1 Introduction and Statement of the Main Results
The second part of the 16th Hilbert’s problem asks for the maximum number H(n) of limit cycles that planar polynomial differential systems of degree n can have, see for instance [7, 8, 11], and the references quoted therein. The problem on the number H(n) remains open, even for n=2.
A weaker problem than the 16th Hilbert’s problem, known now as the weak 16th Hilbert’s problem was proposed by Arnold [2], who asked for the maximum number Z(m,n) of isolated zeros of Abelian integrals of all polynomial 1-forms of degree n over algebraic ovals of degree m, for more details on the weak 16th Hilbert’s problem see [4, 9, 19] and the hundreds of references quoted in these articles. Unfortunately the weak 16th Hilbert’s problem is also extremely hard to study. On the other hand, the weak 16th Hilbert’s problem is a particular case of the problem of studying the maximum number of limit cycles that can bifurcate from the periodic orbits of a center of a polynomial differential system of degree m-1 when it is perturbed inside the class of all polynomial differential systems of degree n. Of course Z(m,n)≤H(max(n,m-1)).
In this paper we provide lower bounds for the maximum number of limit cycles that can bifurcate from the periodic solutions of a polynomial differential uniform isochronous center of degree 3,5 and 7 when it its perturbed inside the class of all polynomial differential systems of the same degree. The main result is based on the averaging theory of first order. But here the main work is to study the maximum number of simple zeros of the obtained averaged functions, because not always the standard study of Extended Chebyshev systems (ET-systems) can be applied (see Appendix B). The study is based on some new results that can be applied when the family of functions that define ℱ is not an ET-system. Some delicate study using qualitative theory on some differential equations is also needed to complete the study.
More precisely, we consider the polynomial differential system
(1.1)
{
x
˙
=
-
y
+
x
2
y
(
x
2
+
y
2
)
n
,
y
˙
=
x
+
x
y
2
(
x
2
+
y
2
)
n
,
of degree 2n+3 with n≥0, having a uniform isochronous center at the origin of coordinates, which in polar coordinates (r,θ), where x=rsinθ and y=rcosθ, becomes
{
r
˙
=
r
2
n
+
3
cos
θ
sin
θ
,
θ
˙
=
1
.
Since θ˙=1, the origin of equation (1.1) is a uniform isochronous center, which taking as independent variable the variable θ writes
d
r
d
θ
=
r
′
=
r
2
n
+
3
cos
θ
sin
θ
.
An easy computation shows that the periodic solutions r(θ,r0) surrounding the center r=0 such that r(0,r0)=r0 are
(1.2)
r
(
θ
,
r
0
)
=
r
0
(
1
-
(
n
+
1
)
r
0
2
(
n
+
1
)
sin
2
θ
)
-
1
2
n
+
2
,
with 0<r0<(n+1)-12n+2. The global phase portraits, in the Poincaré disc, of system (1.1) for n=0,1,2 are shown in Figure 1.
Our purpose is to provide a lower bound for the maximum number of limit cycles that can bifurcate from the periodic solutions r(θ,r0) surrounding the uniform isochronous center at r=0 of degree 3,5,7 when we perturb it inside the class of all polynomial differential systems of degree 3,5,7, respectively. In other words, we study the number of limit cycles of the following three polynomial differential systems:
(1.3)
{
x
˙
=
-
y
+
x
2
y
+
ε
∑
i
+
j
=
0
3
a
i
j
x
i
y
j
,
y
˙
=
x
+
x
y
2
+
ε
∑
i
+
j
=
0
3
b
i
j
x
i
y
j
,
(1.4)
{
x
˙
=
-
y
+
x
2
y
(
x
2
+
y
2
)
+
ε
∑
i
+
j
=
0
5
a
i
j
x
i
y
j
,
y
˙
=
x
+
x
y
2
(
x
2
+
y
2
)
+
ε
∑
i
+
j
=
0
5
b
i
j
x
i
y
j
,
(1.5)
{
x
˙
=
-
y
+
x
2
y
(
x
2
+
y
2
)
2
+
ε
∑
i
+
j
=
0
7
a
i
j
x
i
y
j
,
y
˙
=
x
+
x
y
2
(
x
2
+
y
2
)
2
+
ε
∑
i
+
j
=
0
7
b
i
j
x
i
y
j
,
where ε is a small parameter.
Our main result is the following.
Theorem 1.1
For |ε|≠0 sufficiently small using averaging theory of first order we obtain that
system (1.3) can have up to 3 limit cycles and there are perturbations that only 0,1,2 and 3 limit cycles bifurcate from the center;
system (1.4) can have up to 8 limit cycles and there are perturbations that only 0,1,2,…,7 and 8 limit cycles bifurcate from the center;
there are perturbations that only 0,1,2,…,11 and 12 limit cycles bifurcate from the center of system (1.5).
In fact, in the plane ℝ2 the averaging theory of first order, or the generalized Abelian integrals, or the Melnikov function provide the same information because all these methods are based on the study of the first term in ε of the Poincaré return map. Some concrete applications of that theory to planar differential systems of low degree can be seen in [6, 16]. In higher dimension, the averaging theory can be also used, for example, for the study of the Hopf bifurcation, see [12, 13].
As we will see, by using the averaging theory of first order, the limit cycles of the perturbed system, which emerge from the period annulus of the isochronous center of system (1.1), correspond to the zeros of a linear combination of the functions f0,f1,…,f(n2+7n+6)/2, n=0,1,2. The proof of Theorem 1.1 for the case n=0 is easy and it is done in Section 2. But the difficulty arises evidently as n increases. For n=1, as the collection of functions f0,…,f7 is not an ET-system, part of our efforts have been focused on determining the numbers of simple zeros of Wronskian determinants W6(s) and W7(s), which have the expressions ∑i=0kai(s)Ei(s)Kk-i(s) (k=2,3), where ai is a polynomial of high degree, E and K are respectively the elliptic integrals of the first kind and second kind:
E
(
x
)
=
∫
0
π
/
2
1
-
x
sin
2
θ
𝑑
θ
,
K
(
x
)
=
∫
0
π
/
2
1
1
-
x
sin
2
θ
𝑑
θ
.
The proof is done using qualitative analysis and algebraic calculations. It turns out that all the Wronskian determinants but W6(s) do not vanish and the later has a unique zero which is simple. So the conditions of the classic Chebyshev criterion are not satisfied. According to the result of the recent paper [15], the maximum number of zeros of the linear combination of f0,…,f7 is less than or equal to 8. Consequently, another part of our efforts has been focused on proving that the possible upper bound 8 can be reached. To show this, we construct a function which has a zero of multiplicity 7 as well as an extra simple zero. Then, under suitable perturbation this function possesses 8 simple zeros. This is done in Section 3. For n=2, the corresponding functions f0,f1,…,f12, which contain several hypergeometric functions, is neither an Extended Complete Chebyshev system, nor a system satisfying the condition of [15]. We do not know how to find out the maximum number of zeros of all the possible linear combination of f0,f1,…,f12. Instead, we provide a lower bound for this number or zeros. This is done in Section 4.
3 Proof of Theorem 1.1 (b)
In this section we will study the number of limit cycles of system (1.4) by using averaging theory of first order. We will only prove that this maximum number is 8 because according to the proof, the reader can easily see that system (1.4) can have 0,1,2,…,8 limit cycles. First, let us state and prove the following lemma.
Lemma 3.1
The maximum number of limit cycles of system (1.4) which emerge from the period annulus around the center of system (1.4)ε=0, by using averaging theory of first order, is equal to the maximum number of simple zeros of the function
(3.1)
G
(
s
)
=
b
0
f
0
(
s
)
+
b
1
f
1
(
s
)
+
⋯
+
b
7
f
7
(
s
)
,
s
∈
(
0
,
1
)
,
where b0,b1,…,b7 are independent arbitrary constants and
(3.2)
{
f
0
(
s
)
=
(
1
-
s
)
2
,
f
1
(
s
)
=
(
1
-
s
)
(
1
-
s
2
)
,
f
2
(
s
)
=
(
1
-
s
2
)
2
,
f
3
(
s
)
=
(
1
-
s
)
(
1
-
s
2
)
2
,
f
4
(
s
)
=
(
1
-
s
2
)
3
,
f
5
(
s
)
=
(
1
-
s
2
)
5
/
2
g
1
(
s
)
,
f
6
(
s
)
=
(
1
-
s
2
)
1
/
2
(
g
1
(
s
)
-
g
2
(
s
)
)
-
1
2
(
1
-
s
2
)
g
2
(
s
)
,
f
7
(
s
)
=
(
1
-
s
2
)
3
/
2
(
g
1
(
s
)
-
g
2
(
s
)
)
,
with
(3.3)
g
1
(
s
)
=
2
E
(
1
-
s
2
)
,
g
2
(
s
)
=
2
s
2
K
(
1
-
s
2
)
.
Proof.
Under the polar coordinate transformation, system (1.4) can be changed to
(3.4)
d
r
d
θ
=
R
0
(
θ
,
r
)
+
ε
R
1
(
θ
,
r
)
+
O
(
ε
2
)
,
where R0(θ,r)=r5cosθsinθ and
R
1
(
θ
)
=
(
a
00
C
+
b
00
S
)
+
r
(
a
10
C
2
+
(
a
01
+
b
10
)
C
S
+
b
01
S
2
)
+
r
2
[
a
20
C
3
+
(
a
11
+
b
20
)
C
2
S
+
(
b
11
+
a
02
)
C
S
2
+
b
02
)
S
3
]
+
r
3
[
a
30
C
4
+
(
a
21
+
b
30
)
C
3
S
+
(
a
12
+
b
21
)
C
2
S
2
+
(
a
03
+
b
12
)
C
S
3
+
b
03
S
4
]
+
r
4
[
a
40
C
5
-
b
00
C
2
S
+
(
a
31
+
b
40
)
C
4
S
+
a
00
C
S
2
+
(
a
22
+
b
31
)
C
3
S
2
+
(
a
13
+
b
22
)
C
2
S
3
+
(
a
04
+
b
13
)
C
S
4
+
b
04
S
5
]
+
r
5
[
a
50
C
6
-
b
10
C
3
S
+
(
a
41
+
b
50
)
C
5
S
+
(
a
10
-
b
01
)
C
2
S
2
+
(
a
32
+
b
41
)
C
4
S
2
+
a
01
C
S
3
+
(
a
23
+
b
32
)
C
3
S
3
+
(
a
14
+
b
23
)
C
2
S
4
+
(
a
05
+
b
14
)
C
S
5
+
b
05
S
6
]
+
r
6
[
-
b
20
C
4
S
+
(
a
20
-
b
11
)
C
3
S
2
+
(
a
11
-
b
02
)
C
2
S
3
+
a
02
C
S
4
]
+
r
7
[
-
b
30
C
5
S
+
(
a
30
-
b
21
)
C
4
S
2
+
(
a
21
-
b
12
)
C
3
S
3
+
(
a
12
-
b
03
)
C
2
S
4
+
a
03
C
S
5
]
+
r
8
[
-
b
40
C
6
S
+
(
a
40
-
b
31
)
C
5
S
2
+
(
a
31
-
b
22
)
C
4
S
3
+
(
a
22
-
b
13
)
C
3
S
4
+
(
a
13
-
b
04
)
C
2
S
5
+
a
04
C
S
6
]
+
r
9
[
-
b
50
C
7
S
+
(
a
50
-
b
41
)
C
6
S
2
+
(
a
41
-
b
32
)
C
5
S
3
+
(
a
32
-
b
23
)
C
4
S
4
+
(
a
23
-
b
14
)
C
3
S
5
+
(
a
14
-
b
05
)
C
2
S
6
+
a
05
C
S
7
]
,
with C=cosθ, S=sinθ.
