Various Equivalence Relations    in Global Bifurcation Theory

Goncharuk, Nataliya; Ilyashenko, Yu.

doi:10.1134/s0081543820050065

Cited by 2 publications

(3 citation statements)

References 12 publications

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“…For families with more parameters, classification up to the topological equivalence has functional moduli. This was proved by Roussarie [26] for non-local bifurcations in a threeparameter family, and this example can be easily modified to produce functional invariants for global bifurcations in a two-parameter family, see [11] for details and another way to obtain functional invariants in the latter case.…”

Section: Introductionmentioning

confidence: 80%

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New structurally unstable families of planar vector fields

2021

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We study global bifurcations in generic three-parameter families of vector fields on S 2 . In the recent article by Ilyashenko et al (2018 Invent. Math. 213 461-506), the authors show that three-parameter unfoldings of vector fields with the polycycle 'tears of the heart' are structurally unstable. We consider three-parameter unfoldings of vector fields with separatrix graphs 'ears' and 'glasses', and prove that these families are structurally unstable as well. We also study in more details the classical bifurcation of a saddle loop, and use it as a building block in our main example.

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Section: Introductionmentioning

confidence: 80%

“…There are several different 'natural' definitions of equivalent families of vector fields. Pros and cons of these and some other equivalence relations are discussed in [11]. In this paper we will mostly use the last one.…”

Section: Definition 1 Given An Open Subsetmentioning

confidence: 99%

New structurally unstable families of planar vector fields

2021

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“…Moderate equivalence and other similar equivalences were used in [12,13,14,15,17]. For a comparison of different equivalence relations on the space of families of vector fields see [11].…”

Section: Preliminariesmentioning

confidence: 99%

Families of vector fields with many numerical invariants

Goncharuk¹,

Kudryashov²

2022

DCDS

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<p style='text-indent:20px;'>We study bifurcations in finite-parameter families of vector fields on <inline-formula><tex-math id="M1">\begin{document}$S^2$\end{document}</tex-math></inline-formula>. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable <inline-formula><tex-math id="M2">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of <inline-formula><tex-math id="M3">\begin{document}$(2D+1)$\end{document}</tex-math></inline-formula>-parameter families such that the topological classification of these families has at least <inline-formula><tex-math id="M4">\begin{document}$D$\end{document}</tex-math></inline-formula> numerical invariants and used those examples to construct families with functional invariants of topological classification.</p><p style='text-indent:20px;'>In this paper, we construct locally generic <inline-formula><tex-math id="M5">\begin{document}$4$\end{document}</tex-math></inline-formula>-parameter families with any prescribed number of numerical invariants and use them to construct <inline-formula><tex-math id="M6">\begin{document}$5$\end{document}</tex-math></inline-formula>-parameter families with functional invariants. We also describe a locally generic class of <inline-formula><tex-math id="M7">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families with a tail of an infinite number sequence as an invariant of topological classification.</p>

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Various Equivalence Relations in Global Bifurcation Theory

Cited by 2 publications

References 12 publications

New structurally unstable families of planar vector fields

New structurally unstable families of planar vector fields

Families of vector fields with many numerical invariants

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