# Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

Authors: Junning Cai; Minzhi Wei; Guoping Pang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In the presented paper, the Abelian integralIh of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.


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## Body

## 1. Introduction

A well-known analytic system with planar polynomial differential equation of degreen is of the form:(1)x˙=Pnx,y,y˙=Qnx,y.In 1977, Arnold [1] proposed weak Hilbert’s 16th problem and studied the number of zeros of the Abelian integral:(2)Ih,δ=∮Γhqdx−pdy,h∈J,where p and q are the polynomials of degree n≥2 and Γh are some closed ovals of corresponding Hamiltonian. More precisely, Hx,y is the Hamiltonian function of special form of (1):(3)x˙=Hy+εpx,y,δ,y˙=−Hx+εqx,y,δ,where Hx,y, px,y, and qx,y are the polynomials of x and y, their degrees satisfy maxdegp,degq=n, degH=n+1, and ε is a positive and sufficiently small parameter.More precisely, the following Liénard system of typem,n attracted more and more attentions from mathematicians [2–15]:(4)x˙=y,y˙=fx+εgxy,where fx and gx are the polynomials of degrees m and n, respectively. For example, Wang and Xiao [16] concluded that the number of limit cycles in the system bifurcating from period annulus is at most three. Qi and Zhao [17] considered the Liénard system of type (5, 3), [18], Asheghi and Zangeneh [19] considered the Liénard system of type (5, 4), and Sun [20] studied the limit cycles of type (7, 6) with a heteroclinic loop connecting two nilpotent saddles. In this paper, we intend to study on a following Liénard system that is a small perturbation of the Hamiltonian vector field:(5)x˙=y,y˙=xx+13x+13+εa1x+a2x3+a3x5+x7y,with 0<ε≪1, and a1,a2,a3 are the constants. Equation (5) holds the hyperelliptic Hamiltonian function:(6)H¯x,y=12y2−16x6−23x5−x4−23x3−16x2=12y2+Ax.The level sets (i.e.,H¯x,y=h) of the Hamiltonian function (6) are sketched in Figure 1. H¯x,y=h defines a family of closed orbits of system (5)|ε=0, denoted by Γh. Γ0 is the corresponding orbit to h=0, it incloses an elementary center −1/3,0, and Γ−8/2187 defines two heteroclinic orbits, connecting a nilpotent saddle −1,0 and a hyperbolic saddle 0,0. The Melnikov function on Γh is(7)Ih,δ=∮Γha1x+a2x3+a3x5+x7ydx≡a1I1h+a2I2h+a3I3h+I4h,for h∈−8/2187,0, where δ=a1,a1,a3,1 and Iih=∮Γhx2i−1ydx,i=1,2,3,4. Our main work is to provide a complete description of the number of limit cycles for perturbed system in the whole plane.Figure 1
The level set ofHx,y.


