on the number of limit cycles in near-hamiltonian ... - Semantic Scholar

International Journal of Bifurcation and Chaos, Vol. 17, No. 6 (2007) 2033–2047 c World Scientific Publishing Company

ON THE NUMBER OF LIMIT CYCLES IN NEAR-HAMILTONIAN POLYNOMIAL SYSTEMS* MAOAN HAN† Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China [email protected] GUANRONG CHEN Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China CHENGJUN SUN Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China Department of Biology, McGill University, 1205 ave Docteur Penfield Montreal, Quebec, Canada H3A 1B1 Received June 1, 2005; Revised May 18, 2006 In this paper we study a general near-Hamiltonian polynomial system on the plane. We suppose the unperturbed system has a family of periodic orbits surrounding a center point and obtain some sufficient conditions to find the cyclicity of the perturbed system at the center or a periodic orbit. In particular, we prove that for almost all polynomial Hamiltonian systems the perturbed systems with polynomial perturbations of degree n have at most n(n + 1)/2 − 1 limit cycles near a center point. We also obtain some new results for Lienard systems by applying our main theorems. Keywords: Near-Hamiltonian system; cyclicity; limit cycle; bifurcation.

1. Introduction A near-Hamiltonian polynomial system is a polynomial system in the form of x˙ = Hy + εP (x, y, a) y˙ = −Hx + εQ(x, y, a),

†

1 H(x, y) = (x2 + y 2 ) + O(|x, y|3 ) 2

(1)

where H, P and Q are polynomials in (x, y), with P and Q being C 1 in the vector parameter a ∈ Rm , and ε ≥ 0 is a small real parameter. A general assumption on system (1) is that for ε = 0, there is a family of periodic orbits surrounding an elementary center point. Without ∗

loss of generality, assume that the center point of interest is the origin. Assume also that the polynomial H is in the form of (2)

near the origin. Thus, the family of periodic orbits can be represented as Lh : H(x, y) = h,

h ∈ (0, h∗ ),

where h∗ ∈ (0, +∞] satisfies the property that limh→h∗ Lh = Lh∗ is no longer a periodic orbit of system (1) (with ε = 0). It is likely that h∗ < +∞

This work was supported by Shanghai Leading Academic Discipline Project (T0401). Author for correspondence 2033

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M. Han et al.

and Lh∗ is an unbounded curve, or that h∗ = +∞ and Lh∗ is the equator. If Lh∗ is bounded, which implies h∗ < +∞, then it is a homoclinic or heteroclinic loop (sometimes, called a polycycle). Also, it is clear that limh→0 Lh = 0 (the origin). Since Lh surrounds the origin, it intersects the positive x-axis at a point (x0 (h), 0). Thus, H(x0 (h), 0) = h, x0 (h) > 0 for h ∈ (0, h∗ ). For small ε > 0, the positive orbit of system (1), starting at (x0 (h), 0), will intersect the positive x-axis at a point (x1 (h, ε, a), 0) for the first time. In this case, one has H(x1 , 0) − H(x0 , 0) = ε[M (h, a) + O(ε)] = εF (h, ε, a), where

(3)

Lemma 1.1. For the polynomial system (1), one

has the following: (1) The function M is analytic at h = 0. Hence, for small h ≥ 0, M (h, a) =

M (h, a) = Lh

Qdx − P dy.

(4)

The above Abelian integral M is called the firstorder Melnikov function of system (1). The notion of cyclicity is well known. However, since it is the main topic of the present paper, we state it precisely in the following. Definition 1.1. Let h ∈ [0, h∗ ). System (1) is said

to have cyclicity k at Lh if the following conditions are satisfied: (i) For any given N > 0, there exist ε > 0 and a neighborhood U of Lh such that system (1) has at most k limit cycles in U for all (ε, a) satisfying 0 < |ε| < ε0 , |a| ≤ N . (ii) There exists an a0 ∈ Rm such that for arbitrary δ > 0 and neighborhood U of Lh there exists an (ε, a) satisfying 0 < ε < δ and |a − a0 | < δ such that system (1) has k limit cycles in U . Two remarks are in order. Remark 1.1. System (1) is said to have cyclicity at

most k if the first condition in Definition 1.1 holds. Remark 1.2. If h = 0 and Lh = L0 is the origin in

Definition 1.1, then the cyclicity k at L0 is called the Hopf cyclicity of system (1) at the origin; otherwise, it is called the Poincaré cyclicity. As is well known, the function M given in (4) plays an important role in the study of the number of limit cycles and the cyclicity of system (1) for small ε > 0. For example, one can use the expansion of M at h = 0 to estimate the number of limit cycles in a neighborhood of the origin, known as Hopf bifurcation, based on the following lemma obtained in [Han, 2000].

Bi (a)hi+1 .

i=0

(2) If there exists an a0 ∈ Rm , k ≥ 0 such that Bj (a0 ) = 0, j = 0, . . . , k − 1,

Bk (a0 ) = 0,

then system (1) has at most k limit cycles for all small |ε| + |a − a0 |. If, furthermore, rank

∞

∂(B0 , . . . , Bk−1 ) (a0 ) = k, ∂(a1 , . . . , am )

m ≥ k,

then system (1) has Hopf cyclicity k at the origin. (3) If (B0 (a0), . . . , Bk (a0 )) = 0 and rank(∂(B0 , . . . , Bk )/∂(a1 , . . . , am ))(a0 ) = k + 1 for some a0 ∈ Rm , and system (1) has a center near the origin when Bj = 0, j = 0, . . . , k, then system (1) has Hopf cyclicity k at the origin for all small |ε| + |a − a0 |. As to the number of limit cycles near a nontrivial closed orbit Lh0 with h0 ∈ (0, h∗ ), known as Poincaré bifurcation, the following lemma can be easily proven by Rolle’s theorem and the implicit function theorem. Let h0 ∈ (0, h∗ ) and M (h, a) = ∞ i i=0 Ci (a)(h − h0 ) . If Cj (a0 ) = 0, j = 0, . . . , k − Ck (a0 ) = 0 for some a0 ∈ Rm , then system (1)

Lemma 1.2.

