# Moscow Mathematical Journal

Volume 19, Issue 4, October–December 2019  pp. 709–737.

Global Bifurcations in Generic One-Parameter Families with a Parabolic Cycle on $S^2$

Authors:  N. Goncharuk (1), Yu. Ilyashenko (2), and N. Solodovnikov (3)
Author institution:(1) Department Of Mathematical and Computational Sciences, University of Toronto Mississauga, 3359 Mississauga Road, Deerfield Hall, 3008K, Mississauga, On L5L 1C6
(2) National Research University Higher School of Economics, Russia
Independent University of Moscow
(3) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia

Summary:

We classify global bifurcations in generic one-parameter local families of vector fields on $S^2$ with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.

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

Keywords: Bifurcation, polycycle, structural stability, sparkling saddle connection.