We study the number and distribution of limit cycles of some planar Z4-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.
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 Zq-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 Z2-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 Z2-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 Z2-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 Z4-equivariant system perturbed by quintic Z4-equivariant polynomials having 13 limit cycles. Li et al. [12] studied a quintic system and obtained at least 23 limit cycles for Z2-equivariant case and 17 limit cycles for Z4-equivariant case. In [13], Wu et al. studied a Z4-equivariant system and found 20 limit cycles. Li et al. [14] found that 24 limit cycles existing in a Z6-equivariant quintic system. Yao and Yu [15] studied a Z5-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 Z6-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 Z3-equivariant quintic planar polynomial system.
Our main result is that there can be 24 limit cycles with other distributions for the perturbed quintic Z4-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 Z4-equivariant quintic near-Hamiltonian system will be investigated:
(1)x˙=Hy+εP5(x,y),y˙=-Hx+εQ5(x,y),
where ε is nonnegative and small and the Hamiltonian system is
(2)H(x,y)=2x2+2y2-54x4-54y4+16x6+16y6
with phase portrait of Figure 1. (P5(x,y),Q5(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 (P5(x,y),Q5(x,y)) is, respectively, the real and imaginary parts of the following complex function:
(3)F5,4(z,z¯)=(A0+A1|z|2+A3|z|4)z+(A3+A4|z|2)z¯3+A5z5,
where Ak=ak+ibk, k=0,1,2,…,5, z=x+iy, and z=x-iy. It is direct that
(4)P5=a0x+a1(x3+xy2)+a2(x5+2x3y2+xy4)+a3(x3-3xy2)+a4(x5-2x3y2-3xy4)+a5(x5-10x3y2+5xy4)+b0(-y)+b1(-x2y-y3)+b2(-x4y-2x2y3-y5)+b3(3x2y-y3)+b4(3x4y+2x2y3-y5)+b5(10x2y3-5x4y-y5),Q5=a0y+a1(x2y+y3)+a2(x4y+2x2y3+y5)+a3(y3-3x2y)+a4(y5-3x4y-2x2y3)+a5(5x4y-10x2y3+y5)+b0x+b1(x3+xy2)+b2(x5+2x3y2+xy4)+b3(x3-3xy2)+b4(x5-2x3y2-3xy4)+b5(x5-10x3y2+5xy4).
Our result is the following.
The phase portraits of (1)(ɛ=0).
Theorem 1.
There exist some (a0,a1,a2,a3,a4,a5) 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).
24 limit cycles
24 limit cycles
The rest of this paper is organized as follows. Some useful preliminary theorems will be listed in Section 2. 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
Let H(x,y), p(x,y,δ), and q(x,y,δ) be analytic functions, ɛ positive and small, and δ∈D⊂Rm with D compact; then the following system is a planar Hamiltonian system:
(5)x˙=Hy,y˙=-Hx,
and the below system is usually called near-Hamiltonian system:
(6)x˙=Hy(x,y)+ɛp(x,y,δ),y˙=-Hx(x,y)+ɛq(x,y,δ).
Let system (5) have at least one family of periodic orbits Lh defined by H(x,y)=h which form a periodic annulus {Lh}; then the first-order approximation of the Poincaré map of system (6) is
(7)M(h,δ)=∮Lh(qdx-pdy)
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 {Lh}; see [18].
Let the outer boundary of {Lh} be a homoclinic loop Lβ defined by H(x,y)=β passing through a hyperbolic saddle at the origin; we have the following.
Lemma 2 (see [19]).
(i) The function M(h,δ) has the following expansion:
(8)M(h,δ)=c0(δ)+c1(δ)(h-β)ln|h-β|+c2(δ)(h-β)+c3(δ)(h-β)2ln|h-β|+O(|h-β|2),
for 0<β-h≪1; ci(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+λ2(y2-x2)+∑i+j≥3hijxiyj,λ≠0,p(x,y,δ)=∑i+j≥0aijxiyj,q(x,y,δ)=∑i+j≥0bijxiyj.
Then,
(10)c0(δ)=∮Lβqdx-pdy,c1(δ)=-1|λ|(a10+b01),c2(δ)=∮Lβ(px+qy)dtifc1(δ)=0,c3(δ)=-12|λ|λ{(-3a30-b21+a12+3b03)hh-1λ[(2b02+a11)(3h03-h21)hh+(2a20+b11)(3h30-h12)]},ifc1(δ)=0.
The values c1(δ) and c3(δ) are, respectively, called the first and second local Melnikov coefficients at the saddle O, denoted by c1(O,δ) and c3(O,δ), respectively.
