Local Bifurcation of Critical Periods · the solutions of equation system ... The following...

MM Research Preprints, 39–52No. 15, April 1997. Beijing 39

Local Bifurcation of Critical Periods

in Reversible Cubic Systems1)

Xiaorong Hou Zhenbing Zeng Weinian ZhangLaboratory for Automated Reasoning and Programming

CICA, Academia Sinica610041 Chengdu, P. R. China

Abstract. In this paper we give an inductive algorithm for computing the periodcoefficient polynomials of reversible cubic differential systems, and find the structure ofthe solutions of equation system corresponding to isochrone and weak center of eachfinite order. We show that there are pertubations with k local critical periods bifurcatedfrom a weak center of order k.

1. Weak Center and Critical Period Perturbation

In this section, we give a brief introduction to the general theory of weak centers of planaranalytic vector fields, i.e., the differential systems in the following form:

V(x, y, λ) :

{x = −y + φ(x, y, λ)y = x+ ψ(x, y, λ)

with parameter λ ∈ Rn. We call the origin a nondegenerate center of V(x, y, λ) if the vectorfield does not have a double eigenvalue zero at the origin. Let P (r, λ) denote the minimumperiod of the closed orbit through a nonzero point (r, 0) of a sufficiently small open intervalJ = (−α, α) of the x-axis. The following properties of P (r, λ) have been proved by Chiconeand Jacobs in [?].

Lemma 1 (Period Coefficient Lemma) If P (0, λ) = 2π, for λ ∈ Rn, then for any givenλ∗ ∈ Rn,

P (r, λ) = 2π +∞∑

k=2

pk(λ)rk, (1)

which is analytic for |r| and |λ − λ∗| sufficiently small. Moreover, pk ∈ R[λ1, ..., λn], thenoetherian ring of polynomials; for k ≥ 1, p2k+1 ∈ (p2, p4, ..., p2k), the ideal generated byp2i, i = 1, ..., k over R[λ1, ..., λn]; the first k > 1 such that pk(λ) 6= 0 is even.

We will call polynomials p2, p4, · · · , p2k · · · , in (??) the period coefficient polynomials.For a given parameter λ∗ ∈ Rn, we call the origin a weak center of finite order k if

P (0, λ∗) = 2π, P ′(0, λ∗) = ... = P (2k+1)(0, λ∗) = 0 and P (2k+2)(0, λ∗) 6= 0,

1)This work was supported in part by the Chinese National Science Foundation.

40 X. Hou, Z. Zeng, W. Zhang

where the derivatives indicated are taken with respect to the first argument of P (r, λ). Wecall the origin an isochronous center if P (k)(0, λ∗) = 0 for eack k > 0. Local critical periodis a period corresponding to a critical point of the period function which bifurcates from aweak center. Strictly speaking, we say that k local critical periods bifurcate from the weakcenter corresponding to the parameter λ∗ if

1. for every α > 0, sufficiently small, there exists a neighborhood W of λ∗ such that, forany λ ∈W , P (r, λ) has at most k critical points in U = (0, α); and moreover,

2. any neighborhood of λ∗ contains a point λ1 such that P (r, λ1) has exactly k criticalpoints in U = (0, α).

The following theorem[?] can be used to analysis critical periods bifurcated from weakcenters of finite order or isochrones through corresponding period coefficient polynomials.For a family pα ∈ R[λ1, · · · , λn], α ∈ S of polynomials let V(pα, α ∈ S) denote the algebraicset {λ ∈ Rn|pα(λ) = 0 for all α ∈ S}.

Definition 1 For λ∗ ∈ V(p2, p4, ..., p2k) \V(p2k+2) the period coefficients p2, p4, ..., p2k of Pare said to be independent with respect to p2k+2 at λ∗ when the following conditions aresatisfied:

1. Every neighborhood of λ∗ contains a point λ such that p2k(λ) · p2k+2(λ) < 0;2. For any j = 1, ..., (k − 1), if λ ∈ V(p2, p4, ..., p2j) \ V(p2j+2) then every neighborhood

of λ contains a point σ ∈ V(p2, p4, ..., p2j−2) such that p2j(σ) · p2j+2(λ) < 0.

