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

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.

1. Introduction

Consider a planar system of the form 𝑥̇=𝐻_𝑦+𝜖𝑝𝑥,𝑦,𝛿,𝑦̇=−𝐻_𝑥+𝜖𝑞𝑥,𝑦,𝛿, (1) 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⊂ℝ^𝑚 with 𝐷 bounded. For 𝜖=0, (1) becomes 𝑥̇=𝐻_𝑦,𝑦̇=−𝐻_𝑥, (2) 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 the 𝐶^∞ 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 𝐿_ℎ1 and 𝐿_ℎ2 for ℎ on one side of ℎ=0 and a family of periodic orbits 𝐿_ℎ3 for ℎ on another side of ℎ=0. For the definiteness, let both 𝐿_ℎ1 and 𝐿_ℎ2 exist for 0<−ℎ≪1 and 𝐿_ℎ3 exist for 0<ℎ≪1. Thus, we have three Melnikov functions 𝑀_𝑖(ℎ,𝛿)=_𝐿ℎ𝑖𝑞𝑑𝑥−𝑝𝑑𝑦,𝑖=1,2,3. (3) Let 𝑈_𝑖 denote a closed set with diameter 𝜖₀>0 and with center at 𝑆_𝑖, 𝑖=1,2. See Figure 2(a). And further introduce 𝐿^(𝑗)_ℎ1=𝐿_ℎ1∩𝑈_𝑗𝐿,𝑗=1,2,_(3)ℎ1=Cl.L_ℎ1−²_𝑗=1𝐿^(𝑗)_ℎ1,𝐿⁽¹⁾_ℎ2=𝐿_ℎ2∩𝑈₂,𝐿⁽²⁾_ℎ2=𝐿_ℎ2−𝐿⁽¹⁾_ℎ2,𝐿⁽¹⁾_ℎ3=𝐿_ℎ3∩𝑈₁,𝐿_ℎ3∩𝑈₂=𝐿⁽²⁾_ℎ3∪𝐿⁽³⁾_ℎ3(b),𝐿asshowninFigure2_(4)ℎ3=Cl.𝐿_ℎ3−³_𝑗=1𝐿^(𝑗)_ℎ3. (4) Here the Cl. denotes the closure of a set. Then by (3) and (4), for 𝜖₀ sufficiently small we can write 𝑀₁(ℎ,𝛿)=𝐼₁₁(ℎ,𝛿)+𝐼₁₂(ℎ,𝛿)+𝐼₁₃(ℎ,𝛿)for0≤−ℎ≪1,𝑀₂(ℎ,𝛿)=𝐼₂₁(ℎ,𝛿)+𝐼₂₂(ℎ,𝛿)for0≤−ℎ≪1,𝑀₃(ℎ,𝛿)=𝐼₃₁(ℎ,𝛿)+𝐼₃₂(ℎ,𝛿)+𝐼₃₃(ℎ,𝛿)+𝐼₃₄(ℎ,𝛿)for0≤ℎ≪1, (5)(6)(7) where 𝐼_𝑖𝑗(ℎ,𝛿)=_{𝐿(𝑗)ℎ𝑖}𝑞𝑑𝑥−𝑝𝑑𝑦,𝑖∈{1,2,3},𝑗∈{1,2,3,4}. (8)

(a)

(b)

Figure 2 

By [5], there exist two transformations of the form 𝑥,𝑦^𝑇=𝑇_𝑖(𝑢,𝑣)^𝑇+𝑆_𝑖,𝑖=1,2, (9) where 𝑇_𝑖 is a 2×2 matrix satisfying det𝑇_𝑖=1 such that (1) becomes 𝑢̇=𝐻_𝑖𝑣+𝜖𝑝_𝑖(𝑢,𝑣,𝛿),𝑣̇=−𝐻_𝑖𝑢+𝜖𝑞_𝑖(𝑢,𝑣,𝛿), (10) where 𝐻₁()=𝑣𝑢,𝑣²2+_{𝑘+𝑗≥3}ℎ_𝑘𝑗𝑢^𝑘𝑣^𝑗,𝐻₁𝑢,𝜑₁(𝑢)=ℎ₃𝑢³+𝑂𝑢⁴,ℎ₃𝐻<0,_1𝑣𝑢,𝜑₁(𝑢)=0,𝑝₁(𝑢,𝑣,𝛿)=_{𝑖+𝑗≥0}𝑎_𝑖𝑗𝑢^𝑖𝑣^𝑗,𝑞₁(𝑢,𝑣,𝛿)=_{𝑖+𝑗≥0}𝑏̃_𝑖𝑗𝑢^𝑖𝑣^𝑗,𝐻₂()=𝑣𝑢,𝑣²2+_{𝑘+𝑗≥3}ℎ_𝑘𝑗𝑢^𝑘𝑣^𝑗,𝐻₂𝑢,𝜑₂(𝑢)=ℎ₄𝑢⁴+𝑂𝑢⁵,ℎ₄𝐻<0,_2𝑣𝑢,𝜑₂(𝑢)𝑝=0,₂(𝑢,𝑣,𝛿)=_{𝑖+𝑗≥0}𝑎_𝑖𝑗𝑢^𝑖𝑣^𝑗,𝑞₂(𝑢,𝑣,𝛿)=_{𝑖+𝑗≥0}𝑏_𝑖𝑗𝑢^𝑖𝑣^𝑗, (11)(12) 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 𝐼_𝑖1(ℎ,𝛿)=_{𝐿(1)ℎ𝑖}𝑞₁𝑑𝑢−𝑝₁𝑑𝑣,𝑖=1,3,𝐼₂₁(ℎ,𝛿)=_𝐿(1)ℎ2𝑞₂𝑑𝑢−𝑝₂𝑑𝑣,𝐼₁₂(ℎ,𝛿)=_𝐿(2)ℎ1𝑞₂𝑑𝑢−𝑝₂𝐼𝑑𝑣,_3𝑗(ℎ,𝛿)=_{𝐿(𝑗)ℎ3}𝑞₂𝑑𝑢−𝑝₂𝑑𝑣,𝑗=2,3, (13)(14)(15) where 𝐿⁽¹⁾_ℎ𝑖 denote the image of 𝐿⁽¹⁾_ℎ𝑖 under 𝑇₁ and 𝐿⁽¹⁾_ℎ2, 𝐿⁽²⁾_ℎ1, and 𝐿^(𝑗)_ℎ3 denote the image of 𝐿⁽¹⁾_ℎ2, 𝐿⁽²⁾_ℎ1, and 𝐿^(𝑗)_ℎ3 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,𝐵^∗₀₀,𝐵₁₀,𝐵^∗₁₀ satisfying 𝐵₀₀=35¹₀𝑑𝑣𝑣1−𝑣³=35𝐵×2.4286⋯>0,_∗003=−5¹_−∞𝑑𝑣√1−𝑣³3=−5𝐵×4.2065⋯<0,₁₀3=−7¹₀𝑣^3/2𝑑𝑣√1−𝑣³1+√1−𝑣³𝐵−2>0,_∗10=37₁⁻¹𝑣𝑑𝑣√1−𝑣³−¹₀𝑣^3/2𝑑𝑣√1+𝑣³1+√1+𝑣³−2<0 (16) such that 𝐼₁₁(ℎ,𝛿)=𝐵₀₀𝑐₁𝑆₁,𝛿|ℎ|^5/6+𝐵₁₀𝑐₃𝑆₁,𝛿|ℎ|^7/6−1𝐵11⁰⁰𝑐₄𝑆₁,𝛿|ℎ|^11/6+𝑂ℎ²+𝑁₁₁(ℎ,𝛿) (17) for 0<−ℎ≪1, 𝐼₃₁(ℎ,𝛿)=𝐵^∗₀₀𝑐₁𝑆₁,𝛿ℎ^5/6+𝐵^∗₁₀𝑐₃𝑆₁,𝛿ℎ^7/6+1𝐵11^∗00𝑐₄𝑆₁,𝛿ℎ^11/6+𝑂ℎ²+𝑁₃₁(ℎ,𝛿) (18) for 0<ℎ≪1, where 𝑁_𝑖1(ℎ,𝛿)∈𝐶^𝜔 at ℎ=0 with 𝑁_𝑖1(0,𝛿)=𝑂(𝜖₀),𝑖=1,3, and 𝑐₁𝑆₁,𝛿=2√2ℎ₃^−1/3𝑎₁₀+𝑏̃₀₁,𝑐₃𝑆₁,𝛿=2√2ℎ₃^−5/3×ℎ₃2𝑎₂₀+𝑏̃₁₁+13ℎ²₂₁−2ℎ₄−3ℎ₃ℎ₁₂𝑎₁₀+𝑏̃₀₁,𝑐₄𝑆₁,𝛿=9𝜇₁⁻¹𝛼₀₁−2𝜇₁⁻⁷20𝜇³₂−20𝜇₁𝜇₂𝜇₃+4𝜇²₁𝜇₄𝛼₀₀+4𝜇²₁𝜇₃−10𝜇₁𝜇²₂𝛼₁₀+4𝜇²₁𝜇₂𝛼₂₀−𝜇³₁𝛼₃₀, (19) where 𝜇₁=ℎ₃^1/3,𝜇₂=13ℎ₃^−2/3ℎ₄,𝜇₃=19ℎ₃^−5/33ℎ₃ℎ₅−ℎ²₄,𝜇₄=1ℎ81^3−8/327ℎ²₃ℎ₆−18ℎ₃ℎ₄ℎ₅+5ℎ³₄,ℎ₃=ℎ₃₀,ℎ₄1=−2ℎ²₂₁+ℎ₄₀,ℎ₅=ℎ₁₂ℎ²₂₁−ℎ₂₁ℎ₃₁+ℎ₅₀,ℎ₆=−2ℎ²₁₂ℎ²₂₁−ℎ₀₃ℎ³₂₁+ℎ²₂₁ℎ₂₂+2ℎ₁₂ℎ₂₁ℎ₃₁−12ℎ²₃₁−ℎ₂₁ℎ₄₁+ℎ₆₀,𝛼₀₀=2√2𝑎₁₀+𝑏̃₀₁,𝛼₁₀=2√2−ℎ₁₂𝑎₁₀+𝑏̃₀₁+2𝑎₂₀+𝑏̃₁₁,𝛼₂₀=2√2𝑎₁₀+𝑏̃₀₁3ℎ₀₃ℎ₂₁−ℎ₂₂+32ℎ²₁₂−2ℎ₁₂𝑎₂₀−ℎ₁₂𝑏̃₁₁+3𝑎₃₀+𝑏̃₂₁−𝑎₁₁ℎ₂₁−2𝑏̃₀₂ℎ₂₁,𝛼₃₀=2√2𝑎₁₀+𝑏̃₀₁3ℎ₁₃ℎ₂₁+3ℎ₀₃ℎ₃₁+3ℎ₁₂ℎ₂₂−15ℎ₁₂ℎ₀₃ℎ₂₁−52ℎ³₁₂−ℎ₃₂+3𝑎₁₁+6𝑏̃₀₂ℎ₁₂ℎ₂₁−2𝑏̃₁₂+𝑎₂₁ℎ₂₁−𝑎₁₁+2𝑏̃₀₂ℎ₃₁+4𝑎₄₀+𝑏̃₃₁+3𝑏̃₁₁+6𝑎₂₀ℎ₀₃ℎ₂₁−2𝑎₂₀+𝑏̃₁₁ℎ₂₂+3𝑎₂₀+32𝑏̃₁₁ℎ²₁₂−3𝑎₃₀+𝑏̃₂₁ℎ₁₂,𝛼₀₁=2√223𝑎₁₂+2𝑏̃₀₃−2ℎ₀₃𝑎₁₁−4ℎ₀₃𝑏̃₀₂+𝑎₁₀+𝑏̃₀₁5ℎ²₀₃−2ℎ₀₄. (20)

Lemma 2. Consider system (10) with 𝑖=2 and suppose (12), (15) hold. Then we have 𝐼₁₂(ℎ,𝛿)=𝑐₁𝑆₂,𝛿|ℎ|^3/4−𝑐₂𝑆₂,𝛿ℎln|ℎ|+𝑐₄𝑆₂,𝛿|ℎ|^5/4+𝑐₅𝑆₂,𝛿|ℎ|^7/4−𝑐₆𝑆₂,𝛿ℎ²ln|ℎ|+𝑂ℎ²+𝑁₁₂(ℎ,𝛿) (21) for 0<−ℎ≪1, 𝐼₃₂()=1ℎ,𝛿2𝑐^∗₁𝑆₂,𝛿ℎ^3/4+12𝑐^∗₃𝑆₂,𝛿ℎ^5/4+12𝑐^∗₄𝑆₂,𝛿ℎ^7/4+𝑂ℎ²+𝑁₃₂(),𝐼ℎ,𝛿₃₃()=1ℎ,𝛿2𝑐^∗₁𝑆₂,𝛿ℎ^3/4+12𝑐^∗₃𝑆₂,𝛿ℎ^5/4+12𝑐^∗₄𝑆₂,𝛿ℎ^7/4+𝑂ℎ²+𝑁₃₃(ℎ,𝛿) (22) for 0<ℎ≪1, where 𝑁_𝑖𝑗(ℎ,𝛿)∈𝐶^𝜔 at ℎ=0 with 𝑁_𝑖𝑗(0,𝛿)=𝑂(𝜖₀), (𝑖,𝑗)∈{(1,2),(3,2),(3,3)}, and 𝑐₁𝑆₂,𝛿=−2√2𝐴₀ℎ₄^−1/4𝑎₁₀+𝑏₀₁,𝑐₂𝑆₂√,𝛿=−24ℎ₄^−1/22𝑎₂₀+𝑏₁₁+𝑂𝑎₁₀+𝑏₀₁,𝑐₄𝑆₂,𝛿=𝐴₂2132ℎ²₅−34ℎ₄ℎ₆ℎ₄^−11/4𝛼₀₀+34ℎ₄^−7/4ℎ₅𝛼₁₀+ℎ₄^−3/4𝛼₂₀,𝑐₅𝑆₂=1,𝛿7𝐴₀6𝑑₁⁻¹𝛼₀₁+𝑑₁⁻⁹105𝑑₁𝑑²₂𝑑₃−30𝑑²₁𝑑₂𝑑₄−15𝑑²₁𝑑²₃+5𝑑³₁𝑑₅−70𝑑⁴₂𝛼₀₀+𝑑₁⁻⁸35𝑑³₂−30𝑑₁𝑑₂𝑑₃+5𝑑²₁𝑑₄𝛼₁₀−𝑑₁⁻⁷15𝑑²₂−5𝑑₁𝑑₃𝛼₂₀+5𝑑₁⁻⁶𝑑₂𝛼₃₀−𝑑₁⁻⁵𝛼₄₀,𝑐₆𝑆₂1,𝛿=−32−6𝑑₁⁻³𝑑₂𝛼₀₁+3𝑑₁⁻²𝛼₁₁−𝑑₁⁻¹¹504𝑑₁𝑑³₂𝑑₃−168𝑑²₁𝑑₂𝑑²₃+42𝑑³₁𝑑₃𝑑₄−168𝑑²₁𝑑²₂𝑑₄+42𝑑³₁𝑑₂𝑑₅−252𝑑⁵₂−6𝑑⁴₁𝑑₆𝛼₀₀+𝑑₁⁻¹⁰168𝑑₁𝑑²₂𝑑₃−126𝑑⁴₂−42𝑑²₁𝑑₂𝑑₄−21𝑑²₁𝑑²₃+6𝑑³₁𝑑₅𝛼₁₀−𝑑₁⁻⁹42𝑑₁𝑑₂𝑑₃−56𝑑³₂−6𝑑²₁𝑑₄𝛼₂₀+𝑑₁⁻⁸6𝑑₁𝑑₃−21𝑑²₂𝛼₃₀+6𝑑₁⁻⁷𝑑₂𝛼₄₀−𝑑₁⁻⁶𝛼₅₀,𝑐^∗₁𝑆₂,𝛿=−𝐷₁𝑐₁𝑆₂,𝛿,𝑐^∗₃𝑆₂,𝛿=−𝐷₂𝑐₄𝑆₂,𝑐,𝛿_∗4𝑆₂,𝛿=𝐷₁𝑐₅𝑆₂,𝛿, (23) where 𝐷₁=2|𝐴₀|/|𝐴₀|,𝐷₂=2|𝐴₁|/|𝐴₂|,𝐴₀,𝐴₀,𝐴₁, and 𝐴₂ are constants, given by 