The conjecture E(k)≤k is proved to be true if and only if k=1, 2, 3, where E(k) is the cyclicity of condimension k generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles ex...The conjecture E(k)≤k is proved to be true if and only if k=1, 2, 3, where E(k) is the cyclicity of condimension k generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles except ensembles with "lips" is ≤6. By the way, the methods usually used in the study of cyclicity of polycycles such as derivation division algorithm, Khovanskii procedure and the method of critical point analysis are introduced.展开更多
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-no...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).展开更多
This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node and two hyperbolic saddles, where the hyperbolicity ratio of the saddle （which connects the saddle-node with hp-conn...This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node and two hyperbolic saddles, where the hyperbolicity ratio of the saddle （which connects the saddle-node with hp-connection） is equal to 1 and that of the other saddle is irrational. It is obtained that the cyclicity of this kind of polycycle is no more than 5 if the hp-connection keeps unbroken under the C^∞ perturbations.展开更多
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 ...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.展开更多
The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-s...The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.展开更多
By means of finitely-smooth normal form theory and the method of infinitesimal analysis,it is proved that the cyclicity of a planar codimension 3 polycycle S(2) is three and the complete bifurcation diagram is provided.
文摘The conjecture E(k)≤k is proved to be true if and only if k=1, 2, 3, where E(k) is the cyclicity of condimension k generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles except ensembles with "lips" is ≤6. By the way, the methods usually used in the study of cyclicity of polycycles such as derivation division algorithm, Khovanskii procedure and the method of critical point analysis are introduced.
基金This work was supported by the National Natural Science Foundation of China(Grant No.19901001).
文摘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).
基金Project sponsored by National Science Foundation (19901001)
文摘This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node and two hyperbolic saddles, where the hyperbolicity ratio of the saddle （which connects the saddle-node with hp-connection） is equal to 1 and that of the other saddle is irrational. It is obtained that the cyclicity of this kind of polycycle is no more than 5 if the hp-connection keeps unbroken under the C^∞ perturbations.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10671020)
文摘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.
文摘The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.
基金Project supported by the National Natural Science Foundation of China and the DEPF of China.
文摘By means of finitely-smooth normal form theory and the method of infinitesimal analysis,it is proved that the cyclicity of a planar codimension 3 polycycle S(2) is three and the complete bifurcation diagram is provided.
