Jaume Llibre

The second-order differential equation $\ddot x + a x \dot x + b x^3=0$ with $a,b \in \mathbb{R}$ has been studied by several authors mainly due to its applications. Here, for the first time, we classify all its phase portraits according to its parameters $a$ and $b$. This classification is done in the Poincaré disc in order to control the orbits that escape or come from infinity. We prove that there are exactly six topologically different phase portraits in the Poincaré disc of the first-order differential system associated to the second-order differential equation. Additionally, we show that this system is always integrable, providing explicitly its first integrals.

A tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the in-circle or inscribed circle. In this paper we classify all planar four-body central configurations, where the four bodies are at the vertices of a tangential trapezoid.

For the $4$-body problem there is the following conjecture: Given arbitrary positive masses, the planar $4$-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the $(\ell+2)$-body problem with $\ell \geqslant 3$. In particular, we prove that the symmetric $(2n+1)$-body problem with masses $m_1=\ldots=m_{2n-1}=1$ and $m_{2n}=m_{2n+1}=m$ sufficiently small has at least two classes of convex central configuration when $n=2$, five when $n=3$, and four when $n=4$. We conjecture that the $(2n+1)$-body problem has at least $n $ classes of convex central configurations for $n>4$ and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric $(2n+2)$-body problem with masses $m_1=\ldots=m_{2n}=1$ and $m_{2n+1}=m_{2n+2}=m$ sufficiently small has at least three classes of convex central configuration when $n=3$, two when $n=4$, and three when $n=5$. We also conjecture that the $(2n+2)$-body problem has at least $[(n+1)/2]$ classes of convex central configurations for $n>5$ and we give some numerical evidences that the conjecture can be true.

For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years.