𝐴₀=23^∞₀𝑑𝑣√1+𝑣⁴𝐴≈1.236049785>0,₀2=−3¹₀𝑑𝑣√1−𝑣⁴√=−2𝜋^3/26[Γ(3/4)]²≈−0.8740191847<0,𝐴₁2=−5^∞₀𝑑𝑣√1+𝑣⁴𝑣²+√1+𝑣⁴𝐴≈−0.3388852337<0,₂=251−¹₀𝑣²𝑑𝑣√1−𝑣⁴1+√1−𝑣⁴𝑑≈0.2396280470>0,₁=ℎ₄^1/4,𝑑₂1=−4ℎ₄^−3/4ℎ₅,𝑑₃=132ℎ₄^−7/48ℎ₄ℎ₆−3ℎ²₅,𝑑₄1=−128ℎ₄^−11/47ℎ³₅+32ℎ²₄ℎ₇−24ℎ₄ℎ₅ℎ₆,𝑑₅=12048ℎ₄^−15/4512ℎ³₄ℎ₈−192ℎ²₄ℎ²₆−384ℎ²₄ℎ₅ℎ₇+336ℎ₄ℎ²₅ℎ₆−77ℎ⁴₅,𝑑₆1=−8192ℎ₄^−19/42048ℎ⁴₄ℎ₉+1344ℎ²₄ℎ₅ℎ²₆−1536ℎ³₄ℎ₅ℎ₈−1536ℎ³₄ℎ₆ℎ₇aaaaaaaaaaaaaaaaa+1344ℎ²₄ℎ²₅ℎ₇−1232ℎ₄ℎ³₅ℎ₆+231ℎ⁵₅;𝛼₀₀=2√2𝑎₁₀+𝑏₀₁,𝛼₁₀=2√2−ℎ₁₂𝑎₁₀+𝑏₀₁+2𝑎₂₀+𝑏₁₁,𝛼₂₀=2√2𝑎₁₀+𝑏₀₁3ℎ₀₃ℎ₂₁−ℎ₂₂+32ℎ²₁₂−2ℎ₁₂𝑎₂₀−ℎ₁₂𝑏₁₁+3𝑎₃₀+𝑏₂₁−𝑎₁₁ℎ₂₁−2𝑏₀₂ℎ₂₁,𝛼₃₀=2√2𝑎₁₀+𝑏₀₁×3ℎ₁₃ℎ₂₁+3ℎ₀₃ℎ₃₁+3ℎ₁₂ℎ₂₂−15ℎ₁₂ℎ₀₃ℎ₂₁−52ℎ³₁₂−ℎ₃₂+3𝑎₁₁+6𝑏₀₂ℎ₁₂ℎ₂₁−2𝑏₁₂+𝑎₂₁ℎ₂₁−𝑎₁₁+2𝑏₀₂ℎ₃₁+4𝑎₄₀+𝑏₃₁+3𝑏₁₁+6𝑎₂₀ℎ₀₃ℎ₂₁−2𝑎₂₀+𝑏₁₁ℎ₂₂+3𝑎₂₀+32𝑏₁₁ℎ²₁₂−3𝑎₃₀+𝑏₂₁ℎ₁₂,𝛼₄₀=2√2𝑛₁𝑎₁₀+𝑏₀₁−52ℎ³₁₂−3ℎ₁₂ℎ₂₂+15ℎ₁₂ℎ₀₃ℎ₂₁+ℎ₃₂−3ℎ₁₃ℎ₂₁−3ℎ₀₃ℎ₃₁2𝑎₂₀+𝑏₁₁+32ℎ²₁₂+3ℎ₀₃ℎ₂₁−ℎ₂₂×3𝑎₃₀+𝑏₂₁−𝑎₁₁ℎ₂₁−2𝑏₀₂ℎ₂₁−ℎ₁₂4𝑎₄₀+𝑏₃₁−2𝑎₂₁ℎ₂₁−2𝑏₁₂ℎ₂₁+2𝑎₁₁ℎ₁₂ℎ₂₁+4𝑏₀₂ℎ₁₂ℎ₂₁−𝑎₁₁ℎ₃₁−2𝑏₀₂ℎ₃₁+𝑛₃,𝛼₅₀=2√2𝑛₂𝑎₁₀+𝑏₀₁+𝑛₁2𝑎₂₀+𝑏₁₁−52ℎ³₁₂−3ℎ₁₂ℎ₂₂+15ℎ₁₂ℎ₀₃ℎ₂₁+ℎ₃₂−3ℎ₁₃ℎ₂₁−3ℎ₀₃ℎ₃₁×3𝑎₃₀+𝑏₂₁−𝑎₁₁ℎ₂₁−2𝑏₀₂ℎ₂₁+32ℎ²₁₂+3ℎ₀₃ℎ₂₁−ℎ₂₂×4𝑎₄₀+𝑏₃₁−2𝑎₂₁ℎ₂₁−2𝑏₁₂ℎ₂₁+2𝑎₁₁ℎ₁₂ℎ₂₁+4𝑏₀₂ℎ₁₂ℎ₂₁−𝑎₁₁ℎ₃₁−2𝑏₀₂ℎ₃₁−ℎ₁₂𝑛₃+𝑛₄,𝛼₀₁=2√223𝑎₁₂+2𝑏₀₃−2ℎ₀₃𝑎₁₁−4ℎ₀₃𝑏₀₂+𝑎₁₀+𝑏₀₁5ℎ²₀₃−2ℎ₀₄,𝛼₁₁=2√2𝑎₁₀+𝑏₀₁10ℎ₀₄ℎ₁₂−2ℎ₁₄+10ℎ₀₃ℎ₁₃−35ℎ₁₂ℎ²₀₃+𝑎₁₁+2𝑏₀₂−2ℎ₁₃+10ℎ₁₂ℎ₀₃−2ℎ₁₂𝑎₁₂+3𝑏₀₃+2𝑎₂₀+𝑏₁₁5ℎ²₀₃−2ℎ₀₄−4ℎ₀₃𝑏₁₂+𝑎₂₁+43𝑎₂₂+2𝑏₁₃;ℎ₄1=−2ℎ²₂₁+ℎ₄₀,ℎ₅=ℎ₁₂ℎ²₂₁−ℎ₂₁ℎ₃₁+ℎ₅₀,ℎ₆=−2ℎ²₁₂ℎ²₂₁−ℎ₀₃ℎ³₂₁+ℎ²₂₁ℎ₂₂+2ℎ₁₂ℎ₂₁ℎ₃₁−12ℎ²₃₁−ℎ₂₁ℎ₄₁+ℎ₆₀,ℎ₇=−4ℎ₂₂ℎ²₂₁ℎ₁₂+ℎ₇₀−ℎ₅₁ℎ₂₁−ℎ₁₃ℎ³₂₁+ℎ₃₂ℎ²₂₁+4ℎ³₁₂ℎ²₂₁−ℎ₃₁ℎ₄₁+ℎ₁₂ℎ²₃₁+2ℎ₂₂ℎ₂₁ℎ₃₁+6ℎ₀₃ℎ³₂₁ℎ₁₂−3ℎ₀₃ℎ²₂₁ℎ₃₁−4ℎ²₁₂ℎ₂₁ℎ₃₁+2ℎ₁₂ℎ₂₁ℎ₄₁,ℎ₈=ℎ₈₀−ℎ₂₃ℎ³₂₁−ℎ₆₁ℎ₂₁+ℎ₀₄ℎ⁴₂₁+ℎ₄₂ℎ²₂₁−2ℎ²₁₂ℎ²₃₁−8ℎ⁴₁₂ℎ²₂₁−2ℎ²₂₂ℎ²₂₁−92ℎ²₀₃ℎ⁴₂₁−ℎ₃₁ℎ₅₁+ℎ₂₂ℎ²₃₁−12ℎ²₄₁+8ℎ³₁₂ℎ₂₁ℎ₃₁−4ℎ²₁₂ℎ₂₁ℎ₄₁+12ℎ₂₂ℎ²₂₁ℎ²₁₂−24ℎ₀₃ℎ³₂₁ℎ²₁₂+2ℎ₁₂ℎ₃₁ℎ₄₁+6ℎ₁₂ℎ³₂₁ℎ₁₃−4ℎ₁₂ℎ²₂₁ℎ₃₂+2ℎ₁₂ℎ₂₁ℎ₅₁+6ℎ₂₂ℎ³₂₁ℎ₀₃+2ℎ₂₂ℎ₂₁ℎ₄₁+2ℎ₃₂ℎ₂₁ℎ₃₁−3ℎ₀₃ℎ₂₁ℎ²₃₁−3ℎ₀₃ℎ²₂₁ℎ₄₁−3ℎ₁₃ℎ²₂₁ℎ₃₁−8ℎ₁₂ℎ₂₂ℎ₂₁ℎ₃₁+18ℎ₁₂ℎ₀₃ℎ²₂₁ℎ₃₁,ℎ₉=ℎ₁₂ℎ⁴₂₁45ℎ²₀₃−8ℎ₀₄+6ℎ³₂₁ℎ₂₃−8ℎ₀₃ℎ₂₂+2ℎ²₂₁9ℎ₁₃ℎ₃₁−2ℎ₄₂+6ℎ²₂₂+9ℎ₀₃ℎ₄₁+2ℎ₂₁9ℎ₀₃ℎ²₃₁−4ℎ₄₁ℎ₂₂−4ℎ₃₁ℎ₃₂−4ℎ₂₂ℎ²₃₁+ℎ²₄₁+2ℎ₃₁ℎ₅₁+2ℎ₂₁ℎ₆₁+4ℎ²₁₂ℎ₂₁3ℎ₂₁ℎ₃₂−6ℎ²₂₁ℎ₁₃−ℎ₅₁+6ℎ₃₁ℎ₂₂−18ℎ₃₁ℎ₀₃ℎ₂₁−ℎ₃₁ℎ₄₁+4ℎ³₁₂20ℎ₀₃ℎ³₂₁+2ℎ₄₁ℎ₂₁−8ℎ₂₂ℎ²₂₁+ℎ²₃₁+ℎ₂₁2ℎ₄₂ℎ₃₁−16ℎ⁴₁₂ℎ₃₁−4ℎ²₂₂ℎ₃₁+2ℎ₂₂ℎ₅₁+2ℎ₃₂ℎ₄₁−3ℎ₁₃ℎ²₃₁−ℎ₇₁−6ℎ₀₃ℎ₃₁ℎ₄₁+ℎ²₂₁16ℎ⁵₁₂−4ℎ₃₂ℎ₂₂−3ℎ₃₁ℎ₂₃−3ℎ₀₃ℎ₅₁+ℎ₅₂+18ℎ₂₂ℎ₃₁ℎ₀₃+2ℎ₂₂ℎ₃₁ℎ₄₁+6ℎ₃₂ℎ³₂₁ℎ₀₃+4ℎ₀₄ℎ³₂₁ℎ₃₁−ℎ₄₁ℎ₅₁−ℎ₃₁ℎ₆₁+ℎ₃₂ℎ²₃₁−ℎ₀₃ℎ³₃₁−9ℎ₀₃ℎ⁴₂₁ℎ₁₃−18ℎ₃₁ℎ²₀₃ℎ³₂₁+ℎ₁₄ℎ⁴₂₁−ℎ₃₃ℎ³₂₁+6ℎ₂₂ℎ³₂₁ℎ₁₃+ℎ₉₀;𝑛₁=1812ℎ²₂₂+35ℎ⁴₁₂+24ℎ₁₂ℎ₃₂−60ℎ²₁₂ℎ₂₂−72ℎ₁₂ℎ₀₃ℎ₃₁+324ℎ²₁₂ℎ₀₃ℎ₂₁+108ℎ²₀₃ℎ²₂₁−72ℎ₁₂ℎ₁₃ℎ₂₁−72ℎ₂₂ℎ₀₃ℎ₂₁,𝑛₂1=−8−24ℎ₂₂ℎ₃₂+60ℎ₁₂ℎ²₂₂+60ℎ²₁₂ℎ₃₂−140ℎ³₁₂ℎ₂₂+63ℎ⁵₁₂−180ℎ²₁₂ℎ₀₃ℎ₃₁+780ℎ³₁₂ℎ₀₃ℎ₂₁+72ℎ₂₂ℎ₀₃ℎ₃₁−504ℎ₁₂ℎ₂₂ℎ₀₃ℎ₂₁−216ℎ²₀₃ℎ₂₁ℎ₃₁+972ℎ₁₂ℎ²₀₃ℎ²₂₁+72ℎ₀₃ℎ₂₁ℎ₃₂+72ℎ₂₂ℎ₁₃ℎ₂₁−180ℎ²₁₂ℎ₁₃ℎ₂₁−216ℎ₀₃ℎ²₂₁ℎ₁₃,𝑛₃=5𝑎₅₀+𝑏₄₁−2𝑏₂₂ℎ₂₁−𝑎₁₁ℎ₄₁+2𝑎₁₁ℎ₂₂