Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

Abstract

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.

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

“…These properties has been described previously in (Carmona et al 2008(Carmona et al , 2010(Carmona et al , 2012(Carmona et al , 2014, but it is convenient to remind them here. System (1.2) is divergence free and time reversible with respect to the involution R(x, y, z) = (−x, y, −z).…”

“…After that, a set of conditions for the existence of periodic orbits is introduced. We finish the section by showing, for the sake of completeness, some theoretical results that have been previously proved in (Carmona et al 2012(Carmona et al , 2014. Concretely, the noose bifurcation and the bifurcations that appear on this structure are described.…”

“…In (Carmona et al 2012), the existence of these RP2-orbits was proved. Moreover, the RP4-orbits appear from an RP2-orbit that crosses the separation plane tangentially, as it was proved in (Carmona et al 2014).…”

“…Due to this, it is possible to consider problems that are practically unapproachable in differentiable systems. For example, in (Carmona et al 2008(Carmona et al , 2010(Carmona et al , 2012(Carmona et al , 2014 the analytical proof of the existence of some global connections (homoclinic connections and T-point heteroclinic cycle) and periodic orbits was given in a continuous piecewise linear system.…”

“…Recently, the noose bifurcation has been also reported in the three-dimensional piecewise linear continuous system ⎧ ⎨ ⎩ẋ = y, y = z, z = 1 − y − λ(1 + λ 2 )|x|, (1.2) where the parameter λ is strictly positive, see (Carmona et al 2012(Carmona et al , 2014. This system is formed by two linear systems separated by the plane {x = 0}, called the separation plane, and it can be considered as a continuous piecewise linear version of the Michelson system.…”