1, has Poincaré cyclicity at most k at Lh0 for all small |ε| + |a − a0 |. If, furthermore, rank

∂(C0 , . . . , Ck−1 ) (a0 ) = k, ∂(a1 , . . . , am )

m ≥ k,

then system (1) has Poincaré cyclicity k at Lh0 . If Lh∗ is a homoclinic loop, one can use the expansion of M at h = h∗ to investigate the number of limit cycles near Lh∗ (see, for example [Joyal, 1988; Roussarie, 1986]). If a is used as a constant vector, one can consider the following problem: How many zeros can M (h, a) have in h ∈ (0, h∗ )? This problem was proposed in [Arnold, 1977], which is often referred to as the weak Hilbert’s 16th problem.

On the Number of Limit Cycles in Near-Hamiltonian Polynomial Systems

It follows from Rolle’s theorem that if M (h, a) has at most k zeros in h ∈ (0, h∗ ), then for any given compact set V0 between L0 and Lh∗ , there exists an ε = ε(a, V0 ) > 0 such that for all 0 < ε < ε0 , system (1) has at most k limit cycles in V0 . This conclusion has been widely used to discuss the number of limit cycles for quadratic and cubic systems as well as some polynomial systems of higher degrees (see the survey articles [Li, 2003; Schlomiuk, 1993]). In this paper, we consider the following two problems: (1) What is the cyclicity of system (1) at the origin, or more generally at a nontrivial closed orbit Lh , and generally how does it depend on the degrees of H, P and Q? (2) Is there a relationship between the Hopf cyclicity and the Poincaré cyclicity for system (1)? We state our main results concerning the above two problems in the next section, and present their proofs in Sec. 3.

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Thus, one can write M (h, a) =

k

bj Ij (h),

(7)

j=1

where {I1 , . . . , Ik } ⊂ {Iij |0 ≤ i + j ≤ n − 1} , 2≤k≤

n−1 n(n + 1) = 1. 2 i+j=0

It is clear that if k = n(n + 1)/2 then {b1 , . . . , bk } = {cij |0 ≤ i + j ≤ n − 1} , {I1 , . . . , Ik } = {Iij |0 ≤ i + j ≤ n − 1} .

(8)

Let Jj (h) = Ij (h)/h, j = 1, . . . , k. Then, by (7), one has M (h, a) = h

k

bj Jj (h).

(9)

j=1

2. Main Theoretical Results Let P and Q be polynomials of degree n (n ≥ 2), in the forms of n

P (x, y, a) =

aij xi y j ,

i+j=0 n

Q(x, y, a) =

(5)

Introduce the Wronskian of the functions J1 , . . . , Jk as follows:   ··· Jk J1  J ··· Jk  1   W (h) = det  , · · · · · · · · ·   (k−1)

J1

i j

bij x y ,

···

h ∈ [0, h∗ ).

i+j=0

Px + Qy =



cij xi y j ,

i+j=0

(6)

cij = (i + 1)ai+1,j + (j + 1)bi,j+1 .

M (h, a) = H≤h

(Px + Qy )dxdy =

n−1 i+j=0

where Iij (h) =

xi y j dxdy. H≤h

J1 J1 ···

   W (h) = det    J (k−2)  1 (k) J1

··· ··· ··· ··· ···

Jk Jk ···



   ,  (k−2)  Jk  (k)

Jk

h ∈ [0, h∗ ).

By (4) and Green’s formula, one has

(10)

Then, one has

where aij and bij depend on a ∈ Rm smoothly. It is easy to see that n−1

(k−1)

Jk

cij Iij (h),

(11)

, Jk are said to Recall that the functions J1 , . . . be linearly independent on [0, h∗ ) if ki=1 ci Ji (h) = 0 for ci ∈ R and h ∈ [0, h∗ ) implies ci = 0 for i = 1, . . . , k. We have the following results. Theorem 2.1. Suppose J1 , . . . , Jk are linearly independent on [0, h∗ ). Let |b1 | + · · · + |bk | > 0 for all

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a ∈ Rm . Then,

following system:

(i) W (h) is analytic and has only isolated zeros on [0, h∗ ). (ii) If W (h0 ) = 0, h0 ∈ [0, h∗ ), then system (1) has cyclicity at most k − 1 at Lh0 . (iii) If W (h0 ) = 0, W (h0 ) = 0, h0 ∈ [0, h∗ ), then system (1) has cyclicity at most k at Lh0 . Theorem 2.2. Suppose the following conditions are

satisfied: (a) The vector b = (b1 , . . . , bk ) is linear in a with rank(∂b/∂a) = k and b(a0 ) = 0 for some a0 ∈ Rm . (b) System (1) has a center near the origin when b = 0. (c) There exists an h0 ∈ [0, h∗ ) such that |W (h0 )|+ |W (h0 )| = 0. Then, the following hold: (i) If W (h0 ) = 0, then system (1) has cyclicity k − 1 at Lh0 . (ii) If W (h0 ) = 0, W (h0 ) = 0 and h0 > 0, then system (1) has cyclicity k at Lh0 . (iii) For each h ∈ [0, h∗ ), system (1) has cyclicity at least k − 1 at Lh .

x˙ = Hy (x, y, α) + εP (x, y, a), y˙ = −Hx (x, y, α) + εQ(x, y, a). We have the following result.

Theorem 2.3. If (5) and (12) hold, then there exists

a codimension-one set Σ ⊂ Rq such that for any α ∈ Rq − Σ and h ∈ (0, h∗ ) there exist 0 < h1 < h2 < · · · < hl < h, l ≥ 0, such that (i) For h ∈ [0, h) − {h1 , . . . , hl }, system (13) has cyclicity n(n + 1)/2 − 1 at Lh ; in particular, system (13) has Hopf cyclicity n(n + 1)/2 − 1 at the origin. (ii) For h ∈ {h1 , . . . , hl }, system (13) has cyclicity n(n + 1)/2 at Lh .

