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|Title:||Cocoon bifurcation in three-dimensional reversible vector fields||Authors:||DUMORTIER, Freddy
IBANEZ MESA, Santiago
|Issue Date:||2006||Publisher:||Institute Of Physics||Source:||NONLINEARITY, 19(2). p. 305-328||Abstract:||The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed by Lau (1992 Int. J. Bifurc. Chaos 2 543-58) in the Michelson system, a three-dimensional ODE system describing travelling waves of the Kuramoto-Sivashinsky equation. In this paper, we present an organizing centre of the principal part of the cocoon bifurcation in more general terms in the setting of reversible vector fields on R-3. We prove that in a generic unfolding of an organizing centre called the cusp-transverse heteroclinic chain, there is a cascade of heteroclinic bifurcations with an increasing length close to the organizing Centre, which resembles the principal part of the cocoon bifurcation. We also study a heteroclinic cycle called the reversible Bykov cycle. Such a cycle is believed to occur in the Michelson system, as well as in a model equation of a Josephson Junction (van den Berg et al 2003 Nonlinearity 16 707-17). We conjecture that a reversible Bykov cycle is, in its unfolding, an accumulation point of a sequence of cusp-transverse heteroclinic chains. As a first result in this direction, we show that a reversible Bykov cycle is an accumulation point of reversible generic saddle-node bifurcations of periodic orbits, the main ingredient of the cusp-transverse heteroclinic chain.||Keywords:||KURAMOTO-SIVASHINSKY EQUATION; NILPOTENT SINGULARITY; STEADY SOLUTIONS;; SADDLE-FOCUS; CODIMENSION-3; ORBITS; WAVES; R-3||Document URI:||http://hdl.handle.net/1942/1564||ISSN:||0951-7715||e-ISSN:||1361-6544||DOI:||10.1088/0951-7715/19/2/004||ISI #:||000235461400004||Category:||A1||Type:||Journal Contribution||Validations:||ecoom 2007|
|Appears in Collections:||Research publications|
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