Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 762762, 18 pages
doi:10.1155/2011/762762
Research Article
Sufficient and Necessary Center Conditions for
the Poincaré Systems P 2, 2n n ≤ 5
Jinya Xu1 and Zhengyi Lu1, 2
1
2
Institute of Computer Applications, Academia Sinica, Chengdu 610041, China
Department of Mathematics, Sichuan Normal University, Chengdu 610068, China
Correspondence should be addressed to Zhengyi Lu, zhengyilu@hotmail.com
Received 22 August 2011; Revised 27 October 2011; Accepted 28 October 2011
Academic Editor: Mohamad Alwash
Copyright q 2011 J. Xu and Z. Lu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We obtain sufficient and necessary center conditions for the Poincaré system P 2, 2n n ≤ 5. The
necessity of the condition is derived from the first 2n focal values by symbolic computation with
Maple, and the sufficiency is proved by Volokitin’s method.
1. Introduction
Research on Hilbert’s sixteenth problem 1, 2 in general usually proceeds by the
investigation on specific classes of polynomial systems. Much effort has been devoted in
recent years to the investigation of various systems with cubic or quintic polynomials for
the center problem 3–9. We are interested in a certain family of polynomial systems of the
form
ẋ −y
xR x, y ,
ẏ x
yR x, y ,
1.1
n
with ẋ dx/dt, ẏ dy/dt, and Rx, y i1 Ri x, y, where Ri is a homogeneous
polynomial of degree i. The centers of these systems are called uniformly isochronous centers
10. The case in which Rx, y is of degree i has been investigated in 10, 11.
In the nonhomogeneous case 12, the pioneering studies mainly focus on the systems
0
with Rx, y R1 R2 13, Rx, y R1 R3 14, Rx, y R1 R2 R3 R21 R22 R23 /
0 and only one Ri not equal to zero, for i 1, 2, 3 16.
15, Rx, y R1 R2 R3 R4 R4 /
2
Journal of Applied Mathematics
In this paper, we consider the system 1.1 with Rx, y R2
R2 a2,0 x2
a1,1 xy
R2n a0,2 y2 ,
R2n n > 1, where
2n
b2n−i,i x2n−i yi ,
1.2
i0
and b2,0 , b1,1 , b0,2 ,b2n−i,i i 0, . . . , 2n are real constants. We call the systems above as P 2, 2n.
In 17, 18, the authors prove that in the system P 2, 2n the uniformly isochronous centers
are time reversible, which was done by imposing the existence of a transversal commutator.
However, for any concrete n, it is difficult to obtain explicit form of the conditions for the
origin to be a center for the system P 2, 2n due to increasing expansion of computation
during the management of large expressions. Sufficient and necessary conditions for the
origin to be a center are obtained in Volokitin 19, 20 for n 2 and 3 and in Xu and Lu
21 for n 4.
In this paper, we shall consider the system P 2, 10 and obtain sufficient and necessary
conditions for the origin to be a center. In Section 2, the main result and the basic method
are stated, and the necessary and sufficient center conditions are proved in Sections 3 and
4, respectively. In Section 5, sufficient and necessary center conditions for systems P 2, 2n
n 2, 3 and 4 are given in tight form of polynomial systems.
2. Center Condition
In this section, the technique in 22 is adopted, and a recursive formula for focal values is
obtained.
For n 5, system P 2, 2n takes the form
ẋ −y
x R2 x, y
R10 x, y ,
y R2 x, y
ẏ x
R10 x, y .
2.1
In polar coordinate system, system 2.1 can be written as
ṙ pr 3
qr 11 ,
2.2
θ̇ 1,
where ṙ dr/dθ, p and q are the functions of θ:
p a2,0 cos2 θ
a1,1 cosθ sinθ
a0,2 sin2 θ,
q
10
b10−i,i cos10−i θsini θ.
2.3
i0
Let rθ, c be the solution of 2.2 with r0, c c. For c > 0 small enough, we write
rθ, c ∞
am θcm ,
m1
2.4
Journal of Applied Mathematics
3
where a1 0 1, am 0 0 m > 1. Substituting 2.4 into 2.2 and equating the coefficients
of c, we obtain
ah aj al q
as1 as2 · · · as11 .
ȧm p
2.5
s1 s2 ··· s11 m
h j lm
s1 ,s2 ,...,s11 ≥1
h,j,l≥1
For 2.5, we can easily obtain ȧ1 0, ȧ2k 0 k 1, 2, . . .. Thus we can deduce
⎧
⎪
1,
m 1,
⎪
⎪
⎪
⎪
⎨
m 2k k 1, 2, . . .,
am 0,
⎪
⎪ θ
⎪
⎪
⎪
⎩ Φm dθ, m 2k 1 k 1, 2, . . .,
2.6
0
where
Φm p
ah aj al
q
h j lm
h,j,l≥1
h,j,l is odd
s1 s2 ··· s11 m
s1 ,s2 ,...,s11 ≥1
s1 ,s2 ,...,s11 is odd
as1 as2 · · · as11 .
2.7
The value of a2k 1 2π k 1, 2, . . . is called the kth focal value. Let Lk a2k
then the origin is a center for system 2.1 if and only if
Dk Lk 2π a2k 1 2π 0,
1
k 1, 2, . . .;
2.8
k 1, 2, . . ..
In fact, we have
Lk θ
Ψk dθ
2.9
k 1, 2, . . .,
0
where
Ψk p
Lh Lj Ll
q
Ls1 Ls2 · · · Ls11 ,
2.10
s1 s2 ... s11 k−5
s1 ,s2 ,...,s11 ≥0
h j lk−1
h,j,l≥0
and L0 a1 1.
The main result of the present paper is presented in the following theorem.
