# Limit Cycles Bifurcated from SomeZ

Authors: Simin Qu; Cangxin Tang; Fengli Huang; Xianbo Sun
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study the number and distribution of limit cycles of some planarZ
4-equivariant
quintic near-Hamiltonian systems. By the theories of Hopf and heteroclinic bifurcation, it is proved that the
perturbed system can have 24 limit cycles with some new distributions. The configurations of limit cycles
obtained in this paper are new.


---

## Body

## 1. Introduction

In 1900, Hilbert proposed 23 open mathematical problems [1]; the second part of the 16th problem concerns the maximal number and relative position of limit cycles of the planar polynomial vector fields. Even though there have been many results of obtaining more limit cycles and various configuration patterns of their relative dispositions, it has not been solved completely. To reduce the difficulty one can study the systems with some symmetry. An important symmetry is the Z
q-equivariance which was first introduced in [2]. Here we mention some newer results; for more details, see summary work [3–5]. Li et al. [6] proved a cubic Z
2-equivariant system having 13 limit cycles; Zhao [7] proved that this system has 13 limit cycles with another distribution. Li and Liu [8] proved another cubic Z
2-equivariant system also having 13 limit cycles. Zhang et al. [9] found a quartic system having at least 15 limit cycles. Christopher [10] proved that a Z
2-equivariant system has 22 limit cycles. As to the case of quintic polynomial system, there are more results. Xu and Han [11] studied a cubic Z
4-equivariant system perturbed by quintic Z
4-equivariant polynomials having 13 limit cycles. Li et al. [12] studied a quintic system and obtained at least 23 limit cycles for Z
2-equivariant case and 17 limit cycles for Z
4-equivariant case. In [13], Wu et al. studied a Z
4-equivariant system and found 20 limit cycles. Li et al. [14] found that 24 limit cycles existing in a Z
6-equivariant quintic system. Yao and Yu [15] studied a Z
5-equivariant quintic planar vector fields by normal form theory and proved that the maximal number of small limit cycles bifurcated from such vector fields is 25. Wu et al. [16] proved that a quintic Z
6-equivariant near-Hamiltonian system has 28 limit cycles. In [17], 24 limit cycles are found and two different configurations of them were shown in a Z
3-equivariant quintic planar polynomial system.Our main result is that there can be 24 limit cycles with other distributions for the perturbed quinticZ
4-equivariant systems which are different from the known results. Using the methods of Hopf and heteroclinic bifurcation theories, the number and location of limit cycles of the following Z
4-equivariant quintic near-Hamiltonian system will be investigated:
(1)
x
˙
=
H
y
+
ε
P
5
(
x
,
y
)
,
y
˙
=
-
H
x
+
ε
Q
5
(
x
,
y
)
,
where ε is nonnegative and small and the Hamiltonian system is
(2)
H
(
x
,
y
)
=
2
x
2
+
2
y
2
-
5
4
x
4
-
5
4
y
4
+
1
6
x
6
+
1
6
y
6
with phase portrait of Figure 1. (P
5
(
x
,
y
)
,
Q
5
(
x
,
y
)) is the five-degree polynomial vector invariant under rotation of π
/
2 with respect to the origin O. From [2] we know that (P
5
(
x
,
y
)
,
Q
5
(
x
,
y
)) is, respectively, the real and imaginary parts of the following complex function:
(3)
F
5,4
(
z
,
z
¯
)
=
(
A
0
+
A
1
|
z
|
2
+
A
3
|
z
|
4
)
z
+
(
A
3
+
A
4
|
z
|
2
)
z
¯
3
+
A
5
z
5
,
where A
k
=
a
k
+
i
b
k,  k
=
0,1
,
2
,
…
,
5,  z
=
x
+
i
y, and z
=
x
-
i
y. It is direct that
(4)
P
5
=
a
0
x
+
a
1
(
x
3
+
x
y
2
)
+
a
2
(
x
5
+
2
x
3
y
2
+
x
y
4
)
+
a
3
(
x
3
-
3
x
y
2
)
+
a
4
(
x
5
-
2
x
3
y
2
-
3
x
y
4
)
+
a
5
(
x
5
-
10
x
3
y
2
+
5
x
y
4
)
+
b
0
(
-
y
)
+
b
1
(
-
x
2
y
-
y
3
)
+
b
2
(
-
x
4
y
-
2
x
2
y
3
-
y
5
)
+
b
3
(
3
x
2
y
-
y
3
)
+
b
4
(
3
x
4
y
+
2
x
2
y
3
-
y
5
)
+
b
5
(
10
x
2
y
3
-
5
x
4
y
-
y
5
)
,
Q
5
=
a
0
y
+
a
1
(
x
2
y
+
y
3
)
+
a
2
(
x
4
y
+
2
x
2
y
3
+
y
5
)
+
a
3
(
y
3
-
3
x
2
y
)
+
a
4
(
y
5
-
3
x
4
y
-
2
x
2
y
3
)
+
a
5
(
5
x
4
y
-
10
x
2
y
3
+
y
5
)
+
b
0
x
+
b
1
(
x
3
+
x
y
2
)
+
b
2
(
x
5
+
2
x
3
y
2
+
x
y
4
)
+
b
3
(
x
3
-
3
x
y
2
)
+
b
4
(
x
5
-
2
x
3
y
2
-
3
x
y
4
)
+
b
5
(
x
5
-
10
x
3
y
2
+
5
x
y
4
)
.
Our result is the following.Figure 1
The phase portraits of (1)(
ɛ
=
0
).Theorem 1.
There exist some(
a
0
,
a
1
,
a
2
,
a
3
,
a
4
,
a
5
) such that system (1) can have 24 limit cycles with two different distributions, the distributions of these limit cycles are shown in Figure 2.Two different distributions of 24 limit cycles of system (1).
(a)
24 limit cycles
(b)
24 limit cyclesThe rest of this paper is organized as follows. Some useful preliminary theorems will be listed in Section2. In Section 3, some related coefficients of asymptotic expansions are firstly computed; then using this coefficients and preliminary lemmas we prove the main result.