Now let the outer boundary of {Lh} be an 2-polycycle Γ2:
(11)Γ2=⋃i=12(Li∪Si)
with 2 hyperbolic saddles, S1 and S2, and 2 heteroclinic orbits, L1 and L2, 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[c2j(δ)+c2j+1(δ)(h-β)ln|h-β|]×(h-β)j,
where
(13)c0(δ)=∑i=12∫Liqdx-pdy,c1(δ)=∑i=12c1(Si,δ),c3(δ)=∑i=12c3(Si,δ),
where c1(Si,δ) and c3(Si,δ) are, respectively, the first and the second local Melnikov coefficient at the saddle Si, i=1,2. In particular,
(14)c2(δ)=∮Γ2(px+qy)dt=∑i=12∫Li(px+qy)dt,
if c1(Si,δ)=0, i=1,2.
When the inner boundary of {Lh} is a elementary center (xc,yc) defined by H(xc,yc)=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≥0Bk(h-α)k+1.
If for (x,y) near (xc,yc),
(16)H(x,y)=α+12((x-xc)2+(y-yc)2)+∑i+j≥3hij(x-xc)i(y-yc)j,p(x,y,δ)=∑i+j≥1aij(x-xc)i(y-yc)j,q(x,y,δ)=∑i+j≥1bij(x-xc)i(y-yc)j,
the coefficients Bi 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-x0)+b(y-y0),v=c(x-x0)+d(y-y0)
and time rescaling τ=kt, where D=ad-bc≠0, the system (6) becomes
(18)dudτ=H~v+ɛp~,dvdτ=-H~u+ɛq~,
where H~(u,v)=(D/k)H(x,y), p~(u,v,δ)=(1/k)[ap(x,y,δ)+bq(x,y,δ)], and q~(u,v,δ)=(1/k)[cp(x,y,δ)+dq(x,y,δ)].
(ii) Let
(19)M~(h,δ)=∮Lhq~du-p~dv
which is the Melnikov function of the system (18); then
(20)M(h,δ)=|k|DM~(Dkh,δ).
When systems (5) and (6) are Z4-equivariant, (5) has a compound cycled denoted by Γ4, which consists of 8 hyperbolic saddles S1,…,S8 and 16 heteroclinic orbits L12, L21, L23, L32, L34, L43, L45, L54, L56, L65, L67, L76, L78, L87, L81, and L18 satisfying α(Lij)=Si,ω(Lij)=Sj. Γ4 contains 8 two-polycycles Li(i=1,…,8), where L1=L12∪L21, L2=L23∪L32, L3=L34∪L43, L4=L45∪L54, L5=L56∪L65, L6=L67∪L76, L7=L78∪L87, and L8=L81∪L18. See Figure 3. We suppose that Γ4 is defined by H(x,y)=H(Si)=β, i=1,…,8. There are 8 centers Ci(xi,yi) inside the 2-polycycle LI, with H(C1)=H(C3)=H(C5)=H(C7)=α1 and H(C2)=H(C4)=H(C6)=H(C8)=α2. There are 4 families of periodic orbits Lih inside the 2-polycycle Li, defined by H(x,y)=h for h∈(α1,β), i=1,3,5,7, and 4 families of periodic orbits Ljh inside the 2-polycycle Lj, defined by H(x,y)=h for h∈(α2,β), j=2,4,6,8. Then we have 8 Melnikov functions below:
(21)Mi(h,δ)=∮Lih(qdx-pdy)|ɛ=0,hforh∈(α1,β)i=1,3,5,7.Mj(h,δ)=∮Lih(qdx-pdy)|ɛ=0,hforh∈(α2,β)j=2,4,6,8.
By Z4-equivariance, M1(h,δ)=M3(h,δ)=M5(h,δ)=M7(h,δ) and M2(h,δ)=M4(h,δ)=M6(h,δ)=M8(h,δ), we can only study M1(h,δ) and M8(h,δ). For convenience, the notations are introduced as follows:
(22)c01(δ)=∫L12qdx-pdy,c02(δ)=∫L21qdx-pdy,e01(δ)=∫L81qdx-pdy,e02(δ)=∫L18qdx-pdy,
and di(δ)=(px+qy)(Si,δ),1≤i≤8, where d1=d3=d5=d7 and d2=d4=d6=d8. Letting d1=d2=0, we introduce
(23)c21(δ)=∫L12(px+qy)dt,c22(δ)=∫L21(px+qy)dt,e21(δ)=∫L81(px+qy)dt,e22(δ)=∫L18(px+qy)dt.
The following is directly from Lemma 3.
Lemma 7.
