About the unfolding of a Hopf-zero singularity

Freddy Dumortier; Santiago Ibáñez; Hiroshi Kokubu; Carles Simó

doi:10.3934/dcds.2013.33.4435

Article Contents

2013, Volume 33, Issue 10: 4435-4471. Doi: 10.3934/dcds.2013.33.4435

This issue Previous Article Construction of response functions in forced strongly dissipative systems Next Article Dispersive estimates for matrix Schrödinger operators in dimension two

About the unfolding of a Hopf-zero singularity

1.
Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek
2.
Departamento de Matemáticas, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33007 Oviedo
3.
Department of Mathematics/JST-CREST, Kyoto University, Kyoto 606-8502
4.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08071 Barcelona

Received: August 31, 2012

Revised: December 31, 2012

Published: September 2013

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Abstract

We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension two. They can be written in the following normal form: \begin{eqnarray*} \left\{ \begin{array}{l} x'=-y+\mu x-axz+A(x,y,z,\lambda,\mu) \\ y'=x+\mu y-ayz+B(x,y,z,\lambda,\mu) \\ z'=z^2+\lambda+b(x^2+y^2)+C(x,y,z,\lambda,\mu), \end{array} \right. \end{eqnarray*} with $a>0$, $b>0$ and where $A$, $B$, $C$ are $C^\infty$ or $C^\omega$ functions of order $O(\|(x,y,z,\lambda,\mu)\|^3)$.
Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in the literature, no result valid for arbitrary generic unfoldings is available. In this paper we present new techniques to study global bifurcations from Hopf-zero singularities. They allow us to obtain a general criterion for the existence of Shilnikov homoclinic bifurcations and also provide a detailed description of the bifurcation set. Criteria for the existence of Bykov cycles are also provided. Main tools are a blow-up method, including a related invariant theory, and a novel approach to the splitting functions of the invariant manifolds. Theoretical results are applied to the Michelson system and also to the so called extended Michelson system. Paper includes thorough numerical explorations of dynamics for both systems.

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Mathematics Subject Classification: Primary: 34C23, 34C37; Secondary: 37G10, 34C28.

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