Planar quasi-homogeneous polynomial systems with a given weight degree

Abstract

In this paper, we investigate a class of quasi-homogeneous polynomial systems with a given weight degree. Firstly, by some analytical skills, several properties about this kind of systems are derived and an algorithm can be established to obtain all possible explicit systems for a given weight degree. Then, we focus on center problems for such systems and provide some necessary conditions for the existence of centers. Finally, for a specific quasihomogeneous polynomial system, we characterize its center and prove that the center is not isochronous.

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

“…Actually, if s 2 > s 1 , then one can make a transformation (x, y) → (y, x) such that the resulting system satisfies this assumption; if (s 1 , s 2 ) = ρ = 1, then by Lemma 1 of [García et al, 2013], it is also ( s 1 ρ , s 2 ρ , d−1 ρ + 1)-weighthomogeneous with ( s 1 ρ , s 2 ρ ) = 1. (b) In the paper of [Xiong & Han, 2015], the authors studied the center problem for a quasi-homogeneous polynomial system with (s 1 , s 2 , d) = (3, 1, 3) and also obtained the same results given in Theorem 1(i). But, in this paper, we give a different and new proof.…”

“…(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.…”

“…Aziz et al [2014] considered the center problem for a cubic quasi-homogeneous, but nonhomogeneous polynomial differential system. In the paper [Xiong & Han, 2015], the authors provided an algorithm to obtain all possible explicit forms for system (3) with a given weight degree under some assumptions and studied their center problems. García et al [2013] presented a program to get all possible quasihomogeneous but nonhomogeneous polynomial differential systems of a given degree and researched the integrability of this kind of systems of degrees 2 and 3.…”

“…For the last case k 1 odd and k 2 even, from Lemma 6.4 of [Xiong & Han, 2015], one can find that system (4) has an invariant set {(x, y) | φ(x, y) = 0} filled with singular points of system (4), where…”