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Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 767–784 | Cite as

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

  • Valeriia StarichkovaEmail author
Article
  • 14 Downloads

Abstract

In this paper we prove that generic one-parameter families of vector fields on \(\mathbb{S}^2\) in the neighborhood of the fields of classes AH, SN, HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families.

Keywords

bifurcations equivalence structural stability 

MSC2010 numbers

34C23 37G99 37E35 

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References

  1. 1.
    Andronov, A.A., Leontovich, E.A., Gordon, I. I., and Maier, A.G., Qualitative Theory of Second-Order Dynamic Systems, New York: Wiley, 1973.zbMATHGoogle Scholar
  2. 2.
    Arnold, V. I., Afrajmovich, V. S., Il’yashenko, Yu. S., and Shil’nikov, L.P., Bifurcation Theory and Catastrophe Theory, Encyclopaedia Math. Sci., vol. 5, Berlin: Springer, 1999.Google Scholar
  3. 3.
    Bautin, N.N. and Leontovich, E.A., Methods and Techniques of the Qualitative Study of Dynamical Systems on the Plane, Moscow: Nauka, 1990 (Russian).zbMATHGoogle Scholar
  4. 4.
    Dumortier, F., Roussarie, R., and Rousseau, C., Hilbert’s 16th Problem for Quadratic Vector Fields, J. Differential Equations, 1994, vol. 110, no. 1, pp. 86–133.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fedorov, R. M., Upper Bounds for the Number of Orbital Topological Types of Polynomial Vector Fields on the Plane “Modulo Limit Cycles”, Russian Math. Surveys, 2004, vol. 59, no. 3, pp. 569–570; see also: Uspekhi Mat. Nauk, 2004, vol. 59, no. 3(357), pp. 183–184.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Goncharuk, N., Ilyashenko, Yu., and Solodovnikov, N., Global Bifurcations in Generic One-Parameter Families with a Parabolic Cycle on S 2, arXiv:1707.09779 (2017).Google Scholar
  7. 7.
    Goncharuk, N. and Ilyashenko, Yu., Large Bifurcation Supports, arXiv:1804.04596 (2018).Google Scholar
  8. 8.
    Ilyashenko, Yu., Towards the General Theory of Global Planar Bifurcations, in Mathematical Sciences with Multidisciplinary Applications, B.Toni (Ed.), Springer Proc. Math. Stat., vol. 157, Cham: Springer, 2016, pp. 269–299.Google Scholar
  9. 9.
    Ilyashenko, Yu., Kudryashov, Yu., and Schurov, I., Global Bifurcations in the Two-Sphere: A New Perspective, Invent. Math., 2018, vol. 213, no. 2, pp. 461–506.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ilyashenko, Yu. and Solodovnikov, N., Global Bifurcations in Generic One-Parameter Families with a Separatrix Loop on S 2, Mosc. Math. J., 2018, vol. 18, no. 1, pp. 93–115.MathSciNetGoogle Scholar
  11. 11.
    Ilyashenko, Yu. and Li, W., Nonlocal Bifurcations, Math. Surveys Monogr., vol. 66, Providence,R.I.: AMS, 1999.Google Scholar
  12. 12.
    Malta, I.P. and Palis, J., Families of Vector Fields with Finite Modulus of Stability, in Dynamical Systems and Turbulence (Univ. Warwick, Coventry, 1979/1980), D. A. Rand, L. S.Young (Eds.), Lecture Notes in Math., vol. 898, Berlin: Springer, 1981, pp. 212–229.Google Scholar
  13. 13.
    MacLane, S. and Adkisson, V. W., Extensions of Homeomorphisms on the Sphere, in Lectures in Topology, Ann Arbor,Mich.: Univ. of Michigan, 1941, pp. 223–236.Google Scholar
  14. 14.
    Sotomayor, J., Generic One-Parameter Families of Vector Fields on Two-Dimensional Manifolds, Inst. Hautes Etudes Sci. Publ. Math., 1974, no. 43, pp. 5–46.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Université Paris-Sud 11, Département de Mathématiques, Faculté des Sciences d’OrsayOrsay CedexFrance

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