Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems
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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 quasi-homogeneous polynomial differential system can be transformed into a differential system
of the form u˙=uf(v), v˙=g(v) with f(v) and g(v) polynomials, and vice versa.
Is part ofDiscrete and Continuous Dynamical Systems, 2013, vol. 33, num. 10, p. 4531-4547
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Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Universitat Autònoma de Barcelona, 2014)In this paper we find necessary and suficient 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 ...
Giacomini, Héctor; Giné, Jaume; Grau Montaña, Maite (Rocky Mountain Mathematics Consortium, 2006)In this work we consider rational ordinary differential equations dy/dx = Q(x, y)/P(x, y), with Q(x, y) and P(x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type for ...
Giné, Jaume; Llibre, Jaume; Valls, Claudia (Shanghai Normal University & Wilmington Scientific Publisher, 2017)For the polynomial differential system x˙=−y, y˙=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n≥2 has a ...