## 2. Some Preliminaries

For system (3), some related definitions and significative results are introduced, it can be seen in [21–23] in detail.Definition 1.
Assume thatf0,f1,f2,…,fn−1 are analytic functions on a real open interval J.(i)
The family of setsf0,f1,f2,…,fn−1 is called a Chebyshev system (T-system for short) provided that any nontrivial linear combination k0f0x+k1f1x+⋯+kn−1fn−1x has at most n−1 isolated zeros on J.(ii)
An ordered set ofn functions f0,f1,f2,…,fn−1 is called a complete Chebyshev system (CT-system for short) provided any nontrivial linear combination k0f0x+k1f1x+⋯+kn−1fn−1x has at most i−1 zeros for all i=1,2,…,n. Moreover, it is called an extended complete Chebyshev system (ECT-system for short) if the multiplicities of zeros are taken into account.(iii)
The continuous Wronskian off0,f1,f2,…,fn−1 at x∈R is(8)Wf0,f1,f2,…,fk−1=detfij0≤i,j≤k−1=f0xf1x…fk−1xf0′xf1′x…fk−1′x⋮……⋮f0k−1xf1k−1x…fk−1k−1x,where f′x is the first-order derivative of fx and fijx is the jth order derivative of fix, i≥2. The definitions imply that the function tuple f0,f1,f2,…,fn−1 is an ECT-system on J; therefore, it is an ECT-system on J and then a T-system on J; however, the inverse implications are not true.
LetHx,y=1/2y2+Ax in (5) be an analytic function. The set of ovals Γh=Hx,y=h inside periodic annulus is defined by h∈h1,h2=J. Supposed that P is a punctured neighborhood of the origin foliated by ovals Γh, then the projection of P on the x-axis is an interval xl,xr with xl<0<xr. It is easy to know that xA′x>0, ∀x∈xl,xr\0, such that Ax has a zero of even multiplicity at x=0, and there exists an analytic involution zx, which is defined by Ax=Azx.Lemma 1 (see [22]).
Onxr,xl, supposed that an analytic function fix satisfies(9)Iih=∮Γhfixy2s−1dx,fori=0,1,2,…,n−1,where h∈h1,h2, s∈N, and Ih is the oval surrounding the origin inside the level curve Ax+1/2y2m=h. Setting(10)lix≔fixA′x−fizxA′x.
If the following assumptions are satisfied(i)
Wl0,l1,…,li is nonvanishing on xl,xr for i=0,1,…,n−2(ii)
Wl0,l1,…,ln−1 has k zeros on xl,xr counting with multiplicities(iii)
s>n+k−2then for all nontrivial linear combination of I0,I1,…,In−1 has at most n+k−1 zeros on h1,h2 counting the multiplicities. Meantime, I0,I1,…,In−1 is called a T-system with accuracy k on h1,h2, where Wl0,l1,…,li is Wronskian of l0,l1,…,ln−1.However, the third condition above always not been satisfied, so we usually apply the next lemma to increase the power ofy in Ii.Lemma 2 (see [22]).
LetΓh be an oval inside the level curve Ax+1/2y2m=h, Fx be a function which satisfies Fx/A′x which is analytic at x=0. Hence,(11)∮ΓhFxyk−2dx=∮ΓhGxykdx,∀k∈N,where Gx=1/kFx/A′x′x.