Under the above assumptions, we have the following expansions:
(24)Mi(h,δ)=c0(δ)+c1(δ)(h-β)ln|h-β|+c2(δ)(h-β)+c3(δ)(h-β)2ln|h-β|+⋯,
for 0<h-β≪1, i=1,3,5,7,
(25)Mj(h,δ)=e0(δ)+e1(δ)(h-β)ln|h-β|+e2(δ)(h-β)+e3(δ)(h-β)2ln|h-β|+⋯,
for 0<h-β≪1, j=2,4,6,8, where
(26)c0(δ)=c01(δ)+c02(δ),e0(δ)=e01(δ)+e02(δ),c1(δ)=c1(S1,δ)+c1(S2,δ)=-d1(δ)|λ1|-d2(δ)|λ2|,e1(δ)=c1(S1,δ)+c1(S8,δ)=c1(S1,δ)+c1(S2,δ)=c1(δ),c3(δ)=c3(S1,δ)+c3(S2,δ),e3(δ)=c3(S1,δ)+c3(S8,δ)=c3(S1,δ)+c3(S2,δ)=c3(δ),c2(δ)=c21(δ)+c22(δ),e2(δ)=e21(δ)+e22(δ)
if d1(δ)=d2(δ)=0. λi denotes an eigenvalue of Si for (6) and c1(Si,δ) and c3(Si,δ) are the first and the second Melnikov coefficients at the saddle Si(i=1,2) as defined after Lemma 2.
By Lemma 4, for the expansions of Mi(h,δ) near the center, we have
(27)M1(h,δ)=∑k≥0Bk(δ)(h-α1)k+1for0<h-α1≪1,M8(h,δ)=∑k≥0dk(δ)(h-α2)k+1for0<h-α2≪1.
To obtain more limit cycles, we have the following.
Theorem 8.
Let (24), (25), and (27) hold.
(1) Suppose that there exists δ0∈D such that
(28)c0(δ0)=c1(δ0)=d0(δ0)=d1(δ0)=0,c2(δ0)≠0,d2(δ0)≠0,rank∂(c0,c1,d0,d1)∂(δ1,δ2,…,δm)=4.
Then there exist some (ɛ,δ) near (0,δ0) such that (5) has
(29)4+1-sgn(M1(h1,δ0)M1(h2,δ0))2+1-sgn(M8(h3,δ0)M8(h4,δ0))2
limit cycles in the 2-polycycles L1 and L8, where h1=β-ε0, h2=α1+ε0, h3=β-ε0, and h4=α2+ε0 with ε0 being positive and very small, and the location of these limit cycles is as follows: 2 limit cycles near the 2-polycycle L1, 2 limit cycles near the center C8, (1-sgn(M1(h1,δ0)M1(h2,δ0)))/2 limit cycles between the center C1 and the polycycle L1, and (1-sgn(M8(h3,δ0)M8(h4,δ0)))/2 limit cycles between the center C8 and the polycycle L8.
(2) Suppose that there exists δ0∈D such that
(30)c0(δ0)=c1(δ0)=B0(δ0)=e0(δ0)=0,c2(δ0)≠0,e2(δ0)≠0,B1(δ0)≠0,rank∂(c0,c1,B0,e0)∂(δ1,δ2,…,δm)|δ=δ0=4.
Then there exist some (ɛ,δ) near (0,δ0) such that (5) has
(31)4+1-sgn(M1(h1,δ0)M1(h2,δ0))2+1-sgn(M8(h3,δ0)M8(h4,δ0))2+1+sgn(c2(δ0)e2(δ0))2
limit cycles in the 2-polycycles L1 and L8, where h1=β-ε0, h2=α1+ε0, h3=β-ε0, and h4=α2+ε0 with ε0 being positive and very small, and the location of these limit cycles is the following: 2 limit cycles near the 2-polycycle L1, 1 limit cycles near the center C1, 1+((1+sgn(c2(δ0)e2(δ0)))/2) limit cycles near the 2-polycycle L8, (1-sgn(M1(h1,δ0)M1(h2,δ0)))/2 limit cycles between the center C1, and the polycycle L1 and (1-sgn(M8(h3,δ0)M8(h4,δ0)))/2 limit cycles between the center C8 and the polycycle L8.
Proof.
Because of the similarity, we only prove the last conclusion. For δ=δ0, by continuity, there exist (1-sgn(M1(h1,δ0)M1(h2,δ0)))/2 zeros of M1(h,δ0) between h1 and h2 and (1-sgn(M8(h3,δ0)M8(h4,δ0)))/2 zeros of M8(h,δ0) between h3 and h4. Thus, for all δ near δ0 or δ∈U0={δ∣|δ|<ɛ*,ɛ*isverysmall} there exist (1-sgn(M1(h1,δ0)M1(h2,δ0)))/2 zeros of M1(h,δ) between h1 and h2 and (1-sgn(M8(h3,δ0)M8(h4,δ0)))/2 zeros of M8(h,δ) between h3 and h4.
According to the condition, we can take c0, c1, e0, and B0 as free parameters. Hence, we first take c1 satisfying
(32)|c1|≪|c2|,|c1|≪|e2|,c1c2>0,
so that there is a zero of M1(h,δ) denoted by h~1 near β satisfying h1<h~1<β. On the other hand, if c2e2>0, considering c1=e1, we have |e1|≪|e2| and e1e2>0 which implies that there is a zero of M8(h,δ) denoted by h¯1 near β satisfying h3<h¯1<β. If c2e2<0, we are not sure if M8(h,δ) has a zero. Thus, so far we obtain 1 zero of M1(h,δ) and (1+sgn(c2(δ0)e2(δ0)))/2 zeros of M8(h,δ) for δ∈U1={δ∣|c1|≪|c2|,|c1|≪|e2|,c1c2>0}.
Next, we take c0, e0, and B0 satisfying
(33)|c0|≪|c1|,c0c1<0,|e0|≪|e1|,e0e1<0,|B0|≪|B1|,B0B1<0.
Then M1(h,δ) has two new zeros h~2 near β and h~3 near α2 satisfying h1<h~1<h~2<β and α2<h~3<h4, and M8(h,δ) has a new zero h¯2 near β satisfying h3<h¯2<β. In this step, we get 2 more zeros of M1(h,δ) and 1 more zero of M1(h,δ) for ∈U2={δ∣|c0|≪|c1|, c0c1<0, |e0|≪|e1|,e0e1<0, |B0|≪|B1|, and B0B1<0}. Then totally we have 4+(1-sgn(M1(h1,δ0)M1(h2,δ0)))/2+(1-sgn(M8(h3,δ0)M8(h4,δ0)))/2+(1+sgn(c2(δ0)e2(δ0)))/2 zeros for δ∈U0∩U1∩U2. Therefore, there are 4+(1-sgn(M1(h1,δ0)M1(h2,δ0)))/2+(1-sgn(M8(h3,δ0)M8(h4,δ0)))/2+(1+sgn(c2(δ0)e2(δ0)))/2 limit cycles for some δ near δ0. This completes the proof.
Remark 9.
The signs of M1(h1,δ0), M1(h2,δ0), M8(h3,δ0), and M8(h4,δ0) 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, S1=(1,2), S2=(2,1), S3=(2,-1), S4=(1,-2), S5=(-1,-2), S6=(-2,-1), S7=(-2,1), and S8=(-1,2), as shown in Figure 4. Let Lij, Li, Ljh, c01, c02, e01, e02, c21,c22, e21,e22, and so forth are the same as those for Lemma 7. We can write the compound cycles Γ~1 as
(34)Γ~1=⋃i=18{Si}∪{L12,L21,L23,L32,…,L78,L87,L81,L18}.
The equation H(x,y)=-8/3 defines four centers, C1=(2,2), C3=(2,-2), C5=(-2,-2), and C7=(-2,2), where the equation H(x,y)=-4/3 defines four centers of C2=(2,0), C4=(0,-2), C6=(-2,0), and C8=(0,2). The center Ci is inside the 2-polycycle Li for i=1,…,8.
We first investigate the 2-polycycles L1 and L8. Here, L1h denotes the periodic orbit defined by H(x,y)=h surrounding the unique center C1 and L8h denotes the periodic orbit defined by H(x,y)=h surrounding the unique center C8. Then
(35)M1(h,δ)=∮L1hQ5dx-P5dy=∮L1h(Q5-P5dydx)dx,h∈(-83,-512),M8(h,δ)=∮L8hQ5dx-P5dy=∮L8h(Q5-P5dydx)dx,h∈(-43,-512),
where δ=(a0,a1,…,a5)∈R6. By Lemma 7, we have
(36)M1(h,δ)=c0(δ)+c1(δ)(h+512)ln|h+512|+c2(δ)(h+512)+c3(δ)(h+512)2ln|h+512|+⋯,
for 0<-(h+5/12)≪1,
(37)M8(h,δ)=e0(δ)+e1(δ)(h+512)ln|h+512|+e2(δ)(h+512)+e3(δ)(h+512)2ln|h+512|+⋯,
for 0<-(h+5/12)≪1.
By Lemma 4, for 0<h+8/3≪1, i=1,3,5,7,
(38)Mi(h,δ)=∑k≥0Bk(δ)(h+83)k+1,
and, for 0<h+4/3≪1, j=2,4,6,8,
(39)Mj(h,δ)=∑k≥0dk(δ)(h+43)k+1.
To find the zeros of M1(h,δ) and M8(h,δ), the coefficients in these asymptotic expansions need to be calculated. In order to calculate the coefficients c01(δ), c02(δ), e01(δ), e02(δ), and so forth, the expressions of heteroclinic orbits are found as follows:
(40)L12:y1(x)=122x2+5+-12x4+60x2+33,hhhhh1≤x≤1222,y2(x)=122x2+5--12x4+60x2+33,hhhhh1222≤x≤2,L21:y3(x)=5-x2,2≥x≥1,L81:y4(x)=5-x2,-1≤x≤1,L18:y5(x)=122x2+5+-12x4+60x2+33,hhh1≥x≥-1.
