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Exponentially Small Heteroclinic Breakdown in the Generic Hopf-Zero Singularity

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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.

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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).

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  1. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028, 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 O. Castejón.

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Baldomá, I., Castejón, O. & Seara, T.M. Exponentially Small Heteroclinic Breakdown in the Generic Hopf-Zero Singularity. J Dyn Diff Equat 25, 335–392 (2013). https://doi.org/10.1007/s10884-013-9297-2

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  • DOI: https://doi.org/10.1007/s10884-013-9297-2

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