Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

Abstract

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

1. Introduction and Main Results

As well known, Liénard systems describe the dynamics of systems of one degree of freedom under existence of a linear restoring fore and a nonlinear dumping. In the first half of the last century models based on the Liénard system were important for the development of radio and vacuum tube technology. Nowadays the system is widely used to describe oscillatory processes arising in various studies of mathematical models of physical, biological, chemical, epidemiological, physiological, economical, and many other phenomena (see Glade et al. [1], Llibre [2] and references therein). Further, quadratic systems and some other systems can be transformed into Liénard systems by suitable changes, see for instance Han et al. [3], Cherkas [4], Gasull [5], Giné, and Llibre [6]. As we have seen, the main concern on Liénard systems is the center and focus problem and the number of limit cycles, see [7–17], and references therein. Here, we briefly list some known results related to our study in this paper. Consider a Liénard system of the form ̇(𝑥),̇(𝑥),𝑥=𝑦−𝐹𝑦=−𝑔 (1.1) where we suppose that the functions 𝐹 and 𝑔 satisfy two assumptions below.(I)There exists a positive number 𝜀₀ such that 𝐹 is continuous for |𝑥|<𝜀₀ with 𝐹(0)=0 and 𝑔 is continuous for 0<|𝑥|<𝜀₀ with 𝑥𝑔(𝑥)>0, and lim_𝑥→0⁺𝑔(𝑥),lim_𝑥→0⁻𝑔(𝑥) existing.(II)For |𝑥|<𝜀₀, |𝐹(𝑥)|<𝑎₁𝐺(𝑥), where 0<𝑎₁<√8 and 𝐺(𝑥)=∫^𝑥₀𝑔(𝑢)𝑑𝑢.  Obviously, if 𝑔(0)=0, system (1.1) has a singular point at the origin. By Han and Zhang [12], the origin is a generalized singular point if 𝑔(0)≠0. For example, the following system: 𝑥̇,𝑦̇=𝑦,−1,𝑥≥0,𝑦,1,𝑥<0 (1.2) has a generalized center at the origin.

Under the condition (I), the equation 𝐺(𝑥)=𝑧 has two solutions 𝑥=𝑥₁(𝑧)>0 and 𝑥=𝑥₂(𝑧)<0 for 0<𝑧≪1. Let 𝐹_𝑖(𝑧)=𝐹(𝑥_𝑖(𝑧)),𝑖=1,2. Then in 1952, Filippov proved the following theorem (see Chapter 5 of Ye [15]).

Theorem 1.1. Let (I) and (II) be satisfied. Further suppose 𝑔 in (1.1) is continuous at 𝑥=0. Then system (1.1) has a stable focus (resp., center, unstable focus) if 𝐹₂(𝑧)−𝐹₁(𝑧)<0resp.,≡0,>0for0<𝑧≪1. (1.3)

In 1985, Han [10] obtained the following.

Theorem 1.2. Consider the system ̇𝑦(𝑥),̇(𝑥),𝑥=ℎ−𝐹𝑦=−𝑔 (1.4) where ℎ,𝐹, and 𝑔 are continuous functions satisfying(i)ℎ(𝑦)=(sgny)|𝑦|^𝑚+𝑂(𝑦^𝑚+1) with 𝑚>0,(ii)𝑥𝑔(𝑥)>0 for 0<|𝑥|≪1 and |𝐹(𝑥)|<𝑐[𝐺(𝑥)]^{𝑚/(𝑚+1)}for|𝑥|≪1, (1.5) where 0<𝑐<(𝑚+1)((𝑚+1)/𝑚)^{𝑚/(𝑚+1)}. Let 𝛼(𝑥)=−𝑥+𝑂(𝑥²) satisfy 𝐺(𝛼(𝑥))=𝐺(𝑥) for |𝑥|≪1. Then system (1.4) has a stable focus (resp., center, unstable focus) if 𝐹(𝛼(𝑥))−𝐹(𝑥)<0resp.,≡0,>0 (1.6) for |𝑥|≪1.

For center conditions, in 1976, Cherkas [7] proved the following.

Theorem 1.3. Let the functions 𝐹 and 𝑔 in system (1.1) be polynomials in 𝑥 with 𝐹(0)=𝑔(0)=0,𝑔′(0)>0. (1.7) Then the origin is a center of system (1.1) if and only if the equations 𝐹(𝑥)=𝐹𝑦,𝐺(𝑥)=𝐺𝑦 (1.8) have a unique solution 𝑦=𝛼(𝑥)<0 for 𝑥>0 sufficiently small.

From Cherkas [7], one can see that the above result is also true for analytic system (1.1). In 1998, Gasull and Torregrosa [9] studied the center problem for analytic systems of form (1.4) using the Cherkas’ method, generalized Theorem 1.3 to (1.4) and presented interesting applications to some polynomial systems. Then in 2006, Cherkas and Romanovski [8] gave a necessary and sufficient condition for a Liénard system with nonlinearities of degree six to have a center at the origin. About Theorem 1.2, we have the following two remarks.

