Conditions of infinity to be an isochronous centre for a rational differential system

Mathematical and Computer Modelling 46 (2007) 583–594www.elsevier.com/locate/mcm

Conditions of infinity to be an isochronous centre for a rationaldifferential systemI

Wentao Huanga,b,∗, Yirong Liuc, Weinian Zhanga

a Department of Mathematics, Sichuan University, Chengdu 610064, PR Chinab The Seventh Department, Guilin University of Electronic Technology, Guilin, 541004, PR China

c Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China

Received 5 September 2006; received in revised form 4 November 2006; accepted 21 November 2006

Abstract

In this article we find conditions under which infinity is a centre or an isochronous centre for a rational differential system. Bya transformation, infinity is taken to the origin and therefore properties at infinity can be studied by using the methods developedfor finite critical points. Using the computer algebra system—Mathematica we compute the singular point values and the periodconstants at the origin for the transformed system and then get the necessary conditions of the problems. Finally, we prove that theconditions are sufficient.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Infinity; Centre; Isochronous centre; Singular point value; Period constant

1. Introduction

During the last decades, concern with Hilbert’s 16th problem and the monotonicity of period solutions, has ledto the centre and the isochronous centre problems for differential systems being researched actively. In the case of afinite critical point for that, a lot of work has been done (see, for instance [1–7] and the references therein). For thecase of infinity, several researches are concentrated on the following 2n + 1 degree system:

dxdt

=

2n∑k=0

Xk(x, y) − y(x2+ y2)n

dydt

=

2n∑k=0

Yk(x, y) + x(x2+ y2)n

(1)

where Xk(x, y), Yk(x, y) are homogeneous polynomials of degree k of x and y. For this system, the equator Γ∞ onthe Poincare closed sphere is a trajectory of the system, having no real critical point. Γ∞ is called infinity or the

I Supported by Guangxi Science Foundation # 0575092, and SRFDP # 20050610003.∗ Corresponding author at: Department of Mathematics, Sichuan University, Chengdu 610064, PR China.

E-mail address: [email protected] (W. Huang).

0895-7177/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2006.11.022

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mailto:[email protected]

http://dx.doi.org/10.1016/j.mcm.2006.11.022

584 W. Huang et al. / Mathematical and Computer Modelling 46 (2007) 583–594

equator in this point of view. As far as centre conditions at infinity are concerned, several special systems have beenstudied: cubic systems in [8–10]; fifth degree systems in [11,12]. In [13], we gave the conditions of infinity to be anisochronous centre for a special case of n = 2 of the following real rational systems:

dxdt

=1

(x2 + y2)n

[2n∑

k=0

Xk(x, y) − y(x2+ y2)n

],

dydt

=1

(x2 + y2)n

[2n∑

k=0

Yk(x, y) + x(x2+ y2)n

] (2)

and proved first that the real rational system has an isochronous at infinity. We also discussed the isochronous centreproblem for another case of n = 2 of (2) in the complex field in [14]. In this paper, we study the centre and theisochronous centre conditions at infinity for a special case of n = 1 of system (2), namely:

dxdt

=1

(x2 + y2)(−my + A20x2

+ A11xy + A02 y2− y(x2

+ y2)),

dydt

=1

(x2 + y2)(mx + B20x2

+ B11xy + B02 y2+ x(x2

+ y2)).

(3)

By a time rescaling, system (3) can be transformed into the following real polynomial differential system:

dxdt

= −my + A20x2+ A11xy + A02 y2

− y(x2+ y2),

dydt

= mx + B20x2+ B11xy + B02 y2

+ x(x2+ y2).

(4)

System (3) and (4) have the same phase portraits near the equator in the phase plane, so their centre conditions atinfinity are identical. It is worth pointing that the centre conditions at infinity of (4) have been studied by Blows andRousseau [8]. Our method used in this paper, which is based on the complex concomitant system and the origin, differsfrom that used in [8]. The conditions for infinity to be a centre coincide with that in [8], but the centre conditions setout in this paper are very compact in the expression.

