Journal of Nonlinear Science

, Volume 25, Issue 6, pp 1209–1224 | Cite as

Noose Structure and Bifurcations of Periodic Orbits in Reversible Three-Dimensional Piecewise Linear Differential Systems

  • V. Carmona
  • F. Fernández-Sánchez
  • E. García-Medina
  • A. E. Teruel


The main goal of this work is to describe the periodic behavior of a class of three-dimensional reversible piecewise linear continuous systems. More concretely, we study an interesting structure called the noose bifurcation that was previously detected by Kent and Elgin in the Michelson system. We numerically obtain the curves of periodic orbits that appear from the bifurcations at the noose curve, where other phenomena related to different types of tangencies with the separation plane arise. Besides that, we show that some of these curves of periodic orbits wiggle around global connections when the period increases. The complete structure of periodic orbits, including the stability and bifurcations, coincides with the one observed in the Michelson system. However, we also point out the relevance of the crossing tangency and the small loop that emerges from it in the existence of the noose bifurcation.


Bifurcation Periodic orbit Continuation Piecewise system 

Mathematics Subject Classification

34C14 34C23 34C25 37G15 37G25 



This work has been partially supported by the Ministerio de Economía y Competitividad, Plan Nacional I+D+I cofinanced with FEDER funds, in the frame of the Projects MTM2009-07849, MTM2010-20907-C02-01, MTM2011-22751 and MTM2012-31821 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (TIC-0130, P08-FQM-03770).


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • V. Carmona
    • 1
  • F. Fernández-Sánchez
    • 1
  • E. García-Medina
    • 1
  • A. E. Teruel
    • 2
  1. 1.Dpto. Matemática Aplicada IIUniversidad de SevillaSevilleSpain
  2. 2.Departament de Matemàtiques i InformàticaUniversitat de les Illes BalearsPalma de MallorcaSpain

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