## 3. The Least Upper Bound of Number of Zeros ofIh,δ

MultiplyIih by 2Ax+y2/2h=1, and the following is obtained:(12)Iih=12h∮Γh2Ax+y2x2i−1ydx=12h∮Γh2Axx2i−1ydx+∮Γhx2i−1y3dx,i=0,1,3.Settingk=3 and Fx=2x2i−1Ax and quoting Lemma 2 to ∮Γh2x2i−1Axydx yield(13)∮Γh2x2i−1Axydx=∮ΓhGixy3dx,where Gix=1/32x2i−1Ax/Ax′x=2x2i−13ix2+4ix−x+i/33x+12.By substituting (13) into (12) and multiplying 2Ax+y2/2h=1 again, it changes to(14)Iih=12h∮ΓhGix+x2i−1y3dx=14h2∮Γh2Ax+y2Gix+x2i−1y3dϕ,=14h2∮Γh2Axx2i−1+Gixy3dx+14h2∮Γhx2i−1+Gixy5dx.Quoting Lemma2, setting k=5 and Fx=2Axx2i−1+Gix, the following is obtained:(15)∮Γh2Axx2i−1+Gixy3dx=∮ΓhEixy5dx,where(16)Eix=152Axx2i−1+GixAx′x=2x2i−1rix153x+14,rix=81ix4+18x4i2−24x3+48i2x3+144ix3+44i2x2+84ix2−14x2−4x+16i2x+24ix+3i+2i2.By substituting (15) into (14) and multiplying 2Ax+y2/2h=1 again, the following is obtained:(17)Iih=14h2∮ΓhEix+Gix+x2i−1y5dx=18h3∮Γh2Ax+y2Eix+Gix+x2i−1y5d,=18h3∮Γh2AxEix+Gix+x2i−1y5dx+18h3∮ΓhEix+Gix+x2i−1y7dx.Quoting Lemma2 once more, then(18)∮Γh2AxEix+Gix+x2i−1y5dx=∮ΓhDixy7dx,where(19)Dix=172Axx2i−1+Gix+EixAx′x=2x2i−1gi¯x1053x+16,gi¯x=15i−24x−1008x5−1218x4−672x3−178x2+184ix+3645ix6+8496ix5+7761ix4+3848ix3+1107ix2+3120i2x3+1072i2x2+200i2x+16i2+1296x6i2+4104x5i2+5040x4i2+544i3x3+228i3x2+48i3x+108x6i3+432x5i3+684x4i3+4i3.From the above computation, the following result can be obtained easily.Lemma 3.
(20)8h3Iih=∮Γhfixy7dx≡Ii˜h,where fix=x2i−1+Gix+Eix+Dix. Therefore, I1h,I2h,I3h,I4h is an ECT-system if and only if I1˜h,I2˜h,I3˜h,I4˜h is as well.
Take the following function(21)lix=fiA′x−fiA′zx,where zx is an analytic involution, defined by Ax=Azx on −1,−1/3. Factoring Ax−Azx yields(22)−16x−zqx,z,where(23)qx,z=z+6z3+4z2+4z2x2+6zx2+x+6z2x+4xz3+x5+4x4+6x3+4x2+4xz+4zx3+4z4+x2z3+x4z+xz4+z2x3+z5,which defined zx on −1,0. Hence,(24)ddxlix=ddxfiA′x−ddxfiA′zxdzdx,with dz/dx=−∂qx,z/∂x/∂qx,z/∂z. Suppose that x∈−1,−1/3, then zx∈−1/3,0; in other words,(25)−1<x<−13<z<0.Lemma 4.
The function tuplel1x,l2x,l3x,l4x is an ECT-system for x∈−1,−1/3.Proof.
Taking (24) into consideration, with the aid of Maple 16, we can obtain the 4 following Wronskians:(26)Wl1x=l1x=3x−zw1x,z353x+17x+133z+17z+13,Wl1x,l2x=18x−z3w2x,z125z+163z+113x+163x+113px,z,Wl1x,l2x,l3x=−108x−z6w3x,z42875z+193z+115z4x+193x+115x4p3x,z,Wl1x,l2x,l3x,l4x=−62208x−z10w4x,z300125z+1123z+122z5x+1123x+122x5p6x,z,where(27)px,z=x4+4x3+2x3z+6x2+8x2z+3x2z2+4x+12xz+12xz2+4xz3+1+8z+18z2+16z3+5z4,and w1x,z, w2x,z,w3x,z, and w4x,z are polynomials in x,z of degrees 15, 32, 53, and 73, respectively. In the following, calculating the resultant with respect to z between qx,z and px,z gives(28)Rq,p,z=16x63x+427x3+54x2+27x−4x+110.
From Sturm’s Theorem, we know thatRq,p,z has no root on −1,−1/3, then px,z and qx,z have no common root on −1,−1/3. In the following, we will check whether wix,z and qx,z have common root under the condition (25).(i)
Calculating the resultant with respect toz between qx,z and w1x,z, that is, eliminating from qx,z=0 and w1x,z=0 gives Rq,w1,z=15552x+163x+16φ1x, where φ1x is a polynomial of degree 62 in x. Applying Sturm’s theorem to φ1x, there is not any x such that φ1x=0, so we conclude that W1l1x≠0 on −1,−1/3.(ii)
Calculating the resultant with respect toz between qx,z and w2x,z, that is, eliminating from qx,z=0 and w2x,z=0 gives Rq,w2,z=123834728448x43x+110x+122φ2x, where φ2x is a polynomial of degree 122 in x. Applying Sturm’s theorem to φ2x, there are three points, denoted by x1, x2, and x3, such that φ2x=0, which x1≈−0.5231817697,x2≈−0.5050882970, and x3≈−0.39862246670.(iii)