ℎ₂₁−3𝑎₁₁ℎ₀₃ℎ²₂₁+2𝑎₁₁ℎ₁₂ℎ₃₁−4𝑎₁₁ℎ²₁₂ℎ₂₁−2𝑏₀₂ℎ₄₁+4𝑏₀₂ℎ₂₂ℎ₂₁−6𝑏₀₂ℎ₀₃ℎ²₂₁+4𝑏₀₂ℎ₁₂ℎ₃₁−8𝑏₀₂ℎ²₁₂ℎ₂₁−2𝑏₁₂ℎ₃₁+4𝑏₁₂ℎ₁₂ℎ₂₁−2𝑎₂₁ℎ₃₁+4𝑎₂₁ℎ₁₂ℎ₂₁−3𝑎₃₁ℎ₂₁+3𝑏₀₃ℎ²₂₁+𝑎₁₂ℎ²₂₁,𝑛₄=4𝑏₁₂ℎ₂₂ℎ₂₁+4𝑏₀₂ℎ₃₂ℎ₂₁+6𝑎₃₁ℎ₁₂ℎ₂₁+2𝑎₁₁ℎ₁₂ℎ₄₁+4𝑏₁₂ℎ₁₂ℎ₃₁−4𝑎₁₁ℎ²₁₂ℎ₃₁+4𝑏₀₂ℎ₁₂ℎ₄₁−12𝑏₀₃ℎ²₂₁ℎ₁₂−4𝑎₁₂ℎ²₂₁ℎ₁₂+2𝑎₁₁ℎ₂₂ℎ₃₁+6𝑏₀₃ℎ₂₁ℎ₃₁−3𝑎₁₁ℎ₁₃ℎ²₂₁+6𝑎₆₀+𝑏₅₁−6𝑏₁₂ℎ₀₃ℎ²₂₁+2𝑎₁₁ℎ₃₂ℎ₂₁+4𝑏₀₂ℎ₂₂ℎ₃₁+4𝑎₂₁ℎ₂₂ℎ₂₁+2𝑎₁₂ℎ₂₁ℎ₃₁+4𝑏₂₂ℎ₁₂ℎ₂₁−8𝑏₁₂ℎ²₁₂ℎ₂₁+16𝑏₀₂ℎ³₁₂ℎ₂₁+8𝑎₁₁ℎ³₁₂ℎ₂₁−6𝑏₀₂ℎ₁₃ℎ²₂₁−8𝑎₂₁ℎ²₁₂ℎ₂₁−8𝑏₀₂ℎ²₁₂ℎ₃₁−4𝑎₄₁ℎ₂₁−2𝑏₃₂ℎ₂₁−6𝑎₂₁ℎ₀₃ℎ²₂₁+4𝑎₂₁ℎ₁₂ℎ₃₁+3𝑏₁₃ℎ²₂₁+2𝑎₂₂ℎ²₂₁−6𝑎₁₁ℎ₀₃ℎ₂₁ℎ₃₁+18𝑎₁₁ℎ₀₃ℎ²₂₁ℎ₁₂−8𝑎₁₁ℎ₁₂ℎ₂₂ℎ₂₁−12𝑏₀₂ℎ₀₃ℎ₂₁ℎ₃₁+36𝑏₀₂ℎ₀₃ℎ²₂₁ℎ₁₂−16𝑏₀₂ℎ₁₂ℎ₂₂ℎ₂₁−2𝑏₂₂ℎ₃₁−𝑎₁₁ℎ₅₁−2𝑏₁₂ℎ₄₁−2𝑎₂₁ℎ₄₁−2𝑏₀₂ℎ₅₁−3𝑎₃₁ℎ₃₁. (24)

For convenience, let 𝐿^∗=𝐿₁∪𝐿₂,𝐿^∗₁=𝐿^∗∩𝑈₁,𝐿^∗₂=𝐿^∗∩𝑈₂,𝐿^∗₃=Cl.𝐿^∗−²_𝑖=1𝐿^∗_𝑖,𝐿¹₀=𝐿^∗₁,𝐿₀∩𝑈₂=𝐿²₀∪𝐿³₀(b),𝐿asshowninFigure2₄₀=Cl.𝐿₀−³_𝑗=1𝐿^𝑗₀. (25)

Theorem 3. Assume that system (1) has a compound loop 𝐿₀ as stated before. Then, the functions 𝑀_𝑖(ℎ,𝛿) given in (3) at ℎ=0 have the following expansions: 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|^7/4−1𝐵11⁰⁰𝑐₈(𝛿)|ℎ|^11/6−𝑐₉(𝛿)ℎ²ln|ℎ|+𝑂ℎ²,𝑀₂(ℎ,𝛿)=𝑐₀(𝛿)+𝑐₁(𝛿)|ℎ|^3/4+𝑐₃(𝛿)|ℎ|+ℎln𝑐₃(𝛿)ℎ+𝑐₆(𝛿)|ℎ|^5/4+𝑐₇(𝛿)|ℎ|^7/4+𝑐₉(𝛿)ℎ²ln|ℎ|+𝑂ℎ² (26)(27) for 0<−ℎ≪1, and 𝑀₃(ℎ,𝛿)=𝑐₀̃(𝛿)−𝐷₁𝑐₁(𝛿)ℎ^3/4+𝐵^∗₀₀𝑐₂(𝛿)ℎ^5/6+𝑐₂̃(𝛿)ℎ+𝐵^∗₁₀𝑐₅(𝛿)ℎ^7/6−𝐷₂𝑐₆(𝛿)ℎ^5/4+𝐷₁𝑐₇(𝛿)ℎ^7/4+1𝐵11^∗00𝑐₈(𝛿)ℎ^11/6+𝑂ℎ² (28) for 0<ℎ≪1, where 𝑐₁(𝛿)=𝑐₁𝑆₂,𝛿,𝑐₂(𝛿)=𝑐₁𝑆₁,𝑐,𝛿₃(𝛿)=𝑐₂𝑆₂,𝛿,𝑐₅(𝛿)=𝑐₃𝑆₁,𝑐,𝛿₆(𝛿)=𝑐₄𝑆₂,𝛿,𝑐₇(𝛿)=𝑐₅𝑆₂,𝑐,𝛿₈(𝛿)=𝑐₄𝑆₁,𝛿,𝑐₉(𝛿)=𝑐₆𝑆₂,𝛿,𝑐₀(𝛿)=𝑀₁(0,𝛿)=_𝐿∗𝑞𝑑𝑥−𝑝𝑑𝑦=²_{𝑖=1𝐿𝑖}𝑞𝑑𝑥−𝑝𝑑𝑦,𝑐₀(𝛿)=𝑀₂(0,𝛿)=_𝐿3𝑐̃𝑞𝑑𝑥−𝑝𝑑𝑦,₀(𝛿)=𝑀₃(0,𝛿)=𝑐₀(𝛿)+𝑐₀(𝛿),𝑐₄(𝛿)=_𝐿∗1𝑝_𝑥+𝑞_𝑦+−𝜎𝑑𝑡_𝐿∗2𝑝_𝑥+𝑞_𝑦−𝜂₀−𝜂₁𝑥−𝑥₂+𝑑𝑡_𝐿∗3𝑝_𝑥+𝑞_𝑦𝑑𝑡+𝑡₂𝑐₁(𝛿)+𝑡₃𝑐₂(𝛿)+𝑡₄𝑐₃(𝛿),𝑐₃(𝛿)=_𝐿3𝑝_𝑥+𝑞_𝑦−𝜂₀−𝜂₁𝑥−𝑥₂𝑑𝑡+𝑡₀𝑐₁(𝛿)+𝑡₁𝑐₃(𝛿),𝑐₂̃(𝛿)=_𝐿∗1𝑝_𝑥+𝑞_𝑦−𝜎𝑑𝑡+_{𝐿20∪𝐿30}𝑝_𝑥+𝑞_𝑦−𝜂₀+𝑑𝑡_𝐿40𝑝_𝑥+𝑞_𝑦𝑑𝑡+𝑡₅𝑐₁(𝛿)+𝑡₆𝑐₂(𝛿), (29)(30)(31) where 𝜎=(𝑝_𝑥+𝑞_𝑦)|_{(𝑥1,𝑦1)},𝜂₀=(𝑝_𝑥+𝑞_𝑦)|_{(𝑥2,𝑦2)},𝜂₁=(𝑝_𝑥𝑥+𝑞_𝑦𝑥)|_{(𝑥2,𝑦2)}. In particular, 𝑐₄(𝛿)=²_{𝑖=1𝐿𝑖}𝑝_𝑥+𝑞_𝑦𝑑𝑡,𝑐₃(𝛿)=_𝐿3𝑝_𝑥+𝑞_𝑦𝑐̃𝑑𝑡,₂(𝛿)=³_{𝑖=1𝐿𝑖}𝑝_𝑥+𝑞_𝑦𝑑𝑡 (32) if 𝑐₁(𝛿)=𝑐₂(𝛿)=𝑐₃(𝛿)=0. Here, 𝑡_𝑖,𝑖=0,1,…,6, are constants and 𝐵₀₀,𝐵^∗₀₀,𝐵₁₀,𝐵^∗₁₀ 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 𝑐₀(𝛿),𝑐₃(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 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|^7/4−1𝐵11⁰⁰𝑐₈(𝛿)|ℎ|^11/6−𝑐₉(𝛿)ℎ²ln|ℎ|+𝑂ℎ²+𝑁(ℎ,𝛿) (33) for 0<−ℎ≪1, and 𝑀₃(ℎ,𝛿)=−𝐷₁𝑐₁(𝛿)ℎ^3/4+𝐵^∗₀₀𝑐₂(𝛿)ℎ^5/6+𝐵^∗₁₀𝑐₅(𝛿)ℎ^7/6−𝐷₂𝑐₆(𝛿)ℎ^5/4+𝐷₁𝑐₇(𝛿)ℎ^7/4+1𝐵11^∗00𝑐₈(𝛿)ℎ^11/6+𝑂ℎ²+𝑁^∗(ℎ,𝛿) (34) for 