bifurcation of limit cycles in small perturbation of a class of liénard systems

February 5, 2014 8:45 WSPC/S0218-1274 1450004

International Journal of Bifurcation and Chaos, Vol. 24, No. 1 (2014) 1450004 (23 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127414500047

Bifurcation of Limit Cyclesin Small Perturbation of a Class

of Lienard Systems*

Xianbo Sun† and Hongjian Xi‡Department of Information and Statistics,

Guangxi University of Finance and Economics,Nanning, Guangxi 530003, P. R. China

†[email protected]‡[email protected]

Hamid R. Z. Zangeneh§ and Rasool Kazemi¶Department of Mathematical Sciences,

Isfahan University of Technology,Isfahan, 84156-83111, Iran

§[email protected]¶[email protected]

Received September 1, 2012; Revised February 3, 2013

In this article, we study the limit cycle bifurcation of a Lienard system of type (5, 4) with aheteroclinic loop passing through a hyperbolic saddle and a nilpotent saddle. We study theleast upper bound of the number of limit cycles bifurcated from the periodic annulus insidethe heteroclinic loop by a new algebraic criterion. We also prove at least three limit cycles willbifurcate and six kinds of different distributions of these limit cycles are given. The methods weuse and the results we obtain are new.

Keywords : Melnikov function; Lienard system; limit cycle; heteroclinic loop; nilpotent saddle;bifurcation.

1. Introduction

In many branches of science, such as mechanics,electronics, fluid mechanics, biology, chemistry,astrophysics, etc., one often deals with familiesof special planar differential equations which canmodel different natural phenomena. The exhibitingself-sustained oscillation is an important and inter-esting phenomena, such as beating of a heart, vibra-tions in the wings of an aeroplane and chemicaloscillation of the Brusselator model, can be mod-eled by the limit cycle which is an isolated periodic

orbit of related planar differential equation, there-fore limit cycles attract many scientists to study itsbifurcation and distribution.

In another aspect, the problem of limit cycleis important not only in practical applications, butalso in theoretical studies. The main open problemin the qualitative theory of planar polynomial dif-ferential systems is to determine the number andconfiguration of limit cycles, it is related to thesecond part of Hilbert’s 16th problem which asksfor the maximum number of limit cycles and their

∗This work was supported by the National Natural Science Foundation of China (NSFC) (11101118, 11261013) and ResearchProjects of Guangxi Universities (2013YB216).†Author for correspondence

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http://dx.doi.org/10.1142/S0218127414500047

February 5, 2014 8:45 WSPC/S0218-1274 1450004

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relative locations for the following planar differen-tial system

x = Pn(x, y), y = Qn(x, y), (1)

where Pn(x, y) and Qn(x, y) are all possible realpolynomials of degree n.

The Hilbert’s 16th problem is very difficult andremains unsolved even for n = 2, a weaker version ofthis problem is proposed by Arnold [1988] to studythe zeros of Abelian integrals obtained by inte-grating polynomial 1-forms along ovals of polyno-mial Hamiltonian, that is called weak Hilbert’s 16thproblem. More precisely, let H(x, y), p(x, y) andq(x, y) be real polynomials with deg(H) = m ≥ 2and max(deg(f),deg(g)) = n ≥ 2, where H(x, y)has least one family of ovals defined by {(x, y) |H(x, y) = h, h ∈ J} where J is an open interval,and let ε be a small positive parameter and δ bea vector parameter in some compact set D. Usingthese polynomials and parameters we have the fol-lowing planar near-Hamiltonian system

x = Hy + εp(x, y, δ),

y = −Hx + εq(x, y, δ)(2)

and the unperturbed system of (2)

x = Hy, y = −Hx. (3)

(3) is a Hamiltonian system and has at least onefamily of periodic orbits Γh which is a family ofovals of H(x, y) and form a unique period annulusdenoted by {Γh}. The so-called first order Melnikovfunction or Abelian integral corresponding to sys-tem (2) is

M(h, δ) =∮

Γh

qdx − pdy|ε=0, h ∈ J, (4)

the weak Hilbert’s 16th problem asks for the maxi-mum number of isolated zeros of M(h, δ). We recallthat an isolated zero of M(h, δ) corresponds to alimit cycle of system (2) by Poincare–PontryaginTheorem [Christopher & Li, 2007], thus M(h, δ)plays an important role in studying limit cycles ofsystem (2).

In the process of solving the weak Hilbert’s 16thproblem and practical applications, the followingLienard system of type (m,n) has attracted manymathematicians and other scientists:

x = y, y = Q1(x) + εQ2(x)y, (5)

where Q1 and Q2 are polynomials of degree mand n, respectively. For different m and n therehave been many results for this kind of systems,for instance, in a series of papers [Dumortier &Li, 2001a, 2001b, 2003a, 2003b], the generalizedLienard systems of type (3, 2) are investigated andthe sharp upper bound for the number of limitcycles are obtained by investigating zeros of theAbelian integrals. Li and Roussarie [2006] inves-tigated some Lienard system of type (3, 2) withsymmetry and obtained the sharp upper boundfor number of limit cycles by Poincare bifurcation.Wang and Xiao [2011] investigated some Lienardsystem of type (4, 3) and obtained the maximumnumber of limit cycles bifurcated from the periodicannulus. For the type (5, 4), many Lienard systemswith symmetry are investigated. Zhang et al. [2006]studied some Lienard system of type (5, 4) andproved that the least upper bound of the numberof zeros of the corresponding Abelian integral is 3.For another Lienard system of type (5, 4), Asheghiand Zangeneh proved that the least upper bound forthe number of zeros of the related Abelian integralinside the eye-figure loop is 2 [Asheghi & Zangeneh,2008a] and both inside and outside the eye-figureloop is 4 [Asheghi & Zangeneh, 2008b]. A specialcase of (5) is that the outer boundary of the periodannulus of system (5)ε=0 has degenerated singularpoints, the limit cycle bifurcation for this kind ofsystems has been investigated in [Dumortier et al.,2001; Dumortier et al., 1997; Dumortier et al., 1991;Zhu & Rousseau, 2002].

In this article, we investigate the followingLienard system of type (5, 4) which is a near-Hamiltonian system of form:

x = y,

y = x

(x +

12

)(x − 1)3

+ ε(a + bx + cx3 + x4)y

(6)

Fig. 1. The phase portrait of system (6) for ε = 0.

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Bifurcation of Limit Cycles in Small Perturbation of a Class of Lienard Systems

Fig. 2. The distribution of three limit cycles bifurcated from the annulus of system (6)ε=0.

with Hamiltonian function

H(x, y) =12y2 +

14x2 − 1

6x3

− 38x4 +

12x5 − 1

6x6, (7)

where 0 < ε � 1, a, b and c are real boundedparameters. The level sets of Hamiltonian func-tion (7) are sketched in Fig. 1. It is easy to checkthat Lh = {(x, y) | H(x, y) = h, h ∈ (0, 1

24 )} areclosed clockwise orbits of system (6)ε=0 which forma unique period annulus {Lh}, the outer boundaryof the period annulus is L 1

24which is a heteroclinic

loop connecting a nilpotent saddle (1, 0) to a hyper-bolic saddle (−1

2 , 0). L0 is an elementary center atthe origin. The Melnikov function of system (6) is

M(h, δ) =∮

Lh

(a + bx + cx3 + x4)ydx

≡ aI0(h) + bI1(h) + cI3(h) + I4(h) (8)

for 0 < h < 124 , Ii(h) =

∮Lh

xiydx (i = 0, 1, 3, 4) aregenerating elements of M(h, δ).

System (6) has been investigated in [Kazemiet al., 2012]. It is showed that at most three limitcycles can be bifurcated from system (6), but dueto some mistake in computation the upper boundfound is not correct. In this paper, by using somenew theories developed in [Manosas & Villadelprat,2011; Han et al., 2012; Zang et al., 2008; Sun et al.,2011], we study the bifurcation of limit cycles forsystem (6) and obtain a new result.

Main Theorem. Consider Lienard system (6) andits Melnikov function M(h, δ), then:

(i) For all parameters (a, b, c, 1), system (6) has atmost four limit cycles by Poincare bifurcation.

(ii) Taking ε positive and very small, there existsome (a, b, c, 1) such that system (6) can havethree limit cycles with six kinds of different dis-tributions given in Fig. 2.

We will take M(h, δ) as the main tool to study thelimit cycles of (6). The rest of our paper is organizedas follows: In Sec. 2, we use a new algebraic crite-rion developed in [Manosas & Villadelprat, 2011]to study the least upper bound of the numberof limit cycles bifurcated from the period annu-lus {Lh}. In Sec. 3, we will use the expansions ofM(h, δ) near the center and near the heteroclinicloop L 1

24to find limit cycles and their different

distributions.