In Theorem 2.3, one can take h = h∗ if h∗ < ∞. We note that for ε = 0, system (13) may have families of periodic orbits that are not surrounding the origin, or surrounding the origin and also another singular point. Obviously, the number of these families is finite. Hence, it follows from the proof of Theorem 2.3 that there is a codimension˜ sys˜ ⊂ Rq such that for any α ∈ Rq − Σ one set Σ tem (13) has Poincaré cyclicity n(n + 1)/2 − 1 or n(n+1)/2 on each periodic orbit of the Hamiltonian system

Now, let H have degree p (p ≥ 3). Note that by (2) H is a special case of the following general form:

x˙ = Hy (x, y, α), y˙ = −Hx (x, y, α).

p 1 2 (x + y 2 ) + hij xi y j , 2

As an application, consider the Lienard systems

i+j=3

which contains (p + 1)(p + 2)/2 − 6 terms with degrees larger than 2. Therefore, one can introduce a proper vector parameter α = (α1 , . . . , αq ) ∈ Rq with 2 ≤ q ≤ (p + 1)(p + 2)/2 − 6 and embed H into a family of polynomials H(x, y, α) such that H(x, y) = H(x, y, α0 )

for some α0 ∈ Rq ,

H(x, y, α) = H(x, y, α ) if and only if α = α (12) for some α0 ∈ Rq . In this case, h∗ = h∗ (α) depends on α in general. For example, one can take α = {hij } with q = (p + 1)(p + 2)/2 − 6. Let (5) hold and let a be a vector parameter consisting of all coefficients aij and bij ; namely, let a = (aij , bij ) ∈ R(n+1)(n+2) . Then, consider the

(13)

x˙ = h(y) − ε

2n+1

ai xi ,

y˙ = −g1 (x)

(14)

i=1

and x˙ = h(y) − ε

n

ai x2i+1 ,

i=0

(15)

y˙ = −[g1 (x) + g2 (x)], where h, g1 , g2 are polynomials satisfying h(0) = 0, h (0) > 0, g1 (0) > 0, g2 (0) = 0, g1 (−x) = −g1 (x), g2 (−x) = g2 (x). For ε = 0, the origin is an elementary center of both (14) and (15). Let Lh , h ∈ (0, h∗ ) denote the family of periodic orbits of (14) or (15) surrounding the origin for ε = 0. Then, we have the following result.


Theorem 2.4. The Hopf cyclicity of both (14) and

(15) is n. Furthermore, for any h ∈ (0, h∗ ), there exist 0 < h1 < h2 < · · · < hl < h, l ≥ 0 such that for h ∈ (0, h) − {h1 , . . . , hl } (resp. h ∈ {h1 , . . . , hl }) (14) and (15) have cyclicity n (resp. at least n) at Lh .

is satisfied. As before, if J1 = 0 on [0, h∗ ), then J1 , . . . , Jk are not linearly independent. Let J1 = 0 on [0, h∗ ). Note that for h ∈ [0, h∗ ) and J1 (h) = 0, one has

Ji J1

(j) =

Lemma 3.1. The following statements are equiva-

lent to each other:

j

1

J 1j+1 l=0

3. Proofs of the Main Results Consider system (1). We first discuss some properties of the function W (h).

2037

(j−l)

flj (J1 , J1 , . . . , J 1

j = 1, . . . , k − 1,

(l)

)J i ,

i = 1, . . . , k, (17)

where flj is a homogeneous polynomial of degree j with fjj (J1 ) = J 1j . For example, 1 Ji = 2 (−J1 Ji + J1 Ji ), J1 J1 1 Ji = 3 [(2(J1 )2 − J1 J1 )Ji J1 J1

(i) The function J1 , . . . , Jk are linearly independent on [0, h∗ ). (ii) Any zero, if exists, of the Wronskian W (h) on [0, h∗ ) is isolated. (iii) W (h) ≡ 0 on [0, h∗ ).

− 2J1 J1 Ji + J 12 Ji ]. Proof. (i) → (ii). Let J1 , . . . , Jk be linearly independent on [0, h∗ ). If W (h) has a nonisolated zero on [0, h∗ ), then it follows from its analyticity on [0, h∗ ) that W (h) ≡ 0, h ∈ [0, h∗ ).

(16)

It can be proved that (16) ensures that J1 , . . . , Jk are not linearly independent. We proceed with the proof by induction. First, for k = 2, we have W (h) =

J1 J2

−

J2 J1 .

In general, (17) can be proved by induction in j. Then, by (17), one can solve (j) Ji

=

j l=0

j = 1, . . . , k − 1,

J2 J1

≡0

for h ∈ [0, h∗ ), J1 (h) = 0

which yields J2 /J1 = c or J2 −cJ1 = 0 for J1 (h) = 0. Since both J1 and J2 are analytic on [0, h∗ ), there exists a common constant c such that J2 − cJ1 = 0 on [0, h∗ ). This shows that J1 and J2 are not linearly independent. By induction, suppose the condition holds for k − 1 (k ≥ 2) and let J1 , . . . , Jk be such that (16)

Ji J1

(l) ,

i = 1, . . . , k − 1.

(18)

Substituting (18) into (17) yields

Ji J1

(j) =

j−1 l=0

If J1 (h) ≡ 0 on [0, h∗ ), then J1 and J2 are not linearly independent. Let J1 (h) ≡ 0 on [0, h∗ ). Then, it has only isolated zeros on the interval. It follows from (16) that

(j−l) glj (J1 , J1 , . . . , J 1 )

(j−l) f˜lj (J1 , J1 , . . . , J 1 )

Ji J1

(l)

(j)

+

Ji , j = 1, . . . , k − 1, i = 1, . . . , k. J1 (19)

Obviously,

J1 J1

(j) =

j−1 l=0

(j−l) ) f˜lj (J1 , J1 , . . . , J1 (j)

J + 1 = 0, J1

J1 J1

(l)

j = 1, . . . , k − 1.

