International audienceIn this work, we analyze the existence and stability of canard solutions in a class of planar piecewise linear systems with three zones, using a singular perturbation theory approach. To this aim, we follow the analysis of the classical canard phenomenon in smooth planar slow–fast systems and adapt it to the piecewise-linear framework. We first prove the existence of an intersection between repelling and attracting slow manifolds, which defines a maximal canard, in a non-generic system of the class having a continuum of periodic orbits. Then, we perturb this situation and we prove the persistence of the maximal canard solution, as well as the existence of a family of canard limit cycles in this class of systems. Similarities and differences between the piecewise linear case and the smooth one are highlighted
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.
Abstract. At a two-fold singularity, the velocity vector of a flow switches discontinuously across a codimension one switching manifold, between two directions that both lie tangent to the manifold. Particularly intricate dynamics arises when the local flow curves towards the switching manifold from both sides, a case referred to as the Teixeira singularity. The flow locally performs two different actions: it winds around the singularity by crossing repeatedly through, and passes through the singularity by sliding along, the switching manifold. The case when the number of rotations around the singularity is infinite has been analysed in detail. Here we study the case when the flow makes a finite -but previously unknown -number of rotations around the singularity between incidents of sliding. We show that the solution is remarkably simple: the maximum and minimum number of rotations made anywhere in the flow differs only by one, and increases incrementally with a single parameter: the angular jump in the flow direction across the switching manifold, at the singularity.
We analyze a four-dimensional slow-fast piecewise linear system consisting of two coupled McKean caricatures of the FitzHugh-Nagumo system. Each oscillator is a continuous slow-fast piecewise linear system with three zones of linearity. The coupling is one-way, that is, one subsystem evolves independently and is forcing the other subsystem. In contrast to the original FitzHugh-Nagumo system, we consider a negative slope of the linear nullcline in both the forcing and the forced system. In the forcing system, this lets us, by just changing one parameter, pass from a system having one equilibrium and a relaxation cycle to a system with three equilibria keeping the relaxation cycle. Thus, we can easily control the changes in the oscillation frequency of the forced system. The case with three equilibria and a linear slow nullcline is a new configuration of the McKean caricature, where the existence of the relaxation cycle was not studied previously. We also consider a negative slope of the y-nullcline in the forced system that enables us to reproduce a quasi-steady state called the surge. We analyze not only the qualitative behavior of the four-dimensional system, but also quantitative aspects such as the period, frequency, and amplitude of the oscillations. The system is used to reproduce all the features endowed in a former smooth model and reproduce the secretion pattern of the hypothalamic neurohormone GnRH along the ovarian cycle in different species. Introduction.Piecewise linear (PWL) systems are a family of nonsmooth systems well known for reproducing the dynamics of models coming from applications. The first examples of PWL systems were developed for the modeling of engineering problems (such as mechanical, electronic, and control device problems) [1]. Since then, the applicability of these systems has been fully demonstrated. They have been used to model not only dynamical processes coming from engineering, but also, for instance, social behaviors and financial or biological problems [3,12,13,25]. The analysis of PWL systems revealed that they exhibit rich dynamics as smooth systems, in particular limit cycles and periodic orbits [15,17], homoclinic and heteroclinic connections [5], and strange attractors [25].The celebrated FitzHugh-Nagumo system [14,27] models, in first approximation, the behavior of an excitable system, for example, a neuron. Basically, it is assumed that a neuron behaves as an electronic circuit. The van der Pol oscillator [2] can be considered as a special
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