Centers for the Kukles homogeneous systems with odd degree

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2015Suggested citation
Giné, Jaume;
Llibre, Jaume;
Valls, Claudia;
.
(2015)
.
Centers for the Kukles homogeneous systems with odd degree.
Bulletin of the London Mathematical Society, 2015, vol. 47, p. 315-324.
https://doi.org/10.1112/blms/bdv005.
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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 center at the origin if
and only if its vector field is symmetric about one of the coordinate
axes? (2) Is it true that the origin is an isochronous center of the
previous system with the exception of the linear center only if the
system has even degree? We prove both conjectures for all n odd.
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Bulletin of the London Mathematical Society, 2015, vol. 47, p. 315-324European research projects
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