In this paper we investigate the perturbations from a kind of quartic Hamiltonians under general cubic polynomials. It is proved that the number of isolated zeros of the related abelian integrals around only one center is not more than 12 except the case of global center. It is also proved that there exists a cubic polynomial such that the disturbed vector field has at least 3 limit cycles while the corresponding vector field without perturbations belongs to the saddle loop case.
In this paper, we investigate the number and the distribution of the limit cycles bifurcated from a kind of degenerate planar polycycles through three singular points: a saddle-node P0, a fine saddle P1 with finite order m∈N, a contractive (attracting) saddle P2 with the hyperbolicity ratio q2(0)■Q. The connection between P0 and P1 is of hh-type and the connection between P0 and P2 is of hp-type. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. We obtain that the cyclicity of this polycycle is not more than 3m+1, which is linearly dependent on the order of the resonant saddle P1 We also show that the cyclicity is not more than m+3 if q2(0)>m, and that the nearer q2(0)is close to 1, the more the limit cycles are bifurcated.
Li-qin ZHAO Department of Mathematics, Beijing Normal University, Beijing 100875, China