(20)

Thus, by (19) and (20) and using a property of

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M. Han et al.

determinant, it follows from (10) that 

J1 J1

    J1   J1 k W (h) = J1 det    ···    (k−1) J 1



J1 J1 J1 J1 J1

J2 J1

···

Jk J1

J2 J1

···

Jk J1

···

···

···

···

Jk J1



(k−1)

(k−1)

J2 J1

      k = J1 det    ···    (k−1)  J1

J1 J2 J1 J2 J1

      k = J1 det    ···    (k−1)  J2 J1

J2 J1 J2 J1

J3 J1

    Jk   J1    ···   (k−1)   Jk J1  Jk  J1    Jk   J1    ···   (k−1)   Jk

···

J2 (k−1) J1 J3 J1 J3 J1

··· ··· ··· ···

(k−1)



Jk J1

···

···

            

···

···



···

J1

≡ 0. By the inductive assumption, the function (J2 / J1 ) , . . . , (Jk /J1 ) are not linearly independent on [0, h∗ ) with J1 (h) = 0. Thus, there exist nonzero constants c2 , . . . , ck , such that k Jj cj ≡ 0, for h ∈ [0, h∗ ) and J1 (h) = 0. J1

(ii) ⇒ (iii). This is obvious. (iii) ⇒ (i). Assume that (iii) is satisfied. Then, W (h0 ) = 0 for some h0 ∈ [0, h∗ ). If k

J1 (h) = 0.

j=2

Hence, k

cj Jj ≡ 0,

h ∈ [0, h∗ ).

j=1

This means that J1 , . . . , Jk are not linearly independent on [0, h∗ ). Therefore, the claim follows, which contradicts (i).

h ∈ [0, h∗ ),

j=1

j=2

It then follows that k Jj cj ≡ −c1 , for h ∈ [0, h∗ ) and J1

cj Jj (h) ≡ 0,

then k j=1

(l)

cj Jj (h) ≡ 0,

It then follows  J1  J 1    ··· (k−1)

J1

that J2 J2 ··· (k−1)

J2

h ∈ [0, h∗ ),

··· ··· ··· ···

l = 1, . . . , k − 1.

c  1  c2     ≡0  .    ..   (k−1) Jk ck Jk Jk ···

on [0, h∗ ).


In particular, the above equality holds at h = h0 . Thus, W (h0 ) = 0 implies cj = 0 (j = 1, . . . , k). Hence, J1 , . . . , Jk are linearly independent. This completes the proof of Lemma 3.1. We now prove Theorem 2.1. The first conclusion follows directly from Lemma 3.1. In order to prove the second one, suppose h0 ∈ [0, h∗ ) such that W (h0 ) = 0. It is to show that system (1.1) has cyclicity at most k − 1 at Lh0 . If it is not the case, then there exist N > 0, εn → 0, an ∈ Rm with |an | ≤ N , such that for (ε, a) = (εn , an ), (n) (n) system (1) has k limit cycles Lj with Lj → Lh0 as n → ∞, j = 1, . . . , k. Without loss of generality, assume that an → a∗ , b(an ) → b(a∗ ) = b0 = (b10 , · · · , bk0 ) as n → ∞. One has the following expansion near h = h0 : 1 M (h, a) = b1 + b2 (h − h0 ) + · · · h + bk (h − h0 )k−1 + · · · ,

(21)

where, by (9), (j)

1 M (h, a) bj+1 = j! h h=h0 (j)

(j)

= [b1 J 1 (h0 ) + · · · + bk J k (h0 )]/j! , j = 0, . . . , k − 1.

(22)

W (h0 ) ∂(b1 , . . . , bk ) = = 0. ∂(b1 , . . . , bk ) 1!2! · · · (k − 1)!

(23)

It then follows that det

By assumption given in Theorem 2.1, one has |b0 | > 0. Thus, the above gives (b1 , . . . , bk )|a=a∗ = (b10 , . . . , bk0 ) = 0. One may assume that bj0 = 0, j = 1, . . . , l − 1,

bl0 = 0,

1 ≤ l ≤ k.

For the third one, let W (h0 ) = 0, W (h0 ) = 0. If the conclusion is not true, then, as before, there exist (εn , an ) → (0, a∗ ) such that for (ε, a) = (εn , an ), system (1) has k + 1 limit cycles approaching Lh0 as n → ∞. In this case, one still has (21) with det

∂(b1 , . . . , bk−1 , bk+1 ) ∂(b1 , . . . , bk ) =

W (h0 ) = 0, 1!2! · · · (k − 2)!k!

(b1 , . . . , bk−1 , bk+1 )|a=a∗ = (b10 , . . . , bk−1,0 , bk+1,0 ) = 0. Hence, there exists 1 ≤ l ≤ k + 1, l = k such that (24) holds. This leads to a contradiction in the same way as above. This completes the proof of Theorem 2.1. Next, we prove Theorem 2.2. First, assume that W (h0 ) = 0, h0 ∈ [0, h∗ ), to prove the following two points: (a) there are at most k − 1 limit cycles near Lh0 for small |ε| and bounded a ∈ Rm ; (b) there appear k − 1 limit cycles in any neighborhood of Lh0 for some arbitrarily small || + |a − a0 | with a0 ∈ Rm satisfying b(a0 ) = 0. The conclusions will be shown by contradiction. Suppose that there exist (εn , an ) → (0, a∗ ) such that for (ε, a) = (εn , an ), system (1) has k limit cycles, which approach Lh0 as n → ∞. Let b0 = b(a∗ ). Then, from the proof of Theorem 2.1, we have b0 = 0; otherwise, a contradiction arrives. Hence, one may take a∗ = a0 . Consider the equation b = b(a),

(24)

Then, by Lemma 1.1 (if h0 = 0) or by Lemma 1.2 (if h0 > 0), there exist an ε0 > 0 and a neighborhood U of Lh0 such that for 0 < |ε| < ε0 , |a − a0 | < ε0 , system (1) has at most l−1 limit cycles in U . This contradicts the fact that system (1) has k limit cycles approaching Lh0 for (ε, a) = (εn , an ). Hence, the second conclusion is proved.

(25)

and

Then, by (21), one has 1 M (h, a∗ ) = bl0 (h − h0 )l−1 + O(|h − h0 |l ), h bl0 = 0.

