Advertisement

Inventiones mathematicae

, Volume 213, Issue 2, pp 461–506 | Cite as

Global bifurcations in the two-sphere: a new perspective

  • Yu. Ilyashenko
  • Yu. Kudryashov
  • I. Schurov
Article
  • 301 Downloads

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.

Mathematics Subject Classification

34C23 37G99 37E35 

Notes

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.

References

  1. 1.
    Arnold, V.I., Afrajmovich, V.S., Ilyashenko, Y.S., Shilnikov, L.P.: Dynamical systems V: bifurcation theory and catastrophe theory. In: Encyclopaedia of Mathematical Sciences. Springer (1994).  https://doi.org/10.1007/978-3-642-57884-7. http://www.springer.com/gp/book/9783540181736. Trans. of Arnold, V.I., Afrajmovich, V.S., Ilyashenko, Yu.S., Shilnikov, L.P. Teorija Bifurcacij. Russian, Dinamicheskije sistemy V. 1986
  2. 2.
    Arnold, V.I.: Lectures on bifurcations in versal families. Russ. Math. Surv. 27(5), 54–123 (1972).  https://doi.org/10.1070/RM1972v027n05ABEH001385 CrossRefGoogle Scholar
  3. 3.
    Golenishcheva-Kutuzova, T.I., Kleptsyn, V.A.: Convergence of the Krylov–Bogolyubov procedure in Bowan’s example. Math. Notes 82(5), 608–618 (2007).  https://doi.org/10.1134/S0001434607110041 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Goncharuk, N.B., Ilyashenko Yu. S.: Large bifurcation supports (In preparation)Google Scholar
  5. 5.
    Hubbard, J.H., West, B.H.: Differential equations: a dynamical systems approach. In: Higher-Dimensional Systems. No. 18 in Texts in Applied Mathematics. Springer (1995)Google Scholar
  6. 6.
    Ilyashenko, Yu.S.: Towards the general theory of global planar bifurcations. In: Toni, B. (eds.) Mathematical Sciences with Multidisciplinary Applications. In Honor of Professor Christiane Rousseau and in Recognition of the Mathematics for Planet Earth Initiative, Springer Proceedings in Mathematics and Statistics, vol. 157, pp. 269–299. Springer (2016).  https://doi.org/10.1007/978-3-319-31323-8
  7. 7.
    Ilyashenko, Yu.: Centennial history of Hilbert’s 16th problem. Bull. Am. Math. Soc. 39(3), 301–354 (2002).  https://doi.org/10.1090/S0273-0979-02-00946-1 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ilyashenko, YuS, Yakovenko, S.: Finitely-smooth normal forms of local families of diffeomorphisms and vector fields. Russ. Math. Surv. 46(1), 3–39 (1991)MathSciNetGoogle Scholar
  9. 9.
    Kotova, A., Stanzo, V.: On few-parameter generic families of vector fields on the two-dimensional sphere. In: Concerning the Hilbert 16th Problem, American Mathematical Society Translation Series 2, vol. 165, pp. 155–201. American Mathematical Society (1995)Google Scholar
  10. 10.
    Mourtada, A.: Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale, pp. 272–314. Springer, Berlin (1990).  https://doi.org/10.1007/BFb0085397
  11. 11.
    Roussarie, R.: Weak and continuous equivalences for families on line diffeomorphisms. In: Dynamical Systems and Bifurcation Theory, Proceedings of the Meeting no. 160 in Pitman Research Notes Mathematical Series, pp. 377–385 (1987)Google Scholar
  12. 12.
    Sotomayor, J.: Generic one-parameter families of vector fields on two-dimensional manifolds. Publ. Math. l’IHES 43, 5–46 (1974).  https://doi.org/10.1007/BF02684365 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussian Federation
  2. 2.Cornell UniversityIthacaUSA

Personalised recommendations