Advanced Course: Stability and Bifurcations in Piecewise Linear Systems
January 30, 2012 )
The interest on the analysis of piecewise linear differential systems (PWLS) has increased in the last decades as modern engineering applications require the piecewise linear modeling of a wide range of problems in mechanics, power electronics, control theory, biology and so on. On the one hand piecewise linear systems are the natural extension of the linear ones in order to cope with nonlinear phenomena, for they can reproduce much of the complex behavior observed in smooth nonlinear systems: multi-stability, self-sustained oscillations, hysteretic behavior, homoclinic and heteroclinic connections and of course, chaotic behavior. On the other hand, piecewise linear systems turn out to be the most accurate models for some realistic applications in the quoted fields.
Piecewise linear systems can be classified in two big classes depending on the continuity of the associated vector field. Discontinuous cases constitute nowadays the subject of intense research, and there is not yet a total agreement about basic concepts and definitions. Thus, excepting several simple instances, most of the case studies to be considered in the course will correspond to continuous piecewise linear systems (CPWL, for short). In fact, there are still unsolved issues in the continuous case, even for the seemingly simple problem of stability of the only equilibrium point. Apart from equilibria, it is very important to characterize the periodic orbits of such systems, since they constitute the next step in complexity for observed behavior in practice. We will pay special attention to the study of existence of periodic orbits for piecewise linear systems, following a point of view which is typical in bifurcation theory, that is, we will study degenerated situations and after parameter variations we will look for the appearance of limit cycles.
Contact: Enrique Ponce Nuñez ( email@example.com ), Jorge Galan Vioque ( firstname.lastname@example.org )