Hilbert's 16th problem for classical Liénard equations of even degree

Abstract Classical Lienard equations are two-dimensional vector fields, on the phase plane or on the Lienard plane, related to scalar differential equations x ¨ + f ( x ) x ˙ + x = 0 . In this paper, we consider f to be a polynomial of degree 2 l − 1 , with l a fixed but arbitrary natural number. The related Lienard 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 2 l − 1 . The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.