Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles

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 P 0, a fine saddle P 1 with finite order m ∈ N, a contractive (attracting) saddle P 2 with the hyperbolicity ratio q 2(0) ? Q. The connection between P 0 and P 1 is of hh-type and the connection between P 0 and P 2 is of hp-type. It is assumed that the connections between P 0 to P 2 and P 0 to P 1 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 P 1. We also show that the cyclicity is not more than m + 3 if q 2(0) > m, and that the nearer q 2(0) is close to 1, the more the limit cycles are bifurcated.

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.