# Nine Limit Cycles in a 5-Degree Polynomials Liénard System

Authors: Junning Cai; Minzhi Wei; Hongying Zhu
Journal: Complexity
 (2020)

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## Abstract

In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.


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## Body

## 1. Introduction

Consider the following perturbed Hamiltonian system:(1)x˙=Hyx,y+εpx,y,δ,y˙=−Hxx,y+εqx,y,δ,where Hx,y is a polynomial of degree n+1, px,y,δ and qx,y,δ are polynomials of degree n, ε is the sufficiently small perturbation parameter, and δ∈D⊂ℝN with D compact. Taking ε=0, system (1) becomes a Hamiltonian system:(2)x˙=Hyx,y,y˙=−Hxx,y,and system (1) is usually called a near-Hamiltonian system. We assume that the unperturbed system (2) has a family of periodic orbits Γh defined by x,y|Hx,y=h, and the family forms a periodic annulus. The inner boundary of the periodic annulus is a center which may be an elementary one or a nilpotent one, and the outer boundary of the periodic annulus is usually a homoclinic loop or a heteroclinic loop or a polycycle. Most periodic orbits are broken when system (2) is perturbed and only a finite number of periodic orbits persist as limit cycles of system (1), which is the so-called Poincaré bifurcation. The most efficient tool to study Poincaré bifurcation of system (1) is the following first-order approximation of the Poincaré map:(3)Mh,δ=∮Γhqdx−pdy,which is a continuous integral over the continuous ovals Γh of Hx,y and is usually called the first-order Melnikov function or Abelian integral. The zeros of Mh,δ correspond to the limit cycles by Poincaré bifurcation for system (1) (see [1–3]). It should be noted that studying the maximal number of zeros of Mh,δ is the topic of weak Hilbert’s 16th problem. We note that the original version of Hilbert’s 16th problem asks the maximal number of limit cycles of a general polynomial system:(4)x˙=Pnx,y,y˙=Qnx,y.LetHn denote the maximal number of limit cycles of system (4) and H∗n denote the maximal number of limit cycles of system (1). Most results on the lower bounds of Hn are obtained in the literature as we know, by studying H∗n. However, it should be noted that it is also very difficult to determine H∗n for a given degree n. We refer the works [4] for the relatively new results on the lower bounds of H∗n.In order to reduce the difficulty, researchers usually study some special forms of system (1), such as(5)x˙=y,y˙=gx+εfxy,which is called Liénard system of type m,n having the degree n∗=maxn+1,m, where gx and fx are polynomials of degrees m and n, respectively, and ε is positive and very small. Let Hn,m denote the maximal number of limit cycles of system (5) and In∗ denote the maximal number of limit cycles of system (5) of degree n∗. Dumortier and Li studied four Liénard systems with different portraits of type 3,2 in a series of papers [5–8] and gave the corresponding sharp bound of number of limit cycles by Poincaré bifurcation. The exact bounds of H4,3 and H5,4 for some special Liénard systems were reported in [9–14] and references therein. For results on H7,6 associated with symmetric system (5), the relatively new works are referred [15–17]. It is usually very difficult to give the exact bound; however, lots of lower bounds of Hm,n have been obtained by studying the number of limit cycles near the center, homoclinic loops, and heteroclinic loops (see [18]). In particular, Xu and Li [19] investigated a Liénard system of type 5,n and proved H5,2≥3, H5,4≥5, H5,6≥10, H5,8≥10. In this paper, we study a type 5,4 Liénard system which has a polycycle consisting of a homoclinic loop and a heteroclinic loop:(6)x˙=y,y˙=−17x7x+3x−1x+12+εfxy,with special elliptic Hamiltonian function:(7)H˜x,y=12y2+16x6+27x5−17x4−1021x3−314x2,where 0<ε≪1, fx=∑i=1naixi, ai (i=0,1,…,n) are real bounded parameters. System (6) has the degree n∗=max5,n+1. Let H∗5,n denote the maximum number of limit cycles of (6) and I∗n∗ denote the maximum number of limit cycles of (6) of degree n∗. Our main interest is focused on H∗5,4.The level sets (i.e.,H˜x,y=h) of Hamiltonian function (7) are sketched in Figure 1. H˜x,y=h defines the periodic orbits of system (6). There are a heteroclinic loop and a homoclinic loop for system (6)ε=0 defined by H˜x,y=0, denoted by L1 and L2, respectively; let us in the following say L∗=L1∪L2 is a hetero-homoclinic loop. L1 connects the nilpotent cusp S1−1,0 and the hyperbolic saddle S20,0 of system (6)ε=0, L2 connects a the hyperbolic saddle S20,0 of system (6)ε=0. There are three families of clockwise periodic orbits, one family is L1h=x,y|−25/2688<H˜x,y<0 inside L1 surrounding a center C1−3/7,0 with H−3/7,0=−25/2688, one family is L2h=x,y|−8/21<H˜x,y<0 inside L2 surrounding another center C21,0 with H1,0=−8/21, and the third family is L3h=x,y|0<H˜x,y<+∞ surrounding L1 and L2. Three families of periodic orbits form three periodic annuli, the boundaries of which are L1 and C1, L2 and C2, and L∗.Figure 1
The portrait of system (6) when ε=0.For system (6), we get the following main result.Theorem 1.
There exist somea0, a1, a2, a3 such that system (6) has 9 limit cycles for the type 5,4 system (6). Therefore, H5,4≥H∗5,4≥9, I5≥I∗5≥9.The rest of this paper is organized as follows. In Section2, we present some preliminaries which will be used in the next section. In Section 3, we study the asymptotic expansions of the related Melnikov functions for system (6). The proof of the main result is given in Section 4. Conclusion and discussions are drawn in Section 5.


## 2. Preliminary Lemmas

Consider the Melnikov functionMh,δ for the near-Hamiltonian system (1); we suppose the Hamiltonian system (2) has a bounded periodic annulus Γh denoted by Hx,y=h, and the period orbits in the periodic annulus are clockwise. The boundary of Γh can be a center, a homoclinic loop, and a heteroclinic loop. We suppose the inner boundary of Γh is defined by Hx,y=α and the outer boundary is defined by Hx,y=β; correspondingly, we have(8)Mh,δ=∮Γhqdx−pdy,α<h<β,and Mh,δ can be expanded near its boundary (see [20–23]).When the inner boundary ofΓh is elementary center, we suppose it is located at Cxc,yc with Hxc,yc=α, and for x,y near Cxc,yc,(9)Hx,y=α+12x−xc2+12y−yc2+∑i+j≥3h˜ijx−xciy−ycj,px,y,δ=∑i+j≥0a˜ijx−xciy−ycj,qx,y,δ=∑i+j≥0b˜ijx−xciy−ycj.For the expansion ofMh,δ near Cxc,yc, we have the following.Lemma 1 (see [20]).
Under the condition we suppose above, for the expansion ofMh,δ near the elementary center Cxc,ych=α, we have(10)Mh,δ=∑j≥0bjδh−αj+1,α<h≪1.Under (9), the formulas of bj can be obtained by using the programs in [20].When the outer boundary ofΓh is homoclinic loop denoted by Γβ passing through a hyperbolic saddle Sxs,ys satisfying Hxs,ys=β, for the expansion of Mh,δ near the homoclinic loop, we have the following.Lemma 2.
(i)
(see [21]). Under the condition we suppose above, the function Mh,δ has the following expansion:(11)Mh,δ=c0δ+c1δh−βlnh−β+c2δh−β+c3δh−β2lnh−β+Oh−β2,for0<−h−β≪1, where c1δ and c3δ depend on the coefficients of Hx,y, px,y, and qx,y.(12)c0δ=Mβ,δ=∮Γβqdx−pdy|ε=0,c2δ=∮Γβpx+qy−σdt+bc1δ,forσ=px+qy|x=xs,y=ys is divergence at Sxs,ys, b is a constant.(ii)
(see [21]). In particular, for x,y near the hyperbolic saddle Sxs,ys,(13)Hx,y=β+λ2y−ys2−12x−xs2+∑i+j≥2h¯ijx−xsiy−ysj,px,y,δ=∑i+j≥0a¯ijx−xsiy−ysj,qx,y,δ=∑i+j≥0b¯ijx−xsiy−ysj.
Then,(14)c1δ=−1λa¯10+b¯01,c3δ=−12λλ−3a¯30−b¯21+a¯12+3b¯03−1λ2b¯02+a¯113h¯03−h¯21+2a¯20+b¯11h¯30−h¯12.Definition 1.
The valuesc1δ and c3δ are, respectively, called the first and second local Melnikov coefficients at the saddle Sxs,ys, denoted by c1Ssaddle,δ, c3Ssaddle,δ, respectively.
Note.(i)
The local coefficients are presented in the expansion of a Melnikov function, which are local quantities at a singular point depending on the coefficients of the expansions ofHx,y, px,y, and qx,y at the singular point.(ii)
When there is a nilpotent cuspSxn,yn connecting a homoclinic loop, there are similarly some local coefficients at the cusp Sxn,yn presented in the expansion of the corresponding Melnikov function near the homoclinic loop; we denote these first three local coefficients by c1Scusp,δ, c3Scusp,δ, and c4Scusp,δ (see [22]).Lemma 3 (see [22]).
Suppose there is a nilpotent cuspSxn,yn which is a limit point of a homoclinic loop or a heteroclinic loop, and for x,y near the nilpotent cusp Sxn,yn:(15)Hx,y=β+12y−yn2+∑i+j≥3hijx−xniy−ynj,h30<0,px,y,δ=∑i+j≥0aijx−xniy−ynj,qx,y,δ=∑i+j≥0bijx−xniy−ynj.Then, the first three local coefficients of the corresponding Melnikov functions at the nilpotent cuspSxn,ynare as follows:(16)c1Scusp,δ=22h30−1/3a10+b01,c3Scusp,δ=22h30−5/3h302a20+b11−h12a10+b01+1/3h212−2h40a10+b01,c4Scusp,δ=9μ1−1α01−2μ1−720μ23−20μ1μ2μ3+4μ12μ4α00+4μ12μ3−10μ1μ22α10+4μ12μ2α20−μ13α30,μ1=h303,μ2=−16h30−2/3−2h40+h212,μ3=136h30−5/312h30h50−h31h21+h12h212−4h402+4h40h212−h214,μ4=−1648h30−8/35h216−40h403+30h40h2122h40−h212+144h30h40h50−h31h21+h12h212+72h30h212−h50+h31h21−h12h212+216h302−h60−h22h212+h41h21+h03h213+108h302h312+432h302h12h21h12h21−h31,α00=22a10+b01,α10=22−h12a10+b01+2a20+b11,α20=22a10+b013h03h21−h22+32h122−2h12a20−h12b11+3a30+b21−a11h21−2b02h21,α30=22a10+b013h13h21+3h03h31+3h12h22−15h12h03h21−52h123−h32+3a11+6b02h12h21−2b12+a21h21−a11+2b02h31+4a40+b31+3b11+6a20h03h21−2a20+b11h22+3a20+32b11h122−3a30+b21h12,α01=2223a12+2b03−2h03a11−4h03b02+a10+b015h032−2h04.When the outer boundary ofΓh is heteroclinic loop denoted by Γβ passing through a hyperbolic saddle Sxs,ys and a nilpotent cusp Nxn,yn satisfying Hxs,ys=Hxn,yn=β, for the expansion of Mh,δ near the heteroclinic loop Γβ, we have the following.Lemma 4 (see [23]).
The expansion ofMh,δ near the heteroclinic loop Lβ has the form(17)Mh,δ=c˜0δ+B00c˜1δh−β5/6+c˜2δh−βlnh−β+c˜3δ+b1c˜1δ+b2c˜2δh−β+B10c˜4δh−β7/6+c˜5δh−β2lnh−β−111B00c˜6δh−β11/6+Oh−β2.for 0<−h−β≪1, where B00=3/5∫01dv/v1−v3=3/5×2.4286…, B10=−3/7∫01v3/2dv/1−v31+1−v3−2>0, b1 and b2 are constants, and(18)c˜0δ=Mβ,δ=∮Γβqdx−pdy,c˜1δ=c1Ncusp,δ,c˜2δ=c1Ssaddle,δ,c˜4δ=c3Ncusp,δ,c˜5δ=c3Ssaddle,δ,c˜6δ=c4Ncusp,δ,and c˜3δ∈R. In particular, if c˜1δ=c˜2δ, we have(19)c˜3δ=∮Γβpx+qydt.In many cases, the Hamiltonian function is not of the form supposed in the above lemmas. Then, to apply the lemmas, we need to first introduce suitable linear change of variables which will cause a change of the first-order Melnikov function. The following lemma gives the relationship between the old and new Melnikov functions.Lemma 5 (see [2]).
Under the linear change of variables of the form(20)u=ax−x0+by−y0,v=cx−x0+dy−y0,and time rescaling τ=kt, where D=ad−bc≠0, system (31) becomes(21)dudτ=H˜v+εp˜,dvdτ=−H˜u+εq˜,where(22)H˜u,v=DkHx,y,p˜u,v,δ=1kapx,y,δ+bqx,y,δ,q˜u,v,δ=1kcpx,y,δ+dqx,y,δ.
