# Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle

Authors: Huanhuan Tian; Maoan Han
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the expansions of the first order Melnikov functions for general near-Hamiltonian systems near a compound loop with a cusp and a nilpotent saddle. We also obtain formulas for the first coefficients appearing in the expansions and then establish a bifurcation theorem on the number of limit cycles. As an application example, we give a lower bound of the maximal number of limit cycles for a polynomial system of Liénard type.


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## Body

## 1. Introduction

Consider a planar system of the form(1)x˙=Hy+ϵp(x,y,δ),y˙=-Hx+ϵq(x,y,δ),
where ϵ is a small parameter and H(x,y), p(x,y,δ), and q(x,y,δ) are C∞ functions in (x,y)∈R2 and δ∈D⊂Rm with D bounded. For ϵ=0, (1) becomes
(2)x˙=Hy,y˙=-Hx,
which is a Hamiltonian system. As we know, the system (1) is said to be a near-Hamiltonian system. For (1), the main task is to study the number of limit cycles which are bifurcated from periodic orbits of the unperturbed system (2). On this aspect, the first order Melnikov function of (1) plays an important role. We can use the expansions of it near Hamiltonian values corresponding to a center or an invariant loop to find its zeros and hence the number of limit cycles. See a survey article [1]. There have been many works on this topic. For the study of general near-Hamiltonian systems, see [2–12]; and especially for the system (2) with the elliptic case, one can see [13–17] and references therein. In [2–4], the number of limit cycles of the system (1) near a homoclinic loop with a cusp of order one or two or a nilpotent saddle of order one (for the definition of an order of a cusp or nilpotent saddle, see [5]) was studied. In the heteroclinic case with two hyperbolic saddles, a hyperbolic saddle and a cusp of order one, or two cusps of order one or two, the number of limit cycles of the system (1) was studied in [5, 8, 9], respectively. In this paper, we suppose that the unperturbed system (2) has a compound loop consisting of a cusp S1 of order one, a nilpotent saddle S2 of order one, a homoclinic loop to S2, and two heteroclinic orbits connecting S1 and S2, as shown in Figure 1. We aim to study the number of limit cycles of (1) near the loop for ϵ≠0 small.Figure 1
Compound loop with a cusp and a nilpotent saddle.


## 2. Main Results with Proof

Now consider theC∞ systems (1) and (2). Suppose that (2) has a compound loop denoted by L0=L1∪L2∪L3∪{S1,S2} and defined by equation H(x,y)=0, where S1(x1,y1) is a cusp and S2(x2,y2) is a nilpotent saddle both having order one, L1,L2 are heteroclinic orbits satisfying ω(L1)=α(L2)=S2 and ω(L2)=α(L1)=S1, and L3 is a homoclinic loop to S2. Then, the level curves of H(x,y) define two families of periodic orbits Lh1 and Lh2 for h on one side of h=0 and a family of periodic orbits Lh3 for h on another side of h=0. For the definiteness, let both Lh1 and Lh2 exist for 0<-h≪1 and Lh3 exist for 0<h≪1. Thus, we have three Melnikov functions
(3)Mi(h,δ)=∮Lhiqdx-pdy,i=1,2,3.
Let Ui denote a closed set with diameter ϵ0>0 and with center at Si, i=1,2. See Figure 2(a). And further introduce
(4)Lh1(j)=Lh1∩Uj,j=1,2,Lh1(3)=Cl.(Lh1-⋃j=12‍Lh1(j)),Lh2(1)=Lh2∩U2,Lh2(2)=Lh2-Lh2(1),Lh3(1)=Lh3∩U1,Lh3∩U2=Lh3(2)∪Lh3(3)(asshowninFigure2(b)),Lh3(4)=Cl.(Lh3-⋃j=13‍Lh3(j)).
Here the Cl. denotes the closure of a set. Then by (3) and (4), for ϵ0 sufficiently small we can write
(5)M1(h,δ)=I11(h,δ)+I12(h,δ)+I13(h,δ)aaaaaaaaaaaaaaaafor0≤-h≪1,(6)M2(h,δ)=I21(h,δ)+I22(h,δ)aaaaaafor0≤-h≪1,(7)M3(h,δ)=I31(h,δ)+I32(h,δ)+I33(h,δ)+I34(h,δ)aaaaaaaaaaaaaaaaaaaaaaaaaaafor0≤h≪1,
where
(8)Iij(h,δ)=∫Lhi(j)qdx-pdy,i∈{1,2,3},j∈{1,2,3,4}.(a)
(b)By [5], there exist two transformations of the form
(9)(x,y)T=Ti(u,v)T+Si,i=1,2,
where Ti is a 2×2 matrix satisfying det⁡Ti=1 such that (1) becomes
(10)u˙=Hiv+ϵpi(u,v,δ),v˙=-Hiu+ϵqi(u,v,δ),
where
(11)H1(u,v)=v22+∑k+j≥3‍h~kjukvj,H1(u,φ1(u))=h~3u3+O(u4),h~3<0,H1v(u,φ1(u))=0,p1(u,v,δ)=∑i+j≥0‍a~ijuivj,q1(u,v,δ)=∑i+j≥0‍b~ijuivj,(12)H2(u,v)=v22+∑k+j≥3‍h¯kjukvj,H2(u,φ2(u))=h¯4u4+O(u5),h¯4<0,H2v(u,φ2(u))=0,p2(u,v,δ)=∑i+j≥0‍a¯ijuivj,q2(u,v,δ)=∑i+j≥0‍b¯ijuivj,
for (u,v) near (0,0). Note that qdx-pdy=q1du-p1dv for (x,y) near S1 and qdx-pdy=q2du-p2dv for (x,y) near S2. Then we have
(13)Ii1(h,δ)=∫L~hi(1)‍q1du-p1dv,i=1,3,(14)I21(h,δ)=∫L¯h2(1)‍q2du-p2dv,(15)I12(h,δ)=∫L¯h1(2)‍q2du-p2dv,I3j(h,δ)=∫L¯h3(j)‍q2du-p2dv,j=2,3,
where L~hi(1) denote the image of Lhi(1) under T1 and L¯h2(1), L¯h1(2), and L¯h3(j) denote the image of Lh2(1), Lh1(2), and Lh3(j) under T2, respectively. Then, by using [3, 4] we can obtain the following two lemmas, respectively.Lemma 1.