Theorem 2.1. The origin is a center for system 2.1 if and only if the following conditions are
satisfied:
0 a2,0
a0,2 ,
0 63b0,10
3b6,4
0 126a1,1 b0,10
3b4,6
21a2,0 b3,7
− 3a1,1 b6,4 − 14a1,1 b8,2
7b8,2
7b2,8
15a2,0 b5,5
21a2,0 b7,3
63b10,0 ,
63a2,0 b1,9 − 189b10,0 a1,1
63a2,0 b9,1 ,
7a1,1 b2,8
4
Journal of Applied Mathematics
a21,1 b2,8 − 108a22,0 b10,0 − 4a22,0 b2,8
0 −4a22,0 b8,2
2
a21,1 b8,2
− 108a2,0 b0,10
27a21,1 b10,0
27a21,1 b0,10
3a1,1 a2,0 b3,7 − 3a1,1 a2,0 b7,3 − 18a2,0 a1,1 b9,1
18a2,0 a1,1 b1,9 ,
0 72a22,0 a31,1 b10,0
8a52,0 b7,3
16a42,0 a1,1 b6,4 − 18a32,0 a21,1 b3,7
2a21,1 a32,0 b7,3
96a52,0 b1,9
9a51,1 b0,10 − a51,1 b8,2 − 36a51,1 b10,0
3a41,1 a2,0 b7,3
96a52,0 b9,1
8a52,0 b3,7 − 156a32,0 a21,1 b1,9 − 36a32,0 a21,1 b9,1 − 228a22,0 a31,1 b0,10
− 12a22,0 a31,1 b8,2 − 4a31,1 a22,0 b6,4
768a42,0 a1,1 b0,10
0 a61,1 b8,2
9a2,0 a41,1 b1,9
288a42,0 a1,1 b10,0
64a42,0 a1,1 b8,2
27a41,1 a2,0 b9,1 ,
45a61,1 b10,0 − 3a51,1 a2,0 b7,3 − 36a51,1 a2,0 b9,1 − 28a22,0 a41,1 b0,10
16a22,0 a41,1 b8,2
4a41,1 a22,0 b6,4 − 148a22,0 a41,1 b10,0
76a32,0 a31,1 b9,1 − 28a32,0 a31,1 b1,9
− 16a52,0 a1,1 b9,1
0 −35b10,0 a91,1
4a32,0 a31,1 b7,3
240a42,0 a21,1 b10,0
240a42,0 a21,1 b0,10
16a52,0 a1,1 b1,9 − 64a62,0 b0,10 − 64a62,0 b10,0 ,
35a2,0 b9,1 a81,1 − 30a71,1 a22,0 b8,2
− 340a61,1 a32,0 b9,1
− 600a51,1 a42,0 b10,0
− 288a31,1 a62,0 b8,2
350a71,1 a22,0 b10,0
280a51,1 a42,0 b8,2 − 8a51,1 a42,0 b6,4
20a61,1 a32,0 b7,3
480a41,1 a52,0 b9,1 − 176a41,1 a52,0 b7,3
800a31,1 a62,0 b10,0
128a1,1 a82,0 b8,2 − 640a1,1 a82,0 b10,0
64a31,1 a62,0 b6,4
96a21,1 a72,0 b7,3 − 448a21,1 a72,0 b9,1
128a92,0 b9,1
128a92,0 b1,9 .
2.11
3. The Proof of the Necessity
By simple computation in terms of 2.10 with Maple, we obtain
Ψ1 pL30 ,
Ψ2 3pL20 L1 ,
Ψ3 p 3L20 L2
3L0 L21 ,
Ψ4 p 3L20 L3
6L0 L1 L2
L31 ,
Ψ5 p 3L20 L4
6L0 L1 L3
3L0 L22
Ψ6 p 3L20 L5
6L0 L1 L4
6L0 L2 L3
3L21 L2
3L21 L3
qL11
0 ,
3L1 L22
11qL10
0 L1 ,
Journal of Applied Mathematics
Ψ7 p 3L21 L4
Ψ8 p 3L22 L3
6L1 L2 L4
3L20 L7
Ψ9 p 6L0 L2 L6
3L2 L23
3L20 L6
q 110L90 L1 L3
L22 L5
6L0 L3 L4
q 11L10
0 L3
6L0 L1 L7
3L21 L6
2L1 L24
L32
6L0 L1 L5
6L0 L2 L4
55L90 L21 ,
3L21 L5
3L1 L23
Ψ10 p L20 L9
3L0 L23
6L1 L2 L3
q 11L10
0 L2
5
3L20 L8
3L0 L24
L21 L7
2L0 L1 L8
110L90 L1 L4
165L80 L31 ,
3L22 L4
6L1 L2 L5
330L70 L41
2L0 L2 L7
4L0 L4 L5
11L10
0 L5
11L10
0 L4 ,
4L0 L3 L6
4L2 L3 L4
L33
495L80 L1 L22
495L80 L21 L3
462L60 L51 ,
Ψ11 p 6L0 L3 L7
6L0 L2 L8
3L20 L10
3L2 L24
6L2 L3 L5
3L22 L6
165L80 L32
6L0 L2 L5
6L0 L3 L5
2L1 L2 L6
1320L70 L31 L2
q 11L10
0 L6
6L1 L3 L4
495L80 L21 L2
q 110L90 L2 L3
6L1 L3 L6
110L90 L1 L2
55L90 L22
4L1 L3 L5
6L0 L1 L6
3L0 L25
3L21 L8
6L0 L4 L6
6L1 L4 L5
3L23 L4
6L1 L2 L7
6L0 L1 L9
110L90 L1 L5
55L90 L23
1320L70 L31 L3
110L90 L2 L4
1980L70 L21 L22
495L80 L21 L4
2310L60 L41 L2
990L80 L1 L2 L3
462L50 L61 .
3.1
Substituting 3.1 into 2.9, after a simple calculation, we obtain
L1 1
1
a2,0 cosθ sinθ − a0,2 cosθ sinθ
2
2
and the first focal value D1 L1 2π a2,0 π
1
a1,1
2
1
1
a2,0 θ − a1,1 cos2 θ
2
2
1
a0,2 θ,
2
3.2
a0,2 π. From 2.8, we can see that
0 a2,0
a0,2
3.3
must be satisfied if the origin is a center for system 2.1.
Following the discussion above, we can compute the first kth focal values recursively
by the following algorithm, which consists of three steps.
6
Journal of Applied Mathematics
Step 1 Initialization. Let
L0 1,
pθ a2,0 cos2 θ
qθ k 1,
a1,1 cosθ sinθ − a2,0 sin2 θ,
10
3.4
b10−i,i cos10−i θsini θ.
i0
Step 2 Computation of kth focal value. Lk can be obtained in terms of 2.10 and 3.1 with
Maple. Then the kth focal value Dk can be obtained by simple calculation.
