Abstract
We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below “families” are “families of vector fields in the two-sphere”. This result disproves an Arnold’s conjecture of 1985. Then we construct an open set of six parameter families whose moderate topological classification has a functional invariant. This invariant is an arbitrary germ of a smooth map \((\mathbb {R}_+,a)\rightarrow (\mathbb {R}_+, b)\). More generally, for any positive integers d and \(d'\), we construct an open set of families whose topological classification has a germ of a smooth map \(\left( \mathbb {R}_+^d, a\right) \rightarrow \left( \mathbb {R}_+^{d'}, b\right) \) as an invariant. Any smooth germ of this kind may be realized as such an invariant. These results open a new perspective of the global bifurcation theory in the two sphere. This perspective is discussed at the end of the paper.
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Notes
The notation \({{\mathrm{SPS}}}\) is an abbreviation of “singular points, periodic orbits, and separatrices”.
References
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Acknowledgements
The authors are grateful to Christian Bonatti, Anton Gorodetski and Alexei Klimenko for fruitful suggestions. The authors thank the Referee for many fruitful comments.
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The authors were supported in part by the Grant RFBR 16-01-00748. Research of I.S. was supported in part by Dynasty Foundation.
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Ilyashenko, Y., Kudryashov, Y. & Schurov, I. Global bifurcations in the two-sphere: a new perspective. Invent. math. 213, 461–506 (2018). https://doi.org/10.1007/s00222-018-0793-1
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DOI: https://doi.org/10.1007/s00222-018-0793-1