Integrability of Liénard systems with a weak saddle

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

The Armengol Gasull was supported by a MINECO Grant Number MTM2013-40998-P and by a CIRIT Grant Number 2014SGR568. The Jaume Gin´e was partially supported by a MINECO/ FEDER Grant Number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) Grant Number 2014SGR 1204.