On the centers of the weight-homogeneous polynomial vector fields on the plane

Abstract

We classify all centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.

Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2

“…In [8] the authors classified all centers of a planar weight-homogeneous polynomial differential systems up to weight-degree 4. In particular they proved that the unique family of weight-homogeneous polynomial differential systems with a center with weight-degree 2 is The main goal of this paper is to provide an explicit polynomial whose real positive simple zeros gives the exact number of limit cycles which bifurcate, at first order in the perturbation parameter, from the periodic orbits of the center of the weight-homogeneous polynomial differential system (2).…”

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

“…(1) From references [Li et al, 2009;Llibre & Pessoa, 2009;García et al, 2013;Aziz et al, 2014;Xiong & Han, 2015;Llibre & Zhang, 2002], the system is said to be quasi-homogeneous if there exist positive integers s 1 , s 2 and d such that for any ρ > 0 f (ρ s 1 x, ρ s 2 y) = ρ s 1 +d−1 f (x, y), g(ρ s 1 x, ρ s 2 y) = ρ s 2 +d−1 g(x, y),…”

“…For example, reference papers [Li et al, 2009;Llibre & Pessoa, 2009;Aziz et al, 2014;Xiong & Han, 2015] investigated their centers, [García et al, 2013;Algaba et al, 2011;Giné et al, 2013;Llibre & Zhang, 2002;Cairó & Llibre, 2007;Edneral & Romanovski, 2011] studied their integrability, [Li et al, 2009;Gavrilov et al, 2009] discussed the limit cycle problem, and [Algaba et al, 2010] concerned normal forms, to name but a few. More precisely, in the paper [Li et al, 2009], the authors presented a necessary condition for the existence of a center of (3) at the origin, and when (3) has a center at the origin, they investigated the problem of the maximal number of limit cycles bifurcating from the period annulus surrounding the origin.…”

Centers of quasi-homogeneous polynomial differential equations of degree three

“…From Theorem 2 of [17] we have that there are only two families of cubic polynomial differential homogeneous systems with a center.…”

“…Now we are going to prove Theorem 1(b). The usual forms given in (5) for the cubic homogeneous polynomial differential systems having a center were obtained in Proposition 1 and Theorem 2 of [17]. The phase portrait were classified in [5].…”