2039

(26)

where b is used as a new vector variable. By assumption, one has rank(∂b/∂a) = k. Without loss of generality, assume that det

∂(b1 , . . . , bk ) = 0. ∂(a1 , . . . , ak )

The linear equation (26) has a unique solution in the form of ˜j (b, ak+1 , . . . , am ), aj = a

j = 1, . . . , k,

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F has at most k − 1 zeros near h = h0 for small |ε|+|a−a0 |. This means that system (1) has at most k − 1 limit cycles near Lh0 for small |ε| + |a − a0 |, which is a contradiction as before. Furthermore, by (27), if 0 < |˜bj | |˜bj+1 | |˜bk |,

with a1 (0, ak+1,0 , . . . , am,0 ), . . . , a0 = (˜ a ˜k (0, ak+1,0 , . . . , am,0 ), ak+1,0 , . . . , am,0 ), b(a∗ ) = 0. By (23), one has

˜bj ˜bj+1 < 0,

∂(b1 , . . . , bk ) ∂(b1 , . . . , bk ) = det det ∂(a1 , . . . , ak ) ∂(b1 , . . . , bk ) × det

∂(b1 , . . . , bk ) = 0. ∂(a1 , . . . , ak )

Furthermore, since system (1) has a center near the origin when b = 0, by (21) and (23) it follows that the same is true when (b1 , . . . , bk ) = 0. Thus, by Lemma 1.1, if h0 = 0 then system (1) has Hopf cyclicity k − 1 at the origin for small |ε| + |a − a0 |. This is a contradiction to the existence of k limit cycles near Lh0 . On the other hand, since b(a0 ) = 0, the above implies that k − 1 limit cycles can appear in any neighborhood of the origin for arbitrary small |ε| + |a − a0 |. For the case of h0 > 0, by (3), (9) and (21), one has   k bj Jj (h) + O(ε) F (h, ε, a) = h  j=1

 = h

k

 ˜bj (h − h0 )j−1 + O(|h − h0 |k ) ,

j=1

(27) where ˜bj = bj + O(ε), j = 1, . . . , k. Also, by assumption and (23), it follows that system (1) has a center near the origin for (b1 , . . . , bk ) = 0. This implies that F (h, ε, a) = 0 if (b1 , . . . , bk ) = 0. Thus, (27) can be rewritten as F (h, ε, a) = h

k

˜bj (h − h0 )j−1 [1 + Pj (h, ε, a)]

j=1

(28) where ˜bj = bj + O(ε|b1 , . . . , bk |), j = 1, . . . , k, Pj (h, ε, a) = O(|h − h0 |k−j+1 ), j = 1, . . . , k. Now, using this form of F , similarly to the proof of Theorem 1.3 in [Han, 2000], one can prove that

j = 1, . . . , k − 1,

then F will have k − 1 zeros in h near h = h0 . This means that system (1) can have k − 1 limit cycles in any neighborhood of Lh0 . Hence, the proof of the first conclusion of Theorem 2.2 is completed. For the second one, suppose W (h0 ) = 0, W (h0 ) = 0 with h0 > 0. By (10) and (11), there exist constants α0 , . . . , αk−2 such that (k−1)

(J 1

=

(k−1)

(h0 ), . . . , J k k−2 j=0

(h0 ))

(j)

(j)

αj (J 1 (h0 ), . . . , J k (h0 )).

Hence, by (22), one has bk =

k−1

αj bj

(29)

j=1

for some constants α1 , . . . , αk−1 . Therefore, by (25) and (29), similarly to (28), one obtains  k+1 ˜bj (h − h0 )j−1 (1 + O(|h − h0 |)k−j+2 ) F = h j=1,j=k

 +

k−1



αj bj + O(ε|b1 , . . . , bk−1 , bk+1 |)

j=1



· (h − h0 )k−1  , where ˜bj = bj + O(ε|b1 , . . . , bk−1 , bk+1 |), j = 1, . . . , k − 1, k + 1. Note that bj = ˜bj +O(ε|˜b1 , . . . , ˜bk−1 , ˜bk+1 |), j = 1, . . . , k − 1, k + 1. One furthermore has F =h

k+1

˜bj (h − h0 )j−1 (1 + P˜j )

j=1

= hµ

k+1

δj (h − h0 )j−1 (1 + P˜j ),

(30)

j=1

where P˜j = (αj + O(ε))(h − h0 )k−j ,

j = 1, . . . , k − 1,


P˜k = 0, P˜k+1 = O(|h − h0 |), ˜bk = O(ε˜bk+1 ), 1  2 k+1 2 ˜ bj  , δj = ˜bj /µ. µ= j=1

Using (30) and Lemma 1.2, one can prove that system (1) has at most k limit cycles near Lh0 for small |ε| and bounded a ∈ Rm . To complete the proof, it suffices to prove that k limit cycles can appear in any neighborhood of Lh 0 . In fact, one may take h − h0 = λε, ˜bj = Bj εk+1−j , j = 1, . . . , k + 1, j = k, where Bj are constants such that the polynomial in λ B1 + B1 λ + · · · + Bk−1 λk−2 + Bk+1 λk has k simple zeros λj = 0, j = 1, . . . , k. Then, by (30), one has   k+1 Bj λj−1 + O(ε) = hεk F˜ (λ, ε). F = hεk  j=1,j=k

˜ j = λj + For small ε, F˜ has k simple zeros λ O(ε), j = 1, . . . , k. Thus, the function F has k sim˜ j ε, j = 1, . . . , k, and one ple zeros, hj = h0 + λ obtains k limit cycles near Lh0 . To prove the last conclusion, let h ∈ [0, h∗ ) and U be any neighborhood of Lh . Then, there exists h > h such that W (h) = 0 and Lh ⊂ U . By conclusion (i), system (26) has k − 1 limit cycles in U for some (ε, α). Hence, Lh has cyclicity at least k − 1. This completes the proof of Theorem 2.2.

Corollary 3.1. Suppose that the conditions of Theorem 2.2 hold with h0 > 0. If

(i) W (h0 ) = 0, 1 ≤ l ≤ k − 1, or (ii) W (h0 ) = 0, W (h0 ) = 0, 1 ≤ l ≤ k, l = k − 1, then, there exists a function a = a(ε) such that for a = a(ε), system (1) has a limit cycle of multiplicity l, which approaches Lh0 as ε → 0. Proof.

The equations

˜bl+1 = 1, ˜bj = 0,

j = 1, . . . , k,

j = l + 1

have unique solutions bj = bj (ε) = bj0 + O(ε), j = 1, . . . , k,

2041

which give a vector function b = b∗ (ε),

b∗ (0) ∈ Rk .