2. Preliminaries and the conversions of the system

Definition 2.1 (See [14]). For a real planar differential system, infinity (the equator) Γ∞ is said to be an isochronouscentre if the trajectories of the system in the neighbourhoods of infinity (the equator) are all closed and their periodsare the same constant.

It is obvious that

Theorem 2.1. For differential systems (A) and (B) , if there exists a homeomorphism taking infinity of system (A) tothe origin of system (B) , then infinity of system (A) is a centre (an isochronous centre) if and only if the origin ofsystem (B) is a centre (an isochronous centre).

With the transformation

x =ξ

(ξ2 + η2)2 , y =η

(ξ2 + η2)2 , (5)

system (3) becomes the following polynomial system:

dξ

dt1= −η +

(η2

−ξ2

3

)(A20ξ

2+ A11ξη + A02η

2) −43ξη(B20ξ

2+ B11ξη + B02η

2) − m η (ξ2+ η2)3,

dη

dt1= ξ +

(ξ2

−η2

3

)(B20ξ

2+ B11ξη + B02η

2) −43ξη(A20ξ

2+ A11ξη + A02η

2) + m ξ (ξ2+ η2)3.

(6)

W. Huang et al. / Mathematical and Computer Modelling 46 (2007) 583–594 585

Since transformation (5) is a homeomorphism, by Theorem 2.1, we see that the conditions for centre (isochronouscentre) at infinity of system (3) are the same as that for centre (isochronous centre) at the origin of system (6).

We perform the transformation:

z = ξ + ηi, w = ξ − ηi, T = it, i =√

−1, (7)

to reduce system (6) to a complex system of the form:

dzdT

= z +23

b02z4+

13(2b11 + a20)z3w +

13(a11 + 2b20)z2w2

+13

a02zw3+ m z4w3

= Z(z, w),

dw

dT= −w −

23

a02w4−

13(2a11 + b20)w

3z −13(b11 + 2a20)z2w2

−13

b02wz3− m w4z3

= −W (z, w),

(8)

where

a20 =14(A02i − A11 − A20i − B02 − B11i + B20), b20 = a20;

a02 =14(A02i + A11 − A20i − B02 + B11i + B20), b02 = a02;

a11 =12(−A02i − A20i + B02 + B20), b11 = a11,

(9)

and z, w, T are complex variables.As in [15] we call that system (6) is the concomitant system of (8) and vice versa. Evidently, the coefficients of

system (6) are real if and only if the coefficients of system (8) satisfy the conjugate condition, i.e.

aαβ = bαβ , α ≥ 0, β ≥ 0. (10)

In [15], the authors defined the complex centre and the singular point value, denoted by µk , for the followingcomplex systems:

dzdT

= z +

∞∑k=2

Zk(z, w) = Z(z, w),

dw

dT= −w −

∞∑k=2

Wk(z, w) = −W (z, w),

(11)

with

Zk(z, w) =

∑α+β=k

aαβ zαwβ , Wk(z, w) =

∑α+β=k

bαβwαzβ

and gave

Theorem A. System (11) has a complex centre at the origin if and only if all singular point values at the origin vanish.

Furthermore, the recursive formulas to compute the singular point values of the origin of (11) were given in [16].These results are also restated in [17]. In [6], the authors defined firstly the complex isochronous centre (extendedisochronous centre) for (11).

In [17] Liu and Huang gave the definition of the period constant of the origin for system (11) and its computationalmethod. Now we list some of its results.

