Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems

Abstract

In this paper we find necessary and sufficient conditions in order that a planar quasi-homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasihomogeneous polynomial differential system can be transformed into a differential system of the formu = uf (v),v = g(v) with f (v) and g(v) polynomials, and vice versa.2010 Mathematics Subject Classification. Primary: 34C05, 34A34, 34C20. Key words and phrases. Quasi-homogeneous polynomial differential equations, integrability problem, polynomial first integral, rational first integral.

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

“…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.…”

“…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. Giné et al [2013] found necessary and sufficient conditions in order that a planar quasihomogeneous polynomial differential system has a polynomial or a rational first integral. Cairó and Llibre [2007] classified all quasi-homogeneous planar polynomial differential systems with a weight degree 3 having a polynomial first integral.…”

Integrability of vector fields versus inverse Jacobian multipliers and normalizers

“…Its modern development can be traced back to Liouville [2] and Poincaré [26]. The theory of integrability has important application in mechanics, physics and the dynamical analysis of differential systems, and there are great progress in the past century, see for instance [13,14,16,17,19,20,22,28,30,31] and the references therein.…”

Planar quasi-homogeneous polynomial systems with a given weight degree

“…where f (x, y) and g(x, y) are polynomials in the variables x and y. By [1,6,[8][9][10], if there exists a point (s 1 , s 2 , d) ∈ N + × N + × N + such that for all ρ ∈ R + , f (ρ s1 x, ρ s2 y) = ρ s1+d−1 f (x, y), g(ρ s1 x, ρ s2 y) = ρ s2+d−1 g(x, y),…”