Advertisement

Exponentially Small Heteroclinic Breakdown in the Generic Hopf-Zero Singularity

  • I. Baldomá
  • O. CastejónEmail author
  • T. M. Seara
Article

Abstract

In this paper we prove the breakdown of a heteroclinic connection in the analytic versal unfoldings of the generic Hopf-zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon “beyond all orders”. In this paper we provide a formula for the distance between the corresponding stable and unstable one-dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings.

Keywords

Exponentially small phenomena Splitting of separatrices Hopf-zero singularity Singular perturbation theory 

Notes

Acknowledgments

The authors have been partially supported by the MICINN-FEDER grant MTM2012-31714. T.M. Seara and O. Castejón have been partially supported by the MICINN-FEDER grant MTM2009-06973 and the CUR-DIUE grant 2009SGR859. I. Baldomá has been supported by the Spanish Grant MEC-FEDER MTM2006-05849/Consolider, the Spanish Grant MTM2010-16425 and the Catalan SGR grant 2009SGR859. The research of O. Castejón has been supported by the grant FI-DGR 2011, co-funded by the Secretaria d’Universitats i Recerca (SUR) of the ECO of the Autonomous Government of Catalonia and the European Social Fund (ESF).

References

  1. 1.
    Angenent, S.: A variational interpretation of Melnikov’s function and exponentially small separatrix splitting. In: Lecture Note Series, vol. 192, pp. 5–35. Cambridge University Press, Cambridge (1993)Google Scholar
  2. 2.
    Baldomá, I., Seara, T.M.: Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity. J. Nonlinear Sci. 16(6), 543–582 (2006). doi: 10.1007/s00332-005-0736-z MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baldomá, I., Seara, T.M.: The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discret. Contin. Dyn. Syst. Ser. B 10(2–3), 323–347 (2008)zbMATHGoogle Scholar
  4. 4.
    Broer, H., Vegter, G.: Subordinate Šil’nikov bifurcations near some singularities of vector fields having low codimension. Ergod. Theory Dyn. Syst. 4, 509–525 (1984). doi: 10.1017/S0143385700002613 MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Delshams, A., Seara, T.M.: Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom. Math. Phys. Electron. J. 3(4), 40 (1997)Google Scholar
  6. 6.
    Dumortier, F., Ibánez, S., Kokubu, H., Simó, C.: About the unfolding of a Hopf-zero singularity (2012) (preprint)Google Scholar
  7. 7.
    Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Phys. D 101(3–4), 227–248 (1997). doi:  10.1016/S0167-2789(96)00133-9
  8. 8.
    Guardia, M., Olivé, C., Seara, T.M.: Exponentially small splitting for the pendulum: a classical problem revisited. J. Nonlinear Sci. 20(5), 595–685 (2010). doi:  10.1007/s00332-010-9068-8 Google Scholar
  9. 9.
    Guckenheimer, J.: On a codimension two bifurcation. In: Dynamical Systems and Turbulence, Warwick 1980, pp. 99–142 (1981)Google Scholar
  10. 10.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer, New York (1990)Google Scholar
  11. 11.
    Larreal, O., Seara, T.M.: Cálculos numéricos de la escisión exponencialmente pequeña de una conexión heteroclínica en la singularidad Hopf Zero. In: Actas del XXI Congreso de Ecuaciones Diferenciales y Aplicaciones, XI Congreso de Matemática Aplicada (electronic), pp. 1–8. Ediciones de la Universidad de Castilla-La Mancha (2009)Google Scholar
  12. 12.
    Mel’nikov, V.K.: On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12, 3–52 (1963)Google Scholar
  13. 13.
    Sauzin, D.: A new method for measuring the splitting of invariant manifolds. Ann. Sci. École Norm. Sup. (4) 34(2), 159–221 (2001). doi: 10.1016/S0012-9593(00),01063-6
  14. 14.
    Šil’nikov, L.P.: A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk SSSR 160, 558–561 (1965)MathSciNetGoogle Scholar
  15. 15.
    Šil’nikov, L.P.: The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus. Sov. Math. Dokl. 8, 54–58 (1967)Google Scholar
  16. 16.
    Stokes, G.G.: On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Phil. Soc. 10, 106–128 (1864)Google Scholar
  17. 17.
    Stokes, G.G.: On the discontinuity of arbitrary constants that appear as multipliers of semi-convergent series. Acta Math. 26(1), 393–397 (1902). doi: 10.1007/BF02415503 Google Scholar
  18. 18.
    Takens, F.: A nonstabilizable jet of a singularity of a vector field. In: Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 583–597. Academic Press, New York (1973)Google Scholar
  19. 19.
    Takens, F.: Singularities of vector fields. Publications Mathématiques de l’IHES 43(1), 47–100 (1974)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Treschev, D.V.: Splitting of separatrices for a pendulum with rapidly oscillating suspension point. Russ. J. Math. Phys. 5(1), 63–98 (1997)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations