Nonlinear Analysis: Real World Applications 10 (2009) 1042–1047www.elsevier.com/locate/nonrwa

Bifurcation of limit cycles at the equator for a class of polynomialdifferential system

Qi Zhanga,b,∗, Gui Weihuab, Yirong Liua

a School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075, PR Chinab School of Information Science and Engineering, Central South University, Changsha, Hunan 410075, PR China

Received 18 October 2006; accepted 26 November 2007

Abstract

In this paper, center conditions and bifurcation of limit cycles from the equator for a class of polynomial system of degree sevenare studied. The method is based on converting a real system into a complex system. The recursion formula for the computationof singular point quantities of complex system at the infinity, and the relation of singular point quantities of complex system at theinfinity with the focal values of its concomitant system at the infinity are given. Using the computer algebra system Mathematica,the first 14 singular point quantities of complex system at the infinity are deduced. At the same time, the conditions for the infinityof a real system to be a center and 14 order fine focus are derived respectively. A system of degree seven that bifurcates 13 limitcycles from the infinity is constructed for the first time.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Polynomial system of degree seven; Infinity; Singular point quantity; Bifurcation of limit cycle

1. Introduction

The problem of bifurcation of limit cycles from the infinity is far from being solved in general. Most of the workon this problem is focused on the following 2n + 1 degree symmetric real planar polynomial differential system

dxdt=

2n∑k=0

Xk(x, y)+ (−y + δx)(x2+ y2)n,

dydt=

2n∑k=0

Yk(x, y)+ (x + δy)(x2+ y2)n,

(1)

where n is a natural number, Xk(x, y), Yk(x, y) are homogeneous polynomials of degree k of x, y. The equator Γ∞on the Poincare sphere is a trajectory of this system, which has no real singular point there. We call Γ∞ the infinity ofsystem (1). Due to the difficulty of the problem, the number of limit cycles bifurcating from the infinity is also a openproblem even for n = 1. Recently, there have been some research results on system (1) for n = 1, 2, 3: cubic system,

∗ Corresponding author at: School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075,PR China.

E-mail address: zq8910@mail.csu.edu.cn (Q. Zhang).

1468-1218/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2007.11.021

Q. Zhang et al. / Nonlinear Analysis: Real World Applications 10 (2009) 1042–1047 1043

four limit cycles in [1], five limit cycles in [2], six limit cycles in [3,4], seven limit cycles in [5]; quintic system, fourlimit cycles in [6], eight limit cycles in [7],eleven limit cycles in [8]; seven degree system, eight limit cycles in [9],nine limit cycles in [10]. When n ≥ 3, there are few research results for the planar polynomial system. Following theresearch of [1], the authors researched a class of seven degree polynomial differential system and obtained an exampleof 13 limit cycles bifurcating from the equator.

2. The relation between focal values and singular point quantities

Consider a real planar polynomial system of degree seven:dxdt=

3∑k=1

Xk(x, y)+ X5(x, y)+ (−y + δx)(x2+ y2)3,

dtdt=

3∑k=1

Yk(x, y)+ Y5(x, y)+ (x + δy)(x2+ y2)3,

(2)

where

X1(x, y) = (pB30 − B10)x + (p A30 − A10)y,

X2(x, y) = −(B20 + B11)x2− 2A20xy + (B20 − B11)y2,

X3(x, y) = −(B30 + B21 − 2A30 B30)x3+ (3A2

30 − 3B230 − A21 − 3A30)x2 y

+(3B30 − B21 − 6A30 B30)xy2+ (A30 − A21 − A2

30 + B230)y

3,

X5(x, y) = −(B32x5+ A32x4 y + 2B32x3 y2

+ 2A32x2 y3+ B32xy4

+ A32 y5),

Y1(x, y) = (p A30 + A10)x − (pB30 + B10)y,

Y2(x, y) = (A20 + A11)x2− 2B20xy + (A11 − A20)y2,

Y3(x, y) = (B30 − B21 + 2A30 B30)y3+ (3B2

30 − 3A230 + A21 − 3A30)xy2

−(3B30 + B21 + 6A30 B30)x2 y + (A30 + A21 + A230 − B2

30)x3,

Y5(x, y) = A32x5− B32x4 y + 2A32x3 y2

− 2B32x2 y3+ A32xy4

− B32 y5. (3)

p, Ak j , Bk j (k, j = 0, 1, 2, 3) are real numbers.With the transformation of extended polar coordinates:

x =cos θρ

, y =sin θρ, (4)

system (2) becomes

dρdθ= −

ρ(δ + ρ2ϕ5 + ρ4ϕ3 + ρ

5ϕ2 + ρ6ϕ1)

(1+ ρ2ψ5 + ρ4ψ3 + ρ5ψ2 + ρ6ψ1), (5)

where

ϕk(θ) = cos θXk(cos θ, sin θ)+ sin θYk(cos θ, sin θ),ψk(θ) = cos θYk(cos θ, sin θ)− sin θXk(cos θ, sin θ), k = 1, 2, 3, 5.

