Cyclicity of a kind of degenerate polycycles through three singular points  

作　　者：ZHAO Liqin 

机构地区：[1]Department of Mathematics, Beijing Normal University, Beijing 100875, China

出　　处：《Science China Mathematics》2005年第6期746-756,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.19901001).

摘　　要：This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node P0 and two hyperbolic saddles P1 and P2, where the hyperbolicity ratio of the saddle P1 (which connects the saddle-node with hh-connection) is equal to 1 and that of the other saddle P2 is irrational. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. Then the cyclicity of this kind of polycycle is no more than m + 3 if the saddle P1 is of order m and the hyperbolicity ratio of P2 is bigger than m.Furthermore, the cyclicity of this polycycle is no more than 7 if the saddle P1 is of order 2 and the hyperbolicity ratio of P2 is located in the interval (1, 2).This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node P0 and two hyperbolic saddles P1 and P2, where the hyperbolicity ratio of the saddle P1 (which connects the saddle-node with hh-connection) is equal to 1 and that of the other saddle P2 is irrational. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. Then the cyclicity of this kind of polycycle is no more than m + 3 if the saddle P1 is of order m and the hyperbolicity ratio of P2 is bigger than m. Furthermore, the cyclicity of this polycycle is no more than 7 if the saddle P1 is of order 2 and the hyperbolicity ratio of P2 is located in the interval (1,2).

关 键 词：DEGENERATE polycycle  cyclicity  finitely-smooth NORMAL form  transition map. 

分 类 号：N[自然科学总论]