Figure 5 may be helpful to understand the step to calculate the coefficients in the folowing. By (4), P5(x,y) and P5(x,y) are written as follows:
(41)P5(x,y)=∑i=05p1iai+p2ibi,Q5(x,y)=∑i=05q1iai+q2ibi,
and introduce the following notations:
(42)p1ij=p1i|y=yj(x),p2ij=p2i|y=yj(x),q1ij=q1i|y=yj(x),q2ij=q2i|y=yj(x),hhi,j=1,2,3,4,5,
and f~j=(dy/dx)|y=yi(x)=((-2x+3x3-x5)/(2y-3y3+y5))|y=yi(x). By Lemma 7, we have
(43)c01(δ)=∫L12Q5dx-P5dy=∫122/2(Q5-P5dydx)|y=y1(x)dx+∫22/22(Q5-P5dydx)|y=y2(x)dx=∑i=0aiI1i1+biI2i1+aiI1i2+biI2i2,c02(δ)=∫L21Q5dx-P5dy=∫21(Q5-P5dydx)|y=y3(x)dx=∑i=0aiI1i3+biI2i3,e01(δ)=∫L81Q5dx-P5dy=∫-11(Q5-P5dydx)|y=y4(x)dx=∑i=0aiI1i4+biI2i4,e02(δ)=∫L18Q5dx-P5dy=∫1-1(Q5-P5dydx)|y=y5(x)dx=∑i=0aiI1i4+biI2i4,
where, for i=0,1,2,3,4,5,
(44)I1i1=∫122/2(p1i1+q1i1f~1)dx,I2i1=∫122/2(p2i1+q2i1f~1)dx,I1i2=∫22/22(p1i2+q1i2f~2)dx,I2i2=∫22/22(p2i2+q2i2f~2)dx,I1i3=∫21(p1i3+q1i3f~3)dx,I2i3=∫21(p2i3+q2i3f~3)dx,I1i4=∫-11(p1i4+q1i4f~4)dx,I2i4=∫-11(p2i4+q2i4f~4)dx,I1i5=∫1-1(p1i5+q1i5f~5)dx,I2i5=∫1-1(p2i5+q2i5f~5)dx.
Thus,
(45)c0(δ)=c01(δ)+c02(δ)=∑i=05[(I1i1+I1i2+I1i3)ai+(I2i1+I2i2+I2i2)bi]≡l0a0+l1a1+l2a2+l3a3+l4a4+l5a5,e0(δ)=e01(δ)+e02(δ)=∑i=05[(I1i4+I1i5+I1i3)ai+(I2i4+I2i5+I2i2)bi]≡m0a0+m1a1+m2a2+m4a4+m5a5.
The 2-polycycle L1
The 2-polycycle L8
By (4), the divergence of (1) at S1 and S2 is as follows:
(46)d1(δ)=(dP5dx+dQ5dy)|(1,2)=2a0+20a1+150a2-14a4-70a5-48b4+240b5,d2(δ)=(dP5dx+dQ5dy)|(2,1)=2a0+20a1+150a2-14a4-70a5+48b4-240b5.
Note that
(47)(HyxHyy-Hxx-Hxy)|S1=(02460),(HyxHyy-Hxx-Hxy)|S2=(0-6-240),
which yields
(48)λ1=λ2=12.
By Lemma 3, we have
(49)c1(δ)=e1(δ)=∑i=12-1|λi|(dP5dx+dQ5dy)(Si,δ)=-d1(δ)|λ1|-d2(δ)|λ1|=-13a0-103a1-25a2+73a4+353a5.
In the following by letting b4=5b5, then
(50)d1(δ)=d2(δ),c1(S1,δ)=c1(S2,δ).
Letting c1(δ)=0, then
(51)a0=-10a1-75a2+7a4+35a5.
Under c1(δ)=0, we can apply Lemma 7 to calculate the coefficients c2(δ) and e2(δ). For convenience, the following notations are introduced: h=dx/dt=2y-3y3+y5. For j=1,2,3,4,5, hj=(2y-3y3+y5)|y=yj(x), (dP5/dx+dQ5/dy)≡∑j=05fiai, and
(52)f0j(x)=f0h|y=yj(x)=22y-3y3+y5|y=yj(x),f1j(x)=f1h|y=yj(x)=4x2+4y22y-3y3+y5|y=yj(x),f2j(x)=f2h|y=yj(x)=12y2x2+6x4+6y42y-3y3+y5|y=yj(x),f3j(x)=f3h|y=yj(x)=02y-3y3+y5|y=yj(x)=0,f4j(x)=f4h|y=yj(x)=-12y2x2+2x4+2y42y-3y3+y5|y=yj(x),f5j(x)=f5h|y=yj(x)=-60y2x2+10x4+10y42y-3y3+y5|y=yj(x).
