Sébastien Neukirch Institut Jean le Rond d'Alembert Centre National de la Recherche Scientifique Université Pierre et Marie Curie Paris, France tel: +33 1 44 27 72 61 fax : +33 1 44 27 52 59 e-mail: sebastien.neukirch (-atat-) upmc.fr

Comment on Liénard systems, limit cycles, and Melnikov theory'

Hector Giacomini & Sébastien Neukirch

Physical Review E 59 #2 (1999) 2483

Abstract : In [1] and [2] Li\'enard systems of the form~: $\dot{x} = y-\epsilon F(x,\mu)$ , $\dot{y} = -x$ are studied. In [1] the author compares the results given by Melnikov theory with the results given by the $R_n$ polynomials [2] and conjectures that the roots of the $R_n$ polynomials tend towards the roots of the Melnikov polynomial when $n \to \infty$, for arbitrary values of $\epsilon$. We show here that this is true only when $\epsilon \to 0$ and that this fact strenghten the conjecture proposed in [2].

PACS numbers : 05.45.+b , 02.30.Hq , 02.60.Lj , 03.20.+i

Key words : Liénard equation, limit cycles

[1] M. A. F. Sanjuán, Liénard systems, limit cycles and Melnikov theory'', Physical Review E 57 340 (1998).

[2] H. Giacomini and S. Neukirch, `The number of limit cycles of the Liénard equation'', Physical Review E 56 3809 (1997).