Cristóbal García

The integrability problem for a class of planar systems

In this paper we consider perturbations of quasi-homogeneous planar Hamiltonian systems, where the Hamiltonian function does not contain multiple factors. It is important to note that the most interesting cases (linear saddle, linear centre, nilpotent case, etc) fall into this category. For such kinds of systems, we characterize the integrability problem, by connecting it with the normal form theory.

Quasi-homogeneous normal forms

In this paper, we extend the concepts of the normal form theory for vector fields that are expanded in quasi-homogeneous components of a fixed type (these expansions have been used by some authors in the analysis of the determinacy of a given singularity). Also, the use of reparametrizations in the time are considered. Namely, beyond the use of C∞-conjugation to determine normal forms, we present a method useful to determine how much a vector field can be simplified by using C∞-equivalence. The results obtained are applied in the case of the Bogdanov-Takens singularity, firstly using C∞-conjugation and later, showing the improvements provided by the C∞-equivalence.

On the Formal Integrability Problem for Planar Differential Systems

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems in , and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.

Nilpotent centres via inverse integrating factors

In this paper we are interested in the nilpotent center problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centers. More general objects are considered as the formal inverse integrating factors. However the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centers. For the family studied in this paper it is enough.

A new algorithm for determining the monodromy of a planar differential system

Abstract We give a new algorithmic criterium that determines wether an isolated degenerate singular point of a system of differential equations on the plane is monodromic. This criterium involves the conservative and dissipative parts associated to the edges and vertices of the Newton diagram of the vector field.

Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part

We present the basic ideas of the Normal Form Theory by using quasi-homogeneous expansions of the vector field, where the structure of the normal form is determined by the principal part of the vector field. We focus on a class of tridimensional systems whose principal part is the coupling of a Hamiltonian planar system and an unidimensional system, in such a way that the quoted principal part does not depend on the last variable and has free divergence. Our study is based on several decompositions of quasi-homogeneous vector fields. An application, corresponding to the coupling of a Takens-Bogdanov and a saddle-node singularities, (in fact, it is a triple-zero singularity with geometric multiplicity two), that falls into the class considered, is analyzed.

Local Integrability for Some Degenerate Nilpotent Vector Fields

This work is about the analytic integrability problem around the origin in a family of degenerate nilpotent vector fields. The integrability problem for planar vector fields with first Hamiltonian component having simple factors in its factorization on \(\mathbb {C}[x, y]\) is solved in Algaba et al. (Nonlinearity 22:395–420, 2009) [5]. Nevertheless, when the Hamiltonian function has multiple factors on \(\mathbb {C}[x, y]\) is an open problem. In this second case our problem is framed. More concretely, we study the following degenerate systems:

Algebraic Inverse Integrating Factors for a Class of Generalized Nilpotent Systems

\begin{aligned} \dot{x} = - y (x^{2n}+ny^2)+ \cdots , \quad \dot{y}=x^{2n-1} (x^{2n}+ny^2)+\cdots , \end{aligned}