Crossing limit cycles for some classes of piecewise linear systems

Abstract

The study of piecewise linear di erential systems is relatively recent. Such that the dynamics of the piecewise linear di erential systems started to be studied around 1930, mainly in the book of Andronov et al [1], which is russian version. The contribution of Andronov, Vitt and Khaikin [1] provided the basis for the development of the theory for this system. Many researchers from di erent elds interested this kind of di erential systems, one of the reasons for this interest in the mathematical community is that these systems are widely used to model phenomena appearing in mechanics, electronics, economy, neuroscience,..., and it can be used to model applied problems, such as electronic circuits, biological systems, mechanical devices, etc, see for instance the book [6]. A limit cycle is a periodic orbit of a di erential system in R2 isolated in the set of all periodic orbits of that system. The study of the limit cycles goes back essentially to Poincar e [20] at the end of the 19th century. One of the main problems in the dynamics of the di erential systems in the plane is to control the existence and the number of their limit cycles. This problem restricted to polynomial di erential systems is the famous 16th Hilbert's problem. see more in [12, 14, 13]. The existence of limit cycles became important in the applications to the real world, because many phenomena are related with their existence,see for instance the van der Pol oscillator [22]. Thus in recent years, the theory of piecewise linear di erential systems has been increasingly developed and studied in order to understand the dynamics that such systems may have. In this sense one of the points of greatest interest is to obtain a lower bound for the maximum number of limit cycles that may arise around a single equilibrium point on the discontinuity set (i.e., on the region separating the linear di erential systems). This investigation started with the simplest possible case: the continuous piece- wise linear di erential systems with two zones separated by a staight line. Lum and Chua [18, 19] in 1991 conjectured that such di erential systems have at most one limit cycle. Later this conjecture was proved by Freire, Ponce, Rodrigo and Torres [9] in 1998. while for the planar discontinuous piecewise linear di erential systems in R2. Of course, the simplest piecewise linear di erential systems in R2 are the ones having only two pieces separated by a curve, and when this curve is a straight line. Han and Zhang [11] obtained di erential systems having two limit cycles and conjured that the maximum number of limit cycles of such class of di erential systems is two. But in this last years many authors have studied the limit cycles of discontinuous piecewise linear di erential systems in R2. Thus the limit cycles of these last class of discontinuous piecewise linear di erential systems has been intensively studied, see [2, 3, 7]. Up to know the results of all these papers only provide examples that the discontinuous piecewise linear di erential systems in R2 separated by a straight line can have 3 crossing limit cycles.