Equation (3.4)ε=0 has the periodic solutions r(θ,r0)=r0(1-2r04sin2θ)-1/4 satisfying r0=r(0,r0) for 0<r0<2-1/4. We solve the variational differential equation
d
M
d
θ
=
∂
∂
r
R
0
(
θ
,
r
(
θ
,
r
0
)
)
M
,
with Mr0(0)=1 and get the fundamental solution
M
r
0
(
θ
)
=
(
1
-
2
r
0
4
sin
2
θ
)
-
5
/
4
.
Next, a straightforward calculation leads to
ℱ
(
r
0
)
=
∫
0
2
π
M
r
0
-
1
(
θ
)
R
1
(
θ
,
r
(
θ
,
r
0
)
)
𝑑
θ
=
∫
0
2
π
r
0
(
1
-
2
r
0
4
S
2
)
(
c
00
+
c
02
S
2
+
c
04
S
4
+
c
06
S
6
+
c
40
C
4
+
c
60
C
6
(3.5)
+
c
22
C
2
S
2
+
c
42
C
4
S
2
+
c
62
C
6
S
2
+
c
24
C
2
S
4
+
c
44
C
4
S
4
+
c
26
C
2
S
6
+
Υ
(
C
,
S
)
)
d
θ
,
for r0∈(0,2-1/4), where Υ(C,S)=∑αi,jCiSj is a polynomial in C,S with i or j being an odd number, which leads to
∫
0
2
π
r
0
(
1
-
2
r
0
4
S
2
)
Υ
(
C
,
S
)
𝑑
θ
=
0
and
c
00
=
a
10
:=
e
0
,
c
02
=
-
a
10
+
b
01
:=
e
1
,
c
04
=
b
03
r
2
(
θ
,
r
0
)
:=
e
2
r
2
(
θ
,
r
0
)
,
c
06
=
b
05
r
4
(
θ
,
r
0
)
:=
e
3
r
4
(
θ
,
r
0
)
,
c
40
=
a
30
r
2
(
θ
,
r
0
)
:=
e
4
r
2
(
θ
,
r
0
)
,
c
60
=
a
50
r
4
(
θ
,
r
0
)
:=
e
5
r
4
(
θ
,
r
0
)
,
c
22
=
(
a
12
+
b
21
)
r
2
(
θ
,
r
0
)
+
(
a
10
-
b
01
)
r
4
(
θ
,
r
0
)
:=
e
6
r
2
(
θ
,
r
0
)
-
e
1
r
4
(
θ
,
r
0
)
,
c
24
=
(
a
14
+
b
23
)
r
4
(
θ
,
r
0
)
+
(
a
12
-
b
03
)
r
6
(
θ
,
r
0
)
:=
e
7
r
4
(
θ
,
r
0
)
+
e
8
r
6
(
θ
,
r
0
)
,
c
26
=
(
a
14
-
b
05
)
r
8
(
θ
,
r
0
)
:=
e
9
r
8
(
θ
,
r
0
)
,
c
42
=
(
a
32
+
b
41
)
r
4
(
θ
,
r
0
)
+
(
a
30
-
b
21
)
r
6
(
θ
,
r
0
)
:=
e
10
r
4
(
θ
,
r
0
)
+
e
11
r
6
(
θ
,
r
0
)
,
c
44
=
(
a
32
-
b
23
)
r
8
(
θ
,
r
0
)
:=
e
12
r
8
(
θ
,
r
0
)
,
c
62
=
(
a
50
-
b
41
)
r
8
(
θ
,
r
0
)
:=
e
13
r
8
(
θ
,
r
0
)
.
It is not hard to check that the constants e0,e1,…,e13 are independent. Computing (3.5), we get
ℱ
(
r
0
)
=
I
1
(
r
0
)
+
I
2
(
r
0
)
+
I
3
(
r
0
)
+
I
4
(
r
0
)
,
where
I
1
(
r
0
)
=
α
1
r
0
+
α
2
r
0
5
,
I
2
(
r
0
)
=
1
15
r
0
5
(
(
α
3
+
α
4
r
0
4
+
α
5
r
0
8
)
g
¯
1
(
r
0
)
-
(
α
3
+
(
α
3
+
α
4
)
r
0
4
)
g
¯
2
(
r
0
)
)
,
I
3
(
r
0
)
=
-
π
16
r
0
7
(
(
2
α
6
+
2
α
7
r
0
4
+
α
8
r
0
8
)
g
¯
3
(
r
0
)
+
(
2
α
6
r
0
4
+
(
α
6
+
2
α
7
)
r
0
8
+
(
α
6
+
α
7
+
α
8
)
r
0
12
)
)
,
I
4
(
r
0
)
=
-
1
30
r
0
5
(
(
4
α
9
+
α
10
r
0
4
-
(
7
α
9
+
α
10
)
r
0
8
)
g
¯
1
(
r
0
)
-
(
4
α
9
+
(
4
α
9
+
α
10
)
r
0
4
)
g
¯
2
(
r
0
)
)
,
with
α
1
=
π
(
2
e
0
+
e
1
)
,
α
2
=
π
8
(
-
16
e
0
-
14
e
1
+
5
e
3
+
5
e
5
+
e
7
+
e
10
)
,
α
3
=
-
e
2
-
e
4
+
e
6
,
α
4
=
-
3
e
2
+
7
e
4
-
2
e
6
,
α
5
=
16
e
2
+
6
e
4
+
4
e
6
,
α
6
=
e
9
-
e
12
+
e
13
,
α
7
=
2
e
12
-
4
e
13
,
α
8
=
e
13
,
α
9
=
-
e
8
+
e
11
,
α
10
=
3
e
8
-
13
e
11
,
and
g
¯
1
(
r
0
)
=
E
(
2
r
0
4
)
+
1
-
2
r
0
4
E
(
1
-
1
/
(
1
-
2
r
0
4
)
)
,
g
¯
2
(
r
0
)
=
(
1
-
2
r
0
4
)
K
(
2
r
0
4
)
+
1
-
2
r
0
4
K
(
1
-
1
/
(
1
-
2
r
0
4
)
)
,
g
¯
3
(
r
0
)
=
1
-
2
r
0
4
-
1
.
Using the expression of each αi, one can easily check that α1,α2,…,α10 are independent constants.
To simplify the computation, we let s=(1-2r04)1/2, s∈(0,1). Using the definition of the elliptic functions, we have
s
E
(
1
-
1
/
s
2
)
+
E
(
1
-
s
2
)
=
2
E
(
1
-
s
2
)
,
s
K
(
1
-
1
/
s
2
)
+
s
2
K
(
1
-
s
2
)
=
2
s
2
K
(
1
-
s
2
)
.
Hence we obtain
240
r
0
7
ℱ
(
r
0
)
=
G
(
s
)
=
b
0
f
0
(
s
)
+
b
1
f
1
(
s
)
+
⋯
+
b
7
f
7
(
s
)
,
s
∈
(
0
,
1
)
,
where fi(s), i=0,1,…,7, are the functions defined in (3.2), and the constants b0,b1,…,b7 in (3.1) are independent constants each of which is a linear combination of α1,α2,…,α10.
By Theorem A.1, the lemma is proved. ∎
Next, we denote by Wi(s) the Wronskian determinant for the functions f0,f1,…,fi depending on s:
W
i
(
s
)
=
W
(
f
0
,
…
,
f
i
)
(
s
)
,
i
=
0
,
1
,
…
,
7
.
In what follows we will show that all the Wronskian determinants have no zeros except W6 which vanishes at a unique zero, which is simple.