Thus we will check ifqx and w2x,z have any common roots on −1,−1/3 by using the program with Maple 16 to find all the possible intervals.>withRegularChains;>withChainTools;>withSemiAlgebraicSetTools;>sys:=w2x,z,qx,z;>R2:=PolynomialRingx,z;>dec:=Triangularizesys,R;regular_chain>L:=mapEquations,dec,R;(29)x+1,z,x,z+1,x+1,z+1,3x+1,3z+1,φ1∗x,z,φ2∗x,z,η1∗x,z,η2∗x,z,whereφ1∗x,z=φ11∗zx+φ12∗z, η1∗x,z=η11∗zx+η12∗z, and φ11∗, φ12∗, and η2∗ are polynomials in z of degree 41, 41, and 42, η11∗, η12∗, and η2∗ are polynomials in z of degree 79, 79, and 80, respectively. It is obvious that, all the roots of the regular chains x+1,z, x,z+1, x+1,z+1, and 3x+1,3z+1 do not satisfy (14), the regular chains φ1∗x,z,φ2∗x,z and η1∗x,z,η2∗z are square-free and zero-dimensional (because the number of variables equals the number of polynomials). L51 and L52 represent φ1∗x,z and φ2∗z, and L61 and L62 represent η1∗x,z and η2∗z in Maple, we use the following program to check their common roots.>C:=ChainL52,L51,EmptyR2,R2;regular_chain>RL:=RealRootIsolateC,R2,′abserr′=1/105;box,box>mapBoxValues,RL,R2;(30)x=−68575131072,−3428765536,z=−59409777128373836560319652339976029315340282366920938463463374607431768211456,−118819554256747673120639304679952058629680564733841876926926749214863536422912,x=−174237131072,−4355932768,z=−163983771529936100306969912196931267809936064008805527989890579411376139330301510538742295639337626245683966408394965837152256,−6559350861197444012278796487877250712397442560352221119595623151645504557321206042154969182557350504982735865633579863348609024,x=−572132768,−22883131072,z=−95578876636133688791068498973648060853858107987182687704666362864775460604089535377456991567872,−382315506544534755164273995894592243415432431947730750818665451459101842416358141509827966271488,x=−52249131072,−653116384,z=−34511036383350895369252333113760092596148429267413814265248164610048,−69022072766701790738504666227520175192296858534827628530496329220096,>C:=ChainL62,L61,EmptyR,R2;regular_chain>RL:=RealRootIsolateC,R,′abserr′=1/105;box,box>mapBoxValues,RL,R;(31)x=13355131072,333932768,z=−604723096054972700240230521574793218570238901331974711972621413014756705924586149611790497021399392059392,−2418892384219890800960922086299172874280955605327898747890485652059026823698344598447161988085597568237568,x=−66203131072,−3310165536,z=700202759568719476736,14004055191137438953472.It means that there are 6 pairs of common roots ofw2x,z and qx,z in the listed intervals, respectively. However, there is not any pair of them satisfies the condition (25). It is said that w2x,z≠0 as −1<x<−1/3; therefore, Wl1x,l2x≠0.(iii)
Similarly, we use the same program as (i) and (ii) to find all the possible intervals, which may hold the common roots ofw3x,z and qx,z and then obtain the following regular chains:(32)x,z,x,z+1,x+1,z,x+1,z+1,3x+1,3z+1,u1∗x,z,u2∗x,z,ρ1∗x,z,ρ2∗x,z,whereu1∗x,z=u11∗zx+u12∗z, ρ1∗x,z=ρ11∗zx+ρ12∗z, and u11∗, u12∗, and u2∗ are polynomials in z of degree 57, 57, and 58, respectively, ρ11∗,ρ12∗z, and ρ2∗ are polynomials in z of degree 111, 111, and 112, respectively. Isolating the fifth and sixth regular chains yields(33)x=−54979131072,−2748965536,z=−8562711302028495557623392883223761113823168624554049946785198554013369993333393829974333376885877453834204643052817571560137951281152,−107033891275356194470292411040297013922789607806925624334814981925421249166674228746791672110734681729275580381602196445017243910144,x=−174741131072,−4368532768,z=−30302980390916156789706635829577566892174737847493853797592270907029377629864125874798266065358092198749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296,−606059607818323135794132716591551337843494756949877075951845418140587552597282517495965321307161843717498005798264095394980017816940970922825355447145699491406164851279623993595007385788105416184430592,x=−416316384,−33303131072,z=−361868533714530451234631234914240547837472188608858683748897796324263862718293348820473429344482784628181556388621521298319395315527974912,−14474741348581218049385249396569621913498887544354347349955911852970513450873173395281893717377931138512726225554486085193277581262111899648,x=−4258932768,−170355131072,z=−480531930410139348156667048862506606461576771541323121412001025602246951271904625697166532776746648320380374280103671755200316906558262375061821325312,−9610638608202786963133340977250132129231535430826462428240020