Remark 1.4. By the variable change 𝑢=sgn𝑥2𝐺(𝑥) (1.9) and the scaling of the time 𝑑𝜏=(𝑔(𝑋(𝑢))/𝑢)𝑑𝑡, we can obtain from (1.4) ̇𝑦(𝑋(𝑢,̇𝑢=ℎ−𝐹))𝑦=−𝑢, (1.10) where 𝑋(𝑢) satisfies |𝑢|=2𝐺(𝑋(𝑢)), 𝑢𝑋(𝑢)>0. Thus, Theorem 1.2 is also true if 𝑔(𝑥) is not continuous at 𝑥=0.

Remark 1.5. If 𝐹 and 𝑔 are 𝐶^∞ with 𝑔′(0)>0. Then 𝐹(𝛼(𝑥))−𝐹(𝑥) in (1.6) has the form 𝐹(𝛼(𝑥))−𝐹(𝑥)=_𝑗≥1𝐵_𝑗𝑥^𝑗. (1.11) In this case, Han [11] proved that if 𝐵_𝑗=0 for 𝑗=1,…,2𝑘−1(𝑘≥1), then 𝐵_2𝑘=0, and the origin is a focus of order 𝑘 if 𝐵_2𝑘+1≠0 in addition. Further, if a vector parameter 𝑎 appears in 𝐹, then 𝐵_𝑗=𝐵_𝑗(𝑎) and 𝐵_2𝑘=𝑂𝐵₁,𝐵₃,…,𝐵_2𝑘−1 (1.12) for all 𝑎 and any 𝑘≥1.

In this paper, we consider the following discontinuous Liénard system: ̇𝑦(),̇(),𝑥=ℎ−𝐹𝑥,𝑎𝑦=−𝑔𝑥,𝑏 (1.13) where (𝑎,𝑏)∈𝑅^𝑛1 with 𝑛₁ being an integer, ℎ(𝑦) is a 𝐶^∞ function satisfying ℎ(𝑦)=𝑦+𝑂(𝑦²) and 𝐹()=𝐹𝑥,𝑎⁺()𝐹𝑥,𝑎,𝑥≥0,₋()𝑔(𝑥)=𝑔𝑥,𝑎,𝑥<0,⁺()𝑔𝑥,𝑏,𝑥≥0,₋(𝑥,𝑏),𝑥<0, (1.14) where 𝐹⁺,𝑔⁺ and 𝐹⁻,𝑔⁻ are 𝐶^∞ on [0,𝑥₀], and [−𝑥₀,0], respectively, with 𝑥₀>0. Further, suppose there exist integers 𝑚≥1, 𝑛≥1, 𝑘≥0, 𝑙≥0 such that 𝐹⁺(𝑥,𝑎)=_𝑗≥𝑚𝐹⁺_𝑗(𝑎)𝑥^𝑗,𝑔⁺(𝑥,𝑏)=_𝑗≥𝑘𝑔⁺_𝑗(𝑏)𝑥^𝑗,𝑔⁺_𝑘(𝑏)>0 (1.15) for 0<𝑥≪1 and 𝐹⁻(𝑥,𝑎)=_𝑗≥𝑛𝐹⁻_𝑗(𝑎)𝑥^𝑗,𝑔⁻(𝑥,𝑏)=𝑔⁻_𝑙(𝑏)(−𝑥)^𝑙+_{𝑗≥𝑙+1}𝑔⁻_𝑗(𝑏)𝑥^𝑗,𝑔⁻_𝑙(𝑏)<0 (1.16) for 0<−𝑥≪1.

Let 𝐺(𝑥,𝑏)=^𝑥₀𝑔()𝐺𝑢,𝑏𝑑𝑢=⁺(𝑥,𝑏),0≤𝑥≤𝑥₀,𝐺⁻(𝑥,𝑏),0<−𝑥≤𝑥₀, (1.17) where 𝐺⁺()=𝑔𝑥,𝑏^+𝑘(𝑏)𝑥𝑘+1^𝑘+1+_{𝑗≥𝑘+1}𝑔⁺_𝑗(𝑏)𝑥𝑗+1^𝑗+1,𝐺⁻(𝑥,𝑏)=−𝑔⁻_𝑙(𝑏)()𝑙+1−𝑥^𝑙+1+_{𝑗≥𝑙+1}𝑔⁻_𝑗(𝑏)𝑥𝑗+1^𝑗+1 (1.18) for 0<𝑥≪1 and 0<−𝑥≪1, respectively.

