http://www.gsd.uab.cat THREE LIMIT CYCLES IN DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH TWO ZONES JAUME LLIBRE 1 AND ENRIQUE PONCE 2 Abstract. In this paper we study a planar piecewise linear differential system formed by two regions separated by a straight line so that one system has a real unstable focus and the other a virtual stable focus which coincides with the real one. This system was introduced by S.-M. Huan and X.-S. Yang in [8] who numerically showed that it can exhibit 3 limit cycles surrounding the real focus. This is the first example that a non–smooth piecewise linear differential system with two zones can have 3 limit cycles surrounding a unique equilibrium. We provide a rigorous proof of this numerical result. 1. Introduction and statement of the main result The analysis of piecewise linear differential systems can be traced back to An- dronov and coworkers [1] and still continues to receive attention by researchers. Effectively, in recent years there has been an upsurge of interest from the math- ematical community in understanding their dynamical richness, as such systems are widely used to model many real processes and different modern devices, see for instance [4] and references therein. Recently, they have been shown to be also relevant as idealized models of cell activity, see [3, 11, 12]. The case of continuous piecewise linear systems, when they have only two lin- earity regions separated by a straight line is the simplest possible configuration in piecewise linear systems. We remark that even in this seemingly simple case, only after a thorough analysis it was possible to establish the existence at most of one limit cycle for such systems, see [6]. The reason for that misleading simplicity of piecewise linear systems is twofold. First, even one can easily integrate solutions in any linearity region, the time that each orbit requires to pass from a linearity region to each other is unknown and so the matching of the corresponding solutions is an intricate problem. Second, the number of parameters to consider in order to be sure that one copes with all possible configurations is typically not small, so that the achievement of efficient canonical forms with fewer parameters is crucial. Discontinuous piecewise linear systems with only two linearity regions separated by a straight line have been studied recently in [7, 8], among other papers. In [7] some results about the existence of two limit cycles appeared, so that the authors conjectured that the maximum number of limit cycles for this class of piecewise 2010 Mathematics Subject Classification. Primary 34C05, 34C07, 37G15. Key words and phrases. non–smooth differential system, limit cycle, piecewise linear differen- tial system. * The first author is partially supported by the MICIIN/FEDER grant MTM2008–03437, the Generalitat de Catalunya grant 2009SGR-410 and ICREA Academia. The second author is partially supported by a MICIIN/FEDER grant number MTM2009-07849. 1 This is a preprint of: “Three nested limit cycles in discontinuous piecewise linear differential systems with two zones”, Jaume Llibre, Enrique Ponce, Dynam. Contin. Discrete Impuls. Systems. Ser. B, vol. 19, 325–335, 2012.