Acta Mathematicae Applicatae Sinica, English Series
Vol. 30, No. 3 (2014) 781–800
DOI: 10.1007/s10255-014-0420-x
http://www.ApplMath.com.cn & www.SpringerLink.com
Acta Mathemacae Applicatae Sinica,
English Series
The Editorial Office of AMAS &
Springer-Verlag Berlin Heidelberg 2014
Z2 -Equivariant Cubic System Which Yields 13 Limit
Cycles
Yi-rong LIU1 , Ji-bin LI2,3
1 School
of Mathematics, Central South University, Changsha 410083, China (E-mail: [email protected])
2 Department
3 School
of Mathematics, Zhejiang Normal University, Jinhua 321004, China (E-mail: [email protected])
of Science, Kunming University of Science and Technology, Kunming 650093, China
Abstract
For the planar Z2 -equivariant cubic systems having two elementary focuses, the characterization
of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The
necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in
this paper, we show that under small Z2 -equivariant cubic perturbations, this cubic system has at least 13 limit
cycles with the scheme 1 ⊂ 6 ∪ 6.
Keywords
planar dynamical system; limit cycles; bifurcations; Lyapnov constant; weak focus
2000 MR Subject Classification
1
34C05
Introduction and Preliminary
The second part of Hilbert’s 16th problem, asking for the maximum number H(n) of the
numbers of limit cycles and their relative positions for all polynomial differential systems of
degree n, is still open even for the quadratic case (n = 2) (see [7]).
In order to obtain more limit cycles and various configuration patterns of their relative
dispositions, we indicated in [2–27] that an efficient method is to perturb the symmetric Hamiltonian systems having maximal number of centers, i.e., to study the weakened Hilbert’s 16th
problem for the Zq -equivariant planar polynomial Hamiltonian systems, since bifurcation and
symmetry are closely connected and symmetric systems play pivotal roles as a bifurcation point
in all planar Hamiltonian system class. Using this idea and the method of detection functions,
we proved that H(3) ≥ 11 in [4].
For the Z2 -equivariant planar cubic system with two elementary focuses at (1, 0) and (−1, 0),
an example of existing at least 12 small-amplitude limit cycles was found by Yu and Han[28−30] .
Liu and Huang[27] confirmed their conclusion and gave some shortened expressions of the Liapunov constants for the system. Recently, in [24] and [1], for the above Z2 -equivariant cubic
system having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants have been completely solved. The necessary
and sufficient conditions for the existence of the bi-center are obtained.
On the basis of the above work of the bi-center problem, we consider, in this paper, the
following Z2 −equivariant cubic vector field
dx
= X1 (x, y) + X3 (x, y),
dt
dy
= Y1 (x, y) + Y3 (x, y),
dt
Manuscript received December 18, 2008. Revised December 18, 2009.
Supported by the National Natural Science Foundation of China (No. 11371373 and 10831003).
(1.1)
782
Y.R. LIU, J.B. LI
where Xj (x, y), Yj (x, y), j = 1, 3 are linear and cubic polynomials of x and y, respectively.
Lemma 1.1[24] . Suppose that (1.1) has at least two finite elementary focuses which can be
arranged at (1, 0) and (−1, 0). Then, (1.1) can be reduced to the following form
dx
δ
δ
= − x − (a1 + 1)y + x3 + a1 x2 y + a2 xy 2 + a3 y 3 ,
dt
2
2
dy
1
1 3
= − x + (δ − a4 )y + x + a4 x2 y + a5 xy 2 + a6 y 3 .
dt
2
2
(1.2)
It is easy to see that (±1, 0) are elementary focuses of (1.2), and the linearized system of
(1.2) at (±1, 0) has the form
dx
= δ(x ± 1) − y,
dt
Theorem 1.1.
(±1, 0) are
dy
= (x ± 1) + δy.
dt
(1.3)
If δ = 0, then the first 6 Lyapunov constants of (1.2) at the singular points
1
V3 = (a2 + 2a1 a2 − 2a4 + 2a2 a5 − 2a4 a5 + 3a6 ),
4
1
V5 = [2(6a1 + 6a21 + 5a2 a4 − 2a24 + 6a5 + 6a1 a5 )(2a2 + a1 a2 + 2a4 + a2 a5 + 2a4 a5 )
36
− 3(2 + 2a1 − a3 + 2a5 + 2a1 a5 )(5a1 a2 + 3a4 + 4a1 a4 + 5a2 a5 + 4a4 a5 )],
1
h0 h3 ,
V7 =
(1.4)
864
1
V9 =
h0 h4 ,
45360
1
V11 =
h0 h5 ,
698544
−44
V13 =
a2 (a2 − a4 )(2a2 − a4 )(4a2 − a4 )(5a2 − a4 )(2a2 + a4 )(4a2 + a4 )h0 h6 ,
85562001
where
h0 =6a1 + 6a21 − 3a1 a3 − 4a2 a4 − 2a1 a2 a4 − 4a24 + 6a5
+ 12a1 a5 + 6a21 a5 − 3a3 a5 − 2a2 a4 a5 − 4a24 a5 + 6a25 + 6a1 a25 ,
h3 =30a2 − 54a1 a2 − 84a21 a2 + 70a32 − 105a2a3 + 36a4 + 36a1 a4
− 70a22 a4 − 84a3 a4 − 52a2 a24 + 16a34 + 126a2 a5 + 126a1a2 a5 ,
h4 =108(1 + a1 )2 (5a2 − 8a4 ) − 12(1 + a1 )(525a32 − 420a22 a4 − 110a2a24 + 92a34 )
+ 5(539a52 − 560a32 a24 + 448a22a34 + 96a2 a44 − 64a54 ),
h5 =[−4(1 + a1 )2 (128a2 − 97a4 ) + 4(1 + a1 )(3136a32 − 2401a22 a4 + 440a2 a24 + 94a34 )
− a4 (7007a42 − 784a32 a4 + 2352a22a24 − 1256a2a34 + 16a44 )](9 + 9a1 + 4a24 ),
h6 =9038315a22 + 4146497a2a4 + 191510a24.
Theorem 1.2[24] . If δ = 0, then the necessary and sufficient condition for the singular points
(±1, 0) of (1.2) to be bi-center is that the first 6 Lyapunov constants are all equal to zero.
Z2 -Equivariant
Cubic System Which Yield 13 Limit Cycles
783
Theorem 1.3[24] For δ = 0, the points (±1, 0) of (1.2) are weak focuses of order 6 if and
only if a4 h0 6= 0 and ak = e
ak (a4 , λ), k = 1, 2, 3, 5, 6, where
1
[−18 + (−8 + 154λ + 385λ2 )a24 ],
18
e
a2 (a4 , λ) =λa4 ,
1
e
a3 (a4 , λ) = −
a2 [2880(8 + 74λ + 143λ2 )
64800 4
+ (3578819 + 73223024λ + 158462585λ2)a24 ],
1
e
a5 (a4 , λ) = − [126 + (44 + 320λ + 797λ2 )a24 ],
90
1
e
a6 (a4 , λ) = −
a4 [45(4 − 19λ) + (502 + 6820λ + 16105λ2)a24 ],
675
(1.5)
g(λ) = 20λ3 − 35λ2 − 20λ − 1.
(1.6)
e
a1 (a4 , λ) =
λ is a real root of the cubic algebraic equation
Remark 1.1.
Obviously, the three real roots of g(λ) are
√
97
7
λ1 =
+
cos θ0 = 2.21224585 · · · ,
12 √6
7
97
2π λ2 =
+
cos θ0 −
= −0.05557709 · · ·,
12 √6
3
7
2π 97
λ3 =
+
cos θ0 +
= −0.40666876 · · ·,
12
6
3
where
θ0 =
Remark 1.2.