By Lemma 7, we have
(53)c21(δ)=∫L12(dP5dx+dQ5dy)dt=∑i=05∫L12aifihdx,c22(δ)=∫L21(dP5dx+dQ5dy)dt=∑i=05∫L21aifihdx,e21(δ)=∫L81(dP5dx+dQ5dy)dt=∑i=05∫L81aifihdx,e22(δ)=∫L18(dP5dx+dQ5dy)dt=∑i=05∫L18aifihdx.
Substituting (52) into (53), with f0=2 being considered, we have
(54)c21(δ)=∑i=15∫L12aifihdx+∫L12-20a1-150a2+14a4+70a5hdx=∑i=1aiJi1+∑i=1aiJi2,c22(δ)=∑i=15∫L21aifihdx+∫L21-20a1-150a2+14a4+70a5hdx=∑i=1aiJi3,e21(δ)=∑i=15∫L81aifihdx+∫L81-20a1-150a2+14a4+70a5hdx=∑i=1aiJi4,e22(δ)=∑i=15∫L18aifihdx+∫L18-20a1-150a2+14a4+70a5hdx=∑i=1aiJi5,
where J31=J32=J33=J34=J35=0 and
(55)J11=∫122/2f11(x)+-20h1dx,J12=∫22/22f12(x)+-20h2dx,J13=∫21f13(x)+-20h3dx,J21=∫122/2f21(x)+-150h1dx,J22=∫22/22f22(x)+-150h2dx,J23=∫21f23(x)+-150h3dx,J41=∫122/2f41(x)+14h1dx,J42=∫22/22f42(x)+14h2dx,J43=∫21f43(x)+14h3dx,J51=∫122/2f51(x)+70h1dx,J52=∫22/22f52(x)+70h2dx,J53=∫21f53(x)+70h3dx,J14=∫-11f11(x)+-20h4dx,J15=∫1-1f12(x)+-20h5dx,J24=∫-11f21(x)+-150h4dx,J25=∫1-1f22(x)+-150h5dx,J44=∫-11f41(x)+14h4dx,J45=∫1-1f42(x)+14h5dx,J54=∫-11f51(x)+70h4dx,J55=∫1-1f52(x)+70h5dx.
Applying Lemma 7, we have
(56)c2(δ)=c21(δ)+c22(δ)=∑i=0(Ji1+Ji2+Ji3)ai≡J1a1+J2a2+J4a4+J5a5,e2(δ)=e21(δ)+e22(δ)=∑i=0(Ji4+Ji5+Ji3)ai≡R1a1+R2a2+R4a4+R5a5.
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 coefficient c3(S1,δ) at the saddle S1(1,2) we make a change of variables of the form u=(2/2)(x-1),v=2(y-2) and time rescaling τ=kt,k=1 so that the system (1) becomes
(57)dudτ=H~v+ɛp~(u,v,δ),dvdτ=-H~u+ɛq~(u,v,δ),
where
(58)H~(u,v)=H(x,y)|{x=2u+1,y=(2/2)v+2}=6v2-6u2-1023u3+2526v3+5u4+3516v4+42u5+24v5+43u6+148v6-512,p~(u,v)=22P5(x,y)|{x=2u+1,y=(2/2)v+2},q~(u,v)=2Q5(x,y)|{x=2u+1,y=(2/2)v+2}.
Writing functions p~(u,v) and q~(u,v) as the form
(59)p~(u,v)=∑i+j=05a~ijuivj,q~(u,v)=∑i+j=05b~ijuivj,
then the formula for the second local coefficient at the saddle in Lemma 2 can be applied directly; we have
(60)c3(S1,δ)=7216a1+3572a2+55216a4+275216a5.
Similarly, we have
(61)c3(S2,δ)=7216a1+3572a2+55216a4+275216a5.
Applying Lemma 7, we have
(62)c3(δ)=e3(δ)=c3(S1,δ)+c3(S2,δ)=7108a1+3536a2+55108a4+275108a5.
In order to find Bi(δ), i=0,1,2, we move the center C1=(2,2) into the origin by letting u=26(x-2) and v=26(y-2); that is, x=(6/12)u+2 and y=(6/12)v+2 and make the time rescaling dτ=24dt so that the system (1) becomes
(63)dudτ=dH1cdv+ɛp1(u,v),dvdτ=-dH1cdu+ɛq1(u,v),
where
(64)H1c(u,v)=H(x,y)|{x=(6/12)u+2,y=(6/12)v+2}=-83+12u2+12v2+256432u3+256432v3+352304u4+352304v4+63456u5+63456v5+182944u6+182944v6,p1(u,v)=612p(x,y)|{x=(6/12)u+2,y=(6/12)v+2},q1(u,v)=612q(x,y)|{x=(6/12)u+2,y=(6/12)v+2}.
Let
(65)M~1c(h,δ)=∮H1c(u,v)=hq1du-p1dv=∑k≥0b~k(h+83)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)=16πa0+83πa1+32πa2-323πa4-1603πa5,b~1(δ)|b~0=0=-11108πa1-199πa2+2527πa4+12527πa5,b~2(δ)|b~0=0=-17545839808πa1-2837969984πa2+37445209952πa4+187225209952πa5.