## 2. Preliminary Lemmas

LetH
(
x
,
y
), p
(
x
,
y
,
δ
), and q
(
x
,
y
,
δ
) be analytic functions, ɛ positive and small, and δ
∈
D
⊂
R
m with D compact; then the following system is a planar Hamiltonian system:
(5)
x
˙
=
H
y
,
y
˙
=
-
H
x
,
and the below system is usually called near-Hamiltonian system:
(6)
x
˙
=
H
y
(
x
,
y
)
+
ɛ
p
(
x
,
y
,
δ
)
,
y
˙
=
-
H
x
(
x
,
y
)
+
ɛ
q
(
x
,
y
,
δ
)
.
Let system (5) have at least one family of periodic orbits L
h defined by H
(
x
,
y
)
=
h which form a periodic annulus {
L
h
}; then the first-order approximation of the Poincaré map of system (6) is
(7)
M
(
h
,
δ
)
=
∮
L
h
‍
(
q
d
x
-
p
d
y
)
which is called the Melnikov function or Abel integral. By the Poincaré-Pontryagin-Andronov theorem, an isolated zero of M
(
h
,
δ
) corresponds a limit cycle of system (6). A popular method to find limit cycles of (6) is to find zeros of M
(
h
,
δ
) and an efficient method to find zeros of M
(
h
,
δ
) is to investigate the asymptotic expansion of M
(
h
,
δ
) near the boundaries of {
L
h
}; see [18].Let the outer boundary of{
L
h
} be a homoclinic loop L
β defined by H
(
x
,
y
)
=
β passing through a hyperbolic saddle at the origin; we have the following.Lemma 2 (see [19]).
(i) The functionM
(
h
,
δ
) has the following expansion:
(8)
M
(
h
,
δ
)
=
c
0
(
δ
)
+
c
1
(
δ
)
(
h
-
β
)
ln
⁡
|
h
-
β
|
+
c
2
(
δ
)
(
h
-
β
)
+
c
3
(
δ
)
(
h
-
β
)
2
ln
⁡
|
h
-
β
|
+
O
(
|
h
-
β
|
2
)
,
for 0
<
β
-
h
≪
1; c
i
(
δ
) depends on the parameters of H, p, and q.
(ii) Further suppose that, for(
x
,
y
) near (
0,0
),
(9)
H
(
x
,
y
)
=
β
+
λ
2
(
y
2
-
x
2
)
+
∑
i
+
j
≥
3
‍
h
i
j
x
i
y
j
,
λ
≠
0
,
p
(
x
,
y
,
δ
)
=
∑
i
+
j
≥
0
‍
a
i
j
x
i
y
j
,
q
(
x
,
y
,
δ
)
=
∑
i
+
j
≥
0
‍
b
i
j
x
i
y
j
.
Then,
(10)
c
0
(
δ
)
=
∮
L
β
‍
q
d
x
-
p
d
y
,
c
1
(
δ
)
=
-
1
|
λ
|
(
a
10
+
b
01
)
,
c
2
(
δ
)
=
∮
L
β
‍
(
p
x
+
q
y
)
d
t
if
c
1
(
δ
)
=
0
,
c
3
(
δ
)
=
-
1
2
|
λ
|
λ
{
(
-
3
a
30
-
b
21
+
a
12
+
3
b
03
)
hh
-
1
λ
[
(
2
b
02
+
a
11
)
(
3
h
03
-
h
21
)
hh
+
(
2
a
20
+
b
11
)
(
3
h
30
-
h
12
)
]
}
,
i
f
c
1
(
δ
)
=
0
.
The values c
1
(
δ
) and c
3
(
δ
) are, respectively, called the first and second local Melnikov coefficients at the saddle O, denoted by c
1
(
O
,
δ
) and c
3
(
O
,
δ
), respectively.Now let the outer boundary of{
L
h
} be an 2-polycycle Γ
2:
(11)
Γ
2
=
⋃
i
=
1
2
‍
(
L
i
∪
S
i
)
with 2 hyperbolic saddles, S
1  and S
2, and 2 heteroclinic orbits, L
1 and L
2, connecting them, defined by H
(
x
,
y
)
=
β. The following lemma was proved in [19].Lemma 3 (see [19]).
Under the above assumptions,M
(
h
,
δ
) has the form, for 0
<
β
-
h
≪
1,
(12)
M
(
h
,
δ
)
=
∑
j
≥
0
‍
[
c
2
j
(
δ
)
+
c
2
j
+
1
(
δ
)
(
h
-
β
)
ln
⁡
|
h
-
β
|
]
×
(
h
-
β
)
j
,
where
(13)
c
0
(
δ
)
=
∑
i
=
1
2
‍
∫
L
i
‍
q
d
x
-
p
d
y
,
c
1
(
δ
)
=
∑
i
=
1
2
‍
c
1
(
S
i
,
δ
)
,
c
3
(
δ
)
=
∑
i
=
1
2
‍
c
3
(
S
i
,
δ
)
,
where c
1
(
S
i
,
δ
) and c
3
(
S
i
,
δ
) are, respectively, the first and the second local Melnikov coefficient at the saddle S
i, i
=
1,2. In particular,
(14)
c
2
(
δ
)
=
∮
Γ
2
‍
(
p
x
+
q
y
)
d
t
=
∑
i
=
1
2
‍
∫
L
i
‍
(
p
x
+
q
y
)
d
t
,
if c
1
(
S
i
,
δ
)
=
0, i
=
1,2.When the inner boundary of{
L
h
} is a elementary center (
x
c
,
y
c
) defined by H
(
x
c
,
y
c
)
=
α, the following lemma gives the asymptotic expansion of M
(
h
,
δ
).Lemma 4 (see [20]).
M
(
h
,
δ
) has the form, for 0
<
h
-
α
≪
1,
(15)
M
(
h
,
δ
)
=
∑
k
≥
0
‍
B
k
(
h
-
α
)
k
+
1
.
If for (
x
,
y
) near (
x
c
,
y
c
),
(16)
H
(
x
,
y
)
=
α
+
1
2
(
(
x
-
x
c
)
2
+
(
y
-
y
c
)
2
)
+
∑
i
+
j
≥
3
‍
h
i
j
(
x
-
x
c
)
i
(
y
-
y
c
)
j
,
p
(
x
,
y
,
δ
)
=
∑
i
+
j
≥
1
‍
a
i
j
(
x
-
x
c
)
i
(
y
-
y
c
)
j
,
q
(
x
,
y
,
δ
)
=
∑
i
+
j
≥
1
‍
b
i
j
(
x
-
x
c
)
i
(
y
-
y
c
)
j
,
the coefficients B
i can be obtained by the formulas in [20].Remark 5.
When the inner boundary is a nilpotent center, a new method of limit cycles bifurcated from the annulus near the center can be found in [21]. Lemma 3 has been developed in [22, 23].In many cases the Hamiltonian function is not of the form presented in the above lemmas. Then to apply the lemmas we need first to introduce suitable linear change of variables which will cause a change in the first-order Melnikov function. The following lemma gives the relationship between the old and new Melnikov functions.Lemma 6 (see [24]).
(i) Under the linear change of variables of the form:(17)
u
=
a
(
x
-
x
0
)
+
b
(
y
-
y
0
)
,
v
=
c
(
x
-
x
0
)
+
d
(
y
-
y
0
)
and time rescaling τ
=
k
t, where D
=
a
d
-
b
c
≠
0, the system (6) becomes
(18)
d
u
d
τ
=
H
~
v
+
ɛ
p
~
,
d
v
d
τ
=
-
H
~
u
+
ɛ
q
~
,
where H
~
(
u
,
v
)
=
(
D
/
k
)
H
(
x
,
y
), p
~
(
u
,
v
,
δ
)
=
(
1
/
k
)
[
a
p
(
x
,
y
,
δ
)
+
b
q
(
x
,
y
,
δ
)
],  and  q
~
(
u
,
v
,
δ
)
=
(
1
/
k
)
[
c
p
(
x
,
y
,
δ
)
+
d
q
(
x
,
y
,
δ
)
].
(ii) Let(19)
M
~
(
h
,
δ
)
=
∮
L
h
‍
q
~
d
u
-
p
~
d
v
which is the Melnikov function of the system (18); then
(20)
M
(
h
,
δ
)
=
|
k
|
D
M
~
(
D
k
h
,
δ
)
.When systems (5) and (6) are Z
4-equivariant, (5) has a compound cycled denoted by Γ
4, which consists of 8 hyperbolic saddles S
1
,
…
,
S
8 and 16 heteroclinic orbits L
12, L
21, L
23, L
32, L
34, L
43, L
45, L
54, L
56, L
65, L
67, L
76, L
78, L
87, L
81, and L
18 satisfying α
(
L
i
j
)
=
S
i
,
ω
(
L
i
j
)
=
S
j. Γ
4 contains 8 two-polycycles L
i  (
i
=
1
,
…
,
8
), where L
1
=
L
12
∪
L
21, L
2
=
L
23
∪
L
32, L
3
=
L
34
∪
L
43, L
4
=
L
45
∪
L
54, L
5
=
L
56
∪
L
65, L
6
=
L
67
∪
L
76, L
7
=
L
78
∪
L
87, and L
8
=
L
81
∪
L
18. See Figure 3. We suppose that Γ
4 is defined by H
(
x
,
y
)
=
H
(
S
i
)
=
β, i
=
1
,
…
,
8. There are 8 centers C
i
(
x
i
,
y
i
) inside the 2-polycycle L
I, with H
(
C
1
)
=
H
(
C
3
)
=
H
(
C
5
)
=
H
(
C
7
)
=
α
1 and H
(
C
2
)
=
H
(
C
4
)
=
H
(
C
6
)
=
H
(
C
8
)
=
α
2. There are 4 families of periodic orbits L
i
h inside the 2-polycycle L
i, defined by H
(
x
,
y
)
=
h for h
∈
(
α
1
,
β
), i
=
1,3
,
5,7, and 4 families of periodic orbits L
j
h inside the 2-polycycle L
j, defined by H
(
x
,
y
)
=
h for h
∈
(
α
2
,
β
), j
=
2,4
,
6,8. Then we have 8 Melnikov functions below:
(21)
M
i
(
h
,
δ
)
=
∮
L
i
h
‍
(
q
d
x
-
p
d
y
)
|
ɛ
=
0
,
h
for
h
∈
(
α
1
,
β
)
i
=
1,3
,
5,7
.
M
j
(
h
,
δ
)
=
∮
L
i
h
‍
(
q
d
x
-
p
d
y
)
|
ɛ
=
0
,
h
for
h
∈
(
α
2
,
β
)
j
=
2,4
,
6,8
.
By Z
4-equivariance, M
1
(
h
,
δ
)
=
M
3
(
h
,
δ
)
=
M
5
(
h
,
δ
)
=
M
7
(
h
,
δ
)  and M
2
(
h
,
δ
)
=
M
4
(
h
,
δ
)
= M
6
(
h
,
δ
)
=
M
8
(
h
,
δ
), we can only study M
1
(
h
,
δ
) and M
8
(
h
,
δ
). For convenience, the notations are introduced as follows:
(22)
c
01
(
δ
)
=
∫
L
12
q
d
x
-