(1.7)
36√2319 1
arctan
.
3
4451
Let
κ=
−11
(56794007957132160 − 3379058706136258840a24
354294000000
+ 50225301517577575587a44 + (1161552717525657600
− 69108607749089418880a24 + 1027209339522075478992a44)λ
+ (2512846172532009600 − 149506172061524927560a24
+ 2222214298048672809225a44)λ2 ) .
(1.8)
2
Then V13 = a11
4 κ, when the conditions of Theorem 1.3 hold. Therefore,as a polynomial of a4 , κ
has no positive zero when λ = λ2 and λ = λ3 . When λ = λ1 , κ has exactly two positive zeros
a24 = ω1 and a24 = ω2 , where
ω1 = 0.03274565 · · · ,
ω2 = 0.03453237 · · · .
(1.9)
Thus, the singular points (±1, 0) of (1.2) are weak focuses of order 6 if and only if (1.5) is
satisfied and g(λ) = 0, a4 6= 0. In addition, when λ = λ1 , a24 6= ω1 , ω2 .
In [24], we have proved that the Jacobin determinants of V3 , V5 , V7 , V9 , V11 with respect
to a1 , a2 , a3 , a5 , a6 are not zero when the conditions of Theorem 1.3 hold. Hence, we have
784
Y.R. LIU, J.B. LI
Theorem 1.4[24]. Suppose that the singular points (±1, 0) of (1.2) are weak focuses of order
6. Then, by small perturbations of the coefficients of the right hand sides of (1.2), 6 small
amplitude limit cycles can be created near (1, 0) and (−1, 0), respectively.
We see from Theorem 1.3 and Remark 1.2 that to guarantee the singular points (±1, 0)
of (1.2) to be weak focuses, we are only requested to give the conditions for the parameters
ak , k = 1, 2, 3, 5, 6. The parameter a4 is free. Therefore, when we choose the parameter a4 , we
may have more interesting result than Theorem 1.4.
In Section 5, we shall use the Lyapunov constants to construct an example, such that it
has at least 13 limit cycles. Upon that, we need to use some known results in [23] (or see [1,
Chapter 1]) as follows.
Consider the following real analytic system in the neighborhood of the origin
∞
X
dx
= δx − y +
Xk (x, y) = X(x, y),
dt
k=2
∞
X
dy
= x + δy +
Yk (x, y) = Y (x, y),
dt
(1.10)
k=2
where Xk (x, y) =
P
i+j=k
Aij xi y j , Yk (x, y) =
P
Bij xi y j are k-order homogeneous polynomials
i+j=k
of x and y. By the Poincaré theorem, there exists formal power series F = x2 +y 2 +h.o.t., where
h.o.t. stands for high order terms, such that
∞
X
dF
V2k+1 (x2 + y 2 )k+1 .
= 2δ(x2 + y 2 ) +
dt
(1.11)
k=1
We say that V2k+1 is the kth order Lyapunov constant of (1.10) at the origin. Under the
∞
P
polar coordinates x = r cos θ, y = r sin θ, suppose that r =
vk (θ)hk is a solution of (1.10),
k=1
satisfying the initial condition r|θ=0 = h. Then, we say that v2k+1 (2π) is the k-order focal value
of (1.10) at the origin.
(0)
Theorem 1.5[23] . For any integer m, there are polynomials with rational coefficients ξm ,
(1)
(m−1)
ξm , · · · , ξm
with respect to m values v1 (π), v2 (π), · · · , v2m (π) and v1 (2π), v2 (2π), · · ·,
v2m (2π), such that
(0)
(1)
(m−1)
[1 + v1 (π)]v2m (2π) = ξm
[v1 (2π) − 1] + ξm
v3 (2π) + · · · + ξm
v2m−1 (2π).
(1.12)
Theorem 1.6[23] . For δ = 0 and any positive integer m, v2m+1 (2π) and πV2m+1 are alge(1)
(2)
braically equivalent, i.e., v3 (2π) = πV3 . And when k > 1, there exist polynomials ηm , ηm , · · ·,
(m−1)
ηm
with coefficients Aij , Bij , such that
(1)
(m−1)
v2m+1 (2π) = πV2m+1 + ηm
V3 + · · · + ηm
V2m−1 .
(1.13)
Z2 -Equivariant
2
785
Cubic System Which Yield 13 Limit Cycles
Infinite Singular Point of System (1.2) on the Equator
If δ = 0, then system (1.2) becomes
dx
= −(a1 + 1)y + a1 x2 y + a2 xy 2 + a3 y 3 = X1 (x, y) + X3 (x, y),
dt
dy
1
1
= − x − a4 y + x3 + a4 x2 y + a5 xy 2 + a6 y 3 = Y1 (x, y) + Y3 (x, y).
dt
2
2
(2.1)
To consider the bifurcation condition of limit cycles of (2.1) yielded from the infinity (i.e., the
equator Γ∞ of the Poincare sphere), let
P2 (x, y) =2[xX1 (x, y) + yY1 (x, y)] = −(3 + 2a1 )xy − 2a4 y 2 ,
Q2 (x, y) =2[xY1 (x, y) − yX1 (x, y)] = −x2 − 2a4 xy + 2(1 + a1 )y 2 ,
P4 (x, y) =2[xX3 (x, y) + yY3 (x, y)]
=(1 + 2a1 )x3 y + 2(a2 + a4 )x2 y 2 + 2(a3 + a5 )xy 3 + 2a6 y 4 ,
(2.2)
Q4 (x, y) =2[xY3 (x, y) − yX3 (x, y)]
=x4 + 2a4 x3 y + 2(a5 − a1 )x2 y 2 + 2(a6 − a2 )xy 3 − 2a3 y 4 .
It is well known that on the equator Γ∞ , system (2.1) has no real singular point if and only
if Q4 (x, y) is positive definite.
Suppose that Q4 (x, y) is positive definite. Then, under the transformation
x=
cos θ
,
ρ
y=
sin θ
,
ρ
(2.3)
system (2.1) becomes
dρ
P4 (cos θ, sin θ) + P2 (cos θ, sin θ)ρ2
= −ρ
.
dθ
Q4 (cos θ, sin θ) + Q2 (cos θ, sin θ)ρ2
(2.4)
Q4 (cos θ, sin θ)|θ=0 = 1,
(2.5)
When
we say that ρ = 0 is the infinity of system (2.1).
Clearly, the right hand side of (2.4) is an odd function with respect to ρ. Hence, the solution
of (2.4) satisfying the initial condition ρ|θ=0 = h has the form
ρ=
∞
X
k=0
ve2k+1 (θ)h2k+1 .
(2.6)
Plugging (2.6) into equation (2.4), we have
ve1 (θ) = exp
Z
ve3 (θ) = v̄1 (θ)
θ
0
Z
0
−P4 (cos ϕ, sin ϕ)
dϕ,
Q4 (cos ϕ, sin ϕ)
θ
(2.7)
P4 (cos ϕ, sin ϕ)Q2 (cos ϕ, sin ϕ) − P2 (cos ϕ, sin ϕ)Q4 (cos ϕ, sin ϕ) 2
ve1 (ϕ)dϕ,
Q24 (cos ϕ, sin ϕ)
(2.8)
786
Y.R. LIU, J.B. LI
By (2.7), and since
d
Q4 (cos θ, sin θ),
dθ
(2.9)
R2 = a4 cos2 θ + 2(a1 + a5 ) cos θ sin θ + (a2 + 3a6 ) sin2 θ,
(2.10)
4P4 (cos θ, sin θ) ≡ 2R2 (cos θ, sin θ) −
where
we obtain
ve1 (θ) =
where
p
4
Q4 (cos θ, sin θ) · eG(θ) ,
1
G(θ) = −
2
Z
0
θ
R2 (cos ϕ, sin ϕ)
dϕ.
