The center problem for Z_2-symmetric nilpotent vector fields

View/ Open
Issue date
2018Suggested citation
Algaba, Antonio;
García, Cristóbal;
Giné, Jaume;
Llibre, Jaume;
.
(2018)
.
The center problem for Z_2-symmetric nilpotent vector fields.
Journal of Mathematical Analysis and Applications, 2018, vol. 466, núm. 1, p. 183-198.
https://doi.org/10.1016/j.jmaa.2018.05.079.
Metadata
Show full item recordAbstract
We say that a polynomial differential system ˙x = P(x, y), ˙y = Q(x, y) having the
origin as a singular point is Z2-symmetric if P(−x, −y) = −P(x, y) and Q(−x, −y) =
−Q(x, y).
It is known that there are nilpotent centers having a local analytic first integral, and
others which only have a C∞ first integral. But up to know there are no characterized
these two kinks of nilpotent centers.
Here we prove that the origin of any Z2-symmetric is a nilpotent center if, and
only if, there is a local analytic first integral of the form H(x, y) = y
2 + · · ·, where the
dots denote terms of degree higher than two.