Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/15206
Title: ABOUT THE UNFOLDING OF A HOPF-ZERO SINGULARITY
Authors: DUMORTIER, Freddy 
IBANEZ MESA, Santiago 
Kokubu, Hiroshi
Simo, Carles
Issue Date: 2013
Publisher: AMER INST MATHEMATICAL SCIENCES
Source: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 33 (10), p. 4435-4471
Abstract: We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension two. They can be written in the following normal form: {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) vertical bar lambda vertical bar b(x(2) vertical bar y(2)) vertical bar C(x,y,z,lambda,mu), with a > 0, b > 0 and where A, B, C are C-infinity or C-omega functions of order O(vertical bar vertical bar(x,y,z,lambda,mu)vertical bar vertical bar(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.
Notes: Univ Hasselt, B-3590 Diepenbeek, Belgium. Univ Oviedo, Dept Matemat, Oviedo 33007, Spain. Kyoto Univ, Dept Math JST CREST, Kyoto 6068502, Japan. Univ Barcelona, Dept Matemat Aplicada & Anal, Barcelona 08071, Spain.
Keywords: Hopf-zero singularities; Shilnikov homoclinic orbits; Bykov cycles; splitting functions; Michelson system;Mathematics, Applied; Mathematics
Document URI: http://hdl.handle.net/1942/15206
ISSN: 1078-0947
e-ISSN: 1553-5231
DOI: 10.3934/dcds.2013.33.4435
ISI #: 000317953500005
Category: A1
Type: Journal Contribution
Validations: ecoom 2014
Appears in Collections:Research publications

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