Classification of Phase Portraits of Z2- Equivariant Planar Hamiltonian Vector Field of Degree 7 (Ⅶ)

Yanmei Li

(School of Mathematics and Statistics , Chuxiong Normal University, Yunnan Chuxiong, 675000, China)

Classification of Phase Portraits ofZ2- Equivariant Planar Hamiltonian Vector Field of Degree 7 (Ⅶ)

Yanmei Li

(School of Mathematics and Statistics , Chuxiong Normal University, Yunnan Chuxiong, 675000, China)

In this paper, by the use of the method of qualitative analysis of differential equations, 25 phase portraits of a newZ2- equivariant planar Hamiltonian vector fields of degree 7 are obtained and the parameter space is classified.

Hamiltonian vector field of degree 7;Z2- equivariant property; singular point; phase portrait

In[1―7], the phase portraits of planar Hamiltonian vector fields of degree 7 withZq- equivariant property have been discussed, but there are still many vector fields deserving to be studied. In this paper, we will deliberate a new planar Hamiltonian vector fields as follows and get 25 phase portraits

(1)

wherekis a parameter with k>1.

1 Qualitative Analysis of the Singular Points

The Jacobian of this system is

in which

Discussing the Jacobians of these singular points, we get the following result:

Theorem 1 The singular points (0,0),(±1.2,0),(0,m),(±1,1),(±1.3,1),(±1.2,m),(±1,n)and (±1.3,n) are centers, and the others are saddle points.

2 Phase Portraits of the System (1)

The Hamiltonian of the system is

Itiseasytoget

H(±1,0)=-0.3892333, H(±1.2,0)=-0.385897,H(±1.3,0)=-0.3866327 ,

H(0,1)=-(10.8k2-5.6k+1)/24, H(0,m)=(2.6k4-8.8k3)/24 ,

H(0,n)=0.0486k4-0.324k3,

H(0,m)-H(0,1)=(k-1)3(2.6k-1)/24, H(0,m)-H(0,n)=0.512k3(1.4k-1)/12,

H(0,1)-H(0,n)=(1.8k-1)3(1-0.2k)/24,

andH(±1,0)

ComparingtheHamiltoniansofthesingularpoints,weobtainthefollowingresults.

Theorem2Thereexist25phaseportraitsofsystem(1)showninFig(1),andeveryoneofthemcorrespondstothevalueofkinthefollowingscopes: (1)1

(7)1.17555

(11)1.320463.38523.

ProofBecausethetrainofthoughtissimilar,weonlyprovethefirsttencases.

WeseparatelydenoteH(0,0),H(±1,0),H(±1.2,0),H(±1.3,0),H(0,1),H(0,m),H(0,n),H(±1,1),H(±1,m),H(±1,n),H(±1.2,1),H(±1.2,m),H(±1.2,n),H(±1.3,1),H(±1.3,m)andH(±1.3,n)byh00,h10,hb0,hc0,h01,h0m,h0n,h11,h1m,h1n,hb1,hbm,hbn,hc1,hcmandhcm.

Obviouslyhxy=hx0+h0y, h105, h01

(1)When1

h1n

andthephaseportraitisshownasFig.1(1).

(2)Whenk=1.12831,wehavehc0=h0n,andtheHamiltoniansofthesingularpointssatisfytheinequalities

对于苯、甲苯、环己烷和甲基环己烷等组分的定量分析，由于在色谱图中，苯和环己烷出峰的保留时间在n-C6和n-C7之间，甲苯和甲基环己烷在n-C7和n-C8之间出峰，对这几个组分的定量可采用式(6)计算。

h1n

sothephaseportraitisshownasFig.1(2).

(3)When1.12831

h1n

sothephaseportraitisshownasFig.1(3).

(4)Whenk=1.13101,wegeth10=h0n,andtheHamiltoniansofthesingularpointssatisfytheinequalities

h1n

sothephaseportraitisshownasFig.1(4).

(5)When1.13101

h1n

h1n

h1n

h1n

sothephaseportraitisshownasFig.1(5).

(6)Whenk=1.17555,weobtainh10=h01,andtheHamiltoniansofthesingularpointssatisfytheinequalities

h1n

sothephaseportraitisshownasFig.1(6).

(7)When1.17555

h1n

sothephaseportraitisshownasFig.1(7).

(8)Whenk=1.2025,weobtainhb1=hcm,andtheHamiltoniansofthesingularpointssatisfytheinequalities

h1n

sothephaseportraitofthesystem(1)isshownasFig.1(8).

(9)If1.2025

h1n

sothephaseportraitofthesystem(1)isshownasFig.1(9).

(10)Ifk=1.32046,theHamiltoniansofthesingularpointssatisfytheinequalities

h1n

sothephaseportraitofthesystem(1)isshownasFig.1(10).

Fig.1 The phase portraits of system (1)

[1]YanmeiLi,ZhaoHu.ClassificationofPhasePortraitsofZ2-EquivariantPlanarHamiltonianVectorFieldofDegree7(Ⅰ)[J].JournalofChuxiongNormalUniversity, 2012, 27(6):1-5.

[2]YanmeiLi.ClassificationofPhasePortraitsofZ2-EquivariantPlanarHamiltonianVectorFieldofDegree7(Ⅱ)[J].JournalofChuxiongNormalUniversity, 2012, 27(9):1-5.

[3]YanmeiLi.ClassificationofPhasePortraitsofZ2-EquivariantPlanarHamiltonianVectorFieldofDegree7(Ⅲ)[J].JournalofChuxiongNormalUniversity, 2013, 28(9):1-4.

[4]YanmeiLi.GlobalPhasePortraitsandClassificationofZ2-EquivariantPlanarHamiltonianVectorFieldsofDegree7withinfinitesingularpoints(Ⅰ)[J].JournalofChuxiongNormalUniversity, 2014, 29(3):1-4.

[5]YanmeiLi.ClassificationofPhasePortraitsofZ2-EquivariantPlanarHamiltonianVectorFieldofDegree7(Ⅳ) [J].JournalofChuxiongNormalUniversity, 2014, 29(9):1-5.

[6]YanmeiLi.ClassificationofPhasePortraitsofZ2-EquivariantPlanarHamiltonianVectorFieldofDegree7(Ⅴ) [J].JournalofChuxiongNormalUniversity, 2015, 30(6):1-4.

[7]YanmeiLi.ClassificationofPhasePortraitsofZ2-EquivariantPlanarHamiltonianVectorFieldofDegree7(Ⅵ) [J].JournalofChuxiongNormalUniversity, 2015, 30(9):1-4.

(责任编辑 司民真)

楚雄师范学院国家自然科学基金孵化项目“具有Z-q等变量性质的平面七次哈密顿向量场的相图分类研究”。

2017 - 03 - 25

李艳梅(1966―)，女，楚雄师范学院数学与统计学院教授，研究方向：非线性微分方程。

O175.29

A

1671 - 7406(2017)03 - 0001 - 04

具有Z2-等变性质的平面七次哈密顿向量场的相图分类(Ⅶ)

李艳梅

(楚雄师范学院数学与统计学院，云南 楚雄 675000)

根据微分方程定性理论,本文得到了一类新的具有Z2-等变性质的七次平面哈密顿向量场的25个相图，并对参数空间进行了划分。

七次哈密顿向量场；Z2-等变性质；奇点；相图