Abstract
We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerate differential systems.
Highlights
We consider an autonomous system of the form, x = F(x) = (P (x), Q(x))T, x ∈ R2, (1.1)where F is an analytic planar vector field defined in a neighborhood of the origin U ⊂ R2 having an equilibrium point at the origin, i.e., F(0) = 0 and where P and Q are analytic in U.The so-called monodromy problem consists in characterize when a vector field has a well-defined return map in a neighborhood of an isolated singularity, that is, if the vector field has or not characteristic orbits passing through this isolated singularity, see [1,2,3,4,5]
Where F is an analytic planar vector field defined in a neighborhood of the origin U ⊂ R2 having an equilibrium point at the origin, i.e., F(0) = 0 and where P and Q are analytic in U
Once we know that the singular point is monodromic appears another classic problem called the center problem or stability problem which consists in distinguish if this singularity is a focus or a center, see [8,9,10]
Summary
If the linear part of the vector field at the origin is nondegenerate, the Poincare–Lyapunov method solves the center problem, see the seminal works [11,12,13,14]. In certain monodromic degenerate singular points a geometric method can be applied to determine the stability of singular points with characteristic directions in the homogeneous order, see [21]. The Bautin method [34], introduced to find the maximum number of limit cycles that bifurcate from the origin for quadratic systems with center-type linear part, can be used to degenerate monodromic singular points without characteristic directions. In the present work using this method we have managed to find the first terms of the Poincare map for monodromic singular points without characteristic directions in certain coordinates (using a quasi-homogeneous order which generalized the homogeneous order). For the computation of the generalized focal values we generalize the method of Briskin, Francoise and Yomdin [35], and we compute the integrating factor of the associated Abel equation which gives a linear recursive system of differential equations in order to obtain the focal values
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