This paper is devoted to studying the local property of system (1.13) at the origin and the number of limit cycles bifurcated from the origin as (𝑎,𝑏) varies. The authors [13] considered system (1.13) for the case 𝑚=𝑛=𝑘=𝑙=1 and studied the center focus problem. It is easy to prove that for system (1.13) under (1.15) and (1.16), the condition (ii) of Theorem 1.2 is equivalent to the following: H₁𝑚>((𝑘+1)/2)or𝑚=((𝑘+1)/2),𝐹⁺_𝑚(𝑎)<𝑐𝑔⁺_𝑘(𝑏)/(𝑘+1)^1/2,H₂𝑛>((𝑙+1)/2)or𝑛=((𝑙+1)/2),𝐹⁻_𝑛(𝑎)<𝑐−𝑔⁻_𝑙(𝑏)/(𝑙+1)^1/2, (1.19) where 0<𝑐<2√2. For convenience, introduce 𝑟=𝑘+1=𝑞𝑙+1𝑝, (1.20) where 𝑝,𝑞 are relatively prime numbers, that is, (𝑝,𝑞)=1. Then it is easy to see that there must exist an integer 𝜂∈[0,𝑝−1] such that (𝑛+𝜂)𝑟 is an integer, and our main results are the following.

Theorem 1.6. Let (1.15), (1.16), (H₁), and (H₂) hold. Then(i)𝐹⁻(𝛼(𝑥),𝑎)−𝐹⁺(𝑥,𝑎) can be expressed as 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=_𝑗≥1𝐵_𝑗(𝑎,𝑏)𝑥^𝑘𝑗, (1.21) where 𝛼(𝑥,𝑏)=−[((𝑙+1)𝑔⁺_𝑘(𝑏))/(−(𝑘+1)𝑔⁻_𝑙(𝑏))]^1/(𝑙+1)𝑥^𝑟(1+𝑜(1)) satisfies 𝐺⁻(𝛼(𝑥,𝑏),𝑏)=𝐺⁺(𝑥,𝑏) for 𝑥>0 small and 𝑘_𝜏∈{(𝑛+𝑖)𝑟+𝑗|0≤𝑖≤𝑝−1,𝑖≠𝜂,𝑗≥0}∪{𝑗|𝑗≥min{𝑚,(𝑛+𝜂)𝑟}} for 𝜏≥1 with min{𝑛𝑟,𝑚}=𝑘₁<𝑘₂<⋯.(ii) If there exist (𝑎₀,𝑏₀)∈𝑅^𝑛1 and 𝑠(≥1) such that 𝐵_𝑠+1(𝑎₀,𝑏₀)≠0 and 𝐵_𝑗𝑎₀,𝑏₀𝜕𝐵=0,𝑗=1,…,𝑠,rank¹,𝐵₂,…,𝐵_𝑠𝜕()𝑎𝑎,𝑏⁰,𝑏₀=𝑠, (1.22) where 𝑛₁≥𝑠, then system (1.13) has cyclicity 𝑠 near the origin for all (𝑎,𝑏) near (𝑎₀,𝑏₀).

Theorem 1.7. Let (1.15), (1.16), (H₁), and (H₂) hold. If there exists 𝑠≥1 such that 𝐹⁻(𝛼(𝑥,𝑏),𝑎)≡𝐹⁺(𝑥,𝑎)as𝐵₁=𝐵₂=⋯=𝐵_𝑠=0 (1.23) for 𝑥≥0 small and (1.22) is satisfied, then system (1.13) has cyclicity 𝑠−1 at the origin for |𝑎−𝑎₀|+|𝑏−𝑏₀| small.

In many cases, the function 𝐹 in (1.13) is linear in 𝑎. Then the coefficients 𝐵_𝑗(𝑎,𝑏)(𝑗≥1) in (1.21) are linear in 𝑎. Let there exist 𝑎₀∈𝑅^𝑛2,𝑏₀∈𝑅^𝑛3(𝑛₂+𝑛₃=𝑛₁) and integer 𝑠>0(𝑠≤𝑛₂) such that 𝐵_𝑗𝑎₀,𝑏₀𝜕𝐵=0,𝑗=1,2,…,𝑠,det¹,𝐵₃,…,𝐵_𝑠𝜕𝑎₁,𝑎₂,…,𝑎_𝑠𝑎₀,𝑏₀≠0 (1.24) which implies that the linear equations 𝐵_𝑗=0, 𝑗=1,2,…,𝑠 of 𝑎 have a unique solution of the form 𝑎₁,…,𝑎_𝑠=𝜑𝑎_𝑠+1,…,𝑎_𝑛2,𝑏 (1.25) for 𝑏 near 𝑏₀. Obviously, 𝜑 is linear in 𝑎_𝑠+1,…,𝑎_𝑛2. Further, let 𝐵_𝑠+𝑗∣_{(𝑎1,…,𝑎𝑠)=𝜑(𝑎𝑠+1,…,𝑎𝑛2,𝑏)}=𝐿_𝑗𝑎_𝑠+1,…,𝑎_𝑛2Δ_𝑗(𝑏),𝑗=1,…,𝜎 (1.26) for some integer 𝜎>0. Then, we can obtain the following.