Step 3 Let k k 1, and go to Step 2. Using computer algebra and writing a Maple code
applying the algorithm above, we obtain
Dk 0
k 1, 2, . . . , 4,
1
π63b0,10 3b6,4 63b10,0 3b4,6 7b8,2 7b2,8 ,
128
1
π7a1,1 b4,6 14a2,0 b7,3 10a2,0 b5,5 21b10,0 a1,1 21a1,1 b2,8
D6 128
7a1,1 b8,2 42a2,0 b9,1 5a1,1 b6,4 231a1,1 b0,10 14a2,0 b3,7 42a2,0 b1,9 ,
5
π 132a22,0 b10,0 5a21,1 b6,4 5a21,1 b8,2 20a22,0 b6,4 33a21,1 b2,8 36a22,0 b8,2
D7 512
20a22,0 b4,6 132a1,1 a2,0 b1,9 9a21,1 b4,6 36a2,0 b9,1 a1,1 20a2,0 a1,1 b7,3
D5 20a2,0 a1,1 b5,5
D8 132a22,0 b0,10
5
π 792a32,0 b9,1 360a32,0 b7,3
4096
99a31,1 b4,6 35a31,1 b8,2
540a1,1 a22,0 b8,2
6435a31,1 b0,10
429a31,1 b2,8
210a2,0 a21,1 b7,3
5148a22,0 a1,1 b0,10
1188a22,0 a1,1 b2,8
D9 36a22,0 b2,8
429a21,1 b0,10
9a21,1 b10,0
280a32,0 b5,5
45a31,1 b10,0
360a32,0 b3,7
792a32,0 b1,9
270a21,1 a2,0 b9,1
2574a2,0 a21,1 b1,9
270a2,0 a21,1 b5,5
540a22,0 a1,1 b4,6
420a22,0 a1,1 b6,4
594a21,1 a2,0 b3,7
1188 2 a2,0 b10,0 a1,1
45a31,1 b6,4 ,
35
π 1120a32,0 a1,1 b7,3 1120a32,0 a1,1 b5,5 17160a22,0 a21,1 b0,10
32768
1760a32,0 a1,1 b3,7 1144a2,0 a31,1 b3,7 5720a2,0 a31,1 b1,9
440a2,0 a31,1 b5,5
880a42,0 b2,8
55a41,1 b6,4
143a41,1 b4,6
12155a41,1 b0,10
880a42,0 b8,2
840a22,0 a21,1 b8,2
2288a42,0 b0,10
1320a21,1 a22,0 b10,0
280a31,1 a2,0 b7,3
560a42,0 b4,6
3432a22,0 a21,1 b2,8
840a22,0 a21,1 b6,4
715a41,1 b2,8 ,
1760a32,0 a1,1 b9,1
280a31,1 a2,0 b9,1
2288a42,0 b10,0
4576a1,1 a32,0 b1,9
36a1,1 a2,0 b3,7 ,
35a41,1 b10,0
1320a22,0 a21,1 b4,6
560a42,0 b6,4
35a41,1 b8,2
Journal of Applied Mathematics
D10 7
11
π 2 7b2,8 63b0,10 63b10,0 3b6,4 7b8,2 3b4,6 32768
7
π 3080a31,1 a22,0 b10,0 34320a1,1 a42,0 b0,10 17160b2,8 a31,1 a22,0 11440a1,1 a42,0 b2,8
16384
2520a31,1 a22,0 b8,2 5040a21,1 a32,0 b7,3 24310a41,1 a2,0 b1,9 6160a1,1 a42,0 b8,2
2016a52,0 b5,5
2464a52,0 b3,7
6160a21,1 a32,0 b9,1
143a51,1 b6,4
4576a52,0 b9,1
63a51,1 b10,0
630a41,1 a2,0 b9,1
97240a31,1 a22,0 b0,10
46189a51,1 b0,10
2464a52,0 b7,3
34320a21,1 a32,0 b1,9
231a1,1 b0,10
× 10752a2,0 b7,3 b3,7
9100a1,1 b2,8 b7,3
21b2,8 a1,1
21a1,1 b10,0
1
π
1966080
42a2,0 b1,9 113022a1,1 b7,3 b0,10
23587200a52,0 a1,1 b1,9
663a1,1 b1,9 b4,6
158722200a22,0 a41,1 b0,10
7046a2,0 b5,5 b3,7
9525600a42,0 a21,1 b8,2 − 143332a2,0 b2,8 b10,0 − 129948a2,0 b4,6 b0,10
50388b0,10 a1,1 b3,7 − 268736a2,0 b2,8 b0,10
9525600a42,0 a21,1 b6,4
140a1,1 b8,2 b3,7
234a1,1 b6,4 b3,7
437a1,1 b6,4 b5,5
54145a1,1 b5,5 b0,10
2
− 12318a2,0 b8,2
42b9,1 a2,0
580671a1,1 b0,10 b9,1
9879a1,1 b9,1 b6,4
− 129168a2,0 b6,4 b0,10
11440a32,0 a21,1 b3,7
3080a31,1 a22,0 b6,4
23667a1,1 b9,1 b10,0 − 61296a2,0 b10,0 b4,6
7620480a52,0 a1,1 b7,3
77a51,1 b8,2
4576a52,0 b1,9 ,
126a1,1 b10,0 b3,7
19656000a32,0 a31,1 b3,7
11440a1,1 a42,0 b10,0
770a41,1 a2,0 b7,3
3
π 2 7b2,8 63b10,0 3b6,4 7b8,2 63b0,10 3b4,6 4096
× 7a1,1 b8,2 10b5,5 a2,0 5a1,1 b6,4 14b7,3 a2,0 7a1,1 b4,6
14b3,7 a2,0
4290a2,0 a41,1 b3,7
6160a32,0 a21,1 b5,5
1430a41,1 a2,0 b5,5
5720a31,1 a22,0 b4,6
D11 2431b2,8 a51,1
429a51,1 b4,6
5040a1,1 a42,0 b6,4
6160a42,0 a1,1 b4,6
11007360a52,0 a1,1 b3,7
13759200a42,0 a21,1 b4,6
7862400a62,0 b0,10
2
− 16354a2,0 b2,8
− 11856a2,0 b6,4 b8,2
5012280a2,0 a51,1 b3,7
57330a61,1 b8,2
100245600a21,1 a42,0 b0,10
122850a61,1 b6,4
2
2
4914a2,0 b3,7
− 3198a2,0 b4,6
15555a1,1 b9,1 b4,6
2645370a61,1 b2,8
7802a2,0 b7,3 b5,5
2
3439800a22,0 a41,1 b6,4 29484000a21,1 a42,0 b2,8 78636b9,1
a2,0 39690a61,1 b10,0
2
2540160a62,0 b6,4 − 881790a2,0 b0,10
417690a61,1 b4,6 4199a1,1 b1,9 b2,8
329b8,2 a1,1 b5,5
55552770a61,1 b0,10 − 12220a2,0 b4,6 b8,2
63a1,1 b1,9 b10,0
6350400a32,0 a31,1 b9,1
2
− 2946a2,0 b6,4
195b6,4 a1,1 b1,9
11367a1,1 b9,1 b8,2
476280a2,0 a51,1 b9,1
25061400a22,0 a41,1 b2,8 − 132672a2,0 b10,0 b8,2
385a1,1 b10,0 b5,5
− 14768a2,0 b4,6 b2,8
91a1,1 b8,2 b1,9
3669120a62,0 b2,8
43050a2,0 b9,1 b3,7
1474200a2,0 a51,1 b5,5
2381400a22,0 a41,1 b8,2
21268b1,9 a2,0 b5,5
7620480a52,0 a1,1 b5,5
− 258076a2,0 b0,10 b8,2
30498b1,9 a2,0 b3,7
129024a2,0 b1,9 b9,1
1302b7,3 a1,1 b8,2
687960a2,0 a51,1 b7,3
44766a2,0 b9,1 b7,3
30956b9,1 a2,0 b5,5
3094a1,1 b3,7 b2,8
66830400a32,0 a31,1 b1,9
949a1,1 b4,6 b5,5
7371000a22,0 a41,1 b4,6
8
Journal of Applied Mathematics
32214b1,9 a2,0 b7,3 − 28672a2,0 b8,2 b2,8 − 60516a2,0 b10,0 b6,4
3913a1,1 b5,5 b2,8
2381400a22,0 a41,1 b10,0 − 1290240a2,0 b0,10 b10,0
2
5838a2,0 b7,3
624a1,1 b3,7 b4,6
88179a1,1 b1,9 b0,10
2
− 408450a2,0 b10,0
7862400a62,0 b10,0
2004a1,1 b7,3 b10,0
3669120a62,0 b8,2 2538a1,1 b7,3 b4,6
13759200a42,0 a21,1 b10,0
9172800a32,0 a31,1 b5,5 − 14404a2,0 b2,8 b6,4
2
2560a2,0 b5,5
50267b2,8 b9,1 a1,1 .