Q4 (cos ϕ, sin ϕ)
(2.11)
(2.12)
It follows from the above that the necessary conditions for the infinity to be a weak focus of
(1.2) are that Q4 (x, y) is positive definite and G(2π) = 0.
We see from (2.8), (2.11) and (2.12) that, when G(2π) = 0, ve1 (θ) is a periodic function
with period π and
ve3 (2π) = 2
3
Z
0
π
P4 (cos ϕ, sin ϕ)Q2 (cos ϕ, sin ϕ) − P2 (cos ϕ, sin ϕ)Q4 (cos ϕ, sin ϕ) 2G(ϕ)
p
·e
dϕ.
Q34 (cos ϕ, sin ϕ)
(2.13)
Conditions for the Infinity to be a Weak Focus
In this section, we assume that the points (1, 0) and (−1, 0) are two weak focuses of order 6
of system (2.1). In other words, the conditions of Theorem 1.3 are satisfied. Then, (2.1) is
reduced to
dx
= −(e
a1 + 1)y + e
a1 x2 y + e
a2 xy 2 + e
a3 y 3 ,
dt
(3.1)
dy
1
1 3
2
2
3
= − x − a4 y + x + a4 x y + e
a5 xy + e
a6 y ,
dt
2
2
where a4 6= 0, e
ak = e
ak (a4 , λ), k = 1, 2, 3, 5, 6 given by (1.5), g(λ) = 0 and for λ = λ1 ,
2
a4 6= ω1 , ω2 .
For system (3.1), we have
Q4 (x, y) = x4 + 2a4 x3 y + 2(e
a5 − e
a1 )x2 y 2 + 2(e
a6 − e
a2 )xy 3 − 2e
a3 y 4 .
(3.2)
We shall rigorously prove the following conclusion.
Theorem 3.1. On the equator, system (3.1) has no real singular point and G(2π) = 0 if and
only if a4 = ±a∗4 , λ = λ2 , where a∗4 (= 0.81233628 · · ·) is the largest zero of the function H1
given by (3.19).
To prove this theorem, we need to give a series of lemmas.
It is easy to see that Q4 (x, y) defined by (3.2) is positive definite if and only if there exist
four real numbers α, β and γ1 , γ2 , such that
Q4 (x, y) = [(x + γ1 y)2 + α2 y 2 ][(x + γ2 y)2 + β 2 y 2 ].
(3.3)
Z2 -Equivariant
787
Cubic System Which Yield 13 Limit Cycles
We can expand the right hand side of (3.3) and compare with the coefficients of (3.2). It follows
from (1.5) that α, β, γ1 , γ2 are solutions of the equations f1 = f2 = f3 = f4 = 0, where
f1 =a4 − γ1 − γ2 ,
f2 =36 + 4a24 + 45α2 + 45β 2 + 1090a24 λ + 2722a24λ2
+ 45γ12 + 180γ1 γ2 + 45γ22 ,
f3 =180a4 + 502a34 − 180a4 λ + 6820a34 λ + 16105a34λ2
+ 675β 2 γ1 + 675α2 γ2 + 675γ12γ2 + 675γ1 γ22 ,
(3.4)
f4 =23040a24 + 3578819a44 − 32400α2β 2 + 213120a24λ
+ 73223024a44λ + 411840a24λ2 + 158462585a44λ2
− 32400β 2γ12 − 32400α2γ22 − 32400γ12γ22 .
Lemma 3.1. On the equator, system (3.1) has no real singular point if and only if there exist
α > 0, β > 0, γ ≥ 0 such that
h
2
ih
2
i
1
1
Q4 (x, y) = x + a4 y − a4 γy + α2 y 2
x + a4 y + a4 γy + β 2 y 2 .
(3.5)
2
2
Proof.
Since a4 6= 0, we see from f1 = 0 that there is a γ such that
γ1 =
1
a4 − γa4 ,
2
γ2 =
1
a4 + γa4 .
2
(3.6)
We can show that γ 6= 0. In fact, we see from f1 = f2 = 0 that
1
4
α2 + β 2 = − + 2γ 2 a24 − (143 + 2180λ + 5444λ2 )a24 .
5
90
Since we have g(λ) = 0, (3.7) implies that α2 + β 2 < − 45 + 2γ 2 a24 . Thus, γ 6= 0.
Substituting (3.6) into (3.4), we know that f2 = f3 = 0 if and only if
α2 = f5 ,
where
β 2 = f6 ,
1
(180 − 269a24 − 1080γ − 2145a24γ + 2700a24γ 3
2700γ
+ 360λ + 2710a24λ − 32700a24γλ + 8620a24λ2 − 81660a24γλ2 ),
1
f6 =
(−180 + 269a24 − 1080γ − 2145a24γ + 2700a24γ 3
2700γ
− 360λ − 2710a24λ − 32700a24γλ − 8620a24λ2 − 81660a24γλ2 ).
(3.7)
2
(3.8)
f5 =
(3.9)
Lemma 3.2. On the equator, system (3.1) has no real singular point if and only if there is a
positive number γ such that F1 = 0, f5 > 0, f6 > 0, where
F1 =32400 + 213480a24 + 8910016a44 + 129600λ + 6988320a24λ
+179010340a44λ + 129600λ2 + 15915600a24λ2 + 386328865a44λ2
− 180(6480 − 5760a24 + 255463a44 + 136800a24λ + 5439976a44λ + 465120a24λ2
+ 11877880a44λ2 )γ 2 + 162000a24(72 + 143a24 + 2180a24 λ + 5444a24λ2 )γ 4 − 29160000a44γ 6 .
(3.10)
788
Y.R. LIU, J.B. LI
Proof. Substituting (3.6) and (3.8) into f4 , we have that f4 = g(λ) = 0 if and only if F1 = 0.
By Lemma 3.1 and its proof, we know that when F1 = 0,
2
ih
2
i
h
1
1
x + a4 y + a4 γy + f6 y 2 .
(3.11)
Q4 (x, y) = x + a4 y − a4 γy + f5 y 2
2
2
2
Next we consider the function G(θ) defined by (2.12). From (3.5) and using the method of
integrating by partial fractions, we have
Lemma 3.3.
When θ ∈ (− π2 , π2 ),
G(θ) =
1800αβ(α2
G1 (θ) − G1 (0)
,
− 2αβ + β 2 + 4a24 γ 2 )(α2 + 2αβ + β 2 + 4a24 γ 2 )
(3.12)
where
2a4 (1 − 2γ) + (a24 − 4a24 γ + 4a24 γ 2 + 4α2 ) tan θ
4α
2a4 (1 + 2γ) + (a24 + 4a24 γ + 4a24 γ 2 + 4β 2 ) tan θ
+a4 αn2 arctan
4β
2
2
4α tan θ + (2 + a4 − 2a4 γ tan θ)2
+2αβn3 log 2
,
4β tan2 θ + (2 + a4 + 2a4 γ tan θ)2
G1 (θ) =a4 βn1 arctan
(3.13)
and
n1 =(β 2 − α2 )(−1440 + 943a24 + 900α2 − 4320λ + 31780a24λ + 75700a24λ2 )
−60(3α2 + β 2 )(−72 − 43a24 + 150a24 λ + 376a24 λ2 )γ
+4a24 (−1440 + 943a24 − 675α2 − 225β 2 − 4320λ + 31780a24λ + 75700a24λ2 )γ 2
−240a24(−72 − 43a24 + 150a24 λ + 376a24λ2 )γ 3 − 3600a44γ 4 ,
n2 =−(β 2 − α2 )(−1440 + 943a24 + 900β 2 − 4320λ + 31780a24λ + 75700a24λ2 )
+60(α2 + 3β 2 )(−72 − 43a24 + 150a24 λ + 376a24 λ2 )γ
+4a24 (−1440 + 943a24 − 225α2 − 675β 2 − 4320λ + 31780a24λ + 75700a24λ2 )γ 2
+240a24(−72 − 43a24 + 150a24 λ + 376a24λ2 )γ 3 − 3600a44γ 4 ,
n3 =900a44 γ 3 − 15(α2 − β 2 )(−72 − 43a24 + 150a24 λ + 376a24λ2 )
+a24 (−1440 + 943a24 + 450α2 + 450β 2 − 4320λ + 31780a24λ + 75700a24λ2 )γ.
