Integrability of Liénard systems with a weak saddle

Gasull i Embid, Armengol; Giné, Jaume

doi:https://doi.org/10.1007/s00033-016-0756-6

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Issue date

2017

Author

Gasull i Embid, Armengol

Giné, Jaume

Suggested citation

Gasull i Embid, Armengol; Giné, Jaume; . (2017) . Integrability of Liénard systems with a weak saddle. Zeitschrift für angewandte Mathematik und Physik, 2017, vol. 68, núm. 13, p 1-13. https://doi.org/10.1007/s00033-016-0756-6.

Abstract

We characterize the local analytic integrability of weak saddles for complex Lienard systems, x˙ = y−F(x), y˙ = ax, 0 = a ∈ C, with F analytic at 0 and F(0) = F (0) = 0. We prove that they are locally integrable at the origin if and only if F(x) is an even function. This result implies the well-known

characterization of the centers for real Lienard systems. Our proof is based on finding the obstructions for the existence of a formal integral at the complex saddle, by computing the so-called resonant saddle quantities

URI

http://hdl.handle.net/10459.1/59059

DOI

https://doi.org/10.1007/s00033-016-0756-6

Is part of

Zeitschrift für angewandte Mathematik und Physik, 2017, vol. 68, núm. 13, p 1-13