0<ℎ≪1, where 𝑁(ℎ,𝛿)=𝑁₁₁(ℎ,𝛿)+𝑁₁₂(ℎ,𝛿)+𝐼₁₃(),𝑁ℎ,𝛿_∗(ℎ,𝛿)=𝑁₃₁(ℎ,𝛿)+𝑁₃₂(ℎ,𝛿)+𝑁₃₃(ℎ,𝛿)+𝐼₃₄(ℎ,𝛿). (35) Let 𝑁(ℎ,𝛿)=𝑐₀(𝛿)+𝑐₄(𝛿)ℎ+𝑂ℎ²,𝑁^∗(ℎ,𝛿)=𝑐₀̃(𝛿)+𝑐₂̃(𝛿)ℎ+𝑂ℎ². (36) It follows further that 𝑐₀(𝛿)=𝑁₁₁(0,𝛿)+𝑁₁₂(0,𝛿)+𝐼₁₃(0,𝛿)=lim_𝜖0→0𝑁₁₁(0,𝛿)+𝑁₁₂(0,𝛿)+𝐼₁₃(0,𝛿)=lim_𝜖0→0𝐼₁₃(0,𝛿)=_{𝐿1∪𝐿2}=𝑞𝑑𝑥−𝑝𝑑𝑦_{2𝑖=1𝐿𝑖}𝑞𝑑𝑥−𝑝𝑑𝑦=𝑀₁(),𝑐̃0,𝛿₀(𝛿)=𝑁₃₁(0,𝛿)+𝑁₃₂(0,𝛿)+𝑁₃₃(0,𝛿)+𝐼₃₄(0,𝛿)=lim_𝜖0→0𝑁₃₁(0,𝛿)+𝑁₃₂(0,𝛿)+𝑁₃₃(0,𝛿)+𝐼₃₄(0,𝛿)=lim_𝜖0→0𝐼₃₄(0,𝛿)=_𝐿0=𝑞𝑑𝑥−𝑝𝑑𝑦_{3𝑖=1𝐿𝑖}𝑞𝑑𝑥−𝑝𝑑𝑦=𝑀₃(),𝑐0,𝛿₄(𝛿)+𝑂(ℎ)=𝑁_ℎ(ℎ,𝛿)=𝑀_1ℎ()+3ℎ,𝛿4𝑐₁(𝛿)|ℎ|^−1/4+56𝐵₀₀𝑐₂(𝛿)|ℎ|^−1/6+𝑐₃(𝛿)(ln|ℎ|+1)+𝑂|ℎ|^1/6,𝑐₂̃(𝛿)+𝑂(ℎ)=𝑁^∗_ℎ(ℎ,𝛿)=𝑀_3ℎ()+3ℎ,𝛿4𝐷₁𝑐₁(𝛿)ℎ^−1/4−56𝐵^∗₀₀𝑐₂(𝛿)ℎ^−1/6+𝑂ℎ^1/6. (37) Then by Lemma 3.1.2 in [5], we have 𝑐₄(𝛿)=𝑁_ℎ(0,𝛿)=lim_{ℎ→0−𝐿ℎ1}𝑝_𝑥+𝑞_𝑦3𝑑𝑡+4𝑐₁(𝛿)|ℎ|^−1/4+56𝐵₀₀𝑐₂(𝛿)|ℎ|^−1/6+𝑐₃(𝛿|ℎ|,𝑐̃)(ln+1)₂(𝛿)=𝑁^∗_ℎ(0,𝛿)=lim_{ℎ→0+𝐿ℎ3}𝑝_𝑥+𝑞_𝑦+3𝑑𝑡4𝐷₁𝑐₁(𝛿)ℎ^−1/4−56𝐵^∗₀₀𝑐₂(𝛿)ℎ^−1/6. (38) It is easy to see that _𝐿ℎ1𝑝_𝑥+𝑞_𝑦𝑑𝑡=³_{𝑖=1𝐿(𝑖)ℎ1}𝑝_𝑥+𝑞_𝑦=𝑑𝑡_𝐿(1)ℎ1𝑝_𝑥+𝑞_𝑦−𝜎𝑑𝑡+_𝐿(2)ℎ1𝑝_𝑥+𝑞_𝑦−𝜂₀−𝜂₁𝑥−𝑥₂+𝑑𝑡_𝐿(3)ℎ1𝑝_𝑥+𝑞_𝑦𝑑𝑡+𝜎_𝐿(1)ℎ1𝑑𝑡+𝜂_0𝐿(2)ℎ1𝑑𝑡+𝜂_1𝐿(2)ℎ1𝑥−𝑥₂𝑑𝑡,_𝐿ℎ3𝑝_𝑥+𝑞_𝑦=𝑑𝑡_{4𝑖=1𝐿(𝑖)ℎ3}𝑝_𝑥+𝑞_𝑦𝑑𝑡=_𝐿(1)ℎ3𝑝_𝑥+𝑞_𝑦+−𝜎𝑑𝑡_{𝐿(2)ℎ3∪𝐿(3)ℎ3}𝑝_𝑥+𝑞_𝑦−𝜂₀𝑑𝑡+_𝐿(4)ℎ3𝑝_𝑥+𝑞_𝑦𝑑𝑡+𝜎_𝐿(1)ℎ3𝑑𝑡+𝜂_{0𝐿(2)ℎ3∪𝐿(3)ℎ3}𝑑𝑡. (39) Noting ℎ₃<0,ℎ₄<0, by (19), (23), and (29), we have 𝑐₄(𝛿)=lim_{ℎ→0−𝐿(1)ℎ1}𝑝_𝑥+𝑞_𝑦+−𝜎𝑑𝑡_𝐿(2)ℎ1𝑝_𝑥+𝑞_𝑦−𝜂₀−𝜂₁𝑥−𝑥₂+𝑑𝑡_𝐿(3)ℎ1𝑝_𝑥+𝑞_𝑦𝑑𝑡+lim_ℎ→0−𝜎_𝐿(1)ℎ15√𝑑𝑡−23𝐵₀₀ℎ₃^−1/3|ℎ|^−1/6+lim_ℎ→0−𝜂_0𝐿(2)ℎ13√𝑑𝑡+22𝐴₀ℎ₄^−1/4|ℎ|^−1/4+𝑂(1)(ln|ℎ|+1)+lim_ℎ→0−𝜂_1𝐿(2)ℎ1𝑥−𝑥₂−√𝑑𝑡24ℎ₄^−1/2(|ℎ|,𝑐̃ln+1)₂(𝛿)=lim_{ℎ→0+𝐿(1)ℎ3}𝑝_𝑥+𝑞_𝑦−𝜎𝑑𝑡+_{𝐿(2)ℎ3∪𝐿(3)ℎ3}𝑝_𝑥+𝑞_𝑦−𝜂₀+𝑑𝑡_𝐿(4)ℎ3𝑝_𝑥+𝑞_𝑦𝑑𝑡+lim_ℎ→0+𝜎_𝐿(1)ℎ35√𝑑𝑡+23𝐵^∗₀₀ℎ₃^−1/3ℎ^−1/6+lim_ℎ→0+𝜂_{0𝐿(2)ℎ3∪𝐿(3)ℎ3}𝑑𝑡−3√2𝐴₀ℎ₄^−1/4ℎ^−1/4. (40) Then by the proof of (3.13) in [3], the following equations hold: lim_{ℎ→0−𝐿(1)ℎ1}5√𝑑𝑡−23𝐵₀₀ℎ₃^−1/3|ℎ|^−1/6=𝑇₀,lim_{ℎ→0+𝐿(1)ℎ3}5√𝑑𝑡+23𝐵^∗₀₀ℎ₃^−1/3ℎ^−1/6=𝑇^∗₀,lim_{ℎ→0−𝐿(1)ℎ1}𝑝_𝑥+𝑞_𝑦−𝜎𝑑𝑡=lim_{ℎ→0+𝐿(1)ℎ3}𝑝_𝑥+𝑞_𝑦−𝜎𝑑𝑡=_𝐿∗1𝑝_𝑥+𝑞_𝑦−𝜎𝑑𝑡. (41) Here, 𝑇₀ and 𝑇^∗₀ are constants. By a similar argument used in Theorems 2.2 and 2.4 in [4], one can obtain lim_{ℎ→0−𝐿(2)ℎ1}3√𝑑𝑡+22𝐴₀ℎ₄^−1/4|ℎ|^−1/4+𝑂(1)(ln|ℎ|+1)=𝑇₁,lim_{ℎ→0−𝐿(2)ℎ1}𝑥−𝑥₂√𝑑𝑡−24ℎ₄^−1/2(ln|ℎ|+1)=𝑇^∗₁,lim_{ℎ→0+𝐿(𝑖)ℎ3}3√𝑑𝑡−22𝐴₀ℎ₄^−1/4ℎ^−1/4=𝑇_𝑖,𝑖=2,3,lim_{ℎ→0−𝐿(2)ℎ1}𝑝_𝑥+𝑞_𝑦−𝜂₀−𝜂₁𝑥−𝑥₂=𝑑𝑡_𝐿∗2𝑝_𝑥+𝑞_𝑦−𝜂₀−𝜂₁𝑥−𝑥₂𝑑𝑡,lim_{ℎ→0+𝐿(𝑖)ℎ3}𝑝_𝑥+𝑞_𝑦−𝜂₀𝑑𝑡=_𝐿𝑖0𝑝_𝑥+𝑞_𝑦−𝜂₀𝑑𝑡,𝑖=2,3. (42) Here 𝑇^∗₁, 𝑇_𝑖, 𝑖=1,2,3, are constants. Therefore, we can obtain (31) and (32). Thus we have proved Theorem 3.