p
d
y
,
c
02
(
δ
)
=
∫
L
21
q
d
x
-
p
d
y
,
e
01
(
δ
)
=
∫
L
81
q
d
x
-
p
d
y
,
e
02
(
δ
)
=
∫
L
18
q
d
x
-
p
d
y
,
and d
i
(
δ
)
=
(
p
x
+
q
y
)
(
S
i
,
δ
)
,
1
≤
i
≤
8, where d
1
=
d
3
=
d
5
=
d
7 and d
2
=
d
4
=
d
6
=
d
8. Letting d
1
=
d
2
=
0, we introduce
(23)
c
21
(
δ
)
=
∫
L
12
(
p
x
+
q
y
)
d
t
,
c
22
(
δ
)
=
∫
L
21
(
p
x
+
q
y
)
d
t
,
e
21
(
δ
)
=
∫
L
81
(
p
x
+
q
y
)
d
t
,
e
22
(
δ
)
=
∫
L
18
(
p
x
+
q
y
)
d
t
.
The following is directly from Lemma 3.Figure 3Lemma 7.
Under the above assumptions, we have the following expansions:(24)
M
i
(
h
,
δ
)
=
c
0
(
δ
)
+
c
1
(
δ
)
(
h
-
β
)
ln
⁡
|
h
-
β
|
+
c
2
(
δ
)
(
h
-
β
)
+
c
3
(
δ
)
(
h
-
β
)
2
ln
⁡
|
h
-
β
|
+
⋯
,
for 0
<
h
-
β
≪
1, i
=
1,3
,
5,7,
(25)
M
j
(
h
,
δ
)
=
e
0
(
δ
)
+
e
1
(
δ
)
(
h
-
β
)
ln
⁡
⁡
|
h
-
β
|
+
e
2
(
δ
)
(
h
-
β
)
+
e
3
(
δ
)
(
h
-
β
)
2
ln
⁡
|
h
-
β
|
+
⋯
,
for 0
<
h
-
β
≪
1, j
=
2,4
,
6,8, where
(26)
c
0
(
δ
)
=
c
01
(
δ
)
+
c
02
(
δ
)
,
e
0
(
δ
)
=
e
01
(
δ
)
+
e
02
(
δ
)
,
c
1
(
δ
)
=
c
1
(
S
1
,
δ
)
+
c
1
(
S
2
,
δ
)
=
-
d
1
(
δ
)
|
λ
1
|
-
d
2
(
δ
)
|
λ
2
|
,
e
1
(
δ
)
=
c
1
(
S
1
,
δ
)
+
c
1
(
S
8
,
δ
)
=
c
1
(
S
1
,
δ
)
+
c
1
(
S
2
,
δ
)
=
c
1
(
δ
)
,
c
3
(
δ
)
=
c
3
(
S
1
,
δ
)
+
c
3
(
S
2
,
δ
)
,
e
3
(
δ
)
=
c
3
(
S
1
,
δ
)
+
c
3
(
S
8
,
δ
)
=
c
3
(
S
1
,
δ
)
+
c
3
(
S
2
,
δ
)
=
c
3
(
δ
)
,
c
2
(
δ
)
=
c
21
(
δ
)
+
c
22
(
δ
)
,
e
2
(
δ
)
=
e
21
(
δ
)
+
e
22
(
δ
)
if d
1
(
δ
)
=
d
2
(
δ
)
=
0. λ
i denotes an eigenvalue of S
i for (6) and c
1
(
S
i
,
δ
) and c
3
(
S
i
,
δ
) are the first and the second Melnikov coefficients at the saddle S
i  (
i
=
1,2
) as defined after Lemma 2.By Lemma4, for the expansions of M
i
(
h
,
δ
) near the center, we have
(27)
M
1
(
h
,
δ
)
=
∑
k
≥
0
‍
B
k
(
δ
)
(
h
-
α
1
)
k
+
1
for
0
<
h
-
α
1
≪
1
,
M
8
(
h
,
δ
)
=
∑
k
≥
0
‍
d
k
(
δ
)
(
h
-
α
2
)
k
+
1
for
0
<
h
-
α
2
≪
1
.To obtain more limit cycles, we have the following.Theorem 8.
Let (24), (25), and (27) hold.
(1) Suppose that there existsδ
0
∈
D such that
(28)
c
0
(
δ
0
)
=
c
1
(
δ
0
)
=
d
0
(
δ
0
)
=
d
1
(
δ
0
)
=
0
,
c
2
(
δ
0
)
≠
0
,
d
2
(
δ
0
)
≠
0
,
rank
⁡
∂
(
c
0
,
c
1
,
d
0
,
d
1
)
∂
(
δ
1
,
δ
2
,
…
,
δ
m
)
=
4
.
Then there exist some (
ɛ
,
δ
) near (
0
,
δ
0
) such that (5) has
(29)
4
+
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
2
+
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
2
limit cycles in the 2-polycycles L
1 and L
8, where h
1
=
β
-
ε
0, h
2
=
α
1
+
ε
0, h
3
=
β
-
ε
0, and h
4
=
α
2
+
ε
0 with ε
0 being positive and very small, and the location of these limit cycles is as follows: 2 limit cycles near the 2-polycycle L
1, 2 limit cycles near the center C
8, (
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2 limit cycles between the center C
1 and the polycycle L
1, and (
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2 limit cycles between the center C
8 and the polycycle L
8.
(2) Suppose that there existsδ
0
∈
D such that
(30)
c
0
(
δ
0
)
=
c
1
(
δ
0
)
=
B
0
(
δ
0
)
=
e
0
(
δ
0
)
=
0
,
c
2
(
δ
0
)
≠
0
,
e
2
(
δ
0
)
≠
0
,
B
1
(
δ
0
)
≠
0
,
rank
⁡
∂
(
c
0
,
c
1
,
B
0
,
e
0
)
∂
(
δ
1
,
δ
2
,
…
,
δ
m
)
|
δ
=
δ
0
=
4
.
Then there exist some (
ɛ
,
δ
) near (
0
,
δ
0
) such that (5) has
(31)
4
+
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
2
+
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
2
+
1
+
sgn
⁡
(
c
2
(
δ
0
)
e
2
(
δ
0
)
)
2
limit cycles in the 2-polycycles L
1 and L
8, where h
1
=
β
-
ε
0, h
2
=
α
1
+
ε
0, h
3
=
β
-
ε
0, and h
4
=
α
2
+
ε
0 with ε
0 being positive and very small, and the location of these limit cycles is the following: 2 limit cycles near the 2-polycycle L
1, 1 limit cycles near the center C
1, 1
+
(
(
1
+
sgn
⁡
(
c
2
(
δ
0
)
e
2
(
δ
0
)
)
)
/
2
) limit cycles near the 2-polycycle L
8, (
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2 limit cycles between the center C
1, and the polycycle L
1 and (
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2 limit cycles between the center C
8 and the polycycle L
8.Proof.
Because of the similarity, we only prove the last conclusion. Forδ
=
δ
0, by continuity, there exist (
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2 zeros of M
1
(
h
,
δ
0
) between h
1 and h
2 and (
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2 zeros of M
8
(
h
,
δ
0
) between h
3 and h
4. Thus, for all δ near δ
0 or δ
∈
U
0
=
{
δ
∣
|
δ
|
<
ɛ
*
,
ɛ
*
is
very
small
} there exist (
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2 zeros of M
1
(
h
,
δ
) between h
1 and h
2 and (
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2 zeros of M
8
(
h
,
δ
) between h
3 and h
4.
According to the condition, we can takec
0, c
1, e
0, and B
0 as free parameters. Hence, we first take c
1 satisfying
(32)
|
c
1
|
≪
|
c
2
|
,
|
c
1
|
≪
|
e
2
|
,
c
1
c
2
>
0
,
so that there is a zero of M
1
(
h
,
δ
) denoted by h
~
1 near β satisfying h
1
<
h
~
1
<
β. On the other hand, if c
2
e
2
>
0, considering c
1
=
e
1, we have |
e
1
|
≪
|
e
2
| and e
1
e
2
>
0 which implies that there is a zero of M
8
(
h
,
δ
) denoted by h
¯
1 near β satisfying h
3
<
h
¯
1
<
β. If c
2
e
2
<
0, we are not sure if M
8
(
h
,
δ
) has a zero. Thus, so far we obtain 1 zero of M
1
(
h
,
δ
) and (
1
+
sgn
⁡
(
c
2
(
δ
0
)
e
2
(
δ
0
)
)
)
/
2 zeros of M
8
(
h
,
δ
) for δ
∈
U
1
=
{
δ
∣
|
c
1
|
≪
|
c
2
|
,
|
c
1
|
≪
|
e
2
|
,
c
1
c
2
>
0
}.
Next, we takec
0, e
0, and B
0 satisfying
(33)
|
c
0
|
≪
|
c
1
|
,
c
0
c
1
<
0
,
|
e
0
|
≪
|
e
1
|
,
e
0
e
1
<
0
,
|
B
0
|
≪
|
B
1
|
,
B
0
B
1
<
0
.
Then M
1
(
h
,
δ
) has two new zeros h
~
2 near β and h
~
3 near α
2 satisfying h
1
<
h
~
1
<
h
~
2
<
β and α
2
<
h
~
3
<
h
4, and M
8
(
h
,
δ
) has a new zero h
¯
2 near β satisfying h
3
<
h
¯
2
<
β. In this step, we get 2 more zeros of M
1
(
h
,
δ
) and 1 more zero of M
1
(
h
,
δ
) for ∈
U
2
=
{
δ
∣
|
c
0
|
≪
|
c
1
|,  c
0
c
1
<
0,  |
e
0
|
≪
|
e
1
|
,
e
0
e
1
<
0,  |
B
0
|
≪
|
B
1
|, and  B
0
B
1
<
0
}. Then totally we have 4
+
(
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2
+
(
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2
+
(
1
+
sgn
⁡
(
c
2
(
δ
0
)
e
2
(
δ
0
)
)
)
/
2 zeros for δ
∈
U
0
∩
U
1
∩
U
2. Therefore, there are 4
+
(
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2
+
(
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2
+
(
1
+
sgn
⁡
(
c
2
(
δ
0
)
e
2
(
δ
0
)
)
)
/
2 limit cycles for some δ near δ
0. This completes the proof.Remark 9.
The signs ofM
1
(
h
1
,
δ
0
), M
1
(
h
2
,
δ
0
), M
8
(
h
3
,
δ
0
), and M
8
(
h
4
,
δ
0
) can be determined by using the first nonzero coefficients in their expansions.


