Ricardo Miranda Martins

Reversible-equivariant systems and matricial equations

This paper uses tools in group theory and symbolic computing to classify the representations of finite groups with order lower than, or equal to 9 that can be derived from the study of local reversible-equivariant vector fields in 4 . The results are obtained by solving matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitskii normal form.

Periodic orbits, invariant tori, and cylinders of Hamiltonian systems near integrable ones having a return map equal to the identity

Generically the return map of an integrable Hamiltonian system with two degrees of freedom in a Hamiltonian level foliated by invariant tori is a twist map. If we perturb such integrable Hamiltonian system inside the class of Hamiltonian systems with two degrees of freedom, then the Poincare–Birkhoff theorem allows to determine which periodic orbits of the integrable can be prolonged to the perturbed one, and the KAM theory provides sufficient conditions in order that some invariant tori persist under sufficiently small perturbations. If some power of this return map is the identity, then in general for these degenerate Hamiltonian systems we cannot study which periodic orbits of the integrable can be prolonged to the perturbed one, or if some invariant tori persist. This paper studies the perturbation of integrable Hamiltonian systems with two degrees of freedom having some power of the return map equal to the identity. We show with two different models a way to study the prolongation of periodic orbits ...

The Ricci flow of left-invariant metrics on full flag manifold SU(3)/T from a dynamical systems point of view

In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold SU(3)/T using techniques of the qualitative theory of differential equations, in special the Poincare compactification and Lyapunov exponents. We prove that there are four invariant lines for the Ricci flow equation, each one associated with a singularity corresponding to an Einstein metric. In such manifold, the bi-invariant normal metric is Einstein. Moreover, around each invariant line there is a cylinder of initial conditions such that the limit metric under the Ricci flow is the corresponding Einstein metric; in particular we obtain the convergence of left-invariant metrics to a bi-invariant metric under the Ricci flow.

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.

On Semi-local Structural Stability of Filippov Systems

We introduce the notion of semi-local structural stability which detects if a nonsmooth system is structurally stable around the switching manifold. More specifically, we characterize the semi-local structurally stable systems in a class of Filippov systems on a compact 3-manifold which has a simply connected switching manifold.

On Degenerate Cycles in Planar Filippov Systems

The main objective of this paper is to study bifurcations of a vector field in a neighborhood of a cycle having a homoclinic-like connection at a saddle-regular point. In order to perform such a study it is necessary to analyze how the cycle can be broken, in this way the approach is to look separately at local bifurcations and at the structure of the first return map defined near the cycle.

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.