# Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

Authors: Minzhi Wei; Junning Cai; Hongying Zhu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.


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## Body

## 1. Introduction

In many branches of science, such as mechanics, electronics, fluid mechanics, biology, chemistry, and astrophysics, one often deals with families of special planar differential equations which can model different natural phenomena. The main open problem in the qualitative theory of planar polynomial differential systems is determining the maximum number of limit cycles, which is the well-known second part of Hilbert’s 16th problem.LetH(n) denote the maximal number of limit cycles of polynomial systems of degree n of the form (1)x˙=Pnx,y,y˙=Qnx,y.The problem is still open even for n=2. As the introduction in [1], there are few studies on an upper bound of H(n). However, there have been many interesting results on the lower bound of it for n≥2; see [1–3]. What is more, Chen [4] and Shi [5] proved that H(2)≥4 independently. In 1985, Li and Li [6] found H(3)≥11 by using the method of detection function, and then Han et al. [7, 8] also obtained H(3)≥11 with new different distributions of limit cycles by using the method of stability-changing of homoclinic orbits.Recently,H(3)≥13 had been proved in [9] by Li, and then H(4)≥20, H(5)≥24, H(6)≥35 and H(11)≥121  H(7)≥53, H(9)≥80 had been obtained, respectively; see [10–15].The intersection form of Smale’s problem and weak Hilbert’s 16th problem is studying the number of zeros of Abelian integral corresponding to the following generalised Liénard system:(2)x˙=y,y˙=gx+εfxy.It is called Liénard system of type (m,n) if deg⁡g(x)=m and deg⁡f(x)=n. There are abundant results for weak Hilberts 16th problem restricted to Liénard systems of type (m,n), especially for n=m-1, such as types (3,2) [16], (4,3) [17], (5,4) [18, 19], and (7,6) [20]. More details of the relative researches can be seen in [21–32].In present paper, we consider the following system:(3)x˙=y,y˙=x3x+34x-12+εa0+a1x+a2x2+x3y,with 0<ε≪1, and a0,a1,a2 are constants. Equation (3) holds the hyperelliptic Hamiltonian function (4)Hx,y=12y2-17x7+524x6+110x5-316x4=12y2+Ax.The level sets (i.e., H(x,y)=h) of Hamiltonian function (4) are sketched in Figure 1. (5)Ih,δ=∮Γha0+a1x+a2x2+x3dx≡a0I0h+a1I1h+a2I2h+I3h,for h∈(h∗,0), where δ=(a0,a1,a2,1),h∗=H(-3/4,0)≈-0.2690865654 and Ii(h)=∮Γhxiydx, i=0,1,2,3.Figure 1
The portrait of system (3).The outer boundary ofΓh is a homoclinic loop Γ0 passing through a nilpotent saddles (0,0) defined by H(x,y)=0, and the inner boundary is an elementary center Γ-3/4 at the origin defined by H(x,y)=h∗.


## 2. Some Preliminaries

Some related definitions and useful results are introduced in this section. For more details, refer to [33–35].Definition 1.
Assume thatf0(x),f1(x),f2(x),⋯,fn-1(x) are analytic functions on a real open interval J.
