On my eight articles with the Professor Jorge Sotomayor.

In this note I summarize the eight articles that I wrote with the Professor Jorge Sotomayor and the consequences of some these articles.

Global centers of a family of cubic systems.

Consider the family of polynomial differential systems of degree 3, or simply cubic systems in the plane . An equilibrium point of a planar differential system is a if there is a neighborhood of such that is filled with periodic orbits. When is filled with periodic orbits, then the center is a .

Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation.

The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations x⃛+|x|qx¨+k|x|qxx˙2=0, which has its regularity varying with q, a positive integer. Indeed, for q=1, it is discontinuous on the straight line x=0, whereas for q a positive even integer it is polynomial, and for q>1 a positive odd integer it is continuous but not differentiable on the straight line x=0. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for q=2 and k=3.

Limit cycles in Filippov systems having a circle as switching manifold.

It is known that planar discontinuous piecewise linear differential systems separated by a straight line have no limit cycles when both linear differential systems are centers. Here, we study the limit cycles of the planar discontinuous piecewise linear differential systems separated by a circle when both linear differential systems are centers. Our main results show that such discontinuous piecewise differential systems can have zero, one, two, or three limit cycles, but no more limit cycles than three.

Periods of Morse-Smale diffeomorphisms on , , and .

We study the set of periods of the Morse-Smale diffeomorphisms on the -dimensional sphere , on products of two spheres of arbitrary dimension with , on the -dimensional complex projective space and on the -dimensional quaternion projective space . We classify the minimal sets of Lefschetz periods for such Morse-Smale diffeomorphisms. This characterization is done using the induced maps on the homology.

Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations.

Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations x ¨ + f ( x ) x ˙ + x = 0 , being f ( x ) a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.

Vanishing set of inverse Jacobi multipliers and attractor/repeller sets.

In this paper, we study conditions under which the zero-set of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repeller compact sets. The work generalizes previous results focusing on sink singularities, orbitally asymptotic limit cycles, and monodromic attractor graphics. Taking different flows on the torus and the sphere as canonical examples of attractor/repeller sets with different topologies, several examples are constructed illustrating the results presented.

New classes of polynomial maps satisfying the real Jacobian conjecture in ℝ 2.

We present two new classes of polynomial maps satisfying the real Jacobian conjecture in ℝ 2 . The first class is formed by the polynomials maps of the form (q(x)-p(y), q(y)+p(x)) : R 2 ⟶ R 2 such that p and q are real polynomials satisfying p'(x)q'(x) ≠ 0. The second class is formed by polynomials maps (f, g): R 2 ⟶ R 2 where f and g are real homogeneous polynomials of the same arbitrary degree satisfying some conditions.

Limit cycles created by piecewise linear centers.

In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles play a main role. Up to now, the major part of papers which study the limit cycles of the piecewise linear differential systems consider only two pieces.

The phase portrait of the Hamiltonian system associated to a Pinchuk map.

In this paper we describe the global phase portrait of the Hamiltonian system associated to a Pinchuk map in the Poincaré disc. In particular, we prove that this phase portrait has 15 separatrices, five of them singular points, and 7 canonical regions, six of them of type strip and one annular.

Simultaneity of centres in ℤ -equivariant systems.

We study the simultaneous existence of centres for two families of planar -equivariant systems. First, we give a short review about -equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a -equivariant cubic system and for a -equivariant quintic system.

Periodic solutions of Lienard differential equations via averaging theory of order two.

For ε ≠ 0 sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x'' + f ⁢(x)⁢ x' + n2⁢x + g (x) = ε2p1 ⁢(t) + ε3 ⁢p2(t), where n is a positive integer, f : ℝ → ℝ is a C 3 function, g : ℝ → ℝ is a C 4 function, and p i : ℝ → ℝ for i = 1, 2 are continuous 2π-periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.

A new qualitative proof of a result on the real Jacobian conjecture.

Let F= (f, g) : R2 → R2 be a polynomial map such that det DF(x) is different from zero for all x ∈ R2. We assume that the degrees of f and g are equal. We denote by f and G the homogeneous part of higher degree of f and g, respectively.

On the absence of analytic integrability of the Bianchi Class B cosmological models.

We follow Bogoyavlensky's approach to deal with Bianchi class B cosmological models. We characterize the analytic integrability of such systems.

Generalized van der Waals Hamiltonian: periodic orbits and C1 nonintegrability.

The aim of this paper is to study the periodic orbits of the generalized van der Waals Hamiltonian system. The tool for studying such periodic orbits is the averaging theory. Moreover, for this Hamiltonian system we provide information on its C(1) nonintegrability, i.

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry.

In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured.

Lines of principal curvature on canal surfaces in R3.

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R3. By means of a connection of the differential equations for these curvature lines and real Riccati equations, it is established that canal surfaces have at most two isolated periodic principal lines. Examples of canal surfaces with two simple and one double periodic principal lines are given.