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

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Abstract

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.

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Acknowledgments

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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Authors and Affiliations

  1. Dpto. Matemática Aplicada II, Universidad de Sevilla, Seville, Spain

    V. Carmona, F. Fernández-Sánchez & E. García-Medina

  2. Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, Palma de Mallorca, Spain

    A. E. Teruel

Authors
  1. V. Carmona
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  2. F. Fernández-Sánchez
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  3. E. García-Medina
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  4. A. E. Teruel
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Correspondence to V. Carmona.

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Communicated by Paul Newton.

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Carmona, V., Fernández-Sánchez, F., García-Medina, E. et al. Noose Structure and Bifurcations of Periodic Orbits in Reversible Three-Dimensional Piecewise Linear Differential Systems. J Nonlinear Sci 25, 1209–1224 (2015). https://doi.org/10.1007/s00332-015-9251-z

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