Let(23)M˜h,δ=∮Γ˜hq˜du−p˜dv|ε=0,which is the Melnikov function of system (21). Then,(24)Mh,δ=kDM˜Dkh,δ.Remark 1.
Usually when we apply Lemma5, we always take a linear transformation such that D/k=1 because under this condition, the local coefficients between the old and the new ones, respectively, are the same (if k/D=1) or only different from a symbol “−” (if k/D=−1).
Let us now suppose system (2) has a hetero-homoclinic loop Γ∗=Γ1∪Γ2, where Γ1 is heteroclinic loop connecting a cusp S1x1,y1 and a hyperbolic saddle S2x2,y2 satisfying Hx1,y1=Hx2,y2=β and Γ2 is homoclinic loop passing through the previous hyperbolic saddle Sx2,y2. There exist three families of orbits, the first family of periodic orbits Γh1 surrounding a center C1x3,y3 with Hx3,y3=α1 in Γ1, the second family of periodic orbits Γh2 surrounding another center C2x4,y4 with Hx4,y4=α2 in Γ2, and the third family of periodic orbits Γh∗ surrounding Γ∗. The portrait of Γ∗ is just the same as Figure 1.
Correspondingly, we have three Melnikov functions:(25)Mih,δ=∮Γhiqdx−pdy,forαi<h<β,i=1,2,M3h,δ=∮Γh3qdx−pdy,forβ<h<+∞.
From Lemmas2–4, we have the following.Lemma 6.
Under the condition supposed above, the three Melnikov functions have the following expansions:(26)M1h,δ=e0δ+B00e1δh−β5/6+e2δh−βlnh−β+e3δ+b1e1δ+b2e2δh−β+B10e4δh−β7/6+e5δh−β2lnh−β−111B00e6δh−β11/6+Oh−β2,for 0<−h−β≪1.(27)M2h,δ=c0δ+c1δh−βlnh−β+c2δh−β+c3δh−β2lnh−β+Oh−β2,for 0<−h−β≪1.(28)M3h,δ=c˜0δ+B00∗c˜1δh−β5/6+c˜2δh−βlnh−β+c˜3δ+b3c˜1δ+b4c˜2δh−β+B10∗c˜4δh−β7/6+c˜5δh−β2lnh−β−111B00∗c˜6δh−β11/6+Oh−β2,for 0<h−β≪1, where B00 and B10 are the same as before, B00∗<0 and B10∗<0, b1, b2, b3 and b4 are some constants. and e1δ=c1S1cusp,δ, e2δ=c1S2saddle,δ, e4δ=c3S1cusp,δ,e5δ=c3S2saddleδ, e6δ=c4S1cusp,δ, c1δ=c1S2saddle,δ, c3δ=c3S2saddle,δ, c˜1δ=c1S1cusp,δ, c˜2δ=2c1S2saddle,δ, c˜4δ=c3S1cusp,δ, c˜5δ=2c3S2saddle,δ, c˜6δ=c4S1cusp,δ, where c1S1cusp,δ, c3S1cusp,δ, and c4S1cusp,δ are local coefficients at the nilpotent cusp S1x1,y1 which can be obtained by Lemma 3 and c1S2saddle,δ and c3S2saddle,δ are local coefficients at the hyperbolic saddle S2x2,y2 which can be obtained by Lemma 2, while(29)e0δ=M1β,δ=∮Γ1qdx−pdy,c0δ=M2β,δ=∮Γ2qdx−pdy,c˜0δ=e0+c0,and e3δ∈R; in particular, if c1S2saddle,δ=c1S1cusp,δ=0, we have(30)e3δ=∮Γ1px+qydt,and c2δ=∮Γ2px+qy−a10−b01|ε=0dt+bc1δ∈R; in particular, if c1S2saddle,δ=0, we have(31)c2δ=∮Γ2px+qydt,and c˜3δ=∮Γ1px+qydt+∮Γ2px+qydtδ if c1S2saddle,δ=c1S1cusp,δ=0.