Theorem 1.8. Consider system (1.13), where the function 𝐹 is linear in 𝑎∈𝑅^𝑛2. Suppose (1.15), (1.16), (H₁), and (H₂) hold. Let there exist integers 𝑠>0,𝜎>0 and points 𝑎₀=(𝑎₁₀,…,𝑎_𝑛2,0)∈𝑅^𝑛2 and 𝑏₀∈𝑅^𝑛3 such that (1.24) and (1.26) hold with 𝐿_𝑗𝑎_𝑠+1,0,…,𝑎_𝑛2,0≠0,𝑗=1,…,𝜎,Δ_𝑗𝑏₀=0,𝑗=1,…,𝜎−1. (1.27)(i) If Δ_𝜎(𝑏₀)≠0 and 𝜕Δrank¹…,Δ_𝜎−1𝜕𝑏₁,…,𝑏_𝜎−1𝑏₀=𝜎−1, (1.28) then system (1.13) has cyclicity 𝑠+𝜎−1 limit cycles near the origin for all (𝑎,𝑏) near (𝑎₀,𝑏₀).(ii) If Δ_𝜎(𝑏₀)=0 and 𝐹⁻(𝛼(𝑥),𝑎)≡𝐹⁺(𝑥,𝑎) as (𝑎₁,…,𝑎_𝑠)=𝜑(𝑎_𝑠+1,…,𝑎_𝑛2,𝑏), Δ_𝑗(𝑏)=0, 𝑗=1,…,𝜎 with 𝜕Δrank¹,…,Δ_𝜎𝜕𝑏₁,…,𝑏_𝜎𝑏₀=𝜎, (1.29) then system (1.13) has cyclicity 𝑠+𝜎−1 limit cycles at the origin for all (𝑎,𝑏) near (𝑎₀,𝑏₀).

The conclusion (i) of Theorem 1.8 can be proved in a similar manner to the proof of Theorem 2 of [16] and (ii) can be verified by Theorem 1.7. Thus, we do not verify it here. The proof of Theorems 1.6 and 1.7 is presented in Section 2. In Section 3, we give some applications.

2. Proof of Theorems 1.6 and 1.7

In this section, we verify Theorems 1.6 and 1.7. Before proving them, we need to introduce a bifurcation function 𝑑 of system (1.13) and establish some preliminary lemmas, which will be used in our proof. First, we have the following lemma.

Lemma 2.1. Suppose (1.18) holds. Then the function 𝛼(𝑥,𝑏) defined in Theorem 1.6 has the form 𝛼(𝑥,𝑏)=^𝑝_{𝑖=1𝑗≥0}𝛼_𝑖𝑗𝑥^{𝑖𝑟+𝑗} (2.1) for 0<𝑥≪1 with 𝛼₁₀=−[((𝑙+1)𝑔⁺_𝑘(𝑏))/(−(𝑘+1)𝑔⁻_𝑙(𝑏))]^1/(𝑙+1).

Proof. Consider the equation 𝐺⁻(𝑢,𝑏)=𝐺⁺(𝑥,𝑏) (2.2) for 0<−𝑢≪1 and 0<𝑥≪1. By (1.18), we can obtain [𝐺⁻(𝑢,𝑏)]^1/(𝑙+1)=−𝑔⁻_𝑙(𝑏)𝑙+1^1/(𝑙+1)(−𝑢)1+_{𝑗≥𝑙+1}(𝑙+1)𝑔⁻_𝑗(𝑏)(−1)^𝑙+2(𝑗+1)𝑔⁻_𝑙𝑢(𝑏)^{𝑗−𝑙1/(𝑙+1)},𝐺⁺(𝑥,𝑏)^1/(𝑙+1)=𝑔⁺_𝑘(𝑏)𝑘+1^1/(𝑙+1)𝑥^{(𝑘+1)/(𝑙+1)}1+_{𝑗≥𝑘+1}(𝑘+1)𝑔⁺_𝑗(𝑏)𝑗+1𝑔⁺_𝑘(𝑏)𝑥^{𝑗−𝑘1/(𝑙+1)}=𝑔⁺_𝑘(𝑏)𝑘+1^1/(𝑙+1)𝑥^{(𝑘+1)/(𝑙+1)}_𝑗≥0𝑣_𝑗𝑥^𝑗=_𝑗≥0𝑣_𝑗𝑥^𝑗+𝑟≡𝑣(𝑥), (2.3) where 𝑣₀=((𝑔⁺_𝑘(𝑏))/(𝑘+1))^1/(𝑙+1). The implicit function theorem implies that the equation [𝐺⁻(𝑢,𝑏)]^1/(𝑙+1)=𝑣 has a unique solution 𝑢=𝑢^∗(𝑣)=_𝑖≥1𝑢_𝑖𝑣^𝑖 (2.4) for 𝑣>0 small with 𝑢₁=((𝑙+1)/(−𝑔⁻_𝑙(𝑏)))^1/(𝑙+1). Let 𝛼(𝑥,𝑏)=𝑢^∗(𝑣(𝑥)). Then, 𝑢=𝛼(𝑥,𝑏) for 𝑥>0 sufficiently small is the solution of (2.2). Combining (2.3) and (2.4), the above formula can be represented as 𝛼(𝑥,𝑏)=_𝑖≥1𝑢_𝑖𝑗≥0𝑣_𝑗𝑥^{𝑗+𝑟𝑖}=_{𝑖≥1𝑗≥0}𝛼_𝑖𝑗𝑥^{𝑖𝑟+𝑗}=_{𝑖≥1𝑗≥0}𝛼_𝑖𝑗𝑥^{(𝑞𝑖/𝑝)+𝑗}=^𝑝_{𝑖=1𝑗≥0}𝛼_𝑖𝑗𝑥^{𝑖𝑟+𝑗} (2.5) with 𝛼₁₀=𝑣₀𝑢₁. Thus, the proof is ended.