By means of transformation:

z = reiθ , w = re−iθ , T = it, (12)


system (11) can be transformed into

drdt

= iwZ − zW

2r= ir

∞∑k=1

wZk+1 − zWk+1

2zw=

ir2

∞∑m=1

∑α+β=m+2

(aα,β−1 − bβ,α−1)ei(α−β)θrm,

dθ

dt=

wZ + zW2zw

= 1 +

∞∑k=1

wZk+1 + zWk+1

2zw= 1 +

12

∞∑m=1

∑α+β=m+2

(aα,β−1 + bβ,α−1)ei(α−β)θrm . (13)

For the complex constant h, |h| � 1, we write the solution of (13) associated with the initial condition r |θ=0 = has:

r = r(θ, h) = h +

∞∑k=2

vk(θ)hk (14)

and denote:

τ(ϕ, h) =

∫ ϕ

0

dtdθ

dθ =

∫ ϕ

0

[1 +

12

∞∑m=1

∑α+β=m+2

(aα,β−1 + bβ,α−1)ei(α−β)θ r(θ, h)m

]−1

dθ. (15)

Definition 2.2. For a sufficiently small complex constant h, the origin of system (11) is called a complex centre ifr(2π, h) ≡ h, and it is called a complex isochronous centre if:

r(2π, h) ≡ h, τ (2π, h) ≡ 2π. (16)

Lemma 2.2 ([19]). For system (11), one can derive uniquely the following formal series:

ξ = z +

∞∑k+ j=2

ck j zkw j , η = w +

∞∑k+ j=2

dk, jwk z j , (17)

where ck+1,k = dk+1,k = 0, k = 1, 2, . . ., such that

dξ

dT= ξ +

∞∑j=1

p jξj+1η j ,

dη

dT= −η −

∞∑j=1

q jηj+1ξ j . (18)

Definition 2.3. Let µ0 = 0, µk = pk − qk, τk = pk + qk, k = 1, 2, . . .. For system (11), µk is called the kth singularpoint value of the origin, τk is called the kth period constant of the origin.

Theorem B. For system (11), the origin is a complex isochronous centre if and only if µk = τk = 0, k = 1, 2, . . ..

From [15,17], it can be obtained that if the coefficients of complex system (11) satisfy condition (10), then system(11) has a complex centre (a complex isochronous centre) at the origin if and only if its concomitant real systemhas a centre (an isochronous centre) at the origin. Therefore, the necessary conditions of infinity of (3) (respectively,the origin of (6)) to be a centre and to be an isochronous centre can be derived from the computations of singularpoint values and period constants at the origin of (8). However, its sufficiency is a difficult problem and needs othermethods.

3. Singular point values and centre conditions

In this section we discuss the computation of the singular point values and centre conditions at the origin of thesystem (8).

A recursive formula for singular point values is given in [16]. From [16, Theorem 2.1], we have:


Lemma 3.1. For system (8), one can derive successively the terms of the following formal series:

M(z, w) =

∞∑α+β=0

c(α, β)zαwβ (19)

such that

∂(M Z)

∂z−

∂(MW )

∂w=

∞∑n=1

(n + 1)µn(zw)n (20)

where c(0, 0) = 1, µn is the n-th singular point value at the origin of the system. In addition, for any integer numbersα, β, when α 6= β, c(α, β) is given by the following recursive formula: c(α, β) =

13(−α+β)

((3m(α−β)c(−3+α, −3+

β) + b02(1 + 2α − β)c(−3 + α, β) + (a20(−1 + α − 2β) + b11(1 + 2α − β))c(−2 + α, −1 + β) + (a11(−1 + α −

2β)+ b20(1 + 2α −β))c(−1 +α, −2 +β)+a02(−1 +α − 2β)c(α, −3 +β))); for any positive integer n, µn is givenby the following recursive formula µn =

13 (b02c(−3 + n, n) − (a20 − b11)c(−2 + n, −1 + n) − (a11 − b20)c(−1 +

n, −2 + n) − a02c(n, −3 + n)).

Using Lemma 3.1 and computing with the computer algebra system—Mathematica, we have the following result.

