Jaume Giné

Darboux integrability and the inverse integrating factor

Abstract We mainly study polynomial differential systems of the form dx / dt = P ( x , y ), dy / dt = Q ( x , y ), where P and Q are complex polynomials in the dependent complex variables x and y , and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form H=f 1 λ 1 ⋯f p λ p exp h 1 g 1 n 1 μ 1 ⋯ exp h q g q n q μ q , where the polynomials f i and g j are irreducible, the polynomials g j and h j are coprime, and the λ i and μ j are complex numbers, when i =1,…, p and j =1,…, q . Prelle and Singer proved that these systems have a rational integrating factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincare we define the critical remarkable values of H . Then, we prove that the system has a polynomial inverse integrating factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse integrating factor; and secondly that if the degree of the system is m , the homogeneous part of highest degree of H is a multi-valued function, and the functions exp( h j / g j ) are exponential factors for j =1,…, q , then the system has a polynomial inverse integrating factor of degree m +1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci.

Local analytic integrability for nilpotent centers

Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \dot{x}=y+ X(x,y) , \dot{y}=Y(x,y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following. If X= y f(x,y^2) and Y= g(x,y^2) , then the systemhas a local analytic first integral of the form H=y^2+F(x,y) ,where F starts with terms of order higher than two. If the system has a formal first integral, then it hasa formal first integral of the form H=y^2+F(x,y) , where F starts with terms of order higher than two. In particular, if thesystem has a local analytic first integral defined at the origin,then it has a local analytic first integral of the form H=y^2+F(x,y) , where F starts with terms of order higher than two. As an application we characterize the nilpotent centersfor the differential systems \dot{x}=y+P_3(x,y) , \dot{y}=Q_3(x,y) , which have a local analytic first integral,where P_3 and Q_3 are homogeneous polynomials of degree three.

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

Abstract In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincare–Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincare–Liapunov method.

Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials

In this work we study isochronous centers of two-dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree. A complete classification of the necessary conditions for the time-reversible systems of this class is given in order to have an isochronous center at the origin. An open problem is stated for the sufficient conditions. Moreover, we find two nonreversible isochronous families from the center cases known. All the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods.

Integrability of a linear center perturbed by a fourth degree homogeneous polynomial

In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

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.

A Class of Reversible Cubic Systems with an Isochronous Center

Abstract We study cubic polynomial differential systems having an isochronous center and an inverse integrating factor formed by two different parallel invariant straight lines. Such systems are time-reversible. We find nine subclasses of such cubic systems, see Theorem 8. We also prove that time-reversible polynomial differential systems with a nondegenerate center have half of the isochronous constants equal to zero, see Theorem 3. We present two open problems.

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.

SUFFICIENT CONDITIONS FOR A CENTER AT A COMPLETELY DEGENERATE CRITICAL POINT

Consider the two-dimensional autonomous systems of differential equations of the form where P3(x, y) and Q3(x, y) are homogeneous polynomials of degree 3, and P4(x, y) and Q4(x, y) are homogeneous polynomials of degree 4. The origin is a completely degenerate critical point of this system. In this work we give sufficient conditions in order to have a center at the origin.

Conditions for the existence of a center for the Kukles homogeneous systems

Abstract In this work, we study necessary and sufficient conditions for the existence of centers in two families of linear centers with homogeneous quartic and quintic nonlinearities. Systems of this class are called Kukles homogeneous systems. Systems of this type were studied, for the first time, by Kukles, who studied a linear center with cubic nonhomogeneous nonlinearities.