## 3. Main Result

In this section, we investigate the distributions of limit cycles of system (1). For ɛ
=
0, (1) has two level sets, Γ
~
1 and Γ
~
2, defined by H
(
x
,
y
)
=
-
5
/
12 and H
(
x
,
y
)
=
-
5
/
12 respectively. Γ
~
1 consists of 8 saddles, S
1
=
(
1,2
), S
2
=
(
2,1
), S
3
=
(
2
,
-
1
), S
4
=
(
1
,
-
2
), S
5
=
(
-
1
,
-
2
), S
6
=
(
-
2
,
-
1
), S
7
=
(
-
2,1
), and S
8
=
(
-
1,2
), as shown in Figure 4. Let L
i
j, L
i, L
j
h, c
01, c
02, e
01, e
02, c
21
,
c
22, e
21,e
22,  and so forth are the same as those for Lemma 7. We can write the compound cycles Γ
~
1 as
(34)
Γ
~
1
=
⋃
i
=
1
8
{
S
i
}
∪
{
L
12
,
L
21
,
L
23
,
L
32
,
…
,
L
78
,
L
87
,
L
81
,
L
18
}
.
The equation H
(
x
,
y
)
=
-
8
/
3 defines four centers, C
1
=
(
2,2
), C
3
=
(
2
,
-
2
), C
5
=
(
-
2
,
-
2
), and C
7
=
(
-
2,2
), where the equation H
(
x
,
y
)
=
-
4
/
3 defines four centers of C
2
=
(
2,0
), C
4
=
(
0
,
-
2
), C
6
=
(
-
2,0
), and C
8
=
(
0,2
). The center C
i is inside the 2-polycycle L
i for i
=
1
,
…
,
8.Figure 4We first investigate the2-polycycles L
1 and L
8. Here, L
1
h denotes the periodic orbit defined by H
(
x
,
y
)
=
h surrounding the unique center C
1 and L
8
h denotes the periodic orbit defined by H
(
x
,
y
)
=
h surrounding the unique center C
8. Then
(35)
M
1
(
h
,
δ
)
=
∮
L
1
h
‍
Q
5
d
x
-
P
5
d
y
=
∮
L
1
h
‍
(
Q
5
-
P
5
d
y
d
x
)
d
x
,
h
∈
(
-
8
3
,
-
5
12
)
,
M
8
(
h
,
δ
)
=
∮
L
8
h
‍
Q
5
d
x
-
P
5
d
y
=
∮
L
8
h
‍
(
Q
5
-
P
5
d
y
d
x
)
d
x
,
h
∈
(
-
4
3
,
-
5
12
)
,
where δ
=
(
a
0
,
a
1
,
…
,
a
5
)
∈
R
6. By Lemma 7, we have
(36)
M
1
(
h
,
δ
)
=
c
0
(
δ
)
+
c
1
(
δ
)
(
h
+
5
12
)
ln
⁡
|
h
+
5
12
|
+
c
2
(
δ
)
(
h
+
5
12
)
+
c
3
(
δ
)
(
h
+
5
12
)
2
ln
⁡
|
h
+
5
12
|
+
⋯
,
for 0
<
-
(
h
+
5
/
12
)
≪
1,
(37)
M
8
(
h
,
δ
)
=
e
0
(
δ
)
+
e
1
(
δ
)
(
h
+
5
12
)
ln
⁡
|
h
+
5
12
|
+
e
2
(
δ
)
(
h
+
5
12
)
+
e
3
(
δ
)
(
h
+
5
12
)
2
ln
⁡
|
h
+
5
12
|
+
⋯
,
for 0
<
-
(
h
+
5
/
12
)
≪
1.By Lemma4, for 0
<
h
+
8
/
3
≪
1,  i
=
1,3
,
5,7,
(38)
M
i
(
h
,
δ
)
=
∑
k
≥
0
‍
B
k
(
δ
)
(
h
+
8
3
)
k
+
1
,
and, for 0
<
h
+
4
/
3
≪
1,  j
=
2,4
,
6,8,
(39)
M
j
(
h
,
δ
)
=
∑
k
≥
0
‍
d
k
(
δ
)
(
h
+
4
3
)
k
+
1
.To find the zeros ofM
1
(
h
,
δ
) and M
8
(
h
,
δ
), the coefficients in these asymptotic expansions need to be calculated. In order to calculate the coefficients c
01
(
δ
), c
02
(
δ
), e
01
(
δ
), e
02
(
δ
), and so forth, the expressions of heteroclinic orbits are found as follows:
(40)
L
12