Then, setting aj = 0 for k + 1 ≤ j ≤ m, and ˜j (b∗ (ε), 0), aj = a

j = 1, . . . , k,

where a ˜j are solutions of system (26). Furthermore, letting ˜k (b∗ , 0), 0, . . . , 0), a(ε) = (˜ a1 (b∗ , 0), . . . , a one has, by (27), F (h, ε, a(ε)) = h[(h − h0 )l + O(|h − h0 |k )]. This implies that h = h0 is a zero of F with multiplicity l. Therefore, the corresponding limit cycle has multiplicity l. The conclusion follows under condition (i). Under condition (ii), the conclusion can be proved in the same way by using (30). Thus, the proof of Corollary 3.1 is completed. Next, assume that both (5) and (12) hold. Consider (13). For k = n(n+1)/2, consider the nontrivial function W (h, α), α ∈ Rq . Using polar coordinates, it is easy to verify that all coefficients in the expansion of each Ji (h) in h at h = 0 are polynomials in α ∈ Rq . It then follows that W (0, α) = ω1 (α) is a polynomial in α. (1) Let Σq−1 denote the (q − 1)-dimensional surq face in R defined by ω1 (α) = 0, α ∈ Rq . Let (2) Σq−1 denote the (q − 1)-dimensional surface in Rq defined by W (h, α) = 0, Wh (h, α) = 0, h ∈ [0, h∗ ), α ∈ Rq . (1)

(2)

Then, the set Σ = Σq−1 ∪Σq−1 has dimension (q−1) in Rq and for all α ∈ Rq − Σ one has W (0, α) = 0 and |W (h, α)| + |Wh (h, α)| = 0 for all h ∈ [0, h∗ ). Also, since k = 1/2n(n + 1), system (13) has a center near the origin if b = 0. In fact, system (13) is Hamiltonian in this case. Thus, Theorem 2.3 follows from Theorem 2.2 and Lemma 3.1. The following lemma gives a method to compute W (0). Lemma 3.2. Let M (h, a) = b1 h+b2 h2 +· · ·+bk hk +

. . . , and let v2i+1 (ε, a) = εVi + O(ε2 ) denote the ithorder focus value at the focus of system (1) near the origin, where V0 = (Px + Qy )(0, 0, a). Then, there

M. Han et al.

2042

exist constants Nj > 0, j = 1, 2 such that W (0) = N1 det = N2 det

the above system becomes x˙ = Hy + ε(δ1 x + l1 x2 + m1 xy + n1 y 2 ), y˙ = −Hx + ε(a1 x2 + b1 xy),

∂(b1 , . . . , bk ) ∂(b1 , . . . , bk ) ∂(V0 , . . . , Vk−1 ) . ∂(b1 , . . . , bk )

where

The first equality follows directly from (21) and (23). We now prove the second one. From the formulas (19) and (23) in [Jiang & Han, 1999] and the proof of Theorem B therein, there exist analytic functions hi (β) with hi (0) > 0, i ≥ 0, such that M (h, a) = β 2(i+1) Vi hi (β),

Proof.

i≥0

h = H(β, 0) =

1 1 1 H = − (x2 + y 2 ) + l0 x2 y + n0 y 3 − a0 x3 . 2 3 3 Note that div|(3.16) = ε[δ1 + (b1 + 2l1 )x + m1 y]. One thus has M (h) = δ1 I1 (h) + b1 I2 (h) + m1 I3 (h), b1 = b1 + 2l1 , where

I1 (h) =

1 2 β + O(β 3 ), |β| 1. 2

I3 (h) =

It follows that 2

M (h, a) = 2h0 (0)V0 h + [4h1 (0)V1 + (∗)V0 ]h + · · · + [2k hk−1 (0)Vk−1 + (∗)V0 + · · · + (∗)Vk−2 ]hk + . . . , where (∗) represent constant coefficients. Comparing the above with the expansion of M at h = 0, one obtains b1 = 2h0 (0)V0 , j bj = 2 hj−1 (0)Vj−1 + (∗)V0 + · · · + (∗)Vj−1 , j = 2, . . . , k. ∂(V0 , . . . , Vk−1 ) ∂(b1 , . . . , bk ) = N0 det , ∂(b1 , . . . , bk ) ∂(b1 , . . . , bk )

−H≤h

x dxdy, −H≤h

y dxdy, h ∈ [0, h∗ ).

W0 = εV0 ,

Wi = εVi + O(ε2 ),

i = 1, 2, 3,

where V0 = δ1 , V1 = m1 (n0 + l0 ) − a0 b1 , V2 = 5m1 a20 (n0 + l0 )2 (n0 − 2l0 ) − a20 n0 . ∂(V0 , V1 , V2 ) = −5a30 [(n0 + l0 )2 (n0 − 2l0 ) ∂(δ1 , b1 , m1 ) − a20 n0 ].

N0 > 0.

Therefore, by Lemma 3.1, for (31) one has

We finally show some applications of Lemma 3.2. By results of [Ye et al., 1986], any quadratic system having a focus or center can be changed into the following form: x˙ = −y + δx + lx2 + mxy + ny 2 , y˙ = x(1 + ax + by). The system is Hamiltonian if and only if δ = m = b + 2l = 0. Then, taking δ = εδ1 , m = εm1 , b = −2l0 + εb1 , l = l0 + εl1 , a = a0 + a1 ,

−H≤h

According to [Ye et al., 1986], up to a positive constant, the ith focus value of the origin can be taken as

det

The proof of Lemma 3.2 is thus completed.

dxdy, I2 (h) =

Thus,

Thus, det

(31)

W (0, a0 , l0 , n0 ) = N0 a20 [(n0 + l0 )2 (n0 − 2l0 ) − a20 n0 ], N0 = 0. Hence, by Theorem 2.2 in Sec. 2 and Theorem 6.4 in [Li, 2003], one obtains the following result. The following system is a special case of Eq. (31). x˙ = −y + x2 + ε(δ1 x + 100xy), y˙ = x − 2xy + x2 + ε(100 − δ2 )xy, where ε = 0.01, δ1 = −0.0002, δ2 = 1. For the above, one has 1 1 H(x, y) = − (x2 + y 2 ) + x2 y − x3 . 2 3

(32)


2043

where 1 H = (x2 + y 2 ) + h12 xy 2 + h03 y 3 , 2 aij xi y j , Q(x, y) = bij xi y j . P (x, y) =

0.08

i+j≤3

0.04

0 -0.08

-0.04

Let x = r cos θ, y = r sin θ. Then, H = 1/2(r 2 + R(θ)r 3 ), R(θ) = 2(h12 cos θ sin θ 2 + h03 sin θ 3 ). The equation H = h has a unique solution:

y

0

0.04

0.08

x -0.04

r = ρ + S2 (θ)ρ2 + · · · + S11 (θ)ρ11 + · · · = ϕ(θ, ρ), √ where ρ = 2h, and 1 5 231 4 R, S2 = − R, S3 = R2 , S4 = −R3 , S5 = 2 8 128 7 S6 = − R 5 , 2

-0.08

S9 = Fig. 1.

i+j≤3

S7 = −

1062347 8 R , 32768

Two limit cycles of Eq. (32).