Theorem 1 (Finite Order Bifurcation Theorem From weak centers of finite order k at pa-rameter λ∗ no more than k local critical periods bifurcate. Moreover, there are perturbationswith exactly j critical periods for any j ≤ k, if the coefficients p2, p4, ..., p2k of F are inde-pendent with respect to p2k+2 at λ∗.

The cubic differential system is one of the important object in study of the local bi-furcation of critical periods in the vector fields. Rousseau and Toni ([?]) have studied thesystems with homogeneous nonlinearities of the third degree. The weak center of reversiblecubic system has been investigated in [?]. In this paper, we will give a symbolic solutionto the system of period ceofficient polynomials corresponding to weak centers of each orderin Section 2, in Section 3 we will discuss the problem of critical periods bifurcated fromweak centers in certain reversible cubic system. In the rest part of this section we give analgorithm for computing the period coefficients pk of reversible cubic system.

According to [?], a planar vector field is said to be reversible if it is symmetric with respectto a line. Up to coordinate transformation, any reversible cubic system can be written as{

x = −y + a1x2 + a2y

2 + a3x2y + a4y

3

y = x+ b1xy + b2x3 + b3xy

2 (2)

with parameter λ = (a1, a2, a3, a4, b1, b2, b3) ∈ R7. This system is symmetry with respect tothe y-axis. The symmetry ensures that Eq. (??) has the origin as a center, cf. [?].

Taking polar coordinatex = r cos θ, y = r sin θ, (3)

we have

r = x cos θ + y sin θ = r2G2(θ) + r3G3(θ), (4)θ = (y cos θ − x sin θ)/r = 1 + rH1(θ) + r2H2(θ), (5)

Local Bifurcation of Critical Periods 41

where

G2(θ) = a1 cos3 θ + (a2 + b1) sin2 θ cos θ,G3(θ) = (a3 + b2) cos3 θ sin θ + (a4 + b3) cos θ sin3 θ,

H1(θ) = (b1 − a1) cos2 θ sin θ − a2 sin3 θ,

H2(θ) = (b3 − a3) cos2 θ sin2 θ + b2 cos4 θ − a4 sin4 θ.

Thusdr

dθ=

r2G2(θ) + r3G3(θ)1 + rH1(θ) + r2H2(θ)

. (6)

Lemma 2 . The vector field defined by Eq.( ??) is analytic and

dr

dθ= r2G2 +

∞∑k=3

rk(G2Ak−2 +G3Ak−3) (7)

in a sufficiently small neighborhood of r = 0, where

A0 = 1, A1 = −H1, Ak = −H2Ak−2 −H1Ak−1, ∀k ≥ 2. (8)

Proof. For all θ the functions H1(θ) and H2(θ) are uniformly bounded. Thus forsufficiently small r,

11 + rH1 + r2H2

=∞∑

k=0

rkAk (9)

is analytic. By comparison of coefficients we obtain a recursive relation ( ??), which deter-mines all Ak. The remainder of this proof is simple. 2

Consider the solution of ( ??) with r(0, λ) = r0 > 0 in the form

r(θ, λ) =∞∑

k=1

uk(θ, λ)rk0 . (10)

Then the initial condition implies

u1(0, λ) = 1, uk(0, λ) = 0, ∀k > 1, λ ∈ Rn. (11)

Replacing r in ( ??) with the series ( ??) and comparing coefficients of rk0 , k = 1, 2, ...,

we get the following differential equations

d

dθu1(θ, λ) = 0,

d

dθu2(θ, λ) = G2u

21,

d

dθu3(θ, λ) = (G2A1 +G3)u3

1 + 2u1u2G2,

... ...

(12)


successively. Under the initial conditions ( ??) we can obtain their solutions

u1(θ) = 1,

u2(θ) =∫ θ

0G2(ξ)dξ,

u3(θ) =∫ θ

0(G3 +G2(2u2 −H1))dξ,

... ...

