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Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (II): The Generic Case

Journal of Nonlinear Science volume 28, pages 1489–1549 (2018)Cite this article

Abstract

In this paper, we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singularity. The method consists in obtaining an asymptotic formula for the difference between these manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of the original vector field and which corresponds to the so-called inner equation in singular perturbation theory.

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References

  • Baldomá, I.: The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems. Nonlinearity 19(6), 1415–1446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Baldomá, I., Castejón, O., Seara, T.M.: Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity. J. Dyn. Differ. Equ. 25(2), 335–392 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Baldomá, I., Castejón, O., Seara, T.M.: Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I). preprint (2016)

  • Baldomá, I., Fontich, E., Guardia, M., Seara, T.M.: Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results. J. Differ. Equ. 253(12), 3304–3439 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Baldomá, I., Martín, P.: The inner equation for generalized standard maps. SIAM J. Appl. Dyn. Syst. 11(3), 1062–1097 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Baldomá, I., Seara, T.M.: The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete Contin. Dyn. Syst. Ser. B 10(2–3), 323–347 (2008)

    MathSciNet  MATH  Google Scholar 

  • Castejón, O: Study of invariant manifolds in two different problems: the Hopf-zero singularity and neural synchrony. Ph.D. thesis, UPC (2015)

  • Dumortier, F., Ibáñez, S., Kokubu, H., Simó, C.: About the unfolding of a Hopf-zero singularity. Discrete Contin. Dyn. Syst. 33(10), 4435–4471 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Écalle, J: Les fonctions résurgentes. Tome I, volume 5 of Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81]. Université de Paris-Sud Département de Mathématique, Orsay, (1981). Les algèbres de fonctions résurgentes. [The algebras of resurgent functions], With an English foreword

  • Écalle, J: Les fonctions résurgentes. Tome II, volume 6 of Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81]. Université de Paris-Sud Département de Mathématique, Orsay (1981). Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration]

  • Gelfreich, V.G.: Reference systems for splittings of separatrices. Nonlinearity 10(1), 175–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Gelfreich, V., Naudot, V.: Analytic invariants associated with a parabolic fixed point in \(\mathbb{C}^2\). Ergodic Theory Dyn. Syst. 28(6), 1815–1848 (2008)

    Article  MATH  Google Scholar 

  • Gelfreich, V.G., Sauzin, D.: Borel summation and splitting of separatrices for the Hénon map. Ann. Inst. Fourier (Grenoble) 51(2), 513–567 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Larreal, OJ, Seara, TM: 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

  • Martín, P., Sauzin, D., Seara, T.M.: Resurgence of inner solutions for perturbations of the McMillan map. Discrete Contin. Dyn. Syst. 31(2), 165–207 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Olivé, C, Sauzin, D, Seara, TM.: Resurgence in a Hamilton–Jacobi equation. In: Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002). Ann. Inst. Fourier (Grenoble), 53(4):1185–1235 (2003)

  • Stokes, G.G.: On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Philos. Soc. 10, 106–128 (1864)

    Google Scholar 

  • Stokes, G.G.: On the discontinuity of arbitrary constants that appear as multipliers of semi-convergent series. Acta Math. 26(1), 393–397 (1902) A letter to the editor

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Acknowledgements

The authors are in debt with the two anonymous referees who reviewed the first version of this manuscript. The authors have been partially supported by the Spanish MINECO-FEDER Grant MTM2015-65715-P and the Catalan Grant 2014SGR504. Tere M-Seara is also supported by the Russian Scientific Foundation Grant 14-41-00044. and European Marie Curie Action FP7-PEOPLE-2012-IRSES: BREUDS.

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  1. Universitat Politècnica de Catalunya, Barcelona, Spain

    I. Baldomá, O. Castejón & T. M. Seara

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  1. I. Baldomá
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  2. O. Castejón
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  3. T. M. Seara
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Correspondence to I. Baldomá.

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Communicated by Amadeu Delshams.

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Baldomá, I., Castejón, O. & Seara, T.M. Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (II): The Generic Case. J Nonlinear Sci 28, 1489–1549 (2018). https://doi.org/10.1007/s00332-018-9459-9

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  • DOI: https://doi.org/10.1007/s00332-018-9459-9

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