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The center problem for Z_2-symmetric nilpotent vector fields

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Issue date
2018
Author
Algaba, Antonio
García, Cristóbal
Giné, Jaume
Llibre, Jaume
Suggested 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.
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Abstract
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.
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http://hdl.handle.net/10459.1/72799
DOI
https://doi.org/10.1016/j.jmaa.2018.05.079
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Journal of Mathematical Analysis and Applications, 2018, vol. 466, núm. 1, p. 183-198
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