On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 6, which is a Liénard system of the form x′=y,y′=Q1(x)+εyQ2(x) with Q1Q1 and Q2Q2 polynomials of degree 5 and 4, respectively. It is shown that this system can undergo degenerate Hopf bifurcation and Poincaré bifurcation, from which at most three limit cycles can emerge in the plane for εε sufficiently small. And the limit cycles can only surround an equilibrium inside, i.e. the system can have the configuration (3,0,0)(3,0,0) of limit cycles for some values of the parameters, where (3,0,0)(3,0,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding any of the two or three equilibria.