Following the idea of Han [11], we have the following lemma.

Lemma 2.2. Let (1.15) and (1.16) hold. Then system (1.13) is equivalent to 𝑢̇=𝑣−𝐾(𝑣)𝐹^∗(),̇𝑢,𝑎,𝑏𝑣=−𝑢, (2.6) where 𝐾(𝑣)=(1/√2)+𝑂(𝑣)∈𝐶^∞, and 𝐹^∗()=∑𝑢,𝑎,𝑏^𝑗≥𝑚𝑓⁺_𝑗𝑢^{2𝑗/(𝑘+1)}∑,0≤𝑢≪1,_𝑗≥𝑛𝑓⁻_𝑗|𝑢|^{2𝑗/(𝑙+1)},0<−𝑢≪1 (2.7) with 𝑓⁺_𝑚=𝐹⁺_𝑚((𝑘+1)/(2𝑔⁺_𝑘(𝑏)))^{𝑚/(𝑘+1)} and 𝑓⁻_𝑛=(−1)^𝑛𝐹⁻_𝑛((𝑙+1)/(−2𝑔⁻_𝑙(𝑏)))^{𝑛/(𝑙+1)}.

Proof. Let 𝐻(𝑦)=∫^𝑦₀ℎ(𝑢)𝑑𝑢 and make the transformation 𝑢=𝐺(𝑥,𝑏)sgn𝑥,𝑣=𝐻𝑦sgn𝑦, (2.8) together with the scaling of the time 𝑑𝜏=(𝑔(𝑋(𝑢))ℎ(𝑌(𝑣))/2𝑢𝑣)𝑑𝑡 to system (1.13) so that (1.13) can be changed into 𝑢̇=𝑣−𝐾(𝑣)𝐹^∗(),̇𝑢,𝑎,𝑏𝑣=−𝑢, (2.9) where 𝐾(𝑣)=𝑣/ℎ(𝑌(𝑣))=(1/√2)+𝑂(𝑣)∈𝐶^∞, 𝐹^∗(𝑢,𝑎,𝑏)=𝐹(𝑋(𝑢),𝑎) and 𝑋(𝑢),𝑌(𝑣) denote the inverse of the transformation (2.8). In fact, by (1.18) and similar to the proof of Lemma 2.1, we can obtain 𝑋(𝑢)=𝑋⁺(𝑢)𝑋⁻(𝑢)=_𝑗∑_≥1𝑥⁺_𝑗𝑢^{2𝑗/(𝑘+1)},𝑥⁺₁=𝑘+12𝑔⁺_𝑘(𝑏)^1/(𝑘+1)∑,0≤𝑢≪1,_𝑗≥1𝑥⁻_𝑗|𝑢|^{2𝑗/(𝑙+1)},𝑥⁻₁=−𝑙+1−2𝑔⁻_𝑙(𝑏)^1/(𝑙+1),0<−𝑢≪1. (2.10) For 0≤𝑢≪1, by (1.15) and (2.10), we have 𝐹^∗(𝑢,𝑎,𝑏)=𝐹⁺𝑋⁺(𝑢),𝑎=_𝑗≥𝑚𝐹⁺_𝑗(𝑎)_𝑖≥1𝑥⁺_𝑖𝑢^{2𝑖/(𝑘+1)𝑗}=_𝑗≥𝑚𝑓⁺_𝑗𝑢^{2𝑗/(𝑘+1)}, (2.11) where 𝑓⁺_𝑚=𝐹⁺_𝑚((𝑘+1)/2𝑔⁺_𝑘(𝑏))^{𝑚/(𝑘+1)}. Similarly, for 0<−𝑢≪1, by (1.16) and (2.10), we can obtain 𝐹^∗(𝑢,𝑎,𝑏)=𝐹⁻(𝑋⁻(𝑢),𝑎)=_𝑗≥𝑛𝑓⁻_𝑗|𝑢|^{2𝑗/(𝑙+1)}, (2.12) where 𝑓⁻_𝑛=(−1)^𝑛𝐹⁻_𝑛((𝑙+1)/−2𝑔⁻_𝑙(𝑏))^{𝑛/(𝑙+1)}. This ends the proof.