(i) The family of sets{f0,f1,f2,⋯,fn-1} is called a Chebyshev system (T-system for short) provided that any nontrivial linear combination (6)k0f0x+k1f1x+⋯+kn-1fn-1xhas 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 k0f0(x)+k1f1(x)+⋯+kn-1fn-1(x) 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 of{f0,f1,f2,⋯,fn-1} at x∈R is (7)Wf0,f1,f2,⋯,fk-1=det⁡fij0≤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 f(x) and fij(x) is the jth order derivative of fi(x), i≥2. The definitions imply that the function tuple {f0,f1,f2,⋯,fn-1} is an ECT-system on J; therefore it is a ECT-system on J and then a T-system on J; however, the inverse implications are not true.LetH(x,y)=Φ(x)+1/2y2 be an analytic function. Assume there exists a punctured neighborhood P of the origin foliated by ovals Γh⊆x,y∣Hx,y=h, which corresponds to clockwise periodic orbits of system (2) and forms a period annulus denoted by Γh. The set of ovals Γh inside the period annulus is parameterized by the energy levels h∈(h∗,0). The projection of P on the x-axis is an interval (xl,xr) with xl<0<xr. Under the above assumptions it is easy to verify that xΦ′(x)>0 for all x∈(xl,xr)0; Φ(x) has a zero of even multiplicity at x=0 and has an analytic involution z(x) defined by (8)Φx=Φzx,∀x∈xl,xr.For the number of isolated zeros of nontrivial linear combination of some integrals of special form, the algebraic criterion in [34] (Theorem A) can be stated as follows.Lemma 2 (see [15]).
On(xr,xl), suppose that an analytic function fi(x) satisfies that (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 {A(x)+1/2y2m=h}. Setting that (10)lix≔fixA′x-fizxA′x.If the following assumptions are satisfied(i)
W[l0,l1,⋯,li] is nonvanishing on (xl,xr) for i=0,1,⋯,n-2,(ii)
W[l0,l1,⋯,ln-1] has k zeros on (xl,xr) counting with multiplicities,(iii)
s>n+k-2.Then 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 W[l0,l1,⋯,li] is Wronskian of {l0,l1,⋯,ln-1}.However, the third condition above has not always been satisfied, so we usually apply the next lemma to increase the power ofy in Ii.Lemma 3 (see [34]).
LetΓh be an oval inside the level curve {A(x)+1/2y2m=h}, and let F(x) be a function which satisfies F(x)/A′(x) is analytic at x=0. Hence, (11)∮ΓhFxyk-2dx=∮ΓhGxykdx,∀k∈N,where G(x)=1/k(F(x)/A′(x))′(x).Proposition 4.
{ I 0 ~ ( h ) , I 1 ~ ( h ) , I 2 ~ ( h ) , I 3 ~ ( h ) } is an T-system with accuracy 3, and {I0(h),I1(h),I2(h),I3(h)} is the same. Therefore there are at most 6 zeros for I(h,δ) on h∈(h∗,0).


## 3. The Least Upper Bound of Number of Zeros ofI(h,δ)

MultiplyingIi(h) by 2A(x)+y2/2h=1, it is obtained (12)Iih=12h∮Γh2Ax+y2xiydx=12h∮Γh2Axydx+∮Γhxiy3dx,i=0,1,3Setting k=3 and F(x)=2xiA(x), quoting Lemma 3 to ∮Γh2xiA(x)ydx yields (13)∮Γh2xiAxydx=∮ΓhGixy3dx,where Gi(x)=1/3(2xiA(x)/A(x))′(x)=xigi(x)/630(4x+3)2(x-1)3, and gi(x)=960ix5+960x5-1440x4-1640ix4-1042ix3-458x3+2478ix2+1302x2+189ix+63x-945-945i. Substituting (13) into (12) and multiplying 2A(x)+y2/2h=1 again, it changes to (14)Iih=12h∮ΓhGix+xiy3dx=14h2∮Γh2Ax+y2Gix+xiy5dϕ=14h2∮Γh2Axxi+Gixy3dx+14h2∮Γhxi+Gixy5dx.Quoting Lemma 3, setting k=5 and F(x)=2A(x)(xi+Gi(x)), it is obtained (15)∮Γh2Axxi+Gixy3dx=∮ΓhEixy5dx,where Ei(x)=1/5(2A(x)(xi+Gix)/A(x))′(x)=xiri(x)/661500(4x+3)4(x-1)6, and (16)rix=6251175+7144200i-2024190x+13812876x3-43453368x6-41294736x5+46715382x4-23659209x2+57089920x7+83236160ix7+893025i2+8175520i2x7-6679196i2x6-7598472i2x5+8846208i2x4+2906064i2x3-4647699i2x2-357210i2x+11520000ix10-36768000ix9+7297280ix8+921600x10i2-3148800x9i2+688960x8i2+10598400x10-31795200x9+8429760x8-62137876ix6-69826512ix5+74212866ix4+25527852ix3-36991080ix2-3214890ix.Substituting (15) into (14) and multiplying 2A(x)+y2/2h=1 again, it arrives at (17)Iih=14h2∮ΓhEix+Gix+xiy5dx=18h3∮Γh2Ax+y2Eix+Gix+xiy5d=18h3∮Γh2AxEix+Gix+xiy5dx+18h3∮ΓhEix+Gix+xiy7dx.Quoting Lemma 3 again, then (18)∮Γh2AxEix+Gix+xiy5dx=∮ΓhDixy7dx,where Di(x)=1/7(2A(x)(i+Gi(x)+Ei(x)/A(x))′(x)=xigi-(x)/972405000(4x+3)6(x-1)9, and (19)
g
i
-
x
=
816403507200
x
13
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188227584000
x
15
-
847024128000
x
14
-
80171319375
i
+
884736000
x
15
i
3
-
4534272000
x
14
i
3
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4865126400
x
13
i
3
-
28166321280
x
11
i
3
+
12283379200
x
12
i
3
-
46262732796
i
3
x
7
+
31047818508
i
3
x
6
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22339746702
i
3
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5
-
20419766766