2. The Least Upper Bound of theNumber of Limit Cycles byPoincare Bifurcation

In this section, we discuss the least upper boundof the number of limit cycles bifurcated from theperiod annulus {Γh}, we will show that the leastupper bound is four by a new method. The methodwe will use proposes some criterion functionsdefined directly by Hamiltonian and integrandsof Abelian integrals, through which the problemwhether the basis of the vector space generatedby Abelian integral is a Chebyshev system withaccuracy k could be reduced to the problem whetherthe family of criterion functions form a Chebyshevsystem with accuracy k, since the latter can betackled by checking the nonvanishing properties of

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its Wronskians, this criterion is a pure algebraicmethod which is developed in [Grau et al., 2011;Manosas & Villadelprat, 2011] from the idea intro-duced in [Li & Zhang, 1996]. To check whether therelated Wronskians vanish or not, we need to checkif two polynomials have common roots satisfyingsome condition, to this end, we firstly revisit themethods of triangular decomposition and real rootisolation.

Let k be a field, x1 ≺ x2 ≺ · · · ≺ xn be nordered variables and R = k[x1, . . . , xi] be the poly-nomials ring on k. A nonconstant polynomial f in Rcan be seen as a univariate polynomial in its great-est variable. The greatest variable in f is called itsmain variable, denoted by mvar(f).

Definition 1. A nonempty subset T = {t1, . . . , ts}of R is a triangular set, if the polynomials in T arenonconstant and have distinct main variables.

Hence, a triangular set is finite, and has cardi-nality at most n. The following definition is from[Yang & Zhang, 1994].

Definition 2. Let T = {t1, . . . , ts} be a triangularset such that mvar(t1) ≺ · · · ≺ mvar(ts), hi be theinitial of ti and h be the product of h′

is. Then T isa regular chain if

resultant(h, T )

= resultant(· · · resultant(h, ts), . . . , ti · · ·) �= 0,

where each resultant is computed with respect tothe main variable of ti, respectively.

In broad words, a triangular set is a system ofalgebraic equations that are ready to be solved byevaluating the unknowns one after the other, justlike a triangular linear system, however, there isa difference with the linear case: the back solvingprocess may lead to some degenerated situation,or even to no solutions. Consider, for example, fory ≺ x, the triangular set [yx− 1, y2 − y]. The valuey = 1 leads to x = 1, but the value y = 0 does notlead to a value of x. Regular chains are a particularkind of triangular sets for which the back solvingprocess succeeds in every case.

Regular chains have many interesting compu-tational properties and can translate some complexcomputing problems into simpler ones. Therefore,firstly, many algorithms based on regular chainshave been proposed in various works and embeddedinto the computer algebra system Maple as

the package RegularChains which is a collec-tion of commands for solving systems of algebraicequations symbolically and studying their solutions.Secondly, when one solves problems related to poly-nomials such as polynomial algebra system {f1 = 0,f2 = 0, . . . , fn = 0}, one may try to translate it intoa series of regular chains, therefore in RegularChainspackage the main function is the command Trian-gularize, the algorithm of which is described in thepaper [Maza, 2000], available on the author’s webpage. The command Triangularize computes froma system of polynomials S a list of regular chainsS1, . . . , Sn, which is called a triangular decomposi-tion of the common roots of S. From [Maza, 2000],we know

Lemma 2.1. Let S be a system of polynomials onK with K as the rational number field. Then, Scan be triangular decomposed into a list of regu-lar chains (realized by the command Triangularize),and a point is a solution of S if and only if it is asolution of one of the systems S1, . . . , Sn.

From Lemma 2.1, we know the roots of S can beobtained by computing each regular chain Si. From[Rioboo, 1992] if the regular chain Si is square-freeand zero-dimensional (a system is zero-dimensionalif it has a finite number of solutions in an alge-braically closed extension K of K, in particular, atriangular system is zero-dimensional if the numberof polynomials is equal to the number of variables),then the roots of Si can be obtained by the algo-rithm in [Rioboo, 1992] which has been embeddedinto the computer system Maple and can be realizedby the command RealRootIsolate. The RealRoot-Isolate command returns a list of boxes. Each boxisolates exactly one real root of a regular chain Si.

Therefore, we can use the commands Trian-gularize, RealRootIsolate and other auxiliary com-mands to find all the common roots of twopolynomials on a rational number field, we use thesecommands to construct a simple direct program(shown in Appendix A with its explanation) whichcan achieve our objective. The direct program willbe used to prove two parts of Lemma 2.4 which isthe key conclusion to prove the main result in whatfollows.

Now let us recall the algebraic method whichwe will use. Let H(x, y) = A(x) + 1

2y2 in (3) bean analytic function in some open subset of theplane that has a local minimum at the origin withH(0, 0) = 0. It is also supposed there exists a

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punctured neighborhood P of the origin foliated byovals H(x, y) = h which are the clockwise periodicorbits Γh of system (3) and form a period annu-lus denoted by {Γh}, the period annulus is param-eterized by the energy levels h ∈ (0, h0) for someh0 ∈ (0,+∞]. The projection of P on the x-axis isan interval (xl, xr) with xl < 0 < xr. Under theseassumptions, it is easy to verify that xA′(x) > 0for all x ∈ (xl, xr)\{0}, A(x) has a zero of evenmultiplicity at x = 0 and there exists an analyticinvolution z(x) defined by

A(x) = A(z(x))

for all x ∈ (xl, xr). Let us consider the Abelianintegrals

Ii(h) =∮

Γh

fi(x)y2s−1dx, for h ∈ (0, h0), (9)

where fi(x) for i = 0, 1, . . . , n − 1, are analyticfunctions on (xl, xr) and s ∈ N . We recall thatTheorem A of [Manosas & Villadelprat, 2011] pro-poses an algebraic criterion for the number ofisolated zeros of nontrivial linear combination of{I0, I1, . . . , In−1} which can be stated as follows:

Lemma 2.2. Under the above assumptions, we con-sider the Abelian integrals Ii(h) defined in (9) anddefine

li(x) :=fi(x)A′(x)

− fi(z(x))A′(z(x))

.

If the following conditions are verified :

(i) W [l0, . . . , li] is nonvanishing on (0, xr) fori = 0, 1, . . . , n − 2,

(ii) W [l0, . . . , ln−1] has k zeros on (0, xr) withmultiplicities,

(iii) s > n + k − 2,

then any nontrivial linear combination of {I0,I1, . . . , In−1} has at most n + k − 1 zeros on(0, h0) counted with multiplicities. In this case, {I0,I1, . . . , In−1} is called a Chebyshev system withaccuracy k on (0, h0), where W [l0, . . . , li] denotesWronskian of {l0, . . . , li}.

Usually s is not big enough (i.e. s > n +k − 2 does not hold) such that Lemma 2.2 can-not be applied directly. To overcome this problemwe can use the next result (see [Grau et al., 2011,Lemma 4.1]) to increase the power of y in each Ii.

Lemma 2.3. Let Γh be an oval inside the level curveA(x) + 1

2y2 = h and F (x) be a function such thatF (x)A′(x) is analytic at x = 0. Then, for any k∗ ∈ N,

∮Γh

F (x)yk∗−2dx =∮

Γh

G(x)yk∗dx

where G(x) = 1k∗( F (x)

A′(x)

)′(x).

We now apply the above theories to the gener-ating elements of M(h, δ) for system (6):

Ii(h) =∫

Lh

xiydx, i = 0, 1, 3, 4. (10)

In this case,

A(x) = H(x, y) − y2

2

=14x2 − 1

6x3 − 3

8x4 +

12x5 − 1

6x6,

n = 4, s = 1. However, the condition s > n +k − 2 does not hold even taking k = 1. In orderto overcome this problem we will use Lemma 2.3to increase the power of y in (10). Noting that2A(x)+y2

2h = 1 holds on each orbit Lh, then

Ii(h) =12h

∫Lh

(2A(x) + y2)xiydx

=12h

(∫Lh

2xiA(x)ydx +∫

Lh

xiy3dx

),

i = 0, 1, 2, 3. (11)

Taking k∗ = 3 and F (x) = 2xiA(x) to applyLemma 2.3, then

∫Lh

2xiA(x)ydx =∫

Lh

Gi(x)y3dx, (12)

where Gi(x) = d3dx (2xiA(x)

A′(x) ) = xigi(x)18(2x+1)2(x−1)4 and

gi(x) = (8x6 + 8ix6 − 24x5 − 28ix5 + 26ix4 + 22x4

− 4x3 + 11ix3 − 25ix2 − 3x2

+ 4x + 2ix + 6i + 6).