(13)

one by one. That is what we often do for weak focuses.Finally, we compute the period P (r0, λ) of the closed orbit C(r0) through (r0, 0). From

( ??) and ( ??) we have

P (r0, λ) =∫

C(r0)dt =

∫ 2π

0

11 + rH1 + r2H2

dθ

(14)

=∫ 2π

0(1 +

∞∑k=1

rkAk) dθ

(15)

= 2π +∫ 2π

0

∞∑k=1

rkAk dθ, (16)

meanwhile, from ( ??) we obtain the following power series expansion

∞∑k=1

rkAk = A1u1r0 + (A1u2 +A2u21)r

20 + (A1u3 + 2A2u1u2 +A3u

31)r

30

+ (A1u4 +A2(u22 + 2u1u3) + 3A3u

21u2 +A4u

41)r

40 + ... (17)

Therefore

P (r0, λ) = 2π +∞∑

k=1

pk(λ)rk0 , (18)

where

p1(λ) =∫ 2π

0A1u1 dθ = −

∫ 2π

0H1(θ) dθ = 0,

p2(λ) =∫ 2π

0(A1u2 +A2u

21) dθ,

p3(λ) =∫ 2π

0(A1u3 + 2A2u1u2 +A3u

31) dθ,

p4(λ) =∫ 2π

0(A1u4 +A2(u2

2 + 2u1u3) + 3A3u21u2 +A4u

41) dθ,

... ...

(19)


and Ak, uk, k = 1, 2, ... are determined by ( ??) and ( ??). This gives an inductive algorithmfor computing the period coefficient polynomials. A Maple program for this algorithm isgiven in [?]. We show partial result as follows:

p2 =π

12(−5a1b1 + 4a1

2 − 3b3 + 3a3 + b1 2 − a2b1 + 10a1a2 − 9b2 + 9a4 + 10a22);

p4 =π

1152(−468a1b2a2 − 2a2b1

3 + 126a1a2b12 + b1

4 + 18a2b2 b1 + 700a23b1

−828a12a2b1 − 712b1a1

3 + 369a21b

21 − 58a1b

31 + 198a4a1b1 − · · ·),

p6 =π

1244160(−204930a1b2b3b1 + 914706a1b2a3b1 + 1054296a1

2b2b3

−5238a1a2b3b12 + 576828a1

2a2b3b1 + 13014a3a2b3b1 + · · ·),

The number of terms p2k is 10, 49, 168, 462, 1092, 2310, respectively, for k = 1, 2, · · · , 6.Note in particular,

degree(p2, b3) = 1, coeff(p2, b3) = −π4,

thus let

q2 = primpart(p2)= 4 a1

2 + 3 a3 + b12 − 5 a1 b1 − 3 b3 − a2 b1 − 9 b2 + 10 a1 a2 + 9 a4 + 10 a2

2,

q4 = primpart(prem(p4, p2, b− 3)= 130 a2

3a1 + 51 a12b2 + 3 a2

2b2 − 78 a13a2 + 39 a1

3b1 + 3 a12a3 + 12 a1

2a4

−36 a12a2

2 + 33 a22a3 + 282 a2

2a4 + 42 a23b1 − 12 a1

2b12 + 3 a2

2b12

+18 a3 a4 − 3 b12b2 + a1 b13 + 3 a4 b1

2 + 18 a3 b2 + 81 a42 − 28 a1

4

+120 a24 + 27 b22 + 246 a1 a4 a2 + 48 a1

2b1 a2 + 36 a1 a3 a2

−6 a1 b12a2 − 3 a3 b1 a2 + 69 a4 b1 a2 + 54 a2 b2 a1

+51 a22b1 a1 − 3 a3 b1 a1 + 39 a4 b1 a1 + 6 a1 b2 b1,

where primpart(f),prem(f, g, x) are maple functions stand for primitive part and pseudo-remainder for polynomials, we have proved that the origin is a weak center of order one ofthe reversible cubic system (??) if and only if q2 = 0, q4 6= 0. Generally, let

q2k = primpart(prem(p2k, p2, b3), k = 3, 4, · · · ,

then for a parameter λ ∈ R7 the origin is a weak center of order k(k ≥ 1) if and only if

q2(λ) = · · · = q2k(λ) = 0, q2k+2(λ) 6= 0,

according to Lemma ??.