Theorem 3.2. The first 15 singular point values at the origin of the system (8) are given as follows:

µ3 =13(a11a20 − b11b20),

and either

µ6 = 0, µ9 =172

I1 (g02 − 4m), µ12 = −1

280g02a11b11 I1, µ15 = 0

for a20 = b20 = 0 or

µ6 = −19(−2 + p)(2 + p) I0, µ9 = −

572

(2g02 − 3g20 − 12m)(−2 + p) I0,

µ12 = −1

180(2g2

02 − 61g02g20 + 120g220) I0,

µ15 =11

16 329 600(3748 573g02g2

20 − 7012 560g320 − 132 300g02 J0 + 255 150g20 J0)(−2 + p)I0,

for other a20, b20. Moreover, µk = 0 for k 6= 3i, i < 5, k, i ∈ N. Here:

g02 = a02b02, g20 = a20b20,

I0 = a02a320 − b02b3

20, J0 = a02a320 + b02b3

20,

I1 = a02b311 − a3

11b02,

(21)

and under the conditions µ3 = 0, ai j = bi j and a20b20 6= 0, p is determined by a11 = pb20, b11 = pa20. In theabove expression of µk , it is assumed that: µ1 = µ2 = · · · = µk−1 = 0, k = 2, 3, . . . , 15.

From Theorem 3.2, we get

Theorem 3.3. For system (8), the first 15 singular point values are zero if and only if one of the following threeconditions holds:

KI : a11a20 = b11b20, a02a320 = b02b3

20, a20b20 6= 0; (22)

KII : a11 = 2b20, b11 = 2a20 a02a320 6= b02b3

20, a20b20 6= 0; (23)

KIII : a20 = b20 = 0, a02b311 = b02a3

11. (24)


Proof. The sufficiency is obvious. From the forms of µk, k = 1, 2, . . ., it is easy to see that in order to getthe necessity we only need to prove that H1 := 2g2

02 − 61g02g20 + 120g220 = 0, H2 := 3748 573g02g2

20 −

7012 560g320 − 132 300g02 J0 + 255 150g20 J0 6= 0, if a20b20 6= 0. For a reduction to absurdity, suppose that H2 = 0.

If −132 300g02 + 255 150g20 = 0, then g02 =2714 g20. Substituting the equivalence g02 =

2714 g20 in the relation

H1 = 2g202 − 61g02g20 + 120g2

20, we get g02 = g20 = 0, which contradicts the condition g20 = a20b20 6= 0.

If −132 300g02 + 255 150g20 6= 0, we have from H1 = H2 = 0 that J0 =3748 573g02g2

20−7012 560g220

9450(14g02−27g20), g02 =

14 (61 ±

√2761)g20. We can see from the expressions of I0, J0 that I 2

0 = J 20 − 4g02g3

20. Therefore:

I 20 =

75 614 666 376 819 169 ± 474 512 826 788 651√

276173 156 608 000 000

g420. (25)

Because system (3) is real, we have from (10) that a20 = b20, a02 = b02. Thus I0 = a02a320 − b02b3

20 is zero or a pureimaginary number and leads to I 2

≤ 0. But from the quantity of the right hand of (25) which is greater than zero, weget that the hypothesis does not hold even if the statement holds.

From µ1 = µ2 = · · · = µ15 = 0, it can be derived directly that one of the (22)–(24) or:

a11 = −2b20, b11 = −2a20 (i.e., p = −2), 2g02 − 3g20 − 12m = H1 = H2 = 0 (26)

holds. From the statement above mentioned, (26) does not hold and then we get the conclusion of this theorem. �

In order to make use of the Constructive Theorem of Singular point values ([15, Theorem 2.5] or [18, Theorem 4.15])to get the sufficient conditions of the centre of (8), we need to find out all elementary Lie-invariants of the system.From the technique used in [15], we have the following lemma:

Lemma 3.4. All the elementary Lie-invariants of system (8) are given as follows:

a10, b10, (i.e., m)

a20b20, a11b11, a02b02, a20a11, b20b11,

a320a02, a2

20b11a02, a20b211a02, b3

11a02,

b320b02, b2

20a11b02, b20a211b02, a3

11b02.

