In this paper,the bifurcation problems of twisted and degenerated homoclinic loop for higher dimensional systems are studied.Under the nonresonant condition,the existence,uniqueness,and incoexistence of the 1-homoclin...In this paper,the bifurcation problems of twisted and degenerated homoclinic loop for higher dimensional systems are studied.Under the nonresonant condition,the existence,uniqueness,and incoexistence of the 1-homoclinic loop and 1-periodic orbit near Γ are obtained,and the inexistence of the 2-homoclinic loop and the existence of 2-periodic orbit near Γ are also given.展开更多
This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and...This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and small and large limit cycles are especially investigated.展开更多
In this paper, we make a complete study of the unfolding of a quadratic integrable system with a homoclinic loop. Making a Poincare transformation and using some new techniques to estimate the number of zeros of Abeli...In this paper, we make a complete study of the unfolding of a quadratic integrable system with a homoclinic loop. Making a Poincare transformation and using some new techniques to estimate the number of zeros of Abelian integrals, we obtain the complete bifurcation diagram and all phase portraits of systems corresponding to different regions in the parameter space. In particular, we prove that two is the maximal number of limit cycles bifurcating from the system under quadratic non- conservative perturbations.展开更多
This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients...In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients, and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients. Finally, we present some interesting applications.展开更多
A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurca...A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.展开更多
This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation.With the aid of numeric integral computation provided by Mathematica 4.1,at least 45 limit cycl...This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation.With the aid of numeric integral computation provided by Mathematica 4.1,at least 45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation,Hopf bifurcation and qualitative analysis.The four configurations of 45 limit cycles of the system are also shown.The results obtained are useful to the study of the weakened 16th Hilbert Problem.展开更多
In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polyn...In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polynomials.展开更多
In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving th...In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).展开更多
In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 h...In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.展开更多
This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie^Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bi...This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie^Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14（1） （2010）, 289-306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.展开更多
In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limi...In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.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 are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have ...In this paper, we are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop. Further, we give a condition under which there exist 4 limit cycles.展开更多
基金Supported by the National Natural Science Foundation of China(1 0 0 71 0 0 2 2 ) and the Shanghai PriorityAcademic Discipline
文摘In this paper,the bifurcation problems of twisted and degenerated homoclinic loop for higher dimensional systems are studied.Under the nonresonant condition,the existence,uniqueness,and incoexistence of the 1-homoclinic loop and 1-periodic orbit near Γ are obtained,and the inexistence of the 2-homoclinic loop and the existence of 2-periodic orbit near Γ are also given.
基金Project supported by the National Natural Science Foundation of China (No.10371072) the Ministry of Education of China (No.20010248019, No.20020248010).
文摘This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and small and large limit cycles are especially investigated.
基金Supported by the National Natural Science Foundation of China (10172011)
文摘In this paper, we make a complete study of the unfolding of a quadratic integrable system with a homoclinic loop. Making a Poincare transformation and using some new techniques to estimate the number of zeros of Abelian integrals, we obtain the complete bifurcation diagram and all phase portraits of systems corresponding to different regions in the parameter space. In particular, we prove that two is the maximal number of limit cycles bifurcating from the system under quadratic non- conservative perturbations.
基金Project supported by the National Natural Science Foundation of China (No.10371072)the New Century Excellent Ttdents in University (No.NCBT-04-038)the Shanghai Leading Academic Discipline (No.T0401).
文摘This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
基金the National Natural Science Foundation of China (10671127)
文摘In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients, and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients. Finally, we present some interesting applications.
基金The research is supported by fund of Youth of Jiangsu University(05JDG011)
文摘A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.
基金the fund of Youth of Jiangsu University(05JDG011)the National Nature Science Foundation of China(No:90610031)+1 种基金Outstanding Personnel Program in Six Fields of Jiangsu(No:6-A-029)Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of POE,China(No:2002-383).
文摘This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation.With the aid of numeric integral computation provided by Mathematica 4.1,at least 45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation,Hopf bifurcation and qualitative analysis.The four configurations of 45 limit cycles of the system are also shown.The results obtained are useful to the study of the weakened 16th Hilbert Problem.
文摘In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polynomials.
基金Surported by the Foundation of Shandong University of Technology (2006KJM01)
文摘In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).
基金Supported by Fund of Youth of Jiangsu University (Grant No. 05JDG011)National Natural Science Foundation of China (Grant No. 10771088)
文摘In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.
基金Supported by the National Natural Science Foundation of China(No.11101170)Research Project of the Central China Normal University(No.CCNU12A01007)the State Scholarship Fund of the China Scholarship Council(2011842509)
文摘This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie^Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14（1） （2010）, 289-306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.
基金Supported by the National Ministry of Education(No.20020248010)the National Natural Science Foundation of China(No.10371072)the Shanghai Leading Academic Discipline Project(No.T0401).
文摘In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.
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the National Natural Science Foundation of China under Grant (No.10671127)by Shanghai Shuguang Genzong Project(04SGG05)
文摘In this paper, we are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop. Further, we give a condition under which there exist 4 limit cycles.
