Noose bifurcation and crossing tangency in reversible piecewise linear systems

• Published in: Nonlinearity Vol. 27; no. 3; pp. 585 - 606
• Main Authors: Carmona, V, Fernández-García, S, Fernández-Sánchez, F, García-Medina, E, Teruel, A E
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.03.2014
• Subjects: Bifurcations; crossing tangency; Intersections; Linear systems; Mathematical analysis; Nonlinearity; noose bifurcation; Orbits; periodic orbits; piecewise linear systems; Planes; reversible systems; Separation
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/27/3/585

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.