Noose structure and bifurcations of periodic orbits in reversible three-dimensional piecewise linear differential systems
type: Item
instance of: scholarly article
title: Noose structure and bifurcations of periodic orbits in reversible three-dimensional piecewise linear differential systems
zbMATH Open document ID: 1334.34088
author: V. Carmona
author: E. García-Medina
author: Fernando Fernández-Sánchez
author: A. E. Teruel
published in: Journal of Nonlinear Science
publication date: 2015-12-17T00:00:00Z
review text: This work focuses on the periodic response of a one-parameter family of piecewise-linear ODEs given by \[ x'=y,\quad y'=z,\quad z'=1-y-\lambda(1+\lambda^2)|x|, \] with \(\lambda>0\). Special attention is given to the dynamical phenomenon referred to as noose bifurcation. It occurs when a curve of \(T\)-periodic orbits and a curve of \(2T\)-periodic orbits (connected via a period-doubling bifurcation) come together and annihilate at a fold bifurcation. The authors study in detail the curves of periodic solutions that emerge from the noose structure, which involve pitchfork bifurcations, sequences of period-doubling bifurcations, strong resonances and (crossing and non-crossing) tangential intersections of the periodic orbits with the switching boundary \(x=0\). Using the fact that the system is solvable in closed form in the regions of linearity, the authors construct a system of two equations with three unknowns describing a particular type of periodic solution, which is then solved via reduction techniques and continuation methods. Their numerical investigation also shows that in some cases the noose curve oscillates around a critical parameter value for which the period of the orbit tends to infinity, thus giving rise to a homo- or heteroclinic connection.
reviewed by: Joseph Páez Chávez
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 6521638
zbMATH Keywords: piecewise-linear vector field
zbMATH Keywords: limit cycle
zbMATH Keywords: bifurcation
zbMATH Keywords: connecting orbit
zbMATH Keywords: path-following
MaRDI profile type: Publication
full work available at URL: s00332-015-9251-z
OpenAlex ID: W2038268783
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DOI: 10.1007/S00332-015-9251-Z
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description: scientific article; zbMATH DE number 6521638