Consider system (10) with i=1 and suppose (11), (13) hold. Then there are constants B00,B00*,B10,B10* satisfying
(16)B00=35∫01‍dvv(1-v3)=35×2.4286⋯>0,B00*=-35∫-∞1‍dv1-v3=-35×4.2065⋯<0,B10=-37(∫01‍v3/2dv1-v3(1+1-v3)-2)>0,B10*=37(∫1-1‍vdv1-v3-∫01‍v3/2dv1+v3(1+1+v3)-2)<0
such that
(17)I11(h,δ)=B00c1(S1,δ)|h|5/6+B10c3(S1,δ)|h|7/6-111B00c4(S1,δ)|h|11/6+O(h2)+N11(h,δ)
for 0<-h≪1,
(18)I31(h,δ)=B00*c1(S1,δ)h5/6+B10*c3(S1,δ)h7/6+111B00*c4(S1,δ)h11/6+O(h2)+N31(h,δ)
for 0<h≪1, where Ni1(h,δ)∈Cω at h=0 with Ni1(0,δ)=O(ϵ0),i=1,3, and
(19)c1(S1,δ)=22h~3-1/3(a~10+b~01),c3(S1,δ)=22h~3-5/3×[+13(h~212-2h~4-3h~3h~12)(a~10+b~01)]h~3(2a~20+b~11)aaaaaa+13(h~212-2h~4-3h~3h~12)(a~10+b~01)],c4(S1,δ)=9μ1-1α~01-2μ1-7[(20μ23-20μ1μ2μ3+4μ12μ4)α~00aaaaaaaaa+(4μ12μ3-10μ1μ22)α~10aaaaaaaaa+4μ12μ2α~20-μ13α~30],
where
(20)μ1=h~31/3,μ2=13h~3-2/3h~4,μ3=19h~3-5/3(3h~3h~5-h~42),μ4=181h~3-8/3(27h~32h~6-18h~3h~4h~5+5h~43),h~3=h~30,h~4=-12h~212+h~40,h~5=h~12h~212-h~21h~31+h~50,h~6=-2h~122h~212-h~03h~213+h~212h~22+2h~12h~21h~31-12h~312-h~21h~41+h~60,α~00=22(a~10+b~01),α~10=22[-h~12(a~10+b~01)+2a~20+b~11],α~20=22[(a~10+b~01)(3h~03h~21-h~22+32h~122)-2h~12a~20aaaaaaa(a~10+b~01)(3h~03h~21-h~22+32h~122)-h~12b~11+3a~30+b~21-a~11h~21-2b~02h~21],α~30=22[(a~10+b~01)(3h~13h~21+3h~03h~31+3h~12h~2252h~123-h~32aaaaaaaaaaaaaaaaaaa-15h~12h~03h~21-52h~123-h~32)aaaaaaaaa+(3a~11+6b~02)h~12h~21-2(b~12+a~21)h~21aaaaaaaaa-(a~11+2b~02)h~31+4a~40+b~31aaaaaaaaa+(3b~11+6a~20)h~03h~21-(2a~20+b~11)h~22aaaaaaaaa+(3a~20+32b~11)h~122-(3a~30+b~21)h~12],α~01=22[23a~12+2b~03-2h~03a~11aaaaaaaaaaa23-4h~03b~02+(a~10+b~01)(5h~032-2h~04)].Lemma 2.
Consider system (10) with i=2 and suppose (12), (15) hold. Then we have
(21)I12(h,δ)=c1(S2,δ)|h|3/4-c2(S2,δ)hln⁡|h|+c4(S2,δ)|h|5/4+c5(S2,δ)|h|7/4-c6(S2,δ)h2ln⁡|h|+O(h2)+N12(h,δ)
for 0<-h≪1,
(22)I32(h,δ)=12c1*(S2,δ)h3/4+12c3*(S2,δ)h5/4+12c4*(S2,δ)h7/4+O(h2)+N32(h,δ),I33(h,δ)=12c1*(S2,δ)h3/4+12c3*(S2,δ)h5/4+12c4*(S2,δ)h7/4+O(h2)+N33(h,δ)
for 0<h≪1, where Nij(h,δ)∈Cω at h=0 with Nij(0,δ)=O(ϵ0), (i,j)∈{(1,2),(3,2),(3,3)}, and
(23)c1(S2,δ)=-22|A~0||h¯4|-1/4(a¯10+b¯01),c2(S2,δ)=-24|h¯4|-1/2(2a¯20+b¯11)+O(a¯10+b¯01),c4(S2,δ)=|A~2|[(2132h¯52-34h¯4h¯6)|h¯4|-11/4α¯00aaaaaaaa+34|h¯4|-7/4h¯5α¯10+|h¯4|-3/4α¯20],c5(S2,δ)=17|A~0|[6d1-1α¯01kkkkkkk+d1-9(105d1d22d3-30d12d2d4aaaaaaaaaaaaa-15d12d32+5d13d5-70d24)α¯00aaaaaaaa+d1-8(35d23-30d1d2d3+5d12d4)α¯10aaaaaaa-d1-7(15d22-5d1d3)α¯20aaaaaaa+5d1-6d2α¯30-d1-5α¯40],c6(S2,δ)=-132[-6d1-3d2α¯01+3d1-2α¯11aaaaaaaaaa-d1-11(504d1d23d3-168d12d2d32aaaaaaaaaaaaaaaaaa+42d13d3d4-168d12d22d4aaaaaaaaaaaaaaaaaa+42d13d2d5-252d25-6d14d6)α¯00aaaaaaaaaaa+d1-10(168d1d22d3-126d24aaaaaaaaaaaaaaaaa-42d12d2d4-21d12d32+6d13d5)α¯10aaaaaaaaaaa-d1-9(42d1d2d3-56d23-6d12d4)α¯20aaaaaaaaaaa+d1-8(6d1d3-21d22)α¯30aaaaaaaaaa+6d1-7d2α¯40-d1-6α¯50],c1*(S2,δ)=-D1c1(S2,δ),c3*(S2,δ)=-D2c4(S2,δ),c4*(S2,δ)=D1c5(S2,δ),
where D1=2|A¯0|/|A~0|,D2=2|A¯1|/|A~2|,A¯0,A~0,A¯1, and A~2 are constants, given by