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3
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4
-
5440344021
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3
x
3
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6537478815
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3
x
2
+
506345175
i
3
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-
4618335840
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10
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3
+
51278343992
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9
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3
-
19457293464
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8
i
3
+
39101099625
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+
214769664000
i
x
15
-
1005677568000
i
x
14
+
968597760000
i
x
13
+
27426816000
x
15
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2
-
134295552000
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14
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136300646400
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64980964125
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352197950769
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6
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1490338869138
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406865226285
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3541922901216
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7
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3434948889600
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11
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1675062681600
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12
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4743445072640
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x
11
-
726053239680
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11
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2
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2325343232000
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336369907200
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639315670644
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479698587018
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403391228058
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114429330099
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125562351285
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10745769825
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7435484615536
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9
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2267138615952
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10
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1224971389992
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9
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4832826270720
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3378434698296
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2614804657542
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609719998671
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x
3
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622215704565
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x
2
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58004652825
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x
.From the computation above, the following result easily can be obtained.Lemma 5.
(20) 8 h 3 I i h = ∮ Γ h f i x y 7 d x ≡ I i ~ h ,where fi(x)=xi+Gi(x)+Ei(x)+Di(x).Therefore,{I0(h),I1(h),I2(h),I3h)} is a T-system with accuracy k if and only if {I0~(h),I1~(h),I3~(h)} is the same as well.Taking the following function(21)lix=fiA′x-fiA′zx,where z(x) is an analytic involution, defined by A(x)=A(z(x)) on (-3/4,0). Factoring A(x)-A(z(x)) yields (22)-11680x-zqx,z,whereq(x,z)=240x6-350x5 + 240zx5-168x4-350zx4 + 240z2x4+315x3-168zx3-350z2x3 + 240x3z3+315zx2-168z2x2-350x2z3 + 240x2z4+315z2x-168xz3-350xz4 + 240xz5+315z3-168z4-350z5+240z6, which defined z(x) on (-1,0). Hence, (23)ddxlix=ddxfiA′x-ddxfiA′zxdzdx,with dz/dx=-∂qx,z/∂x/∂q(x,z)/∂z. Suppose that x∈(-3/4,0), and then z(x)∈(xr,-3/4), where xr is the projection of Γh0 on x-axis and xr≈-0.9073535571; in other words, (24)xr<z<-34<x<0.Lemma 6.
The function tuplel0x,l1x,l2x,l3x is an T-system with accuracy 3 for x∈(-1,-1/3).Proof.