In Theorem 3.3, if (22) holds, from a11a20 = b11b20, a20b20 6= 0, there is a constant p, such that a11 = p b20, b11 =

pa20. Hence, from a02a320 = b02b3

20 we see that a220b11a02 = b2

20a11b02, a20b211a02 = b20a2

11b02, b311a02 = a3

11b02. Wealso notice that a10 = b10 = m and a20b20, a11b11, a02b02 are self-symmetry. Therefore, let g = f (aαβ , bαβ), g∗ =

f (bαβ , aαβ), under the condition (22), if g is any Lie-invariant of the system (8), then g = g∗. According to theConstructive Theorem of Singular point values (see ([15, Theorem 2.5] or [18, Theorem 4.15])), we get that allµk = 0, k = 1, 2, . . .. So, the origin of (8) is a complex center. Similarly, if condition (24) holds, the origin of (8) isalso a centre.

If condition (23) holds, (8) has an analytical first integral F(z, w) in the neighbourhood of the origin with the form:

F(z, w) =6z6w6

−3 − 2a02w3 − 6b20w2z − 6a20wz2 − 2b02z3 − 6mw3z3 .

Therefore, we have from the above discussion that the following result holds:

Theorem 3.5. For system (8), all the singular point values of the origin are zero if and only if the first 15 singularpoint values of the origin are zero, i.e. one of the three conditions of Theorem 3.3 holds. Correspondingly, the origincan be a complex centre if and only if one of the three conditions given by Theorem 3.3 holds.

Corollary 3.6. The origin of (6) (resp. infinity of (3) or (4)) is a centre if and only if one of the three conditions givenby Theorem 3.3 holds.

From the course of proof of Theorems 3.3 and 3.5, we have:


Corollary 3.7. The origin of system (6) (resp. infinity of system (3)) to be a weak focus with the highest order ifand only if a20b20 6= 0, a11 = −2b20, b11 = −2a20, 2g02 − 3g20 − 12m = 2g2

02 − 61g02g20 + 120g220 = 0, here

g02 = a02b02, g20 = a20b20.

As in [8], by performing suitable perturbations to the coefficients of system (3), we can obtain that (3) has five largeamplitude limit cycles bifurcating from infinity (the equator).

4. Period constants and isochronous centre conditions

We now discuss the conditions for isochronous centres.In [17], we have given the general recursive formulas to compute complex periods. From [17, Theorem 3.1] we

have:

Lemma 4.1. For system (8), the period constant τ j can be determined by the following a series of recursive formulas:

c(k, j) = −1

3(1 + j − k)(3( j − k)mc(−3 + k, −3 + j) + b02(6 + j − 2k)c(−3 + k, j)

+ (b11(3 + j − 2k) + a20(2 j − k))c(−2 + k, −1 + j) + (b20( j − 2k)

+ a11(−3 + 2 j − k))c(−1 + k, −2 + j) + a02(−6 + 2 j − k)c(k, −3 + j));

d(k, j) = −1

3(1 + j − k)(3( j − k)md(−3 + k, −3 + j) + a02(6 + j − 2k)d(−3 + k, j)

+ (a11(3 + j − 2k) + b20(2 j − k))d(−2 + k, −1 + j) + (a20( j − 2k)

+ b11(−3 + 2 j − k))d(−1 + k, −2 + j) + b02(−6 + 2 j − k)d(k, −3 + j));

p( j) =13(3mc(−2 + j, −3 + j) + b02(−4 + j)c(−2 + j, j) − (a20 − b11)(−1 + j)c(−1 + j, −1 + j)

− (a11(−4 + j) − b20(2 + j))c( j, −2 + j) − a02(−7 + j)c(1 + j, −3 + j));

q( j) =13(3md(−2 + j, −3 + j) + a02(−4 + j)d(−2 + j, j) + (a11 − b20)(−1 + j)d(−1 + j, −1 + j)

+ (−b11(−4 + j) + a20(2 + j))d( j, −2 + j) − b02(−7 + j)d(1 + j, −3 + j));

τ j = p( j) + q( j).