2. Solution of Period Coefficient Polynomials

In this section we analyze the isochrone and weak centers in the following class of re-versible cubic systems

Sλ :

{x = −y − ax2 + ay2 + a3x

2y + a4y3

y = x− 2axy + b2x3 + b3xy

2,(20)


where λ = (a, a3, a4, b2, b3) ∈ R5 denotes the bifurcation parameters. The period coefficients,reduced by modulo p2, of system Sλ are given as follows:

p2 = −b3 + a3 + 3a4 − 3b2,p4 = 3b22 + 2a3b2 + 2a3a4 + 9a4

2,

p6 = 45b23 + 44a2b22 + 204a2a4

2 − 50a32a4 − 325a4

2a3 − 15a3b22 − 30a3

2b2

+165a42b2 + 40a2a3b2 − 8a2a4b2 + 40a2a3a4 + 20a3a4b2 − 450a4

3,

p8 = −1200a2a3b22 + 1575b24 − 3360a2a3

2b2 + 1920a4b2a3 + 60480a3a43 + 59535a4

4

+840a4b22a3 − 1480a3

2a4b2 + 3690a42b2

2 − 37260a43b2 − 21120a3a4

2b2

−600a3b23 + 1080a3

3b2 + 520a32b2

2 + 2520a33a4 + 540a4b2

3 + 2304a2b23

+1920a4b22 − 43632a2a4

3 + 21840a32a4

2 + 64a2a3a4b2 − 768a4b2a4 + 18912a2a42b2

−4464a2a4b22 − 4320a2a3

2a4 + 1920a4a3a4 − 30864a2a3a42 + 10368a4a4

2,

p10 = −15410544a2b2a42a3 − 218880a6b2a4 − 1310400a4a3

2a4 + 34912224a2a3a43

−9718560a4a42a3 − 23073336a2b2a4

3 + 32446575b2a44 + 28775250b2a4

3a3

+352800a33b2a4 − 372750a3

2b22a4 − 122850a3b2

3a4 − 13906080a4a43

+29089908a2a44 + 2246400a6a4

2 + 6796800a4b2a42 − 24277050a32a4

3

+403200a6a4a3 + 13159080a2a32a4

2 + 1436400a2a33a4 − 61425a3b2

4

+791040a4b23 − 6993000b22a4

3 − 35724a2b24 + 103950b24a4

−103950b23a42 − 176400b2a3

4 + 7915950a32b2a4

2 − 47661075a3a44

−36316350a45 − 5821200a3

3a42 + 68250a3

2b23 − 3307500a3b2

2a42

−529200a34a4 − 109200b22a3

3 + 403200a6b2a3 + 932400a2a33b2

−24000a4b2a3a4 − 91680a2a32b2a4 + 391680a6b2

2 + 3695544a2b22a4

2

−1947360a4b22a4 − 1108800a4a3

2b2 − 1056896a2a3b23 + 1474416a2b2

2a4a3

−1477992a2a4b23 − 232200a2a3

2b22 − 270240a4b2

2a3 + 278775b25

· · ·

Now the isochronous centers of system Sλ can be identified by the theorem below:

Theorem 2 Sλ(λ ∈ R5, a 6= 0) is an isochrone system if and only if λ lies in one of thefollowing two sets:

SI = {λ ∈ R5|a3 = b3 = −3a4 = −3b2},SII = {λ ∈ R5|a3 = b3, a4 = b2 = 0},

Proof: For necessary condition, we need the following properties: for any polynomialsf1, f2, f3, · · · ∈ R[x],

V(f1, f2, f3, · · ·) = V(f1,prem(f2, f1, x),prem(f3, f1, x), · · ·)⋃V(f1, lcoeff(f1, x), f2, f3, · · ·);

V(f1, f2, f3, · · ·) ⊂ V(f1, resultant(f2, f1, x), resultant(f3, f1, x), · · ·)⋃V(f1, lcoeff(f1, x), f2, f3, · · ·).