By direct calculation we obtain
(3.6)
{
W
0
(
s
)
=
(
1
-
s
)
2
,
W
1
(
s
)
=
(
1
-
s
)
4
,
W
2
(
s
)
=
2
(
1
-
s
)
6
,
W
3
(
s
)
=
-
12
(
1
-
s
)
8
,
W
4
(
s
)
=
288
(
1
-
s
)
10
,
W
5
(
s
)
=
Y
5
(
s
)
(
Z
50
(
s
)
g
2
(
s
)
+
Z
51
(
s
)
g
1
(
s
)
)
,
W
6
(
s
)
=
Y
6
(
s
)
(
Z
60
(
s
)
g
2
2
(
s
)
+
Z
61
(
s
)
g
2
(
s
)
g
1
(
s
)
+
Z
62
(
s
)
g
1
2
(
s
)
)
,
W
7
(
s
)
=
Y
7
(
s
)
(
Z
70
(
s
)
g
2
3
(
s
)
+
Z
71
(
s
)
g
2
2
(
s
)
g
1
(
s
)
+
Z
72
(
s
)
g
2
(
s
)
g
1
2
(
s
)
+
Z
73
(
s
)
g
1
3
(
s
)
)
,
where
Y
5
(
s
)
=
288
s
-
3
(
1
-
s
)
15
/
2
(
1
+
s
)
-
5
/
2
,
Z
50
(
s
)
=
1
-
20
s
-
33
s
2
-
20
s
3
+
s
4
,
Z
51
(
s
)
=
-
2
(
1
-
5
s
+
10
s
2
+
5
s
3
+
10
s
4
-
5
s
5
+
s
6
)
,
Y
6
(
s
)
=
432
s
-
6
(
1
-
s
)
2
(
1
+
s
)
-
8
,
Z
60
(
s
)
=
1
-
s
2
(
-
330
-
761
s
+
3720
s
2
+
25036
s
3
+
63490
s
4
+
100713
s
5
+
102410
s
6
+
66145
s
7
+
23800
s
8
+
3760
s
9
-
770
s
10
-
269
s
11
)
-
2
(
210
+
637
s
-
2490
s
2
-
22210
s
3
-
67910
s
4
-
129477
s
5
-
160950
s
6
-
135498
s
7
-
74890
s
8
-
25715
s
9
-
2910
s
10
+
728
s
11
+
240
s
12
)
,
Z
61
(
s
)
=
1
-
s
2
(
660
+
1544
s
-
2430
s
2
-
6587
s
3
+
18350
s
4
+
65033
s
5
+
107880
s
6
+
106430
s
7
+
64350
s
8
+
11893
s
9
-
4790
s
10
-
2177
s
11
+
1580
s
12
+
536
s
13
)
+
4
(
210
+
644
s
-
150
s
2
-
2133
s
3
+
1670
s
4
+
9718
s
5
+
23630
s
6
+
29805
s
7
+
27970
s
8
+
14822
s
9
+
3430
s
10
-
2877
s
11
-
300
s
12
+
736
s
13
+
240
s
14
)
,
Z
62
(
s
)
=
s
1
-
s
2
(
-
44
-
1410
s
+
405
s
2
+
6880
s
3
-
6789
s
4
-
63230
s
5
-
111946
s
6
-
111330
s
7
-
57735
s
8
-
6880
s
9
+
7555
s
10
-
270
s
11
-
1146
s
12
-
80
s
13
+
4
s
14
)
-
2
s
(
28
+
1590
s
+
1873
s
2
-
8750
s
3
-
4018
s
4
+
81350
s
5
+
209051
s
6
+
273750
s
7
+
208594
s
8
+
80650
s
9
-
4307
s
10
-
8050
s
11
+
1712
s
12
+
1560
s
13
+
32
s
14
)
,
Y
7
(
s
)
=
-
3888
s
-
9
(
1
-
s
)
-
7
/
2
(
1
+
s
)
-
27
/
2
,
Z
70
(
s
)
=
1
-
s
2
(
12600
+
126840
s
+
317850
s
2
+
52804
s
3
-
990100
s
4
-
1480411
s
5
+
1111220
s
6
+
6912788
s
7
+
11963620
s
8
+
12059608
s
9
+
7333520
s
10
+
1499322
s
11
-
1510040
s
12
-
1134068
s
13
+
73250
s
14
+
339916
s
15
+
124820
s
16
+
15361
s
17
+
60
s
18
)
-
8
1
-
s
2
(
1
-
s
2
)
3
s
9
(
1
+
s
)
10
(
3150
+
31710
s
+
95130
s
2
+
77561
s
3
-
209720
s
4
-
618425
s
5
-
351890
s
6
+
1120612
s
7
+
3107560
s
8
+
4115235
s
9
+
3350990
s
10
+
1407368
s
11
-
195310
s
12
-
611035
s
13
-
252580
s
14
+
65059
s
15
+
103320
s
16
+
36240
s
17
+
3600
s
18
)
,
Z
71
(
s
)
=
3
1
-
s
2
(
-
8400
-
85120
s
-
232660
s
2
-
173008
s
3
+
437450
s
4
+
811463
s
5
-
2067190
s
6
-
10578234
s
7
-
22263600
s
8
-
30680904
s
9
-
30782260
s
10
-
22474150
s
11
-
10890340
s
12
-
2297236
s
13
+
798740
s
14
+
530744
s
15
-
174630
s
16
-
249833
s
17
-
82630
s
18
-
10282
s
19
-
80
s
20
)
-
24
(
2100
+
21280
s
+
69900
s
2
+
98437
s
3
-
32560
s
4
-
271689
s
5
+
257650
s
6
+
2845680
s
7
+
7627330
s
8
+
12561491
s
9
+
14784900
s
10
+
12847024
s
11
+
8032670
s
12
+
3146115
s
13
+
365000
s
14
-
280461
s
15
-
58190
s
16
+
94628
s
17
+
76050
s
18
+
24320
s
19
+
2400
s
20
)
,
Z
72
(
s
)
=
3
s
1
-
s
2
(
1120
+
43540
s
+
298396
s
2
+
428670
s
3
-
487118
s
4
-
1615600
s
5
+
1598463
s
6
+
14431800
s
7
+
34113716
s
8
+
48793180
s
9
+
48574944
s
10
+
34190000
s
11
+
15087426
s
12
+
1926380
s
13
-
1797616
s
14
-
499770
s
15
+
470882
s
16
+
285160
s
17
+
48663
s
18
-
1040
s
19
+
84
s
20
+
80
s
21
)
+
24
s
(
280
+
13530
s
+
98799
s
2
+
200770
s
3
-
31141
s
4
-
611730
s
5
+
35248
s
6
+
4717630
s
7
+
13875311
s
8
+
23495150
s
9
+
27804240
s
10
+
24063400
s
11
+
14574499
s
12
+
5157020
s
13
+
149657
s
14
-
658770
s
15
-
66644
s
16
+
208280
s
17
+
108756
s
18
+
14970
s
19
+
320
s
20
)
,
Z
73
(
s
)
=
s
2
1
-
s
2
(
480
-
10712
s
-
177820
s
2
-
762730
s
3
-
87610
s
4
+
2806841
s
5
+
909430
s
6
-
14409046
s
7
-
40794440
s
8
-
62886616
s
9
-
63914340
s
10
-
42197910
s
11
-
14285100
s
12
+
1389476
s
13
+
2447740
s
14
-
233714
s
15
-
691850
s
16
-
196931
s
17
+
7070
s
18
+
1990
s
19
-
760
s
20
-
8
s
21
)
-
8
s
2
(
270
+
538
s
+
49880
s
2
+
295952
s
3
+
320270
s
4
-
754291
s
5
-
1276480
s
6
+
4132165
s
7
+
18820380
s
8
+
36641394
s
9
+
45116550
s
10
+
37467621
s
11
+
19916970
s
12
+
4809620
s
13
-
1224770
s
14
-
907079
s
15
+
291730
s
16
+
324433
s
17
+
58270
s
18
+
1472
s
19
+
180
s
20
)
.
Lemma 3.2
Let g1 and g2 be the two functions defined in (3.3) and let h(s)=g1(s)/g2(s). Then h(s)>0, h′(s)<0, s∈(0,1) and
(3.7)
lim
s
→
0
+
h
(
s
)
=
+
∞
,
lim
s
→
1
-
h
(
s
)
=
1
,
lim
s
→
0
+
h
′
(
s
)
=
-
∞
,
lim
s
→
1
-
h
′
(
s
)
=
-
1
.
Proof.
It follows directly from the definition of the elliptic integral that gi(s)>0 (i=1,2), s∈(0,1) and hence h(s)>0, s∈(0,1). A direct computation shows that
g
1
(
s
)
=
1
-
1
2
s
2
log
s
+
1
4
s
2
(
4
log
2
-
1
)
+
o
(
s
2
)
,
g
2
(
s
)
=
-
s
2
log
s
+
2
s
2
log
2
+
o
(
s
2
)
,
where s→0+. Thus the first and the third equalities of (3.7) hold.
Similarly, we find that
(3.8)
{
g
1
(
s
)
=
π
2
(
1
-
1
2
(
1
-
s
)
+
1
16
(
1
-
s
)
2
+
O
(
(
1
-
s
)
3
)
)
,
g
2
(
s
)
=
π
2
(
1
-
3
2
(
1
-
s
)
+
5
16
(
1
-
s
)
2
+
O
(
(
1
-
s
)
3
)
)
,
as s→1-. This verifies the second and the fourth equalities of (3.7).
Next we go to prove that h′(s)<0, s∈(0,1). By straightforward calculation we find
d
h
/
d
s
=
(
1
-
2
h
+
h
2
s
2
)
/
(
s
-
s
3
)
.
Hence h(s) is a solution of system
(3.9)
h
˙
=
s
2
h
2
-
2
h
+
1
,
s
˙
=
s
-
s
3
.
System (3.9) has two invariant straight lines s=0 and s=1 as well as two singularities at S1(0,1/2) and S2(1,1), where S1 is a saddle and S2 is a saddle-node of system (3.9). Moreover, system (3.9) has two horizontal isocline curves
(3.10)
h
+
(
s
)
=
1
1
-
1
-
s
2
and
h
-
(
s
)
=
1
1
+
1
-
s
2
,
satisfying
h
+
′
(
s
)
<
0
,
h
-
′
(
s
)
>
0
,
h
+
(
0
)
=
+
∞
,
h
-
(
0
)
=
1
2
,
h
+
(
1
)
=
h
-
(
1
)
=
1
.
Obviously,
(3.11)
h
′
(
s
)
=
s
2
(
h
(
s
)
-
h
+
(
s
)
)
h
(
s
)
-
h
-
(
s
)
s
-
s
3
.
In view of (3.8) and (3.10), it follows that
(3.12)
h
-
(
s
)
<
h
(
s
)
<
h
+
(
s
)
,
s
→
1
-
.
We assert that
(3.13)
h
-
(
s
)
<
h
(
s
)
<
h
+
(
s
)
,
s
∈
(
0
,
1
)
.
Indeed, if there exists some point s0∈(0,1) such that h(s0)≥h+(s0), then by (3.11) we find h′(s0)≥0. By the monotonicity of h+(s) we know that h(s)>h+(s) for all s0<s<1. This contradicts (3.12). Hence h(s)<h+(s) for s∈(0,1). If there exists some point s0∈(0,1) such that h(s0)=h-(s0), then by (3.11) we know that h′(s0)=0. Since h-′(s0)>0, it follows h(s)<h-(s) for s→s0+. Using this fact, we find that the curve h=h(s) cannot go across the curve h=h-(s) at any point s1>s0 because once h(s1)=h-(s1), it must hold that h(s)<h-(s) for s→s1+. This also contradicts (3.12). Hence h(s)>h-(s), s∈(0,1).
Finally, combining (3.13) and (3.11), we conclude that dh/ds<0, s∈(0,1). ∎
Lemma 3.3
The function W5(s) does not vanish in the open interval (0,1).
Proof.
Using Sturm’s theorem (see [18]) and Z51(0)=-2, we find that Z51(s)<0 for all s∈(0,1). Hence we have
(3.14)
W
5
(
s
)
=
Y
5
(
s
)
Z
51
(
s
)
g
2
(
s
)
(
Z
50
(
s
)
Z
51
(
s
)
+
g
1
(
s
)
g
2
(
s
)
)
,
s
∈
(
0
,
1
)
.
A direct computation leads to Z50(s0)=0, where
s
0
=
5
+
3
15
/
2
-
231
+
60
15
/
2
≈
0.0463551
.
Again, by Sturm’s theorem we find that Z50(s)>0 for s∈(0,s0) and Z50(s)<0 for s∈(s0,1). Further,
d
d
s
(
Z
50
(
s
)
Z
51
(
s
)
)
=
2
p
9
(
s
)
Z
51
2
(
s
)
,
where
p
9
(
s
)
=
15
+
86
s
-
290
s
2
-
364
s
3
-
575
s
4
-
274
s
5
+
190
s
6
+
68
s
7
-
65
s
8
+
2
s
9
.