## 3. The Expansions of Melnikov Functions of System (6)

Liénard system (6) is a special form of near-Hamiltonian system (1) with Hx,y=H˜x,y, px,y=0,qx,y=fxy=∑i=04aixiy. In this section, we take n=4; then, system (6) is a Liénard system of type 5,4. There are three Melnikov functions corresponding to three periodic annuli of system (6)ε=0:(32)M1h,δ=∮L1hfxydx,for−252688<h<0,M2h,δ=∮L2hfxydx,for−821<h<0,M3h,δ=∮L3hfxydx,for0<h<+∞.By Theorem1, we obtain(33)M1h,δ=e0δ+B00e1δh5/6+e2δhlnh+e3δ+b1e1δ+b2e2δh+B10e4δh7/6+e5δh2lnh−111B00e6δh11/6+Oh2,for 0<−h≪1.(34)M2h,δ=c0δ+c1δhlnh+c2δh+c3δh2lnh+Oh2,for 0<−h≪1.(35)M3h,δ=c˜0δ+B00∗c˜1δh5/6+c˜2δhlnh+c˜3δ+b3c˜1δ+b4c˜2δh+B10∗c˜4δh7/6+c˜5δh2lnh+111B00∗c˜6δh11/6+Oh2,for 0<h≪1.By Lemma1, for 0<h+25/2688≪1,(36)M1h,δ=∑i=0biδh+252688i+1,and for 0<h+8/21≪1,(37)M2h,δ=∑i=0diδh+821i+1.In the following, we use the preliminary lemmas to compute the coefficients of the above expansions forM1h,δ, M2h,δ, and M3h,δ. Firstly, we have(38)e0δ=M10,δ=∮L1fxydx=2∫−10fx−2Axdx=∑i=09kiai,where(39)k0=741214802−25632401π+51232401arcsin18,k1=11912112005−25632401π+51232401arcsin18,k2=1746721168070−4352350421π+8704350421arcsin18,k3=382302214117715−744963823543π+1489923823543arcsin18,k4=1152596721115296020−50611235764801π+101222435764801arcsin18.Secondly,(40)c0δ=M20,δ=∮L2fxydx=2∫09/7fx−2Axdx=∑i=09Iiai,where(41)I0=741214802+25632401π+51232401arcsin18,I1=11912112005+25632401π+51232401arcsin18,I2=1746721168070+4352350421π+8704350421arcsin18,I3=382302214117715+744963823543π+1489923823543arcsin18,I4=1152596721115296020+50611235764801π+101222435764801arcsin18.Therefore,(42)c˜0δ=e0δ+c0δ=∑i=19I˜iai,where I˜i=ki+Ii.In order to get the local coefficientsc1S1cusp,δ at the cusp S1−1,0, we introduce the transformation u≔x+1,v≔y; then, system (6) becomes(43)u˙=v,v˙=−u5+257u4−267u3+87u2+εq¯u,v,with Hamiltonian function H¯u,v=1/6u6−5/7u5+13/14u4−8/21u3+1/2v2, h30=−8/21, q¯u,v=fxy|x=u−1,y=v, q¯v|u=0,v=0=a0−a1+a2−a3+a4.Applying