Centers of quasi-homogeneous polynomial differential equations of degree three

Abstract

Abstract. We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.

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.…”

“…Llibre and Pessoa [2009] classified all centers of system (3) with weight degrees d = 1, 2, 3, 4. 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.…”

Planar Semi-quasi Homogeneous Polynomial Differential Systems with a Given Degree

“…Next consider vector field X 0,0,0,1,1,2 . Taking transformation (X, Y, T ) = |a −1 0,3 b 3 1,2 | 1/2 a −1 2,0 x, b 1,2 a −1 2,0 y, |a 0,3 b −3 1,2 | 1/2 t , X 0,0,0,1,1,2 becomes dX dT = ±Y 3 + X, dY dT = XY 2 , with w m = (3,2,4,6). This yields the canonical form (G ± 1,2 ) of Proposition 14.…”

Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

“…Smooth Quasi-homogeneous polynomial differential systems have been intensively studied by a great deal of authors from different views. We refer readers to see for example the integrability [2,17,19,21,29], the centers and limit cycles [1,15,18,24], the algorithm to compute quasi-homogeneous systems with a given degree [14], the characterization of centers or topological phase portraits for quasi-homogeneous equations of degrees 3-5 respectively [5,26,32] and the references therein.…”