(24)A¯0=23∫0∞‍dv1+v4≈1.236049785>0,A~0=-23∫01‍dv1-v4=-2π3/26[Γ(3/4)]2≈-0.8740191847<0,A¯1=-25∫0∞‍dv1+v4[v2+1+v4]≈-0.3388852337<0,A~2=25[1-∫01‍v2dv1-v4(1+1-v4)]≈0.2396280470>0,d1=|h¯4|1/4,d2=-14|h¯4|-3/4h¯5,d3=132|h¯4|-7/4(8h¯4h¯6-3h¯52),d4=-1128|h¯4|-11/4(7h¯53+32h¯42h¯7-24h¯4h¯5h¯6),d5=12048|h¯4|-15/4(512h¯43h¯8-192h¯42h¯62aaaaaaaaaaaaaaa-384h¯42h¯5h¯7+336h¯4h¯52h¯6-77h¯54),d6=-18192|h¯4|-19/4(2048h¯44h¯9+1344h¯42h¯5h¯62aaaaaaaaaaaaaaaaa-1536h¯43h¯5h¯8-1536h¯43h¯6h¯7aaaaaaaaaaaaaaaaa+1344h¯42h¯52h¯7-1232h¯4h¯53h¯6aaaaaaaaaaaaaaaaa+231h¯55);α¯00=22(a¯10+b¯01),α¯10=22[-h¯12(a¯10+b¯01)+2a¯20+b¯11],α¯20=22[(a¯10+b¯01)(3h¯03h¯21-h¯22+32h¯122)aaaaaaa-2h¯12a¯20-h¯12b¯11+3a¯30+b¯21aaaaaaa(3h¯03h¯21-h¯22+32h¯122)-a¯11h¯21-2b¯02h¯21],α¯30=22[(a¯10+b¯01)(3a¯20+32b¯11)aaaaaaa×(3h¯13h¯21+3h¯03h¯31+3h¯12h¯22-15h¯12h¯03h¯21-52h¯123-h¯32aaaaaaaaaaa-15h¯12h¯03h¯21-52h¯123-h¯32)aaaaaaaaa+(3a¯11+6b¯02)h¯12h¯21-2(b¯12+a¯21)h¯21aaaaaaaaa-(a¯11+2b¯02)h¯31+4a¯40+b¯31aaaaaaaaa+(3b¯11+6a¯20)h¯03h¯21aaaaaaaaa-(2a¯20+b¯11)h¯22+(3a¯20+32b¯11)h¯122aaaaaaaa(3a¯20+32b¯11)-(3a¯30+b¯21)h¯12],α¯40=22[n1(a¯10+b¯01)(52h¯123-3h¯12h¯22+15h¯12h¯03h¯21+h¯32aaaaaaaa-(52h¯123-3h¯12h¯22+15h¯12h¯03h¯21+h¯3252aaaaaaaaaaaa-3h¯13h¯21-3h¯03h¯31)(2a¯20+b¯11)aaaaaaaa+(32h¯122+3h¯03h¯21-h¯22)aaaaaaaa×(3a¯30+b¯21-a¯11h¯21-2b¯02h¯21)aaaaaaaa-h¯12(h¯214a¯40+b¯31-2a¯21h¯21aaaaaaaaaaaaaaaa-2b¯12h¯21+2a¯11h¯12h¯21+4b¯02h¯12h¯21aaaaaaaaaaaaaaa(52h¯123-3h¯12h¯22+15h¯12h¯03h¯21+h¯32-a¯11h¯31-2b¯02h¯31)+n3],α¯50=22[n2(a¯10+b¯01)+n1(2a¯20+b¯11)(52h¯123-3h¯12h¯22+15h¯12h¯03h¯21+h¯32aaaaaaaa-(52h¯123-3h¯12h¯22+15h¯12h¯03h¯21+h¯32aaaaaaaaaaa52-3h¯13h¯21-3h¯03h¯31)aaaaaaa×(3a¯30+b¯21-a¯11h¯21-2b¯02h¯21)aaaaaaaa+(32h¯122+3h¯03h¯21-h¯22)aaaaaaa×(4a¯40+b¯31-2a¯21h¯21aaaaaaaaaaaa-2b¯12h¯21+2a¯11h¯12h¯21aaaaaaaaaaaa+4b¯02h¯12h¯21-a¯11h¯31-2b¯02h¯31)aaaaaaa(52h¯123-3h¯12h¯22+15h¯12h¯03h¯21+h¯32-h¯12n3+n4],α¯01=22[23a¯12+2b¯03-2h¯03a¯11-4h¯03b¯02aaaaaaa23+(a¯10+b¯01)(5h¯032-2h¯04)],α¯11=22[(a¯10+b¯01)(10h¯04h¯12-2h¯14+10h¯03h¯13-35h¯12h¯03243aaaaaaaaaaaaaaaaa+10h¯03h¯13-35h¯12h¯032)aaaaaaaaa+(a¯11+2b¯02)(-2h¯13+10h¯12h¯03)aaaaaaaa-2h¯12(a¯12+3b¯03)aaaaaaaaa+(2a¯20+b¯11)(5h¯032-2h¯04)aaaaaaaaa-4h¯03(b¯12+a¯21)+43a¯22+2b¯13];h¯4=-12h¯212+h¯40,h¯5=h¯12h¯212-h¯21h¯31+h¯50,h¯6=-2h¯122h¯212-h¯03h¯213+h¯212h¯22+2h¯12h¯21h¯31-12h¯312-h¯21h¯41+h¯60,h¯7=-4h¯22h¯212h¯12+h¯70-h¯51h¯21-h¯13h¯213+h¯32h¯212+4h¯123h¯212-h¯31h¯41+h¯12h¯312+2h¯22h¯21h¯31+6h¯03h¯213h¯12-3h¯03h¯212h¯31-4h¯122h¯21h¯31+2h¯12h¯21h¯41,h¯8=h¯80-h¯23h¯213-h¯61h¯21+h¯04h¯214+h¯42h¯212-2h¯122h¯312-8h¯124h¯212-2h¯222h¯212-92h¯032h¯214-h¯31h¯51+h¯22h¯312-12h¯412+8h¯123h¯21h¯31-4h¯122h¯21h¯41+12h¯22h¯212h¯122-24h¯03h¯213h¯122+2h¯12h¯31h¯41+6h¯12h¯213h¯13-4h¯12h¯212h¯32+2h¯12h¯21h¯51+6h¯22h¯213h¯03+2h¯22h¯21h¯41+2h¯32h¯21h¯31-3h¯03h¯21h¯312-3h¯03h¯212h¯41-3h¯13h¯212h¯31-8h¯12h¯22h¯21h¯31+18h¯12h¯03h¯212h¯31,h¯9=h¯12[h¯214(45h¯032-8h¯04)+6h¯213(h¯23-8h¯03h¯22)aaaaaaaa+2h¯212(9h¯13h¯31-2h¯42+6h¯222+9h¯03h¯41)aaaaaaaa+2h¯21(9h¯03h¯312-4h¯41h¯22-4h¯31h¯32)aaaaaaa-4h¯22h¯312+h¯412+2h¯31h¯51+2h¯21h¯61]+4h¯122[h¯21(3h¯21h¯32-6h¯212h¯13-h¯51aaaaaaaaaaaaah¯21(3h¯21h¯32-6h¯212h¯13-h¯51+++l+3h¯21h¯32-6h¯212h¯13-h¯516h¯31h¯22-18h¯31h¯03h¯21)-h¯31h¯41]+4h¯123(20h¯03h¯213+2h¯41h¯21-8h¯22h¯212+h¯312)+h¯21(2h¯42h¯31-16h¯124h¯31-4h¯222h¯31+2h¯22h¯51aaaaaaaa+2h¯32h¯41-3h¯13h¯312-h¯71-6h¯03h¯31h¯41)+h¯212(16h¯125-4h¯32h¯22-3h¯31h¯23-3h¯03h¯51+h¯52+18h¯22h¯31h¯03aaaaaaa16h¯125-4h¯32h¯22-3h¯31h¯2316h¯125-4h¯32h¯22-3h¯31h¯23-3h¯41h¯13-3h¯03h¯51+h¯52+18h¯22h¯31h¯03)+2h¯22h¯31h¯41+6h¯32h¯213h¯03+4h¯04h¯213h¯31-h¯41h¯51-h¯31h¯61+h¯32h¯312-h¯03h¯313-9h¯03h¯214h¯13-18h¯31h¯032h¯213+h¯14h¯214-h¯33h¯213+6h¯22h¯213h¯13+h¯90;n1=18(12h¯222+35h¯124+24h¯12h¯32aaaaaaa-60h¯122h¯22-72h¯12h¯03h¯31+324h¯122h¯03h¯21aaaaaaa+108h¯032h¯212-72h¯12h¯13h¯21-72h¯22h¯03h¯21),n2=-18(-24h¯22h¯32+60h¯12h¯222+60h¯122h¯32+72h¯22h¯13h¯21-180h¯122h¯13h¯21-216h¯03h¯212h¯13aaaaaaaa-140h¯123h¯22+63h¯125aaaaaaaa-180h¯122h¯03h¯31+780h¯123h¯03h¯21aaaaaaaaa+72h¯22h¯03h¯31-504h¯12h¯22h¯03h¯21aaaaaaaaa-216h¯032h¯21h¯31+972h¯12h¯032h¯212aaaaaaaa+72h¯03h¯21h¯32+72h¯22h¯13h¯21aaaaaaaa18-180h¯122h¯13h¯21-216h¯03h¯212h¯13),n3=5a¯50+b¯41-2b¯22h¯21-a¯11h¯41+2a¯11h¯22h¯21-3a¯11h¯03h¯212+2a¯11h¯12h¯31-4a¯11h¯122h¯21-2b¯02h¯41+4b¯02h¯22h¯21-6b¯02h¯03h¯212+4b¯02h¯12h¯31-8b¯02h¯122h¯21-2b¯12h¯31+4b¯12h¯12h¯21-2a¯21h¯31+4a¯21h¯12h¯21-3a¯31h¯21+3b¯03h¯212+a¯12h¯212,n4=4b¯12h¯22h¯21+4b¯02h¯32h¯21+6a¯31h¯12h¯21+2a¯11h¯12h¯41+4b¯12h¯12h¯31-4a¯11h¯122h¯31+4b¯02h¯12h¯41-12b¯03h¯212h¯12-4a¯12h¯212h¯12+2a¯11h¯22h¯31+6b¯03h¯21h¯31-3a¯11h¯13h¯212+6a¯60+b¯51-6b¯12h¯03h¯212+2a¯11h¯32h¯21+4b¯02h¯22h¯31+4a¯21h¯22h¯21+2a¯12h¯21h¯31+4b¯22h¯12h¯21-8b¯12h¯122h¯21+16b¯02h¯123h¯21+8a¯11h¯123h¯21-6b¯02h¯13h¯212-8a¯21h¯122h¯21-8b¯02h¯122h¯31-4a¯41h¯21-2b¯32h¯21-6a¯21h¯03h¯212+4a¯21h¯12h¯31+3b¯13h¯212+2a¯22h¯212-6a¯11h¯03h¯21h¯31+18a¯11h¯03h¯212h¯12-8a¯11h¯12h¯22h¯21-12b¯02h¯03h¯21h¯31+36b¯02h¯03h¯212h¯12-16b¯02h¯12h¯22h¯21-2b¯22h¯31-a¯11h¯51-2b¯12h¯41-2a¯21h¯41-2b¯02h¯51-3a¯31h¯31.For convenience, let(25)L*=L1∪L2,L1*=L*∩U1,L2*=L*∩U2,L3*=Cl.(L*-⋃i=12‍Li*),L01=L1*,L0∩U2=L02∪L03(asshowninFigure2(b)),L04=Cl.(L0-⋃j=13‍L0j).Theorem 3.
Assume that system (1) has a compound loop L0 as stated before. Then, the functions Mi(h,δ) given in (3) at h=0 have the following expansions:
(26)M1(h,δ)=c0(δ)+c1(δ)|h|3/4+B00c2(δ)|h|5/6-c3(δ)hln⁡|h|+c4(δ)h+B10c5(δ)|h|7/6+c6(δ)|h|5/4+c7(δ)|h|7/4-111B00c8(δ)|h|11/6-c9(δ)h2ln⁡|h|+O(h2),(27)M2(h,δ)=c¯0(δ)+c1(δ)|h|3/4+c3(δ)hln⁡|h|+c¯3(δ)h+c6(δ)|h|5/4+c7(δ)|h|7/4+c9(δ)h2ln⁡|h|+O(h2)
for 0<-h≪1, and
(28)M3(h,δ)=c~0(δ)-D1c1(δ)h3/4+B00*c2(δ)h5/6+c~2(δ)h+B10*c5(δ)h7/6-D2c6(δ)h5/4+D1c7(δ)h7/4+111B00*c8(δ)h11/6+O(h2)
for 0<h≪1, where
(29)c1(δ)=c1(S2,δ),c2(δ)=c1(S1,δ),c3(δ)=c2(S2,δ),c5(δ)=c3(S1,δ),c6(δ)=c4(S2,δ),c7(δ)=c5(S2,δ),c8(δ)=c4(S1,δ),c9(δ)=c6(S2,δ),(30)c0(δ)=M1(0,δ)=∮L*‍qdx-pdy=∑i=12‍∫Li‍qdx-pdy,c¯0(δ)=M2(0,δ)=∮L3‍qdx-pdy,c~0(δ)=M3(0,δ)=c0(δ)+c¯0(δ),(31)c4(δ)=∫L1*‍(px+qy-σ)dt+∫L2*‍[px+qy-η0-η1(x-x2)]dt+∫L3*‍(px+qy)dt+t2c1(δ)+t3c2(δ)+t4c3(δ),c¯3(δ)=∮L3‍[px+qy-η0-η1(x-x2)]dtaaaaaaaaaaaa+t0c1(δ)+t1c3(δ),c~2(δ)=∫L1*‍(px+qy-σ)dt+∫L02∪L03‍(px+qy-η0)dt+∫L04‍(px+qy)dt+t5c1(δ)+t6c2(δ),
where σ=(px+qy)|(x1,y1),η0=(px+qy)|(x2,y2),η1=(pxx+qyx)|(x2,y2). In particular,
(32)c4(δ)=∑i=12‍∫Li‍(px+qy)dt,c¯3(δ)=∮L3‍(px+qy)dt,c~2(δ)=∑i=13‍∫Li‍(px+qy)dt
if c1(δ)=c2(δ)=c3(δ)=0. Here, ti,i=0,1,…,6, are constants and B00,B00*,B10,B10* are given in Lemma 1.Proof.
First, by (6), (10) with i=2, (12), (14), (29), and Theorem 2.2 in [4], we directly obtain (27) with c¯0(δ),c¯3(δ) given by (30) and (31), respectively. Then we study the expansions of M1 and M3.
By (5), (7), (29), and Lemmas 1 and 2, we have
(33)M1(h,δ)=c1(δ)|h|3/4+B00c2(δ)|h|5/6-c3(δ)hln⁡|h|+B10c5(δ)|h|7/6+c6(δ)|h|5/4+c7(δ)|h|7/4-111B00c8(δ)|h|11/6-c9(δ)h2ln⁡|h|+O(h2)+N(h,δ)
for 0<-h≪1, and
(34)M3(h,δ)=-D1c1(δ)h3/4+B00*c2(δ)h5/6+B10*c5(δ)h7/6-D2c6(δ)h5/4+D1c7(δ)h7/4+111B00*c8(δ)h11/6+O(h2)+N*(h,δ)
for 0<h≪1, where
(35)N(h,δ)=N11(h,δ)+N12(h,δ)+I13(h,δ),N*(h,δ)=N31(h,δ)+N32(h,δ)+N33(h,δ)+I34(h,δ).