Taking (23) into consideration, with the aid of Maple 16, we can obtain the 4 following Wronskians: (25)Wl1x=l1x=2x-zw1x,z165375z2x24x+35x-184z+35z-18,Wl1x,l2x=2x-z3w2x,z27348890625z-1164z+39x-1164x+39x4z4px,z,Wl1x,l2x,l0x=-4x-z6w3x,z4522822787109375z-1233z+112z9x-1233x+112x9p3x,z,Wl1x,l2x,l0x,l3x=32x-z10w4x,z831068687131347656254z+315z-1304x+315x-130x12z12p6x,z,where p(x,z) = 240x5-350x4 + 480x4z-168x3-700x3z + 720x3z2+315x2-336x2z-1050x2z2 + 960x2z3+630zx-504xz2-1400xz3 + 1200xz4+945z2-672z3-1750z4 + 1440z5, and w1(x,z), w2(x,z),w3(x,z), and w4(x,z) are polynomials in {x,z} of degrees 24, 50, 79, and 110, respectively. On the following, calculating the resultant with respect to z between q(x,z) and p(x,z) gives (26)Rq,p,z=104552985600000x630720x5-90880x4+97536x3-54864x2+27432x-10287x-12×240x4+370x3+222x2+111x+372240x3-350x2-168x+3153.From Sturm’s Theorem, we know that R(q,p,z) has two roots x1=x2=-0.6455780606∈(-3/4,0). Thus we will check if q(x) and w2(x,z) have any common roots on (-1,-1/3) by using the program with Maple 16 to find all the possible intervals:>with(RegularChains);>with(ChainTools);>with(SemiAlgebraicSetTools);>sys≔[p(x,z),q(x,z)];>R2≔PolynomialRing([x,z]);>dec≔Triangularize(sys,R2);[regular _ chain]>L≔map(Equations,dec,R2); (27)x,z,240x3-350x2-168x+315,z,x-1,z-1,240x4+370x3+222x2+111x+37,z-1,30720x5-90880x4+97536x3-54864x2+27432x-10287,4z+3.
It is obvious that all the roots of the five regular chains do not satisfy (24), so we conclude that p(x,z) and q(x,z) have no common root on xr<z<-3/4<x<1.
(i) Calculating the resultant with respect toz between q(x,z) and w1(x,z), that is, eliminating from q(x,z)=0 and w1(x,z)=0 gives R(q,w1,z)=2097152x4(4x+3)4(x-1)14φ1(x), where φ1(x) is a polynomial of degree 122 in x. Applying Sturm’s theorem to φ1(x), there is a point, denoted by x0 such that φ1(x)=0, with x0≈-0.6831595756. Thus we will check if q(x) and w1(x,z) have any common roots on (-3/4,0) by using the program with Maple 16 to find all the possible intervals:>with(RegularChains);>with(ChainTools);>with(SemiAlgebraicSetTools);>sys≔[w1(x,z),q(x,z)];>R2≔PolynomialRing([x,z]);>dec≔Triangularize(sys,R2);[regular _ chain]>L≔map(Equations,dec,R2); (28)[x,z,x-1,z-1,4x+3,4z+3,η1∗x,z,η2∗x,z]
whereη1∗(x,z)=η11∗(z)x+η12∗(z), and η11∗, η12∗, and η2∗ are polynomials in z of degrees 108, 110, and 122, respectively. It is obvious that all the roots of the regular chains [x,z],[x-1,z-1] and [4x+3,4z+3] do not satisfy (24), and the regular chains [η1∗(x,z),η2∗(x,z)] is square-free and zero-dimensional (because the number of variables equals the number of polynomials). L[4][1] and L[4][2] represent φ1∗(x,z) and φ2∗(z) in Maple; we use the following program to check their common roots:>C≔Chain([L[4][2],L[4][1]],Empty(R2),R2);[regular _ chain]>RL≔RealRootIsolate(C,R2,a′bserr′=1/105);[box,box]>map(BoxValues,RL,R2); (29)x=7505565536,150111131072,z=-8660076460171029500504398346511267650600228229401496703205376,-433003823008551475025219917325633825300114114700748351602688,x=-1119316384,-89543131072,z=5107994732592040953184192754843493623737173144601490397061246283071436545296723011960832,10215989465184081906368385509686987247474346389202980794122492566142873090593446023921664.