In this case, one has cij M (h) =

Corollary 3.2. For any (a0 , l0 , n0 ) ∈ R3 satisfying

a0 [(n0 + l0 )2 (n0 − 2l0 ) − a20 n0 ] = 0,

(i) the Hopf cyclicity of (31) at the origin is 2; (ii) any closed curve Lh defined by H(x, y) = −h, h ∈ (0, h∗ ) has cyclicity 2.

1 Iij = i+j+2

2π

xi y j dxdy =

cij Iij

0≤i+j≤2

cosi θ sinj θϕi+j+2 (θ, ρ)dθ,

0

By using the Maple 7 software, we computed ∂ l Iij (0), ∂ρl

(33)

143 9 R , 2

i + j ≤ 2.

Consider the following cubic system: x˙ = Hy + εP (x, y), y˙ = −Hx + εQ(x, y),

S10 = −

H≤h

0≤i+j≤2

S8 = −15R7 ,

13764979 10 R . 262144

S11 = The phase portraits near the origin are shown in Fig. 1.

7293 6 R , 1024

l = 1, 2, . . . , 12,

and obtained M (h) = b1 h + · · · + b6 h6 + . . . , where

b1 = 2c00 π,

3 2 15 2 h + h b2 = c00 − c10 h12 − 3c01 h03 + c20 + c02 π, 2 03 2 12 315 2 2 35 4 5 3 1155 4 35 2 h + h h + h12 π + c10 − h03 h12 − h12 π b3 = c00 8 03 4 03 12 8 2 2 35 63 105 3 105 2 h03 − h03 h212 π + c20 h03 − h212 π + c01 − 2 2 32 32 105 2 735 2 105 c11 h03 h12 π + c02 h03 + h π, + 16 32 32 12

2044

M. Han et al.

b4 = 16π c00 + c10

225225 4 2 45045 2 4 1155 6 255255 6 h03 + h03 h12 + h03 h12 + h 1024 1024 1024 1024 12

3465 2 3 315 5 15015 4 h03 h12 − h03 h12 − h − 512 256 512 12

15015 3 2 3465 45045 5 h03 − h03 h12 − h03 h412 + c01 − 512 256 512 945 2 2 175 4 315 1155 4 1155 3 3 h + h h + h h h12 + h03 h12 + c11 + c20 512 03 256 03 12 512 12 128 03 128 3465 2 2 315 4 15015 4 h + h h + h , + c02 512 03 256 03 12 512 12

b5 = 32π c00

45045 8 20369349 6 2 66927861 8 h + h + h03 h12 16384 03 16384 12 4096

765765 2 6 14549535 4 4 h03 h12 + h h ) 8192 4096 03 12 969969 6 765765 4 3 135135 2 5 3003 7 h − h h12 − h h − h h + c10 − 2048 12 2048 03 2048 03 12 2048 03 12 2909907 5 2 765765 3 4 45045 2909907 7 6 h − h h − h h − h03 h12 + c01 − 2048 03 2048 03 12 2048 03 12 2048 135135 4 2 45045 2 4 1617 6 51051 6 h + h h + h h + h + c20 2048 03 2048 03 12 2048 03 12 2048 12 45045 3 3 9009 153153 5 5 h h12 + h h + h03 h12 + c11 1024 03 512 03 12 1024 765765 4 2 135135 2 4 3003 6 969969 6 h + h h + h h + h , + c02 2048 03 2048 03 12 2048 03 12 2048 12 +

b6 = 64π c00

(34)

2400219594045 2 8 228706638461145 10 354889611405225 8 2 h03 h12 + h h + h03 8589934592 8589934592 03 12 8589934592

92008417771725 6 4 18401683554345 4 6 22859234229 10 h03 h12 + h03 h12 + h + 4294967296 4294967296 8589934592 12 557732175 8 255255 9 156165009 6 3 101846745 4 5 4849845 2 7 h h − h03 h12 − h − h03 h12 − h03 h12 + c10 − 16384 03 12 65536 65536 12 16384 32768 557732175 7 2 468495027 5 4 33948915 3 6 1673196525 9 4849845 8 h03 − h03 h12 − h03 h12 − h03 h12 − h h + c01 − 65536 65536 16384 32768 16384 03 12 269325 8 40596255 6 2 48328875 4 4 3561075 2 6 44462565 8 h + h + h h + h h + h h + c20 131072 03 131072 12 32768 03 12 65536 03 12 32768 03 12


+ c11 + c02 Thus,

44462565 7 40596255 5 3 9665775 3 5 508725 h03 h712 + h03 h12 + h03 h12 + h h 16384 16384 16384 16384 03 12

2045

508725 8 311237955 6 2 202981275 4 4 9665775 2 6 1111564125 8 h03 + h12 + h03 h12 + h03 h12 + h h 131072 131072 32768 65536 32768 03 12

.

∂ b1 , b2 , b3 , b4 , b5 , b6 det ∂(c00 , c10 , c01 , c20 , c11 , c02 ) 1014 14 8 25861873 12 10 16307 16 6 6 h03 h12 + 18051355 h03 h12 + 10661360 h h = −π 21110907 18754 3385 2835426 03 12 407 10 12 23874 8 14 79857 6 16 h h + 12597475 h h + 259711 h h 6352 03 12 50237 03 12 842352 03 12 3141 4 18 2082 2 20 h h − 6193 h h − 26838 15378 03 12 45831 03 12 + 45529928