Let

I4 = lcoeff(p4, a3) = a4 + b2, (21)G6 = prem(p6, p4, a3), G8 = prem(p8, p4, a3), G10 = prem(p10, p4, a3), (22)J6 = primpart(gcd(G6, G8, G10)) = a4 − b2, (23)g6 = G6/J6, G8 = G8/J6, G10 = G10/J6 (24)I6 = lcoeff(g6, b2) = 15a4 + 8a2, (25)H8 = resultant(g8, g6, b2), H10 = resultant(g10, g6, b2), (26)J8 = primpart(gcd(H8,H10)) = −a15

4 a2, (27)

h8 = H8/J8, h10 = H10/J8, (28)Q10 = primpart(resultant(h8, h10, a4) = −a154. (29)

Then we have,

V(p2, p4, p6, p8, p10) = V(p2, p4, G6, G8, G10)⋃

V(p2, p4, I4, p6, p8, p10)

= V(p2, p4, g6, g8, g10)⋃

V(p2, p4, J6)⋃

V(p2, p4, I4, p6, p8, p10)

⊂ V(p2, p4, g6,H8,H10)⋃

V(p2, p4, p6, I6, g8, g10)⋃V(p2, p4, J6)

⋃V(p2, p4, I4, p6, p8, p10)

= V(p2, p4, g6, h8, h10)⋃

V(p2, p4, g6, J8)⋃

V(p2, p4, p6, I6, g8, g10)⋃V(p2, p4, J6)

⋃V(p2, p4, I4, p6, p8, p10).

It is easy to check

V(p2, p4, I4) = V(−b3 + a3 − 3b2 + 3a4, 9a24 + 3b2, a4 + b2)

= V(a4, b2, a3 − b3) ⊂ SI ,

V(p2, p4, J6) = V(−b3 + a3 − 3b2 + 3a4, a4(a3 + 3a4), a4 − b2)= V(a4, b2, a3 − b3)

⋃V(a3 − b3, a3 + 3a4, a4 − b2) ⊂ SI

⋃SII ;

and if a 6= 0,

V(p2, p4, p6, I6) = V(p2, p4,primpart(prem(p6, I6, a4)), I6)= V(p2, p4,−a4(2025b22 − 240a2b2 + 64a4), I6)= V(p2, p4, 2025b22 − 240a2b2 + 64a4, I6) = ∅,

V(p2, p4, g6, J8) = V(p2, p4, b2, a4) = V(a4, b2, a3 − b3) ⊂ SII ,

V(p2, p4, g6, h8, h10) = V(p2, p4, g6, h8, a) = ∅.

Together we have proved

V(p2, p4, · · · , p10) \V(a) ⊂ SI

⋃SII . (30)

LetSIII = R5 \ SI \ SII .


We say that (Sλ)λ∈R5 has a center of type I (respectively II, III) if the system ( ??) isnonlinear and λ ∈ SI (respectively SII , SIII).

We are in the position to prove the other side of this theorem. For a center of type I thecorresponding system ( ??) has the following form{

x = −y − ax2 + ay2 − 3bx2y + by3

y = x− 2axy + bx3 − 3bxy2.(31)

This system satisfies Cauchy-Riemann conditions, i.e., it can also be written into

dt =dz

iz − az2 + ibz3

= dz

(−iz

+O(z)),

where z = x+ i y. By the residue theorem the period is constant. Thus a system which hasa center of type I is an isochrone system.

For λ∗ ∈ SII , the corresponding system can be written as follows:{x = −y − ax2 + ay2 + bx2yy = x− 2axy + bx3 + bxy2.

(32)

The polar coordinate form of this system is{r = −a cos(θ) + b sin(θ) cos(θ)r2

θ = 1− ar sin(θ).

Therefore, we haved r

d θ=r2 cos(θ)(−a+ br sin(θ))

1− ar sin(θ). (33)

Letr =

ρ

a sin(θ), b = a2β.

Then the equation (??) can be transformed into the following form:

sin(θ)d ρ =

(ρ2(1− β ρ)

1− ρ+ ρ

)d sin(θ),

which follows immediatelyρ√

β ρ2 − 2ρ+ 1= k sin(θ), (34)

where k is a constant.To prove the original system is isochrone, we need to compute the integral∮

Cd t =

∫ 2π

0

1θd θ =

∫ 2π

0

11− ar sin(θ)

d θ =∫ 2π

0

11− ρ

d θ.


