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

Algebraic approximations to bifurcation curves of limit cycles for the Liénard equation

Hector Giacomini & Sébastien Neukirch

Physics Letters A 244 (1998) 53-58

Abstract : In this paper, we study the bifurcation of limit cycles in Li\'enard systems of the form $\frac{d x}{d t}=y-F(x) , \frac{d y}{d t}=-x$ , where $F(x)$ is an odd polynomial that contains, in general, several free parameters. By using a method introduced in a previous paper, we obtain a sequence of algebraic approximations to the bifurcation sets, in the parameter space. Each algebraic approximation represents an exact lower bound to the bifurcation set. This sequence seems to converge to the exact bifurcation set of the system. The method is non perturbative. It is not necessary to have a small or a large parameter in order to obtain these results.

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

Key words : Bifurcation, Liénard equation, limit cycles.