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Various Equivalence Relations in Global Bifurcation Theory

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Abstract

We discuss various definitions of equivalence for bifurcations of vector fields on the sphere and give a large number of examples (both known and new) that illustrate the advantages and disadvantages of different definitions. In addition to the classical definitions of strong and weak equivalence, we consider new notions of Sing-equivalence and moderate equivalence. These definitions seem to be more relevant to and consistent with the intuitive notion of equivalent bifurcations. They were introduced and used to describe the structural instability of some finite-parameter families of vector fields on the sphere and to study invariants of their classification.

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Notes

  1. B. L. Shleifman, “Topology of deformations of complex parabolic germs” (in preparation).

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Funding

This work was supported by the HSE Laboratory of Dynamical Systems and Applications (under grant 075-15-2019-1931 of the Ministry of Science and Higher Education of the Russian Federation) and by the Russian Foundation for Basic Research (project no. 20-01-00420).

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Authors and Affiliations

  1. Department of Mathematical and Computational Sciences, University of Toronto Mississauga, 3359 Mississauga Rd., Mississauga, ON L5L 1C6, Canada

    N. B. Goncharuk

  2. National Research University Higher School of Economics, ul. Myasnitskaya 20, Moscow, 101000, Russia

    Yu. S. Ilyashenko

  3. Independent University of Moscow, Bol’shoi Vlas’evskii per. 11, Moscow, 119002, Russia

    Yu. S. Ilyashenko

  4. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Yu. S. Ilyashenko

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  1. N. B. Goncharuk
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Correspondence to N. B. Goncharuk.

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Goncharuk, N.B., Ilyashenko, Y.S. Various Equivalence Relations in Global Bifurcation Theory. Proc. Steklov Inst. Math. 310, 78–97 (2020). https://doi.org/10.1134/S0081543820050065

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  • DOI: https://doi.org/10.1134/S0081543820050065

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