Let1

1− ρ= 1 + τ, i.e. ρ =

τ

1 + τ.

Substituting it into (??), we obtain

τ = ± sin(θ)

√k

1 + (1− β)k sin2(θ),

and therefore, ∫ 2π

0

11− ρ

d θ =∫ 2π

0(1 + τ)d θ

= 2π +∫ 2π

0τ d θ = 2π

since τ(π + θ) = τ(θ). This proves that the system Slambda is isochrone if λ ∈ SII .The proof of Theorem ?? is completed.Note that relation (??) also implies that any weak center of type III has order at most

four. LetΛk := V(p2, · · · , p2k) \V(p2k+2), k = 1, 2, 3, 4.

Then the origin is a weak center of order k if and only if λ ∈ Λk for each k = 1, 2, 3, 4. It isclear that Λ1 6= ∅, and all solutions form a subspace of co-dimension 1 in R5. In what followswe are devoted to find all explicit solutions of period coefficient polynomials correspondingto each order of weak centers. Our purpose is to express Λk as the set of zeros of a descendingchain of polynomials. Let

q2 = primpart(p2)= −b3 + a3 − 3 b2 + 3 a4 ,

q4 = primpart(p4)= 2 (b2 + a4 )a3 + 9 a4

2 + 3 b22,

q6 = primpart(g6)

=(15 a4 + 8 a2

)b2

3 −(150 a4

2 − 28 a4 a2)b2

2

+(15 a4

3 + 32 a42a2)b2 + 12 a2a4

3,

q8 = primpart(h8)= −132890625 a4

7 − 9071831250 a2a46 + 15239407500 a4a4

5

+2404621800 a6a44 + 724745440 a8a4

3 − 17954688 a10a42

−1544704 a12a4 + 184320 a14,

q10 = a,

be those polynomials defined in (??). Since

J6 q6 = primpart(prem(p6, p4, a3)),


there exist a constant c and a polynomial A(λ) such that

c J6 q6 = Is4p6 +A(λ) p4, s = degree(p6, a3)− degree(p4, a3) + 1.

Thus, restricted to λ ∈ SIII , i.e.,

a 6= 0, I4(λ) = a4 + b2 6= 0, J6(λ) = a4 − b2 6= 0,

we haveΛ2 = V (p2, p4) \V(p6) = V(q2, q4) \V(q6),

andV (p2, p4, p6) = V(q2, q4, q6), V(p2, p4, p6, p8) ⊂ V(q2, q4, q6, q8).

Using computer algebra software we can factorize polynomial q6(a, a4, b2) over the alge-braic number field defined by q8(a, a4):

q6 = c I6 q61 q62, where c is a constant andq61 = 3746202104402426801664b2a10 + 34356956307359754240a12

+9868333828993569248896a4a10 − 126247317157437110913600a4

2a8

−385618752578317761603400a43a6 − 3152714195511025778467500a4

4a4

+1836456407049386749556250a45a2 + 26915872598239575234375a4

6,

q62 = 15812719082682643529823744b22a10 − 171779299033382856944640b2a12

+13964703867236240994311808b2a4a10 + 134722493678814861034696320a4

2b2a8

+2269046043892102760675124600a43b2a

6 + 11755986639288601072792612500a44b2a

4

−7054770266034922117289193750a45b2a

2 − 103322646851786605091953125a46b2

−35705585130063460392960a14 + 212432326142658183068672a4a12

+7874351200222330330176768a42a10 − 79491326764530658001004800a4

3a8

+659963871107286922200912000a44a6 − 334146449149816562899200000a4

5a4

−4915493478588710193750000a46a2.

