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Browsing by Author "Baldoma, Inmaculada"

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    Item type:Article, Access status: Open Access ,
    Breakdown of Heteroclinic Connections in the Analytic Hopf-Zero Singularity: Rigorous Computation of the Stokes Constant
    (2023) Capiński, Maciej J.; Baldoma, Inmaculada; Guardia, Marcel; Seara, Tere M.
    Wydział Matematyki Stosowanej
    Consider analytic generic unfoldings of the three- dimensional conservative Hopf-zero singularity. Under open conditions on the parameters determining the singularity, the unfolding possesses two saddle-foci when the unfolding parameter is small enough. One of them has one-dimensional stable manifold and two-dimensional unstable manifold, whereas the other one has one- dimensional unstable manifold and two- dimensional stable manifold. Baldomá et al. (J Dyn Differ Equ 25(2):335–392, 2013) gave an asymptotic formula for the distance between the one-dimensional invariant manifolds in a suitable transverse section. This distance is exponentially small with respect to the perturbative parameter, and it depends on what is usually called a Stokes constant. The nonvanishing of this constant implies that the distance between the invari- ant manifolds at the section is not zero. However, up to now there do not exist analytic techniques to check that condition. In this paper we provide a method for obtaining accurate rigorous computer-assisted bounds for the Stokes constant. We apply it to two concrete unfoldings of the Hopf-zero singularity, obtaining a computer-assisted proof that the constant is nonzero.