Integrability of Liénard systems with a weak saddle

Abstract

We characterize the local analytic integrability of weak saddles for complex Liénard systems.

On the integrability of Liénard systems with a strong saddle

“…The result obtained in [8] is the following: Theorem 2. Consider the analytic differential systems in C 2 of the form (1)ẋ = y + F (x),ẏ = ax, with 0 ̸ = a ∈ C and where F (x) is an analytic function without linear and constant terms.…”

“…For more information on the second mentioned problem, see [1]. Liénard differential systems appeared in electrical circuits with nonlinear elements (see [18,19]), but later many other situations have been modeled by these type of differential equations, see [8,14,15] and references therein.…”

“…In [8] Theorem 2 is generalized to get the characterization of the real analytic integrable weak saddles for general Liénard systems in R 2 , that is, for systems of the form…”

“…Moreover in [8], using Lüroth theorem as in [4], an effective characterization of real integrable polynomial Liénard systems with a weak saddle is obtained. In [12] some other integrable Liénard-like complex systems with a weak saddle are also characterized.…”

On a conjecture on the integrability of Liénard systems

“…When a planar differential system has a (local) first integral we say that it is (locally) integrable. In [4] the authors left open the following problem (see the last sentence of their paper):…”