(6)

For sufficiently small h, the solution of (5) satisfying condition ρ|θ=0 = h can be written as

ρ = ρ(θ, h) =∞∑

k=1

vk(θ, δ)hk . (7)

It is evident that

v1(0, δ) = e−δθ , vk(0, δ) = 0, k = 2, 3, . . . . (8)

We call v1(2π, δ)− 1 (that is, e−2πδ− 1) the 0th focal value of system (2) at the infinity. k = 1, 2, . . . .

1044 Q. Zhang et al. / Nonlinear Analysis: Real World Applications 10 (2009) 1042–1047

Definition 1. For system (2), in Expression (7), if v1(2π) 6= 1, then the infinity is called a rough focus. If v1(2π) = 1,and v2(2π) = v3(2π) = · · · = v2k(2π) = 0, v2k+1(2π) 6= 0, then the infinity is called a fine focus of order k andv2k+1(2π) is called the kth focal value at the infinity (k = 1, 2, . . .). If v1(2π) = 1, and for any positive integer k,v2k+1(2π) = 0, then the infinity is called a center.

System (2), by transformation

z = x + iy, w = x − iy, T = it, i =√−1 (9)

becomesdzdT= a10z + pb30w + a20z2

+ a11zw + a30z3+ a21z2w + b2

30w3+ a32z3w2

+ z4w3(1− iδ),

dwdT= −b10w − pa30z − b20w

2− b11zw − b30w

3− b21w

2z − a230z3− b32w

3z2− w4z3(1+ iδ).

(10)

Here ak j = Ak j + i Bk j , bk j = ak j (k = 1, 2, 3, j = 0, 1, 2) (bk j is the conjugation of ak j ), the infinity of system(2) becomes the infinity of system (10). Systems (10) and (2) are said to be concomitant.

For simplicity we suppose

a01 = pb30, b01 = pa30, a11 6= 3b20, b11 6= 3a20. (11)

From Theorem 3.2′ in [1], we have

Lemma 1. For system (10)δ=0, we can derive successively the following formal series

F(z, w) =1

zw

∞∑k=0

f7k(z, w)(zw)4k , (12)

such that

dFdT=

∞∑m=1

µm/(zw)m−2, (13)

where f7k(z, w) =∑α+β=7k cαβ zαwβ are homogeneous polynomials of degree 7k of z, w;

c00 = 1, ck,k = 0 (k ≥ 1), when α 6= β,

cαβ =∑k, j

[(3α − 4β − 7k + 21)ak, j−1 − (3β − 4α − 7 j + 21)b j,k−1]cα+3k+4 j−28,β+3 j+4k−28/[7(β − α)],

(14)

µm =∑k, j

[(3− k − m)ak, j−1 − (3− j − m)b j,k−1]c3k+4 j+7m−28,3 j+4k+7m−28, (15)

and v2m+1(2π, 0)A lg∼ iπµm, m = 1, 2, . . . . (16)

In Lemma 1, v2m+1(2π, 0) is the mth focal value of system (2)δ=0 at the infinity, “A lg∼ ” is the symbol of algebraic

equivalence, v2m+1(2π, 0)A lg∼ iπµm means that there exist ξ (k)m (k = 1, 2, . . . ,m − 1), polynomial functions of the

coefficients of system (10)δ=0, such that

v2m+1(2π, 0) = iπ

(µm +

m−1∑k=1

ξ (k)m µk

). (17)

Definition 2. We call µm (m = 1, 2, 3, . . .) determined by (13) the mth singular point quantity of system (10)δ=0 atthe infinity. If for all m, µm = 0, we call the infinity a generalized complex center of system (10)δ=0.

