Maurício Firmino Silva Lima

Global dynamics of the Rössler system with conserved quantities

The subject of this paper concerns with a class of Rossler systems that admits conserved quantities. For this class of systems a complete description of the global dynamics in the Poincare sphere is provided.

LIMIT CYCLES AND INVARIANT CYLINDERS FOR A CLASS OF CONTINUOUS AND DISCONTINUOUS VECTOR FIELD IN DIMENTION 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.

On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones

Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincare map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.

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.

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.

Normally Hyperbolic Sets in Discontinuous Dry Friction Oscillator

In this paper, we study a four-parameters piecewise-smooth dry friction oscillator from Control theory. Using Filippovs convention, we prove the existence of a codimension-1 bifurcation which gives rise to a normally hyperbolic set composed by a family of attracting cylinders. This bifurcation exhibits interesting discontinuous oscillation phenomena. We also present consistent numerical simulations.