Conveniently, we say that system (8) has a complex centre of type KI (resp. KII, KIII) if condition (22) (resp. (23)and (24)) holds. From the centre conditions, we investigate the following three cases.

4.1. Centre of type KI

Since a20b20 6= 0, from (22) we can let

a11 = sb20, b11 = sa20, a02 = hb320, b02 = ha3

20, (27)

where h, s are arbitrary complex constants. Putting expression (27) into the recursive formulas given by Lemma 4.1,we get the following result, after careful computations:

Theorem 4.2. For system (8), the first 15 period constants of the origin are as follows:

τ3 = −23(3g20 + g3

20h2− 3 m + 3 g20 s),

τ6 =154

(9g20 + g320h2

− 3m)F1,

Case 1 : when 9g20 + g320h2

− 3m = 0,

τ9 =1

486g3

20 F2,

τ12 = 0,

τ15 = −1

48g6

20hF3,


Case 2 : when 9g20 + g320h2

− 3m 6= 0, F1 = 0,

τ9 =g4

20h(1 + g20h)

8505(−2 + g20h)4(6 + g20h)4 F4,

τ12 = −g5

20h(1 + g20h)

48 620 250(−2 + g20h)6(6 + g20h)6 F5,

τ15 = 0,

where

g20 = a20b20,

F1 = −12g220h − 9g3

20h2+ 4g4

20h3+ g5

20h4+ 36m − 12g20hm − 3g2

20h2m,

F2 = 1458 − 8505g20h + 486g220h2

+ 1890g320h3

+ 54g420h4

+ 2g620h6,

F3 = 8001 − 56 598g20h + 5124g220h2

+ 12 768g320h3

+ 245g420h4,

F4 = 587 865 600 − 713 276 928g20h + 45 886 176g220h2

+ 185 053 248g320h3

− 16 793 244g420h4

− 26 318 196g520h5

+ 1188 513g620h6

+ 1704 510g720h7

+ 37 659g820h8

− 36 790g920h9

− 870g1020h10

+ 525g1120h11

+ 35g1220h12,

F5 = −190 756 758 974 745 600 + 228 403 212 765 977 088g20h − 8842 240 148 849 856g220h2

− 62 588 254 899 657 888g320h3

+ 4185 316 351 234 584g420h4

+ 9203 353 227 091 116g520h5

− 137 356 698 226 878g620h6

− 626 896 745 242 155g720h7

− 44 057 530 518 924g820h8

+ 13 122 395 414 455g920h9

+ 2005 933 935 140g1020h10

+ 77 721 990 850g1120h11

and τk = 0, k 6= 3i, i < 5, i ∈ N. In the above expression of τk , we have already let τ1 = · · · = τk−1 = 0, k =

2, 3, . . . 15.

Remark: If F1 = 0, we get 36 − 12g20h − 3g220h2

= −3(−2 + g20h)(6 + g20h) 6= 0, and then (−2 + g20h)(6 + g20h)

can be used as the denominator of µk in Theorem 4.2. Because either −2 + g20h = 0 or 6 + g20h = 0 leads to F1 6= 0if −3(−2 + g20h)(6 + g20h) = 0.

By Theorem 4.2, we have

Theorem 4.3. Under condition (22), the first 15 complex period constants of the origin of system (8) are zero if andonly if one of the following expressions holds:

m = h = 0, s = −1, (28)i.e. m = 0, a02 = b02 = 0, a11 = −b20, b11 = −b20, a20b20 6= 0;

m = 0, h = −1

a20b20, s = −

43, (29)

i.e., m = 0, a02 =b2

20a20

, b02 =a2

20b20

, a11 = −43

b20, b11 = −43

b20, a20b20 6= 0.

