Classification of Phase Portraits of Z2－Equivariant Planar Hamiltonian Vector Fields of Degree 7(Ⅱ)*

LI Yanmei

(Department of Mathematics，Chuxiong Normal University，Yunnan Chuxiong，675000，China)

The phase portraits of planar Zq－equivariant Hamiltonian vector fields of degree 7 are much more complicated than that of degree 5，and only a few vector fields of degree 7 have been discussed［1—5］.In this paper，we will classify the phase portraits of a new planar Z2－ equivariant Hamiltonian vector field of degree 7

where k is a parameter with k＞0.

1 Qualitative Analysis of the Singular Points

Because the system(1)has the property of Z2－ equivariant，namely，the phase portrait is the same when it rotates π clockwise，so we only need to study the singular points in the first and second quadrants.

The Jacobian of this system is

in which

Discussing the Jacobians of these singular points，we can obtain the following results without difficulty:

Obviously，the function H(x，y)satisfies the equality H(x，y)=H(x，0)+H(0，y)，and we can readily obtain

2 Phase Portraits of the System(1)

The Hamiltonian of the system(1)is

H(x，y)= ［3x8－ (k2+2k+2.36)x6+(0.51k2+1.02k+0.645)x4－ 0.0675(k+1)2x2

Comparing the Hamiltonians of the singular points，we get the following results.

Theorem 2

(1)If0＜k＜0.239066，the phase portrait of the system(1)can be shown as Fig.1(1).

(2)If k=0.239066，the phase portrait of the system(1)can be shown as Fig.1(2).

(3)If 0.239066＜k＜0.255214，the phase portrait of the system(1)can be shown as Fig.1(3).

(4)If k=0.255214，the phase portrait of the system(1)can be shown as Fig.1(4).

(5)If 0.255214＜k＜0.2806248，the phase portrait of the system(1)can be shown as Fig.1(5).

(6)If k=0.2806248，the phase portrait of the system(1)can be shown as Fig.1(6).

(7)If 0.2806248＜k＜0.28217，the phase portrait of the system(1)can be shown as Fig.1(7).

(8)If k=0.28217，the phase portrait of the system(1)can be shown as Fig.1(8).

(9)If 0.28217＜k＜0.282535，the phase portrait of the system(1)can be shown as Fig.1(9).

(10)If k=0.282535，the phase portrait of the system(1)can be shown as Fig.1(10).

(11)If k ＞0.282535，the phase portrait of the system(1)can be shown as Fig.1(11).

Proof

We denote H(0，0)，H(± a，0)，H(± b，0)，H(± c，0)，H(0，l)，H(0，m)，H(0，n)，H(± a，l)，H(± a，m)，H(± a，n)，H(± b，l)，H(± b，m)，H(± b，n)，H(± c，l)，H(± c，m)，and H(± c，n)by h00，ha0，hb0，hc0，h0l，h0m，h0n，hal，ham，han，hbl，hbm，hbn，hcl，hcmand hcm，respectively.Obviously，we have hxy=hx0+h0y，h0l=h0n，h0m=0.

(1)If k=0.229005，then hcl=ha0.Hence，when 0 ＜ k ＜ 0.239066，the Hamiltonians of the singular points satisfy the relations hal=han＜ha0=ham≤hcl=hcn＜hc0=hcm＜hbl=hbn＜hb0=hbm＜h0l=h0n＜h00=h0m，so the phase portrait can be shown as Fig.1(1).

(2)When k=0.239066，we have hcm=hbl，and the Hamiltonians of the singular points satisfy the relations hal=han＜hcl=hcn＜ha0=ham＜hc0=hcm=hbl=hbn＜hb0=hbm＜h0l=h0n＜h00=h0m，so the phase portrait can be shown as Fig.1(2).

(3)When 0.239066＜k＜0.255214，the Hamiltonians of the singular points satisfy the relations hal=han＜hcl=hcn＜ha0=ham＜hbl=hbn＜hc0=hcm＜hb0=hbm≤h0l=h0n＜h00=h0m，so the phase portrait can be shown as Fig.1(3).

(4)When k=0.255214，we get ha0=hbl，and the Hamiltonians of the singular points satisfy the relations hal=han＜hcl=hcn＜ha0=ham=hbl=hbn＜hc0=hcm＜h0l=h0n＜hb0=hbm＜h00=h0m，so the phase portrait can be shown as Fig.1(4).

(5)When0.255214＜k＜0.2806248，the Hamiltonians of the singular points satisfy the relations hal=han＜hcl=hcn＜hbl=hbn＜ha0=ham＜hc0=hcm＜h0l=h0n＜hb0=hbm＜h00=h0m，so the phase portrait can be shown as Fig.1(5).

(6)When k=0.2806248，we obtain aa0=hc0，and the Hamiltonians of the singular points satisfy the relations hal=han=hcl=hcn＜hbl=hbn＜ha0=ham=hc0=hcm＜h0l=h0n＜hb0=hbm＜h00=h0m，so the phase portrait can be shown as Fig.1(6).

(7)When 0.2806248＜k＜0.28217，the Hamiltonians of the singular points satisfy the relations hcl=hcn＜hal=han＜hbl=hbn＜hc0=hcm＜ha0=ham＜h0l=h0n＜hb0=hbm＜h00=h0m，so the phase portrait can be shown as Fig.1(7).

(8)When k=0.28217，we obtain ha0=hol，and the Hamiltonians of the singular points satisfy the relations hcl=hcn＜hal=han＜hbl=hbn＜hc0=hcm＜ha0=ham=h0l=h0n＜hb0=hbm＜h00=h0m，so the phase portrait of the system(1)can be shown as Fig.1(8).

(9)If 0.28217＜k＜0.282535，the Hamiltonians of the singular points satisfy the relations hcl=hcn＜hal=han＜hbl=hbn＜hc0=hcm＜h0l=h0n＜ha0=ham＜hb0=hbm＜h00=h0m，so the phase portrait of the system(1)can be shown as Fig.1(9).

(10)If k=0.282535，we get h0n=hc0，and the Hamiltonians of the singular points satisfy the relations hcl=hcn＜hal=han＜hbl=hbn＜hc0=hcm=h0l=h0n＜ha0=ham＜hb0=hbm＜h00=h0m，so the phase portrait of the system(1)can be shown as Fig.1(10).

(11)If k ＞0.282535，the Hamiltonians of the singular points satisfy the relations hcl=hcn＜hal=han＜hbl=hbn＜h0l=h0n＜hc0=hcm＜ha0=ham＜hb0=hbm＜h00=h0m，so the phase portrait of the system(1)can be shown as Fig.1(11).

Fig.1(1)～(11)The phase portraits of system(1)

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