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Substituting (12) into (11), then

Ii(h) =12h

∫Lh

(xi + Gi(x))y3dx

=1

4h2

∫Lh

(2A(x) + y2)(xi + Gi(x))y3dx

=1

4h2

∫Lh

2A(x)(xi + Gi(x))y3dx

+1

4h2

∫Lh

(xi + Gi(x))y5dx. (13)

Taking k∗ = 5 and F (x) = 2A(x)(xi + Gi(x)) toapply Lemma 2.3, then∫

Lh

2A(x)(xi + Gi(x))y3dx =∫

Lh

Gi(x)y5dx, (14)

where

Gi(x) =d

5dx

(2A(x)(xi + Gi(x))

A′(x)

)

=xigi(x)

540(2x + 1)4(x − 1)8

and

gi(x) = 144 + 180i + 144x − 3814x6 − 870x5 + 1863x4 − 328x3 − 318x2 + 120ix − 6935ix6 − 3973ix5

+ 4500ix4 + 1224x8 + 5472x7 + 190ix3 − 1282ix2 − 1916ix8 + 12828ix7 + 36i2 + 24i2x

− 1195i2x6 − 782i2x5 + 981i2x4 + 32i2x3 − 296i2x2 − 340i2x8 + 2004i2x7 − 11056ix9

+ 704ix12 − 4544ix11 + 11184ix10 + 1200x10i2 − 1280x9i2 + 64x12i2 − 448x11i2

− 9056x9 + 640x12 − 3840x11 + 8928x10. (15)

Substituting (14) into (13), then

Ii(h) =1

4h2

∫Lh

(xi + Gi(x) + Gi(x))y5dx

=1

8h3

∫Lh

(2A(x) + y2)(xi + Gi(x) + Gi(x))y5dx

=1

8h3

∫Lh

2A(x)(xi + Gi(x) + Gi(x))y5dx +1

8h3

∫Lh

(xi + Gi(x) + Gi(x))y7dx. (16)

Again taking k∗ = 7 and F (x) = A(x)(xi + Gi(x) + Gi(x)) to apply Lemma 2.3, we have∫Lh

2A(x)(xi + Gi(x) + Gi(x))y5dx =∫

Lh

Gi(x)y7dx, (17)

where

Gi(x) =d

7dx

(2A(x)(xi + Gi(x) + Gi(x))

A′(x)

)=

xigi(x)22680(2x + 1)6(x − 1)12

and

gi(x) = 5184 + 7344i + 6624x − 617081x6 − 10164x5 + 152210x4 − 26592x3 − 20016x2 + 6912ix

− 1429543ix6 − 285842ix5 + 427198ix4 + 1236024x8 + 631288x7 − 8696ix3 − 73056ix2

+ 1956110ix8 + 2127095ix7 + 2376i2 + 2232i2x − 466581i2x6 − 92538i2x5 + 154434i2x4

− 3900i2x3 − 27420i2x2 + 576585i2x8 + 632823i2x7 − 6426810ix9 − 9913232ix12

+ 7847976ix11 + 2154528ix10 + 6979264ix14 + 1184608ix13 + 3557120ix16 − 7308160ix15

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+ 533718x10i2 − 1711905x9i2 − 2151744x12i2 + 1806864x11i2 + 1248576x14i2 + 293664x13i2

+ 575232x16i2 − 1248768x15i2 − 2874664x9 + 5510400x14 − 333184x13 − 5243904x12

+ 4701600x11 + 335808x10 + 2799616x16 − 5591552x15 + 81920x18 − 737280x17 + 216i3

− 898048ix17 + 95232ix18 + 13824x18i2 − 137472x17i2 − 40987i3x6 − 7170i3x5 + 14550i3x4

− 604i3x3 − 2628i3x2 + 45483i3x8 + 49287i3x7 + 33858x10i3 − 119149x9i3 − 119104x12i3

+ 107280x11i3 + 57792x14i3 + 16800x13i3 + 23808x16i3 − 54784x15i3 + 512x18i3

− 5376x17i3 + 216xi3. (18)

By (16) and (17) we have

8h3Ii(h) =∮

Lh

fi(x)y7dx ≡ Ii(h),

where fi(x) = xi + Gi(x) + Gi(x) + Gi(x). It isclear that {I0, I1, I3, I4} is a Chebyshev system withaccuracy 1 on (0, 1

24) if and only if {I0, I1, I3, I4}is as well. Now we can apply Lemma 2.2, becauses = 4 and the condition s > n + k − 2 holds fork = 1. We set

li(x) =(

fi

A′

)(x) −

(fi

A′

)(z(x)),

here z(x) is an analytic involution defined byA(x) = A(z(x)) with A(1) = A(−1

2), which isequivalent to

− 124

(x − z)(2x3 − 3x2 + 2 − 3z2 + 2z3)q(x, z) = 0

with

q(x, z) = 2x2 − 3x + 2xz − 3z + 2z2. (19)

It is not difficult to find that z(x) is defined byq(x, z) = 0 implicitly on {Lh}, therefore

d

dxli(x) =

d

dx

(fi

A′

)(x) −

[d

dz

(fi

A′

)(z(x))

]· dz

dx,

where dzdx = −∂q(x,z)

∂x /∂q(x,z)∂z . If we restrict x ∈

(0, 1), then −12 < z(x) < 0. We shall rely on

symbolic computations using Maple.15 to applySturm’s Theorem and analyze carefully to check if{I0, I1, I3, I4} is a Chebyshev system with accuracy1 or not. We have

Lemma 2.4. The following holds

(i) W [l0](x) �= 0, x ∈ (0, 1);(ii) W [l0, l1](x) �= 0, x ∈ (0, 1);(iii) W [l0, l1, l3](x) �= 0, x ∈ (0, 1);(iv) W [l0, l1, l3, l4](x) has a unique simple root x∗

in (0, 1).

In fact, the above four Wronskians are all functionsof x and z, and (x, z) satisfies q(x, z) = 0 with

−12

< z < 0 < x < 1. (20)

Therefore, in order to check if each of the Wron-skians vanishes on (0, 1), we only restrict x ∈ (0, 1)and check if each of the Wronskians and q(x, z) havecommon roots that satisfy (20).

Proof [Proof of Lemma 2.4]. Using Maple.15 wehave

W [l0](x) = l0(x) =(x − z)w0(x, z)

11340xz(2x + 1)7(x − 1)15(2z + 1)7(z − 1)15,

W [l0, l1](x) =−(x − z)3w1(x, z)

128595600x2z2(2x − 3 + 4z)(z − 1)30(2z + 1)13(x − 1)30(2x + 1)13,

W [l0, l1, l3](x) =−(x − z)6w3(x, z)

729137052000x3z3(2x − 3 + 4z)3(x − 1)45(2x + 1)18(z − 1)45(2z + 1)18,

W [l0, l1, l3, l4](x) =(x − z)10w4(x, z)

689034514140000x4z4(2x − 3 + 4z)6(x − 1)59(2x + 1)22(z − 1)59(2z + 1)22,

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where w0(x, z), w1(x, z), w3(x, z) and w4(x, z) arelong expression polynomials of degree 40, 78, 115and 148, respectively. It is not difficult to obtainthat 2x− 3+4z and q(x, z) have no common roots,then it is clear that W [l0, l1](x),W [l0, l1, l3](x) andW [l0, l1, l3, l4](x) are well defined on −1

2 < z <0 < x < 1. Hence, we only need to check if eachwi(x, z)(i = 0, 1, 3, 4) and q(x, z) have commonroots satisfying (20), we will check these problemsin turns.

(i) Firstly, calculating the resultant with respect toz between q(x, z) and w0(x, z) (i.e. eliminating zfrom q(x, z) = 0 and w0(x, z) = 0) gives R(q, w0,z)= 2097152(x− 1)28(2x+ 1)14p0(x), where p0(x)is a long polynomial of degree 38. By applyingSturm’s Theorem we obtain p0(x) �= 0 for all x ∈(0, 1). Hence, w0(x, z) and q(x, z) have no commonroots, which implies that W [l0] �= 0 for all x ∈ (0, 1).

(ii) Calculating the resultant with respect to zbetween q(x, z) and w1(x, z) gives R(q, w1, z) =

2199023255552(2x + 1)26(x − 1)54p1(x), wherep1(x) is a polynomial of degree 76. By applyingSturm’s Theorem to p1(x) on (0, 1) we can assertthat p1(x) �= 0 for all x ∈ (0, 1). Hence, w1(x, z)and q(x, z) have no common roots, which impliesthat W [l0, l1] �= 0 for all x ∈ (0, 1).

(iii) Calculating the resultant with respect toz between q(x, z) and w3(x, z) gives R(q, w3,z) = 129703669268270284800(2x + 1)36(x − 1)78

p3(x), where p3(x) is a polynomial of degree 116 ofx. By applying Sturm’s Theorem, there is a uniqueroot x0 ≈ 0.5845202748 ∈ (0, 1), applying Sturm’sTheorem to another resultant R(q, w3, x), we obtainthat it has roots on (−1

2 , 0), therefore the methodbefore is defeated for this case.

