Durval José Tonon

PIECEWISE SMOOTH REVERSIBLE DYNAMICAL SYSTEMS AT A TWO-FOLD SINGULARITY

This paper focuses on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in the presence of symmetries.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

Symmetric periodic orbits for the collinear charged 3-body problem

In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each body possesses mass and an electric charge. The main technique applied in this study is the continuation method of Poincare.

The chaotic behavior of piecewise smooth differential equations on two dimensional torus and sphere

ABSTRACT This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippovs convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.

The Chaotic Behavior of Piecewise Smooth Dynamical Systems on Torus and Sphere

In this work we discuss the appearance of minimal trajectories for the flow of piecewise smooth dynamical systems defined in the two dimensional torus and sphere in such a way that the switching manifold breaks the manifold into two connected components. We show that the number of pseudo-singularities of the sliding vector field is an invariant for the structural stability and study global bifurcations. Using a generic normal form, we prove that these systems can present chaotic behavior.

Bifurcations of Piecewise Smooth Vector Fields via Geometric Singular Perturbations

We deal with piecewise smooth vector fields on the plane and prove that the analysis of their local behavior around certain typical singularities can be treated via singular perturbation theory. In fact, after a regularization of a such system and a blow−up we are able to bring out some results that bridge the space between piecewise smooth vector fields presenting typical singularities and singularly perturbed smooth systems.