Canonical Discontinuous Planar Piecewise Linear Systems

Abstract

Abstract. The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Liénard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts one's attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonical form will be a useful tool in the systematic study of planar discontinuous piecewise linear systems, in which this paper is a first step.Key words. Filippov systems, normal form, limit cycle, piecewise linear differential systems AMS subject classifications. 34C05, 34C07, 37G15DOI. 10.1137/11083928X1. Introduction. Piecewise linear (PWL) systems constitute a class of differential systems which are widely used to model many real processes and different devices; see, for instance, [1,5,17]. The case of continuous PWL systems with two linearity regions separated by a straight line is the simplest possible configuration in piecewise smooth systems. Such a family of systems was completely studied in a previous paper [8], where in particular the existence of at most one limit cycle was established.Enforced by modern nonlinear engineering problems and mathematical biology (see [5]), some recent works in the literature on planar PWL systems deal with vector fields where continuity at the common boundary is not assumed. After the pioneering work of Filippov [7], one must cite in this context the papers of Coll, Gasull, and Prohens [3], Giannakopoulos and Pliete [9], Huan and Yang [11], Kuznetsov, Rinaldi, and Gragnani [13], Llibre, Ponce, and Torres [15], Shui, Zhang, and Li [18], and the recent thorough work of Guardia, Seara, and Teixeira [6], among others.As mentioned in the recent work of Huan and Yang [11], the study of discontinuous PWL systems is a difficult task because of the lack of a canonical form that can cope with a sufficiently broad class of systems, in contrast to what can be done for the continuous case; see [2]. In fact, only very particular cases are thoroughly analyzed in the available literature.

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

“…Without loss of generality we can consider the case T L = T C and ℓ = 1, because the other case can be studied in a similar way. Hence in this section we will work with the system (13) x ′ = F (x) − y, and y…”

“…Since the system becomes purely linear, it is easy to see, see for instance [13] that P L is a linear function given by…”

Piecewise linear differential systems without equilibria produce limit cycles?

“…Up to now we know that there are discontinuous systems with at least three limit cycles, see for instance [2,4,3,6,8,9,10,11,12,13,14,15,22,17,18,19,20,22,24].…”

“…Similarly, let −t − be the time that the solution (x − (t), y − (t)), starting at the point (0, y 0 ) when t = 0, enters in backward time in the half-plane x < 0 and reaches by first time the straight line x = 0, in case that such solution exists. Therefore, the piecewise linear differential system (1) and (2) has limit cycles if the system (8) x + (t + ) = 0,…”

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

“…In smooth systems there is a well known mechanism to search for the occurrence of limit cycles, the Hopf bifurcation theorem, see [13,19]. There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6,7,26,27], and for the case of discontinuous systems see [1,8,11,12,14,18]. In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2,3,4,10,15,16,17,20,22,23,24].…”