Hilbert's 16th problem for classical Lienard 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)double over dot> + f(x)<(x) over dot> + x = 0. In this paper, we consider f to be a polynomial of degree 2l - 1, with I 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 2l - 1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l. (c) 2007 Elsevier Inc. All rights reserved.