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The book is devoted to the qualitative study of differential equations defined by piecewise linear (PWL) vector fields, mainly continuous, and presenting two or three regions of linearity. The study focuses on the more common bifurcations that PWL differential systems can undergo, with emphasis on those leading to limit cycles. Similarities and differences with respect to their smooth counterparts are considered and highlighted. Regarding the dimensionality of the addressed problems, some general results in arbitrary dimensions are included. The manuscript mainly addresses specific aspects in PWL differential systems of dimensions 2 and 3, which are sufficinet for the analysis of basic electronic oscillators.
Dynamical systems are relevant whenever one wants to describe evolution problems with respect to time, usually given by ordinary or partial differential equations, or by the application of discrete transformations. Typically, it is very useful to consider such evolution not only in the phase space but also while studying the dependence of the system with respect to parameters, which leads to the concept of bifurcations.
Within the realm of nonlinear dynamical systems, piecewise-linear differential systems (PWL systems, for short) constitute an interesting class from several points of view. First, they naturally appear in realistic nonlinear engineering models, as certain devices are accurately modeled by piecewise linear vector fields. In fact, these kinds of models are frequent in applications taken from simple electronic oscillators (to be the main field chosen in this book for illustrating the theoretical results) and nonlinear control systems, where piecewise linear systems turn out to be very accurate models. They are used in mathematical biology as well, where they constitute rough approximate models. Thus, they represent an interesting yet elementary subclass of piecewise-smooth dynamical systems.
Furthermore, since non-smooth piecewise linear characteristics can be considered as the uniform limit of smooth nonlinearities, the global dynamics of smooth models has been sometimes approximated by piecewise linear models and vice versa, obtaining a good qualitative agreement between both modelling approaches. Note that, in practice, any nonlinear characteristic usually exhibits a saturated part, which is difficult to be approximated by polynomial models. Therefore, this possibility of what we could call global linearization by means of a finite number of linear pieces emphasizes the usefulness of PWL systems, frequently being the most natural extension of linear systems in order to capture nonlinear phenomena. In fact, it is a widely extended feeling among researchers in the field that the richness of dynamical behaviour to be found in piecewise linear systems covers almost all the instances of dynamics exhibited by general smooth nonlinear systems: limit cycles, homoclinic and heteroclinic orbits, strange attractors…
The consideration of this class as an alternative to smooth nonlinear systems is gaining popularity due to the fact that solutions can be written in closed form when they are restricted to a region of the phase space where the system becomes linear. Nevertheless, the analysis of the global dynamics is far from being trivial since one must match the different solutions in every linearity region. Such matching typically requires the explicit knowing of different flight times, that is, the times employed by the orbit in each linearity zone, which is true only by exception. On the other hand, standard families of PWL systems have a non-small number of parameters, so that the complete analysis of possible dynamical behaviours is usually a formidable task. In this sense, the disposal of good canonical forms is a preliminary aim of great relevance, as it will be evident throughout this book.
Summarizing, this book is organized as follows. In Part I, after some introductory chapter to emphasize the differences between linear and piecewise linear systems, we review some terminology and results related to canonical forms in the study of PWL systems along with certain techniques that are useful for the bifurcation analysis of their periodic orbits, see Chap. 2. First, we develop a three zones Lienard canonical form able to represent also systems with two zones, to facilitate the subsequent study on existence and uniqueness of limit cycles. Next, we work in arbitrary dimension to review general results although we will later deal only with systems in dimensions 2 and 3.
Part II is completely devoted to planar PWL systems. We exploit and extend recent results achieved in, which allows us to pave the way for a shorter proof of the Lum–Chua conjecture. We also give all the details about the focus-center-limit cycle bifurcation (FCLC bifurcation, for short) in planar systems with only two linear zones, see Chap. 3. Such a bifurcation will be studied in other contexts along the book, since it is the analogue for the PWL setting to the Poincare–Andronov–Hopf bifurcation for smooth systems.
Other general results for existence and uniqueness of limit cycles in 3CPWL2 systems, that is planar systems with three linear zones, appear in Chap. 4, along with the symmetric version of the FCLC bifurcation, just mentioned. Other mechanisms able to generate limit cycles, as the boundary equilibrium bifurcations (BEB, for short), are explored in Chap. 5. As a consequence, cases with two limit cycles surrounding the only equilibrium point are detected. At this point, we are in a position to apply the developed theory in basic realistic circuits coming from nonlinear electronics, by analyzing the bifurcation set of quasi-symmetric Wien bridge oscillators.
In a different direction of research, in Chap. 6, a family of algebraically computable piecewise linear nodal oscillators is introduced, and some real electronic devices that belong to the family are shown. The outstanding feature of this family makes it an exceptional benchmark for testing approximate methods of analysis of oscillators.
Another contribution included in this part is the study of a specific bifurcation that can only appear in PWL systems, as is the focus-saddle bifurcation, see Chap. 7, which can also involve the appearance of periodic orbits in generic configuration of parameters, and homoclinic connections in some non-generic cases.
Part III represents a particular incursion in PWL systems of dimension three, basically by studying the FCLC bifurcation and its possible degeneration in three different contexts. In this sense, some results involving 2CPWL3 vector fields, that is in PWL systems with only two zones, are offered in Chap. 8. These results are also of interest in systems with three linear zones, and we show this by analysing the celebrated Chua’s circuit, by considering two adjacent zones where the oscillations take place. The symmetric case for the results of Chap. 8 is tackled in Chap. 9.
We want to emphasize that there is much to be done in 3D systems, since for instance, nowadays, their boundary equilibrium bifurcations are not completely well understood. Pursuing the aim of filling in the catalog of possible bifurcations leading to limit cycles, we study some unfoldings of the analogous to Hopf-pitchfork bifurcations in PWL systems, see Chap. 10. Our theorems predict the simultaneous bifurcation of 3 limit cycles, but we also formulate a natural, strongly numerically based conjecture on the simultaneous bifurcation of up to 5 limit cycles. We finish by applying most of the 3D results of Part III to the analysis of some circuits with rather complex dynamics such analysis is merely illustrative of the usefulness of previous results and is very far from obtaining the complete bifurcation set, which surely involves an infinity of bifurcation curves