By Lemma 6,
(67)M1(h,δ)=M~1c(h,δ).
Therefore
(68)B0(δ)=b~0(δ)=16πa0+83πa1+32πa2-323πa4-1603πa5,B1(δ)|B0=0=b~1(δ)|b~0=0=-11108πa1-199πa2+2527πa4+12527πa5,B2(δ)|B0=0=b~2(δ)|b~0=0=-17545839808πa1-2837969984πa2+37445209952πa4+187225209952πa5.
Similarly, the expressions of dk(δ) are obtained as follows:
(69)d0(δ)=66πa0+463πa1+86πa2+863πa4+4063πa5,d1(δ)|d0=0=-6108πa1+618πa2-35654πa4-175654πa5,d2(δ)|d0=0=535104976π6a1+194917496π6a2-1205552488π6a4-6027552488π6a5.
We will use the coefficients c0(δ), c1(δ), c2(δ), c3(δ), B0(δ), B1(δ), B2(δ), e0(δ), e1(δ), e2(δ), e3(δ), d0(δ), d1(δ), and d2(δ) obtained above to study the limit cycle bifurcation.
(1) Solving the equations
(70)c0(δ)=c1(δ)=d0(δ)=d1(δ)=0
gives
(71)a0=μ1a1,a2=μ2a1,a4=μ3a1,a5=μ4a1,
where
(72)μ1=-23273,μ2=-1631752,μ3=1175239l5+27840l0-8760l1+815l2-l5+5l4,μ4=-117525568l0-1752l1+163l2+39l4-l5+5l4.
Approximate computation using Maple.13 gives
(73)a0=μ1a1=-3.1780821917808219178a1≡a0*,a2=μ2a1=-0.093036529680365296804a1≡a2*,a4=μ3a1=1.3418001549779299848×108a1≡a4*,a5=μ4a1=-2.6836003104010654490×107a1≡a5*.
We can easily find that
(74)rank∂(c0,c1,d0,d1)∂(a0,a2,a4,a5)=4.
We fix a1>0, a3≠0, and take δ=(a0,a2,a4,a5), δ0=(a0*,a2*,a4*,a5*). Inserting δ0 into c2(δ), d2(δ), e0(δ) and B0(δ) we have
(75)c2(δ0)=-1.6154178146383561644a1,d2(δ0)=-0.000155025650574300610π6a1,e0(δ0)=0.001040156096575342a1,B0(δ0)=-0.6027397260050228310πa1.
Taking h1=-5/12-ε1, h2=-8/3+ε1, h3=-5/12-ε1, and h4=-4/3+ε1 with ε1>0 small enough, we have
(76)M1(h1,δ0)=c2(δ0)(-ε1)+O((-ε1)2ln|ε1|)>0,M1(h2,δ0)=B0(δ0)(ε1)+O(ε12)<0,M8(h3,δ0)=e0(δ0)+O((ε1)2ln|ε1|)>0,M8(h4,δ0)=d2(δ0)(ε1)3+O((ε1)4)<0,
which yield (1-sgn(M1(h1,δ0)M1(h2,δ0)))/2=1 and (1-sgn(M8(h3,δ0)M8(h4,δ0)))/2=1. Hence, by Theorem 8, (1) can have 6 limit cycles inside the 2-polycycles L1 and L8, 2 limit cycles near the 2-polycycle L1, 2 limit cycles near the center C8, 1 limit cycle between the center C1 and the polycycle L1, and 1 limit cycle between the center C8 and the polycycle L8. Considering that the system (1) is Z4-equivariant, the system (1) can have at least 24 limit cycles. See Figure 2(a) for their distribution.
(2) Further let
(77)c0(δ)=c1(δ)=B0(δ)=e0(δ)=0,
to obtain
(78)a0=w1a1,a2=w2a1,a4=w3a1,a5=w4a1,
where w1=-16(55m2l4-11m2l5-55m4l2+15m5l4+11m5l2-15m4l5+360m4l1-72m5l1-360m1l4+72m1l5)/(5760m4l0-95m4l2-1152m5l0+39m5l4+19m5l2-5760m0l4+1152m0l5-39m4l5+95m2l4-19m2l5), w2=-(-2m4l5+880m4l0-95m4l1-176m5l0+2m5l4+19m5l1-880m0l4+176m0l5+95m1l4-19m1l5)/(5760m4l0-95m4l2-1152m5l0+39m5l4+19m5l2-5760m0l4+1152m0l5-39m4l5+95m2l4-19m2l5), w3=(240m5l0-2m2l5+2m5l2-240m0l5-39m5l1+39m1l5-5760l0m1-880l2m0+95l2m1+880l0m2+5760l1m0-95l1m2)/(5760m4l0-95m4l2-1152m5l0+39m5l4+19m5l2-5760m0l4+1152m0l5-39m4l5+95m2l4-19m2l5), and w4=(240m0l4-39m1l4+2m2l4-19l2m1-240m4l0+176l2m0+1152l0m1-2m4l2-176l0m2+39m4l1+19l1m2-1152l1m0)/(5760m4l0-95m4l2-1152m5l0+39m5l4+19m5l2-5760m0l4+1152m0l5-39m4l5+95m2l4-19m2l5). Approximate computation using Maple.13 gives
(79)a0=-9.4485863716681452021a1≡a0*,a2=0.0030582821716100337153a1≡a2*,a4=8655624.6835808229498a1≡a4*,a5=-1731124.9144080276985a1≡a5*.