Lemma 3.4.
G(2π) =
where
−a4 π(αf7 + βf8 )
,
900αβ(α2 + 2αβ + β 2 + 4a24 γ 2 )
(3.14)
f7 =1440 − 943a24 + 900β 2 + 4320γ + 2580a24γ + 900a24 γ 2
+ 4320λ − 31780a24λ − 9000a24γλ − 75700a24λ2 − 22560a24γλ2 ,
f8 =1440 − 943a24 + 900α2 − 4320γ − 2580a24γ + 900a24 γ 2
+ 4320λ − 31780a24λ + 9000a24γλ − 75700a24λ2 + 22560a24γλ2 .
(3.15)
Z2 -Equivariant
Cubic System Which Yield 13 Limit Cycles
789
Proof. The integrand of the right hand side of (2.12) is a periodic function with period π.
From (3.13), we have
π
π
G(2π) =2[G( ) − G(− )]
2
2
2[G1 ( π2 ) − G1 (− π2 )]
1800αβ(α2 − 2αβ + β 2 + 4a24 γ 2 )(α2 + 2αβ + β 2 + 4a24 γ 2 )
2πa4 (βn1 + αn2 )
.
=
1800αβ(α2 − 2αβ + β 2 + 4a24 γ 2 )(α2 + 2αβ + β 2 + 4a24 γ 2 )
=
This gives the conclusion of Lemma 3.4.
(3.16)
2
Remark 3.1. When αf7 + βf8 6= 0, the function G(θ) has jump discontinuous points at
θ = π2 ± kπ, k = 0, 1, 2 · · ·. When αf7 + βf8 = 0, G(θ) is a continuous periodic function in the
interval (−∞, ∞).
Lemma 3.5. If system (3.1) has no real singular point on the equator, then G(2π) = 0 if and
only if f7 f8 ≤ 0 and F2 = 0, where
F2 =777600a24(48600 − 99630a24 − 4001235a44 + 142751540a64 + 194400λ − 1590300a24λ
− 78937845a44λ + 2920922183a64λ − 3877200a24λ2 − 170418660a44λ2 + 6319248215a64λ2 )γ 4
− 216(104976000 + 1138989600a24 − 83650309200a44 − 949863385800a64 + 68019121524943a84
+ 419904000λ + 24686856000a24λ − 1705595216400a44λ − 19423812453480a64λ
+ 1391129313186520a84λ + 51858144000a24λ2 − 3691578067200a44λ2 − 42021926420400a64λ2
+ 3009500477048005a84λ2 )γ 2 + 1133740800 + 6858432000a24 − 3027453103440a44
− 30040565308416a64 + 2789782868404321a84 + 15116544000λ + 170222083200a24λ
− 61921065003840a44λ − 614438279223480a64λ + 57056708189236408a84λ
− 133959025108800a44λ2 + 30233088000λ2 + 376625894400a24λ2
− 1329260609063880a64λ2 + 123433679684533387a84λ2 .
(3.17)
Proof. By Lemma 3.2, if system (3.1) has no real singular point on the equator, we have
F1 = 0. We see from (3.14) that G(2π) = 0 if and only if f7 f8 ≤ 0 and α2 f72 − β 2 f82 = 0. Using
(3.7) and (3.8), we can eliminate the terms of α and β with power exponents larger than 1 in
α2 f72 − β 2 f82 . Furthermore,we can use F1 = 0 and g(λ) = 0 to eliminate the terms of γ and λ
with power exponents larger than 4 and 2 in α2 f72 − β 2 f82 , respectively. Finally, we obtain
α2 f72 − β 2 f82 =
1
F2 .
30375a24γ 3
Thus, Lemma 3.5 holds.
(3.18)
2
Lemma 3.6. If F1 = F2 = g(λ) = 0, a4 γ 6= 0, then, a4 is a real zero of the following 15-order
polynomial H1 with respect to a24 :
H1 =24488801280000000000 + 770943744000000000000a24
−469421964289260000000000a44 + 36075095205512305500000000a64
−1143110740438000496812500000a84 + 11013872157343419644770312500a10
4
790
Y.R. LIU, J.B. LI
14
+67294307690668435658116875000a12
4 + 8216045042989819669497109375a4
18
−455712654622496257745066187500a16
4 − 817172022465840200592407725000a4
22
−57549976589616052075594587500a20
4 + 1286475949345038306506007073750a4
26
+1637228153622181244360199321500a24
4 + 934842729588870230115343355500a4
30
+263170086745751773461987484900a28
4 + 29615860952895797456782793171a4 .
(3.19)
Proof. It follows from the polynomials theory that Resultant [F1 , F2 , γ] = 0 when F1 = F2 = 0.
Using Mathematica program, we obtain
6 4
Resultant [Resultant [F1 , F2 , γ], g(λ), λ] = a36
4 H0 H1 ,
(3.20)
H0 = −28800 − 1389200a24 + 1293760a44 + 5630539a64.
(3.21)
where
Consequently, when the conditions of Lemma 3.5 are satisfied, we have that H0 H1 = 0.
Next we prove that H0 6= 0 by contradiction. Suppose that H0 = 0. Under the conditions
of Lemma 3.6, we have
M1 = Resultant [H0 , F1 , a4 ] = 0,
M2 = Resultant [H0 , F2 , a4 ] = 0.
By using Mathematica, we know that M1 , M2 are two polynomials with respect to γ and λ.
The highest common factor of Resultant [M1 , g(λ), λ)] and Resultant [M2 , g(λ), λ)] with respect
to γ is γ 12 . Thus, when H0 = F1 = F2 = g(λ) = 0, we have γ = 0. This contradicts the
conditions of Lemma 3.6. Thus Lemma 3.6 holds.
2
Lemma 3.7. When F1 = F2 = H1 = g(λ) = 0, a4 6= 0, λ is a 14-order polynomial with
respect to a24 , i.e.,
14
1 X
λ = H2 =
bk a2k
(3.22)
4 ,
m
k=0
where m and b0 , b1 , · · · , b14 can be seen in Appendix.
p
Proof. By using Mathematica, we see that Resultant [F1 , F2 , γ] is a polynomial with respect
to a4 , λ. Doing Euclidean algorithm to this polynomial and g(λ) with respect to λ, we can get
3
a84 (F3 − F4 λ) = 0. Hence , λ = F
F4 when F1 = F2 = H1 = g(λ) = 0 and a4 6= 0, where F3
2
and F4 are two polynomials of a4 with rational coefficients. The highest common factor of F4
and H1 is 1. By the polynomials theory, there exist two polynomials F5 and F6 of a24 with
rational coefficients, such that F4 F5 + H1 F6 ≡ 1. Therefore, when F1 = F2 = H1 = g(λ) = 0
and a4 6= 0, we have that λ = F3 F5 . Using H1 = 0 to eliminate the terms of a4 with power
exponents larger than 28 in the expansion of F3 F5 , we obtain the conclusion of this lemma. 2
Remark 3.2.
Using Mathematica, we can get
g(H2 ) = H1 F7 ,
(3.23)
where F7 is a polynomial of a24 . Thus, when H1 = 0, we must have g(H2 ) = 0.