Proof. The sufficiency is obvious. Let us consider the necessity. In Case 1 of Theorem 4.2, g20 = a20b20 6= 0,µ9 = µ15 = 0 leads to F2 = F3 = 0 or F2 = h = 0. Let the resultant of the polynomials F and G with respect to thevariable x be denoted by resultant (F, G, x). Then:

resultant (F2, F3, h) = −12 787 784 685 876 147 565 203 275 046 075g2420 < 0,

which indicates that F2 and F3 have no common zero. Consider h = 0 implies F2 = 1458 > 0, so µ1 = µ2 = · · · =

µ15 = 0 cannot appear in Case 1.For Case 2 of Theorem 4.2, from µ9 = µ12 = 0, we have that h = 0 or h = −

1g20

or F4 = F5 = 0. But,

resultant (F4, F5, h) = −Ag13220 < 0,


where A is a positive integer number with 203 digits. It implies that F4 and F5 have no common zero. Therefore,µ9 = µ12 = 0 only leads to h = 0 or h = −

1g20

. Applying h = 0, h = −1

g20respectively to F1 = 0 and µ3 = 0, it is

easy to get the conclusion of this theorem. �

Theorem 4.3 indicates that (28) or (29) is the necessary condition of the origin to be an isochronous centre. Let usprove that they also give the sufficient condition.

If (28) holds, system (8) becomes:

dzdT

= z +b20w

2z2

3−

a20wz3

3dw

dT= −

(w −

b20w3z

3+

a20w2z2

3

).

(30)

Let written in polar coordinates as z = reiθ , w = re−iθ , (30) can be:

θ =12i

(log z − log w). (31)

Differentiating both sides of (31) with respect to T along the trajectories of system (30), we obtain:

dθ

dT=

12i

(1z

dzdT

+1w

dw

dT

)= −i (32)

namely,

dθ

dt= i

dθ

dT= 1. (33)

So system (30) has a complex isochronous centre at the origin.If (29) holds, system (8) becomes

dzdT

= z −b20

2w3z3a20

+2b20w

2z2

9−

5a20wz3

9−

2a202z4

3b20

dw

dT= −

(w −

2b202w4

3a20−

5b20w3z

9+

2a20w2z2

9−

a202wz3

3b20

).

(34)

We perform transformations: z = ( 3a20b20

)23 b20u, w = ( 3

a20b20)

23 a20v. It brings system (34) to the form:

dudT

= u − 6u4− 5u3v + 2u2v2

− 3uv3

dv

dT= −(v − 3u3v + 2u2v2

− 5uv3− 6v4).

(35)

Being that the origin is a centre, system (35) has an integral factor M(u, v) with M(0, 0) = 1. Under thetransformations:

u =M(u, v)

14 u(1 − 6uv2

− 6v3)1124

(1 − 6u3 − 6u2v)124

, v =M(u, v)

14 v(1 − 6u3

− 6u2v)1124

(1 − 6uv2 − 6v3)1

24, (36)

system (35) takes the form:

dudT

= u,dv

dT= −v. (37)

Then dθdT =

12i (

1u

dudT +

1v

dvdT ) = −i, i.e., dθ

dt = i dθdT = 1. So the origin of system (34) is a complex isochronous centre.