In order to make sure if there exist a com-mon root of w3(x, z) and q(x, z) satisfying (20), wecan use algebra symbolic computation method tocompute the intervals in which all common rootsexist, the direct program in Appendix A realizesthe objective.

> with(RegularChains):> with(ChainTools):> with(SemiAlgebraicSetTools):> sys := [w_3(x, z), q(x, z)]:> R := PolynomialRing([x, z]):> dec := Triangularize(sys, R);[regular_chain, regular_chain, regular_chain, regular_chain]

> L := map(Equations, dec, R);

[[p1(x, z), p2(z)], [x − 1, 2z + 1], [x − 1, z − 1], [2x + 1, z − 1]]

where p1(x, z) = p11(z)x + p12(z), p11 is a polynomial in z of degree 60, p12 is a polynomial in z of degree61, p2 is a polynomial in z of degree 116, shown in Appendix B. It is obvious that, all the roots of theregular chains [x− 1, 2z + 1], [x− 1, z − 1] and [2x + 1, z − 1] do not satisfy (20), and the first regular chain[p1(x, z), p2(z)] is square-free and zero-dimensional (because the number of variables equals the numberof polynomials). L[1][1] and L[1][2] represent p1 and p2 in Maple and they have two and one variables,respectively. Therefore we need to change their order in the first regular chain.

> C := Chain([L[1][2], L[1][1]], Empty(R), R);regular_chain

> RL := RealRootIsolate(C, R, ’abserr’ = 1/10^5);[box, box, box, box, box, box]

> map(BoxValues, RL, R)[[x =

[122238192

,195569131072

], z =

[− 7038861

67108864,− 1759715

16777216

]],

[x =

[175387131072

,4384732768

], z =

[− 6887004807311

17592186044416,−13774009614621

35184372088832

]],

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[x =

[− 3437

32768,− 13747

131072

], z =

[73883693208782647906181043594951760157141521099596496896

,472855636536208946599558678977316912650057057350374175801344

]],

[x =

[− 51313

131072,−3207

8192

],

z =[217119208257943625924646696781729162259276829213363391578010288128

,868476833031774503698586787126917649037107316853453566312041152512

]],

[x =

[3830765536

,76615131072

],

z =[2678629726661883514594428785168920282409603651670423947251286016

,1339314863330941757297214392584510141204801825835211973625643008

]],

[x =

[8655165536

,173103131072

], z =

[4211912467230802572057594037927936

,2105956233615401336028797018963968

]]]

> evalf(%);[[x = [1.492065430, 1.492073059], z = [-0.1048872024, -0.1048871875]],[ x = [1.338096619, 1.338104248], z = [-0.3914808990, -0.3914808990]],[ x = [-0.1048889160, -0.1048812866], z = [1.492069302, 1.492069302]],[ x = [-0.3914871216, -0.3914794922], z = [1.338100431, 1.338100431]],[ x = [0.5845184326, 0.5845260620], z = [1.320666419, 1.320666419]],[x = [1.320663452, 1.320671082], z = [0.5845202749, 0.5845202749]]]

From the result of the program, there are sixpairs of common roots of w3(x, z) and q(x, z)in the above six pairs of intervals, but no pairsatisfies (20). Therefore, W [l0, l1, l3] �= 0 for allx ∈ (0, 1).

(iv) Calculation of the resultant with respectto z between q(x, z) and w4(x, z) gives R(q, w4,z) = 5206409047363783848099840000(2x + 1)44 ×

(x − 1)98p4(x), where p4(x) is a polynomial ofdegree 154 in x. By applying Sturm’s Theorem,there is a unique root x∗ ≈ 0.7074986036 ∈ (0, 1),substituting x = x∗ into q(x, z), and solvingq(x∗, z) we have z = z∗ ≈ −0.4509259346. In orderto make sure if (x∗, z∗) is the common root ofw4(x, z) and q(x, z), it is necessary only to substi-tute [w3(x, z), q(x, z)] by [w4(x, z), q(x, z)] in theMaple program of the previous stage to obtain:

> L := map(Equations, dec, R);

[[r1(x, z), r2(z)], [x − 1, 2z + 1], [x − 1, z − 1], [2x + 1, z − 1]]

where r1(x, z) = r11(z)x + r12(z), r11 is a polynomial in z of degree 81, r12 is a polynomial in z of degree82, r2 is a polynomial in z of degree 154, shown in Appendix B. It is obvious that, all roots of the regularchains [x − 1, 2z + 1], [x − 1, z − 1] and [2x + 1, z − 1] do not satisfy (20), and the first regular chain[r1(x, z), r2(z)] is square-free and zero-dimensional (because the number of variables is equal to the numberof polynomials). L[1][1] and L[1][2] represent r1 and r2 in Maple and they have two and one variables,respectively. Therefore we need to change their order in the first regular chain.

> C := Chain([L[1][2], L[1][1]], Empty(R), R);regular_chain


> map(BoxValues, RL, R);

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[x =

[92733131072

,4636765536

], z =

[−16636230623269898825

36893488147419103232,−2079528827908737353

4611686018427387904

]],

[x =

[159595131072

,3989932768

], z =

[−266291259437046015

576460752303423488,−1065165037748184059

2305843009213693952

]],

[x =

[142845131072

,7142365536

], z =

[−291756897068721941421

590295810358705651712,− 583513794137443882841

1180591620717411303424

]],

[x =

[−15137

32768,− 60547

131072

],

z =[868920112304961028798074403415106990885527765713623846352979940529142984724747568191373312

,

17378402246099220575961488068302139817710555311427247692705959881058285969449495136382746624

]],

[x =

[−4049

8192,− 64783

131072

],

z =[398193970344963844795981922305352848417944651667365375409332725729550921208179070754913983135744

,

9954849258624096119899548057633821210448616291791343852333181432387730302044767688728495783936

]],

[x =

[−1847

4096,− 59103

131072

],

z =[459192847138396775501674967043953649037107316853453566312041152512

,9183856942767935510033499340879071298074214633706907132624082305024

]]> evalf(%);[[x = [0.7074966431, 0.7075042725], z = [-0.4509259346, -0.4509259346]],[x = [1.217613220, 1.217620850], z = [-0.4619416992, -0.4619416992]],[x = [1.089820862, 1.089828491], z = [-0.4942554088, -0.4942554088]],[x = [-0.4619445801, -0.4619369507], z = [1.217616419, 1.217616419]],[x = [-0.4942626953, -0.4942550659], z = [1.089821483, 1.089821483]],[x = [-0.4509277344, -0.4509201050], z = [0.7074986036, 0.7074986036]]].

From the result of the program, there are six pairs of common roots of w4(x, z) and q(x, z) in the abovesix pairs of intervals respectively. Only the first pair denoted by (x∗, z∗) satisfies (20). Therefore, there isa unique x∗ ∈ (0, 1) such that W [l0, l1, l3, l4](x∗) = 0. Now we will show that x∗ is a simple root. Let usdenote W [l0, l1, l3, l4](x) by W4(x, z(x)) and calculate its derivative,

dW4

dx=

∂W4

∂x+

∂W4

∂z× dz

dx

=(x − z)10w5(x, z)

689034514140000x4(x − 1)59(2x + 1)22z4(z − 1)59(2z + 1)22(2x − 3 + 4z)6,

where w5(x, z) is a polynomial of degree 156 with 7566 terms. Taking w5(x, z) in place of p2(x, z) in theabove program, we obtain ten common roots of w5(x, z) and q(x, z), but none inside[

x =[

92733131072

,4636765536

], z =

[−16636230623269898825

36893488147419103232,−2079528827908737353

4611686018427387904

]]

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(in which (x∗, z∗) exists). Because q(x∗, z∗) =q(x∗, z(x∗)) = 0, then w5(x∗, z(x∗)) �= 0, hence x∗ isa simple root of W4(x, z(x)). Then we have proved(iv) of Lemma 2.4.

By Lemmas 2.2 and 2.4, {I0, I1, I3, I4} is aChebyshev system with accuracy 1 on (0, 1

24 ) and{l0, l1, l3, l4} is as well. Hence, M(h, δ) has at mostfour zeros which implies system (6) has at most fourlimit cycles by Poincare bifurcation. (i) of the MainTheorem is proved. �

Remark 2.1. We prove the last two parts ofLemma 2.4 with the aid of symbolic computationwhich is based on polynomial theories, but notnumerical computation except for eval f(%) whichtranslates the intervals with fraction form into deci-mal form. This direct program obtains precise inter-vals, the fraction endpoints which are computedfrom the coefficients of two polynomials by sym-bolic computation based on regular chain theories.

Remark 2.2. In [Kazemi et al., 2012], authorsfind the roots x∗ = 0.7776101980 and z∗ =−0.4707903354 for p4(x) and q(x∗, z) by Maple,respectively. After substituting x = x∗ andz = z∗ in w4(x, z) they obtained w4(x∗, z∗) =1.008710694 × 1019 (here p4(x) and w4(x, z) arep4(x) and w4(x, z) of [Kazemi et al., 2012], respec-tively, but not of this article), they conclude that(x∗, z∗) is not a root of w4(x, z). However thisis not true, the error comes from the fact thatcoefficients of monomials in x and z in w4(x, z)are very large, for example, the coefficient of x40z30

is 25187224466678935151766954360686247936 andif we use the value of x∗ and z∗ accurately up to100 digits instead of ten digits used in [Kazemiet al., 2012] we get a completely different result,i.e. w4(x, z) = 8.59721809779×10−72 . In fact, usingthe above Maple program, it can show that (x∗, z∗)is a common root of w4(x, z) and q(x, z).

3. Three Limit Cycles of System (6)

In this section, we use the Melnikov function to findlimit cycles of system (6) and obtain that system (6)can have three limit cycles with five different distri-butions. We first give some preliminaries which areused in the following discussion.

Consider system (2), (3) and suppose the annu-lus {Γh} of (3) is bounded, then the inner boundaryof {Γh} may be an elementary center or a nilpotentcenter and the outer boundary may be a homoclinic

loop or a heteroclinic loop. The Melnikov functionM(h, δ) can be expanded near the boundaries, see[Roussarie, 1986; Schlomiuk, 1993; Han et al., 2008;Hou & Han, 2006; Han et al., 2009; Sun et al., 2011;Zang et al., 2008; Han et al., 2012], some results ofwhich are the following lemmas that we will use.

We assume (3) has an elementary center C(0, 0)with H(0, 0) = 0 at the origin and

H(x, y) =12(x2 + y2) +

∑i+j≥3

hijxiyj ,

p(x, y, δ) =∑

i+j>0

aijxiyj,

q(x, y, δ) =∑

i+j>0

bijxiyj,

(21)

for (x, y) near C(0, 0). Then M(h, δ) near the ele-mentary center C(0, 0) has the following expansion(see [Han et al., 2009])

M(h, δ) =∑j≥0

bj(δ)hj+1, 0 < h � 1. (22)

Under (21), the formulas of bj , j = 0, 1, 2, 3 canbe found in [Hou & Han, 2006] and more coeffi-cients can be obtained by using the programs in[Han et al., 2009].

Lemma 3.1 [Han et al., 2008]. Assume system (3)has a homoclinic loop Γhs defined by H(x, y) = hs

passing through a hyperbolic saddle S1(x1, y1) and

H(x, y) = hs +λ

2(y − y1)2 − 1

2(x − x1)2

+∑

i+j≥2

hij(x − x1)i(y − y1)j,

p(x, y, δ) =∑

i+j>0

aij(x − x1)i(y − y1)j ,

q(x, y, δ) =∑

i+j>0

bij(x − x1)i(y − y1)j ,

(23)

for (x, z) near S1(x1, y1). Then near the homoclinicloop Γhs has the form

M(h, δ) =∑j≥0

(c2j + c2j+1(h − hs) ln|h − hs|)hj

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for 0 < −(h − hs) � 1, where

c0(δ) = M(hs, δ) =∮

Γhs

qdx − pdy,

c1(δ) = −a10 + b01

|λ| ,

c2(δ) =∮

Γhs

(px + qy − a10 − b01)|ε=0dt + bc1(δ),

c3(δ) =−1λ|λ|(a21 + b12)

− 1λ

[h12(2a20 + b11) + h21(a11 + 2b02)]

+ bc1(δ),

where b and b are constants.

As in [Han et al., 2008], c1(δ) and c3(δ) arecalled the local homoclinic coefficients of M at thehyperbolic saddle S1, we denote them by c1(S1hsδ)and c3(S1hs, δ), respectively.

Lemma 3.2 [Han et al., 2012]. Assume system (3)has a homoclinic loop Γhs defined by H(x, y) = hs

passing through a nilpotent saddle S2(x2, y2) and

H(x, y) = hs +12(y − y2)2

+∑

i+j≥4

hij(x − x2)i(y − y2)j ,

h40 < 0,

p(x, y, δ) =∑

i+j>0

aij(x − x2)i(y − y2)j ,

q(x, y, δ) =∑

i+j>0

bij(x − x2)i(y − y2)j

(24)

for (x, y) near the nilpotent saddle S2(x2, x2). ThenM(h, δ) near the homoclinic loop Γhs has theform

M(h, δ) = c0(δ) + c1(δ)|h − hs| 34+ c2(δ)(h − hs) ln |h − hs|

+ c3(δ)(h − hs) + c4(δ)|h − hs| 54

+ c5(δ)|h − hs|74 + c6(δ)h2 ln |h|

+ O(h2)

for 0 < −(h − hs) � 1, where

c0(δ) = M(hs, δ) =∮

Γhs

qdx − pdy,

c1(δ) = 2√

2A0|h4|−1/4a0,

c2(δ) = −√

24

|h4|−1/2a1 + O1(a0),

c3(δ) =∮

Γhs

[(px + qy)|ε=0 − a0 − a1x]dt

+ O1(c1) + O1(c2),

c4(δ) = A2r20,

c5(δ) = −A0

(67r01 − 1

7r40

),

c6(δ) = −18

(34r11 − 1

4r50

),

(25)

where

a0 = (px + qy)|x=xs,y=ys ,

a1 = (pxx + qyx)|x=xs,y=ys

and r20, r01, r40, r11 and r50 can be found in [Hanet al., 2012], depending on the coefficients of (24).O1(c) denotes c times a constant, and A0, A2, A0

and A1 are constants with

A0 ≈ −0.8740191847 < 0,

A2 ≈ 0.2396280470 > 0,

A0 ≈ 1.236049785 > 0,

A1 ≈ −0.3513961008 < 0.

By [Han et al., 2010], the condition h40 < 0means that the saddle S2(x2, y2) is a nilpotentsaddle of order one. As in [Han et al., 2012],c1(s2), c2(s2), c4(s2), c5(s2) and c6(s2) are thelocal coefficients of Melnikov function M(h, δ) atthe nilpotent saddle S2(x2, y2), we denote themby c1(S2nsδ), c2(S2nsδ), c4(S2nsδ), c5(S2nsδ), andc6(S2nsδ), respectively.

After giving the definitions and formulas ofthe local coefficients of Melnikov function at ahyperbolic saddle S1(x1, y1) and a nilpotent sad-dle S2(x2, y2), we consider the expansion of M(h, δ)near a special heteroclinic loop. In the following,we assume system (3) has a heteroclinic loop Γhs

defined by H(x, y) = hs connecting a hyperbolic

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saddle S1(x1, y1) to a nilpotent saddle S2(x2, y2), asthe outer boundary of {Γh}, and the inner bound-ary of {Γh} is an elementary center. Then we have

Theorem 1. Under the condition we suppose,M(h, δ) near the heteroclinic loop Γhs has the form

M(h, δ) = c0(δ) + c1(δ)|h − hs| 34+ c2(δ)(h − hs) ln |h − hs|+ c3(δ)(h − hs) + c4(δ)|h − hs|

54

+ c5(δ)|h − hs| 74 + c6(δ)h2 ln |h|+ O(h2) (26)

for 0 < −(h − hs) � 1, where

c0(δ) = M(hs, δ) =∮

Γhs

qdx − pdy,

c1(δ) = c1(S2ns, δ),

c2(δ) = −c2(S2ns, δ) + c1(S1hs, δ),

c4(δ) = c4(S2ns, δ),

c5(δ) = c5(S2ns, δ),

c6(δ) = c6(S2ns, δ) + c3(S1hs, δ),

(27)

and c3(δ) ∈ R. In particular, if c1(S1hs, δ) =c1(S2ns, δ) = c2(S2ns, δ) = 0, we have

c3(δ) =∮

Γhs

(px + qy)dt.

To obtain more limit cycles we consider thelimit cycles bifurcated from the annulus not onlynear the center C(0, 0) but also near the hetero-clinic loop Γhs , based on the following theorem.

Theorem 2. Consider the expansions (22) and(26) of M(h, δ), suppose that there exists δ0 ∈ R

N

such that

c0(δ0) = c1(δ0)

= · · · = cm−1(δ0) = 0, cm(δ0) �= 0,

b0(δ0) = b1(δ0)

= · · · = bk−1(δ0) = 0, bk(δ0) �= 0

(28)

and

rank∂(c0, c1, . . . , cm−1, b0, b1, . . . , bk−1)

∂δ= m + k.