3.5
Usually Wu’s method 23 is used to do the triangular sets reduction for the focal values, that
is, D5 , . . . , D11 . The method provides a standard algorithm 24 to handle the reduction of
polynomials; however, we here just take use the idea of Wu’s method and do the reduction
by substitution method due to the speciality of our case. Thus we can obtain the relations
2.11. Therefore, the necessary part of Theorem 2.1 is proved.
4. The Proof of the Sufficiency
Case 1 a1,1 0. From 2.11, we have that either
i a0,2 a2,0 0, 63b0,10
3b6,4
3b4,6
7b8,2
7b2,8
63b10,0 0, or
ii a0,2 a2,0 0, b0,10 b10,0 0, b1,9 b9,1 0, b2,8 b8,2 0, b3,7 b7,3 0, b4,6 b6,4 0, b5,5 0.
For Case 1, system 2.1 takes the form
ẋ −y
xR10 x, y ,
2π
0
The center condition of system 4.1 is
63b0,10
3b6,4
yR10 x, y .
ẏ x
4.1
R10 sinθ, cosθdθ 0 see 10. That is,
3b4,6
7b8,2
63b10,0 0.
7b2,8
4.2
For Case 2, system 2.1 takes the form
ẋ −y
xT,
ẏ x
yT,
4.3
where T a0,2 y2 − x2 b0,10 y10 − x10 b1,9 xy9 − x9 y b2,8 x2 y8 − x8 y2 b3,7 x3 y7 −
x7 y3 b4,6 x4 y6 − x6 y4 . For system 4.3, the following linear transformation:
⎛
1
⎜ 2
x, y u, v⎜
⎝ 1
−
2
⎞
1
2⎟
⎟
1⎠
2
4.4
leads to
u̇ −v
u2 vJu, v,
v̇ u
uv2 Ju, v,
4.5
Journal of Applied Mathematics
where Ju, v J0,0
J8,0 u8
J6,2 u6 v2
9
J4,4 u4 v4
J2,6 u2 v6
J0,8 v8 ,
J0,0 a0,2 ,
J8,0 1
3b2,8
256
J6,2 1
24b1,9 − 4b3,7
256
J4,4 1
−14b2,8
256
J2,6 1
4b3,7 − 24b1,9
256
60b0,10 − 4b4,6
J0,8 1
−4b1,9
256
b4,6
b4,6
5b0,10
60b0,10
126b0,10
3b2,8
4b1,9
2b3,7 ,
4b2,8 − 4b4,6 ,
4.6
6b4,6 ,
4b2,8 ,
5b0,10 − 2b3,7 .
Obviously, in this case, system 4.5 satisfies the condition of symmetry principle 25, page
135; thus the origin is a center for the system.
Case 2 a1,1 / 0, a2,0 0. From 2.11, we have
b10,0 b8,2 b6,4 b4,6 b2,8 b0,10 0.
4.7
Thus system 2.1 takes the form
ẋ −y
x2 ya1,1
S,
ẏ x
xy2 a1,1
S,
4.8
where S b9,1 x8 b7,3 x6 y2 b5,5 x4 y4 b3,7 x2 y6 b1,9 y8 . Obviously, system 4.8 satisfies the
condition of symmetry principle; thus the origin is a center for the system.
Case 3 a2,0 /
0, a1,1 /
0. In this case, 2.11 takes the form
a0,2 −a2,0 ,
7
7
b4,6 − b2,8 − 21b0,10 − b6,4 − b8,2 − 21b10,0 ,
3
3
1
b5,5 −
−14a1,1 b8,2 21a2,0 b7,3 63a2,0 b9,1 − 3a1,1 b6,4 − 189a1,1 b10,0
15a2,0
21a2,0 b3,7
b2,8 −
126a1,1 b0,10
63a2,0 b1,9
1
3a1,1 a2,0 b3,7 − 18a1,1 a2,0 b9,1
−4a22,0 a21,1
7a1,1 b2,8 ,
a21,1 b8,2
27a21,1 b10,0
− 4a22,0 b8,2 − 108a22,0 b10,0 − 3a1,1 a2,0 b7,3 − 108a22,0 b0,10
18a1,1 a2,0 b1,9 ,
27a21,1 b0,10
10
Journal of Applied Mathematics
b3,7 1
2a32,0 −4a22,0
9a21,1
× 3a41,1 a2,0 b7,3
9a51,1 b0,10
16a1,1 a42,0 b6,4 − a51,1 b8,2
768a1,1 a42,0 b0,10 − 36a21,1 a32,0 b9,1 − 228a31,1 a22,0 b0,10
96a52,0 b1,9
288a1,1 a42,0 b10,0 − 4a31,1 a22,0 b6,4
b0,10 − 12a31,1 a22,0 b8,2
2a21,1 a32,0 b7,3
72a31,1 a22,0 b10,0
64a1,1 a42,0 b8,2
9a41,1 a2,0 b1,9
27a41,1 a2,0 b9,1
96a52,0 b9,1 − 36a51,1 b10,0 ,
1
4a22,0 7a41,1 − 60a21,1 a22,0
× −148a41,1 a22,0 b10,0
−36a51,1 a2,0 b9,1
16a42,0
76a31,1 a32,0 b9,1
− 28a31,1 a32,0 b1,9
b1,9 8a52,0 b7,3 − 156a21,1 a32,0 b1,9
a61,1 b8,2
4a41,1 a22,0 b6,4
16a1,1 a52,0 b1,9 − 3a51,1 a2,0 b7,3
240a21,1 a42,0 b10,0
4a31,1 a32,0 b7,3
16a41,1 a22,0 b8,2 − 16a1,1 a52,0 b9,1
45a61,1 b10,0 − 64a62,0 b10,0 ,
1
−280a51,1 a42,0 b8,2 − 20a61,1 a32,0 b7,3 − 480a41,1 a52,0 b9,1
128a92,0
− 64a31,1 a62,0 b6,4 − 800a31,1 a62,0 b10,0 − 128a1,1 a82,0 b8,2 − 96a21,1 a72,0 b7,3
176a41,1 a52,0 b7,3 − 35a81,1 a2,0 b9,1
− 128a92,0 b9,1
448a21,1 a72,0 b9,1
640a1,1 a82,0 b10,0
30a71,1 a22,0 b8,2
288a31,1 a62,0 b8,2
600a51,1 a42,0 b10,0
8a51,1 a42,0 b6,4
340a61,1 a32,0 b9,1
35a91,1 b10,0 − 350a71,1 a22,0 b10,0 .
