Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I)
arXiv:1608.01115
Abstract
In this paper we study a beyond all order 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.
References in corpus (4)
- Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example
- Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity
- Asymptotic series for the splitting of separatrices near a Hamiltonian bifurcation
- Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case