4.2. Centre of type KII

Applying (23) to the recursive formulas given by Lemma 4.1 we find that the first 21 period constants are asfollows:

τ3 = −23(g02 + 9g20 − 3m),

τ6 = 0,

τ9 =136

(−945g20 J0 + 210J0m + 4m3),

τ12 = 0,

τ15 =F6

264 600(9g20 − 2m)2 ,

τ18 = 0,

τ21 = −m4 F7

4767 562 800 000(9g20 − 2m)3 ,

τk = 0, when k 6= 3i, i < 7, k, i ∈ N

where g02, g20, I0, J0 are given by (21), and

F6 = 68 959 390 500g720 − 69 725 605 950g6

20m + 26 136 460 350g520m2

− 4296 244 050g420m3

+ 259 003 710g320m4

− 1633 905g220m5

+ 840 390g20m6− 84 616m7,

F7 = 74 926 326 264 139 350g620 − 82 512 320 778 812 925g5

20m + 35 512 398 502 459 200g420m2

− 7634 032 938 458 955g320m3

+ 892 635 252 567 240g220m4

− 62 126 438 657 500g20m5

+ 2583 297 384 128m6.

In the above expression of τk , we have already let τ1 = · · · = τk−1 = 0, k = 2, 3, . . . 21.

Remark: When computing µ15, we have let µ3 = µ9 = 0. So, in this case we get 9g20 − 2m 6= 0. Otherwise, it can

lead to µ9 =81g3

204 > 0.

From resultant(F6, F7, m) = −Bg4220 < 0, here B is a positive integer with 143 digits, it can be obtained that

F6 and F7 have no common zero. Then, µ15 = µ21 = 0 if and only if m = 0, F6 = 0. But m = 0 leads toF6 = 68 959 390 500g7

20 = 0, which contradicts with g20 = a20b20 > 0. Therefore, it is clear that if µ15 = 0 thenµ21 6= 0.

From the above discussion, we have that a centre of type KII is a finite weak centre and isn’t isochronous.

4.3. Centre of type KIII

Substituting a20 = b20 = 0 to the recursive formulas given by Lemma 4.1 we find that:

τ3 = −23(a02b02 − 3m),

τ6 =16

a02b02a11b11,

τ9 =a02b02

1944(−27a3

11b02 + 8a202b2

02 − 27a02b311)

τk = 0 for other k.

It is obvious that τ3 = τ6 = τ9 = 0 if and only if:

m = a02 = b02 = 0. (38)


Under conditions (24) and (38) system (8) becomes:

dzdT

= z +23

b11z3w +13

a11z2w2

dw

dT= −

(w +

23

a11w3z +

13

b11w2z2)

.

(39)

By means of transformations:

u =z√

1 + a11 w2 z√1 + b11 w z2

, v =w√

1 + b11 w z2√1 + a11 w2 z

(40)

system (39) can be transformed into:

dudT

=z√

1 + a11w2z(6 − a11w2z + b11wz2)

6√

1 + b11wz2= U (u, v)

dv

dT= −

w√

1 + b11wz26 + a11w2z − b11wz2

6√

1 + a11w2z= −V (u, v).

(41)

For convenience in computation, we still regard the function U, V in (41) as functions in z, w. Similar to (32), forsystem (41), we have:

dθ

dT=

12i

(1u

dudT

+1v

dv

dT

)=

12i

(Uu

−Vv

)= −i, (42)

i.e., dθdt = i dθ

dT = 1. Hence, system (39) has a complex isochronous centre at the origin.Synthesizing all cases in above subsections, and considering the centre conditions, we have the following two main

conclusions:

Theorem 4.4. System (8) (resp. (6)) has a complex isochronous centre (resp. an isochronous centre) at the origin ifand only if one of the following three conditions holds:

m = a02 = b02 = 0, a11 = −b20, b11 = −b20, a20b20 6= 0; (43)

m = 0, a02 =b2

20a20

, b02 =a2

20b20

, a11 = −43

b20, b11 = −43

b20, a20b20 6= 0; (44)

m = a20 = b20 = a02 = b02 = 0. (45)

Theorem 4.5. Infinity (the equator) of system (3) is an isochronous centre if and only if one of the three conditionsgiven by Theorem 4.4 holds.

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Conditions of infinity to be an isochronous centre for a rational differential system

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