Let
(36)N(h,δ)=c0(δ)+c4(δ)h+O(h2),N*(h,δ)=c~0(δ)+c~2(δ)h+O(h2).
It follows further that
(37)c0(δ)=N11(0,δ)+N12(0,δ)+I13(0,δ)=lim⁡ϵ0→0[N11(0,δ)+N12(0,δ)+I13(0,δ)]=lim⁡ϵ0→0I13(0,δ)=∮L1∪L2‍qdx-pdy=∑i=12‍∫Li‍qdx-pdy=M1(0,δ),c~0(δ)=N31(0,δ)+N32(0,δ)+N33(0,δ)+I34(0,δ)=lim⁡ϵ0→0[N31(0,δ)+N32(0,δ)aaaaaaa+N33(0,δ)+I34(0,δ)]=lim⁡ϵ0→0I34(0,δ)=∮L0‍qdx-pdy=∑i=13‍∫Li‍qdx-pdy=M3(0,δ),c4(δ)+O(h)=Nh(h,δ)=M1h(h,δ)+34c1(δ)|h|-1/4+56B00c2(δ)|h|-1/6+c3(δ)(ln⁡|h|+1)+O(|h|1/6),c~2(δ)+O(h)=Nh*(h,δ)=M3h(h,δ)+34D1c1(δ)h-1/4-56B00*c2(δ)h-1/6+O(h1/6).
Then by Lemma 3.1.2 in [5], we have
(38)c4(δ)=Nh(0,δ)=lim⁡h→0-[∮Lh1‍(px+qy)dt+34c1(δ)|h|-1/4aaaaaaaaaa∮Lh1‍(px+qy)+56B00c2(δ)|h|-1/6+c3(δ)(ln⁡|h|+1)],c~2(δ)=Nh*(0,δ)=lim⁡h→0+[∮Lh3‍(px+qy)dtaaaaaaaa∮Lh3‍(px+qy)+34D1c1(δ)h-1/4-56B00*c2(δ)h-1/6].
It is easy to see that
(39)∮Lh1(px+qy)dt=∑i=13‍∫Lh1(i)‍(px+qy)dt∮Lh1(px+qy)dt=∫Lh1(1)‍(px+qy-σ)dt∮Lh1(px+qy)dt+∫Lh1(2)‍[px+qy-η0aaaagaaa∮Lh1(px+qy)dt-η1(x-x2)]dt∮Lh1(px+qy)dt+∫Lh1(3)‍(px+qy)dt+σ∫Lh1(1)‍dt∮Lh1(px+qy)dt+η0∫Lh1(2)‍dt+η1∫Lh1(2)‍(x-x2)dt,∮Lh3‍(px+qy)dt=∑i=14‍∫Lh3(i)‍(px+qy)dt=∫Lh3(1)‍(px+qy-σ)dt+∫Lh3(2)∪Lh3(3)‍(px+qy-η0)dt+∫Lh3(4)‍(px+qy)dt+σ∫Lh3(1)‍dt+η0∫Lh3(2)∪Lh3(3)‍dt.
Noting h~3<0,h¯4<0, by (19), (23), and (29), we have
(40)c4(δ)=lim⁡h→0-[∫Lh1(1)‍(px+qy-σ)dtaaaaaaaaaa+∫Lh1(2)‍[px+qy-η0-η1(x-x2)]dtaaaaaaaaa+∫Lh1(3)‍(px+qy)dt]+lim⁡h→0-σ[∫Lh1(1)‍dt-523B00|h~3|-1/3|h|-1/6]+lim⁡h→0-η0[∫Lh1(2)‍dt+322A~0|h¯4|-1/4|h|-1/4aaaaaaaaaaa∫Lh1(2)‍+O(1)(ln⁡|h|+1)]+lim⁡h→0-η1[∫Lh1(2)‍(x-x2)dtaaaaaaaaaaaa-24|h¯4|-1/2(ln⁡|h|+1)],c~2(δ)=lim⁡h→0+[∫Lh3(1)‍(px+qy-σ)dtaaaaaaaaa+∫Lh3(2)∪Lh3(3)‍(px+qy-η0)dtaaaaaaaa+∫Lh3(4)‍(px+qy)dt]+lim⁡h→0+σ[∫Lh3(1)‍dt+523B00*|h~3|-1/3h-1/6]+lim⁡h→0+η0[∫Lh3(2)∪Lh3(3)‍dt-32A¯0|h¯4|-1/4h-1/4].
Then by the proof of (3.13) in [3], the following equations hold:
(41)lim⁡h→0-[∫Lh1(1)‍dt-523B00|h~3|-1/3|h|-1/6]=T0,lim⁡h→0+[∫Lh3(1)‍dt+523B00*|h~3|-1/3h-1/6]=T0*,lim⁡h→0-∫Lh1(1)‍(px+qy-σ)dt=lim⁡h→0+∫Lh3(1)‍(px+qy-σ)dt=∫L1*‍(px+qy-σ)dt.
Here, T0 and T0* are constants. By a similar argument used in Theorems 2.2 and 2.4 in [4], one can obtain
(42)lim⁡h→0-[∫Lh1(2)‍dt+322A~0|h¯4|-1/4|h|-1/4+O(1)(ln⁡|h|+1)]=T1,lim⁡h→0-[∫Lh1(2)‍(x-x2)dt-24|h¯4|-1/2(ln⁡|h|+1)]=T1*,lim⁡h→0+[∫Lh3(i)‍dt-322A¯0|h¯4|-1/4h-1/4]=Ti,i=2,3,lim⁡h→0-∫Lh1(2)‍[px+qy-η0-η1(x-x2)]dt=∫L2*‍[px+qy-η0-η1(x-x2)]dt,lim⁡h→0+∫Lh3(i)‍(px+qy-η0)dt=∫L0i‍(px+qy-η0)dt,aaaaaaaaaaaaai=2,3.
Here T1*, Ti, i=1,2,3, are constants. Therefore, we can obtain (31) and (32). Thus we have proved Theorem 3.In the following we use Theorem3 to study the problem of limit cycle bifurcation near L0. For the sake of convenience, we say that (1) has a distribution (i,j)+k of i+j+k limit cycles if there are i and j limit cycles near the inside of L* and L3, respectively, and k limit cycles near the outside of L0. Then we can prove the following theorem.Theorem 4.