512044939025411809251394333065553493296640760748560207343510400633813116524750123642650624,x=−69625131072,−870316384,z=−2968006133719558087077106477110092074248769964722835963083295358096932575511191922182123945984,−1484003066859779043538553238555046037124384982311417981541647679048466287755595961091061972992,x=−2269965536,−45397131072,z=−249398274372099944349741313215654480864719460648955187072209578355573530071658587684226515959365500928,−12469913718604997217487065660782724043235973032447793536104789177786765035829293842113257979682750464x=15487131072,1211024,z=−6711863274178644481642066700063144267843414280194808598930711903933818936488952451897137590064188545819787018382342682267975428761855001222473056385648716020711424,−167796581854466112041051667501578606696085357004870214973267797598345473412223811474284397516047136454946754595585670566993857190463750305618264096412179005177856,x=−1159332768,−46371131072,z=2724659906283478332305843009213693952,435945585005356532936893488147419103232.It means that there are 8 pairs of common roots ofw3x,z and qx,z in the listed intervals, respectively. However, there is not any pair of them satisfies the condition (25). It is said that w3x,z≠0 as −1<x<−1/3; therefore, Wl1x,l2x,l3x≠0.(iv)
Similarly, we use the same program as (ii) and (iii) to find all the possible intervals, which may hold the common roots ofw4x,z and qx,z and then obtain the following regular chains:(34)x,z,x+1,z,x,z+1,x+1,z+1,3x+1,3z+1,v1∗x,z,v2∗x,z,ω1∗x,z,ω2∗x,z,wherev1∗x,z=v11∗zx+v12∗z,ω1∗x,z=ω11∗zx+ω12∗z, and v11∗, v12∗, and v2∗ are polynomials in z of degree 75, 75, and 76, respectively, ω11∗, ω12∗, and ω2∗ are polynomials in z of degree 145, 145, and 146, respectively. Isolating the fifth and sixth regular chains yields>C:=ChainL62,L61,EmptyR,R2;regular_chain>RL:=RealRootIsolateC,R,′abserr′=1/105;box,box,box,box,box,box,box,box,box,box,box,box,box,box>mapBoxValues,RL,R;>evalf%;(35)x=−1.113883972,−1.113876343,z=−0.01488446846,−0.01488446846,x=−1.277610779,−1.277603149,z=−0.1301213260,−0.1301213260,x=−0.5849761963,−0.5849685669,z=−0.1345142627,−0.1345142627,x=−1.329246521,−1.329238892,z=−0.2715335562,−0.2715335562,x=−0.3662719727,−0.3662643433,z=−0.3014530812,−0.3014530812,x=−1.333335876,−1.333328247,z=−0.3333242893,−0.3333242893,x=−0.3014602661,−0.3014526367,z=−0.3662645453,−0.3662645453,x=−0.1345214844,−0.1345138550,z=−0.5849732646,−0.5849732646,x=−1.078308105,−1.078300476,z=−0.9149922188,−0.9149922188,x=−0.9149932861,−0.9149856567,z=−1.078306231,−1.078306231,x=−0.01488494873,−0.01487731934,z=−1.113876799,−1.113876799,x=−0.1301269531,−0.1301193237,z=−1.277609231,−1.277609231,x=−0.2715377808,−0.2715301514,z=−1.329244729,−1.329244729,x=−0.3333282471,−0.3333206177,z=−1.333333333,−1.333333333,>C:=ChainL72,L71,EmptyR,R2;regular_chain>RL:=RealRootIsolateC,R,′abserr′=1/105;box,box,box,box,box,box>mapBoxValues,RL,R;>evalf%;(36)x=0.1184310913,0.1184387207,z=−0.3332116462,−0.3332116462,x=0.1063232422,0.1063308716,z=−0.4784850961,−0.4784850961,x=0.5298614502,0.5299377441,z=−0.7129253486,−0.7129253486,x=−0.3332138062,−0.3332061768,z=0.1184337894,0.1184337894,x=−0.4784851074,−0.4784774780,z=0.1063252093,0.1063252093,x=−0.7129287720,−0.7129211426,z=0.5298903953,0.5298903953.
It means that there are 20 pairs of common roots ofw4x,z and qx,z in the listed intervals, respectively. However, there is not any pair of them satisfies the condition (25). It is said that w4x,z≠0 as −1<x<−1/3; therefore, Wl1x,l2x,l3x,l4x≠0.
Based on Lemma1 and Lemma 4, we obtain the following proposition:Proposition 1.
I1˜h,I2˜h,I3˜h,I4˜h is an ECT-system and I1h,I2h,I3h,I4h is as well. Therefore there are at most 3 zeros for Ih,δ on h∈−8/2187,0.


## 4. Conclusion

In this work, we study the Poincare´ bifurcation of the Lie´nard system with the form (5), and we prove 4 is the least upper bound of the number of limit cycles by the Poincare´ bifurcation.



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