the formula forc1S1cusp,δ, c3S1cusp,δ, c4S1cusp,δ in Lemma 3, in accordance with Lemma 6, we have(44)e1δ=c˜1δ=c1S1cusp,δ=−211/32a0−a1+a2−a3+a4,e4δ=c˜4δ=c3S1cusp,δ=−212/321613a0−5a1−3a2+11a3−19a4,e6δ=c˜6δ=c4S1cusp,δ=212132409611669a0−4765a1+1189a2+83a3−75a4.In order to get the local coefficientc1S2saddle,δ near the hyperbolic saddle S20,0, we introduce the transformation u:=3/7x,v:=y,τ=3/7t; then, system (6) becomes(45)dudτ=v,dvdτ=−34327u5−702721u4+289u3+10921u2+u,with Hamiltonian system H^u,v=343/162u6+14/2721u5−7/9u4−10/2721u31/2u2+1/2v2, and q^u,v=3/7∑i=0aixiy|x=7/3,y=v. Applying the formula for c1S2saddle,δ and c3S2saddle,δ in Lemma 2, in accordance with Lemma 5, we have(46)e2δ=c1δ=c˜2δ=c1S2saddle,δ=−1321a0,c˜2δ=2c1S2saddle,δ=−2321a0,e5δ=c3δ=c3S2saddle,δ=−7215410a1−3a2,c˜5δ=2c3S2saddle,δ=−7212710a1−3a2.Lete1δ=e2δ=0; we have(47)a0=0,a1=a2−a3+a4.Under this case,(48)e3δ=∮L1fxdt=∮L1fxydx=∮L1fx−2Axdx=2∫−10fx−2Axdx.Taking (47) into (48), we have with the help of Maple 13,(49)e3δ=∑i=29liai,where(50)l2=23arcsin18−3π,l3=−1273arcsin18+673π+6217,l4=150349arcsin18−75349π−332149.Letc1δ=0, i.e., a0=0; we have(51)c2δ=∮L1fxydx=∮L2fx−2Axdx=2∫09/7fx−a0−2Axdx.Direct computation gives(52)c2δ=∑i=19Jiai,where(53)J1=3/421,J2=23arcsin12−3421+3π,J3=−1237arcsin18+452128−637π,J4=150349arcsin18−27921196+75349π.Then, by (49) and (51) and Lemma 6, we have(54)c˜3δ=∑i=19J˜iai,where J˜1=J1,J˜i=li+Ji,i=2,3,…,9.Last we computeb0, b1, d0, and d1 in (36) and (37). Taking a linear transformation u=430/49x+3/7, v=y with time scaling dτ=430/49dt and applying Lemma 5 and program in [20], we obtain(55)b0=30π4960a0−720a1+320a2−9140a3+27980a4,b1=30π93966256399532800a0−829307801165888000a1+25400419155296000a2−7178269718432000a3+176305436144000a4.Taking a linear transformationu=40/7x−1, v=y with time scaling dτ=40/7dt and applying Lemma 5 and program in [20], we obtain(56)d0=110π70a0+a1+a2+a3+a4,d1=77064000π327a0+111a1−25a2−81a3−57a4.