Thus we obtain two descending chains of polynomials

DC1 :

q2 = q2(a, a4, b2, a3, b3)q4 = q4(a, a4, b2, a3)q61 = q61(a, a4, b2)q8 = q8(a, a4)

and DC2

q2 = q2(a, a4, b2, a3, b3)q4 = q4(a, a4, b2, a3)q62 = q62(a, a4, b2)q8 = q8(a, a4)

It can be verified that the successive resultant

resultant({q2, q4, q62, q8}, p8, {a4, b2, a3, b3})= resultant({q4, q62, q8}, resultant(q2, p8, b3), {a4, b2, a3})= · · ·= a616,


up to an integer factor, which implies

V(q2, q4, q62, q8, p8) \V(a) = ∅;

Meanwhile, the successive pseudo-remainder

prem({q2, q4, q62, q8}, p8, {a4, b2, a3, b3})= prem({q4, q61, q8},prem(q2, p8, b3), {a4, b2, a3})= · · ·= 0

which impliesV(q2, q4, q61, q8, p8) ⊂ V(q2, q4, q61, q8, p8);

Together with the following fact

V (q2, q4, q6, I6) = V (p2, p4, p6, I6) = ∅

we have

V(p2, p4, p6, p8) = V(q2, q4, q6, q8, p8)= V(q2, q4, q6, I6, q8, p8)

⋃V(q2, q4, q61, q8, p8)

⋃V(q2, q4, q62, q8, p8)

= V(q2, q4, q61, q8).

Finally, we have proved the following theorem.

Theorem 3 For k = 1, 2, 3, 4, Λk ⊂ R5 is an algebraic surface of co-dimension k, definedby

Λ1 = V(q2) \V(q4),Λ2 = V(q2, q4) \V(q6),Λ3 = V(q2, q4, q6) \V(q61, q8),Λ4 = V(q2, q4, q61, q8),

respectively.

3. Bifurcation of Critical Periods

In this section, we discuss the bifurcation behavior of critical periods in the system (??)with λ ∈ Λk for each k = 1, 2, 3, 4. We prove the following result.

Theorem 4 For k = 1, 2, 3, 4, any λ ∈ Λk and each j = 1, · · · , k, there exist perturbationwith exactly j critical periods.

Proof: For brief we only give the proof for Λ2 and Λ4.Part 1: For Λ2. From the construction of q2, q4, q6 it can be seen that, up to a constant,

q2 = p2, (35)q4 = p4 +A21(λ) p2, (36)

J6q6 = Is4p6 +A32(λ) p4 +A31(λ) p2, (37)


for some polynomials Aji(λ) and positive integer s. For any parameter λ∗ ∈ Λ2, we mayassume

λ∗ = (a, a4, b2, a3, b3) ∈ V(p2, p4) \V(p6) = V(q2, q4) \V(q6),

It is clear thatI4(λ∗) 6= 0, J6(λ∗) 6= 0, q6(λ∗) 6= 0.

Let λ = (a, a4, b3, a3 + ε, b3 + ε). Then

I4(λ) = I4(λ∗), J6(λ) = J6(λ∗), q6(λ) = q6(λ∗),

and

p2(λ) = 0,p4(λ) = q4(λ) = 2I4(λ)ε,

p6(λ) =J6(λ)I4(λ)s

q6(λ)− A32(λ)I4(λ)s

p4(λ)

=J6(λ∗)I4(λ∗)s

q6(λ∗)− 2A32(λ)I4(λ∗)s−1

ε,

hence,

p4(λ)p6(λ) = 2J6(λ∗)I4(λ∗)s−1

p6(λ∗)ε− o(ε)

which means that in any neighborhood of λ∗ there exists λ such that p4(λ)p6(λ) < 0. Thisproves the first condition of the independence.

To verify the second condition, assume λ = (a, a4, b2, a3, b3) ∈ V(p2) \ V(p4). Let σ =(a, a4, b2, a3, b3 + ε), then

p2(σ) p4(λ) = −p4(λ)ε.

Thus we can choose ε sufficiently small such that σ contained in any given neighborhood ofλ and p4(σ)p6(λ) < 0. This proves that p2, p4 is independent with respect to p6 at any pointin Λ2.

According to Theorem ?? there are perturbations with exactly j critical periods forj = 1, 2.

