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Abstract. In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. We provide sufficient conditions to ensure the existence of a limit cycle bifurcating from the infinity. The main tools used are the Bendixson transformation and the averaging theory.

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Abstract. We consider a n-dimensional piecewise smooth vector fields with two zones separated by a hyperplane Σ which admits an invariant hyperplane Ω transversal to Σ containing a period-annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n = 3 we provide normal forms in the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the normal forms established.

We present a global dynamical analysis of the following quadratic differential systeṁ x=a(y−x),ẏ = dy − xz,ż =−bz + f x 2 + gx y, where (x, y, z) ∈ R 3 are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in R 3 , alike Lorenz, Rössler, Chen and other. By using the Poincaré compactification for a polynomial vector field in R 3 , we study the dynamics of this system on the Poincaré ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.

A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

In this paper, some ergodic aspects of non-smooth vector fields are studied. More specifically, the concepts of recurrence and invariance of a measure by a flow are discussed, and two versions of the classical Poincaré Recurrence Theorem are presented. The results allow us to soften the hypothesis of the classical Poincaré Recurrence Theorem by admitting non-smooth multivalued flows. The methods used in order to prove the results involve elements from both measure theory and topology.