Using Sturm’s theorem, we get that p9(s)>0, s∈(0,1/5). This fact, combined with Z50(0)/Z51(0)=-1/2, yields that
(3.15)
Z
50
(
s
)
Z
51
(
s
)
>
0
for
s
∈
(
s
0
,
1
)
and
Z
50
(
s
)
Z
51
(
s
)
∈
(
-
1
2
,
0
)
for
s
∈
(
0
,
s
0
)
.
Since by Lemma 3.2 we have g1(s)/g2(s)>1, it follows from (3.14) and (3.15) that W5(s)≠0 for all s∈(0,1). ∎
Next, we will determine the sign of the functions W6(s) and W7(s). In order to make the computation easier we need to make the transformation of variable r=((1-s)/(1+s))1/2 or equivalently, s=(1-r2)/(1+r2). We also need the following lemma. Let
(3.16)
h
¯
(
r
)
=
g
1
(
s
)
g
2
(
s
)
|
s
=
(
1
-
r
2
)
/
(
1
+
r
2
)
,
r
∈
(
0
,
1
)
.
Lemma 3.4
The function h=h¯(r) is the solution of the differential system
(3.17)
h
˙
=
(
(
r
-
1
)
2
h
-
r
2
-
1
)
(
(
r
+
1
)
2
h
-
r
2
-
1
)
,
r
˙
=
r
(
r
4
-
1
)
,
satisfying h¯′(r)>0 for r∈(0,1), h¯(0)=1, and limr→1-h¯(r)=+∞.
Proof.
The conclusion follows from the proof of Lemma 3.2 by direct calculation. ∎
Lemma 3.5
The function W6(s) has a unique zero in (0,1) and this zero is simple.
Proof.
Let s=(1-r2)/(1+r2) for 0<r<1. Then it follows from the definition of W6(s) that
W
¯
6
(
r
)
:=
W
6
(
1
-
r
2
1
+
r
2
)
=
Y
¯
6
(
r
)
g
¯
2
2
(
r
)
(
C
60
(
r
)
+
C
61
(
r
)
h
¯
(
r
)
+
C
62
(
r
)
h
¯
2
(
r
)
)
,
where h¯(r) is the function defined in (3.16) and
Y
¯
6
(
r
)
=
2
Y
6
(
s
)
(
1
+
r
2
)
16
|
s
=
(
1
-
r
2
)
/
(
1
+
r
2
)
,
g
¯
2
(
r
)
=
g
2
(
s
)
|
s
=
(
1
-
r
2
)
/
(
1
+
r
2
)
,
C
60
(
r
)
=
-
(
1
+
r
2
)
4
(
-
620235
-
386944
r
+
63082
r
2
+
4352
r
3
+
1114260
r
4
+
747808
r
5
-
136770
r
6
-
21280
r
7
+
349425
r
8
+
201312
r
9
-
5852
r
10
+
97568
r
11
+
90000
r
12
+
59232
r
13
+
29692
r
14
-
39392
r
15
-
39225
r
16
-
6880
r
17
-
10350
r
18
+
18912
r
19
+
24540
r
20
-
1152
r
21
-
1242
r
22
+
2304
r
23
+
2835
r
24
)
,
C
61
(
r
)
=
2
(
1
+
r
2
)
2
(
107415
+
181136
r
-
33728
r
2
+
11072
r
3
-
399945
r
4
-
731304
r
5
+
149536
r
6
-
47384
r
7
+
1468275
r
8
+
1717984
r
9
-
340256
r
10
+
125584
r
11
+
2454435
r
12
+
1436304
r
13
-
95680
r
14
+
188016
r
15
+
156165
r
16
-
122704
r
17
+
100096
r
18
-
61184
r
19
-
145275
r
20
-
30376
r
21
-
23136
r
22
+
33384
r
23
+
39345
r
24
-
1632
r
25
-
2592
r
26
+
4464
r
27
+
5985
r
28
)
,
C
62
(
r
)
=
(
1
-
r
2
)
(
-
835065
-
346016
r
-
836827
r
2
-
363808
r
3
+
3987923
r
4
+
1598928
r
5
+
4009041
r
6
+
1699296
r
7
-
8764869
r
8
-
3404464
r
9
-
8875247
r
10
-
3660192
r
11
+
1910023
r
12
+
319232
r
13
+
2054749
r
14
+
561472
r
15
+
3501349
r
16
+
1423232
r
17
+
3542143
r
18
+
1373728
r
19
-
275927
r
20
+
31056
r
21
-
405789
r
22
-
8224
r
23
+
15321
r
24
+
43728
r
25
+
44043
r
26
+
4512
r
27
+
2493
r
28
+
6624
r
29
+
9135
r
30
)
.
Define
w
¯
6
(
r
,
h
)
=
C
60
(
r
)
+
C
61
(
r
)
h
+
C
62
(
r
)
h
2
.
We will show that on the curve C:={(r,h)∣w¯6(r,h)=0,r∈(0,1)}, there is a unique point P at which vector field (3.17) is tangent to C. We call P the contact point with the vector field (3.17). In fact, by direct computation we obtain
D
(
r
,
h
)
:=
(
∂
w
¯
6
/
∂
r
,
∂
w
¯
6
/
∂
h
)
⋅
(
r
˙
,
h
˙
)
=
-
2
∑
i
=
0
3
d
i
(
r
)
h
i
,
where
d
0
(
r
)
=
(
1
+
r
2
)
4
(
-
107415
+
12336
r
+
2451586
r
2
+
1530176
r
3
-
4561843
r
4
-
2896872
r
5
-
3880856
r
6
-
2845432
r
7
+
4366665
r
8
+
2321984
r
9
-
1985574
r
10
-
1627056
r
11
-
175627
r
12
-
500432
r
13
-
524832
r
14
+
797392
r
15
+
938867
r
16
+
960688
r
17
+
476566
r
18
-
543968
r
19
-
529425
r
20
-
119176
r
21
-
195912
r
22
+
200040
r
23
+
275163
r
24
-
24288
r
25
-
27378
r
26
+
31248
r
27
+
39375
r
28
)
,
d
1
(
r
)
=
-
(
1
+
r
2
)
2
(
-
1049895
-
889424
r
-
726170
r
2
-
1522240
r
3
+
7709074
r
4
+
7536696
r
5
+
1889478
r
6
+
6432616
r
7
-
31052618
r
8
-
28741640
r
9
-
6577218
r
10
-
16058648
r
11
-
6172022
r
12
+
328016
r
13
-
22042946
r
14
-
12811872
r
15
+
33763160
r
16
+
23233568
r
17
-
4741326
r
18
+
5413264
r
19
+
1388182
r
20
-
2485928
r
21
+
3589522
r
22
-
1915736
r
23
-
3612822
r
24
-
869704
r
25
-
768726
r
26
+
678984
r
27
+
822606
r
28
-
77664
r
29
-
113814
r
30
+
122832
r
31
+
170415
r
32
)
,
d
2
(
r
)
=
-
(
1
-
r
4
)
(
1777545
+
1046176
r
+
3311818
r
2
+
1457408
r
3
-
16459052
r
4
-
8289520
r
5
-
15874018
r
6
-
7006192
r
7
+
62475516
r
8
+
28826544
r
9
+
35342330
r
10
+
15708784
r
11
-
49795012
r
12
-
19466016
r
13
-
8698050
r
14
-
3515776
r
15
-
24983266
r
16
-
12853376
r
17
-
14258354
r
18
-
5372832
r
19
+
31346828
r
20
+
11380816
r
21
+
2668682
r
22
+
500624
r
23
-
4087764
r
24
-
691664
r
25
-
471570
r
26
+
404016
r
27
+
455268
r
28
-
51264
r
29
-
120294
r
30
+
84960
r
31
+
121905
r
32
)
,
d
3
(
r
)
=
-
(
1
-
r
2
)
3
(
-
835065
-
346016
r
-
836827
r
2
-
363808
r
3
+
3987923
r
4
+
1598928
r
5
+
4009041
r
6
+
1699296
r
7
-
8764869
r
8
-
3404464
r
9
-
8875247
r
10
-
3660192
r
11
+
1910023
r
12
+
319232
r
13
+
2054749
r
14
+
561472
r
15
+
3501349
r
16
+
1423232
r
17
+
3542143
r
18
+
1373728
r
19
-
275927
r
20
+
31056
r
21
-
405789
r
22
-
8224
r
23
+
15321
r
24
+
43728
r
25
+
44043
r
26
+
4512
r
27
+
2493
r
28
+
6624
r
29
+
9135
r
30
)
.
Using Sturm’s theorem, we find d3(r)>0, r∈(0,1). Further, the resultant of w¯6(r,h) and D(r,h) with respect to h is a polynomial in the variable r of degree 166, which, applying again Sturm’s theorem, can be proved to have a unique simple zero r0∈(0,1) with 9/10<r0<91/100. Hence there exists a unique h0 such that
w
¯
6
(
r
0
,
h
0
)
=
D
(
r
0
,
h
0
)
=
0
.
This confirms that on the curve C there is a unique point (r0,h0) at which the vector field (3.17) is tangent to C.
By direct computation we have
C
61
2
-
4
C
60
C
62
=
3600
(
1
+
r
2
)
4
p
56
(
r
)
,
where p56(r) is a polynomial of degree 56. Again, we can apply Sturm’s theorem to prove that p56(r)>0 and C62(r)<0 in r∈(0,1). Let
C
-
=
{
h
=
h
¯
-
(
r
)
=
-
C
61
-
C
61
2
-
4
C
60
C
62
2
C
62
}
and
C
+
=
{
h
=
h
¯
+
(
r
)
=
-
C
61
+
C
61
2
-
4
C
60
C
62
2
C
62
}
be the two branches of the curve C in the (r,h)-plane. A calculation shows that
h
¯
-
(
0
)
=
1
,
h
¯
+
(
0
)
=
-
179
241
,
h
¯
(
0
)
=
1
,
h
¯
-
′
(
0
)
=
64
231
,
h
¯
+
′
(
0
)
=
96160
1916673
,
h
¯
′
(
0
)
=
0
,
h¯+(1)=1/2 and when r→1-,
h
¯
-
(
r
)
=
15
1
-
r
+
⋯
,
h
¯
(
r
)
=
1
(
log
4
-
log
(
1
-
r
)
)
(
r
-
1
)
2
+
⋯
,
where the dots denote the terms which are infinitesimal being compared to the former one.
It follows that
h
¯
+
(
r
)
<
h
¯
(
r
)
<
h
¯
-
(
r
)
as
r
→
0
+
,
h
¯
+
(
r
)
<
h
¯
-
(
r
)
<
h
¯
(
r
)
as
r
→
1
-
.
Obviously, the curve Γ={h=h¯(r)} intersects C- in at least one point (r∗,h∗). By an observation on the direction of vector field (3.17) at the two endpoints of the segment of curve {(r,h)∣h=h¯-(r),r∈(0,r∗]}, we find that there exists a point P at which the vector field (3.17) is tangent to the curve C- (see Figure 2). Since the contact point P is unique, the curve Γ cannot intersect C- in other points. Moreover, the curve Γ has no common point with C+, otherwise a second contact point would emerge. Therefore the function w¯6(r,h¯(r)) has a unique zero in the interval (0,1). This yield that W¯6(r) has a unique zero in the interval (0,1).
