CARMONA CENTENO , Victoriano

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The intrinsic nature of a problem usually suggests a first suitable method to deal with it. Unfortunately, the apparent ease of application of these initial approaches may make their possible flaws seem to be inherent to the problem and often no alternative ways to solve it are searched for. For instance, since linear systems of differential...

In this work we deal with the canard regime as a part of a canard explosion taking place in a PWL version of the van der Pol equation having a flat critical manifold. The proposed analysis involves the identification of two specific canard cycles, one at the beginning and the other at the end of the canard regime, here called birth and...

In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining...

We close the problem of the existence of crossing period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness. In fact, a characterization for the existence of such objects is provided by means of a few basic operations on the parameters.

This paper deals with fundamental properties of Poincaré half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincaré half-maps, their series expansions (Taylor and Newton–Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincaré half maps and...

In this article, we prove the existence of a saddle-node bifurcation of limit cycles in continuous piecewise linear systems with three zones. The bifurcation arises from the perturbation of a non-generic situation, where there exists a linear center in the middle zone. We obtain an approximation of the relation between the parameters of the...

The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum–Chua's work, it has remained an open question whether this uniform upper...

The already proved Lum–Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincaré half-maps in planar linear systems. This proof is very short and...

Presentamos en esta comunicación, en primer lugar, un mecanismo para explicar la aparición de un ciclo límite bizonal en un circuito en puente de Wien polarizado de forma asimétrica modelado mediante funciones lineales a trozos. Damos expresiones para la amplitud y el periodo de la oscilación bizonal y las comparamos con las obtenidas...

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...

We study saddle-node bifurcations of canard limit cycles in PWL systems by using singular perturbation theory tools. We distinguish two cases: the subcritical and the supercritical. In the subcritical case, we find saddle-node bifurcations of canard cycles both with head and without head. Moreover, we detect a transition between them. In the...

By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The...

Presentamos en esta comunicación una técnica para probar de forma analítica la existencia de conexiones globales en sistemas dinámicos continuos lineales a trozos. Más concretamente, utilizamos esta técnica para demostrar la existencia de dos conexiones homoclinas directas (aquellas que cortan al plano de separación exactamente dos veces) y un...

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,...

En este trabajo se considera la existencia de conos invariantes en sistemas dinámicos continuos tridimensionales lineales a trozos, dada la relevancia que estas variedades invariantes tienen en la determinación de la estabilidad del origen en tales sistemas. Se recogen varios resultados de existencia de conos invariantes y se analiza una...