Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

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.

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Abstract. Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.

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We show that a planar slow-fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these quasi-canard cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of ε to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call super-explosion.

The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincaré maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found.In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S.-M. Huan and X.-S. Yang [13].

Abstract. In this paper we study the non-existence and the uniqueness of limit cycles for the Liénard differential system of the form x − f (x)ẋ + g(x) = 0 where the functions f and g satisfy xf (x) > 0 and xg(x) > 0 for x = 0 but they can be discontinuous at x = 0.In particular our results allow first to prove the non-existence of limit cycles under suitable assumptions, and second to prove the existence and uniqueness of a limit cycle in a class of discontinuous Liénard systems which are relevant in engineering applications.