Assume that system (1) has a compound loop L0 as stated before and (26)–(28) hold. Define c4*(δ)=c4(δ)|c1=c2=c3=0,  c¯3*(δ)=c¯3(δ)|c1=c3=0,c~2*(δ)=c4*(δ)+c¯3*(δ),  c3*(δ)=c3(δ)|c1=0. Let there exist δ0∈Rm, such that (c0,c¯0,c1,c2,c3*,c4*,c¯3*)(δ0)=(0,0,0,0,0,0,0).(1)
Ifcl(δ0)≠0,cj(δ0)=0,j=5,…,l-1, and
(43)rank⁡∂(c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,…,cl-1)∂(δ1,δ2,δ3,δ4,δ5,δ6,δ7,δ8,…,δm)=l+2,then (1) can have 2l+1 limit cycles near L0 for some (ϵ,δ) near (0,δ0), where l=6,7 or 9.(2)
Ifc5(δ0)c6(δ0)<0, and
(44)rank⁡∂(c0,c¯0,c1,c2,c3*,c4*,c¯3*)∂(δ1,δ2,δ3,δ4,…,δm)=7,then (1) can have 11 limit cycles near L0 for some (ϵ,δ) near (0,δ0).(3)
Ifc8(δ0)c9(δ0)>0,cj(δ0)=0,j=5,6,7, and
(45)rank⁡∂(c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,c6,c7)∂(δ1,δ2,δ3,δ4,δ5,δ6,δ7,δ8,…,δm)=10,then (1) can have 18 limit cycles near L0 for some (ϵ,δ) near (0,δ0).Proof.
(1) Because of the similarity in the proof, we only prove the conclusion forl=9 and omit the rest. By our assumptions, there exists δ0∈Rm such that c¯0(δ0)=c¯3*(δ0)=c3*(δ0)=c4*(δ0)=0,cj(δ0)=0,j=0,1,2,5,6,7,8,c9(δ0)≠0, and
(46)rank⁡∂(c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,…,c8)∂(δ1,δ2,δ3,δ4,δ5,δ6,δ7,δ8,…,δm)=11.
By the implicit function theorem, we can take c¯0,c¯3*,c3*,c4*,cj,j=0,1,2,5,6,7,8 as free parameters varying near zero. Obviously, for these parameters varying near zero we have |c9|≥|(1/2)c9(δ0)|>0. In the following we proceed the process by 9 steps.
Step 1. Fix (c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,c6,c7)=(0,0,0,0,0,0,0,0,0,0) and vary c8 near 0.
First, forc8=0, we have by (26) c9M1>0 for 0<-h≪1.
Let0<|c8|≪1. Then c9M1<0 for 0<-h≪1 if c8c9>0. Thus, M1 has a zero. Hence, for 0<|c8|≪1,
(1) the conditionc8c9>0 implies a distribution (1,0)+0 of one limit cycle.
Step 2. Fix (c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,c6)=(0,0,0,0,0,0,0,0,0), c8c9>0 and vary c7 near 0.
First, forc7=0, we have by (26), (27), and (28) c8M1<0,c9M2<0 for 0<-h≪1, and c8M3<0 for 0<h≪1.
Let0<|c7|≪|c8|. Then c8M1>0,c9M2>0 for 0<-h≪1, and c8M3>0 for 0<h≪1 if c7c8>0,c7c9>0. Thus, M1,M2, and M3 each gets a zero and the zero of M1 got in Step 1 still exists. Hence, for 0<|c7|≪|c8|≪1,
(2) the conditionsc7c8>0,c8c9>0 imply a distribution (2,1)+1 of 4 limit cycles.
Step 3. Fix (c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5)=(0,0,0,0,0,0,0,0), c7≠0 and vary c6 near 0.
First, forc6=0, we have by (26), (27), and (28) c7M1>0,c7M2>0 for 0<-h≪1, and c7M3>0 for 0<h≪1.
Let0<|c6|≪|c7|. Then c7M1<0,c7M2<0 for 0<-h≪1, and c7M3>0 for 0<h≪1 if c6c7<0. Thus, M1 and M2 each gets a new zero and the zeros got in above steps still exist. Hence, for 0<|c6|≪|c7|≪|c8|≪1,
(3) the conditionsc6c7<0,c7c8>0,c8c9>0 imply a distribution (3,2)+1 of 6 limit cycles.
Step 4. Fix (c0,c¯0,c1,c2,c3*,c4*,c¯3*)=(0,0,0,0,0,0,0), c6≠0 and vary c5 near 0.
First, forc5=0, we have by (26) and (28) c6M1>0 for 0<-h≪1, and c6M3<0 for 0<h≪1.
Let0<|c5|≪|c6|. Then c6M1<0 for 0<-h≪1, and c6M3>0 for 0<h≪1 if c5c6<0. Thus, M1 and M3 each has a new zero and the zeros got in above steps still exist. Hence, for 0<|c5|≪|c6|≪|c7|≪|c8|≪1,
(4) the conditionsc5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (4,2)+2 of 8 limit cycles.
Step 5. Fix (c0,c¯0,c1,c2,c3*)=(0,0,0,0,0), c5c6<0 and vary (c¯3*,c4*) near (0,0) with c~2*(=c¯3*+c4*)≠0.
First, for(c¯3*,c4*)=(0,0), we have by (26), (27), and (28) c5M1>0,c6M2>0 for 0<-h≪1, and c5M3<0 for 0<h≪1.
Let0<|c¯3*,c4*|≪|c5|. Then
(47)c5M1<0,c6M2<0for0<-h≪1,c5M3>0for0<h≪1
if c4*c5>0,c¯3*c6>0,c~2*c5>0, and
(48)c5M1<0,c6M2<0for0<-h≪1,c5M3<0for0<h≪1
if c4*c5>0,c¯3*c6>0,c~2*c5<0. Thus, M1,M2, and M3 each has one more zero in the first case and M1 and M2 each has a new zero in the second case. And the zeros got in above steps still exist. Hence, for 0<|c¯3*,c4*|≪|c5|≪|c6|≪|c7|≪|c8|≪1, (5i)
the conditionsc¯3*c4*<0,c¯3*c~2*<0,c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (5,3)+3 of 11 limit cycles, and(5ii)
the conditionsc¯3*c4*<0,c¯3*c~2*>0,c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (5,3)+2 of 10 limit cycles.
Step 6. Fix (c0,c¯0,c1,c2)=(0,0,0,0), c4*c¯3*<0 with c~2*≠0 and vary c3* near 0.
First, forc3*=0, we have by (26) and (27) c4*M1<0 and c¯3*M2<0 for 0<-h≪1.
Let0<|c3*|≪|c4*,c¯3*|. Then c4*M1>0 and c¯3*M2>0 for 0<-h≪1, if c3*c4*<0,c¯3*c3*>0. Thus, M1 and M2 each gets a new zero and the zeros got in above steps still exist. Hence, for 0<|c3*|≪|c4*,c¯3*|≪|c5|≪|c6|≪|c7|≪|c8|≪1, (6i)
the conditionsc3*c4*<0,c¯3*c4*<0,c¯3*c~2*<0,c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (6,4)+3 of 13 limit cycles,(6ii)
the conditionsc3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0,c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (6,4)+2 of 12 limit cycles.
Step 7. Fix (c0,c¯0,c1)=(0,0,0), c3*c~2*≠0 and vary c2 near 0.
First, forc2=0, we have by (26) and (28) c3*M1<0 for 0<-h≪1 and c~2*M3>0 for 0<h≪1.