Lemma 3.8.
If γ 6= 0, g(λ) = H1 = 0, λ =
F3
F4 ,
then F1 = F2 = 0 if and only if H2 = 0.
Z2 -Equivariant
Proof.
791
Cubic System Which Yield 13 Limit Cycles
By using Mathematica, it is easy to verify that when g(λ) = 0, we have
48q02 F1 + (1800a24q0 γ 2 − q1 )F2 = H3 (a4 , λ)γ 2 + H4 (a4 , λ),
(3.24)
where
q0 =48600 − 99630a24 − 4001235a44 + 142751540a64 + 194400λ − 1590300a24λ
−78937845a44λ + 2920922183a64λ − 3877200a24λ2 − 170418660a44λ2 + 6319248215a64λ2
q1 = − 17496000 − 42573600a24 + 11777807700a44 − 640423404750a64 + 11135043548357a84
− 69984000λ − 1567836000a24λ + 240482822400a44λ − 13097306243370a64λ
+ 227734631514230a84λ − 3316464000a24λ2 + 520536499200a44λ2
− 28333646277600a64λ2 + 492670092977045a84λ2 .
In (3.24), H3 (a4 , λ) and H4 (a4 , λ) are two polynomials of a4 , λ with rational coefficients, for
which the highest power exponent of λ is 2. We have
H3 (a4 , H2 ) = H1 F8 ,
H4 (a4 , H2 ) = H1 F9 ,
where F8 and F9 are two polynomials of a24 . Thus, when H1 = 0, λ = H2 , we have H3 = H4 = 0.
Therefore, (3.24) becomes
48q02 F1 + (1800a24q0 γ 2 − q1 )F2 = 0.
(3.25)
Again using Mathematica, we know that Resultant [q0 , H1 , a4 ],the polynomial of λ and g(λ),
are relatively prime. It follows that q0 6= 0 when H1 = g(λ) = 0. From (3.25), we obtain the
conclusion of this lemma.
2
Remark 3.3. Note that all the operations in the above lemmas are rational operations
through using Mathematica to polynomials of a4 , λ, γ with rational coefficients. Hence, they
have no any rounding error.
Now we prove our main theorem of this section.
Proof of Theorem 3.1. It follows from Lemmas 3.5–3.7 that if system (3.1) has no real singular
point on the equator and G(2π) = 0, then we have H1 = F2 = 0, λ = H2 . We can show that
H1 |a24 =ζ has exact four zeros a24 = ζk , k = 1, 2, 3, 4, where
ζ1 = 0.65989023 · · · ,
ζ2 = 0.37330788 · · · ,
ζ3 = 0.03359416 · · · ,
ζ4 = 0.01780119 · · ·
(3.26)
and ζ1 = (a∗4 )2 . By Lemma 3.6, we know that when H1 = 0, λ = H2 , the following 4 conditions
hold:
C1 : a24 = ζ1 ,
λ = λ2 ,
C2 : a24 = ζ2 ,
λ = λ2 ,
a24
a24
= ζ3 ,
λ = λ1 ,
= ζ4 ,
λ = λ1 .
C3 :
C4 :
(3.27)
From (3.25) we have that F1 = 0 when
γ2 =
q1
.
1800a24q0
(3.28)
792
Y.R. LIU, J.B. LI
By (3.25), we have the following computational results:

−0.08092371 · · · ,
if a24



 0.01951104 · · · ,
if a24
γ2 =

0.22561074 · · · ,
if a24



190.45459958 · · ·,
if a24
= ζ1 , λ = λ2 ,
= ζ2 , λ = λ2 ,
= ζ3 , λ = λ1 ,
(3.29)
= ζ4 , λ = λ1 .
q1
,
1800a24 q0
Clearly, when one of the conditions C2 , C3 and C4 holds, if taking γ 2 =
2
then F1 =
0, γ > 0. By (3.7), when (3.28) holds, we have
4
q1
1
α2 + β 2 = − +
− (143 + 2180λ + 5444λ2)a24 .
5 900q0 90
(3.30)
Thus, we obtain


 −0.94577861 · · · ,
2
2
α + β = −12.58341029 · · ·,


−0.27130003 · · · ,
if a24 = ζ2 , λ = λ2 ,
if a24 = ζ3 , λ = λ1 ,
a24
if
(3.31)
= ζ4 , λ = λ1 .
In (3.31), α2 + β 2 is negative. By Lemma 3.2 and Remark 3.1, we obtain the necessary conditions.
Next we prove the sufficiency of the theorem.
When the condition C1 holds, we see from F2 = 0, (3.8) and (3.15) that
α2 = 0.02182871 · · · ,
γ = 0.95518279 · · · ,
f7 = 8243.65696364 · · ·,
β 2 = 0.09886413 · · · ,
f8 = −3873.59848007 · · ·.
(3.32)
By using Lemma 3.2 and Lemma 3.5, we obtain the conclusion.
4
The Existence of Global Limit Cycle of (3.1) Bifurcating
from the Infinity
It is easy to see that Q4 (x, y) is positive definite when λ = λ2 and a4 changes in a small
neighborhood of a∗4 or −a∗4 . By Lemma 3.1 and Lemma 3.2, Q4 (x, y) can be written as (3.5),
where α, β and γ satisfy
α2 = f5 ,
β 2 = f6 ,
F1 = 0.
(4.1)
It means that α, β and γ are continuous functions of a24 . If the condition
dβ
dγ
dα
da4 , da4 , da4
we can calculate
(3.14) and (3.18) that
G(2π) = H5 F2 ,
H5 =
by using (4.1). From (3.14) we can get
∂F1
∂γ 6=
dG(2π)
da4 .
0 holds, then
We see from
−π
,
27337500a4αβγ 3 (α2 + 2αβ + β 2 + 4a24 γ 2 )(αf7 − βf8 )
(4.2)
We can also see from (3.32) that when λ = λ2 and a4 changes in a small neighborhood of a∗4 or
−a∗4 , we have αf7 − βf8 > 0.
Lemma 4.1.
When F1 = H1 = g(λ) = 0, we have
∂F1
∂γ
6= 0.
1
Proof. By using Mathematica, we know that Resultant[F1 , ∂F
∂γ , γ] is a polynomial of a4 and
λ. And Resultant [Resultant [F1 , M3 , γ], g(λ), λ], the resultant of this polynomial and g(λ) with
Z2 -Equivariant
793
Cubic System Which Yield 13 Limit Cycles
1
respect to λ, is a polynomial of a4 . Since the greatest common factor of Resultant F1 , ∂F
∂γ , γ
and H1 is 1, this lemma holds.
2
Lemma 4.2.
dG(2π) da4 a4 =±a∗4 ,
λ=λ2
> 0.
(4.3)
Proof. By Lemma 3.2, we have F1 = 0. When λ = λ2 and a4 changes in a small neighborhood
of a∗4 or −a∗4 . Thus, we see from Lemma 4.1 that
dγ
∂F1 . ∂F1
=−
.
(4.4)
da4
∂a4 ∂γ
By (4.2) and (4.4), when λ = λ2 , at a4 = ±a∗4 we have
. ∂F
dG(2π)
dF2
1
= H5
= (H5 H6 )
,
da4
da4
∂γ
where
H6 =
∂F1 ∂F2
∂F2 ∂F1
−
.
∂γ ∂a4
∂γ ∂a4
(4.5)
(4.6)
Using (3.32), (4.5) and (4.6), we obtain
dG(2π) = 17.00901058 · · · > 0.
da4 a4 =±a∗4 ,λ=λ2
(4.7)
2
Remark 4.1. Similar to the proof of Lemma 4.1, we can exactly prove that when F1 = H1 =
g(λ) = 0, we have H6 6= 0.
