Electron. J. Differential Equations,
Vol. 2018 (2018), No. 184, pp. 123.
Center problem for generalized lambdaomega differential systems
Jaume Llibre, Rafael Ramirez, Valentin Ramirez
Abstract:

differential systems are the real planar
polynomial differential equations of degree m of the form
where
and
are polynomials of degree at most m1 such that
.
A planar vector field with linear type center can be written as a

system if and only if the PoincareLiapunov
first integral is of the form
.
The main objective of this article is to study the center problem for

systems of degree m with
,
and
,
where
are constants and
is a homogenous polynomial of degree
,
for .
We prove the following results. Assuming that
and
the

system has a weak center at the origin
if and only if these systems after a linear change of variables
are invariant under the transformations
.
If
and
then the origin is a weak center. We observe that the main difficulty
in proving this result for m>6 is related to the huge computations.
Submitted July 9, 2018. Published November 14, 2018.
Math Subject Classifications: 34C05, 34C07.
Key Words: Linear type center; Darboux first integral; weak center;
PoincareLiapunov theorem; Reeb integrating factor.
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Jaume Llibre
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra,
Barcelona, Catalonia, Spain
email: jllibre@mat.uab.cat


Rafael Ramírez
Departament d'Enginyeria Informática i Matemátiques
Universitat Rovira i Virgili
Avinguda dels Pa&imul;sos Catalans 26
43007 Tarragona, Catalonia, Spain
email: rafaelorlando.ramirez@urv.cat


Valentín Ramírez
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra
Barcelona, Catalonia, Spain
email: valentin.ramirez@ecampus.uab.cat

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