:
y
1
(
x
)
=
1
2
2
x
2
+
5
+
-
12
x
4
+
60
x
2
+
33
,
hh
hhh
1
≤
x
≤
1
2
22
,
y
2
(
x
)
=
1
2
2
x
2
+
5
-
-
12
x
4
+
60
x
2
+
33
,
h
h
hhh
1
2
22
≤
x
≤
2
,
L
21

:
y
3
(
x
)
=
5
-
x
2
,
2
≥
x
≥
1
,
L
81

:
y
4
(
x
)
=
5
-
x
2
,
-
1
≤
x
≤
1
,
L
18

:
y
5
(
x
)
=
1
2
2
x
2
+
5
+
-
12
x
4
+
60
x
2
+
33
,
hhh
1
≥
x
≥
-
1
.Figure5 may be helpful to understand the step to calculate the coefficients in the folowing. By (4), P
5
(
x
,
y
) and P
5
(
x
,
y
) are written as follows:
(41)
P
5
(
x
,
y
)
=
∑
i
=
0
5
‍
p
1
i
a
i
+
p
2
i
b
i
,
Q
5
(
x
,
y
)
=
∑
i
=
0
5
‍
q
1
i
a
i
+
q
2
i
b
i
,
and introduce the following notations:
(42)
p
1
i
j
=
p
1
i
|
y
=
y
j
(
x
)
,
p
2
i
j
=
p
2
i
|
y
=
y
j
(
x
)
,
q
1
i
j
=
q
1
i
|
y
=
y
j
(
x
)
,
q
2
i
j
=
q
2
i
|
y
=
y
j
(
x
)
,
hh
i
,
j
=
1,2
,
3,4
,
5
,
and f
~
j
=
(
d
y
/
d
x
)
|
y
=
y
i
(
x
)
=
(
(
-
2
x
+
3
x
3
-
x
5
)
/
(
2
y
-
3
y
3
+
y
5
)
)
|
y
=
y
i
(
x
). By Lemma 7, we have
(43)
c
01
(
δ
)
=
∫
L
12
‍
Q
5
d
x
-
P
5
d
y
=
∫
1
22
/
2
‍
(
Q
5
-
P
5
d
y
d
x
)
|
y
=
y
1
(
x
)
d
x
+
∫
22
/
2
2
‍
(
Q
5
-
P
5
d
y
d
x
)
|
y
=
y
2
(
x
)
d
x
=
∑
i
=
0
‍
a
i
I
1
i
1
+
b
i
I
2
i
1
+
a
i
I
1
i
2
+
b
i
I
2
i
2
,
c
02
(
δ
)
=
∫
L
21
‍
Q
5
d
x
-
P
5
d
y
=
∫
2
1
‍
(
Q
5
-
P
5
d
y
d
x
)
|
y
=
y
3
(
x
)
d
x
=
∑
i
=
0
‍
a
i
I
1
i
3
+
b
i
I
2
i
3
,
e
01
(
δ
)
=
∫
L
81
‍
Q
5
d
x
-
P
5
d
y
=
∫
-
1
1
‍
(
Q
5
-
P
5
d
y
d
x
)
|
y
=
y
4
(
x
)
d
x
=
∑
i
=
0
‍
a
i
I
1
i
4
+
b
i
I
2
i
4
,
e
02
(
δ
)
=
∫
L
18
‍
Q
5
d
x
-
P
5
d
y
=
∫
1
-
1
‍
(
Q
5
-
P
5
d
y
d
x
)
|
y
=
y
5
(
x
)
d
x
=
∑
i
=
0
‍
a
i
I
1
i
4
+
b
i
I
2
i
4
,
where, for i
=
0,1
,
2,3
,
4,5,
(44)
I
1
i
1
=
∫
1
22
/
2
‍
(
p
1
i
1
+
q
1
i
1
f
~
1
)
d
x
,
I
2
i
1
=
∫
1
22
/
2
‍
(
p
2
i
1
+
q
2
i
1
f
~
1
)
d
x
,
I
1
i
2
=
∫
22
/
2
2
‍
(
p
1
i
2
+
q
1
i
2
f
~
2
)
d
x
,
I
2
i
2
=
∫
22
/
2
2
‍
(
p
2
i
2
+
q
2
i
2
f
~
2
)
d
x
,
I
1
i
3
=
∫
2
1
‍
(
p
1
i
3
+
q
1
i
3
f
~
3
)
d
x
,
I
2
i
3
=
∫
2
1
‍
(
p
2
i
3
+
q
2
i
3
f
~
3
)
d
x
,
I
1
i
4
=
∫
-
1
1
‍
(
p
1
i
4
+
q
1
i
4
f
~
4
)
d
x
,
I
2
i
4
=
∫
-
1
1
‍
(
p
2
i
4
+
q
2
i
4
f
~
4
)
d
x
,
I
1
i
5
=
∫
1
-
1
‍
(
p
1
i
5
+
q
1
i
5
f
~
5
)
d
x
,
I
2
i
5
=
∫
1
-
1
‍
(
p
2
i
5
+
q
2
i
5
f
~
5
)
d
x
.
Thus,
(45)
c
0
(
δ
)
=
c
01
(
δ
)
+
c
02
(
δ
)
=
∑
i
=
0
5
‍
[
(
I
1
i
1
+
I
1
i
2
+
I
1
i
3
)
a
i
+
(
I
2
i
1
+
I
2
i
2
+
I
2
i
2
)
b
i
]
≡
l
0
a
0
+
l
1
a
1
+
l
2
a
2
+
l
3
a
3
+
l
4
a
4
+
l
5
a
5
,
e
0
(
δ
)
=
e
01
(
δ
)
+
e
02
(
δ
)
=
∑
i
=
0
5
‍
[
(
I
1
i
4
+
I
1
i
5
+
I
1
i
3
)
a
i
+
(
I
2
i
4
+
I
2
i
5
+
I
2
i
2
)
b
i
]
≡
m
0
a
0
+
m
1
a
1
+
m
2
a
2
+
m
4
a
4
+
m
5
a
5
.(a)
The 2-polycycleL
1
(b)
The 2-polycycleL
8By (4), the divergence of (1) at S
1 and S
2 is as follows:
(46)
d
1
(
δ
)
=
(
d
P
5
d
x
+
d
Q
5
d
y
)
|
(
1,2
)
=
2
a
0
+
20
a
1
+
150
a
2
-
14
a
4
-
70
a
5
-
48
b
4
+
240
b
5
,
d
2
(
δ
)
=
(
d
P
5
d
x
+
d
Q
5
d
y
)
|
(
2,1
)
=
2
a
0
+
20
a
1
+
150
a
2
-
14
a
4
-
70
a
5
+
48
b
4
-
240
b
5
.
Note that
(47)
(
H
y
x
H
y
y
-
H
x
x
-
H
x
y
)
|
S
1
=
(
0
24
6
0
)
,
(
H
y
x
H
y
y
-
H
x
x
-
H
x
y
)
|
S
2
=
(
0
-
6
-
24
0
)
,
which yields
(48)
λ
1
=
λ
2
=
12
.
By Lemma 3, we have
(49)
c
1
(
δ
)
=
e
1
(
δ
)
=
∑
i
=
1
2
‍
-
1
|
λ
i
|
(
d
P
5
d
x
+
d
Q
5
d
y
)
(
S
i
,
δ
)
=
-
d
1
(
δ
)
|
λ
1
|
-
d
2
(
δ
)
|
λ
1
|
=
-
1
3
a
0
-
10
3
a
1
-
25
a
2
+
7
3
a
4
+
35
3
a
5
.
In the following by letting b
4
=
5
b
5, then
(50)
d
1
(
δ
)
=
d
2
(
δ
)
,
c
1
(
S
1
,
δ
)
=
c
1
(
S
2
,
δ
)
.
Letting c
1
(
δ
)
=
0, then
(51)
a
0
=
-
10
a
1
-
75
a
2
+
7
a
4
+
35
a
5
.
Under c
1
(
δ
)
=
0, we can apply Lemma 7 to calculate the coefficients c
2
(
δ
) and e
2
(
δ
). For convenience, the following notations are introduced: h
=
d
x
/
d
t
=
2
y
-
3
y
3
+
y
5. For j
=
1,2
,
3,4
,
5,  h
j
=
(
2
y
-
3
y
3
+
y
5
)
|
y
=
y
j
(
x
), (
d
P
5
/
d
x
+
d
Q
5
/
d
y
)
≡
∑
j
=
0
5
f
i
a
i, and
(52)
f
0
j
(
x
)
=
f
0
h
|
y
=
y
j
(
x
)
=
2
2
y
-
3
y
3
+
y
5
|
y
=
y
j
(
x
)
,
f
1
j
(
x
)
=
f
1
h
|
y
=
y
j
(
x
)
=
4
x
2
+
4
y
2
2
y
-
3
y
3
+
y
5
|
y
=
y
j
(
x
)
,
f
2
j
(
x
)
=
