BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A CLASS OF HYPER-ELLIPTIC HAMILTONIAN SYSTEMS OF DEGREE 5 WITH A CUSP

Abstract

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li e nard system of the form \(\dot{x}=y,\) \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar e bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.