4.9
Substituting 4.9 into 2.1, we get
ẋ −y
a1,1 xy − a2,0 y2 H x, y ,
x a2,0 x2
y a2,0 x2
a1,1 xy − a2,0 y2 H x, y ,
128b10,0 a82,0 x8
−128b10,0 a1,1 a72,0
ẏ x
4.10
where
H x, y 1
128a92,0
128a92,0
128a21,1 b10,0 a62,0 − 128a1,1 b9,1 a72,0
128a21,1 b9,1 a62,0
128b10,0 a82,0
128b9,1 a82,0 yx7
128a82,0 b8,2 y2 x6
128b9,1 a82,0 − 256b10,0 a1,1 a72,0 − 128a1,1 3 b10,0 a52,0
128a82,0 b7,3 − 128a72,0 a1,1 b8,2 y3 x5
−256a1,1 b9,1 a72,0 − 128a31,1 b9,1 a52,0
128a41,1 b10,0 a42,0
128a82,0 b8,2
384a21,1 b10,0 a62,0
128b10,0 a82,0
128a62,0 a21,1 b8,2
128a82,0 b6,4 − 128a72,0 a1,1 b7,3 y4 x4
Journal of Applied Mathematics
11
−880a41,1 b9,1 a42,0 − 512a31,1 b10,0 a52,0
880a51,1 b10,0 a32,0
128b9,1 a82,0 − 256a72,0 a1,1 b8,2
768a52,0 a31,1 b8,2
−544a62,0 a21,1 b7,3 − 384b10,0 a1,1 a72,0
192a62,0 a21,1 b6,4 − 800a51,1 b9,1 a32,0
− 384a1,1 b9,1 a72,0
384a21,1 b9,1 a62,0
128a82,0 b7,3
256a72,0 a1,1 b6,4 y5 x3
832a31,1 b9,1 a52,0
192a72,0 a1,1 b7,3
128b10,0 a82,0 − 1040a41,1 b10,0 a42,0
688a42,0 a41,1 b8,2
800a61,1 b10,0 a22,0 − 464a52,0 a31,1 b7,3
128a82,0 b8,2
768a21,1 b10,0 a62,0 − 512a62,0 a21,1 b8,2 y6 x2
−384a21,1 b9,1 a62,0 − 160a42,0 a41,1 b7,3
240a2,0 3 a51,1 b8,2
96a62,0 a21,1 b7,3
64a52,0 a31,1 b6,4 − 256a52,0 a31,1 b8,2 − 480a51,1 b10,0 a32,0 − 512b10,0 a1,1 a72,0
128a72,0 a1,1 b8,2
−280a61,1 b9,1 a22,0
280a71,1 b10,0 a2,0
640a31,1 b10,0 a52,0
16a52,0 a31,1 b7,3 − 80a31,1 b9,1 a52,0
400a41,1 b9,1 a42,0
128b9,1 a82,0 y7 x
60a51,1 b9,1 a32,0
120a41,1 b10,0 a42,0
− 20a32,0 a51,1 b7,3 − 70a61,1 b10,0 a22,0
30a22,0 a61,1 b8,2
− 160a21,1 b10,0 a62,0 − 35a71,1 b9,1 a2,0
8a42,0 a41,1 b6,4
64a1,1 b9,1 a72,0 − 40a42,0 a41,1 b8,2
35a81,1 b10,0 y8 .
32a62,0 a21,1 b8,2
128b10,0 a82,0
4.11
For system 4.10, we consider the following linear transformation:
t −1
,
x, y u, v
1 t
4.12
where the diagonal elements of the transformation matrix satisfy the equation a2,0 t2 − a1,1 t −
a2.0 0. Applying the transformation 4.12 to 4.10, we obtain
u̇ −v
u2 vM · Iu, v,
v̇ u
uv2 M · Iu, v,
4.13
where M 4a2,0 t−a1,1 a1,1 t2 /128a92,0 , Iu, v I0,0 I8,0 u8 I6,2 u6 v2 I4,4 u4 v4 I2,6 u2 v6 I0,8 v8 ,
and I0,0 , I8,0 , I6,2 , I4,4 , I2,6 , I0,8 are given in the appendix.
Obviously, system 4.13 satisfies the conditions of symmetry principle, thus the origin
is a center for the system. Therefore, the sufficient part of Theorem 2.1 is proved.
5. Center Conditions of P 2, 2n for n 2, 3 and 4
By similar steps discussed above, we can obtain the following theorem.
Theorem 5.1. The origin is a center for system P 2, 8 if and only if the following conditions are
satisfied:
0 a2,0
0 5b6,2
a0,2 ,
35b0,8
5b2,6
3b4,4
35b8,0 ,
12
Journal of Applied Mathematics
0 a1,1 b2,6
3a2,0 b5,3
14a1,1 b0,8
3a2,0 b3,5 − a1,1 b6,2
7a2,0 b1,7
0 7a2,0 b1,7 a1,1 − 14a2,0 a1,1 b7,1 − 3a2,0 a1,1 b5,3
a21,1 b6,2
7a2,0 b7,1 − 14a1,1 b8,0 ,
21b8,0 a21,1
7a21,1 b0,8 − 2a22,0 b6,2 − 56a22,0 b0,8 − 2a22,0 b2,6 − 56a22,0 b8,0 ,
0 −3a21,1 a2,0 b5,3 − 48a1,1 a22,0 b8,0
18a31,1 b8,0 − 11a21,1 a2,0 b7,1
− 64a22,0 a1,1 b0,8 − 4a1,1 a22,0 b6,2
0 8a32,0 a31,1 b7,1
10a31,1 b0,8
16a42,0 a21,1 b8,0 − 5a51,1 a2,0 b7,1
− 16a62,0 b0,8
a31,1 b6,2
10a21,1 a2,0 b1,7 − 4a32,0 b1,7 − 4a32,0 b7,1 ,
5a61,1 b8,0 − 16a62,0 b8,0
4a22,0 a41,1 b6,2 − 2a32,0 a31,1 b5,3 − 8a52,0 a1,1 b7,1 − 4a42,0 a21,1 b6,2 − 10a22,0 a41,1 b8,0 .
5.1
The relations above give a complete set of relations in the sense given in 21, that is,
this set of relations cover all the cases of the conditions which are sufficient and necessary for
the system to be of center type.
Theorem 5.2. The origin is a center for system P 2, 6 if and only if the following conditions are
satisfied:
0 a2,0
a0,2 ,
0 5b0,6
b2,4
b4,2
5b6,0 ,
0 3a2,0 b3,3 − a1,1 b4,2
0 a21,1 b0,6
5a1,1 b0,6
5a2,0 b1,5
b6,0 a21,1 − 4a22,0 b0,6 − 4a22,0 b6,0
5a2,0 b5,1 − 10a1,1 b6,0 ,
a2,0 a1,1 b1,5 − a2,0 a1,1 b5,1 ,
5.2
0 18a22,0 a31,1 b6,0 − 16a32,0 a21,1 b5,1 − 24a42,0 a1,1 b6,0 − 3a51,1 b6,0
3a41,1 a2,0 b5,1
8a52,0 b1,5
8a52,0 b5,1 − 2a31,1 a22,0 b4,2
8a1,1 a42,0 b4,2 .