f
2
h
|
y
=
y
j
(
x
)
=
12
y
2
x
2
+
6
x
4
+
6
y
4
2
y
-
3
y
3
+
y
5
|
y
=
y
j
(
x
)
,
f
3
j
(
x
)
=
f
3
h
|
y
=
y
j
(
x
)
=
0
2
y
-
3
y
3
+
y
5
|
y
=
y
j
(
x
)
=
0
,
f
4
j
(
x
)
=
f
4
h
|
y
=
y
j
(
x
)
=
-
12
y
2
x
2
+
2
x
4
+
2
y
4
2
y
-
3
y
3
+
y
5
|
y
=
y
j
(
x
)
,
f
5
j
(
x
)
=
f
5
h
|
y
=
y
j
(
x
)
=
-
60
y
2
x
2
+
10
x
4
+
10
y
4
2
y
-
3
y
3
+
y
5
|
y
=
y
j
(
x
)
.
By Lemma 7, we have
(53)
c
21
(
δ
)
=
∫
L
12
‍
(
d
P
5
d
x
+
d
Q
5
d
y
)
d
t
=
∑
i
=
0
5
‍
∫
L
12
‍
a
i
f
i
h
d
x
,
c
22
(
δ
)
=
∫
L
21
‍
(
d
P
5
d
x
+
d
Q
5
d
y
)
d
t
=
∑
i
=
0
5
‍
∫
L
21
‍
a
i
f
i
h
d
x
,
e
21
(
δ
)
=
∫
L
81
‍
(
d
P
5
d
x
+
d
Q
5
d
y
)
d
t
=
∑
i
=
0
5
‍
∫
L
81
‍
a
i
f
i
h
d
x
,
e
22
(
δ
)
=
∫
L
18
‍
(
d
P
5
d
x
+
d
Q
5
d
y
)
d
t
=
∑
i
=
0
5
‍
∫
L
18
‍
a
i
f
i
h
d
x
.
Substituting (52) into (53), with f
0
=
2 being considered, we have
(54)
c
21
(
δ
)
=
∑
i
=
1
5
‍
∫
L
12
‍
a
i
f
i
h
d
x
+
∫
L
12
‍
-
20
a
1
-
150
a
2
+
14
a
4
+
70
a
5
h
d
x
=
∑
i
=
1
‍
a
i
J
i
1
+
∑
i
=
1
‍
a
i
J
i
2
,
c
22
(
δ
)
=
∑
i
=
1
5
‍
∫
L
21
‍
a
i
f
i
h
d
x
+
∫
L
21
‍
-
20
a
1
-
150
a
2
+
14
a
4
+
70
a
5
h
d
x
=
∑
i
=
1
‍
a
i
J
i
3
,
e
21
(
δ
)
=
∑
i
=
1
5
‍
∫
L
81
‍
a
i
f
i
h
d
x
+
∫
L
81
‍
-
20
a
1
-
150
a
2
+
14
a
4
+
70
a
5
h
d
x
=
∑
i
=
1
‍
a
i
J
i
4
,
e
22
(
δ
)
=
∑
i
=
1
5
‍
∫
L
18
‍
a
i
f
i
h
d
x
+
∫
L
18
‍
-
20
a
1
-
150
a
2
+
14
a
4
+
70
a
5
h
d
x
=
∑
i
=
1
‍
a
i
J
i
5
,
where J
3
1
=
J
3
2
=
J
3
3
=
J
3
4
=
J
3
5
=
0 and
(55)
J
1
1
=
∫
1
22
/
2
‍
f
1
1
(
x
)
+
-
20
h
1
d
x
,
J
1
2
=
∫
22
/
2
2
‍
f
1
2
(
x
)
+
-
20
h
2
d
x
,
J
1
3
=
∫
2
1
‍
f
1
3
(
x
)
+
-
20
h
3
d
x
,
J
2
1
=
∫
1
22
/
2
‍
f
2
1
(
x
)
+
-
150
h
1
d
x
,
J
2
2
=
∫
22
/
2
2
‍
f
2
2
(
x
)
+
-
150
h
2
d
x
,
J
2
3
=
∫
2
1
‍
f
2
3
(
x
)
+
-
150
h
3
d
x
,
J
4
1
=
∫
1
22
/
2
‍
f
4
1
(
x
)
+
14
h
1
d
x
,
J
4
2
=
∫
22
/
2
2
‍
f
4
2
(
x
)
+
14
h
2
d
x
,
J
4
3
=
∫
2
1
‍
f
4
3
(
x
)
+
14
h
3
d
x
,
J
5
1
=
∫
1
22
/
2
‍
f
5
1
(
x
)
+
70
h
1
d
x
,
J
5
2
=
∫
22
/
2
2
‍
f
5
2
(
x
)
+
70
h
2
d
x
,
J
5
3
=
∫
2
1
‍
f
5
3
(
x
)
+
70
h
3
d
x
,
J
1
4
=
∫
-
1
1
‍
f
1
1
(
x
)
+
-
20
h
4
d
x
,
J
1
5
=
∫
1
-
1
‍
f
1
2
(
x
)
+
-
20
h
5
d
x
,
J
2
4
=
∫
-
1
1
‍
f
2
1
(
x
)
+
-
150
h
4
d
x
,
J
2
5
=
∫
1
-
1
‍
f
2
2
(
x
)
+
-
150
h
5
d
x
,
J
4
4
=
∫
-
1
1
‍
f
4
1
(
x
)
+
14
h
4
d
x
,
J
4
5
=
∫
1
-
1
‍
f
4
2
(
x
)
+
14
h
5
d
x
,
J
5
4
=
∫
-
1
1
‍
f
5
1
(
x
)
+
70
h
4
d
x
,
J
5
5
=
∫
1
-
1
‍
f
5
2
(
x
)
+
70
h
5
d
x
.
Applying Lemma 7, we have
(56)
c
2
(
δ
)
=
c
21
(
δ
)
+
c
22
(
δ
)
=
∑
i
=
0
‍
(
J
i
1
+
J
i
2
+
J
i
3
)
a
i
≡
J
1
a
1
+
J
2
a
2
+
J
4
a
4
+
J
5
a
5
,
e
2
(
δ
)
=
e
21
(
δ
)
+
e
22
(
δ
)
=
∑
i
=
0
‍
(
J
i
4
+
J
i
5
+
J
i
3
)
a
i
≡
R
1
a
1
+
R
2
a
2
+
R
4
a
4
+
R
5
a
5
.
The integrals in (44) and (55) and the coefficients in (45) and (56) can be obtained by numeral computation on Maple 13; see Appendix.In order to find the local coefficientc
3
(
S
1
,
δ
) at the saddle S
1
(
1,2
) we make a change of variables of the form u
=
(
2
/
2
)
(
x
-
1
)
,
v
=
2
(
y
-
2
) and time rescaling τ
=
k
t
,
k
=
1 so that the system (1) becomes
(57)
d
u
d
τ
=
H
~
v
+
ɛ
p
~
(
u
,
v
,
δ
)
,
d
v
d
τ
=
-
H
~
u
+
ɛ
q
~
(
u
,
v
,
δ
)
,
where
(58)
H
~
(
u
,
v
)
=
H
(
x
,
y
)
|
{
x
=
2
u
+
1
,
y
=
(
2
/
2
)
v
+
2
}
=
6
v
2
-
6
u
2
-
10
2
3
u
3
+
25
2
6
v
3
+
5
u
4
+
35
16
v
4
+
4
2
u
5
+
2
4
v
5
+
4
3
u
6
+
1
48
v
6
-
5
12
,
p
~
(
u
,
v
)
=
2
2
P
5
(
x
,
y
)
|
{
x
=
2
u
+
1
,
y
=
(
2
/
2
)
v
+
2
}
,
q
~
(
u
,
v
)
=
2
Q
5
(
x
,
y
)
|
{
x
=
2
u
+
1
,
y
=
(
2
/
2
)
v
+
2
}
.
Writing functions p
~
(
u
,
v
) and q
~
(
u
,
v
) as the form
(59)
p
~
(
u
,
v
)
=
∑
i
+
j
=
0
5
‍
a
~
i
j
u
i
v
j
,
q
~
(
u
,
v
)
=
∑
i
+
j
=
0
5
‍
b
~
i
j
u
i
v
j
,
then the formula for the second local coefficient at the saddle in Lemma 2 can be applied directly; we have
(60)
c
3
(
S
1
,
δ
)
=
7
216
a
1
+
35
72
a
2
+
55
216
a
4
+
275
216
a
5
.
Similarly, we have
(61)
c
3
(
S
2
,
δ
)
=
7
216
a
1
+
35
72
a
2
+
55
216
a
4
+
275
216
a
5
.
Applying Lemma 7, we have
(62)
c
3
(
δ
)
=
e
3
