Bifurcaciones de Órbitas Periódicas y Conjuntos Invariantes en Sistemas Dinámicos Lineales a Trozos

Abstract

This year is the 100th anniversary of the death of Jules Henri Poincaré (Nancy, France, 29 April 1854-Paris, France, 17 July 1912), the founding father of Dynamical Systems theory. In mathematics, he is known as The Last Universalist, since he excelled in all fields of the discipline during his life. As it is well-known, the motivation of part of his work was the Celestial Mechanics [82, 83] and more specifically, the three-body problem. Within these more than 100 years, the Dynamical Systems theory has become one of the most important topics of interest for the scientific community. This is mainly due to the broad field of application. Although the first applications of Poincaré"™s ideas were in engineering, more concretely in electronic circuits and control theory in the 20"™s (Appleton and Van der Pol [3, 4, 93], Cartan [27], Liénard [69], Andronov and Pontryaguin [1, 2]), nowadays the applications go from engineering to biomathematics, (such as neural networks [63, 95]) passing through financial problems and social behaviors [84]. Among dynamical systems, in the last years we have attended to the expansion of the field of Piecewise-Smooth dynamical systems. First examples of the use of piecewise-smooth functions (in particular, piecewise linear) are found in the 1937 book of Andronov, Vitt and Khaikin [1], where they used it to model electronic, mechanical and control systems (saturation functions, impacts, switching...). Since then, the capability of piecewise-smooth systems to model a multitude of phenomena has been proven. In the 2008 published book of Mario di Bernardo et al. [35] they revise the state of the art of piecewise-smooth systems and we can find a huge number of references. In the framework of piecewise-smooth dynamical systems, there exists a class that is worth mentioning: the Piecewise Linear (PWL) Systems. As we have just said, the first examples of their use can be found in [1]. The importance of PWL systems is due, inter alia, to their ability to model faithfully real applications (neuron models [5, 33, 86], Chua"™s circuit [81], Colpitts"™s oscillator [74], Wien-Bridge oscillator [62, 76]), to reproduce bifurcations of differential systems and to show new behaviors, impossible to obtain under differentiability hypothesis (the behavior around the Teixeira singularity [90, 91], the continuous matching of two stable linear systems can be unstable [26]...). Furthermore, although the system can be integrated in each zone of linearity, which allows us to obtain explicitly some geometric and dynamical basic elements, it is not possible to obtain the general solution of the system and the classical theory of differential systems cannot be applied to PWL systems. Therefore, it is necessary creating a new theory to tackle PWL systems. The first step to analyze PWL systems is their simplification and reduction to a canonical form [13, 22, 45, 60, 61]. In this thesis, we focus our attention, mainly, on two-zonal three-dimensional continuous and planar discontinuous PWL systems. The work is split into six chapters. In the first section of the introductory Chapter 1, we show the canonical forms of the systems object of study along this work and we do it by classifying the systems from the point of view of the control notions (observability and controllability), as it was performed in [13, 22]. In a second step, the dynamical behavior must be studied. The analysis begins usually by finding the equilibrium states, i.e., the equilibrium points of the system and their stabilities. After that, the objective is the searching of periodic behaviors, that is, periodic orbits. This is neither an easy task in a differential system in general, nor in a piecewise-smooth system. A usual technique is the construction of the so-called Poincaré map, which in the case of PWL systems is defined through the composition of some transition maps, the Poincaré half-maps [56, 59, 72], defined in each zone of linearity. The second section of Chapter 1 is devoted to defining the Poincaré map for continuous PWL systems. As we have just commented, the analysis of periodic orbits in piecewise-smooth systems is not obvious. One of the aims of this essay is to shed light on this problem, by using different techniques to analyze the existence, bifurcations and stabilities of periodic orbits in planar and three-dimensional PWL systems. To find periodic orbits in planar smooth systems it is well-known the Melnikov theory [8], sometimes called Malkin-Loud theory [73, 75, 78]. The most important property that a family of systems must fulfill to apply the Melnikov theory is the existence of a system of the family having a continuum of periodic orbits, homoclinic connections or heteroclininic cycles. The Melnikov method was generalized to planar continuous piecewise-smooth systems in [13]. In Chapter 2 of this thesis, we generalize the Melnikov theory to hybrid systems (mixture between a flow and a map [35]) and we apply it to discontinuous PWL systems with two zones of linearity and to continuous PWL systems with three zones.