Part 2: For Λ4. First we need to check that given any λ∗ ∈ V(p2, , p8) \V(p10), Everyneighborhood contains a point λ such that p8(λ) p10(λ) < 0. In fact, we can prove thefollowing result: Assume λ = (λi) = (a, a4, b2, a3, b3) and

ri = resultant({q2, · · · , q8},∂p8

∂λi, {a4, b2, a3, b3}), i = 1, · · · , 5,

Ifa 6= 0, I4(λ) 6= 0, J6(λ) 6= 0, I6(λ) 6= 0, J8(λ) 6= 0,

then

V (p2, · · · , p8,∂p8

∂λ1, · · · , ∂p8

∂λ5) ⊂ V (q2, q4, q6, q8,

∂p8

∂λ1, · · · , ∂p8

∂λ5)

⊂ V(q2, ..., q8, r1, · · · , r5) = ∅.


Thus if λ = λ∗ + (ε1, · · · , ε5), then there exists i(1 ≤ i ≤ 5) such that∂p8

∂λi(λ∗) 6= 0,

p8(λ) = p8(λ∗) +∂p8

∂λ1(λ∗) ε1 + · · ·+ ∂p8

∂λ5(λ∗) ε5 + o(ε),

and p10(λ) 6= 0 for sufficiently small ε, therefore we can choose ε such that λ contained inany given neighborhood of λ∗ such that

p8(λ)p10(λ) =(∂p8

∂λ1(λ∗) ε1 + · · ·+ ∂p8

∂λ5(λ∗) ε5

)p10(λ) + o(ε) < 0.

Then we need to check that for any j = 1, 2, 3 and any λ ∈ V(p2, · · · , p2j)\V(p2j+2), everyneighborhood of λ contains a point σ ∈ V(p2, · · · , p2j−2)\V(p2j) such that p2j−2(σ) p2j(λ) <0. For j = 1, 2 this can be done in a similar way as in Part 1. For j = 3, it is easy to prove

V (q2, q4, q6,∂q6∂a

,∂q6∂a4

,∂q6∂b2

) ⊂ V (q6,∂q6∂a

,∂q6∂a4

,∂q6∂b2

) = ∅,

provided λ ∈ SIII . Given any

λ = (a, a4, b3, a2, b3) ∈ V(p2, p4, p6) \V(p8) = V(q2, q4, q6) \V(p8),

we may assumeI4(λ) 6= 0, J6(λ) 6= 0,

and at least one of∂q6∂a

(λ),∂q6∂a4

(λ),∂q6∂b2

(λ) is not zero.

Let σ = (a+ ε1, a4 + ε2, b3 + ε3, a′2, b

′3) with ε1, ε2, ε3 sufficiently small such that

I4(σ) 6= 0, J6(σ) 6= 0

and

q4(σ) = q4(a+ ε1, a4 + ε2, b3 + ε3, a′2) = 0,

q2(σ) = q2(a+ ε1, a4 + ε2, b3 + ε3, a′2, b

′3) = 0,

Then by (??) we obtain

p2(σ) = 0,p4(σ) = 0,

p6(σ) =J6(σ)I4(σ)s

q6(σ),

and therefore,

p6(σ) p8(λ) =J6(σ)I4(σ)s

q6(σ)p8(λ)

= q6(a+ ε1, a4 + ε2, b2 + ε3)J6(σ)I4(σ)s

p8(λ)

=(∂q6∂a

(λ) ε1 +∂q6∂a4

(λ) ε2∂q6∂b2

(λ) ε3)J6(σ)I4(σ)s

p8(λ) + o(ε),

Thus we can choose ε sufficiently small such that σ contained in any given neighborhood ofλ and p6(σ)p8(λ) < 0. This has verified the second condition of independence and completedthe proof for Λ4.


References

[1] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Amer. Math.Soc. Transactions, 312(1989): 433-486.

[2] P. Mardesic, C. Rousseau and B. Toni, Linearization of Isochronous Centers, Preprint(November1993), University of Montreal, Canada.

[3] Zhifen Zhang, et al., Qualitative Theory of Differential Equations, Thansl. Math. Monographs,Amer. Math. Soc., 1992.

[4] C. Rousseau and B. Toni, Local bifurcation of critical periods in vector fields with homogeneousnonlinearities of the third degree, Canad. Math. Bull., 36(1993):473-484.

[5] W. Zhang, X. Hou and Z. Zeng, Weak centres for reversible cubic systems, Research ReportIMS-80, December 1996, Institute of Mathematical Sciences Academia Sinica, China.

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