## 4. Proof of Theorem1

In this section, we prove Theorem1. Solving the equations c0δ=e0δ=c1S1cusp,δ=c1S2saddle,δ=0 gives(57)a0=0,a1=−2756,a2=−2728,a3=2956.We takeδ=a0,a1,a2,a3,1 and δ0=0,−27/56,−27/28,29/56,1; then,(58)e3δ0=−331968π−16arcsin18+157≈−1.665254593<0,c2δ0=−33196−8π−16arcsin18+157≈−0.3326673051<0,c˜3δ0=−3398−16arcsin18+157≈−1.997921898<0.Therefore, there existh10, h20, and h30, satisfying 0<−h10≪1, 0<−h20≪1, and 0<h30≪1, respectively, such that(59)M1h10,δ0=e3δ0h10+Oh107/6>0,M2h20,δ0=c2δ0h20+Oh202lnh20>0,M3h30,δ0=c˜3δ0h30+Oh307/6<0,while taking h=100,(60)M3100,δ0=∮L1003fx,δ0ydx=∮L1003fx,δ0200−Axdx≈1678.486757>0,and thus there exists a z30 between h=h30 and h=100, such that(61)M3z30,δ0=0.It is easy to find(62)Rank∂c0δ,e0δ,c1S1cusp,δ,c1S2saddle,δ∂a0,a1,a2,a3=4.Therefore,e0δ, c0δ, c1S1cusp,δ, and c1S2saddle,δ can be taken as free parameters; we denote them by e0, c0, c1S1cusp, and c1S2saddle, respectively. Next, we take these free parameters to perturb 8 zeros.We takec1S2saddle,c1S1cusp,c0,e0, in turn satisfying(63)c1S2saddle<0,c1S1cusp>0,c0>0,e0<0,0<|e0|≪|c0|≪c1S1saddle≪c1S2cusp≪1.We takeδ1 in U1δ0,ϵ≜δ|e0,c0,c1S1cusp,c1S2saddlesatisfying62 which is a very small neighborhood of δ0.Therefore,e0<0, e1>0, e2<0, e0≪e1≪e2≪e3δ0 for M1h,δ1 by Lemma 6; then, there exist 3 zeros of h11, h12 and h13 of M1h,δ1 satisfying 0<−h13<−h12<−h11<−h10≪1.Under (63), c0>0, c1<0, c0≪c1≪c2δ0 for M2h,δ1 by Lemma 6; then, there exist 2 zeros h21, h22 of M2h,δ1 satisfying 0<−h22<−h21<−h20≪1.Under (63), c˜0>0, c˜1>0, c˜2<0, c˜0≪c˜1≪c˜2≪c˜3δ0 for M3h,δ1 by Lemma 6; then, there exist 3 zeros h31, h32, and h33 of M3h,δ1 satisfying 0<h33<h32<h31<h30≪1.Note that(64)M3h,δ1=M3h,δ0+Oϵ,and in accordance with (59), we obtain a z30∗ near z30, such that M3z30∗,δ1=0; therefore, there is another zero of M3h,δ1 near z30.The above 9 zeros are isolated real zeros; therefore, they correspond to 9 limit cycles of system (6). The main theorem is proved.


## 5. Conclusion and Discussion

In the paper, we have applied the theories of Hopf bifurcation, homoclinic loop bifurcation, and heteroclinic loop bifurcation to detect the limit cycles near the center and polycycle locally. The main routine is to prove the independence of the coefficients of the asymptotic expansions of the Melnikov functions and then treat them as free perturbation parameters. The computational analysis shows that there exist at least 9 limit cycles in the suitably damped system. The result gives a relative larger lower bound on the number of limit cycles for the Liénard system of degree 5. It is interesting to show that the asymptotic expansions of the Melnikov functions not only are the efficient tools to detect limit cycles such as a complicated investigation in [24] but also have been successfully applied to study the existence of periodic traveling waves and the coexistence of periodic solitary traveling waves (see [25, 26]). It may be more interesting to investigate the periodic traveling waves in external perturbation considered in the model given in [27]. It should also be pointed that there exist more questions left to solve for system (5), for example, what is the maximal number of limit cycles, or limit cycles by Poincaré bifurcation, or limit cycles by Hopf , or limit cycles by Hopf bifurcation, how about the global dynamics, and what is the period function of the undamped system. However, it is not easy at all to solve these questions.



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