Polynomial Lienard equations near infinity

Abstract

We study the behaviour near infinity of the (generalized) Lienard equations (x) over dot = y, (y) over dot = -Sigma(k=0)(m) a(k)x(k) - y Sigma(k=0)(n) b(k)x(k) providing a complete classification using Poincare compactification as well as Poincare-Lyapunov compactification. We show that all the necessary information is contained in a(m) and b(n), except for the so called "center-focus" problem occuring in case m greater than or equal to 2n + 1 with m, n odd, and where the behaviour also depends on the value of the other coefficients a(i) and b(j). We also indicate how to take care about obtaining uniform information near infinity. (C) 1999 Academic Press.