It means that there are 2 pairs of common roots ofw2(x,z) and q(x,z) in the listed intervals, respectively. However, there is not any pair of listed common root satisfing (24), so we conclude that W1[l1(x)]≠0 on (-3/4,0).
(ii) Calculating the resultant with respect toz between q(x,z) and w2(x,z), that is, eliminating from q(x,z)=0 and w2(x,z)=0, gives R(q,w2,z)=136853837364162723840000x12(4x+3)6(x-1)28φ2(x), where φ2(x) is a polynomial of degree 254 in x. Applying Sturm’s theorem to φ2(x), there are five points, denoted by x1 and x2, such that φ2(x)=0, in which x1≈-0.6362217374, and x2≈-0.6326476723. Thus we will check if q(x) and w2(x,z) have any common roots on (-3/4) by using the program with Maple 16 to find all the possible intervals:>with(RegularChains);>with(ChainTools);>with(SemiAlgebraicSetTools);>sys≔[w2(x,z),q(x,z)];>R2≔PolynomialRing([x,z]);>dec≔Triangularize(sys,R2);[regular _ chain]>L≔map(Equations,dec,R2); (30)[x,z,x-1,z-1,4x+3,4z+3,φ1∗x,z,φ2∗x,z]
whereφ1∗(x,z)=φ11∗(z)x+φ12∗(z), and φ11∗, φ12∗, and φ2∗ are polynomials in z of degrees 228, 229, and 254, respectively. It is obvious that all the roots of the regular chains [x,z],[x-1,z-1] and [4x+3,4z+3] do not satisfy (24); the regular chains [φ1∗(x,z),φ2∗(x,z)] are square-free and zero-dimensional (because the number of variables equals the number of polynomials). L[4][1] and L[4][2] represent φ1∗(x,z) and φ2∗(z) in Maple, we use the following program to check their common roots:>C≔Chain([L[4][2],L[4][1]],Empty(R2),R2);[regular _ chain]>RL≔RealRootIsolate(C,R2,′abserr′=1/105);[box,box,box,box,box,box,box,box]>map(BoxValues,RL,R2); (31)x=-2729732768,-109187131072,z=-18492312177733628062823020497019494150223459275512923003274661805836407369665432566039311865085952,-1479384974218690245025841639761559532017876742040723384026197294446691258957323460528314494920687616,x=5766965536,115339131072,z=-17735273570380596931788539750094371487913392787593149816327892691964784081045188247552,-35470547140761193863577079500188742975826775575186299632655785383929568162090376495104,x=153921131072,7696165536,z=-50003316692032326563420860327001281205778687482541170559792388176582018229284824168619876730229402019930943462534319453394436096,-156260364662601020510690188521879003768058398382941157999351213205688069665150755269371147819668813122841983204197482918576128,x=3562132768,142485131072,z=-25829826454534583657276070060013025365226199509871273840633138550867693340381917894711603833208051177722232017256448,-51659652909069167314552140120026050730452399019742547681256277101735386680763835789423207666416102355444464034512896,x=-82923131072,-4146165536,z=-54830720393478399907489352613061393597232611465857291111161460616582018229284824168619876730229402019930943462534319453394436096,-13707680098369599976872338153265348399308152866464322777790365151645504557321206042154969182557350504982735865633579863348609024,x=-99575131072,-4978765536,z=7300257202083707792106826374829209495215637023192606911994400469363967779260271260032962165404551223330269422781018352605012557018849668464680057997111644937126566671941632,3650128601041853896053413187414604747607818511596303455997200234681983889630135630016531082702275611665134711390509176302506278509424834232340028998555822468563283335970816,x=-107871131072,-5393565536,z=135155975133267206773823547400634720703666289249903506522392637683797752303075845772421124330809102446660538845562036705210025114037699336929360115994223289874253133343883264,270311950266534413547647094801269441407332578499807013044785275367595504606151691544843248661618204893321077691124073410420050228075398673858720231988446579748506266687766528,x=-333565524288,-166781262144,z=741367729863243853116477867614961473411743117702864065072716925117842498333348457493583344221469363458551160763204392890034487820288,59309418389059508249318229409196917872939449416229125205817354009376739986666787659948666753771754907668409286105635143120275902562304.