The relations above is another form as the result of paper 20.
Theorem 5.3. The origin is a center for system P 2, 4 if and only if the following conditions are
satisfied:
0 a2,0
a0,2 ,
0 3b0,4
b2,2
0 a1,1 b0,4
3b4,0 ,
a2,0 b1,3
a2,0 b3,1 − a1,1 b4,0 ,
0 −2a22,0 b0,4 − a1,1 a2,0 b3,1
5.3
a21,1 b4,0 − 2a22,0 b4,0 .
The relations above is another form of center condition for the system P 2, 4, which
has been discussed by Volokitin in paper 19.
Journal of Applied Mathematics
13
Appendix
I0,0 , I8,0 , I6,2 , I4,4 , I2,6 , I0,8
I0,0 128a92,0 ,
I8,0 128b10,0 a82,0 t8
128a72,0 b10,0 a1,1 − 128a82,0 b9,1 t7
128b10,0 a82,0 − 128a1,1 b9,1 a72,0
128a62,0 a21,1 b10,0
256a72,0 b10,0 a1,1 − 128a82,0 b9,1
−128a82,0 b7,3
128a52,0 a31,1 b10,0 − 128a62,0 a21,1 b9,1
128a72,0 a1,1 b8,2 t5
128b10,0 a82,0 − 256a1,1 b9,1 a72,0
384a62,0 a21,1 b10,0
128a82,0 b8,2
128a82,0 b8,2 t6
128a62,0 a21,1 b8,2
128a42,0 a41,1 b10,0 − 128a72,0 a1,1 b7,3
128a82,0 b6,4 − 128a52,0 a31,1 b9,1 t4
−256a72,0 a1,1 b6,4 − 880a51,1 b10,0 a32,0 − 128a82,0 b7,3
512a52,0 a31,1 b10,0
− 384a62,0 a21,1 b9,1 − 768a52,0 a31,1 b8,2 − 128a82,0 b9,1
256a72,0 a1,1 b8,2
384a72,0 b10,0 a1,1
880a41,1 b9,1 a42,0
544a62,0 a21,1 b7,3 t3
128a82,0 b8,2 − 800a32,0 a51,1 b9,1 − 512a62,0 a21,1 b8,2
800a22,0 a61,1 b10,0
192a72,0 a1,1 b7,3
−384a1,1 b9,1 a72,0
832a52,0 a31,1 b9,1
688a42,0 a41,1 b8,2
768a62,0 a21,1 b10,0
128b10,0 a82,0 − 464a52,0 a31,1 b7,3
192a62,0 a21,1 b6,4 − 1040a42,0 a41,1 b10,0 t2
−128a72,0 a1,1 b8,2 − 64a52,0 a31,1 b6,4 − 640a52,0 a31,1 b10,0 − 280a71,1 b10,0 a2,0
− 128a82,0 b9,1
512a72,0 b10,0 a1,1 − 240a32,0 a51,1 b8,2
− 400a41,1 b9,1 a42,0
384a62,0 a21,1 b9,1
160a42,0 a41,1 b7,3
480a51,1 b10,0 a32,0 − 96a62,0 a21,1 b7,3
256a52,0 a31,1 b8,2
30a22,0 a61,1 b8,2 − 20a32,0 a51,1 b7,3
280a61,1 b9,1 a22,0 t
128b10,0 a82,0
8a42,0 a41,1 b6,4 − 70a22,0 a61,1 b10,0 − 35a71,1 b9,1 a2,0 − 40a42,0 a41,1 b8,2
60a32,0 a51,1 b9,1 − 80a52,0 a31,1 b9,1
64a1,1 b9,1 a72,0 − 160a62,0 a21,1 b10,0
120a42,0 a41,1 b10,0
35a81,1 b10,0
16a52,0 a31,1 b7,3
32a62,0 a21,1 b8,2 ,
A.1
I6,2 128a62,0 a21,1 b10,0
−384a82,0 b7,3
128b10,0 a82,0 − 128a1,1 b9,1 a72,0
384a52,0 a31,1 b10,0
−384a62,0 a21,1 b9,1
768a62,0 a21,1 b10,0
128a2,0 8 b8,2 t8
384a72,0 a1,1 b8,2 − 128a72,0 b10,0 a1,1
512a82,0 b9,1 t7
768a42,0 a41,1 b10,0 − 768a52,0 a31,1 b9,1
768a82,0 b6,4
14
Journal of Applied Mathematics
768a62,0 a21,1 b8,2 − 768a82,0 b8,2 − 768a72,0 a1,1 b7,3 t6
2816b10,0 a82,0
−1920a62,0 a21,1 b9,1 − 2048a82,0 b9,1
5440a62,0 a21,1 b7,3
640a72,0 a1,1 b8,2 − 8800a51,1 b10,0 a32,0
2688a72,0 b10,0 a1,1 − 7680a52,0 a31,1 b8,2
8800a41,1 b9,1 a42,0 − 2560a72,0 a1,1 b6,4
640a82,0 b7,3
3200a52,0 a31,1 b10,0 t5
14528a52,0 a31,1 b9,1 − 12000a32,0 a51,1 b9,1 − 9728a62,0 a21,1 b8,2 − 2048a82,0 b6,4
12000a22,0 a61,1 b10,0
7296a62,0 a21,1 b10,0
4928a72,0 a1,1 b7,3
1792b10,0 a82,0
10320a42,0 a41,1 b8,2 − 17648a42,0 a41,1 b10,0 − 3584a1,1 b9,1 a72,0
1792a82,0 b8,2
−6960a52,0 a31,1 b7,3
2880a62,0 a21,1 b6,4 t4
−1344a52,0 a31,1 b6,4
640a82,0 b7,3 − 19840a52,0 a31,1 b10,0
− 21600a41,1 b9,1 a42,0
12544a62,0 a21,1 b9,1 − 5248a72,0 a1,1 b8,2
16896a52,0 a31,1 b8,2 − 2048a82,0 b9,1
3360a42,0 a41,1 b7,3
− 5880a71,1 b10,0 a2,0 − 10176a62,0 a21,1 b7,3
−11560a22,0 a61,1 b10,0
4864a1,1 b9,1 a72,0
23280a51,1 b10,0 a32,0
5880a61,1 b9,1 a22,0 − 5040a32,0 a51,1 b8,2 t3
7552a72,0 b10,0 a1,1
− 560a32,0 a51,1 b7,3
3840a72,0 a1,1 b6,4
840a22,0 a61,1 b8,2
980a81,1 b10,0
7808a62,0 a21,1 b8,2
2816b10,0 a82,0
16608a42,0 a41,1 b10,0
6016a52,0 a31,1 b7,3
224a42,0 a41,1 b6,4 − 2304a62,0 a21,1 b6,4 − 12992a52,0 a31,1 b9,1
768a2,0 8 b6,4 − 11392a62,0 a21,1 b10,0
