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



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.


Exponentially small splitting Hopf-zero bifurcation Inner equation Stokes constant 

Mathematics Subject Classification

34E10 E4E15 37C29 37G99 



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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Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

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