It means that there are 8 pairs of common roots ofw2(x,z) and q(x,z) in the listed intervals, respectively. It is obvious that the fifth pair of interval satisfies (24), and then w2(x,z) and q(x,z) have a common root, denoted by (x0,z0), on the fifth listed interval. Noted that W2(x,z(x))=W[l1(x),l2(x)]; it follows (32)dW2dx=∂W2∂x+∂W2∂z·dzdx=2x-z2w5x,z27348890625x-z9z-1164z+39x-1174x+310x5z4p2x,z,where w5(x,z) is a polynomial of degree 58. Furthermore, by using Sturm’s Theorem, there is no root inside the interval(33)x=-82923131072,-4146165536,z=-54830720393478399907489352613061393597232611465857291111161460616582018229284824168619876730229402019930943462534319453394436096,-13707680098369599976872338153265348399308152866464322777790365151645504557321206042154969182557350504982735865633579863348609024for R(q,w5,z)=0. Therefore, x0 is a simple root of W2(x,z(x)).
(iii) Similarly, we use the same program as (i) and (ii) to find all the possible intervals, which may hold the common roots ofw3(x,z) and q(x,z) and then obtain the following regular chains (34)x,z,x-1,z-1,4x+3,4z+3,u1∗x,z,u2∗x,zwhere u1∗(x,z)=u11∗(z)x+u12∗(z), and u11∗, u12∗, and u2∗ are polynomials in z of degrees 355, 356, and 398, respectively. Isolating the last regular chain yields(35)x=104117131072,5205965536,z=-111466044818115533912177175496075352645970762329182687704666362864775460604089535377456991567872,-1393325560226444173902214693700941908074634529122835963083295358096932575511191922182123945984,x=125839131072,503359524288,z=-518350530247069932093764379100104309467418691903308988506881803469022129495137770981046170581301261101496891396417650688,-10367010604941398641875287582002086189348373838066179770137611606938044258990275541962092341162602522202993782792835301376,x=-102675131072,-5133765536,z=-2111559835917280439894835840299991194671842134543900777641211658976219210862187529642774844752946028434172162224104410437116074403984394101141506025761187823616,-1055779917958640219947417920149995597335921067271950388820605829488109605431093714821387422376473014217086081112052205218558037201992197050570753012880593911808,x=154007131072,1925116384,z=-5375484483043515759498119503476091096224184790521552523252627362934186127773268212244968146646834078002771671831749689734737838152978190216899892655911508785116799651230841339877765150252188079784691427704832,-2687742241521757879749059751738045548112092395260776261626313681467093063886634106122484073323417039001335835915874844867368919076489095108449946327955754392558399825615420669938882575126094039892345713852416,x=-1167116384,-93367131072,z=-97393815505800945325611199158748538576449415152622778709458752564592956991085918059907124330809102446660538845562036705210025114037699336929360115994223289874253133343883264,-4869690775290047266280559957937426928822470757631138935472937628229647849554295902995362165404551223330269422781018352605012557018849668464680057997111644937126566671941632,x=150763131072,603055524288,z=-362185356272529668611375873156871094801239634207606269872625924155478862647452312848583266388373324160190187140051835877600158453279131187530910662656,-181092678136264834305687936578435547400619817103803134936312962077739431323226156424291633194186662080095093570025917938800079226639565593765455331328,x=6775165536,542011524288,z=-1254050684711709807099188198381597949975593036839272079732945704661173425727457612915177100720513508366558296147058741458143803430094840009779784451085189728165691392,-4898635487155116433981203899928116992092160300153406561456819158832708694247881359285549689505892056868344324448208820874232148807968788202283012051522375647232,x=105089131072,5254565536,z=-14879549415043235026008970762082986308718342193805808637647097905226841091766847064778