11280a32,0 a51,1 b9,1 − 9376a42,0 a41,1 b8,2
−980a71,1 b9,1 a2,0 − 768a82,0 b8,2 − 3072a72,0 a1,1 b7,3 t2
5440a41,1 b9,1 a42,0
1680a32,0 a51,1 b8,2 − 1120a42,0 a41,1 b7,3
448a52,0 a31,1 b6,4
6016a52,0 a31,1 b10,0 − 3840a62,0 a21,1 b9,1 − 6000a51,1 b10,0 a32,0
− 384a82,0 b7,3
1960a71,1 b10,0 a2,0 − 4096a52,0 a31,1 b8,2 − 1960a61,1 b9,1 a22,0
−768a72,0 a1,1 b6,4 − 2432a72,0 b10,0 a1,1
2304a62,0 a21,1 b7,3
1664a72,0 a1,1 b8,2 t
128a82,0 b8,2 − 800a32,0 a51,1 b9,1 − 512a62,0 a21,1 b8,2
800a22,0 a61,1 b10,0
192a72,0 a1,1 b7,3
512a82,0 b9,1
832a52,0 a31,1 b9,1
− 464a52,0 a31,1 b7,3 − 384a1,1 b9,1 a72,0
768a62,0 a21,1 b10,0
128b10,0 a82,0
688a42,0 a41,1 b8,2
192a62,0 a21,1 b6,4
− 1040a42,0 a41,1 b10,0 ,
A.2
I4,4 128b10,0 a82,0
128a82,0 b6,4 − 128a72,0 a1,1 b7,3
128a62,0 a21,1 b8,2
−128a52,0 a31,1 b9,1 t8
128a42,0 a41,1 b10,0
384a62,0 a21,1 b10,0 − 256a1,1 b9,1 a72,0
128a82,0 b8,2
Journal of Applied Mathematics
15
640a72,0 a1,1 b8,2 − 1280a62,0 a21,1 b9,1 − 3840a52,0 a31,1 b8,2 − 1280a72,0 a1,1 b6,4
− 4400a51,1 b10,0 a32,0
1920a52,0 a31,1 b10,0
640a72,0 b10,0 a1,1
4400a41,1 b9,1 a42,0
2720a62,0 a21,1 b7,3 t7
14528a52,0 a1,1 3 b9,1 − 12000a32,0 a51,1 b9,1 − 9728a62,0 a21,1 b8,2
− 2048a82,0 b6,4
7296a62,0 a21,1 b10,0
4928a72,0 a1,1 b7,3
12000a22,0 a61,1 b10,0
10320a42,0 a41,1 b8,2
1792b10,0 a82,0 − 17648a42,0 a41,1 b10,0 − 3584a1,1 b9,1 a72,0 − 6960a52,0 a31,1 b7,3
1792a82,0 b8,2
2880 a21,1 b6,4 t6
9800a61,1 b9,1 a22,0 − 9800a71,1 b10,0 a2,0 − 19680a62,0 a21,1 b7,3
9600a72,0 b10,0 a1,1
43200a51,1 b10,0 a32,0 − 33920a52,0 a31,1 b10,0 − 8400a32,0 a51,1 b8,2
21120a62,0 a21,1 b9,1
5600a42,0 a41,1 b7,3 − 8320a72,0 a1,1 b8,2 − 40400a41,1 b9,1 a42,0
−2240a52,0 a31,1 b6,4
7680a72,0 a1,1 b6,4 t5
−43488a52,0 a31,1 b9,1
19680a52,0 a31,1 b7,3
560a42,0 a41,1 b6,4 − 1400a32,0 a51,1 b7,3
4608a82,0 b6,4
32000a52,0 a31,1 b8,2
27328a62,0 a21,1 b8,2
12288b10,0 a82,0 − 5632a82,0 b8,2
2450a81,1 b10,0 − 36900a22,0 a61,1 b10,0 − 12288a72,0 a1,1 b7,3
15744a1,1 b9,1 a72,0
− 2450a71,1 b9,1 a2,0 − 7680a62,0 a21,1 b6,4 − 33216a62,0 a21,1 b10,0
36200a32,0 a51,1 b9,1 − 30320a42,0 a41,1 b8,2
19680a62,0 a21,1 b7,3
2240a52,0 a31,1 b6,4
− 9600a72,0 b10,0 a1,1
54608a42,0 a41,1 b10,0
2100a22,0 a61,1 b8,2 t4
9800a71,1 b10,0 a2,0 − 5600a42,0 a41,1 b7,3
8320a72,0 a1,1 b8,2 − 9800a61,1 b9,1 a22,0
− 43200a51,1 b10,0 a32,0 − 7680a72,0 a1,1 b6,4
8400a32,0 a51,1 b8,2
33920a52,0 a31,1 b10,0
40400a41,1 b9,1 a42,0
−21120a62,0 a21,1 b9,1 − 32000a52,0 a1,1 b8,2 t3
14528a52,0 a31,1 b9,1 − 12000a32,0 a51,1 b9,1 − 9728a62,0 a21,1 b8,2
− 2048a82,0 b6,4
7296a62,0 a21,1 b10,0
4928a72,0 a1,1 b7,3
12000a22,0 a61,1 b10,0
10320a42,0 a41,1 b8,2
1792b10,0 a82,0 − 17648a42,0 a41,1 b10,0 − 3584a1,1 b9,1 a72,0 − 6960a52,0 a31,1 b7,3
1792a82,0 b8,2
2880a62,0 a21,1 b6,4 t2
3840a52,0 a31,1 b8,2
1280a72,0 a1,1 b6,4
1280a62,0 a21,1 b9,1 − 640a72,0 a1,1 b8,2
4400a51,1 b10,0 a32,0 − 640a72,0 b10,0 a1,1 − 2720a62,0 a21,1 b7,3 − 4400a41,1 b9,1 a42,0
−1920a52,0 a31,1 b10,0 t
128a42,0 a41,1 b10,0
128b10,0 a82,0
128a62,0 a21,1 b8,2
128a82,0 b6,4 − 128a72,0 a1,1 b7,3
384a62,0 a21,1 b10,0 − 256a1,1 b9,1 a72,0
128a82,0 b8,2 − 128a52,0 a31,1 b9,1 ,
A.3
16
Journal of Applied Mathematics
I2,6 128a82,0 b8,2
128b10,0 a82,0
− 464a52,0 a31,1 b7,3
768a62,0 a21,1 b10,0
832a52,0 a31,1 b9,1
800a22,0 a61,1 b10,0 − 512a62,0 a21,1 b8,2
− 1040a42,0 a41,1 b10,0 − 800a32,0 a51,1 b9,1
688a42,0 a41,1 b8,2
192a62,0 a21,1 b6,4
192a72,0 a1,1 b7,3
−384a1,1 b9,1 a72,0 t8
3840a62,0 a21,1 b9,1 − 6016a52,0 a31,1 b10,0 − 512a82,0 b9,1
4096a52,0 a31,1 b8,2 − 448a52,0 a31,1 b6,4
384a82,0 b7,3
1120a42,0 a41,1 b7,3 − 5440a41,1 b9,1 a42,0