(29)

Then system (2) can have m + k +1−sgn(M(h1,δ0)M(h2,δ0))

2 limit cycles for some (ε, δ)near (0, δ0), m limit cycles of which are near theheteroclinic loop Γhs , k limit cycles are near thecenter C(0, 0) and another 1−sgn(M(h1,δ0)M(h2,δ0))

2limit cycle surrounds the center C(0, 0), whereh1 = hs − ε1, h2 = 0 + ε2 with ε1 and ε2 arepositive and very small.

Remark 3.1. Theorem 1 is proved in [Kazemi et al.,2012] for the local coefficients of Melnikov functionat the nilpotent saddle S2(x2, y2) based on the resultof [Zang et al., 2008], while we get the local coeffi-cients of Melnikov function at the nilpotent saddleS2(x2, y2) by Lemma 2.2 of [Han et al., 2012] andwe omit the proof here. Theorem 2, can be provedsimilarly as Theorem 2.1 proved in [Yang & Han,2011] by using implicit function theorem, then herewe omit its proof for the sake of brevity.

Now, we apply the above theorems to Lienardsystem (6) which is a special form of near-Hamiltonian system (2) with H(x, y) = H(x, y),p(x, y) = 0 and q(x, y) = (a + bx + cx3 + x4)y.We denote the hyperbolic saddle and the nilpotentsaddle of system (6)ε=0 respectively by S1(−1

2 , 0),and S2(1, 0). By Theorem 1, we have the followingexpansion of Melnikov function (8) for system (6)near the heteroclinic loop L 1

24:

M(h, δ) = c0(δ) + c1(δ)∣∣∣∣h − 1

24

∣∣∣∣34

+ c2(δ)(

h − 124

)ln∣∣∣∣h − 1

24

∣∣∣∣+ c3(δ)

(h − 1

24

)+ O

(∣∣∣∣h − 124

∣∣∣∣54

)

(30)

for 0 < −(h − 124) � 1. Taking δ = (a, b, c, 1), we

have

c0(δ) =∮

L 124

(a + bx + cx3 + x4)ydx

= 2∫ 1

− 12

(a + bx + cx3 + dx4)

√112

− 2A(x)dx.

Then

c0(δ) =√

3(

932

a +9

320b +

9896

c +279

35840

).

(31)

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In order to get the local coefficients c1(S2, δ) andc2(S2, δ) near the nilpotent saddle S2(1, 0), we intro-duce the transformation u := x − 1, v := y, thensystem (6) becomes

u = v, v =(

u − 32

)(u − 1)u3 + εq(u, v) (32)


H(u, v) =124

+12v2 − 3

8u4

−12u5 − 1

6u6, h40 = −3

8,

q(u, v) = (a + b + c + 1 + bu + 3cu + 4u

+ 3cu2 + 6u2 + cu3 + 4u3 + u4)v,

α0 = qy|x=1,y=0 = qv|u=0,v=0 = a + b + c + 1,

α1 = qyx|x=1,y=0 = qvu|u=0,v=0 = b + 3c + 4.

H(x, y) has the same form as (24), applying theformula for c1(S2ns, δ) and c2(S2ns, δ), we have

c1(S2ns, δ) =43

4√

54A0(a + b + c + 1),

c2(S2ns, δ) = −√

33

(b + 3c + 4) + O(α0).

(33)

In order to get the local coefficient c1(S1hs, δ) nearthe hyperbolic saddle S1(−1

2 , 0), we introduce thetransformation u := x + 1

2 , v := y, then system (6)becomes

u = v, v = u

(u +

12

)(u +

32

)3

+ εq(u, v) (34)


H(u, v) =124

+12v2 − 1

2u2 +

1627

√3u3 − 64

81u4

+10246561

√3u5 − 2048

59049u6,

q(u, v) =√

326244

(−5832b − 1458c + 11664a

+ 729 − 2592√

3u + 5184√

3bu

+ 3888√

3cu − 10368cu2 + 10368u2

+ 3072√

3cu3 − 6144√

3u3 + 4096u4)v,

H(x, y) has the same form as (23), applying theformula for c1(S1, δ), we have

c1(S1, δ) = −√

336

(16a − 8b − 2c + 1).

By (27), we have

c1(δ) = c1(S2ns, δ) =43

4√

54A0(a + b + c + 1),

c2(δ) = −c2(S2ns, δ) + c1(S1hs, δ)

= −√

336

(16a − 20b − 38c − 47) + O(α0).

(35)

Let c1(S1hs, δ) = c1(S2ns, δ) = c2(S2ns, δ) = 0, i.e.

a = 0, b =12, c = −3

2. (36)

Then we have

c3(δ) =∮

L 124

qydt =∮

L 124

qy

ydx

= 2∫ 1

− 12

a + bx + cx3 + dx4

√3

6(x − 1)2(2x + 1)

dx

=3√

34

(37)

if a = 0, b = 12 , c = −3

2 .For the expansion of Melnikov function (8) of

system (6) near the center L0, we have

M(h, δ) =∑j≥0

bjhj+1.

By using the programs in [Han et al., 2009], we have

b0(δ) = 2√

2πa, b1(δ) =376

√2πa + 2π

√2b,

b2(δ) =√

2(

12745216

πa +2059

πb +203

πc + 4π)

,

b3(δ) =√

2π(

1201028515552

a +140665

432b

+234518

c +1752

). (38)

Next, we will use the coefficients given in (31), (35)and (38) to discuss distributions of limit cycles ofsystem (6) by Theorem 2.

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(1) Firstly, solving b0(δ) = c0(δ) = 0 gives a = 0,b = − 31

112 − 514c. If taking δ0 = (0,− 31

112 − 514c, c, 1),

then

b1(δ0) = 2π√

2(− 31

112− 5

14c

),

c1(δ0) =43

4√

54A0

(81112

+914

c

).

Obviously, if we fix c ∈ (−3140 ,−9

8 ), then b1(δ0) ×c1(δ0) < 0 and 1−sgn(M(h1,δ0)M (h2,δ0))

2 = 1 for h1 =ε1 and h2 = 1

24 −ε2 with ε1 and ε2 positive and verysmall. Noting that

rank(

∂(b0, c0)∂(a, b, c, 1)

)= 2,

by Theorem 2, there exists some (a, b, c, 1) near(0,− 31

112 − 514c, c, 1) for c ∈ (−31

40 ,−98) and ε posi-

tive and very small such that system (6) has threelimit cycles, one limit cycle is near the heteroclinicloop L 1

24, one limit cycle is near the center L0 and

another one limit cycle surrounds the center L0

(between the center L0 and the heteroclinic loopL 1

24), see Fig. 2(a).

(2) Solving b0(δ) = b1(δ) = 0 gives a = 0, b = 0. Iftaking δ0 = (0, 0, c, 1), then

b2(δ0) =4√

23

π(5c + 3),

c0(δ0) =9√

335840

(40c + 31).

Obviously, if we fix c ∈ (−3140 ,−3

5 ), then b2(δ0) ×c0(δ0) < 0 and 1−sgn(M(h1,δ0)M (h2,δ0))

2 = 1 for h1 =ε1 and h2 = 1

24 −ε2 with ε1 and ε2 positive and verysmall. Noting that

rank(

∂(b0, b1)∂(a, b, c, 1)

)= 2,

by Theorem 2 there exists some (a, b, c, 1) near(0, 0, c, 1) for c ∈ (−31

40 ,−35 ) and ε positive and very

small such that system (6) has three limit cycles,two limit cycles of which are near the center L0

and another one limit cycle surrounds the centerL0 (between the center L0 and the heteroclinic loopL 1

24), see Fig. 2(b).

(3) Solving c0(δ) = c1(δ) = 0 gives a = 9112 +

114c, b = −121

112 − 1514c. If taking δ0 = (0, 9

112 + 114c,

−121112 − 15

14c, c, 1), then

c2(δ0) = −3√

3112

(25 + 16, c),

b0(δ0) =√

2π56

(9 + 8c).

If c ∈ (−∞,−2516 )∪(−9

8 ,+∞), then c2(δ0)b0(δ0) < 0

and 1−sgn(M(h1,δ0)M(h2,δ0))2 = 1 for h1 = ε1 and

h2 = 124 −ε2 with ε1 and ε2 positive and very small.

Noting

rank(

∂(c0, c1)∂(a, b, c, 1)

)= 2,

then by Theorem 2, there exists some (a, b, c, 1)near (0, 9

112 + 114c,−121

112 − 1514c, c, 1) with c ∈ (−∞,

−14396 ) ∪ (−9

8 ,+∞) for ε positive and very small,such that system (6) has three limit cycles, two limitcycles are near L 1

24, one limit cycle is surrounding

the center L0 (between the center L0 and the hete-roclinic loop L 1

24), see Fig. 2(c).