By using (2.13), (3.12) and (3.13) to compute, we have
(4.8)
ve3 (2π)a =±a∗ , λ=λ ≈ ±5.36546 × 1011 .
4
4
2
On the basis of the results in Section 3.1, we see from (4.7) and (4.8) that when λ = λ2 , a4 =
(or a4 = −a∗4 ), the equator is an unstable (a stable) limit cycle of system (3.1) in the inner
side. When λ = λ2 , a4 = a∗4 (1 − σ) (or a4 = −a∗4 (1 − σ)) and 0 < σ ≪ 1, the equator is a
stable ( an unstable) limit cycle of system (3.1) in the inner side. Thus, we obtain the following
conclusion.
a∗4
Theorem 4.1. When λ = λ2 , a4 = a∗4 (1 − σ) (or a4 = −a∗4 (1 − σ)) and 0 < σ << 1, near
the equator, equation (3.1) has at least an unstable (a stable) limit cycle.
5
An Example of Existence of 13 Limit Cycles
We know from Remark 1.2 that when the points (±1, 0) are two weak focuses of order 6 of system
11 12
(1.2), the 6th Lyapunov is a11
4 κ, where κ is given by (1.8). When δ = 259200κa4 ε , (a1 , a2 , a3 ,
a5 , a6 ) = (A1 , A2 , A3 , A5 , A6 ), system (1.2) reduces to
dx
12
11
12 3
2
2
3
= −129600a11
4 κε x − (A1 + 1)y + 129600a4 κε x + A1 x y + A2 xy + A3 y ,
dt
dy
1
1
12
= − x + (259200a10
− 1)a4 y + x3 + a4 x2 y + A5 xy 2 + A6 y 3 ,
4 κε
dt
2
2
(5.1)
794
Y.R. LIU, J.B. LI
where
1
(−18 − 8a24 + 154a24 λ + 385a24λ2 )
18
4004
+
(594561331 + 12159853840λ + 26305992745λ2)a64 ε4 ,
939195
104
A2 =λa4 +
(13 + 220λ + 460λ2 )a34 ε2 ,
45
1
A3 = −
a2 [2880(8 + 74λ + 143λ2 )
64800 4
+ (3578819 + 73223024λ + 158462585λ2)a24 ]
A1 =
(5.2)
+ c1 ε 2 + c2 ε 4 + c3 ε 6 + c4 ε 8 ,
1
A5 = − [126 + (44 + 320λ + 797λ2 )a24 ] + c5 ε2 + c6 ε4 + c7 ε6 + c8 ε8 ,
90
1
A6 = −
a4 [45(4 − 19λ) + (502 + 6820λ + 16105λ2)a24 ]
675
+ c9 ε2 + c10 ε4 + c11 ε6 + c12 ε8 + c13 ε10
and c1 , c2 , · · · , c13 (see Appendix). When ε = 0, system (5.1) is just system (3.1).
Remark 5.1.
If a4 = a∗4 , λ = λ2 , ε = 0, then
A1 = −1.56346116 · · · ,
A2 = −0.04514729 · · · ,
A5 = −1.61026387 · · · ,
A6 = −0.41096727 · · · .
A3 = −0.118614487 · · ·,
Suppose that under the polar coordinates x ± 1 = r cos θ, y = r sin θ, r =
∞
P
(5.3)
vk (θ)hk is
k=1
the solution of (5.1) satisfying r|θ=0 = h in the neighborhoods of (±1, 0). From Theorem 1.1
and Theorem 1.6, we have
Lemma 5.1.
At the points (±1, 0), the focal values of system (5.1) are as follows:
12
v1 (2π) − 1 = 518400a11
+ o(ε12 ),
4 κπε
10
v3 (2π) = −773136a11
+ o(ε10 ),
4 κπε
6
6
v7 (2π) = −44473a11
4 κπε + o(ε ),
2
2
v11 (2π) = −91a11
4 κπε + o(ε ),
8
8
v5 (2π) = 296296a11
4 κπε + o(ε ),
4
4
v9 (2π) = 3003a11
4 κπε + o(ε ),
(5.4)
v13 (2π) = a11
4 κπ + o(1).
Theorem 5.1. There exists a ε0 > 0 such that when 0 < |ε| < ε0 , system (5.1) has 12
limit cycles at the neighborhoods of (±1, 0). These limit cycles lie near (x ± 1)2 + y 2 = k 2 ε2 ,
k = 1, 2, 3, 4, 5, 6, respectively.
Proof. Let △(h) = r(2π, h) − h be the succession function of the P oincaré map of system
(5.1) at the points (±1, 0). We have
△(εη) = (v1 (2π) − 1)εη +
13
X
vk (2π)εk η k + o(ε13 ).
(5.5)
k=2
We know from Theorem 1.5 and (5.4) that
2
4
6
8
10
△(εη) =a11
+ η 12 )ε13 + o(ε13 )
4 κπη(518400 − 773136η + 296296η − 44473η + 3003η − 91η
2
2
2
2
2
2
2
2
2
2
2
2 13
=a11
+ o(ε13 ).
4 κπη(η − 1 )(η − 2 )(η − 3 )(η − 4 )(η − 5 )(η − 6 )ε
(5.6)
Z2 -Equivariant
Cubic System Which Yield 13 Limit Cycles
795
From (5.5) and the implicit function theorem, we obtain the conclusion of this theorem.
2
∗
When λ = λ2 , a4 = ±a4 , we have κ = 0.00037809 · · · > 0. Therefore, to sum up, the
following main result follows from Theorem 4.1, Lemma 5.1 and Theorem 5.1.
Theorem 5.2. (i) When ε = 0, λ = λ2 , a4 = a∗4 (or a4 = −a4 ∗), the points (±1, 0) are
unstable (stable) weak focuses of order 6. The equator is an unstable (a stable) limit cycle of
(5.1) in the inner side.
(ii) When ε = 0, 0 < σ << 1, λ = λ2 , a4 = a∗4 (1 − σ) (or a4 = −a∗4 (1 − σ)), the points
(±1, 0) are unstable (stable) weak focuses of order 6. In a neighborhood of the equator there
exists at least one unstable (stable) limit cycle of (5.1).
(iii) When 0 < |ε| << σ << 1, λ = λ2 , a4 = a∗4 (1 − σ) (or a4 = −a∗4 (1 − σ)), in the
neighborhoods of (±1, 0) there exist 6 limit cycles. And in the inner neighborhood of the equator,
there exists at least one unstable (stable) limit cycle of (5.1). Altogether, there exist 13 limit
cycles of (5.1) with the relative position h1h6 ∐ 6ii. In addition, the equator Γ∞ is a inner stable
(inner unstable) limit cycle.