384329583297500742918515827483896875618958121606201292619776,-371988735376080875650224269052074657717958554845145215941177447630671027441711766194596082395824375185729628956870974218904739530401550323154944,⋯⋯x=-110383131072,-5519165536,z=1246055440087652294798999757908599996977470263257522210585044567657926236797330079896115541351137805832567355695254588151253139254712417116170014499277911234281641667985408,2492110880175304589597999515817199993954940526515044421170089135315852473594660159792331082702275611665134711390509176302506278509424834232340028998555822468563283335970816,x=-79973131072,-1999332768,z=376748219254335143997799742969614591943547941993129791307144488089186725253022769474284397516047136454946754595585670566993857190463750305618264096412179005177856,188374109627167571998899871484807295971773970996564895653572244044593362626511385237142198758023568227473377297792835283496928595231875152809132048206089502588928.
It means that there are fourteen pairs of common roots ofw3(x,z) and q(x,z) in the listed intervals, respectively, and we know that the fifth pairs of interval satisfy (24), and then w3(x,z) and q(x,z) have a common root, denoted by (x#,z#), on the first listed interval. Noted that W3(x,z(x))=W[l1(x),l2(x),l0(x)]; it follows (36)dW3dx=∂W3∂x+∂W3∂z·dzdx=-4x-z5w6x,z4522822787109375z-1234x+313z9x-1244z+312x10p4x,z,where w6(x,z) is a polynomial of degree 87 in x. Furthermore,(37)Rq,w6,z=125516666719925656132877164868220165759187808157696×1017x364x+37x-144ψx,with ψ(x) being a polynomial in z of degree 435, and ψ(x)≠0on(38)x=-77037131072,-1925932768,z=-92575072482333894229188992679733785697931696898287454081973172991196020261297061888,-46287536241166947114594496339866892848965348449143727040986586495598010130648530944.Therefore, x# is a simple root of W3(x,z(x)).
(iv) Similarly, we use the same program as (i) and (ii) to find all the possible intervals, which may hold the common roots ofw4(x,z) and q(x,z) and then obtain the following regular chains (39)x,z,x-1,z-1,4x+3,4z+3,v1∗x,z,v2∗x,zwhere v1∗(x,z)=v11∗(z)x+v12∗(z), and v11∗, v12∗, and v2∗ are polynomials in z of degrees 492, 492, and 552, respectively. Isolating the fourth regular chains yields the following 16 pairs of common roots of w4(x,z) and q(x,t) in the listed intervals, (40)x=92901131072,4645165536,z=-8527736783370307578941148342474772255746725420980011496577676626844588240573268701473812127674924007424,-5329835489606442236838217714046732659841703388112593536104789177786765035829293842113257979682750464,x=-2807932768,-112315131072,z=-62097811964485368974018846032164624263763104161396101637820261927217107839786668602559178668060348078522694548577690162289924414440996864,-248391247857941475896075384128658497055052416645584406551281047708867431359146674410236714672241392314090778194310760649159697657763987456,x=129907131072,3247732768,z=-598017798453319089682467176741126562956892774310466556363290822430262768537665926336713898529563388567880069503262826159877325124512315660672063305037119488,-23920711938132763587298687069645062518275710972418662254531632897210510741506593705346855594118253554271520278013051304639509300498049262642688253220148477952,x=132241131072,528967524288,z=-149506956041959422693109279692314028903728674040828851288804660177139154251063231584178474632390847141970017375815706539969331281128078915168015826259279872,-299013912083918845386218559384628057807457348081657702577609320354278308502125463168356949264781694283940034751631413079938662562256157830336031652518559744,x=7408165536,148163131072,z=-970920108972836127454018227738948097891023316537137303226285020612568108992114474011154664524427946373126085988481658748083205070504932198000989141204992,-75853133513502822457345174042105320147