(4) Solving b0(δ) = b1(δ) = c0 = 0, gives a = 0, b =0, c = −31

40 . If taking δ0 = (0, 0,−3140 , 1), then

b2(δ0) = −7√

26

π < 0, c1(δ0) =310

4√

54A0 < 0

and 1−sgn(M(h1,δ0)M(h2,δ0))2 = 0 for h1 = 0 + ε1,

h2 = 124 −ε2 with ε1 and ε2 positive and very small.

Noting

rank(

∂(b0, b1, c0)∂(a, b, c, 1)

)= 3,

then there exists some (a, b, c, 1) near (0, 0,−3140 , 1)

for ε positive and very small, such that system (6)has three limit cycles, one limit cycle of which isnear L 1

24, two limit cycles are near the center L0,

see Fig. 2(d).

Remark 3.2. Using the similar method as above wecan obtain that: (i) there exists some (a, b, c, 1)such that system (6) has three limit cycles, twolimit cycles of which are near L 1

24, and one limit

cycle is near the center L0. See Fig. 2(e); (ii) thereexists some (a, b, c, 1) such that system (6) hasthree limit cycles near the center L0. See Fig. 2(f);

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(iii) however, we cannot obtain three limit cyclesnear the heteroclinic loop L 1

24, because when we

choose δ0 such that c0(δ0) = c1(δ0) = c2(δ0) = 0,then we cannot calculate c3(δ0), because it is adivergent improper integral.

Remark 3.3. In this paper, we investigate sys-tem (6) without the perturbative term x2, thatis because ∮

Lh

xydx =∮

Lh

x2ydx,

therefore x2 plays an equivalent role to the pertur-bative term x. Here we give a short proof.

Firstly, computing 2(x−1)2(2x+1)

from H(x, y) =h gives

2(x − 1)2(2x + 1)

= s(y)

where

s(y) =8n

(n23 + 1 + n

13 )2(n

23 + 1 + 2n

13 )

which is an even function of y, with

n = −1 + 2√

1 + 12y2 − 24h

+ 2√

−√

1 + 12y2 − 24h + 1 + 12y2 − 24h,

it is obvious that∮Lh

x2ydx −∮

Lh

xydx

=∮

Lh

(x2 − x)ydx

dydy

=∮

Lh

(x2 − x)y2

x

(x +

12

)(x − 1)3

dy

=∮

Lh

2y2

(x − 1)2(2x + 1)dy

=∮

Lh

s(y)y2dy.

Hence,

∮Lh

x2ydx −∮

Lh

xydx

=∫ y∗

0s(y)y2dy +

∫ 0

y∗s(y)y2dy

+∫ −y∗

0s(y)y2dy +

∫ 0

−y∗s(y)y2dy = 0

by symmetry and s(y) is an even function, whereH(0, y∗) = H(0,−y∗) = h.

Acknowledgments

The authors are thankful to the referees for theirhelpful comments and suggestion for this article.The authors also thank Dr. Yong Yao in School ofComputer Science and Engineering of University ofElectronic Science and Technology of China, andDr. Chenqi Mou and Jing Yang in School of Math-ematics and Systems Science of Beihang University,they do well in Triangularize and RealRootIsolate,they helped us to construct the direct program.

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Appendix A

> with(RegularChains):> with(ChainTools):> with(SemiAlgebraicSetTools):> sys := [p_1(x, z), p_2(x, z)]:> R := PolynomialRing([x, z]):

we use the package RegularChains , the subpackage ChainTools and SemiAlgebraicSetTools, define a set ofequations, and a polynomial ring.

> dec := Triangularize(sys, R);

[regular−chain, regular−chain, regular−chain, regular−chain]

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The Triangularize command decomposes the set sys into a list of regular chains, in other words, itdecomposes the set of common solutions of sys into lists of points by using symbolic expressions. Sinceregular chains may contain large expressions, their output form is just a word “regular−chain”. To viewtheir members, use the Equations command as follows:

> L := map(Equations, dec, R);[[r_{11}, r_{12}], ... , [r_{n1}, r_{n2}]]

As above n lists of regular chains are obtained. For a general regular chain, because it is a triangularset it is obvious that if the number of its polynomials equals the number of variables, then it has finitenumber of solutions, therefore it is zero-dimensional. Further suppose ri1 and ri2 are square-free, before weuse the command RealRootIsolate to find the roots of regular chain [ri1, ri2], we need to assure ri1 and ri2

are ordered as the ascending number of their variables by the command Chain(lp, rc,R) which returns theregular chain obtained by extending regular chain rc with polynomials lp. Let rij and ris have one variableand two variables respectively, Empty(R) denotes the empty regular chain of R.

> C := Chain([r[i][j], r[i][s]], Empty(R), R);regular_chain


> map(BoxValues, RL, R);

then we obtain some closed intervals with fraction endpoints, and the length of intervals is equal to orless than the map(BoxValues ,RL, R) retrieves the box value 1

105 . For convenience, we translate fractionintervals into decimal form.

> evalf(%).

Appendix B

p11(z) = 20076080736816332800z60 − 602282422104489984000z59

+ 8481231562180499865600z58 − 73980843371725659832320z57

+ 442761579372626439045120z56 − 1889045730962753338736640z55

+ 5678502821491329504116736z54 − 10694456575969835228332032z53

+ 3979912981804419964207104z52 + 51279126890157950497718272z51

− 202826936348718191886729216z50 + 431584067394706544472883200z49

− 538054642811926344974204928z48 + 176138038584145305069944832z47

+ 769819560257030434760687616z46 − 1688944319837783380586397696z45

+ 1410292248081927574767796224z44 + 535411610203152876111396864z43

− 2710221900848079414379151360z42 + 2651824436013617089582399488z41

+ 236270569257881879690674176z40 − 3211608484535977302669656064z39

+ 2906531219410675393737498624z38 + 457079855532617472189935616z37

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− 3019344355086081200131627008z36 + 2012173898973840221932560384z35

+ 901165474172214706663378944z34 − 2124594846111659493073936640z33

+ 753684074294255041270338816z32 + 936338437534820737589342592z31

− 1002669209462415171582281760z30 − 5710726075760952460021008z29

+ 552685655787654790218335760z28 − 259436216043009953400084168z27

− 153358673678080060261667202z26 + 183375703771676683672087656z25

− 5938747061691717431424848z24 − 72556737238525374190279434z23

+ 27542140042416267953050224z22 + 17231640144491316281402778z21

− 14666848580423560866223014z20 − 1283149160385649119138528z19

+ 4752773068448344874282562z18 − 843333403396050585532692z17

− 1052339079823882508156454z16 + 440646687355169996227240z15

+ 150005193428374715899734z14 − 122453122445290291060830z13

− 7694301229276457541663z12 + 23397949112869582846704z11

− 2247268434116139873216z10 − 3210656701459981347072z9

+ 702651711688270271568z8 + 311096034287933017104z7

− 105592245590518375680z6 − 19924815831573314304z5

+ 9795733305594504192z4 + 722766478457281536z3

− 544567112855543808z2 − 9539732624449536z + 14258446728560640.

p12(z) = 20076080736816332800z61 − 682586745051755315200z60

+ 10991654203406942208000z59 − 111197790586723337502720z58

+ 789636425761484596838400z57 − 4158572320315048093286400z56

+ 16683137107654508046974976z55 − 51268850990519685067309056z54

+ 118168281211737410126217216z53 − 188014695812369997828718592z52

+ 137019139192852169074147328z51 + 234435573840921537879736320z50

− 971841895936325845436596224z49 + 1591070482864753495196565504z48

− 1082443239183150340940759040z47 − 1118411073084628243011600384z46

+ 3728600583463603546663944192z45 − 3816389560933145855245418496z44

− 191313756227965207986176000z43 + 5394472269926720060151627776z42

− 6113835334221015870160502784z41 + 426422327405627935973670912z40

+ 6158769814898061038753808384z39 − 6251425361338152137438232576z38

− 272178721029553250247954432z37 + 5721281877069803506239461376z36

− 4186557226024486788024331776z35 − 1402657536822434144395616768z34

1450004-19

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X. Sun et al.

