Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (I)



In this paper we study a beyond all orders phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of this singularity at any order. The results in this paper are twofold: on the one hand, we give results for generic unfoldings which lead to sharp exponentially small upper bounds of the difference between these manifolds. On the other hand, we provide asymptotic formulas for this difference by means of the Melnikov function for some non-generic unfoldings.


Exponentially small splitting Hopf-zero bifurcation Melnikov function Borel transform 

Mathematics Subject Classification

34E10 E4E15 37C29 37G99 



The three authors are really grateful to the two anonymous referees who carefully reviewed the manuscript. Specially the comments and suggestions about the introduction have greatly improved the quality of the final version. The authors are in debt with Jordi Villanueva for his help in the proof of Theorem 2.9. 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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Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

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