Analytic integrability of a class of planar polynomial differential systems

Jaume Llibre; Claudia  Valls

doi:10.3934/dcdsb.2015.20.2657

Article Contents

2015, Volume 20, Issue 8: 2657-2661. Doi: 10.3934/dcdsb.2015.20.2657

This issue Previous Article Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system Next Article Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays

Analytic integrability of a class of planar polynomial differential systems

Jaume Llibre^1, and
Claudia Valls^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
Departamento de Matemática, Instituto Superior Técnico , Universidade Técnica de Lisboa, Av. Rovisco Pais 1049-001, Lisboa

Received: September 30, 2014

Revised: December 31, 2014

Published: September 2015

Abstract Related Papers Cited by

Abstract

In this paper we find necessary and sufficient conditions in order that the differential systems of the form $\dot x = x f(y)$, $\dot y =g(y)$, with $f$ and $g$ polynomials, have a first integral which is analytic in the variable $x$ and meromorphic in the variable $y$. We also characterize their analytic first integrals in both variables $x$ and $y$.
These polynomial differential systems are important because after a convenient change of variables they contain all quasi--homogeneous polynomial differential systems in $\mathbb{R}^2$.

Keywords:

Planar polynomial systems,
quasi--homogeneous polynomial differential systems,
analytic first integrals,
pseudo--meromorphic first integrals.

Mathematics Subject Classification: Primary: 34A34, 34C14.

Citation:

Related Papers

Cited by

References

[1]	J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete and Continuous Dynamical Systems, Series A, 33 (2013), 4531-4547.doi: 10.3934/dcds.2013.33.4531.
[2]	E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover Publications, Inc., New York, 1994.
[3]	J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.doi: 10.1088/0951-7715/15/4/313.