+ 3992328476763440497148637440z33 − 1577233255732519736695555392z32

− 1627516698416754705380272128z31 + 1867526048397104267944133904z30

− 54454192135900045728137928z29 − 978087543164554730042303004z28

+ 483169429277218249763250414z27 + 256483225230908915165619168z26

− 323839529256745276235178644z25 + 16333724975951391139520006z24

+ 124075200363333453349426668z23 − 48733313648870629572371562z22

− 28529562598265214612115710z21 + 25003260602273122778669436z20

+ 1890645725528007077815986z19 − 7918465559774257602940296z18

+ 1456925592073384145952738z17 + 1719171427678953837850204z16

− 730615906452206564024242z15 − 239787353298689482847028z14

+ 198966791897184350136897z13 + 11552876910025103072880z12

− 37420771181330970052992z11 + 3681133747127119372896z10

+ 5062983217891184644848z9 − 1114242927059313005040z8

− 484117927162657899264z7 + 164661397444830534912z6

+ 30609515761694309376z5 − 15067431231894365184z4

− 1096021599366979584z3 + 827181729767153664z2

+ 14258446728560640z − 21398641075814400.

p2(z) = 100762254437791957024289070210088960000z116

− 5844210757391933507408766072185159680000z115

+ 164943230412190988559805558430724259840000z114

− 3016153499851484939687989233127719436288000z113

+ 40136669725663537830504971709386233741312000z112−, . . . ,

+ 150817052899752034770718996561920z4 − 45075119245463782065935493365760z3

− 3862358979395134985905530470400z2 + 625793125761184654231496294400z

+ 58691031724378396645392384000.

r11 = 17104178353183937422950400z81 − 692719223303949465629491200z80

+ 13231053784503297411278438400z79 − 157101360060256582621441556480z78

+ 1280181624412131888640787742720z77 − 7367125177122274138843345059840z76

+ 28578444995835831882451775389696z75 − 52975351145120206463381903769600z74

−183208349672022684231869078175744z73 + 2134297330432808821042044974661632z72

− 10848461682380664237812522472701952z71 + 38230087762031486216118101556068352z70

− 101049930353429692862869595823800320z69 + 199591361710646279960561074997035008z68

1450004-20

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February 5, 2014 8:45 WSPC/S0218-1274 1450004


− 268268002545839463434937280952795136z67 + 135003459540200157898869155727147008z66

+ 399854379182294748929741577001107456z65 − 1269443312313288115483952156107603968z64

+ 1798798493107795668799134028777127936z63 − 877769739315150108477677793473200128z62

− 1896755242007454933331108658329681920z61 + 4817594283805097097913091993964118016z60

− 4452637482279053474841206930226020352z59 − 990647642983113162085404167381188608z58

+ 7982793406917407164350292119636672512z57 − 9081290657137449240758381353164079104z56

+ 711822182255291542689319654420119552z55 + 10449374688657170201429260337728716800z54

− 12326411211729047512085023075615113216z53 + 1187619795472513722944578762652516352z52

+ 11849592042073174669551061766003228672z51 − 12257116885028861866325440928120045568z50

− 442921253969517100059364927887114240z49 + 11504394495709237932900055098155859968z48

− 8823582060755360742192719502472249344z47 − 2772784850839298443479062933792784384z46

+ 8955376681542570438018312799203172352z45 − 4099445744769761123657108171515207680z44

− 3732334019550373612526520596277891072z43 + 5155784224900074660245154529817984000z42

− 636959917478999753470211595981490176z41 − 2873611692477483199193930350251289344z40

+ 1950438646485318465519909955300732160z39 + 637185846301834697890724352357532032z38

− 1391825810338906028405569054627885728z37 + 325794066037991484683186970068004416z36

+ 555461339389108115185426816971948672z35 − 399236913715253480978626962971781360z34

− 91715054498883963132761640020722300z33 + 213855051631371400444167036657340968z32

− 42295827622461713094646930119086868z31 − 71728417195734793905631950263618800z30

+ 41439317352336910301467211466386100z29 + 12955684917454510481525844378341256z28

− 18887239048472384355111370799628212z27 + 1331751544232817758319896200460280z26

+ 5747182857469147674420736664824500z25 − 2005128263536368325177859921558084z24

− 1164035728492254275351469746885670z23 + 855156848195314023938702007157230z22

+ 104842126703798838493729369749130z21 − 239590020310530891800330466446172z20

+ 26725189891893473026302271166751z19 + 48526567739126244705500026721028z18

− 15045076006838073600264471452076z17 − 6941220196436967647884356158832z16

+ 3997560992204708709673231984230z15 + 571438535418057995177383683846z14

− 734121340249101185069409902670z13 + 15796206880770545441551564686z12

+ 99716196575011399564995330624z11 − 13723591070731667202657332928z10

− 10057913748155386043642558592z9 + 2406363151653591998479664256z8

+ 730242390330959571299377152z7 − 254157559941579431676936192z6

− 35471569115647564742873088z5 + 17516212028505486454247424z4

+ 983098550385306256932864z3 − 740627788437100467191808z2

− 9939787349502401445888z + 14909137018345707798528.

1450004-21

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r12 = 17104178353183937422950400z82 − 803896382599645058878668800z81

+ 18010699805902686106366771200z80 − 255705393341057146961348526080z79

+ 2577152907588081906043259453440z78 − 19550253613510868606717609902080z77

+ 115268580097343304691108802461696z76 − 536064537951024395046777880838144z75

+ 1962444639430870358791549668556800z74 − 5495683233607213692508091189297152z73

+ 10625546670506766815502528766017536z72 − 7611165627973272870696864517718016z71

− 36786611508132536306297727449300992z70 + 184978313660072110631349902743961600z69

− 481386362605138295162461106730958848z68 + 811003839396946464703296813620264960z67

− 740716236171483395179157732926160896z66 − 356696477857707704784279692434735104z65

+ 2577591630495903667060235793613193216z64 − 4488724747052179269714267508622491648z63

+ 3263956923509633812447403307253628928z62 + 2708214790599016811871398672652566528z61

− 10165457838952698807751123444896366592z60 + 11162721643972635354425287856413999104z59

− 451891981285255205229293619852083200z58 − 15494900070506813112301657301186510848z57

+ 20329683509944213340374123772192489472z56 − 4516024343921111392124429957488181248z55

− 19542166360801255285843572172961873920z54 + 26134213038414168124605918423114842112z53

− 5234662102861168720604556349424533504z52 − 22002775589389287415493676427847925760z51

+ 25243675365578490933620697973704622080z50 − 1302216188924697882153622988901842944z49

− 21436871427368297649107356581047959552z48 + 17930028614815824263834439295409012736z47

+ 3991171959670395706216820567143129088z46 − 16726854584958176125742689850304937984z45

+ 8397664633381557480840356420697919488z44 + 6394526050975536695471097652629791744z43

− 9628585467388706410347701550138537472z42 + 1537886187098357597443233603246206208z41

+ 5098227096864416655434691418897361024z40 − 3653594221043336533292747405855673664z39

− 1018493646998210799581436343451072544z38 + 2488334522652847858447527443651040320z37

− 637475386000048441602200912381668816z36 − 956877820728261886705869374556224232z35

+ 716186998969643648448536068082240900z34 + 144654273753600087393565160215763168z33

− 371623312649831892243427761693071460z32 + 78897856875281416561261450503154424z31

+ 121396660078634163191052009857730740z30 − 72241201537057651603390586726642976z29

− 21004670314640852795716343507685812z28 + 32102617727335249137118754840152528z27

− 2588275705187796886617754617046764z26 − 9579761873574180945738912749154056z25

+ 3424719847998434738849993552781454z24 + 1899624690983838930999726511549266z23

− 1425056314996221101769813370638386z22 − 162242523334974335289929855745206z21

+ 392763325377317248439672147847255z20 − 45745644532264500211454121081300z19

− 78422017605864481176627238233876z18 + 24640676963927503496384757245892z17

1450004-22

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February 5, 2014 8:45 WSPC/S0218-1274 1450004


+ 11053703354270458460416133314122z16 − 6436121406876471661818793572942z15

− 890048766159935066455154158338z14 + 1167034053212818366426997248782z13

− 27719922589919953315653301056z12 − 156781092934647017280716681664z11

+ 21759280660134300092327019648z10 + 15654040015916236642173076608z9

− 3754406757769706093077831680z8 − 1125657405031870550754385920z7

+ 392061798407397202238361600z6 + 54173753129908727090860032z5

− 26754730621380743859929088z4 − 1487879370031081394405376z3

+ 1120877372390971390820352z2 + 14909137018345707798528z

− 22363599955943604879360.

r2 = 9142278660547687174528234597529501073020026880000z154

− 708526596192445756025938181308536333159052083200000z153

+ 26901674406312736993275155771105416538104214323200000z152

− 666693104907788077509520641968600619485206006464512000z151

+ 12122102218128624589831305310173939216977179896184832000z150

− 172327054857736570167148876615443329998270637109411840000z149

+ · · · + 6117647888796850727000116606020618289152z2

− 357614818522263004884872289219961159680z

− 73780014657912552064667012929274511360

1450004-23

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bifurcation of limit cycles in small perturbation of a class of liénard systems

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