736196604463851814553517235356883515113078212145816597093331040047546785012958969400039613319782796882727665664,x=3849132768,153965131072,z=-7934069000259293230339900193595999462143913221474197483771010920406420034171829505932967566311591067993517960455041197510853084776057301352261178326384973520803911109862890320275011481043468288,-31736276001037172921359600774383997848575652885896789935084043681625680136687318023731870265246354271974071841820164790043412339104229205409044713305539894083215644439451561281100045924173873152,x=-1886932768,-75475131072,z=-491325446537020304627744682442542192028966769185746026660275055103311732197187093329245736369326316631281573374653997517877902705223825521735199141247292070280934397209846730719022121202017504638277531421638656,-78612071445923248740439149190806750724634683069719364265644008816529877151549934932679317819092210661004959173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496,x=90165131072,4508365536,z=-8788064464986802426857977386447097113426520111131128669827057830675844234346894153846159171018517988167243043134222844204689080525734196832968125318070224677190649881668353091698688,-35152257859947209707431909545788388453706080444524514679308231322703376937387576615384636674074071952668972172536891376818756322102936787331872501272280898708762599526673412366794752,x=721332768,28853131072,z=-18032702246771773005116595615827710116494253682796810994126423384054859133097138282620211989292945639146568621528992587283360401824603189390869761855907572637988050133502132224,-36065404493543546010233191231655420232988507365593621988252846768109718266194276565240413978585891278293137243057985174566720803649206378781739523711815145275976100267004264448,⋯⋯x=-113093131072,-2827332768,z=13050533785057456157295832511770523562505162740258073053842618568582459060074376291897137590064188545819787018382342682267975428761855001222473056385648716020711424,652526689252872807864791625588526178125258137012903652692130928429122953003718815948568795032094272909893509191171341133987714380927500611236528192824358010355712,x=-37134096,-118815131072,z=425782184818719608816274319342813113834066795298816,1703128739274878435265097377371252455336267181195264.
The seventh interval satisfies (13); it is said that there exists a common root of w4(x,z) and q(x,z), denoted (x∗,z∗). Noted that W4(x,z(x))=W[l1(x),l2(x),l0(x),l3(x)], it follows (41)dW4dx=∂W4∂x+∂W4∂z·dzdx=-32x-z9w7x,z83106868713134765625x13x-1314x+316z-1304z+315z12p7x,z,where w7(x,z) is a polynomial of degree 118 in x. Furthermore, (42)Rq,w7,z=A×1029x544x+37x-156ψ-x,with(43)A=33264629888493003881538379168484242740236178671706338400101606065709062155191451648,and ψ-(x) is a polynomial in z of degree 591, and ψ-(x)≠0  on(44)x=-1886932768,-75475131072,z=-491325446537020304627744682442542192028966769185746026660275055103311732197187093329245736369326316631281573374653997517877902705223825521735199141247292070280934397209846730719022121202017504638277531421638656,-78612071445923248740439149190806750724634683069719364265644008816529877151549934932679317819092210661004959173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496.Therefore, x∗ is a simple root of W4(x,z(x)). It is said that w4(x,z)≠0.
Based on Lemmas2 and 6, we obtain Proposition 4


## 4. Conclusion

In this work, we have studied the limit cycle bifurcation of the strongly nonlinear oscillator. The main tool is the first approximation of the Poincaré map with some classical and new methods in bifurcation theory of dynamical systems. Some results on the possible maximal number of limit cycles are obtained by an algebraic analysis on the corresponding period annulus.



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*Source: 6943563-2019-06-02.xml*