一类扰动的超椭圆Hamilton系统的极限环分布情况
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摘要
在定性分析理论指导下,运用判定函数和数值探测方法,研究了一类具有幂零鞍点的超椭圆Hamilton系统在多项式扰动下的极限环个数和分布问题,这里的多项式扰动共有3个任意参数.证明了该系统在无界周期环域中最多分出3个极限环,并运用数值模拟得到了3个极限环的准确位置.该研究成果有助于进一步研究希尔伯特的第16个问题.
Under the guidance of qualitative analysis theory,using the detection function and numerical exploration method,we study the limit cycles number and distribution problem of a class of hyper-elliptic Hamiltonian systems with power zero saddle points under polynomial perturbation.Here,the polynomial perturbation has three arbitrary parameters.This paper proves that the system can divide up to three limit cycles in the unbounded periodic ring domain,and the exact position of the three limit cycles by numerical simulation is obtained.The research results can further study Hilbert′s 16 th problem.
Under the guidance of qualitative analysis theory,using the detection function and numerical exploration method,we study the limit cycles number and distribution problem of a class of hyper-elliptic Hamiltonian systems with power zero saddle points under polynomial perturbation.Here,the polynomial perturbation has three arbitrary parameters.This paper proves that the system can divide up to three limit cycles in the unbounded periodic ring domain,and the exact position of the three limit cycles by numerical simulation is obtained.The research results can further study Hilbert′s 16 th problem.
引文
[1] DUMORTIER F,LI C.Perturbations from an elliptic Hamiltonian of degree four[J].J Differential Equations,2001,176:114-157.
[2] DUMORTIER F,LI C.Perturbations from an elliptic Hamiltonian of degree four[J].J Differential Equations,2001,175:209-243.
[3] DUMORTIER F,LI C.Perturbations from an elliptic Hamiltonian of degree four[J].J Differential Equations,2003,188:512-554.
[4] QI M,ZHAO L.Bifurcations of limit cycles from a quintic Hamiltonian system with a figure double-fish[J].Internat J Bifur Chaos Appl Sci Engrg,2013,23:116-124.
[5] ASHEGHI R,ZANGENEH H R Z.Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop[J].Nonlinear Anal,2008,68:2957-2976.
[6] ASHEGHI R,ZANGENEH H R Z.Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop[J].Nonlinear Anal,2008,69:4143-4162.
[7] ASHEGHI R,ZANGENEH H R Z.Bifurcations of limit cycles from quintic Hamiltonian system with a double cuspidal loop[J].Comput Math Appl,2010,59:1409-1418.
[8] ZHAO L.The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle[J].Nonlinear Anal,2014,95:374-387.
[9] KAZEMI R,ZANGENEH H R Z,ATABAIGI A.On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems[J].Nonlinear Anal,2012,75:574-587.
[10] WANG J,XIAO D.On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle[J].J Differential Equations,2011,250:2227-2243.
[11] LI J B,LI C F.Distribution of limit cycles for planar cubic Hamiltonian systems[J].Acta Math Sinica,1985,28:509-521.
[12] 王颜超,邓亚莉.Hamilton体系的辛算法[J].赤峰学院学报(自然科学版),2013(12):6-8.
[13] 张申贵.一类超二次Hamilton系统的无穷多周期解[J].西北民族大学学报(自然科学版),2008,29(4):4-10.
[14] 洪晓春,谢绍龙.一类三次Hamilton系统的相图[J].西南民族大学学报(自然科学版),2004,30(2):127-130.
[2] DUMORTIER F,LI C.Perturbations from an elliptic Hamiltonian of degree four[J].J Differential Equations,2001,175:209-243.
[3] DUMORTIER F,LI C.Perturbations from an elliptic Hamiltonian of degree four[J].J Differential Equations,2003,188:512-554.
[4] QI M,ZHAO L.Bifurcations of limit cycles from a quintic Hamiltonian system with a figure double-fish[J].Internat J Bifur Chaos Appl Sci Engrg,2013,23:116-124.
[5] ASHEGHI R,ZANGENEH H R Z.Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop[J].Nonlinear Anal,2008,68:2957-2976.
[6] ASHEGHI R,ZANGENEH H R Z.Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop[J].Nonlinear Anal,2008,69:4143-4162.
[7] ASHEGHI R,ZANGENEH H R Z.Bifurcations of limit cycles from quintic Hamiltonian system with a double cuspidal loop[J].Comput Math Appl,2010,59:1409-1418.
[8] ZHAO L.The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle[J].Nonlinear Anal,2014,95:374-387.
[9] KAZEMI R,ZANGENEH H R Z,ATABAIGI A.On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems[J].Nonlinear Anal,2012,75:574-587.
[10] WANG J,XIAO D.On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle[J].J Differential Equations,2011,250:2227-2243.
[11] LI J B,LI C F.Distribution of limit cycles for planar cubic Hamiltonian systems[J].Acta Math Sinica,1985,28:509-521.
[12] 王颜超,邓亚莉.Hamilton体系的辛算法[J].赤峰学院学报(自然科学版),2013(12):6-8.
[13] 张申贵.一类超二次Hamilton系统的无穷多周期解[J].西北民族大学学报(自然科学版),2008,29(4):4-10.
[14] 洪晓春,谢绍龙.一类三次Hamilton系统的相图[J].西南民族大学学报(自然科学版),2004,30(2):127-130.