The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the wellknown Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameterized three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.In the piecewise system, the noose bifurcation involves reversible periodic orbits that intersect the separation plane at two or four points. This work is focused on those reversible periodic orbits that intersect the separation ✩ This work was partially supported by the Ministerio de Ciencia e Innovación, under the projects MTM2009-07849, MTM2010-20907-C02-01 and MTM2011-22751 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (TIC-0130, P08-FQM-03770). The second author is supported by Ministerio de Educación, grant AP2008-02486.* Corresponding author Email addresses: vcarmona@us.es (V. Carmona), soledad@us.es (S. Fernández-García), fefesan@us.es (F. Fernández-Sánchez), egarme@us.es (E. García-Medina), antonioe.teruel@uib.es (A. E. Teruel) Preprint submitted to Nonlinear Analysis: Theory, Methods & ApplicationsFebruary 16, 2012 This is a preprint of: "Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations", Victoriano Carmona, Soledad Fernández-García, Fernando Fernández-Sánchez, Elisabeth Garcia-Medina, Antonio E. Teruel, Nonlinear Anal., vol. 75, 5866-5883, 2012. DOI: [10.1016/j.na.2012 plane twice (RP2-orbits). It is established that for every T between 2π and a critical point there exists a unique value of the parameter for which the system has a RP2-orbit with period T . Moreover, this critical value, that separates periodic orbits with two or four intersection points with the separation plane, corresponds to a RP2-orbit that crosses tangentially the separation plane.It is also proved that in a bifurcation diagram parameter versus period, the curve of this family of periodic orbits has a unique maximum point, which corresponds to the saddle-node bifurcation of periodic orbits that appears in the noose bifurcation.

The structure of the periodic orbits analysed in this paper was first reported in the Michelson system by Kent and Elgin (1991 Nonlinearity 4 1045-61), who gave it the name noose bifurcation. Recently, it has been found in a piecewise linear system with two linearity zones separated by a plane, which is called the separation plane. In this system, the orbits that take part in the noose bifurcation have two and four points of intersection with the separation plane, and they are arranged in two curves that are connected by a point where the periodic orbit has a crossing tangency with the separation plane. In this work, we analytically prove the local existence of the curve of periodic orbits with four intersections that emerges from the point corresponding to the crossing tangency. Moreover, we add a numerical study of the stability and bifurcations of the periodic orbits involved in the noose curve for the piecewise linear system and check that they exhibit the same configuration as that of the Michelson system.

Numerical methods are often used to put in evidence the existence of global connections in differential systems. The principal reason is that the corresponding analytical proofs are usually very complicated. In this work we give an analytical proof of the existence of a pair of homoclinic connections in a continuous piecewise linear system, which can be considered to be a version of the widely studied Michelson system. Although the computations developed in this proof are specific to the system, the techniques can be extended to other piecewise linear systems.

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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