6
Appendix
m =41768861099732811507005593611646374236805368532722796304266557197195552700000000
40269749660833784102730754868241173225875313540958021030644938927835937920050666
60140089967669946316611433155969040351460015344542405511035570117375152062710,
b0 =133044732241688250129897645400021066090787419135379646332915211181851004248541
33074443255206144160082848298871381787422856569544217176386954421990273292015801
94110039187275928082176088793039126831249241415685346698100477058442570825000000,
b1 =1934234333754748946841567262351339056161062881680376463975517652845723302
37593455612044682944712048056992395396957872838664842053837667629212299915007276
5450055602803114392713055900512548736710402698508094834731289270886348800000000,
b2 =−1304086348652361001504528329700875532056818601220393247517143398575230655319
66015172812800179804978560701304020511967849367060216459026837192352946746385599
72851827031031632566019361322811477543817721009100136779391416121070540800000000,
b3 =103572979665013063817681177641403590462440193902902884676511988412878390186393
27908024921638778609056804200520176323188746642895922493614282349832731909920992
12062565245562426201672460230309648002075930824242396963357840881994478840000000,
b4 =−3460089866550368464115993377149611861085983900612920426835200721074487360806731
35640227253191954709488674460537317215542982703744822887815376493192915183887486
81693099420924226663330813424207204782943964064431644448241366153766864679000000,
b5 =40268480405535264666026335762209781098632654229856321754438831540885455299110807
84536121208490652906224184294417903710712261498631604275499765696065112488974511
13576747645034682498705532813119661391531685620417255350204783974586683544725000,
b6 =78039454039961170914384772040880996940137141355112920288216945991979382573568304
796
Y.R. LIU, J.B. LI
96396716256962035053198167397759898221743273266600626539983493606941561946111039
47424392967563838044014685779506651507583047675051175402270906818823761126621875,
b7 =−300182457140231178604251863371151864149881728448383375632262387574260513198215899
20016392408324507364145794241993059100657117059767940037273233977641093481750995
47202927563542202059490306652133862900553259677827175137705363782242284278652500,
b8 =−869565851019757275424312259473714626201935566018902874848586658678384836299613261
78584564266834474549631419735132815626720746305252067760582202899623888879219632
10388043070402781239360014924074377800878165544067697215604197082137278595275000,
b9 =−351215895944591709669958029992291472177924645159423648585456748610439038468724173
31876868416434077877520832728387553549296093296068600637174930021814620752528369
19760268964065707016059698892649273598719535138119499434467174672913633103496500,
b10 =1136734450218184171868703323402624899193778935067822782343810554037144092784693946
19694936062270258789013022233978282296183597310914589599475309120050094795980126
3978360199546544363857996470132228766300966062769477324868138447442219392301110,
b11 =1758537836625989707675920419880598705835165713131162670862091838470166751372796326
4954563538615811784215607181115902004652848158254485655426947783252095775914348
36976364035424660761708522472839528481644522771064844337498003568656527343117700,
b12 =1090207524559549387874861711346072433102837433983516445965949895646153252938941281
87679233530160151431933195777501859766129895098057209186046743779570574334805468
12913648850418781253711176096485628875358010200191617158012607460091024125124080,
b13 =322651898840812916854837680011336850966749180188399810590009633200100408087458236
41291757383448775615551522083856805273297884172070469517508008409636013814501143
41821263609286230076704935433258418799768104803836971289380534968847770021460476,
b14 =37581541338721502665448857904407239421659394133860592481017760576209065297804531
14149594176635059879272781147765426412700898207260332189646547073028908102374075
37509832177899639544708691594205316929398289424810073677303066024011374998517151.
104
c1 = −
(−1074510 + 452297684a24 − 21857085λ + 9250222925a24λ
455625
− 47246400λ2 + 20011444745a24λ2 )a44 ,
13
c2 =−
(2558716387449552 + 1013924199420034321a24 + 52332524917779600λ
31697831250
+20736808593910992256a24λ + 113214060155936160λ2 + 44860994915155953859a24λ2 )a64 ,
−26
c3 =
(−2505649578712461828360 + 1435447422350937850454657a24
24962042109375
−51245617654399769951760λ + 29357812780720622598192416a24λ
−110862255246939470854440λ2 + 63511252100124300505455635a24λ2 )a84 ,
−13
c4 =
×
11329547209494127560905908185258052295450390625
(−31834460290713919579083281019136199779853763043200000
+50280260932461639855623872865520590812238414266800000a24
Z2 -Equivariant
Cubic System Which Yield 13 Limit Cycles
+6365021324691796815400929707842167502489513617516480000a44
+57259561083570503256091720196239707527029823176530770400a64
−5746283610376483087751237895938940357206343623688256986400a84
+649846631231070052107661520445798338133894874027033648917807a10
4
−651171308515396915028594335554609014646473804062080000λ
+1028322466139960504393080463711907756405729048913680000a24λ
+130177523057425237007563600201174967602039292098802208000a44λ
+1171074244239424146532781705576096546132899285293528528000a64λ
−117523160886860981291242154839583128481670374217405899553280a84λ
+13290682359031702403896437369325210492560868821295597635210000a10
4 λ
−1408739766249350098044568988501631958183010730094080000λ2
+2224618732623284121083329748456342776528481557535040000a24λ2
+281619644127217978175922220374204092025950234649115680000a44λ2
+2533444581900216017380135702052534033344828405678997364000a64λ2
−254243841462741792188480943393426556725577682816903219855200a84λ2
2 2
+28752410275787415274115592423234227686652051162404968443078445a10
4 , λ )a4 ,
−104
c5 =
(180 + 18317a24 − 2025λ + 373439a24λ − 6660λ2 + 807620a24λ2 )a24 ,
3375
52
c6 =
(−14640825890 + 277794403047a24 − 297642955430λ
65221875
+5682054029130a24λ − 643354561400λ2 + 12292426157715a24λ2 )a44 ,
−416
c7 =
(−20183385331212507 + 2971382709418174442a24
184904015625
− 412796851484198355λ + 60770804241575838161a24λ
− 893026165172960340λ2 + 131468576418175907831a24λ2 )a64 ,
1664
c8 =
511645004860105000294937342896376559375
· (108860103505395151810421890871440964979720
+174103681879414520761136159458903752571815a24
−10557357616993051187063587884723483133192010a44
+156861433129261054037861541885554846857378251a64
+2311942086469272701580432648879926909975400λ
+3621108841362360016619289562154317939678125a24λ
−215846749820364988164980114966120038295542310a44λ
+3208158102172947586570870728190270495607385813a64λ
+5027394357647431888386160966577548423111200λ2
+7851739732470236473683967444019078955305700a24λ2
−466930391365533503019328911089191159220412440a44λ2
+6940378973092320074653506846195325346427210480a64λ2 ),
−104
c9 =
(−2679 + 67652a24 − 53790λ + 1389194a24λ − 117060λ2 + 3007532a24λ2 )a34 ,
10125
797
798
Y.R. LIU, J.B. LI
−104
(152217275580 + 25959189233645a24 + 3106111900860λ
1760990625
+ 530914901675948a24λ + 6717064481280λ2 + 1148554667524115a24λ2 )a54
−208
=
(−55324412125423200 + 27911714110836463963a24
924520078125
− 1131514912456026120λ + 570851179670703914920a24λ
c10 =
c11
c12
− 2447869640821698600λ2 + 1234951436665000410505a24λ2 )a74
1664
=
83922571922178722673377097668578165151484375
×(−15583426332122373999281701317970676482828104000
−23891149373767040074636852963343188844405944500a24
+1398494859372864280452777486185639782423058722200a44
−20793724322697563749127889076387518827736461552550a64
−9707002224940525802338273507828750607471648608651812a84
+1429035705368691279125794047999381103393371923503049563a10
4
−308838558171646666927676025307026981981932112000λ
−481657725556160268425252765418742742402603296500a24λ
+28610443258382367246628444191252339953749095009000a44λ
−425271149947718077830876419109038170441002139083300a64λ
−198527896338605010832955718278602835685067281710568300a84λ
+29226680152648068103911007398642534109910980164052081012a10
4 λ
−665120046911974977664090336426461197656217280000λ2
−1039909056612189336615325693624189379525811690000a24λ2
+61896970814722727872213287892247516133181626834000a44λ2
−920009919003345718714045444351200198668529238142550a64λ2
−429485513310802642360546161171960366525002618072581080a84λ2
c13
2
+63227566203473001110005154354345428715380731301618997941a10
4 λ )a4
13
=
×
33569028768871489069350839067431266060593750
(−4040205092075217603862475032779911919096132362403840
−6313720500235766190675753060783427329906110269870080a24
+375651463192764333261425365487437865341361202842787840a44
−5583567470484805513994084581188403174298092037520826368a64
+4693766296112163253393898683177878958618737696194359680a84
−279263824441095674891306422385346838792741503249616883320a10
4
+4150892602142403983926935824603530249787675757718894268751a12
4
−82632819838886726884258829868258612352273189240012800λ
−129130109212039115957602442092652287080925644033945600a24λ
+7682832680274963868683889895540070630201612896542924800a44λ
−114195286313436287529831721901137693721362327782148632576a64λ
Z2 -Equivariant
Cubic System Which Yield 13 Limit Cycles
799
+95997046040395840702680712182933461502967213072165324800a84λ
−5711511926905246144144229474804367855377698640154882394240a10
4 λ
+84894177226224001992666441721005962217714575442125870700816a12
4 λ
−178764503729383404203403601980269512002134207034163200λ2
−279354249526317031945953522931124592655758960850944000a24λ2
+16620662751872805587639269962363364226394854431066030080a44λ2
−247044481054784797022208066454576065774106626431621836800a64λ2
+207675300550152899924828413787343853013720775912715420800a84λ2
2
−12356004739316987035148856963617616057492861928103728203880a10
4 λ
2 3
+183655898748902198758426425776820926875135138559382371030925a12
4 λ )a4
Last Remark. The main result of this paper have been presented in two international workshops in 2007 and 2008. One is “The workshop on dynamical systems and differential equations”, Shanghai Normal University, June 10-21, 2007, and the other is “The workshop of classical problems on planar polynomial vector fields,” Nov.23–28, 2008, The Banff International
Research Station for Mathematical Innovation and Discovery (BIRS), Banff, Canada.
