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

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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 ′ = Q 1 ( x ) + ε y Q 2 ( x ) with Q 1 and Q 2 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 ε sufficiently small. And the limit cycles can only surround an equilibrium inside, i.e. the system can have the configuration ( 3 , 0 , 0 ) of limit cycles for some values of the parameters, where ( 3 , 0 , 0 ) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding any of the two or three equilibria.