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

Abstract

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard 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 Liénard equation is of degree 2 l. 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.