Perturbation from an elliptic Hamiltonian of degree four - III global centre

Abstract

The paper deals with Lienard equations of the form <(x)over dot> = y, (y) over circle = P(x) + yQ(x) with P and Q polynomials of degree, respectively, 3 and 2. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree four, exhibiting a global centre. It is proven that the least upper bound of the number of zeros of the related elliptic integral is four, and this is a sharp one. This result permits to prove the existence of Lienard equations of type (3,2) with a quadruple limit cycle, with both a triple and a simple limit cycle, with two semistable limit cycles, with one semistable and two simple limit cycles or with four simple limit cycles. (C) 2002 Elsevier Science (USA). All rights reserved.