From Lemma 1, the focal values of system (2)δ=0 at the infinity can be deduced from the singular point quantitiesof system (10)δ=0 at the infinity. It is obvious that the infinity of system (2)δ=0 is a center if and only if that of system(10)δ=0 is a generalized complex center.

Q. Zhang et al. / Nonlinear Analysis: Real World Applications 10 (2009) 1042–1047 1045

3. Singular point quantities and center conditions at the infinity

Theorem 1. The first 14 singular point quantities of system (10)δ=0 at the infinity are as follows:

µ1 = b32 − a32,

µ2 = b21 − a21,

µ3 = b10 − a10,

µ4 = a12a30 − b12b30,

µ5 = a11a20 − b11b20,

µ6 = 0,

µ7 = (2a211a30 − 2b2

11b30 + 3a30b220 − 3b30a2

20 + 7a20b11b30 − 7b20a11a30 + 3b12b220

−3a12a220 + a12a20b11 − b12b20a11)/2,

µ8 = p(a211a30 − b2

11b30 + 4a30b220 − 4b30a2

20 + 4a20b11b30 − 4b20a11a30),

µ9 = −3a30b30(10a30b220 − 10b30a2

20 + 3a211a30 − 3b2

11b30 + 11a20b11b30 − 11b20a11a30)/2,

µ10 = −8a30b30(a30b220 − b30a2

20)(a32 + b32)/27,

µ11 = −2a30b30(a30b220 − b30a2

20)(21a21 + 21b21 + a30b30)/81,

µ12 = −(a30b220 − b30a2

20)(a30b220 + b30a2

20 + 6a10a30b30 + 6b10a30b30)/9,

µ13 = −a230b2

30(83 247a30b30 − 744 800)(a30b220 − b30a2

20)/714 420,

µ14 = −29 792 000(186 200a10 + 186 200b10 + 471 733a20b20)(a30b220 − b30a2

20)/168 400 531 1187. (18)

In the above expression of µk , we have already let µ1 = µ2 = · · · = µk−1 = 0

(k = 2, 3, . . . , 14).

Theorem 2. The first 14 singular point quantities of system (10)δ=0 at the infinity are all zero if and only if one of thefollowing conditions holds:

(1) a10 = b10, a21 = b21, a32 = b32, a20b20 = 0, a30b30 = 0; (19)(2) a10 = b10, a21 = b21, a32 = b32, a11a20 = b11b20, a12a2

20 = b12b220,

a20b20 6= 0, a30b30 = 0; (20)(3) a10 = b10, a21 = b21, a32 = b32, a12a30 = b12b30,

a30a211 = b30b2

11, a30b30 6= 0, a20b20 = 0; (21)(4) a10 = b10, a21 = b21, a32 = b32, a12a30 = b12b30, a11a20 = b11b20,

a30b220 = b30a2

20, a30b30 6= 0, a20b20 6= 0; (22)(5) a10 = b10, a21 = b21, a32 = b32, a11 = 2b20, b11 = 2a20, a12 = 3b30, b12 = 3a30, a20b20 6= 0,

a30b30 6= 0. (23)

From the definition of elementary Lie-invariants in [1], we can obtain

Lemma 2. All the elementary Lie-invariants of system (10)δ=0 are as follows:

a21 b21 a10 b10 a32 b32 a11a20

a11b20 a11b11 a20b20 a30b30 a12b12 a12a30 b12b30

a20b11b30 b20a11a30 a20b11a12 b20a11b12 a30b220 b30a2

20 a211a30

b211b30 a12a2

20 b12b220 a2

11b12 b211a12

1046 Q. Zhang et al. / Nonlinear Analysis: Real World Applications 10 (2009) 1042–1047

Theorem 3. All the singular point quantities of system (10)δ=0 at the infinity are zero (that is, the infinity of system(10)δ=0 is a generalized complex center) if and only if one of the conditions of Theorem 2 holds.

Proof. The necessity is evident. We need only prove the sufficiency. If condition (1) in Theorem 2 holds, then frombk j = ak j (k = 1, 2, 3, j = 0, 1, 2), we know that a20 = b20 = a30 = b30 = 0, and system (10)δ=0 becomes

dzdT= a10z + a11zw + a21z2w + a32z3w2

+ z4w3,

dwdT= −b10w − b11zw − b21w

2z − b32w3z2− w4z3.

