The center problem for a family of systems of differential equations having a nilpotent singular point

Abstract

We study the analytic system of differential equations in the planewhere p, q ∈ N, p q, s = (n + 1)p − q > 0, n ∈ N, and F i = (P i , Q i ) t are quasi-homogeneous vector fields of type t = (p, q) and degree i, with F q−p = (y, 0) t and Q q−p+2s (1, 0) < 0. The origin of this system is a nilpotent and monodromic isolated singular point. We prove for this system the existence of a Lyapunov function and we solve theoretically the center problem for such system. Finally, as an application of the theoretical procedure, we characterize the centers of several subfamilies.

A note on analytic integrability of planar vector fields

“…• Algaba et al [3] study the centre problem of the analytic system of differential equations on the plane whose origin is a nilpotent singular poinṫ…”

“…Others papers related to the analytic integrability problem of nilpotent centre are [7-10 and 15] and related to the nilpotent centres are [3] and [11].…”

Periodic orbits for perturbations of piecewise linear systems

“…After that, if possible, some systems in the family are described as perturbations of the non-generic systems studied, and then, the dynamical behavior of the perturbed systems is analyzed [4,13,25,29,30]. Some non-generic systems present a continuum of periodic orbits (this is related with the center problem [1,18]). …”

“…are invariant manifolds for system (1), and its dynamical behavior can be studied from the onedimensional equation…”

“…Theorem 1. If λ + > 0 and λ − < 0, then, the only equilibria for (1) are the points (1/λ − , 1, 0) which is stable and (1/λ + , 1, 0), which is unstable; the segment joint the equilibria is a heteroclinic orbit, and the system has one bounded invariant manifold foliated by periodic orbits. Moreover, every periodic orbit has period 2π .…”