In the following we use Theorem 3 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 𝑐^∗₄(𝛿)=𝑐₄(𝛿)|_{𝑐1=𝑐2=𝑐3=0},  𝑐^∗₃(𝛿)=𝑐₃(𝛿)|_{𝑐1=𝑐3=0},𝑐^∗₂̃(𝛿)=𝑐^∗₄(𝛿)+𝑐^∗₃(𝛿),  𝑐^∗₃(𝛿)=𝑐₃(𝛿)|_𝑐1=0. Let there exist 𝛿₀∈ℝ^𝑚, such that (𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃)(𝛿₀)=(0,0,0,0,0,0,0).(1)If 𝑐_𝑙(𝛿₀)≠0,𝑐_𝑗(𝛿₀)=0,𝑗=5,…,𝑙−1, and 𝜕𝑐rank⁰,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅,…,𝑐_𝑙−1𝜕𝛿₁,𝛿₂,𝛿₃,𝛿₄,𝛿₅,𝛿₆,𝛿₇,𝛿₈,…,𝛿_𝑚=𝑙+2, (43) then (1) can have 2𝑙+1 limit cycles near 𝐿₀ for some (𝜖,𝛿) near (0,𝛿₀), where 𝑙=6,7 or 9.(2)If 𝑐₅(𝛿₀)𝑐₆(𝛿₀)<0, and 𝜕𝑐rank⁰,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃𝜕𝛿₁,𝛿₂,𝛿₃,𝛿₄,…,𝛿_𝑚=7, (44) then (1) can have 11 limit cycles near 𝐿₀ for some (𝜖,𝛿) near (0,𝛿₀).(3)If 𝑐₈(𝛿₀)𝑐₉(𝛿₀)>0,𝑐_𝑗(𝛿₀)=0,𝑗=5,6,7, and 𝜕𝑐rank⁰,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅,𝑐₆,𝑐₇𝜕𝛿₁,𝛿₂,𝛿₃,𝛿₄,𝛿₅,𝛿₆,𝛿₇,𝛿₈,…,𝛿_𝑚=10, (45) 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 for 𝑙=9 and omit the rest. By our assumptions, there exists 𝛿₀∈ℝ^𝑚 such that 𝑐₀(𝛿₀)=𝑐^∗₃(𝛿₀)=𝑐^∗₃(𝛿₀)=𝑐^∗₄(𝛿₀)=0,𝑐_𝑗(𝛿₀)=0,𝑗=0,1,2,5,6,7,8,𝑐₉(𝛿₀)≠0, and 𝜕𝑐rank⁰,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅,…,𝑐₈𝜕𝛿₁,𝛿₂,𝛿₃,𝛿₄,𝛿₅,𝛿₆,𝛿₇,𝛿₈,…,𝛿_𝑚=11. (46) By the implicit function theorem, we can take 𝑐₀,𝑐^∗₃,𝑐^∗₃,𝑐^∗₄,𝑐_𝑗,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,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅,𝑐₆,𝑐₇)=(0,0,0,0,0,0,0,0,0,0) and vary 𝑐₈ near 0.
First, for 𝑐₈=0, we have by (26) 𝑐₉𝑀₁>0 for 0<−ℎ≪1.
Let 0<|𝑐₈|≪1. Then 𝑐₉𝑀₁<0 for 0<−ℎ≪1 if 𝑐₈𝑐₉>0. Thus, 𝑀₁ has a zero. Hence, for 0<|𝑐₈|≪1,
(1) the condition 𝑐₈𝑐₉>0 implies a distribution (1,0)+0 of one limit cycle.
Step 2. Fix (𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅,𝑐₆)=(0,0,0,0,0,0,0,0,0), 𝑐₈𝑐₉>0 and vary 𝑐₇ near 0.
First, for 𝑐₇=0, we have by (26), (27), and (28) 𝑐₈𝑀₁<0,𝑐₉𝑀₂<0 for 0<−ℎ≪1, and 𝑐₈𝑀₃<0 for 0<ℎ≪1.
Let 0<|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 conditions 𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (2,1)+1 of 4 limit cycles.
Step 3. Fix (𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅)=(0,0,0,0,0,0,0,0), 𝑐₇≠0 and vary 𝑐₆ near 0.
First, for 𝑐₆=0, we have by (26), (27), and (28) 𝑐₇𝑀₁>0,𝑐₇𝑀₂>0 for 0<−ℎ≪1, and 𝑐₇𝑀₃>0 for 0<ℎ≪1.
Let 0<|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 conditions 𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (3,2)+1 of 6 limit cycles.
Step 4. Fix (𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃)=(0,0,0,0,0,0,0), 𝑐₆≠0 and vary 𝑐₅ near 0.
First, for 𝑐₅=0, we have by (26) and (28) 𝑐₆𝑀₁>0 for 0<−ℎ≪1, and 𝑐₆𝑀₃<0 for 0<ℎ≪1.
Let 0<|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 conditions 𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (4,2)+2 of 8 limit cycles.
Step 5. Fix (𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃)=(0,0,0,0,0), 𝑐₅𝑐₆<0 and vary (𝑐^∗₃,𝑐^∗₄) near (0,0) with 𝑐^∗₂̃(=𝑐^∗₃+𝑐^∗₄)≠0.
First, for (𝑐^∗₃,𝑐^∗₄)=(0,0), we have by (26), (27), and (28) 𝑐₅𝑀₁>0,𝑐₆𝑀₂>0 for 0<−ℎ≪1, and 𝑐₅𝑀₃<0 for 0<ℎ≪1.
Let 0<|𝑐^∗₃,𝑐^∗₄|≪|𝑐₅|. Then 𝑐₅𝑀₁<0,𝑐₆𝑀₂𝑐<0for0<−ℎ≪1,₅𝑀₃>0for0<ℎ≪1 (47) if 𝑐^∗₄𝑐₅>0,𝑐^∗₃𝑐₆>0,𝑐^∗₂̃𝑐₅>0, and 𝑐₅𝑀₁<0,𝑐₆𝑀₂𝑐<0for0<−ℎ≪1,₅𝑀₃<0for0<ℎ≪1 (48) if 𝑐^∗₄𝑐₅>0,𝑐^∗₃𝑐₆>0,𝑐^∗₂̃𝑐₅<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,𝑐^∗₄|≪|𝑐₅|≪|𝑐₆|≪|𝑐₇|≪|𝑐₈|≪1, (5i)the conditions 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃<0,𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (5,3)+3 of 11 limit cycles, and(5ii)the conditions 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0,𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (5,3)+2 of 10 limit cycles.
Step 6. Fix (𝑐₀,𝑐₀,𝑐₁,𝑐₂)=(0,0,0,0), 𝑐^∗₄𝑐^∗₃<0 with 𝑐^∗₂̃≠0 and vary 𝑐^∗₃ near 0.
