Resumen de Bifurcaciones de órbitas periódicas y conjuntos invariantes en sistemas dinámicos lineales a trozos = Bifurcations of periodic orbits and invariant sets in piecewise linear dynamical systems
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.
On the other hand, we will consider three-dimensional homogeneous continuous PWL systems. In such systems, when only equilibrium point is the origin, the most significant question is the stability of this equilibrium point. This is not at all an easy question and the answer can be counterintuitive. In fact, the continuous matching of two stable linear systems can be unstable [26]. As we explain in the last section of Chapter 1, the stability of the system is related to the existence of a type of invariant objects called invariant cones. The analysis of their existence, bifurcations and stabilities is other objective of this thesis. Specifically, in Chapter 3, we analyze the invariant cones of a family of observable three-dimensional continuous PWL linear systems by studying the periodic orbits of a family of planar hybrid PWL systems, where the Melnikov theory of Chapter 2 can be applied.
The original Melnikov theory was developed for smooth planar systems, and in Chapter 2 we have generalized it to a class of non-smooth systems. In Chapter 4 we adapt it to a class of continuous three-dimensional systems. As it has been commented, one of the properties that a family of systems must fulfill to be able to adapt the ideas of the Melnikov theory is the existence of a system of the family having a continuum of periodic orbits. We have found an appropriate family of non-controllable systems in Chapter 4 and we have applied the ideas of the Melnikov theory to analyze the existence of periodic orbits.
The existence of invariant cones in observable PWL systems has been analyzed, inter alia, in [18, 21, 25]. However, as far as we know, the problem has not been tackled for the non-observable case. Chapter 5 begins with the analysis of invariant cones in three-dimensional continuous homogeneous non-observable PWL systems. Among these systems, we set the conditions for the existence of a system having a cone foliated by periodic orbits. Beyond the Melnikov theory, there exist other methods to find periodic orbits. For instance, in Chapter 5, we have adapted the method of Chapter 14 of [31] for analyzing the periodic orbits of the continuum that remain after a perturbation which makes the system observable and non-homogeneous.
The analysis of the existence of periodic orbits or invariant cones by the usual techniques in PWL systems, for instance, by computing the fixed points of the Poincaré map, needs the transversal intersection of the orbits (respectively, cones) with the separation boundary between the linearity zones. With the theory and methods that has been developed in this thesis, we have been able to study the existence of periodic orbits and invariant cones with tangential intersection to the separation boundary. Between the periodic orbits with tangential intersection with the separation boundary, we can distinguish two different cases. In a first case, the orbit remain locally in the same zone of differentiability. This case has been analyzed along the work in chapters 3 and 4. The other posible case is when the tangent orbit crosses the separation boundary. In Chapter 6, the dynamical richness that this situation fosters has been shown by analyzing a three-dimensional continuous PWL system with two zones of linearity, which is considered a PWL version of the well-known Michelson system [77], which possesses a periodic orbit with two intersection points with the separation plane, which crosses it with tangential intersection. In particular, it will be shown that the presence of this orbit is the necessary germ for the existence of the so-called noose bifurcation [58]. The noose bifurcation occurs when the curve of the family of periodic orbits that appears from a period-doubling bifurcation and the curve of the original family of periodic orbits come together and annihilate at a saddle-node bifurcation. Therefore, two of the most common bifurcations of periodic orbits (saddle-node and period-doubling) are connected by a noose-shaped curve.
To conclude the thesis, we will write a short summary and we will consider the open problems that arise from this study.