Finally, since r0<91/100 and h¯(91/100)≈30.54045135<h-(91/100)≈35.81140037, it follows that r0<r∗. This means that (r∗,h∗) is not the contact point of C- with the vector field. Therefore, the unique zero of W¯6(r) is simple and thus the required conclusion holds. ∎
Lemma 3.6
The function W7(s) does not vanish in the open interval (0,1).
Proof.
By taking transformation s=(1-r2)/(1+r2),0<r<1, we obtain from
W
7
(
s
)
=
Y
7
(
s
)
g
2
3
(
s
)
(
Z
70
(
s
)
+
Z
71
(
s
)
g
1
(
s
)
g
2
(
s
)
+
Z
72
(
s
)
g
1
2
(
s
)
g
2
2
(
s
)
+
Z
73
(
s
)
g
1
3
(
s
)
g
2
3
(
s
)
)
that
W
¯
7
(
r
)
:=
W
7
(
s
)
|
s
=
(
1
-
r
2
)
/
(
1
+
r
2
)
=
Y
¯
7
(
r
)
g
¯
2
3
(
r
)
w
7
(
r
)
,
with
w
7
(
r
)
=
C
70
(
r
)
+
C
71
(
r
)
h
¯
(
r
)
+
C
72
(
r
)
h
¯
2
(
r
)
+
C
73
(
r
)
h
¯
3
(
r
)
,
where h¯(r) is the function defined in (3.16) and
Y
¯
7
(
r
)
=
16
r
Y
7
(
s
)
(
1
+
r
2
)
45
|
s
=
(
1
-
r
2
)
/
(
1
+
r
2
)
,
g
¯
2
(
r
)
=
g
2
(
s
)
|
s
=
(
1
-
r
2
)
/
(
1
+
r
2
)
,
C
70
(
r
)
=
4
r
(
2301810
+
59701740
r
2
+
727558755
r
4
-
364073500959
r
6
+
3428595727383
r
8
-
10544549722741
r
10
+
3730074158113
r
12
+
49576965802069
r
14
-
40961684822285
r
16
-
396767446632771
r
18
+
1609108209115716
r
20
-
3203874112868486
r
22
+
4488939441796380
r
24
-
4090928490421940
r
26
+
3532364976473268
r
28
-
553179108109916
r
30
+
1013150621056664
r
32
+
1960313990634764
r
34
+
589939846153370
r
36
+
1620210743086006
r
38
+
485382645874034
r
40
+
531165876327274
r
42
+
41364680501774
r
44
-
94823180435898
r
46
-
177772367419966
r
48
-
169241753304026
r
50
-
122994867094516
r
52
-
75347197109816
r
54
-
38883154071124
r
56
-
17219939255972
r
58
-
6420781546876
r
60
-
1958359101836
r
62
-
448781050130
r
64
-
53625628864
r
66
+
12582124635
r
68
+
10801790529
r
70
+
4279675055
r
72
+
1249653091
r
74
+
292463001
r
76
+
55639117
r
78
+
8432979
r
80
+
963045
r
82
+
73560
r
84
+
2790
r
86
)
,
C
71
(
r
)
=
-
3
(
1
+
r
2
)
23
(
62214075
+
32618040
r
+
57649725
r
2
+
31801560
r
3
-
311083290
r
4
-
161979460
r
5
-
284045278
r
6
-
156667936
r
7
+
1118662427
r
8
+
566796672
r
9
+
1015532421
r
10
+
545264968
r
11
+
1483781736
r
12
+
882038312
r
13
+
1421172920
r
14
+
856116840
r
15
+
32362070
r
16
-
26403192
r
17
+
175436346
r
18
+
24968536
r
19
-
2404284
r
20
-
55354400
r
21
+
31425964
r
22
-
122819896
r
23
-
129246866
r
24
-
82418248
r
25
-
147649550
r
26
-
83415400
r
27
-
50931800
r
28
-
40785832
r
29
-
66284296
r
30
-
5366728
r
31
-
2020081
r
32
-
3163552
r
33
-
1758167
r
34
+
617456
r
35
+
624838
r
36
+
127460
r
37
+
489090
r
38
+
19640
r
39
+
50775
r
40
+
14760
r
41
+
40425
r
42
)
,
C
72
(
r
)
=
3
(
-
1
+
r
2
)
(
1
+
r
2
)
21
(
-
113149575
-
48972840
r
-
219419550
r
2
-
96894000
r
3
+
524903715
r
4
+
226248140
r
5
+
1216381628
r
6
+
542101276
r
7
-
1447999469
r
8
-
628726896
r
9
-
3888634006
r
10
-
1779992536
r
11
-
2876628303
r
12
-
1259091264
r
13
-
2141852592
r
14
-
754743200
r
15
+
1112715114
r
16
+
583398240
r
17
+
4296755236
r
18
+
1852262832
r
19
+
2357760878
r
20
+
907216248
r
21
+
687064168
r
22
+
234254008
r
23
+
208154838
r
24
+
151069392
r
25
-
302393084
r
26
-
153692000
r
27
-
210572686
r
28
-
204720480
r
29
-
215766192
r
30
-
104223744
r
31
-
131334803
r
32
-
41765176
r
33
-
20357126
r
34
-
11754496
r
35
-
11547809
r
36
+
729436
r
37
+
904508
r
38
+
33740
r
39
+
444615
r
40
+
42400
r
41
+
114450
r
42
+
18360
r
43
+
50925
r
44
)
,
C
73
(
r
)
=
-
(
-
1
+
r
2
)
2
(
1
+
r
2
)
20
(
164085075
+
58271640
r
+
318975300
r
2
+
115256400
r
3
-
920074365
r
4
-
320089980
r
5
-
2079306888
r
6
-
746616756
r
7
+
2728744989
r
8
+
941220096
r
9
+
7172532884
r
10
+
2616432392
r
11
+
779348149
r
12
+
546885328
r
13
-
4995512160
r
14
-
1385204368
r
15
-
4685978914
r
16
-
1569573552
r
17
-
4525683128
r
18
-
2222726784
r
19
+
166924638
r
20
-
650651016
r
21
+
4249280720
r
22
+
1445886696
r
23
+
2224407802
r
24
+
1065394144
r
25
+
814165128
r
26
+
244092272
r
27
+
320088074
r
28
-
95701872
r
29
-
346329440
r
30
-
166958928
r
31
-
180571809
r
32
-
99999912
r
33
-
42394764
r
34
-
27896336
r
35
-
30911889
r
36
+
462756
r
37
+
365688
r
38
-
212820
r
39
+
107865
r
40
+
50400
r
41
+
137700
r
42
+
21960
r
43
+
61425
r
44
)
.
The number of zeros of W¯7(r) in (0,1) equals the number of intersection points of the curve
C
=
{
C
70
(
r
)
+
C
71
(
r
)
h
+
C
72
(
r
)
h
2
+
C
73
(
r
)
h
3
=
0
}
with the curve Γ={h=h¯(r)} in the (r,h)-plane. In what follows we will study the relative positions of C and Γ. To this end, since Γ is not an algebraic curve, we need to establish another auxiliary algebraic curve which is easier for computation.
First, using Sturm’s theorem, we find that C73(r)≠0, r∈(0,1). This means that
w
¯
7
(
r
,
h
)
:=
C
70
(
r
)
+
C
71
(
r
)
h
+
C
72
(
r
)
h
2
+
C
73
(
r
)
h
3
is a cubic polynomial of h for each fixed r∈(0,1). Let
A
=
C
72
2
-
3
C
71
C
73
,
B
=
C
71
C
72
-
9
C
70
C
73
,
C
=
C
71
2
-
3
C
70
C
72
,
and Δ=B2-4AC. It is not hard to see that Δ has exactly two zeros r1,r2 in (0,1) with 39/50<r1<79/100, 91/100<r2<23/25. If r∈(0,r1)∪(r2,1) then Δ>0; if r∈(r1,r2) then Δ<0. Therefore, the curve C has three branches C1 (the lower branch), C2 (the middle branch) and C3 (the upper branch) with the property that C2 and C3 have the same endpoints E1(r1,h1) and E2(r2,h2). See Figure 3.
Second, we claim that C2∪C3 lies over the curve Γ. To show this we introduce an auxiliary algebraic curve Υ={h=Φ(r)} with
Φ
(
r
)
=
1
2
(
5
+
4
r
2
+
6
r
4
+
8
r
6
+
10
r
8
+
12
r
10
+
13
r
12
+
15
r
14
+
16
r
16
+
18
r
18
+
20
r
20
+
21
r
22
+
22
r
24
+
38
r
26
+
25
r
28
+
26
r
30
+
28
r
32
+
30
r
34
+
30
r
36
+
32
r
38
+
34
r
40
+
34
r
42
+
36
r
44
+
38
r
46
+
38
r
48
+
40
r
50
)
,
where r∈(0,1). By direct computations as well as by applying Sturm’s theorem we obtain
w
¯
7
(
r
,
Φ
(
r
)
)
=
p
238
(
r
)
<
0
,
where p238(r) is a polynomial of degree 238. Thus the curve C does not intersect Υ. Moreover, since the straight line r=9/10 intersects the curve C and Υ at the points (9/10,c1∗),(9/10,c2∗),(9/10,c3∗)∈C and (9/10,ϕ1∗)∈Υ, respectively, where
c
1
∗
≈
0.1592878
,
c
2
∗
≈
30.7373179
,
c
3
∗
≈
40.3056908
,
ϕ
1
∗
≈
26.9337561
,
we conclude that C2∪C3 lies over the curve Υ, and C1 lies below the curve Υ. See Figure 3.
On the other hand, using (3.17), we obtain by computation that
(
h
˙
-
Φ
′
(
r
)
r
˙
)
|
h
=
Φ
(
r
)
=
1
4
(
-
9
+
42
r
2
+
3
r
4
-
4
r
8
-
8
r
10
+
18
r
12
-
2
r
14
+
14
r
16
-
6
r
18
-
36
r
20
+
10
r
22
+
43
r
24
-
826
r
26
+
280
r
28
+
748
r
30
-
23
r
32
-
82
r
34
+
70
r
36
+
62
r
38
-
132
r
40
+
102
r
42
+
9
r
44
-
124
r
46
+
103
r
48
+
42
r
50
+
4248
r
52
+
4446
r
54
-
81
r
56
+
82
r
58
+
115
r
60
+
184
r
62
-
20
r
64
+
124
r
66
+
108
r
68
+
72
r
70
+
128
r
72
+
116
r
74
+
1308
r
78
-
1056
r
80
+
64
r
82
+
136
r
84
+
152
r
86
-
12
r
88
+
144
r
90
+
156
r
92
-
8
r
94
+
152
r
96
+
160
r
98
-
4
r
100
+
160
r
102
-
1600
r
104
)
,
which has a unique zero in the interval r0∈(0,1) with 0<r0<1/2. Therefore, there exists a unique contact point on the curve Υ with the vector field (3.17). Taking this into account and noting the fact that
Φ
(
0
)
-
h
¯
(
0
)
>
0
,
Φ
(
1
/
2
)
-
h
¯
(
1
/
2
)
>
1
>
0
,
Φ
(
23
/
25
)
-
h
¯
(
23
/
25
)
>
2
/
5
>
0
,
it is clear that the curve Υ|r∈(0,23/25) lies over Γ|r∈(0,23/25), otherwise there would exist at least two contact points on Υ|r∈(0,23/25) with the vector field (3.17), which leads to a contradiction.