For a relation between the function 𝐹(𝑥,𝑎) in (1.13) and the function 𝐹^∗(𝑢,𝑎,𝑏) in (2.6) we have

Lemma 2.3. Suppose (1.15) and (1.16) hold. Then we have(i)𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎) has the form (1.21) for 0≤𝑥≪1;(ii)let 𝐹₀(𝑢,𝑎,𝑏)=𝐹^∗(−𝑢,𝑎,𝑏)−𝐹^∗(𝑢,𝑎,𝑏). Then 𝐹₀ can be expressed as 𝐹₀(𝑢,𝑎,𝑏)=_𝑗≥1𝐴_𝑗(𝑎,𝑏)𝑢^{2𝑘𝑗/(𝑘+1)},for0≤𝑢≪1, (2.13) where 𝐴₁(𝑎,𝑏)=𝐵₁(𝑎,𝑏)𝑁₁𝑥⁺₁,𝐴_𝑗(𝑎,𝑏)=𝐵_𝑗(𝑎,𝑏)𝑁_𝑗𝑥⁺₁+𝑂𝐵₁,𝐵₂,…,𝐵_𝑗−1,𝑗≥2 (2.14) with 𝑁_𝑗(𝑗≥1) are positive 𝐶^∞ functions in 𝑥⁺₁, and 𝑘_𝑗,𝑗≥1, are as appeared in (1.21).

