Polynomial and rational first integrals for planar homogeneous polynomial differential systems

Abstract

In this paper we find necessary and sufficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems.

Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

“…We see that, by hypothesis, an open interval with boundary the origin of coordinates and belonging to {(x, y) ∈ R 2 : x > 0, y = 0} is a transversal section for system (1) on the whole period annulus P. Therefore there exists an ε sufficiently small such that this interval is also a transversal section for system (3). We consider the Poincaré return map associated to this transversal section and if we denote by ϕ ε (θ; r 0 ) the solution of equation (9) with initial condition ϕ ε (0; r 0 ) = r 0 , the Poincaré return map is ϕ ε (2π; r 0 ).…”

“…The proof of this lemma is an extension "mutatis-mutandi" of the proof of Lemma 2.1 of [3] for systems with P and Q not coprime, see also [1]. We give here its proof for completeness.…”

“…Another contribution of this paper is that we give the explicit function whose simple zeros provide the periodic solutions of the quasihomogeneous center in (1) which for ε sufficiently small persist as limit cycles for system (3). This function is…”

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

“…Recently in [10] the polynomial and rational integrability of homogeneous polynomial differential systems has been characterized. In the present paper we give the characterization of polynomial or rational integrability for quasi-homogeneous polynomial differential systems.…”

“…For the sake of simplicity, we assume for the rest of the paper that system (1) is not linear, that is, n > 1. The linear case is included in the results of [10] where all the homogeneous systems were studied. Let N denote the set of positive integers.…”