In this paper, we show, by means of a linear scaling in time and coordinates, that the Chen system, given by x=a(y-x), y=(c-a)x+cy-xz, ż=-bz+xy, is, generically (c≠0), a special case of the Lorenz system. First, we infer that it is enough to consider two parameters to study its dynamics. Furthermore, we prove that there exists a homothetic transformation between the Chen and the Lorenz systems and, accordingly, all the dynamical behavior exhibited by the Chen system is present in the Lorenz system (since the former is a special case of the second). We illustrate our results relating Hopf bifurcations, periodic orbits, invariant surfaces, and chaotic attractors of both systems. Since there has been a large literature that has ignored this equivalence, the aim of this paper is to review and clarify this field. Unfortunately, a lot of the previous papers on the Chen system are unnecessary or incorrect.

Abstract. The proof of the existence of a global connection in differential systems is generally a difficult task. Some authors use numerical techniques to show this existence, even in the case of continuous piecewise linear systems. In this paper we give an analytical proof of the existence of a reversible T-point heteroclinic cycle in a continuous piecewise linear version of the widely studied Michelson system. The principal ideas of this proof can be extended to other piecewise linear systems.

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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.