Proof. By (1.15), (1.16), and Lemma 2.1, we can have 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=_𝑠≥𝑛𝐹⁻_𝑠(𝑎)^𝑝_𝑖=1𝑥^𝑖𝑟_𝑗≥0𝛼_𝑖𝑗𝑥^𝑗𝑠−_𝑠≥𝑚𝐹⁺_𝑠(𝑎)𝑥^𝑠=_𝑠≥𝑛𝐹⁻_𝑠(𝑎)𝑥^{(𝑞/𝑝)𝑠𝑝−1}_𝑖=0𝑥^{𝑞𝑖/𝑝}_𝑗≥0𝛼_𝑖+1,𝑗𝑥^𝑗𝑠−_𝑠≥𝑚𝐹⁺_𝑠(𝑎)𝑥^𝑠=^𝑝−1_𝑖=0𝑥^{(𝑛+𝑖)𝑟}_𝑗≥0𝐵_𝑖𝑗(𝑎,𝑏)𝑥^𝑗−_𝑠≥𝑚𝐹⁺_𝑠(𝑎)𝑥^𝑠. (2.15) Since 𝑟(𝑛+𝜂) is an integer, the above formula can be written as 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=^𝑝−1_{𝑖=0,𝑖≠𝜂𝑗≥0}𝐵_𝑖𝑗(𝑎,𝑏)𝑥^{(𝑛+𝑖)𝑟+𝑗}+_𝑗≥𝑇𝐵_𝑗(𝑎,𝑏)𝑥^𝑗, (2.16) where 𝑇=min{𝑚,𝑟(𝑛+𝜂)}. Then we express (2.16) with respect to the power of 𝑥 in ascending order, yielding the form of (1.21). In addition, for 𝑘_𝜏 defined in Theorem 1.6 there must exist some integers 𝑖_𝜏 and 𝑗_𝜏 such that 𝑘_𝜏=(𝑛+𝑖_𝜏)𝑟+𝑗_𝜏. Hence, we can obtain 𝐵_𝜏(𝑎,𝑏)=𝐵_{𝑖𝜏𝑗𝜏}if𝑖_𝜏≠𝜂,𝐵_𝜏(𝑎,𝑏)=𝐵_𝑘𝜏if𝑖_𝜏=𝜂. (2.17) This completes the proof of the conclusion (i).
Now we prove the conclusion (ii). For 𝑢≥0 sufficiently small, (2.8) and (2.10) imply that 𝐺⁺𝑋⁺(𝑢),𝑏=𝐺⁻(𝑋⁻(−𝑢),𝑏)=𝑢². (2.18) Then noting that 𝐺⁻(𝛼(𝑥,𝑏),𝑏)=𝐺⁺(𝑥,𝑏), we can get 𝑋⁻(−𝑢)=𝛼𝑋⁺(𝑢). (2.19) By (2.11), (2.12) and the above formula, we have for 𝑢≥0 sufficiently small 𝐹₀(𝑢,𝑎,𝑏)=𝐹^∗(−𝑢,𝑎,𝑏)−𝐹^∗(𝑢,𝑎,𝑏)=𝐹⁻(𝑋⁻(−𝑢),𝑎)−𝐹⁺𝑋⁺(𝑢)=𝐹,𝑎₋(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)∣_𝑥=𝑋⁺_(𝑢). (2.20) Let 𝜑𝜇=𝑋⁺𝜇^(𝑘+1)/2. (2.21) Combining (2.10) and (2.21) gives that 𝜑𝜇=_𝑗≥1𝑥⁺_𝑗𝜇^𝑗=𝑥⁺₁𝜇1+𝑂𝜇. (2.22) Substituting the above formula into (1.21) or (2.16) yields that 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)∣_{𝑥=𝜑(𝜇)}=_𝑗≥1𝐵_𝑗(𝑎,𝑏)𝑥⁺₁^𝑘𝑗𝜇^𝑘𝑗𝜇=1+𝑂_{𝑝−1𝑖=0,𝑖≠𝜂𝑗≥0}𝐵_𝑖𝑗(𝑎,𝑏)_𝑠≥1𝑥⁺_𝑠𝜇^{𝑠(𝑛+𝑖)𝑟+𝑗}+_𝑗≥𝑇𝐵_𝑗(𝑎,𝑏)_𝑠≥1𝑥⁺_𝑠𝜇^𝑠𝑗. (2.23) Note that ^𝑝−1_{𝑖=0,𝑖≠𝜂𝑗≥0}𝐵_𝑖𝑗(𝑎,𝑏)_𝑠≥1𝑥⁺_𝑠𝜇^{𝑠(𝑛+𝑖)𝑟+𝑗}=^𝑝−1_{𝑖=0,𝑖≠𝜂𝑗≥0}𝐵_𝑖𝑗(𝑎,𝑏)𝑥⁺₁^{(𝑛+𝑖)𝑟+𝑗}𝜇^{(𝑛+𝑖)𝑟+𝑗}1+_𝑠≥1𝑥⁺_𝑠+1𝑥⁺₁𝑢^{𝑠(𝑛+𝑖)𝑟+𝑗}=^𝑝−1_{𝑖=0,𝑖≠𝜂𝑗≥0}𝐵_𝑖𝑗(𝑎,𝑏)𝑥⁺₁^{(𝑛+𝑖)𝑟+𝑗}𝜇^{(𝑛+𝑖)𝑟+𝑗}1+_𝑠≥1𝑥_𝑠𝜇^𝑠=^𝑝−1_{𝑖=0,𝑖≠𝜂𝑗≥0}𝐵^∗_𝑖𝑗(𝑎,𝑏)𝜇^{(𝑛+𝑖)𝑟+𝑗}, (2.24) where 𝐵^∗_𝑖0=𝑥⁺₁^{(𝑛+𝑖)𝑟}𝐵_𝑖0,𝐵^∗_𝑖𝑗=𝑥⁺₁^{(𝑛+𝑖)𝑟+𝑗}𝐵_𝑖𝑗+𝑂𝐵_𝑖0,𝐵_𝑖1,…,𝐵_{𝑖,𝑗−1},𝑗≥1 (2.25) with 𝑖=0,…,𝑝−1 and 𝑖≠𝜂. Similarly, _𝑗≥𝑇𝐵_𝑗(𝑎,𝑏)_𝑠≥1𝑥⁺_𝑠𝜇^𝑠𝑗=_𝑗≥𝑇𝐵^∗_𝑗(𝑎,𝑏)𝜇^𝑗, (2.26) where 𝐵^∗_𝑇=𝑥⁺₁^𝑇𝐵_𝑇,𝐵^∗_𝑗=𝑥⁺₁^𝑗𝐵_𝑗+𝑂𝐵_𝑇,𝐵_𝑇+1,…,𝐵_𝑗−1,𝑗≥𝑇+1. (2.27) Combining (2.23), (2.24), and (2.26), we can obtain 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)∣_{𝑥=𝜑(𝜇)}=^𝑝−1_{𝑖=0,𝑖≠𝜂𝑗≥0}𝐵^∗_𝑖𝑗(𝑎,𝑏)𝜇^{(𝑛+𝑖)𝑟+𝑗}+_𝑗≥𝑇𝐵^∗_𝑗(𝑎,𝑏)𝜇^𝑗. (2.28) Similar to (1.21) and (2.16), the above formula can be rewritten as 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)∣_{𝑥=𝜑(𝜇)}=_𝑗≥1𝐴_𝑗(𝑎,𝑏)𝜇^𝑘𝑗, (2.29) where from (2.17) 𝐴_𝜏(𝑎,𝑏)=𝐵^∗_{𝑖𝜏𝑗𝜏}if𝑖_𝜏≠𝜂,𝐴_𝜏(𝑎,𝑏)=𝐵^∗_𝑘𝜏if𝑖_𝜏=𝜂. (2.30) From (2.25), (2.27), and the above equation, 𝐴_𝑗 in (2.29) can be rewritten as the form of (2.14). Further, combining (2.17), (2.20), and (2.29) yields (2.13). Thus, the proof is ended.