(
δ
)
=
c
3
(
S
1
,
δ
)
+
c
3
(
S
2
,
δ
)
=
7
108
a
1
+
35
36
a
2
+
55
108
a
4
+
275
108
a
5
.In order to findB
i
(
δ
),  i
=
0,1
,
2, we move the center C
1
=
(
2,2
) into the origin by letting u
=
2
6
(
x
-
2
) and v
=
2
6
(
y
-
2
); that is, x
=
(
6
/
12
)
u
+
2  and y
=
(
6
/
12
)
v
+
2 and make the time rescaling d
τ
=
24
d
t so that the system (1) becomes
(63)
d
u
d
τ
=
d
H
1
c
d
v
+
ɛ
p
1
(
u
,
v
)
,
d
v
d
τ
=
-
d
H
1
c
d
u
+
ɛ
q
1
(
u
,
v
)
,
where
(64)
H
1
c
(
u
,
v
)
=
H
(
x
,
y
)
|
{
x
=
(
6
/
12
)
u
+
2
,
y
=
(
6
/
12
)
v
+
2
}
=
-
8
3
+
1
2
u
2
+
1
2
v
2
+
25
6
432
u
3
+
25
6
432
v
3
+
35
2304
u
4
+
35
2304
v
4
+
6
3456
u
5
+
6
3456
v
5
+
1
82944
u
6
+
1
82944
v
6
,
p
1
(
u
,
v
)
=
6
12
p
(
x
,
y
)
|
{
x
=
(
6
/
12
)
u
+
2
,
y
=
(
6
/
12
)
v
+
2
}
,
q
1
(
u
,
v
)
=
6
12
q
(
x
,
y
)
|
{
x
=
(
6
/
12
)
u
+
2
,
y
=
(
6
/
12
)
v
+
2
}
.
Let
(65)
M
~
1
c
(
h
,
δ
)
=
∮
H
1
c
(
u
,
v
)
=
h
‍
q
1
d
u
-
p
1
d
v
=
∑
k
≥
0
‍
b
~
k
(
h
+
8
3
)
k
+
1
which is the Melnikov function of the new system (63). Applying the formula for the Hopf coefficients b
~
0
(
δ
), b
~
1
(
δ
), and b
~
2
(
δ
) directly in [20], we have
(66)
b
~
0
(
δ
)
=
1
6
π
a
0
+
8
3
π
a
1
+
32
π
a
2
-
32
3
π
a
4
-
160
3
π
a
5
,
b
~
1
(
δ
)
|
b
~
0
=
0
=
-
11
108
π
a
1
-
19
9
π
a
2
+
25
27
π
a
4
+
125
27
π
a
5
,
b
~
2
(
δ
)
|
b
~
0
=
0
=
-
17545
839808
π
a
1
-
28379
69984
π
a
2
+
37445
209952
π
a
4
+
187225
209952
π
a
5
.
By Lemma 6,
(67)
M
1
(
h
,
δ
)
=
M
~
1
c
(
h
,
δ
)
.
Therefore
(68)
B
0
(
δ
)
=
b
~
0
(
δ
)
=
1
6
π
a
0
+
8
3
π
a
1
+
32
π
a
2
-
32
3
π
a
4
-
160
3
π
a
5
,
B
1
(
δ
)
|
B
0
=
0
=
b
~
1
(
δ
)
|
b
~
0
=
0
=
-
11
108
π
a
1
-
19
9
π
a
2
+
25
27
π
a
4
+
125
27
π
a
5
,
B
2
(
δ
)
|
B
0
=
0
=
b
~
2
(
δ
)
|
b
~
0
=
0
=
-
17545
839808
π
a
1
-
28379
69984
π
a
2
+
37445
209952
π
a
4
+
187225
209952
π
a
5
.
Similarly, the expressions of d
k
(
δ
) are obtained as follows:
(69)
d
0
(
δ
)
=
6
6
π
a
0
+
4
6
3
π
a
1
+
8
6
π
a
2
+
8
6
3
π
a
4
+
40
6
3
π
a
5
,
d
1
(
δ
)
|
d
0
=
0
=
-
6
108
π
a
1
+
6
18
π
a
2
-
35
6
54
π
a
4
-
175
6
54
π
a
5
,
d
2
(
δ
)
|
d
0
=
0
=
535
104976
π
6
a
1
+
1949
17496
π
6
a
2
-
12055
52488
π
6
a
4
-
60275
52488
π
6
a
5
.We will use the coefficientsc
0
(
δ
), c
1
(
δ
), c
2
(
δ
), c
3
(
δ
), B
0
(
δ
), B
1
(
δ
), B
2
(
δ
), e
0
(
δ
), e
1
(
δ
), e
2
(
δ
), e
3
(
δ
), d
0
(
δ
), d
1
(
δ
), and d
2
(
δ
) obtained above to study the limit cycle bifurcation.(1) Solving the equations(70)
c
0
(
δ
)
=
c
1
(
δ
)
=
d
0
(
δ
)
=
d
1
(
δ
)
=
0
gives
(71)
a
0
=
μ
1
a
1
,
a
2
=
μ
2
a
1
,
a
4
=
μ
3
a
1
,
a
5
=
μ
4
a
1
,
where
(72)
μ
1
=
-
232
73
,
μ
2
=
-
163
1752
,
μ
3
=
1
1752
39
l
5
+
27840
l
0
-
8760
l
1
+
815
l
2
-
l
5
+
5
l
4
,
μ
4
=
-
1
1752
5568
l
0
-
1752
l
1
+
163
l
2
+
39
l
4
-
l
5
+
5
l
4
.
Approximate computation using Maple.13 gives
(73)
a
0
=
μ
1
a
1
=
-
3.1780821917808219178
a
1
≡
a
0
*
,
a
2
=
μ
2
a
1
=
-
0.093036529680365296804
a
1
≡
a
2
*
,
a
4
=
μ
3
a
1
=
1.3418001549779299848
×
1
0
8
a
1
≡
a
4
*
,
a
5
=
μ
4
a
1
=
-
2.6836003104010654490
×
1
0
7
a
1
≡
a
5
*
.
We can easily find that
(74)
rank
⁡
∂
(
c
0
,
c
1
,
d
0
,
d
1
)
∂
(
a
0
,
a
2
,
a
4
,
a
5
)
=
4
.
We fix a
1
>
0, a
3
≠
0, and take δ
=
(
a
0
,
a
2
,
a
4
,
a
5
), δ
0
=
(
a
0
*
,
a
2
*
,
a
4
*
,
a
5
*
). Inserting δ
0 into c
2
(
δ
), d
2
(
δ
), e
0
(
δ
) and B
0
(
δ
) we have
(75)
c
2
(
δ
0
)
=
-
1.6154178146383561644
a
1
,
d
2
(
δ
0
)
=
-
0.000155025650574300610
π
6
a
1
,
e
0
(
δ
0
)
=
0.001040156096575342
a
1
,
B
0
(
δ
0
)
=
-
0.6027397260050228310
π
a
1
.
Taking h
1
=
-
5
/
12
-
ε
1, h
2
=
-
8
/
3
+
ε
1, h
3
=
-
5
/
12
-
ε
1, and h
4
=
-
4
/
3
+
ε
1 with ε
1
>
0 small enough, we have
(76)
M
1
(
h
1
,
δ
0
)
=
c
2
(
δ
0
)
(
-
ε
1
)
+
O
(
(
-
ε
1
)
2
ln
⁡
|
ε
1
|
)
>
0
,
M
1
(
h
2
,
δ
0
)
=
B
0
(
δ
0
)
(
ε
1
)
+
O
(
ε
1
2
)
<
0
,
M
8
(
h
3
,
δ
0
)
=
e
0
(
δ
0
)
+
O
(
(
ε
1
)
2
ln
⁡
|
ε
1
|
)
>
0
,
M
8
(
h
4
,
δ
0
)
=
d
2
(
δ
0
)
(
ε
1
)
3
+
O
(
(
ε
1
)
4
)
<
0
,
which yield (