Let0<|c2|≪|c3*|≪|c4*,c¯3*|. Then
(49)c3*M1>0for0<-h≪1,c~2*M3<0for0<h≪1
if c2c3*>0,c~2*c2>0, and
(50)c3*M1>0for0<-h≪1,c~2*M3>0for0<h≪1
if c2c3*>0,c~2*c2<0. Thus, M1 and M3 each gets one more zero in the first case and only M1 has a new zero in the second case. And the zeros got in above steps still exist. Hence, for 0<|c2|≪|c3*|≪|c4*,c¯3*|≪|c5|≪|c6|≪|c7|≪|c8|≪1, (7i)
the conditionsc2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0,c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (7,4)+3 of 14 limit cycles,(7ii)
the conditionsc2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*<0,c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (7,4)+3 of 14 limit cycles.
Step 8. Fix (c0,c¯0)=(0,0), c2c3*>0 and vary c1 near 0.
First, forc1=0, we have by (26), (27), and (28) c2M1>0,c3*M2>0 for 0<-h≪1, and c2M3<0 for 0<h≪1.
Let0<|c1|≪|c2|. Then c2M1<0,c3*M2<0 for 0<-h≪1, and c2M3>0 for 0<h≪1 if c1c2<0. Thus, M1,M2, and M3 each gets a new zero and the zeros got in above steps still exist. Hence, for 0<|c1|≪|c2|≪|c3*|≪|c4*,c¯3*|≪|c5|≪|c6|≪|c7|≪|c8|≪1,
(8) the conditionsc1c2<0,c2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0),c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (8,5)+4 of 17 limit cycles.
Step 9. Fix c1≠0 and vary (c0,c¯0) near (0,0).
First, for(c0,c¯0)=(0,0), we have by (26), (27), and (28) c1M1>0,c1M2>0, for 0<-h≪1, and c1M3<0 for 0<h≪1.
Let0<|c0,c¯0|≪|c1|. Then
(51)c1M1<0,c1M2<0,for0<-h≪1,c1M3<0for0<h≪1
if c0c1<0,c¯0c1<0,
(52)c1M1<0,c1M2>0,for0<-h≪1,c1M3>0for0<h≪1
if c0c1<0,c¯0c1>0 and (c0+c¯0)c1>0, and
(53)c1M1>0,c1M2<0,for0<-h≪1,c1M3>0for0<h≪1
if c0c1>0,c¯0c1<0 and (c0+c¯0)c1>0. Thus, we have correspondingly (a) M1 and M2 each has a new zero, (b) M1 and M3 each has a new zero, or (c) M2 and M3 each has a new zero. And the zeros got in above steps still exist.
Hence, for0<|c0,c¯0|≪|c1|≪|c2|≪|c3*|≪|c4*,c¯3*|≪|c5|≪|c6|≪|c7|≪|c8|≪1, (9i)
the conditionsc0c¯0>0,c0c1<0,c1c2<0,c2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0),c4*c5>0,c5c6<0,c6c7<0,c7c8>0, c8c9>0 imply a distribution (9,6)+4 of 19 limit cycles,(9ii)
the conditionsc0c¯0<0, c0c1<0,(c0+c¯0)c1>0, c1c2<0,c2c3*>0,c3*c4*<0, c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0),c4*c5>0,c5c6<0, c6c7<0,c7c8>0,c8c9>0 imply a distribution (9,5)+5 of 19 limit cycles,(9iii)
the conditionsc0c¯0<0,c0c1>0,(c0+c¯0)c1>0,c1c2<0,c2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0), c4*c5>0,c5c6<0,c6c7<0,c7c8>0,c8c9>0 imply a distribution (8,6)+5 of 19 limit cycles.
Thus we get the conclusion forl=9.
(2) By our assumptions in case (2) and the implicit function theorem we can takec0,c¯0,c1,c2,c3*,c4*,c¯3* as free parameters varying near zero. Obviously, for these parameters varying near zero we have c5c6<0. By a similar argument in the above proof, we can prove that for 0<|c0,c¯0|≪|c1|≪|c2|≪|c3*|≪|c4*,c¯3*|≪1, (i)
the conditionsc0c¯0>0,c0c1<0,c1c2<0,c2c3*>0,c3*c4*<0,c¯3*c4*<0, c¯3*c~2*>0(orc¯3*c~2*<0), c4*c5>0 imply a distribution (5,4)+2 of 11 limit cycles,(ii)
the conditionsc0c¯0<0,c0c1<0, (c0+c¯0)c1>0, c1c2<0, c2c3*>0, c3*c4*<0,c¯3*c4*<0, c¯3*c~2*>0(orc¯3*c~2*<0),c4*c5>0 imply a distribution (5,3)+3 of 11 limit cycles,(iii)
the conditionsc0c¯0<0, c0c1>0,(c0+c¯0)c1>0,c1c2<0,c2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0), c4*c5>0 imply a distribution (4,4)+3 of 11 limit cycles.
(3) By our assumptions in case (3) and the implicit function theorem we can takec0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,c6,andc7 as free parameters varying near zero. Obviously, for these parameters varying near zero we have c8c9>0. By a similar argument used in proving case (1), we can prove that for 0<|c0,c¯0|≪|c1|≪|c2|≪|c3*|≪|c4*,c¯3*|≪|c5|≪|c6|≪|c7|≪1, (i)
the conditionsc0c¯0>0,c0c1<0,c1c2<0,c2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0),c4*c5>0,c5c6<0,c6c7<0, c7c8>0 imply a distribution (8,6)+4 of 18 limit cycles,(ii)
the conditionsc0c¯0<0,c0c1<0,(c0+c¯0)c1>0, c1c2<0, c2c3*>0, c3*c4*<0, c¯3*c4*<0, c¯3*c~2*>0(orc¯3*c~2*<0), c4*c5>0, c5c6<0,c6c7<0,c7c8>0 imply a distribution (8,5)+5 of 18 limit cycles,(iii)
the conditionsc0c¯0<0, c0c1>0,(c0+c¯0)c1>0,c1c2<0, c2c3*>0,c3*c4*<0,c¯3*c4*<0,c¯3*c~2*>0(orc¯3*c~2*<0), c4*c5>0,c5c6<0,c6c7<0, c7c8>0 imply a distribution (7,6)+5 of 18 limit cycles.This completes the proof.


## 3. An Application

Consider a Liénard system of the form(54)x˙=y,y˙=-(x+1)2x3(x2-14x-12)-ϵf(x,δ)y,
where
(55)f(x,δ)=∑j=0n‍ajxj,δ=(a0,a1,…,an),aaaaaaaaaaaaaaaaaaaaaaaaaaaaa8≤n≤12.
System (54) |ϵ=0 is Hamiltonian with
(56)H(x,y)=12y2+18(x+1)3x4(x-1).
We have the following theorem.Theorem 5.
LetC(n) denote the maximal number of limit cycles of the system (54) for ϵ small and all δ. Then we have C(11)≥18, C(n)≥2n-5(n=8,9,10,12).Proof.
