Javier Chavarriga

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.

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.

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 a Result of Darboux

This paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of new, simple proofs of two known results. First, it is shown that an irreducible invariant algebraic curve of degree n > 1 without multiple points for a polynomial vector field of degree m satisfies n ≤ m + 1. Second, a proof is given that quadratic polynomial vector fields have no algebraic limit cycles of degree 3.

A family of non-darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree

Abstract We show that the system x dot = y dot = 2n+2xy+y 2 has the algebraic solution h(x, y) = Hn(x)y + 2nHn−1(x), where Hn(x) is the Hermite polynomial of degree n, and the system is not Darboux integrable and has no Darboux integrating factor for any n ϵ N .

An improvement to Darboux integrability theorem for systems having a center

Abstract We consider planar polynomial differential systems of degree m with a center at the origin and with an arbitrary linear part. We show that if the system has m(m + 1) 2 − [ (m + 1) 2 ] algebraic solutions or exponential factors then it has a Darboux integrating factor. This result is an improvement of the classical Darboux integrability theorem and other recent results about integrability.

Integrable systems via polynomial inverse integrating factors

Abstract We study the integrability of two-dimensional autonomous systems in the plane of the form x =−y+X s (x,y), y =x+Y s (x,y), where X s ( x , y ) and Y s ( x , y ) are homogeneous polynomials of degree s with s ⩾2. Writing this system in polar coordinates, we study the existence of polynomial inverse integrating factors and we give some related invariants, from which we can compute a formal first integral for the system. Finally, we give a family of systems with s =4 and with a centre at the origin, via inverse integrating factors, in which radial and angular coefficients do not independently vanish in Lyapunov constants.