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Mosc. Math. J., 2018, Volume 18, Number 1, Pages 93–115 (Mi mmj664)  

This article is cited in 4 scientific papers (total in 4 papers)

Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$

Yu. Ilyashenkoab, N. Solodovnikova

a National Research University Higher School of Economics, 119048, Usacheva 6, Moscow, Russia
b Independent University of Moscow

Abstract: 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.

Key words and phrases: bifurcation, separatrix loop, sparkling saddle connection.

DOI: https://doi.org/10.17323/1609-4514-2018-18-1-93-115

Full text: http://www.mathjournals.org/.../2018-018-001-005.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 34C23, 37G99, 37E35
Language:

Citation: Yu. Ilyashenko, N. Solodovnikov, “Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$”, Mosc. Math. J., 18:1 (2018), 93–115

Citation in format AMSBIB
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\by Yu.~Ilyashenko, N.~Solodovnikov
\paper Global bifurcations in generic one-parameter families with a~separatrix loop on $S^2$
\jour Mosc. Math.~J.
\yr 2018
\vol 18
\issue 1
\pages 93--115
\mathnet{http://mi.mathnet.ru/mmj664}
\crossref{https://doi.org/10.17323/1609-4514-2018-18-1-93-115}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85044049505}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Valeriia Starichkova, “Global Bifurcations in Generic One-parameter Families on $\mathbb{S}^2$”, Regul. Chaotic Dyn., 23:6 (2018), 767–784  mathnet  crossref
    2. N. Goncharuk, Yu. Ilyashenko, N. Solodovnikov, “Global bifurcations in generic one-parameter families with a parabolic cycle on $S^2$”, Mosc. Math. J., 19:4 (2019), 709–737  mathnet  crossref  mathscinet
    3. Yu. Ilyashenko, “First steps of the global bifurcation theory in the plane”, Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemporary Mathematics, 734, eds. P. Kuchment, E. Semenov, Amer. Math. Soc., 2019, 145–158  crossref  mathscinet  zmath  isi  scopus
    4. N. B. Goncharuk, Yu. S. Ilyashenko, “Various Equivalence Relations in Global Bifurcation Theory”, Proc. Steklov Inst. Math., 310 (2020), 78–97  mathnet  crossref  crossref  mathscinet  isi  elib
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