(24)

If a11 6= b11, then system (24) has integrating factor F1(z, w) = (zw)a11−2b11b11−a11 . If a11 = b11, then system (24) has

first integral F2(z, w) = zw; When one of (2)–(4) in Theorem 2 holds, from Lemma 2 we can easily prove that system(10)δ=0 satisfies the extended symmetric principle in [1]; when condition (5) in Theorem 2 holds, system (10)δ=0 isHamiltonian. This completes the proof of the sufficiency. �

Corollary 1. The infinity of system (2)δ=0 is a center if and only if one of the five conditions in Theorem 2 holds.

4. Bifurcation of limit cycles at the infinity

Theorem 4. The infinity of system (10)δ=0 is a 14 order singular point, that is, µ1 = µ2 = · · · = µ13 = 0, µ14 6= 0,if and only if the following conditions hold:

b10 = a10, b21 = a21 = −53 200/249 741, a11 = 5b20/3, b11 = 5a20/3,a30b30 = 744 800/83 247, a32 = b32 = 0, a12 = 7b30/3, b12 = 7a30/3,

a30b220 + b30a2

20 = −1489 600(a10 + b10)/27 749, a30b220 − b30a2

20 6= 0.

(25)

Corollary 2. The infinity of system (2)δ=0 is a 14 order fine focus if and only if (25) in Theorem 4 holds.

Theorem 5. If the coefficients of system (10) satisfy:

δ = ε91/2, a10 = 17/310+ iε55/2, b10 = a10, p = −ε21,

a20 = b20 = 1, a11 = 5/3+ ε15− iε36/2, b11 = a11,

a21 = (−744 800+ 83 247ε − 3496 374ε6− 1748 187iε66)/3496 374, b21 = a21,

a30 = −(p1 + i√

p2)/25 806 570, b30 = a30.

a12 = (−7i− 6iε15− 3iε28

+ 3ε45)(p1i+√

p2)/77 419 710, b12 = a12,

p1 = −75 969 600+ 8491 194ε + 12 903 285ε3,

p2 = 3(62 349 070 000 000+ 208 055 389 413 300ε − 24 033 458 515 212ε2

+653 504 933 424 000ε3− 73 042 864 114 860ε4

− 55 498 254 597 075ε6),

a32 = ε10+ iε78/2, b32 = a32,

(26)

accordingly, the coefficients of system (2) are determined by (11). Then, when ε = 0, the infinity of system (2) is astable 14 order fine focus; when 0 < ε << 1, there are 13 limit cycles in a small enough neighborhood of the infinityof system (2).

Proof. Being a polynomial differential system, the solutions of system (2) satisfy

v2m+1(2π, δ) = v2m+1(2π, 0)+ δϕ2m+1(aαβ , bαβ , δ), (27)

where ϕ2m+1(aαβ , bαβ , δ) are analytic about δ. From Theorems 1 and 2 and v1(2π, δ) − 1 = e−2πδ− 1,

v2m+1(2π, 0)A lg∼ iπµm (m = 1, 2, . . .). Using the computer algebra system Mathematica for computation, we obtain

Q. Zhang et al. / Nonlinear Analysis: Real World Applications 10 (2009) 1042–1047 1047

v1(2π, δ)− 1 = −πε91+ o(ε91), v3(2π, δ) = πε78

+ o(ε78),

v5(2π, δ) = −πε66+ o(ε66), v7(2π, δ) = πε55

+ o(ε55),

v9(2π, δ) = −1489 600πε45/83 247+ o(ε45), v11(2π, δ) = πε36+ o(ε36),

v15(2π, δ) = −2800p0πε28/2580 657+ o(ε28),

v17(2π, δ) = 1400p0πε21/7741 971+ o(ε21),

v19(2π, δ) = −521 360 000p0πε15/23 870 217 031+ o(ε15),

v21(2π, δ) = 16 683 520 000p0πε10/1933 487 579 511+ o(ε10),

v23(2π, δ) = −29 196 160 000p0πε6/1933 487 579 511+ o(ε6),

v25(2π, δ) = 1400p0πε3/7741 971+ o(ε3),

v27(2π, δ) = −792 467 200 000p0πε/52 204 164 646 797+ o(ε),

v29(2π, δ) = 212 114 314 988 800 000p0π/14 969 004 769 434 356 181+ O(ε),

(28)

where p0 =√

1272 430/3. From (28) and [1] Th. 6.6 we complete the proof. �

Acknowledgement

This project was supported by the Postdoctoral Science Foundation of Central South University and the NationalNature Science Foundation of China (number 60634020).

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