First, for 𝑐^∗₃=0, we have by (26) and (27) 𝑐^∗₄𝑀₁<0 and 𝑐^∗₃𝑀₂<0 for 0<−ℎ≪1.
Let 0<|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|. Then 𝑐^∗₄𝑀₁>0 and 𝑐^∗₃𝑀₂>0 for 0<−ℎ≪1, if 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₃>0. Thus, 𝑀₁ and 𝑀₂ each gets a new zero and the zeros got in above steps still exist. Hence, for 0<|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|≪|𝑐₅|≪|𝑐₆|≪|𝑐₇|≪|𝑐₈|≪1, (6i)the conditions 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃<0,𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (6,4)+3 of 13 limit cycles,(6ii)the conditions 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0,𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (6,4)+2 of 12 limit cycles.
Step 7. Fix (𝑐₀,𝑐₀,𝑐₁)=(0,0,0), 𝑐^∗₃𝑐^∗₂̃≠0 and vary 𝑐₂ near 0.
First, for 𝑐₂=0, we have by (26) and (28) 𝑐^∗₃𝑀₁<0 for 0<−ℎ≪1 and 𝑐^∗₂̃𝑀₃>0 for 0<ℎ≪1.
Let 0<|𝑐₂|≪|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|. Then 𝑐^∗₃𝑀₁𝑐̃>0for0<−ℎ≪1,_∗2𝑀₃<0for0<ℎ≪1 (49) if 𝑐₂𝑐^∗₃>0,𝑐^∗₂̃𝑐₂>0, and 𝑐^∗₃𝑀₁𝑐̃>0for0<−ℎ≪1,_∗2𝑀₃>0for0<ℎ≪1 (50) if 𝑐₂𝑐^∗₃>0,𝑐^∗₂̃𝑐₂<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|≪|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|≪|𝑐₅|≪|𝑐₆|≪|𝑐₇|≪|𝑐₈|≪1, (7i)the conditions 𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0,𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (7,4)+3 of 14 limit cycles,(7ii)the conditions 𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃<0,𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (7,4)+3 of 14 limit cycles.
Step 8. Fix (𝑐₀,𝑐₀)=(0,0), 𝑐₂𝑐^∗₃>0 and vary 𝑐₁ near 0.
First, for 𝑐₁=0, we have by (26), (27), and (28) 𝑐₂𝑀₁>0,𝑐^∗₃𝑀₂>0 for 0<−ℎ≪1, and 𝑐₂𝑀₃<0 for 0<ℎ≪1.
Let 0<|𝑐₁|≪|𝑐₂|. Then 𝑐₂𝑀₁<0,𝑐^∗₃𝑀₂<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|≪|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|≪|𝑐₅|≪|𝑐₆|≪|𝑐₇|≪|𝑐₈|≪1,
(8) the conditions 𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0),𝑐^∗₄𝑐₅>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,𝑐₀) near (0,0).
First, for (𝑐₀,𝑐₀)=(0,0), we have by (26), (27), and (28) 𝑐₁𝑀₁>0,𝑐₁𝑀₂>0, for 0<−ℎ≪1, and 𝑐₁𝑀₃<0 for 0<ℎ≪1.
Let 0<|𝑐₀,𝑐₀|≪|𝑐₁|. Then 𝑐₁𝑀₁<0,𝑐₁𝑀₂𝑐<0,for0<−ℎ≪1,₁𝑀₃<0for0<ℎ≪1 (51) if 𝑐₀𝑐₁<0,𝑐₀𝑐₁<0, 𝑐₁𝑀₁<0,𝑐₁𝑀₂𝑐>0,for0<−ℎ≪1,₁𝑀₃>0for0<ℎ≪1 (52) if 𝑐₀𝑐₁<0,𝑐₀𝑐₁>0 and (𝑐₀+𝑐₀)𝑐₁>0, and 𝑐₁𝑀₁>0,𝑐₁𝑀₂𝑐<0,for0<−ℎ≪1,₁𝑀₃>0for0<ℎ≪1 (53) if 𝑐₀𝑐₁>0,𝑐₀𝑐₁<0 and (𝑐₀+𝑐₀)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, for 0<|𝑐₀,𝑐₀|≪|𝑐₁|≪|𝑐₂|≪|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|≪|𝑐₅|≪|𝑐₆|≪|𝑐₇|≪|𝑐₈|≪1, (9i)the conditions 𝑐₀𝑐₀>0,𝑐₀𝑐₁<0,𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0),𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0, 𝑐₈𝑐₉>0 imply a distribution (9,6)+4 of 19 limit cycles,(9ii)the conditions 𝑐₀𝑐₀<0, 𝑐₀𝑐₁<0,(𝑐₀+𝑐₀)𝑐₁>0, 𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0, 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0),𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0, 𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (9,5)+5 of 19 limit cycles,(9iii)the conditions 𝑐₀𝑐₀<0,𝑐₀𝑐₁>0,(𝑐₀+𝑐₀)𝑐₁>0,𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0), 𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0,𝑐₈𝑐₉>0 imply a distribution (8,6)+5 of 19 limit cycles.
Thus we get the conclusion for 𝑙=9.
(2) By our assumptions in case (2) and the implicit function theorem we can take 𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,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,𝑐₀|≪|𝑐₁|≪|𝑐₂|≪|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|≪1, (i)the conditions 𝑐₀𝑐₀>0,𝑐₀𝑐₁<0,𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0, 𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0), 𝑐^∗₄𝑐₅>0 imply a distribution (5,4)+2 of 11 limit cycles,(ii)the conditions 𝑐₀𝑐₀<0,𝑐₀𝑐₁<0, (𝑐₀+𝑐₀)𝑐₁>0, 𝑐₁𝑐₂<0, 𝑐₂𝑐^∗₃>0, 𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0, 𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0),𝑐^∗₄𝑐₅>0 imply a distribution (5,3)+3 of 11 limit cycles,(iii)the conditions 𝑐₀𝑐₀<0, 𝑐₀𝑐₁>0,(𝑐₀+𝑐₀)𝑐₁>0,𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0), 𝑐^∗₄𝑐₅>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 take 𝑐₀,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,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,𝑐₀|≪|𝑐₁|≪|𝑐₂|≪|𝑐^∗₃|≪|𝑐^∗₄,𝑐^∗₃|≪|𝑐₅|≪|𝑐₆|≪|𝑐₇|≪1, (i)the conditions 𝑐₀𝑐₀>0,𝑐₀𝑐₁<0,𝑐₁𝑐₂<0,𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0),𝑐^∗₄𝑐₅>0,𝑐₅𝑐₆<0,𝑐₆𝑐₇<0, 𝑐₇𝑐₈>0 imply a distribution (8,6)+4 of 18 limit cycles,(ii)the conditions 𝑐₀𝑐₀<0,𝑐₀𝑐₁<0,(𝑐₀+𝑐₀)𝑐₁>0, 𝑐₁𝑐₂<0, 𝑐₂𝑐^∗₃>0, 𝑐^∗₃𝑐^∗₄<0, 𝑐^∗₃𝑐^∗₄<0, 𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0), 𝑐^∗₄𝑐₅>0, 𝑐₅𝑐₆<0,𝑐₆𝑐₇<0,𝑐₇𝑐₈>0 imply a distribution (8,5)+5 of 18 limit cycles,(iii)the conditions 𝑐₀𝑐₀<0, 𝑐₀𝑐₁>0,(𝑐₀+𝑐₀)𝑐₁>0,𝑐₁𝑐₂<0, 𝑐₂𝑐^∗₃>0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₄<0,𝑐^∗₃𝑐^∗₂̃>0(or𝑐^∗₃𝑐^∗₂̃<0), 𝑐^∗₄𝑐₅>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 𝑥̇=𝑦,𝑦̇=−(𝑥+1)²𝑥³𝑥²−141𝑥−2−𝜖𝑓(𝑥,𝛿)𝑦, (54) where 𝑓(𝑥,𝛿)=^𝑛_𝑗=0𝑎_𝑗𝑥^𝑗,𝛿=𝑎₀,𝑎₁,…,𝑎_𝑛,8≤𝑛≤12. (55) System (54) |_𝜖=0 is Hamiltonian with 𝐻=1𝑥,𝑦2𝑦²+18(𝑥+1)³𝑥⁴(𝑥−1). (56) We have the following theorem.

Theorem 5. Let 𝐶(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 loop 𝐿₀=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 for 𝑛=11 and omit the rest of the proof.