It is easy to verify that the unperturbed system has a compound loopL0=L1∪L2∪L3∪{S1,S2} with a cusp S1(-1,0) of order one and a nilpotent saddle S2(0,0) of order one, L1,L2 are heteroclinic orbits satisfying ω(L1)=α(L2)=S2 and ω(L2)=α(L1)=S1, and L3 is a homoclinic loop to S2. Inside L*=L1+L2 (L3, resp.), there is a center C1((1/8)(1-33),0) (C2((1/8)(1+33),0), resp.).
Because of the similarity in the proof, here we only prove the case forn=11 and omit the rest of the proof.
Letn=11. By Theorem 3, we obtain
(57)c0(δ)=M1(0,δ)=-∮L*‍f(x,δ)ydx=-∑j=011‍ajI1j,c¯0(δ)=M2(0,δ)=-∮L3‍f(x,δ)ydx=-∑j=011‍ajI2j,
where
(58)I1j=∮L*‍xjydx=2∫-10‍xjydx=∫-10‍xj+2(x+1)1-x2dx,j=0,1,…,11,I2j=∮L3‍xjydx=2∫01‍xjydx=∫01‍xj+2(x+1)1-x2dx,j=0,1,…,11.
Therefore,
(59)c0(δ)=-a0(π16-215)-a1(π32-215)-a2(π32-8105)-a3(5π256-8105)-a4(5π256-16315)-a5(7π512-16315)-a6(7π512-1283465)-a7(21π2048-1283465)-a8(21π2048-2569009)-a9(33π4096-2569009)-a10(33π4096-102445045)-a11(429π65536-102445045),c¯0(δ)=-a0(π16+215)-a1(π32+215)-a2(π32+8105)-a3(5π256+8105)-a4(5π256+16315)-a5(7π512+16315)-a6(7π512+1283465)-a7(21π2048+1283465)-a8(21π2048+2569009)-a9(33π4096+2569009)-a10(33π4096+102445045)-a11(429π65536+102445045).
Note that S2 is a nilpotent saddle of order one and h¯4=-1/8. By (23), we have
(60)c1(δ)=c1(S2,δ)=29/4|A~0|a0,c3(δ)=c2(S2,δ)=a1+O1(a0),c6(δ)=c4(S2,δ)=23/4|A~2|(-21a0+12a1-8a2),c7(δ)=c5(S2,δ)i=21/4|A~0|7(20354a0-310a1+180a2-80a3+32a4),c9(δ)=c6(S2,δ)i=3334a0-2074a1+29a2-15a3+6a4-2a5.
Making the transformationx=u-1,y=v, system (54) becomes
(61)u˙=v,v˙=-u2(u-1)3(u2-94u+34)-ϵf~(u,δ)v.
Then we have
(62)H(u,v)=v22+18(u8-6u7+14u6-16u5+9u4-2u3),h~3=-14,f~(u,δ)=∑j=011‍aj(u-1)j=∑j=011‍(-1)jaj+∑j=111‍(-1)(j-1)jaju+∑j=211‍(-1)(j-2)Cj2aju2+∑j=311‍(-1)(j-3)Cj3aju3+∑j=411‍(-1)(j-4)Cj4aju4+∑j=511‍(-1)(j-5)Cj5aju5+∑j=611‍(-1)(j-6)Cj6aju6+∑j=711‍(-1)(j-7)Cj7aju7+∑j=811‍(-1)(j-8)Cj8aju8+∑j=911‍(-1)(j-9)Cj9aju9+(a10-11a11)u10+a11u11.
By (19), we have
(63)c2(δ)=c1(S1,δ)=213/6∑j=011‍(-1)jaj,c5(δ)=c3(S1,δ)=217/6[∑j=111‍(-1)j(j-3)aj-3a0],c8(δ)=c4(S1,δ)=213/6(-13163a0+272a1-4603a2+2243a3aaaaaaaa-28a4+163a5+43a6-43a8aaaaaaaa-163a9+28a10-2243a11).
Note that
(64)c1(δ)=c1(S2,δ)=29/4|A~0|a0,c2(δ)=c1(S1,δ)=213/6∑j=011‍(-1)jaj,c3(δ)=c2(S2,δ)=a1+O1(a0).
We have c1(δ)=c2(δ)=c3(δ)=0 if and only if a0=a1=0, and a11=∑i=210(-1)iai. It implies further that
(65)c4(δ)=∮L*‍(px+qy)dt=-∮L*‍f(x,δ)dt=-∮L*‍f(x,δ)ydx=-2∫-10‍f(x,δ)ydx=-2∫-10‍1y(∑i=011‍aixi)dx=-4∫-10‍1x2(x+1)1-x2aaaaaa×(∑i=210‍aixi+∑i=210‍(-1)iaix11)dx=-4∑i=210‍ai∫-10‍fi(x)dx,
where
(66)fi(x)=xi-2[1+(-1)ix(11-i)](x+1)1-x2,i=2,3,…,10.
Similarly,
(67)c¯3(δ)=∮L3‍(px+qy)dt=-∮L3‍f(x,δ)dt=-∮L3‍f(x,δ)ydx=-2∫01‍f(x,δ)ydx=-4∑i=210‍ai∫01‍fi(x)dx.
Therefore,
(68)c4(δ)=-4a2(315π256+9335)+4a3(187π256+9335)-4a4(187π256+5835)+4a5(123π256+5835)-4a6(123π256+104105)+4a7(75π256+104105)-4a8(75π256+1635)+4a9(35π256+1635)-4a10(35π256),c¯3(δ)=-4a2(315π256-9335)+4a3(187π256-9335)-4a4(187π256-5835)+4a5(123π256-5835)-4a6(123π256-104105)+4a7(75π256-104105)-4a8(75π256-1635)+4a9(35π256-1635)-4a10(35π256).
Letc4*(δ)=c4(δ)|c1=c2=c3=0,c¯3*(δ)=c¯3(δ)c1=c3=0, c3*(δ)=c3(δ)|c1(δ)=0. Furthermore, one sees that equations c0(δ)=c¯0(δ)=c1(δ)=c2(δ)=c3*(δ)=c4*(δ)=c¯3*(δ)=c5(δ)=c6(δ)=c7(δ)=0 have the solution a0=a1=a2=0,a3=(13/74)a9-(1/2)a7,a4=(65/148)a9-(5/4)a7,a5=-(16/37)a9,a6=(7/4)a7-(255/148)a9,a8=(96/37)a9,a10=-(53/37)a9,a11=-(32/37)a9, which gives c8(δ)=(55/37)231/6a9,c9(δ)=(32/37)a9. And further, c8(δ)c9(δ)>0 if a9≠0. Thus, fix a9≠0 and take δ0=(0,0,0,(13/74)a9-(1/2)a7,(65/148)a9-(5/4)a7,-(16/37)a9,(7/4)a7-(255/148)a9,a7,(96/37)a9,a9,-(53/37)a9,-(32/37)a9).
Then we have(69)rank⁡∂(c0,c¯0,c1,c2,c3*,c4*,c¯3*,c5,c6,c7)∂(a0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11)=10.
Hence by Theorem 4(3), we know that there are 18 limit cycles near L0 for some δ near δ0. This ends the proof.



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*Source: 819798-2014-07-16.xml*