Isaac A. García

Isochronicity into a family of time-reversible cubic vector fields

In this work, we study necessary and sufficient conditions for the existence of isochronous centers into a family of cubic time-reversible systems. This class of reversible systems is characterized by the existence of an inverse integrating factor which is a certain power of an invariant straight line.

The inverse integrating factor and the Poincaré map

This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincare return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincare return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.

Generalized cofactors and nonlinear superposition principles

It is known from Lies works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of variables, the Riccati equation. For planar complex polynomial differential systems, the classical Darboux integrability theory exists based on the fact that a sufficient number of invariant algebraic curves permits the construction of a first integral or an inverse integrating factor. In this paper, we present a generalization of the Darboux integrability theory based on the definition of generalized cofactors.

ON INTEGRABILITY OF DIFFERENTIAL EQUATIONS DEFINED BY THE SUM OF HOMOGENEOUS VECTOR FIELDS WITH DEGENERATE INFINITY

The paper deals with polynomials systems with degenerate infinity from different points of view. We show the utility of the projective techniques for such systems, and a more detailed study in the quadratic and cubic cases is carried out. On the other hand, some results on Darboux integrability in the affine plane for a class of systems are given. In short we show the explicit form of generalized Darboux inverse integrating factors for the above kind of systems. Finally, a short proof of the center cases for arbitrary degree homogeneous systems with degenerate infinity is given, and moreover we solve the center problem for quartic systems with degenerate infinity and constant angular speed.

ON THE INTEGRABILITY OF QUASIHOMOGENEOUS AND RELATED PLANAR VECTOR FIELDS

In this work we consider planar quasihomogeneous vector fields and we show, among other qualitative properties, how to calculate all the inverse integrating factors of such C1 systems. Additionally, we obtain a necessary condition in order to have analytic inverse integrating factors and first integrals for planar positively semi-quasihomogeneous vector fields which are related to the existence of polynomial inverse integrating factors and first integrals for the quasihomogeneous cut. Examples are given and their relationship with Kovalevskaya exponents is shown.

A necessary condition in the monodromy problem for analytic differential equations on the plane

Abstract In this paper we give a very easy to compute necessary condition in the monodromy problem for all singular point of analytic differential systems in the real plane. Our main tool is considering the analytic function, angular speed , and studying its limit through straight lines to the singular point.

On Lie's symmetries for planar polynomial differential systems

In this paper we show that any polynomial planar vector field which is either polynomial or rational integrable possesses a polynomial or rational infinitesimal generator of a Lie symmetry, respectively. Moreover, if all the critical points of the vector field are strong and there exists a polynomial inverse integrating factor that vanishes at all the critical points we show that, independently of the class of their first integral, there exists a polynomial infinitesimal generator of a Lie symmetry.

Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

Abstract The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G 2 cosmologies. By using Darboux’s theory in order to study ordinary differential equations in the complex projective plane ℂℙ2 we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases.

On the periodic orbit bifurcating from a zero Hopf bifurcation in systems with two slow and one fast variables

The Hopf bifurcation in slow-fast systems with two slow variables and one fast variable has been studied recently, mainly from a numerical point of view. Our goal is to provide an analytic proof of the existence of the zero Hopf bifurcation exhibited for such systems, and to characterize the stability or instability of the periodic orbit which borns in such zero Hopf bifurcation. Our proofs use the averaging theory.

Formal Inverse Integrating Factor and the Nilpotent Center Problem

We are interested in deepening the knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields 𝒳. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors V of 𝒳. Although by the existence of V it is not possible to describe all nilpotent centers strata, we simplify, improve and also extend previous results on the relationship between these concepts. We use in the performed analysis the so-called Andreev number n ∈ ℕ with n ≥ 2 associated to 𝒳 which is invariant under orbital equivalency of 𝒳. Besides the leading terms in the (1,n)-quasihomogeneous expansions that V can have, we also prove the following: (i) If n is even and there exists V then 𝒳 has a center; (ii) if n = 2, the existence of V characterizes all the centers; (iii) if there is a V with minimum “vanishing multiplicity” at the singularity then, generically, 𝒳 has a center.