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2
=
1 and (
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2
=
1. Hence, by Theorem 8, (1) can have 6 limit cycles inside the 2-polycycles L
1 and L
8, 2 limit cycles near the 2-polycycle L
1, 2 limit cycles near the center C
8, 1 limit cycle between the center C
1 and the polycycle L
1, and 1 limit cycle between the center C
8 and the polycycle L
8. Considering that the system (1) is Z
4-equivariant, the system (1) can have at least 24 limit cycles. See Figure 2(a) for their distribution.(2) Further let(77)
c
0
(
δ
)
=
c
1
(
δ
)
=
B
0
(
δ
)
=
e
0
(
δ
)
=
0
,
to obtain
(78)
a
0
=
w
1
a
1
,
a
2
=
w
2
a
1
,
a
4
=
w
3
a
1
,
a
5
=
w
4
a
1
,
where w
1
=
-
16
(
55
m
2
l
4
-
11
m
2
l
5
-
55
m
4
l
2
+
15
m
5
l
4
+
11
m
5
l
2
-
15
m
4
l
5
+
360
m
4
l
1
-
72
m
5
l
1
-
360
m
1
l
4
+
72
m
1
l
5
)
/
(
5760
m
4
l
0
-
95
m
4
l
2
-
1152
m
5
l
0
+
39
m
5
l
4
+
19
m
5
l
2
-
5760
m
0
l
4
+
1152
m
0
l
5
-
39
m
4
l
5
+
95
m
2
l
4
- 19
m
2
l
5
), w
2
=
-
(
-
2
m
4
l
5
+
880
m
4
l
0
-
95
m
4
l
1
- 176
m
5
l
0
+
2
m
5
l
4
+
19
m
5
l
1
-
880
m
0
l
4
+
176
m
0
l
5
+
95
m
1
l
4
-
19
m
1
l
5
)
/
(
5760
m
4
l
0
-
95
m
4
l
2
-
1152
m
5
l
0
+
39
m
5
l
4
+
19
m
5
l
2
-
5760
m
0
l
4
+
1152
m
0
l
5
-
39
m
4
l
5
+ 95
m
2
l
4
-
19
m
2
l
5
), w
3
=
(
240
m
5
l
0
-
2
m
2
l
5
+
2
m
5
l
2
-
240
m
0
l
5
-
39
m
5
l
1
+
39
m
1
l
5
- 5760
l
0
m
1
-
880
l
2
m
0
+
95
l
2
m
1
+
880
l
0
m
2
+
5760
l
1
m
0
-
95
l
1
m
2
)
/
(
5760
m
4
l
0
-
95
m
4
l
2
-
1152
m
5
l
0
+
39
m
5
l
4
+
19
m
5
l
2
-
5760
m
0
l
4
+
1152
m
0
l
5
-
39
m
4
l
5
+
95
m
2
l
4
-
19
m
2
l
5
), and w
4
=
(
240
m
0
l
4
-
39
m
1
l
4
+
2
m
2
l
4
-
19
l
2
m
1
-
240
m
4
l
0
+
176
l
2
m
0
+
1152
l
0
m
1
-
2
m
4
l
2
- 176
l
0
m
2
+
39
m
4
l
1
+
19
l
1
m
2
-
1152
l
1
m
0
)
/
(
5760
m
4
l
0
-
95
m
4
l
2
- 1152
m
5
l
0
+
39
m
5
l
4
+
19
m
5
l
2
-
5760
m
0
l
4
+
1152
m
0
l
5
-
39
m
4
l
5
+
95
m
2
l
4
-
19
m
2
l
5
). Approximate computation using Maple.13 gives
(79)
a
0
=
-
9.4485863716681452021
a
1
≡
a
0
*
,
a
2
=
0.0030582821716100337153
a
1
≡
a
2
*
,
a
4
=
8655624.6835808229498
a
1
≡
a
4
*
,
a
5
=
-
1731124.9144080276985
a
1
≡
a
5
*
.
As before, we have
(80)
rank
⁡
∂
(
c
0
,
c
1
,
B
0
,
e
0
)
∂
(
a
0
,
a
2
,
a
4
,
a
5
)
=
4
,
fixing a
1
>
0, a
3
≠
0, and taking δ
=
(
a
0
,
a
2
,
a
4
,
a
5
) and δ
0
=
(
a
0
*
,
a
2
*
,
a
4
*
,
a
5
*
). Noting that
(81)
c
2
(
δ
0
)
=
-
0.45572159725036392404
a
1
,
B
1
(
δ
0
)
=
-
0.0050298137906989600656
π
a
1
,
e
2
(
δ
0
)
=
-
0.85071588097
a
1
,
d
0
(
δ
0
)
=
0.0804770206482136030
π
6
a
1
,
we have
(82)
M
1
(
h
1
,
δ
0
)
=
c
2
(
δ
0
)
(
-
ε
3
)
+
O
(
-
ε
3
ln
⁡
|
ε
3
|
)
>
0
,
M
1
(
h
2
,
δ
0
)
=
B
1
(
δ
0
)
(
ε
3
2
)
+
O
(
ε
3
3
)
<
0
,
M
8
(
h
3
,
δ
0
)
=
e
2
(
δ
0
)
(
-
ε
3
)
+
O
(
(
-
ε
3
)
ln
⁡
|
ε
3
|
)
>
0
,
M
8
(
h
4
,
δ
0
)
=
d
0
(
δ
0
)
ε
3
+
O
(
(
-
ε
3
)
2
)
>
0
,
where h
1
=
-
5
/
12
-
ε
3, h
2
=
-
8
/
3
+
ε
3, h
3
=
-
5
/
12
-
ε
3, and h
4
=
-
4
/
3
+
ε
3 with ε
1
>
0being small. Hence, noting that (
1
-
sgn
⁡
(
M
1
(
h
1
,
δ
0
)
M
1
(
h
2
,
δ
0
)
)
)
/
2
=
1, (
1
-
sgn
⁡
(
M
8
(
h
3
,
δ
0
)
M
8
(
h
4
,
δ
0
)
)
)
/
2
=
0, and  (
1
+
sgn
⁡
(
c
2
(
δ
0
)
e
2
(
δ
0
)
)
)
/
2
=
1, by Theorem 8 again, we can obtain 6 limit cycles inside the 2-polycycles L
1 and L
8. By Z
4-equivariance, the system (1) can have 24 limit cycles. See Figure 2(b). Then Theorem 1 is proved.


## 4. Conclusion

In this paper, we proved that aZ
4-equivalent quintic near-Hamiltonian system can also have 24 limit cycles compared to a z
6 and z
3-equivalent quintic near-Hamiltonian system having 24 limit cycles. Certainly, the distributions of 24 limit cycles obtained in this paper are new. The method we use is the expansions of the corresponding Melnikov functions, which is different from the methods of detect function and normal form, which are the main methods of the previous work on z
q-equivalent quintic near-Hamiltonian system.



---

*Source: 792439-2014-03-03.xml*