It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions f...It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.展开更多
Using the methods of dynamical systems for two generalized Boussinesq systems, the existence of all possible solitary wave solutions and many uncountably infinite periodic wave solutions is obtained. Exact explicit pa...Using the methods of dynamical systems for two generalized Boussinesq systems, the existence of all possible solitary wave solutions and many uncountably infinite periodic wave solutions is obtained. Exact explicit parametric representations of the travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.展开更多
This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations o...This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation. It can be concluded that H(5) ? 25 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert problem.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
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671179)the Natural Science Foundation of Yunnan Province (Grant No. 2005A0013M)
文摘It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10671179, 10772158)
文摘Using the methods of dynamical systems for two generalized Boussinesq systems, the existence of all possible solitary wave solutions and many uncountably infinite periodic wave solutions is obtained. Exact explicit parametric representations of the travelling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.
基金Supported by the Fund of Youth of Jiangsu University(Grant No.05JDG011)the National Natural Science Foundation of China(Nos.90610031,10671127)+1 种基金the Outstanding Personnel Program in Six Fields of Jiangsu Province(Grant No.6-A-029)Shanghai Shuguang Genzong Project(Grant No.04SGG05)
文摘This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation. It can be concluded that H(5) ? 25 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert problem.
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.