In summary, according to the relative positions of Υ and Γ as well as the relative positions of Υ and C2∪C3, we find that the claim is true.
Third, we claim that C1 lies below the curve Γ. This claim is easy to confirm due to the following facts:
Γ lies over the straight line h=1 (by Lemma 3.4);
C1 does not intersect the line h(r)=1 because C70(r)+C71(r)+C72(r)+C73(r)≠0,r∈(0,1) (by applying Sturm’s theorem);
C1 is a continuous curve passing through the point (0,0) (because C70(0)+C71(0)h+C72(0)h2+C73(0)h3=0 implies h=0).
Finally, taking into account the above results, we conclude that the curve C has no common points with the curve Γ. Thus W¯7(r)≠0, i.e., W7(s)≠0. ∎
Proof of Theorem 1.1 (b).
It follows from equation (3.6), Lemma 3.3, Lemma 3.5 and Lemma 3.6 that wi(s)≠0, i=0,1,2,3,4,5,7 and w6(s) has a simple zero in (0,1). Very recently, Novaes and Torregrosa [15] proved that if the analytical functions f0,f1,…,fn:I→ℝ satisfy
all Wronskian determinants Wi(s)=W(f0,…,fi)(s) but Wn-1 have no zero in the interval I, and
Wn-1 has a unique simple zero in I,
then any linear combination of f0,f1,…,fn can possess at most n+1 zeros in I, counting with multiplicities. But Novaes and Torregrosa do not prove that the upper bound can be reached in the general cases.
In what follows we will show that the upper bound 8 can be reached in our system.
Let s0=200/10001, E0=E(1-s02), K0=K(1-s02) and
k
=
4224932006353520086857838671137556
(
A
E
0
2
+
B
E
0
K
0
+
C
K
0
2
)
,
where
A
=
162632756824646343526934358039550191813769181219039360950399
,
B
=
-
26652192151547736499563618692313149392412558977363562257000
,
C
=
5303055781903243156556160927943006794119698485540000000
.
Consider the function
(3.18)
G
(
s
)
=
a
0
f
0
(
s
)
+
a
1
f
1
(
s
)
+
⋯
+
a
6
f
6
(
s
)
+
k
f
7
(
s
)
,
s
∈
(
0
,
1
)
.
By direct calculation we get the power series of G around the point s0:
G
(
s
)
=
e
0
+
e
1
(
s
-
s
0
)
+
⋯
+
e
6
(
s
-
s
0
)
6
+
e
7
(
s
-
s
0
)
7
+
⋯
,
where ei is the linear combination of a0,a1,…,a6. We solve the equations
e
0
=
0
,
e
1
=
0
,
…
,
e
6
=
0
,
and find the values of a0,a1,…,a6 which have the form
a
i
=
∑
j
=
0
3
q
i
j
E
0
j
K
0
3
-
j
,
i
=
0
,
1
,
…
,
6
,
where each qij is an integer which occupies many digits. We will not write down here the explicit expression of ai for the sake of brevity. By the way, we would like to point out that our purpose of choosing such a k in (3.18) is to make the expression of ai to be relative simple. It turns out that
(3.19)
G
(
s
)
=
e
7
(
s
-
s
0
)
7
+
O
(
(
s
-
s
0
)
8
)
,
s
→
s
0
,
where
e
7
=
-
1000600150020001500060001
264828047171937480000
(
A
0
E
0
3
-
A
1
E
0
2
K
0
+
A
2
E
0
K
0
2
-
A
3
K
0
3
)
,
with
A
0
=
9434365215900096702757086133640555232723441933485420521056595192874
214200423786458280868229
,
A
1
=
1732705885903853693884640417850773768026603157714652167368522666300
981344222008726672767200
,
A
2
=
1550165493805896510978302557036974594023078666779498920543561841897
54798606490809063800000
,
A
3
=
3092940994614887237716575160316676355701108658523634572794702093798
4340166396000000000
,
and
e
7
≈
-
8.7875569
×
10
97
.
On the other hand, at the endpoint s=1 we have
(3.20)
G
(
s
)
=
B
(
1
-
s
)
+
o
(
(
1
-
s
)
)
,
where
B
=
12800263824853240746592933248429440000532
π
⋅
(
5792157212337693345948517518844704378216313299609167
E
0
2
-
1020168724968577415929102393676106281813032946881000
E
0
K
0
+
203589533638142582450403592018967753470820000000
K
0
2
)
≈
1.6008470
×
10
91
.
Equations (3.19) and (3.20) mean that (i) G has a zero at s0 with multiplicity 7, (ii) there exists an ε0 with s0<ε0<1 such that G(s) is negative in (s0,ε0], and (iii) G(s) is positive near the endpoint s=1.
Fixing the numbers a0,a1,…,a6 and k, we consider the function
(3.21)
G
ε
(
s
)
=
G
(
s
)
+
∑
i
=
0
7
ε
i
f
i
(
s
)
,
s
∈
(
0
,
1
)
.
We note that fi can be extended analytically to (0,1]. Thus there exists an M>0 such that
G
ε
(
ε
0
)
<
1
2
G
(
ε
0
)
<
0
,
G
ε
(
s
)
>
1
2
B
(
1
-
s
)
,
when
s
→
1
-
,
for all |εi|<M, i=0,1,…,7. Moreover, near s0 we find
∑
i
=
0
7
ε
i
f
i
(
s
)
=
μ
0
+
μ
1
(
s
-
s
0
)
+
⋯
+
μ
7
(
s
-
s
0
)
7
+
⋯
,
where μi=μi(ε0,ε1,…,ε7) is linear combination of ε0,ε1,…,ε7. One can directly check that the matrix of the coefficients of μ0,μ1,…,μ7 with respect to ε0,ε1,…,ε7 has rank 8, and hence μ0,μ1,…,μ7 are independent.
Consequently, since fi is analytic at s0 and G has a zero at s0 with multiplicity 7, it follows that there exists some small |εi|≪M (i=0,1,…,7) (and hence μi is small) such that Gε has exactly 7 simple zeros in a neighborhood of s0. In view of (3.21) G has an extra zero in (ε0,1). According to the result of [15], this zero is simple. That is to say, Gε has 8 simple zeros.
Finally, using Lemma 3.1 and averaging theory of first order, we see that systems (1.4) have at most 8 limit cycles, and the upper bound can be reached. The proof is finished. ∎
4 Proof of Theorem 1.1 (c)
The goal of this section is to investigate the number of limit cycles of system (1.5) which bifurcate from the period annulus of the isochronous center. Before we prove our result, we should first recall the concept of hypergeometric function.
Let H(a,b,c,z) be the ordinary hypergeometric function which is defined for |z|<1 by the power series
(4.1)
H
(
a
,
b
,
c
,
z
)
=
∑
k
=
0
∞
(
a
)
k
(
b
)
k
(
c
)
k
z
k
k
!
,
where
(
a
)
k
=
{
1
,
k
=
0
,
a
(
a
+
1
)
⋯
(
a
+
k
-
1
)
,
k
>
0
.
It is undefined (or infinite) if c equals a non-positive integer. Many of the common mathematical functions can be expressed in terms of the hypergeometric function. For example, (1-z)-a=H(a,1,1,z) and
H
(
1
2
,
1
2
,
1
,
m
)
=
2
K
(
m
)
π
,
H
(
-
1
2
,
1
2
,
1
,
m
)
=
2
E
(
m
)
π
.
For more information on hypergeometric functions, the reader is refereed to [1, Chapter 15].
Lemma 4.1
The maximum number of limit cycles of system (1.5) which emerge from the period annulus of center of system (1.5)ε=9, by using averaging theory of first order, is equal to the maximum number of the simple zeros of the function
G
(
s
)
=
c
0
g
0
(
s
)
+
c
1
g
1
(
s
)
+
⋯
+
c
12
g
12
(
s
)
,
where c0,c1,…,c12 are independent arbitrary constants, and
g
0
(
s
)
=
s
,
g
1
(
s
)
=
s
2
,
g
2
(
s
)
=
s
1
-
s
,
g
3
(
s
)
=
s
H
(
1
,
1
/
2
,
2
,
s
)
,
g
4
(
s
)
=
s
H
(
1
,
5
/
2
,
4
,
s
)
,
g
5
(
s
)
=
s
H
(
1
,
3
/
2
,
3
,
s
)
,
g
6
(
s
)
=
s
4
/
3
H
(
-
2
/
3
,
1
/
2
,
1
,
s
)
,
g
7
(
s
)
=
s
5
/
3
H
(
-
1
/
3
,
1
/
2
,
1
,
s
)
,
g
8
(
s
)
=
s
5
/
3
H
(
-
1
/
3
,
3
/
2
,
2
,
s
)
,
g
9
(
s
)
=
s
5
/
3
H
(
-
1
/
3
,
5
/
2
,
3
,
s
)
,
g
10
(
s
)
=
s
4
/
3
H
(
-
2
/
3
,
3
/
2
,
2
,
s
)
,
g
11
(
s
)
=
s
4
/
3
(
-
(
1
-
s
)
H
(
1
3
,
5
2
,
3
,
s
)
-
2
H
(
-
2
3
,
5
2
,
3
,
s
)
)
,
g
12
(
s
)
=
s
2
/
3
(
(
4
-
9
s
)
H
(
2
3
,
3
2
,
3
,
s
)
-
(
1
-
s
)
(
4
+
33
s
)
H
(
2
3
,
5
2
,
3
,
s
)
)
.
Proof.
