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In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \]parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.

This paper deals with the period function of the reversible quadratic centers where . Compactifying the vector field to , the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of that tend to in the Hausdorff sense as . This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all outside the segments and . With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all . In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result.

In this paper we study planar polynomial Kolmogorov's differential systems X μ { x ˙ = f ( x , y ; μ ) , y ˙ = g ( x , y ; μ ) , with the parameter μ varying in an open subset Λ ⊂ R N . Compactifying X μ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ , that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all μ ∈ Λ . We are interested in the cyclicity of Γ inside the family { X μ } μ ∈ Λ , i.e., the number of limit cycles that bifurcate from Γ as we perturb $\mu.$ In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N = 3 and N = 5 , and in both cases we are able to determine the cyclicity of the polycycle for all μ ∈ Λ , including those parameters for which the return map along Γ is the identity.

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension $7$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq3$ there are closed orbits in $\mathbf F(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{B}(\mathcal{F})$ of a foliation $\mathcal{F}\in\mathbf{F}(d)$ as the set of $\mathcal{G}\in\mathbf{F}(d)$ such that $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ We show that the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1)$, resp. $\dim \mathbf{F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, D\'eserti, Garba Belko and Meziani.

In this work we introduce the notion of Galois foliations on P2 , defined as those folations whose Gauss applications restricted to a Zariski open subset is a Galois covering. We also present some examples and acriterium for identifying such foliations. En el presente trabajo introducimos la nocion de foliaciones Galois sobre P2C, definidas como aquellas cuya aplicacion de Gauss restringida aun abierto Zariski es un recubrimiento Galois. Asimismo, presentamo salgunos ejemplos y un criterio para identicar este tipo de foliaciones.

Un k−web W viene dado por una ecuación diferencial ordinaria de primer orden definida de forma implícita por un polinomio de grado k que puede entenderse como una estructura geométrica descrita localmente por k−foliaciones en posición general. La geometría de webs es el estudio de invariantes de familias finitas de foliaciones y fue iniciado por Blaschke y su escuela a inicios de la década de 1920 en Hamburgo. Uno de los resultados emblemáticos obtenido por él junto con Dubordieu, es el que caracteriza la equivalencia local de un germen de un 3−web W en el plano complejo con el 3−web definido por dx · dy · d(x+ y) a través del anulamiento de un covariante diferencial: la curvatura K(W) del web W, que es una 2−forma meromorfa con polos en su discriminante ∆(W), este último conjunto es el lugar donde las tangentes a las hojas de las foliaciones que conforman el web dejan de ser transversales. La estructura local de un k−web no es rígida como sucede en los casos k = 1, 2 sino que admiten invariantes analíticos no triviales: el rango de un web, que no es sino la dimensión de un espacio que relaciona las integrales primeras de las foliaciones que definen a un web, y su curvatura. El estudio de webs desde el punto de vista local ha sido tratado por diferentes autores, ver [2, 11]. Un ejemplo de un k−web proveniente de la geometría algebraica proyectiva es obtenido al considerar una curva algebraica reducida C sobre P 2 C de grado k, la curva dual Cˇ ⊂ Pˇ2 C de C es la curva formada por las tangentes a C. Como Cˇ es de clase k entonces por un punto genérico ℓ ∈ Pˇ2 C pasan exactamente k tangentes a Cˇ. Podemos considerar estas k rectas como hojas de foliaciones sobre un abierto Zariski de Pˇ2 C , de esta manera obtenemos un k−web, llamado web algebraico asociado a la curva C, denotado por WC. Como consecuencia de un teorema clásico de Abel, el rango del k−web WC es maximal, en el sentido que coincide con la cota superior (k − 1)(k − 2)/2. Para un k−web con k > 3 la curvatura es definida como la suma de las curvaturas de todos los 3−subwebs extraídos de un web W. Miháileanu obtiene un resultado donde demuestra que el anulamiento de la curvatura de un k−web es una condición necesaria para la maximalidad del rango de W, ver [32]. Los webs de rango máximo que no son localmente equivalentes a ningún web algebraico WC han sido denominados excepcionales. En [25] los autores demuestran que para cada k > 4 existe una familia infinita de k−webs excepcionales contenidos en el espacio de k−webs de grado 1. Tesis