Breakdown of heteroclinic connections in analytic unfoldings of the Hopf-zero singularity
speaker: Oriol Castejon (Universitat Politecnica de Catalunya)abstract: If one considers conservative (i.e. one-parameter) unfoldings of the so-called Hopf-zero singularity, one can see that the truncation of the normal form at any finite order possesses two saddle-focus critical points with a one- and a two-dimensional heteroclinic connection. The same happens for non-conservative (i.e. two-parameter) unfoldings when the parameters lie on a certain curve.
However, considering the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of $\mathcal{C}\infty$ unfoldings, this was proved by Broer and Vegter during the 80's, but for analytic unfoldings it is still an open problem. Recently, it has been seen that the last step to prove the existence of Shilnikov bifurcations for the analytic case requires a precise knowledge of how the heteroclinic connections are broken.
Thus, our study concerns the splittings of the one and two-dimensional heteroclinic connections. These cannot be detected in the truncation of the normal form at any order, and hence they are expected to be exponentially small with respect to one of the perturbation parameters. We shall present asymptotic formulas of these splittings, putting emphasis on the differences between the conservative and non-conservative cases.
timetable:
Mon 5 May, 11:30 - 12:30, Aula Dini
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