As usual we take the polar coordinate transformation to change system (1.5) to
(4.2)
d
r
d
θ
=
R
0
(
θ
,
r
)
+
ε
R
1
(
θ
,
r
)
+
O
(
ε
2
)
,
where R0(θ,r)=r7cosθsinθ and R1(θ,r)=R11(θ,r)+R12(θ,r) with
R
11
(
θ
,
r
)
=
r
(
a
10
C
2
+
b
01
S
2
)
+
r
3
[
a
30
C
4
+
(
a
12
+
b
21
)
C
2
S
2
+
b
03
S
4
]
+
r
5
[
a
50
C
6
+
(
a
32
+
b
41
)
C
4
S
2
+
(
a
14
+
b
23
)
C
2
S
4
+
b
05
S
6
]
+
r
7
[
a
70
c
8
+
(
a
10
-
b
01
)
C
2
S
2
+
(
a
52
+
b
61
)
C
6
S
2
+
(
a
34
+
b
43
)
C
4
S
4
+
(
a
16
+
b
25
)
C
2
S
6
+
b
07
S
8
]
+
r
9
[
(
a
30
-
b
21
)
C
4
s
2
+
(
a
12
-
b
03
)
C
2
S
4
]
+
r
11
[
(
a
50
-
b
41
)
C
6
S
2
+
(
a
32
-
b
23
)
C
4
S
4
+
(
a
14
-
b
05
)
C
2
S
6
]
+
r
13
[
(
a
70
-
b
61
)
C
8
S
2
+
(
a
52
-
b
43
)
C
6
S
4
+
(
a
34
-
b
25
)
C
4
S
6
+
(
a
16
-
b
07
)
C
2
S
8
]
,
and R12(θ,r) is a polynomial of degree 13 in r of the form ∑i,j,kdi,j,kCiSjrk where i or j are odd numbers. As before, here C=cosθ, S=sinθ. We do not write down the explicit expression of R12(θ,r) because it is too long and, as we will see, it does not play any role in further calculation.
The equation (4.2)ε=0 has the periodic solutions r(θ,r0)=r0(1-3r06sin2θ)-1/6 satisfying r0=r(0,r0) for 0<r0<3-1/6. The corresponding variational differential equation
d
M
d
θ
=
∂
∂
r
R
0
(
θ
,
r
(
θ
,
r
0
)
)
M
,
with Mr0(0)=1 has the fundamental solution
M
r
0
(
θ
)
=
(
1
-
3
r
0
6
sin
2
θ
)
-
7
/
6
.
Next we go to study the maximum number of zeros of the function
ℱ
(
r
0
)
=
∫
0
2
π
M
r
0
-
1
(
θ
)
R
1
(
θ
,
r
(
θ
,
r
0
)
)
𝑑
θ
=
r
0
7
∫
0
2
π
r
-
7
(
θ
,
r
0
)
R
1
(
θ
,
r
(
θ
,
r
0
)
)
𝑑
θ
,
when r0∈(0,3-1/6).
One can check directly that
∫
0
2
π
r
-
7
(
θ
,
r
0
)
R
12
(
θ
,
r
(
θ
,
r
0
)
)
𝑑
θ
=
0
.
It turns out that
ℱ
(
r
0
)
=
r
0
7
∫
0
2
π
r
-
7
(
θ
,
r
0
)
R
11
(
θ
,
r
(
θ
,
r
0
)
)
𝑑
θ
,
r
0
∈
(
0
,
3
-
1
/
6
)
.
Further, taking the transformation r0=(s/3)1/6, we have
ℱ
¯
(
s
)
:=
ℱ
(
r
0
)
=
∫
0
2
π
(
1
-
s
sin
2
θ
)
7
/
6
R
11
(
θ
,
r
¯
(
θ
,
s
)
)
𝑑
θ
,
s
∈
(
0
,
1
)
,
where r¯(θ,s)=3-1/6(s/(1-ssin2θ))1/6.
Using an algebraic manipulator, we obtain after a long calculation that
(4.3)
ℱ
¯
(
s
)
=
c
¯
0
f
¯
0
(
s
)
+
c
¯
1
f
¯
1
(
s
)
+
⋯
+
c
¯
15
f
¯
15
(
s
)
s
17
/
6
(
1
-
s
)
2
/
3
,
where
f
¯
0
(
s
)
=
s
3
(
1
-
s
)
2
/
3
,
f
¯
1
(
s
)
=
s
4
(
1
-
s
)
2
/
3
,
f
¯
2
(
s
)
=
s
3
(
1
-
s
)
7
/
6
,
f
¯
3
(
s
)
=
s
2
(
(
1
-
s
)
7
/
6
-
(
1
-
s
)
2
/
3
)
,
f
¯
4
(
s
)
=
8
(
1
-
s
)
7
/
6
+
(
1
-
s
)
2
/
3
(
s
2
+
4
s
-
8
)
,
f
¯
5
(
s
)
=
2
s
(
1
-
s
)
7
/
6
-
(
1
-
s
)
2
/
3
(
2
s
-
s
2
)
,
f
¯
6
(
s
)
=
s
10
/
3
(
1
-
s
)
2
/
3
H
(
-
2
3
,
1
2
,
1
,
s
)
,
f
¯
7
(
s
)
=
s
11
/
3
(
1
-
s
)
2
/
3
H
(
-
1
3
,
1
2
,
1
,
s
)
,
f
¯
8
(
s
)
=
s
11
/
3
(
1
-
s
)
2
/
3
H
(
-
1
3
,
3
2
,
2
,
s
)
,
f
¯
9
(
s
)
=
s
11
/
3
(
1
-
s
)
2
/
3
H
(
-
1
3
,
5
2
,
3
,
s
)
,
f
¯
10
(
s
)
=
s
14
/
3
(
(
1
-
s
)
2
/
3
H
(
2
3
,
3
2
,
3
,
s
)
+
H
(
2
3
,
3
2
,
3
,
s
s
-
1
)
)
,
f
¯
11
(
s
)
=
s
13
/
3
(
(
1
-
s
)
2
/
3
H
(
1
3
,
3
2
,
3
,
s
)
+
(
1
-
s
)
1
/
3
H
(
1
3
,
3
2
,
3
,
s
s
-
1
)
)
,
f
¯
12
(
s
)
=
s
14
/
3
(
2
H
(
2
3
,
3
2
,
3
,
s
s
-
1
)
+
(
1
-
s
)
2
/
3
H
(
2
3
,
5
2
,
4
,
s
)
-
H
(
2
3
,
5
2
,
4
,
s
s
-
1
)
)
,
f
¯
13
(
s
)
=
s
10
/
3
(
1
-
s
)
2
/
3
(
9
Γ
(
-
2
3
)
Γ
(
7
6
)
H
(
-
2
3
,
3
2
,
-
1
6
,
1
-
s
)
-
10
3
π
3
/
2
H
(
-
2
3
,
3
2
,
2
,
s
)
+
18
(
1
-
s
)
7
/
6
Γ
(
-
7
6
)
Γ
(
8
3
)
H
(
1
2
,
8
3
,
13
6
,
1
-
s
)
)
,
f
¯
14
(
s
)
=
s
10
/
3
(
1
-
s
)
1
/
3
(
243
(
1
-
s
)
1
/
3
Γ
(
-
2
3
)
Γ
(
7
6
)
H
(
-
2
3
,
5
2
,
-
1
6
,
1
-
s
)
-
360
3
π
3
/
2
(
1
-
s
)
1
/
3
H
(
-
2
3
,
5
2
,
3
,
s
)
+
40
3
π
3
/
2
s
H
(
1
3
,
3
2
,
3
,
s
s
-
1
)
+
20
3
π
3
/
2
s
(
(
1
-
s
)
1
/
3
H
(
1
3
,
5
2
,
4
,
s
)
-
H
(
1
3
,
5
2
,
4
,
s
s
-
1
)
)
+
324
(
1
-
s
)
3
/
2
Γ
(
-
7
6
)
Γ
(
11
3
)
H
(
1
2
,
11
3
,
13
6
,
1
-
s
)
)
,
f
¯
15
(
s
)
=
s
11
/
3
(
-
240
(
1
-
s
)
2
/
3
H
(
-
1
3
,
7
2
,
4
,
s
)
+
s
(
16
H
(
2
3
,
3
2
,
3
,
s
s
-
1
)
-
16
H
(
2
3
,
5
2
,
4
,
s
s
-
1
)
+
5
(
1
-
s
)
2
/
3
H
(
2
3
,
7
2
,
5
,
s
)
+
5
H
(
2
3
,
7
2
,
5
,
s
s
-
1
)
)
)
.
Here Γ(z) is the Gamma function defined by Γ(z)=∫0∞tz-1e-tdt and c¯i, for i=0,1,2,…,15, is the linear combination of aij and bij. We do not give the explicit expressions of c¯i (i=0,1,2,…,15) because they are too long. We can check by direct calculation that c¯1,c¯2,…,c¯15 are independent.
From (4.1) we have
f
¯
3
(
s
)
=
s
3
(
1
-
s
)
2
/
3
1
-
s
-
1
s
=
-
1
2
s
3
(
1
-
s
)
2
/
3
H
(
1
,
1
2
,
2
,
s
)
,
f
¯
4
(
s
)
=
-
1
2
s
3
(
1
-
s
)
2
/
3
8
1
-
s
+
s
2
+
4
s
-
8
-
s
3
/
2
-
1
2
s
3
(
1
-
s
)
2
/
3
H
(
1
,
5
2
,
4
,
s
)
,
f
¯
5
(
s
)
=
-
1
4
s
3
(
1
-
s
)
2
/
3
s
-
2
+
2
1
-
s
-
s
2
/
4
=
-
1
4
s
3
(
1
-
s
)
2
/
3
H
(
1
,
3
2
,
3
,
s
)
.
Using Pfaff transformation (see [1, Chapter 15])
(
1
-
z
)
a
H
(
a
,
b
,
c
,
z
)
=
H
(
a
,
c
-
b
,
c
,
z
/
(
z
-
1
)
)
,
as well as Gauss’ contiguous relation
a
b
z
c
H
(
a
+
1
,
b
+
1
,
c
+
1
,
z
)
=
a
(
H
(
a
+
1
,
b
,
c
,
z
)
-
H
(
a
,
b
,
c
,
z
)
)
=
b
(
H
(
a
+
1
,
b
,
c
,
z
)
-
H
(
a
,
b
,
c
,
z
)
)
=
(
c
-
1
)
(
H
(
a
,
b
,
c
-
1
,
z
)
-
H
(
a
,
b
,
c
,
z
)
)
=
(
c
-
a
)
H
(
a
-
1
,
b
,
c
,
z
)
+
(
a
-
c
+
b
z
)
H
(
a
,
b
,
c
,
z
)
1
-
z
=
(
c
-
b
)
H
(
a
,
b
-
1
,
c
,
z
)
+
(
b
-
c
+
a
z
)
H
(
a
,
b
,
c
,
z
)
1
-
z
(4.4)
=
z
(
c
-
a
)
(
c
-
b
)
H
(
a
,
b
,
c
+
1
,
z
)
+
c
(
a
+
b
-
c
)
H
(
a
,
b
,
c
,
z
)
c
(
1
-
z
)
,
we obtain that
f
¯
10
(
s
)
=
2
s
14
/
3
(
1
-
s
)
2
/
3
H
(
2
3
,
3
2
,
3
,
s
)
,
f
¯
11
(
s
)
=
2
s
13
/
3
(
1
-
s
)
2
/
3
H
(
1
3
,
3
2
,
3
,
s
)
,
f
¯
12
(
s
)
=
2
s
14
/
3
(
1
-
s
)
2
/
3
H
(
2
3
,
5
2
,
4
,
s
)
,
f
¯
14
(
s
)
=
3
s
10
/
3
(
1
-
s
)
2
/
3
(
81
Γ
(
-
2
3
)
Γ
(
7
6
)
H
(
-
2
3
,
5
2
,
-
1
6
,
1
-
s
)
+
40
3
π
3
/
2
(
-
H
(
-
2
3
,
5
2
,
3
,
s
)
-
2
(
1
-
s
)
H
(
1
3
,
5
2
,
3
,
s
)
)
+
108
(
1
-
s
)
7
/
6
Γ
(
-
7
6
)
Γ
(
11
3
)
H
(
1
2
,
11
3
,
13
6
,
1
-
s
)
)
,
f
¯
15
(
s
)
=
216
35
s
8
/
3
(
1
-
s
)
2
/
3
(
(
4
-
9
s
)
H
(
2
3
,
3
2
,
3
,
s
)
-
(
1
-
s
)
(
4
+
33
s
)
H
(
2
3
,
5
2
,
3
,
s
)
)
.
