Abstract

This talk deals with current research work in collaboration with F. Dumortier. It deals with Hilbert′s 16th problem for classical Liénard equations, which are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations $x\prime \prime +f(x)x\prime +x=0$. In this talk, we consider $f$ to be a polynomial of degree $2l-1$, with $l$ a fixed but arbitrary natural number. The related Liénard equation is of degree $2l$. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict $f$ to some compact set of polynomials of degree exactly $2l-1$. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most $l+1$.