Nonlinear Analysis 73 (2010) 1318–1327
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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
Rational integrability of two-dimensional quasi-homogeneous
polynomial differential systems
A. Algaba, C. García, M. Reyes
∗
Department of Mathematics, Facultad de Ciencias Experimentales, Campus del Carmen, University of Huelva, Spain
article info
Article history:
Received 25 March 2008
Accepted 28 April 2010
Keywords:
Quasi-homogeneous vector field
Rational integrability
Kowalevskaya exponents
abstract
We characterize, in terms of the conservative–dissipative decomposition of a vector field,
the two-dimensional quasi-homogeneous polynomial differential systems which have a
rational first integral. We obtain the Kowalevskaya exponents of these vector fields and
relate the rational integrability of these fields to their Kowalevskaya exponents.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction and statement of main results
In this paper, we deal with polynomial differential systems
( ˙ x, ˙ y)
T
= F
r
= (P , Q )
T
, (1)
where F
r
is a polynomial vector field of degree r ≥ 0 with respect to the type t = (t
1
, t
2
) fixed. In the particular case that
t = (1, 1), (1) is a homogeneous polynomial differential system of degree r + 1. The purpose of our approach is to know
when (1) is a rationally integrable system.
We recall that a function of two variables f is a quasi-homogeneous function of degree k ∈ Z with respect to the type
t = (t
1
, t
2
) if f (ε
t
1
x,ε
t
2
y) = ε
k
f (x, y) (it will be called t-function of degree deg
t
(f )). We will denote P
t
k
the vector space of
t-polynomials of degree k ≥ 0. A two-dimensional vector field F = (P , Q )
T
is quasi-homogeneous of degree k with respect
to the type t if P ∈ P
t
k+t
1
and Q ∈ P
t
k+t
2
(it will be called t-vector field of degree k). The vector space of polynomial t-vector
fields of degree k will be denoted by Q
t
k
.
A function H is a first integral of system (1) in an open subset U of R
2
if H is a nonconstant function in U which is constant
on each solution curve of system (1). If there exists a rational first integral of (1) it is said that (1) is rationally integrable.
Clearly, if H =
f
g
, with f , g polynomials, is a first integral of system (1) then the Lie derivative of H by F
r
is zero in the open
subset Ω
g
={(x, y) ∈ R
2
: g (x, y) = 0}, i.e. L
F
r
H :=
∂ H
∂ x
P +
∂ H
∂ y
Q ≡ 0 in Ω
g
.
The integrability problem consists in determining if the planar vector field has a first integral. In a general framework,
the integrability is an important question because the existence of a first integral determines completely its phase portrait.
It is known that we can always calculate a first integral explicitly. It is enough to make the change of variables (x, y) →
(u,v) according to x = v
t
1
, y = v
t
2
u which transforms the differential equation
dy
dx
=
Q (x,y)
P (x,y)
into a linear equation
v
t
1
du
dv
+
t
2
t
1
u =
Q (1,u)
P (1,u)
easy to integrate. But this first integral usually has a huge algebraic expression and, therefore, it is
difficult to show whether it is rational or not.
∗
Corresponding author.
E-mail address: colume@uhu.es (M. Reyes).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2010.04.059