Further, applying the formula
H
(
a
,
b
,
c
,
z
)
=
Γ
(
c
)
Γ
(
c
-
a
-
b
)
Γ
(
c
-
a
)
Γ
(
c
-
b
)
H
(
a
,
b
,
a
+
b
-
c
+
1
,
1
-
z
)
+
(
1
-
z
)
c
-
a
-
b
Γ
(
c
)
Γ
(
a
+
b
-
c
)
Γ
(
a
)
Γ
(
b
)
H
(
c
-
a
,
c
-
b
,
c
-
a
-
b
+
1
,
1
-
z
)
,
for |arg(1-z)|<π, we obtain that
f
¯
13
(
s
)
=
s
10
/
3
(
1
-
s
)
2
/
3
(
9
Γ
(
-
2
3
)
Γ
(
8
3
)
Γ
(
1
2
)
-
10
3
π
3
/
2
)
H
(
-
2
3
,
3
2
,
2
,
s
)
=
π
s
10
/
3
(
1
-
s
)
2
/
3
(
9
Γ
(
-
2
3
)
Γ
(
8
3
)
-
10
3
π
)
H
(
-
2
3
,
3
2
,
2
,
s
)
=
-
20
3
π
3
/
2
s
10
/
3
(
1
-
s
)
2
/
3
H
(
-
2
3
,
3
2
,
2
,
s
)
,
f
¯
14
(
s
)
=
3
2
s
10
/
3
(
1
-
s
)
2
/
3
(
(
81
Γ
(
-
2
3
)
Γ
(
11
3
)
Γ
(
1
2
)
-
80
3
π
3
/
2
)
H
(
-
2
3
,
5
2
,
3
,
s
)
-
160
3
π
3
/
2
(
1
-
s
)
H
(
1
3
,
5
2
,
3
,
s
)
)
=
3
π
2
s
10
/
3
(
1
-
s
)
2
/
3
(
(
81
Γ
(
-
2
3
)
Γ
(
11
3
)
-
80
3
π
)
H
(
-
2
3
,
5
2
,
3
,
s
)
+
160
3
π
(
s
-
1
)
H
(
1
3
,
5
2
,
3
,
s
)
)
=
-
240
3
π
3
/
2
s
10
/
3
(
1
-
s
)
2
/
3
(
(
1
-
s
)
H
(
1
3
,
5
2
,
3
,
s
)
+
2
H
(
-
2
3
,
5
2
,
3
,
s
)
)
.
By the above equalities, we obtain from (4.3) that
(4.5)
ℱ
¯
(
s
)
=
s
-
5
/
6
(
c
~
0
f
0
(
s
)
+
c
~
1
f
1
(
s
)
+
⋯
+
c
~
15
f
15
(
s
)
)
,
where the functions fi modulo a nonzero constant are f¯i/(s2(1-s)2/3):
f
0
(
s
)
=
s
,
f
1
(
s
)
=
s
2
,
f
2
(
s
)
=
s
1
-
s
,
f
3
(
s
)
=
s
H
(
1
,
1
2
,
2
,
s
)
,
f
4
(
s
)
=
s
H
(
1
,
5
2
,
4
,
s
)
,
f
5
(
s
)
=
s
H
(
1
,
3
2
,
3
,
s
)
,
f
6
(
s
)
=
s
4
/
3
H
(
-
2
3
,
1
2
,
1
,
s
)
,
f
7
(
s
)
=
s
5
/
3
H
(
-
1
3
,
1
2
,
1
,
s
)
,
f
8
(
s
)
=
s
5
/
3
H
(
-
1
3
,
3
2
,
2
,
s
)
,
f
9
(
s
)
=
s
5
/
3
H
(
-
1
3
,
5
2
,
3
,
s
)
f
10
(
s
)
=
s
8
/
3
H
(
2
3
,
3
2
,
3
,
s
)
,
f
11
(
s
)
=
s
7
/
3
H
(
1
3
,
3
2
,
3
,
s
)
,
f
12
(
s
)
=
s
8
/
3
H
(
2
3
,
5
2
,
4
,
s
)
,
f
13
(
s
)
=
s
4
/
3
H
(
-
2
3
,
3
2
,
2
,
s
)
,
f
14
(
s
)
=
-
s
4
/
3
(
(
1
-
s
)
H
(
1
3
,
5
2
,
3
,
s
)
+
2
H
(
-
2
3
,
5
2
,
3
,
s
)
)
f
15
(
s
)
=
s
2
/
3
(
(
4
-
9
s
)
H
(
2
3
,
3
2
,
3
,
s
)
-
(
1
-
s
)
(
4
+
33
s
)
H
(
2
3
,
5
2
,
3
,
s
)
)
,
It is also not hard to see that in (4.5), c~0,c~1,…,c~15 are independent constants.
Further, applying repeatedly Gauss’ contiguous relation (4.4), we find that
(4.6)
f
10
=
12
f
7
-
12
f
8
,
f
11
=
6
f
6
-
6
f
13
,
f
12
=
36
f
8
-
36
f
9
.
It follows from (4.5) and (4.6) that
ℱ
(
r
0
)
=
ℱ
¯
(
s
)
=
s
-
5
/
6
∑
i
=
0
12
c
i
g
i
(
s
)
=
s
-
5
/
6
G
(
s
)
,
where
g
0
=
f
0
,
g
1
=
f
1
,
…
,
g
9
=
f
9
,
g
10
=
f
13
,
g
11
=
f
14
,
g
12
=
f
15
,
and c0,c1,…,c12 are independent constants.
Finally, by Theorem A.1, we obtain the required conclusion. ∎
Proof of Theorem 1.1 (c).
First, we claim that the functions g0,g1,…,g12 in Lemma 4.1 are linear independent. To show this, we write
G
¯
(
u
)
=
d
0
g
0
(
u
3
)
+
d
1
g
1
(
u
3
)
+
⋯
+
d
12
g
12
(
u
3
)
,
u
>
0
.
Then, near the point u=0 we have
8957952
G
¯
(
u
)
u
3
=
8957952
(
d
0
+
d
2
+
d
3
+
d
4
+
d
5
)
+
8957952
(
d
10
-
3
d
11
+
d
6
)
u
-
(
348364800
d
12
-
8957952
(
d
7
+
d
8
+
d
9
)
)
u
2
+
1119744
(
8
d
1
-
4
d
2
+
2
d
3
+
5
d
4
+
4
d
5
)
u
3
-
1492992
(
3
d
10
-
11
d
11
+
2
d
6
)
)
u
4
+
82944
(
1400
d
12
-
3
(
6
d
7
+
9
d
8
+
10
d
9
)
)
u
5
-
559872
(
2
d
2
-
2
d
3
-
7
d
4
-
5
d
5
)
u
6
-
124416
(
5
d
10
-
20
d
11
+
3
d
6
)
u
7
+
6912
(
5390
d
12
-
3
(
18
d
7
+
30
d
8
+
35
d
9
)
)
u
8
-
139968
(
4
d
2
-
5
d
3
-
7
(
3
d
4
+
2
d
5
)
)
u
9
-
6912
(
35
d
10
-
147
d
11
+
20
d
6
)
u
10
+
2880
(
6860
d
12
-
3
(
20
d
7
+
35
d
8
+
42
d
9
)
)
u
11
-
69984
(
5
d
2
-
7
d
3
-
33
d
4
-
21
d
5
)
u
12
-
14112
(
9
d
10
-
39
d
11
+
5
d
6
)
u
13
+
6720
(
1870
d
12
-
3
(
5
d
7
+
9
d
8
+
11
d
9
)
)
u
14
-
4374
(
56
d
2
-
3
(
28
d
3
+
143
d
4
+
88
d
5
)
)
u
15
-
1008
(
77
d
10
-
341
d
11
+
42
d
6
)
u
16
+
176
(
50050
d
12
-
378
d
7
-
693
d
8
-
858
d
9
)
u
17
-
2187
(
84
d
2
-
11
(
12
d
3
+
65
d
4
+
39
d
5
)
)
u
18
+
O
(
u
19
)
:
=
α
0
+
α
1
u
+
⋯
+
α
18
u
18
+
O
(
u
19
)
.
If G¯(u)≡0, then we have α0=α1=⋯=α18=0. By direct calculation, we get d0=d1=⋯=d12=0. (By the way, it is remarkable that we cannot get from α0=α1=⋯=α12=0 that d0=⋯=d12=0.) This shows that our claim holds.
Since gi is an analytic function on (0,1) for i=0,1,…,12, we know that, by applying [5, Lemma 4.5], by suitable choice of c0,c1,…,c12, c0g0(s)+c1g1(s)+⋯+c12g12(s) can have 0,1,2,…,12 simple zeros in (0,1).
Consequently, according to Theorem A.1, there exist some coefficients aij, bij (i+j=0,1,…,7) such that system (1.5) has 0,1,2,…,12 limit cycles. This completes the proof. ∎
Remark 4.2
It seems very hard to find the smallest upper bound of the number of limit cycles of system (1.5) which emerge from the period annulus of the isochronous center for the case n=2. In fact, the expressions of the Wronskian determinants Wk for k≥9 are too complicated to determine the number of simple zeros of them. On the other hand, by using the same method as the one in the proof of case n=1, we can find some coefficients c0,c1,…,c12 such that c0g0(s)+c1g1(s)+⋯+c12g12(s) has 12 simple zeros in a small interval (s0-ε,s0+ε) and has extra zero in (0,s0-ε) with s0=7/10. However, we cannot prove that the extra zero is simple.
