Global Bifurcations in Generic One-parameter Families on 
                
                  
                
                $$\mathbb{S}^2$$
                
                  
                    
                      
                        S
                      
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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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“…This was accomplished in [12,18,29]. In particular, generic one-parameter families of vector fields on the sphere are structurally stable.…”

Section: One-parameter Families: Fully Classify Them Up To the Weak E...mentioning

confidence: 99%

New structurally unstable families of planar vector fields

2021

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“…Families with few parameters. On the one hand, a generic one-parameter family of vector fields on the sphere is structurally stable, see [12,18,22,29]. On the other hand, classification of locally generic three-parameter families can have a numerical invariant arising from a robustly invariant function, see [14,17], or a non-robust invariant with values in a larger space, see [13] and Theorem 3.2.…”

Section: 1mentioning

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

Goncharuk

Ilyashenko

2020

Proc. Steklov Inst. Math.

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Global Bifurcations in Generic One-parameter Families on $$\mathbb{S}^2$$ S 2

Cited by 4 publications

References 5 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

Various Equivalence Relations in Global Bifurcation Theory

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