≡ K(h12 , h03 ). Therefore, by Theorem 2.2 the following result can be obtained. Corollary 3.3. Let K(h12 , h03 ) = 0. Then, the fol-

lowing hold: (i) The Hopf cyclicity of (33) at the origin is 5. (ii) For any h ∈ (0, h∗ ), the cyclicity of (33) at Lh is at least 5. In order to give a concrete system of the form (33), we introduce a function = h6 − 43h5 + 186h4 − 652h3 + 968h2 − 480h. ˜ where µ > 0 is small. Then, we Further, let h = µh, have M (h) = b1 h + b2 h2 + b3 h3 + b4 h4 + b5 h5 + b6 h6 + · · · ˜ + b2 µ−4 h ˜ 2 + b3 µ−3 h ˜ 3 + b4 µ−2 h ˜4 = µ6 [b1 µ−5 h ˜ 5 + b6 h ˜ 6 + O(µ)]. + b5 µ−1 h

b4 µ−2 = 186,

b2 = 968µ4 = 968 × 10−8 , b3 = −652µ3 = −652 × 10−6 , b4 = 186µ2 = 186 × 10−4 ,

(36)

b6 = 1. For definiteness, take h03 =1, h12 = 1 so that K < 0. Also, take P (x, y) = 3i+j=1,i≥1 aij xi y j and Q = 0 in (33), where aij (i + j = 1, 2, 3, i ≥ 1) are to be determined. In this case, (34) becomes b1 = 2πc00 , b2 = π(9c00 − c10 − 3c01 + c20 + c02 ), b3 = π(227.5c00 − 20c10 − 70c01 + 1.3125c20

If we take b2 µ−4 = 968,

b1 = −480µ5 = −480 × 10−10 ,

b5 = −43µ = −43 × 10−2 ,

ϕ(h) = h(h − 1)(h − 2)(h − 4)(h − 6)(h − 10)

b1 µ−5 = −480,

respectively, for sufficiently small µ > 0. Hence, ˜ j , j = 1, . . . , 5. M (h) has five positive zeros, hj = µh −2 Note that for µ = 10 , (35) is equivalent to

b3 µ−3 = −652,

b5 µ−1 = −43,

b6 = 1, (35)

+ 6.5625c11 + 26.25c02 ), b4 = 16π(514.3359c00 − 43.4766c10 − 153.3984c01 + 6.2891c20 + 11.4844c11 + 43.4766c02 ),

then the function M (h) becomes ˜ + O(µ)] = µ6 M ˜ ˜ (h). M (h) = µ6 [ϕ(h)

b5 = 32π(11024c00 − 914.9766c10 − 3237.6c01

˜ must have five pos˜ (h) By the definition of ϕ, M ˜ ˜ ˜ ˜ 5 , near 1, 2, 4, 6, 10, ˜ itive zeros, h1 , h2 , h3 , h4 , h

b6 = 64π(32590c00 − 21450c10 − 76016c01

+ 113.6953c20 + 246.3398c11 + 914.9766c02 ), + 2426.3c20 + 5812.6c11 + 21375c02 ).

M. Han et al.

2046

For (b1 , . . . , b6 ) given by (36), solving the above equations we obtain c00 = −7.6395 × 10−9 , c10 = 0.3755, c01 = −0.1068, c20 = 0.0032, c11 = −0.2031, c02 = 0.0520, a10 = −7.6395 × a11 = −0.1068, a21 = −0.1016,

a30 a12

a20 = 0.1878, = 0.0011, = 0.0520.

N1 = det

(37)

∂(b1 , b2 , . . . , bk ) > 0. ∂(B1 , B3 , . . . , B2k−1 )

Hence, by Lemma 3.2, one has

which together with (6) give 10−9 ,

It follows from the proof of Theorem 3 in [Han, 2001] that

Proof.

W (0) = N1 det (38)

For the above choice of the coefficients aij , we obtained the phase portraits shown in Fig. 2.

= N1 N1 det

x˙ = h(y) − F (x, a), y˙ = −g(x),

This completes the proof of Lemma 3.3.

H1 ≤h

i=1

where h (0) > 0,

∂(B1 , B3 , . . . , B2k−1 ) . ∂(b1 , b2 , . . . , bk )

Finally, we prove Theorem 2.4. First, consider (14). One has 2n+1 iai xi−1 dxdy M (h) = −

Lemma 1. Consider the following equations:

h(0) = g(0) = 0,

∂(b1 , b2 , . . . , bk ) ∂(b1 , b2 , . . . , bk )

g (0) > 0.

=−

−x + O(x2 )

satisfy G(α) = G(x), where Let α(x) = x G(x) = 0 g(u)du. Suppose that Bi xi . F (α, a) − F (x, a) = i≥1

n+1

(2j − 1)a2j−1

x2(j−1) dxdy H1 ≤h

j=1

since g1 is odd, where x g1 (x)dx + H1 (x, y) = 0

Then,

y

h(y)dy.

0

Comparing it with (7), one has

W (0) = N3 det

∂(B1 , B3 , . . . , B2k−1 ) , ∂(b1 , b2 , . . . , bk )

N3 > 0.

bj = −(2j − 1)a2j−1 ,

j = 1, . . . , n + 1.

On the other hand, for (14), one has F (−x, a) − F (x, a) = −2

n+1

a2j−1 x2j−1 .

j=1

0.04

x -0.08

-0.04

0

0.04

0.08

0

It then follows from Lemma 3.3 that W (0) = 0. Therefore, the conclusion follows from Theorem 2.2 and Lemma 3.1. For (15), similarly one has n (2i + 1)ai x2i dxdy M (h) = − H≤h

i=0

y

bi = −(2i − 1)ai−1 , where

-0.04

H(x, y) = 0

i = 1, . . . , n + 1.

x

(g1 + g2 )dx +

y

h(y)dy. 0

Since α(x) = −x + O(x2 ), for (15) one has -0.08

F (α, a) − F (x, a) = −2 Fig. 2.

Five limit cycles near the origin under (38).

n i=0

ai x2i+1 (1 + O(x)).


It follows that B1 = −2a0 , B3 = −2a1 + O(a0 ), ··· ······ B2n+1 = −2an + O(a0 , a1 , . . . , an+1 ). Therefore, by Lemma 3.3, one has W (0) = 0. This finally completes the proof of Theorem 2.4.

Acknowledgments The authors thank the referees for their valuable comments and suggestions, and in particular, thank Prof. Pei Yu and Dr. Chengqing Li for helping in numerical simulations.

References Arnold, V. I. [1977] “Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields,” Funct. Anal. Appl. 11, 85–92. Han, M. [2000] “On Hopf cyclicity of planar systems,” J. Math. Anal. Appl. 245, 404–422.

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