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

作　　者：Li-qin ZHAO Department of Mathematics, Beijing Normal University, Beijing 100875, China 

出　　处：《Science China Mathematics》2007年第3期334-344,共11页中国科学：数学（英文版）

基　　金：This work was supported by the National Natural Science Foundation of China(Grant No.10671020)

摘　　要：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.

关 键 词：degenerate polycycle CYCLICITY finitely-smooth normal form transition map 34C05 34C23 58F14 

分 类 号：O177[理学—数学]