João Carlos da Rocha Medrado

Original article: Piecewise linear differential systems with two real saddles

In this paper we study piecewise linear differential systems formed by two regions separated by a straight line so that each system has a real saddle point in its region of definition. If both saddles are conveniently situated, they produce a transition flow from a segment of the splitting line to another segment of the same line, and this produces a generalized singular point on the line. This point is a focus or a center and there can be found limit cycles around it. We are going to show that the maximum number of limit cycles that can bifurcate from this focus is two. One of them appears through a Hopf bifurcation and the second when the focus becomes a node by means of the sliding.

Bifurcation of limit cycles from a centre in R-4 in resonance 1:N

For every positive integer N ≥ 2 we consider the linear differential centre in ℝ4 with eigenvalues ±i and ±Ni. We perturb this linear centre inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree N, i.e. where every component of F(x) is a linear polynomial plus a homogeneous polynomial of degree N. Then if the displacement function of order ϵ of the perturbed system is not identically zero, we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential centre.

Symmetric singularities of reversible vector fields in dimension three

Abstract For a large class of reversible vector fields on the three-dimensional space we present all the topological types and their respective normal forms of the symmetric singularities of codimension 0, 1.

On the invariant hyperplanes for d-dimensional polynomial vector fields

We deal with polynomial vector fields X of the formd=1 Pk(x1 ,...,x d )∂/∂xk with d 2. Let mk be the degree of Pk. We call (m1 ,...,m d ) the degree of X. We provide the best upper bounds for the polynomial vector field X in the function of its degree (m1 ,...,m d ) of (1) the maximal number of invariant hyperplanes, (2) the maximal number of parallel invariant hyperplanes, and (3) the maximal number of invariant hyperplanes that pass through a single point. Moreover, if mi = m, i = 1 ,...,d , we show that these best upper bounds are reached taking into account the multiplicity of the invariant hyperplanes.

On the limit cycles of a class of piecewise linear differential systems in ℝ4 with two zones

We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems with two zones. Our main result shows that three is an upper bound for the number of limit cycles that bifurcate from a center, up to first order expansion of the displacement function. Moreover, this upper bound is reached. The main technique used is the averaging method.

On the limit cycles of a class of piecewise linear differential systems in R-4 with two zones

We study the bifurcation of limit cycles from the periodic orbits of a four-dimensional center in a class of piecewise linear differential systems with two zones. Our main result shows that three is an upper bound for the number of limit cycles that bifurcate from a center, up to first order expansion of the displacement function. Moreover, this upper bound is reached. The main technique used is the averaging method.

Codimension-two Singularities of Reversible Vector Fields in 3D

This paper is concerned with the dynamics near an equilibrium point of reversible systems. For a large class of reversible vector fields on the three dimensional space we present all the topological types and their respective normal forms of the codimension-two symmetric singularities. Such classification comes from useful new results, also proved here, on dynamical systems defined on manifolds with boundary.

Limit cycles of continuous and discontinuous piecewise–linear differential systems in R3

Abstract We study the limit cycles of two families of piecewise-linear differential systems in R 3 with two pieces separated by a plane Σ . In one family the differential systems are continuous on the plane Σ , and in the other family they are discontinuous on the plane Σ . The usual tool for studying these limit cycles is the Poincare map, but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems. All the computations have been done with the algebraic manipulator Mathematica.

Propriedade do divergente para campos vetoriais não diferenciáveis em duas zonas

Neste trabalho estendemos a propriedade do divergente para os campos vetoriais nao diferenciaveis definidos em duas zonas.

The monodromic singularity to piecewise linear vector fields

Abstract: Consider in R2 the semi-planes N = { y > 0} and S = { y < 0} having as common bo u nda r y the s t r a i ght line D = { y = 0} . In N and S are de fi ned l i nea r vecto r fi elds X and Y , respectively, leading to a discontinuous polynomial vector field Z = ( X, Y ) . If the vector fields X and Y satisfy suitable conditions, they produce a transition flow from a segment of the splitting line to another segment and this produces a generalized singular point on the line. This point can be a focus or a center. In this paper we give necessary and sufficient conditions to a D -singular point be a monodromic.