On the Poincaré map and limit cycles for a class of continuous piecewise linear differential systems with three zones

Abstract

Due to the encouraging increase in their applications, control theory [5] and [9], design of electric circuits [2], neurobiology [4] and [8] piecewise linear differential systems were studied early from the point of view of qualitative theory of ordinary differential equations [1]. Nowadays, a lot of papers are being devoted to these differential systems. On the other hand, starting from linear theory, in order to capture nonlinear phenomena, a natural step is to consider piecewise linear systems. As local linearizations are widely used to study local behavior, global linearizations (achieved quite naturally by working with models which are piecewise linear) can help to understand the richness of complex phenomena observed in the nonlinear world. The study of piecewise linear systems can be a difficult task that is not within the scope of traditional nonlinear systems analysis techniques. In particular, a sound bifurcation theory is lacking for such systems due to their nonsmooth character. In this work, we study the existence of limit cycles for the class of continuous piecewise linear differential systems x′ = X(x), (2) where x = (x, y) ∈ R2, and X is a continuous piecewise linear vector field. We will consider the following situation, that we will name the three-zone case. We have two parallel straight lines L− and L+ symmetric with respect to the origin dividing the phase plane in three closed regions: R−, Ro and R+ with (0, 0) ∈ Ro and the regions R− and R+ have as boundary the straight lines L− and L+ respectively. We will denote by X− the vector field X restrict to R−, by Xo the vector field X restricted to Ro and by X+ the vector field X restrict to R+. We suppose that the restriction of the vector field to each one of these zones are linear systems with constant coefficients that are glued continuously at the common boundary. Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 1, 2015. Trabalho apresentado no XXXV CNMAC, Natal-RN, 2014.