Paulo Ricardo da Silva

Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R � , � ≥ 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

GLOBAL DYNAMICS OF THE LORENZ SYSTEM WITH INVARIANT ALGEBRAIC SURFACES

In this paper by using the Poincare compactification of ℝ3 we describe the global dynamics of the Lorenz system \begin{eqnarray*} \dot{x}= s(-x+y),\quad \dot{y} = rx-y-xz, \quad \dot{z} =-bz+xy, \end{eqnarray*} having some invariant algebraic surfaces. Of course (x, y, z) ∈ ℝ3 are the state variables and (s, r, b) ∈ ℝ3 are the parameters. For six sets of the parameter values, the Lorenz system has invariant algebraic surfaces. For these six sets, we provide the global phase portrait of the system in the Poincare ball (i.e. in the compactification of ℝ3 with the sphere 𝕊2 of the infinity).

Closed poly-trajectories and Poincaré index of non-smooth vector fields on the plane

This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a class of non-smooth vector fields we provide necessary and sufficient conditions for the existence of closed poly-trajectorie. By means of a regularization process we prove that hyperbolic closed poly-trajectories are limit sets of a sequence of limit cycles of smooth vector fields. In our approach the Poincaré Index for non-smooth vector fields is introduced.

Global dynamics in the Poincare ball of the Chen system having invariant algebraic surfaces

In this paper, we perform a global analysis of the dynamics of the Chen system where (x, y, z) ∈ ℝ3 and (a, b, c) ∈ ℝ3. We give the complete description of its dynamics on the sphere at infinity. For six sets of the parameter values, the system has invariant algebraic surfaces. In these cases, we provide the global phase portrait of the Chen system and give a complete description of the α- and ω-limit sets of its orbits in the Poincare ball, including its boundary 𝕊2, i.e. in the compactification of ℝ3 with the sphere 𝕊2 of infinity. Moreover, combining the analytical results obtained with an accurate numerical analysis, we prove the existence of a family with infinitely many heteroclinic orbits contained on invariant cylinders when the Chen system has a line of singularities and a first integral, which indicates the complicated dynamical behavior of the Chen system solutions even in the absence of chaotic dynamics.

Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds

We consider a differential equation , with discontinuous right-hand side and discontinuities occurring on a set Σ. We discuss the dynamics of the sliding mode which occurs when, for any initial condition near p ∈ Σ, the corresponding solution trajectories are attracted to Σ. Firstly we suppose that Σ = H−1(0), where H is a smooth function and is a regular value. In this case Σ is locally diffeomorphic to the set . Secondly we suppose that Σ is the inverse image of a non-regular value. We focus our attention to the equations defined around singularities as described in Gutierrez and Sotomayor (1982 Proc. Lond. Math. Soc 45 97–112). More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of in terms of implicit functions and immersions are broken in a stable manner. In this case Σ is locally diffeomorphic to one of the following algebraic varieties: (double crossing); (triple crossing); (cone) or (Whitneys umbrella).

Regularization of discontinuous foliations: Blowing up and sliding conditions via Fenichel theory

Abstract We study the regularization of an oriented 1-foliation F on M ∖ Σ where M is a smooth manifold and Σ ⊂ M is a closed subset, which can be interpreted as the discontinuity locus of F . In the spirit of Filippovs work, we define a sliding and sewing dynamics on the discontinuity locus Σ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions.

Asymptotic Expansion of the Heteroclinic Bifurcation for the Planar Normal Form of the 1:2 Resonance

We consider the family of planar differential systems depending on two real parameters ẋ = y,ẏ = δ1x + δ2y + x3 − x2y. This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve δ2 = c(δ1) = −1 5δ1 + O(δ12), δ1 < 0. We significantly improve the knowledge of this curve in a neighborhood of the origin.