2432a72,0 b10,0 a1,1 − 1664a2,0 7 a1,1 b8,2 − 1680a32,0 a51,1 b8,2
− 2304a62,0 a21,1 b7,3
1960a61,1 b9,1 a22,0 − 1960a71,1 b10,0 a2,0
768a72,0 a1,1 b6,4
6000a51,1 b10,0 a32,0 t7
768a82,0 b6,4 − 768a82,0 b8,2
11280a32,0 a51,1 b9,1
− 11392a62,0 a21,1 b10,0 − 980a71,1 b9,1 a2,0
− 2304a62,0 a21,1 b6,4
980a81,1 b10,0
16608a42,0 a41,1 b10,0 − 3072a72,0 a1,1 b7,3
4864a1,1 b9,1 a72,0 − 12992a2,0 5 a31,1 b9,1 − 560a32,0 a51,1 b7,3
224a42,0 a41,1 b6,4
6016a52,0 a31,1 b7,3
2816b10,0 a82,0
840a22,0 a61,1 b8,2 − 11560a22,0 a61,1 b10,0 − 9376a42,0 a41,1 b8,2 t6
7808a62,0 a21,1 b8,2
21600a41,1 b9,1 a42,0 − 7552a72,0 b10,0 a1,1
5248a72,0 a1,1 b8,2
− 3840a72,0 a1,1 b6,4 − 23280a51,1 b10,0 a32,0 − 640a82,0 b7,3
1344a52,0 a31,1 b6,4
10176a62,0 a21,1 b7,3
− 5880a61,1 b9,1 a22,0 − 12544a62,0 a21,1 b9,1 − 3360a42,0 a41,1 b7,3 − 16896a52,0 a31,1 b8,2
5040a32,0 a51,1 b8,2
19840a52,0 a31,1 b10,0
5880a71,1 b10,0 a2,0
2048a82,0 b9,1 t5
−2048a82,0 b6,4 − 6960a52,0 a31,1 b7,3 − 12000a32,0 a51,1 b9,1
7296a62,0 a21,1 b10,0 − 9728a62,0 a21,1 b8,2
14528a52,0 a31,1 b9,1
1792b10,0 a82,0
4928a72,0 a1,1 b7,3 t4
2048a82,0 b9,1 − 640a82,0 b7,3 − 5440a62,0 a21,1 b7,3
− 3200a52,0 a1,1 3 b10,0 − 8800a41,1 b9,1 a42,0
1920a62,0 a21,1 b9,1
8800a51,1 b10,0 a32,0
−768a52,0 a31,1 b9,1 − 768a82,0 b8,2
768a82,0 b6,4
12000a22,0 a61,1 b10,0
2880a62,0 a21,1 b6,4 − 3584a1,1 b9,1 a72,0
10320a42,0 a41,1 b8,2 − 17648a42,0 a41,1 b10,0
−640a72,0 a1,1 b8,2
1792a82,0 b8,2
7680a52,0 a31,1 b8,2 − 2688a72,0 b10,0 a1,1
2560a72,0 a1,1 b6,4 t3
768a62,0 a21,1 b10,0
768a62,0 a21,1 b8,2 − 768a72,0 a1,1 b7,3
2816b10,0 a82,0
768a42,0 a41,1 b10,0 t2
384a82,0 b7,3 − 384a72,0 a1,1 b8,2 − 384a52,0 a31,1 b10,0
384a62,0 a21,1 b9,1 − 512a82,0 b9,1
128a62,0 a21,1 b10,0
128a72,0 b10,0 a1,1 t
128b10,0 a82,0 − 128a1,1 b9,1 a72,0
128a82,0 b8,2 ,
A.4
Journal of Applied Mathematics
17
I0,8 16a52,0 a31,1 b7,3 − 35a71,1 b9,1 a2,0
35a81,1 b10,0 − 40a42,0 a41,1 b8,2
120a42,0 a41,1 b10,0
128b10,0 a2,0 8 − 70a22,0 a61,1 b10,0
32a62,0 a21,1 b8,2
−80a52,0 a31,1 b9,1 − 20a3 2,0 a51,1 b7,3
30a22,0 a61,1 b8,2 − 160a62,0 a21,1 b10,0
60a32,0 a51,1 b9,1
64a1,1 b9,1 a72,0
128a72,0 a1,1 b8,2 − 256a52,0 a31,1 b8,2
240a32,0 a51,1 b8,2
8a42,0 a41,1 b6,4 t8
640a52,0 a31,1 b10,0
128a82,0 b9,1 − 160a42,0 a41,1 b7,3 − 480a51,1 b10,0 a32,0
96a62,0 a21,1 b7,3
400a41,1 b9,1 a42,0 − 512a72,0 b10,0 a1,1 − 280a61,1 b9,1 a22,0
280a71,1 b10,0 a2,0
64a52,0 a31,1 b6,4 − 384a62,0 a21,1 b9,1 t7
128a82,0 b8,2
128b10,0 a82,0
− 464a52,0 a31,1 b7,3
768a62,0 a21,1 b10,0
832a52,0 a31,1 b9,1
800a22,0 a61,1 b10,0 − 512a62,0 a21,1 b8,2
3
− 1040a42,0 a41,1 b10,0 − 800a2,0 a51,1 b9,1
688a42,0 a41,1 b8,2
192a62,0 a21,1 b6,4
192a72,0 a1,1 b7,3
−384a1,1 b9,1 a72,0 t6
−512a52,0 a31,1 b10,0
384a62,0 a21,1 b9,1 − 256a72,0 a1,1 b8,2
128a82,0 b7,3 − 544a62,0 a21,1 b7,3
768a52,0 a31,1 b8,2
256a72,0 a1,1 b6,4
128a82,0 b9,1
880a51,1 b10,0 a32,0 − 384a72,0 b10,0 a1,1 − 880a41,1 b9,1 a42,0 t5
128b10,0 a82,0 − 256a1,1 b9,1 a72,0
128a62,0 a21,1 b8,2
128a82,0 b8,2
128a42,0 a41,1 b10,0
384a62,0 a21,1 b10,0 − 128a72,0 a1,1 b7,3 − 128a52,0 a31,1 b9,1 t4
−128a52,0 a31,1 b10,0 − 256a72,0 b10,0 a1,1
128a82,0 b9,1
128a82,0 b6,4
128a62,0 a21,1 b9,1
128a82,0 b7,3 − 128a72,0 a1,1 b8,2 t3
128a62,0 a21,1 b10,0
128b10,0 a82,0 − 128a1,1 b9,1 a72,0
128a82,0 b9,1 − 128a72,0 b10,0 a1,1 t
128a82,0 b8,2 t2
128b10,0 a82,0 .
A.5
Acknowledgments
The authors are very grateful to the referees for the useful remarks and suggestions.
This paper is partially supported by a National Key Basic Research of China Grant no.
2004CB318000.
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