As before, we have
(80)rank∂(c0,c1,B0,e0)∂(a0,a2,a4,a5)=4,
fixing a1>0, a3≠0, and taking δ=(a0,a2,a4,a5) and δ0=(a0*,a2*,a4*,a5*). Noting that
(81)c2(δ0)=-0.45572159725036392404a1,B1(δ0)=-0.0050298137906989600656πa1,e2(δ0)=-0.85071588097a1,d0(δ0)=0.0804770206482136030π6a1,
we have
(82)M1(h1,δ0)=c2(δ0)(-ε3)+O(-ε3ln|ε3|)>0,M1(h2,δ0)=B1(δ0)(ε32)+O(ε33)<0,M8(h3,δ0)=e2(δ0)(-ε3)+O((-ε3)ln|ε3|)>0,M8(h4,δ0)=d0(δ0)ε3+O((-ε3)2)>0,
where h1=-5/12-ε3, h2=-8/3+ε3, h3=-5/12-ε3, and h4=-4/3+ε3 with ε1>0being small. Hence, noting that (1-sgn(M1(h1,δ0)M1(h2,δ0)))/2=1, (1-sgn(M8(h3,δ0)M8(h4,δ0)))/2=0, and (1+sgn(c2(δ0)e2(δ0)))/2=1, by Theorem 8 again, we can obtain 6 limit cycles inside the 2-polycycles L1 and L8. By Z4-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 a Z4-equivalent quintic near-Hamiltonian system can also have 24 limit cycles compared to a z6 and z3-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 zq-equivalent quintic near-Hamiltonian system.
Appendix
In this section, by numeral computation using Maple.13, we give the approximate values of the integrals in (44) and (55) and the coefficients in (45) and (56) as
(A.1)I101=3.309003029,I102=1.730303665,I103=-3.217505541,I111=28.51638338,I112=12.79094042,I113=-16.08752770,I121=254.3849935,I122=97.6750154,I123=-80.43763849,I131=-13.03562364,I132=1.03562364,I133=12,I141=-141.0435372,I142=-0.5116118346,I143=60,I151=-317.8639878,I152=-149.9117575,I153=60,I104=4.636476089,I105=-3.114725692,I114=23.18238044,I115=-10.86431923,I124=115.9119022,I125=-39.42284767,I134=12,I135=-12.00000000,I144=60,I145=-43.62567211,I154=60,I155=21.87163945,I201=94,I202=-94,I203=0,I204=0,I205=0,I211=26116,I212=-26116,I213=0,I214=0,I215=0,I221=195316,I222=-195316,I223=0,I224=0,I225=0,I231=-31516,I232=31516,I233=0,I234=0,I235=0,I241=-238516,I242=238516,I243=0,I244=0,I245=0,I251=-54916,I252=54916,I253=0,I254=0,I255=0,J11=1.831700353,J12=1.100027834,J13=0,J21=36.46355425,J22=20.04837845,J23=0,J41=-14.15186229,J42=-7.471228579,J43=-10.29601773,J51=-70.75931146,J52=-37.35614289,J53=-51.48008865,J14=0,J15=-3.964132938,J24=0,J25=-49.77854225,J44=14.83672348,J45=14.44096924,J54=74.18361742,J55=72.20484619,
and l3=m3=0,
(A.2)l0=1.821801153,l1=25.21979610,l2=271.6223704,l4=-81.55514903,l5=-407.7757453,m0=1.521750397,m1=12.31806121,m2=76.48905453,m4=16.37432789,m5=81.87163945,J1=2.931728187,J2=56.51193270,J3=0,J4=-31.91910860,J5=-159.5955430,R1=-3.964132938,R2=-49.77854225,R3=0,R4=29.27769272,R5=146.3884636.
Conflict of Interests
The authors declare that there is no conflict of interests for any of the authors in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) (11161051), the Natural Science foundation of Guangxi Province in China (2012GXNSFDA 276040) supervised by Professor Hongjian Xi, and the Research Project of GUFE (2013YB216, 2012C08). The authors thank Professor Zhuoen Nong and Professor Qingguang He at Department of Information and Statistics of GUFE for their encouragement in research of differential equations.
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