References
[1] Liu, Y.R., Li, J.B., Huang W.T. Singular Point Values, Center Problem and Bifurcations of Limit Cycles
of Two Dimensional Differential Autonomous Systrms. Science Press, Beijing, 2008, 192–209
[2] Li, J.B., Li, C.F. Planar cubic Hamiltonian systems and distribution of limit cycles of (E3 ). Acta Math.
Sinica, 28(4): 509–521 (1985)
[3] Li, J.B., Chen X.Q. Poincaré bifurcations in the quadratic differential system. Kexue Tongbao (English
Ed.), 32(10): 655–660 (1987)
[4] Li, J.B., Huang, Q.M. Bifurcations of limit cycles forming compound eyes in the cubic system. Chin. Ann.
Math., 8B: 391–403 (1987)
[5] Li, J.B., Ou, Y.H. Global bifurcations and chaotic behavior in a disturbed quadratic system with two
centers. Acta Math. Appl. Sinica, 11(3): 312–323 (1988)
[6] Li, J.B., Zhao, X.H. Rotation symmetry groups of planar Hamiltonian systems. Ann. Diff. Equs., 5:
25–33 (1989)
[7] Li, J.B., Wan, S.D. Global bifurcations ina disturbed Hamiltonian vector field approaching a 3:1 resonant
Poincaré map. Acta. Math. Appl. Sinica (Eng. Ser.), 7: 80–89 (1991)
[8] Li, J.B., Liu, Z.R. Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian
system. Publications Mathmatiques, 35: 487–506 (1991)
[9] Li, J.B., Lin Y.P. Normal form and critical points of the period of closed orbits for planar autonomous
systems. Acta Mathematica Sinica, 34: 490–501 (1991) (in Chinese)
[10] Li, J.B., Liu, Z.R. Bifurcation set and compound eyes in a perturbed cubic Hamiltonian system. In
Ordinary and Delay Differential Equations, π Pitman Research Notes in Math., Vol.272, Longman, England,
1992, 116–128
[11] Li, J.B., Lin, Y.R. Global bifurcations in a perturbed cubic system with Z2 -symmetry. Acta Math. Appl.
Sinica (English Ser.), 8(2): 131–143 (1992)
[12] Li, C.Z., Llibre, J., Zhang Z F. Abelian intrgrals of quadratic Hamiltonian vector fields with a invariant
straight line. Publ. Math., 39(2): 355–366 (1995)
[13] Li, J.B., Feng, B.Y. Stability, Bifurcations and Chaos. Science and Technology Press of Yunnan, Kunming,
1995 (in Chinese)
[14] Li, J.B., Chan, H.S.Y., Chung, K.W. Bifurcations of limit cycles in a Z6 -equivariant planar vector field of
degree 5. Sci. in China (Ser. A), 45(7): 817–826 (2002)
[15] Li, J.B., Chan, H.S.Y., Chung K.W. Investigations of bifurcations of limit cycles in a Z2 -equivariant planar
vector field of degree 5. Int. J. Bifur. Chaos, 12: 2137–2157 (2002)
[16] Li, J.B., Chan, H.S.Y., Chung, K.W. Some lower bounds for H(n) in Hilbert’s 16th problem. Qualitative
Theory of Dynamical Systems, 44(3): 345–360 (2003)
[17] Li, J.B. Hilbert’s 16th problem and Bifurcations of Planar Polynomial vector Fields. Int. J. Bifur. Chaos,
13(1): 47–106 (2003)
800
Y.R. LIU, J.B. LI
[18] Li, J.B., Zhang, M.Q. Bifurcations of limit cycles of Z8 -equivariant planar vector field of degree 7. J.
Dynamics and Differentil Equations, 16(4): 1123–1139 (2004)
[19] Li, J.B., Zhou, H.X. On the Control of Parameters of Distributions of Limit Cycles for a Z2 -Equivariant
Perturbed Planar Hamiltonian Polynomial Vector Field. Int. J. Bifur. Chaos, 15(1): 137–155 (2005)
[20] Li, J.B., Zhang, M.J. Bifurcations of limit cycles in a Z2 -equivariant planar polynomial vector field of
degree 7. Int. J. Bifur. Chaos, 16: 925–943 (2006)
(3)
[21] Liu, Y.R. Formulas of focal values, center conditions and center integrals for the system (E3 ). Chinese
Science Bulletin, 33(5): 357–359 (1988)
[22] Liu, Y.R., Li, J.B. Theory of values of singular point in complex autonomous differential systems. Sci. in
China 33(1): 10–23 (1989)
[23] Liu, Y.R. Theory of center-focus in a class of high order singular points and infinity. Sci. in China, 31(1):
37–48 (2001)
[24] Liu, Y.R., Li, J.B. Complete Study on a Bi-Center Problem for the Z2 Equivariant Cubic Vector Fields.
Acta Mathematica Sinica (English Series), 27(7): 1379–1394 (2011)
[25] Liu, Y.R., Li, J.B. Center problem and multiple Hopf bifurcation for the Z6 equivariant planar polynomial
vector fields of degree 5. Int. J. Bifur. Chaos, 19(5): 1741–1749 (2009)
[26] Liu, Y.R., Li, J.B. Center problem and multiple Hopf bifurcation for the Z5 equivariant planar polynomial
vector fields of degree 5. Int. J. Bifur. Chaos, 19(6): 2115–2121 (2009)
[27] Liu, Y.R., Huang, W.T. A cubic system with twelve small amplitude limit cycles. Bull. Sci. Math., 129:
83–98 (2005)
[28] Yu, P., Han, M.A. Twelve limit cycles in 3rd-planar system with Z2 symmetry. Communications on Appl.
Pure Anal., 3: 515–526 (2004)
[29] Yu, P., Han, M.A. Small limit cycles bifurcating from fine focus points in cubic order Z2 -equivariant vector
fields. Chaos, Solitons and Fractals, 24: 329–348 (2005)
[30] Yu, P., Han, M.A. Twelve limit cycles in a cubic case of the 16th Hilbert problem. Int. J. Bifur. Chaos,
15: 2192–2205 (2005)