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

On the number of limit cycles of the Liénard equations

Hector Giacomini & Sébastien Neukirch

Proceedings of the Third Catalan Days on Applied Mathematics, edited by J. Chavarriga and J. Giné, Institut d'Estudis Ilerdencs, Lleida, Spain (1996)

Abstract : We present preliminary results of work in progress about the number of limit cycles of the Li\'enard equations: \begin{eqnarray} \dot{x} & = & y-F(x) \nonumber \\ \dot{y} & = & -x \label{lie} \end{eqnarray} where $F(x)$ is an odd polynomial. We propose a function $f_n (x,y)=y^n+g_1(x) y^{n-1}+g_{n-2}(x) y^{n-2}+...+g_0(x)$, where $n$ is an even integer and $g_j(x)$ (with $0 \leq j \leq n-1$) are arbitrary functions of $x$. These functions $g_j(x)$ can be choosen in such a way that $\dot{f_n}=(y-F(x)) \frac{\partial f}{\partial x} - x \frac{\partial f}{\partial y}=R_n(x)$, where $R_n(x)$ is an even polynomial.

AMS classification :

Key words : limit cycle, polynomials