Claudio Pessoa

About Limit Cycles in Continuous Piecewise Linear Differential Systems

In 2012, Lima and Llibre in [3] have studied a class of planar continuous piecewise linear vector fields with three zones. This class can be separated in four other classes and they proved, using the Poincare map, that this particular class admits always a unique hyperbolic limit cycle. Here, we extended this study for other classes. We proved that some of them also admit always a unique hyperbolic limit cycle, moreover, we find a class that does not have limit cycles and prove the appearance of two limit cycles with one of these cycles appear by perturbations of a period annulus.

A hybrid symbolic-numerical approach to the center-focus problem

We propose a new hybrid symbolic-numerical approach to the center-focus problem. The method allowed us to obtain center conditions for a three-dimensional system of differential equations, which was previously not possible using traditional, purely symbolic computational techniques.

Limit Cycles Bifurcating from a Period Annulus in Continuous Piecewise Linear Differential Systems with Three Zones

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincare map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

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

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.

The monodromic singularity to piecewise linear vector fields

Abstract: Consider in R2 the semi-planes N = { y > 0} and S = { y < 0} having as common bo u nda r y the s t r a i ght line D = { y = 0} . In N and S are de fi ned l i nea r vecto r fi elds X and Y , respectively, leading to a discontinuous polynomial vector field Z = ( X, Y ) . If the vector fields X and Y satisfy suitable conditions, they produce a transition flow from a segment of the splitting line to another segment and this produces a generalized singular point on the line. This point can be a focus or a center. In this paper we give necessary and sufficient conditions to a D -singular point be a monodromic.