Let 𝑛=11. By Theorem 3, we obtain 𝑐₀(𝛿)=𝑀₁(0,𝛿)=−_𝐿∗𝑓(𝑥,𝛿)𝑦𝑑𝑥=−¹¹_𝑗=0𝑎_𝑗𝐼^𝑗₁,𝑐₀(𝛿)=𝑀₂(0,𝛿)=−_𝐿3𝑓(𝑥,𝛿)𝑦𝑑𝑥=−¹¹_𝑗=0𝑎_𝑗𝐼^𝑗₂, (57) where 𝐼^𝑗₁=_𝐿∗𝑥^𝑗𝑦𝑑𝑥=2⁰₋₁𝑥^𝑗=𝑦𝑑𝑥₀₋₁𝑥^𝑗+2(𝑥+1)√1−𝑥²𝐼𝑑𝑥,𝑗=0,1,…,11,_𝑗2=_𝐿3𝑥^𝑗𝑦𝑑𝑥=2¹₀𝑥^𝑗𝑦𝑑𝑥=¹₀𝑥^𝑗+2(𝑥+1)√1−𝑥²𝑑𝑥,𝑗=0,1,…,11. (58) Therefore, 𝑐₀(𝛿)=−𝑎₀𝜋−21615−𝑎₁𝜋−23215−𝑎₂𝜋−832105−𝑎₃5𝜋−8256105−𝑎₄5𝜋−25616315−𝑎₅7𝜋−51216315−𝑎₆7𝜋−5121283465−𝑎₇21𝜋−20481283465−𝑎₈21𝜋−20482569009−𝑎₉33𝜋−40962569009−𝑎₁₀33𝜋−4096102445045−𝑎₁₁429𝜋−655361024,45045𝑐₀(𝛿)=−𝑎₀𝜋+21615−𝑎₁𝜋+23215−𝑎₂𝜋+832105−𝑎₃5𝜋+8256105−𝑎₄5𝜋+25616315−𝑎₅7𝜋+51216315−𝑎₆7𝜋+5121283465−𝑎₇21𝜋+20481283465−𝑎₈21𝜋+20482569009−𝑎₉33𝜋+40962569009−𝑎₁₀33𝜋+4096102445045−𝑎₁₁429𝜋+655361024.45045 (59) Note that 𝑆₂ is a nilpotent saddle of order one and ℎ₄=−1/8. By (23), we have 𝑐₁(𝛿)=𝑐₁𝑆₂,𝛿=2^9/4𝐴₀𝑎₀,𝑐₃(𝛿)=𝑐₂𝑆₂,𝛿=𝑎₁+𝑂₁𝑎₀,𝑐₆(𝛿)=𝑐₄𝑆₂,𝛿=2^3/4𝐴₂−21𝑎₀+12𝑎₁−8𝑎₂,𝑐₇(𝛿)=𝑐₅𝑆₂,𝛿=2^1/4𝐴₀720354𝑎₀−310𝑎₁+180𝑎₂−80𝑎₃+32𝑎₄,𝑐₉(𝛿)=𝑐₆𝑆₂=,𝛿3334𝑎₀−2074𝑎₁+29𝑎₂−15𝑎₃+6𝑎₄−2𝑎₅. (60)
Making the transformation 𝑥=𝑢−1,𝑦=𝑣, system (54) becomes 𝑢̇=𝑣,𝑣̇=−𝑢²(𝑢−1)³𝑢²−943𝑢+4−𝜖𝑓(𝑢,𝛿)𝑣. (61) Then we have 𝐻()=𝑣𝑢,𝑣²2+18𝑢⁸−6𝑢⁷+14𝑢⁶−16𝑢⁵+9𝑢⁴−2𝑢³,ℎ₃1=−4,𝑓(𝑢,𝛿)=¹¹_𝑗=0𝑎_𝑗(𝑢−1)^𝑗=¹¹_𝑗=0(−1)^𝑗𝑎_𝑗+¹¹_𝑗=1(−1)^(𝑗−1)𝑗𝑎_𝑗𝑢+¹¹_𝑗=2(−1)^(𝑗−2)𝐶²_𝑗𝑎_𝑗𝑢²+¹¹_𝑗=3(−1)^(𝑗−3)𝐶³_𝑗𝑎_𝑗𝑢³+¹¹_𝑗=4(−1)^(𝑗−4)𝐶⁴_𝑗𝑎_𝑗𝑢⁴+¹¹_𝑗=5(−1)^(𝑗−5)𝐶⁵_𝑗𝑎_𝑗𝑢⁵+¹¹_𝑗=6(−1)^(𝑗−6)𝐶⁶_𝑗𝑎_𝑗𝑢⁶+¹¹_𝑗=7(−1)^(𝑗−7)𝐶⁷_𝑗𝑎_𝑗𝑢⁷+¹¹_𝑗=8(−1)^(𝑗−8)𝐶⁸_𝑗𝑎_𝑗𝑢⁸+¹¹_𝑗=9(−1)^(𝑗−9)𝐶⁹_𝑗𝑎_𝑗𝑢⁹+𝑎₁₀−11𝑎₁₁𝑢¹⁰+𝑎₁₁𝑢¹¹. (62) By (19), we have 𝑐₂(𝛿)=𝑐₁𝑆₁,𝛿=2^13/611_𝑗=0(−1)^𝑗𝑎_𝑗,𝑐₅(𝛿)=𝑐₃𝑆₁,𝛿=2^17/611_𝑗=1(−1)^𝑗𝑗−3𝑎_𝑗−3𝑎₀,𝑐₈(𝛿)=𝑐₄𝑆₁,𝛿=2^13/6−13163𝑎₀+272𝑎₁−4603𝑎₂+2243𝑎₃−28𝑎₄+163𝑎₅+43𝑎₆−43𝑎₈−163𝑎₉+28𝑎₁₀−2243𝑎₁₁. (63) Note that 𝑐₁(𝛿)=𝑐₁𝑆₂,𝛿=2^9/4𝐴₀𝑎₀,𝑐₂(𝛿)=𝑐₁𝑆₁,𝛿=2^13/611_𝑗=0(−1)^𝑗𝑎_𝑗,𝑐₃(𝛿)=𝑐₂𝑆₂,𝛿=𝑎₁+𝑂₁𝑎₀. (64) We have 𝑐₁(𝛿)=𝑐₂(𝛿)=𝑐₃(𝛿)=0 if and only if 𝑎₀=𝑎₁=0, and 𝑎₁₁=∑¹⁰_𝑖=2(−1)^𝑖𝑎_𝑖. It implies further that 𝑐₄(𝛿)=_𝐿∗𝑝_𝑥+𝑞_𝑦𝑑𝑡=−_𝐿∗𝑓(𝑥,𝛿)𝑑𝑡=−_𝐿∗𝑓(𝑥,𝛿)𝑦𝑑𝑥=−2⁰₋₁𝑓(𝑥,𝛿)𝑦𝑑𝑥=−2⁰₋₁1𝑦¹¹_𝑖=0𝑎_𝑖𝑥^𝑖𝑑𝑥=−4⁰₋₁1𝑥²(𝑥+1)√1−𝑥²×¹⁰_𝑖=2𝑎_𝑖𝑥^𝑖+¹⁰_𝑖=2(−1)^𝑖𝑎_𝑖𝑥¹¹𝑑𝑥=−4¹⁰_𝑖=2𝑎_𝑖⁰₋₁𝑓_𝑖(𝑥)𝑑𝑥, (65) where 𝑓_𝑖(𝑥)=𝑥^𝑖−21+(−1)^𝑖𝑥^(11−𝑖)(𝑥+1)√1−𝑥²,𝑖=2,3,…,10. (66) Similarly, 𝑐₃(𝛿)=_𝐿3𝑝_𝑥+𝑞_𝑦𝑑𝑡=−_𝐿3𝑓(𝑥,𝛿)𝑑𝑡=−_𝐿3𝑓(𝑥,𝛿)𝑦𝑑𝑥=−2¹₀𝑓(𝑥,𝛿)𝑦𝑑𝑥=−4¹⁰_𝑖=2𝑎_𝑖¹₀𝑓_𝑖(𝑥)𝑑𝑥. (67) Therefore, 𝑐₄(𝛿)=−4𝑎₂315𝜋+2569335+4𝑎₃187𝜋+2569335−4𝑎₄187𝜋+2565835+4𝑎₅123𝜋+2565835−4𝑎₆123𝜋+256104105+4𝑎₇75𝜋+256104105−4𝑎₈75𝜋+2561635+4𝑎₉35𝜋+2561635−4𝑎₁₀35𝜋,256𝑐₃(𝛿)=−4𝑎₂315𝜋−2569335+4𝑎₃187𝜋−2569335−4𝑎₄187𝜋−2565835+4𝑎₅123𝜋−2565835−4𝑎₆123𝜋−256104105+4𝑎₇75𝜋−256104105−4𝑎₈75𝜋−2561635+4𝑎₉35𝜋−2561635−4𝑎₁₀35𝜋.256 (68)
Let 𝑐^∗₄(𝛿)=𝑐₄(𝛿)|_{𝑐1=𝑐2=𝑐3=0},𝑐^∗₃(𝛿)=𝑐₃(𝛿)_{𝑐1=𝑐3=0}, 𝑐^∗₃(𝛿)=𝑐₃(𝛿)|_{𝑐1(𝛿)=0}. Furthermore, one sees that equations 𝑐₀(𝛿)=𝑐₀(𝛿)=𝑐₁(𝛿)=𝑐₂(𝛿)=𝑐^∗₃(𝛿)=𝑐^∗₄(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)2^31/6𝑎₉,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 𝜕𝑐rank⁰,𝑐₀,𝑐₁,𝑐₂,𝑐^∗₃,𝑐^∗₄,𝑐^∗₃,𝑐₅,𝑐₆,𝑐₇𝜕𝑎₀,𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₅,𝑎₆,𝑎₇,𝑎₈,𝑎₉,𝑎₁₀,𝑎₁₁=10. (69) Hence by Theorem 4(3), we know that there are 18 limit cycles near 𝐿₀ for some 𝛿 near 𝛿₀. This ends the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project was supported by National Natural Science Foundation of China (11271261) and FP7-PEOPLE-2012-IRSES-316338.