashenko (1) and N. Solodovnikov (2)
Author institution:(1) National Research University Higher School of Economics, Russia
Independent University of Moscow
(2) National Research University Higher School of Economics, 119048, Usacheva 6, Moscow, Russia


Global bifurcations in the generic one-parameter families that unfold a vector field with a separatrix loop on the two-sphere are described. The sequence of bifurcations that occurs is in a sense in one-to-one correspondence with finite sets on a circle having some additional structure on them. Families under study appear to be structurally stable. The main tool is the Leontovich–Mayer–Fedorov (LMF) graph, analog of the separatrix sceleton and an invariant of the orbital topological classification of the vector fields on the two-sphere. Its properties and applications are described.

2010 Math. Subj. Class. 34C23, 37G99, 37E35.

Keywords: Bifurcation, separatrix loop, sparkling saddle connection.

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