Next, we establish the bifurcation function 𝑑 of system (1.13). For the purpose, let 𝑢=𝑟cos𝜃,𝑣=𝑟sin𝜃. Then (2.6) can be translated into 𝑑𝑟=()𝐹𝑑𝜃cos𝜃𝐾𝑟sin𝜃^∗(𝑟cos𝜃,𝑎,𝑏)1−sin𝜃𝐾(𝑟sin𝜃)𝐹^∗(𝑟cos𝜃,𝑎,𝑏)/𝑟≡𝑅^∗(𝜃,𝑟,𝑎,𝑏). (2.31) In a similar way to the proof of Lemma 2.3 of [13], it is easy to prove that ()𝐹1−sin𝜃𝐾𝑟sin𝜃^∗(𝑟cos𝜃,𝑎,𝑏)𝑟>0 (2.32) for 𝑟>0 small under (H₁) and (H₂), which shows that 𝑅^∗(𝜃,𝑟,𝑎,𝑏) in (2.31) is well defined. Introduce 𝛽=(𝑘+1)(𝑙+1),𝜌=𝑟^1/𝛽. (2.33) Then by (2.31) and (2.33), we can obtain 𝑑𝜌=𝜌𝑑𝜃𝛽^1−𝛽×cos𝜃𝐾𝜌^𝛽sin𝜃𝐹^∗𝜌^𝛽cos𝜃,𝑎,𝑏1−sin𝜃𝐾𝜌^𝛽sin𝜃𝐹^∗𝜌^𝛽cos𝜃,𝑎,𝑏/𝜌^𝛽=𝜌𝛽×cos𝜃𝐾𝜌^𝛽sin𝜃𝐹₁𝜃,𝜌,𝑎,𝑏1−sin𝜃𝐾𝜌^𝛽sin𝜃𝐹₁,𝜃,𝜌,𝑎,𝑏≡𝑅𝜃,𝜌,𝑎,𝑏 (2.34) where 𝐹₁(𝜃,𝜌,𝑎,𝑏)=𝜌^−𝛽𝐹^∗(𝜌^𝛽cos𝜃,𝑎,𝑏). Clearly, (2.34) is a 2𝜋-periodic equation in 𝜃. By (2.7), we have 𝐹₁=∑𝜃,𝜌,𝑎,𝑏^𝑗≥𝑚𝑓⁺_𝑗(cos𝜃)^{2𝑗/(𝑘+1)}𝜌^{(2𝑗/(𝑘+1))𝛽−𝛽}∑,cos𝜃≥0,_𝑗≥𝑛𝑓⁻_𝑗|cos𝜃|^{2𝑗/(𝑙+1)}𝜌^{(2𝑗/(𝑙+1))𝛽−𝛽},cos𝜃<0. (2.35) From (2.33) and (2.35), it is easy to see that 𝐹₁ is 𝐶^∞ in 𝜌 for any given 𝜃. Let 𝜌(𝜃,𝜌₀,𝑎,𝑏) denote the solution of (2.34) satisfying 𝜌(𝜋/2,𝜌₀,𝑎,𝑏)=𝜌₀. For convenience, let 𝜌−𝜋2,𝜌₀,𝑎,𝑏=𝜌₁,𝜌3𝜋2,𝜌₀,𝑎,𝑏=𝜌₂. (2.36) Define a function as follows: 𝑑𝜌₀−𝜋,𝑎,𝑏=𝜌2,𝜌₀,𝑎,𝑏−𝜌3𝜋2,𝜌₀,𝑎,𝑏=𝜌₁−𝜌₂. (2.37) It is obvious that system (1.13), (2.6), (2.31), or (2.34) has a periodic orbit near the origin if and only if 𝑑(𝜌₀,𝑎,𝑏) in (2.37) has a zero in 𝜌₀ for 𝜌₀>0 sufficiently small. See Figure 1.

Figure 1 

Hence, the function 𝑑 in (2.37) is called a displacement function or bifurcation function of system (1.13), (2.6), (2.31), or (2.34). About the bifurcation function 𝑑, we have.

Lemma 2.4. Let (1.15), (1.16), (H₁), and (H₂) hold. Then the bifurcation function 𝑑 in (2.37) has the form 𝑑𝜌₀,𝑎,𝑏=_𝑗≥1𝐶_𝑗(𝑎,𝑏)𝜌^𝑟𝑗₀1+𝑑_𝑗𝜌₀, (2.38) where 𝑟_𝑗=2(𝑙+1)𝑘_𝑗−𝑘𝑙−𝑘−𝑙 are positive integers satisfying 1≤𝑟₁<𝑟₂<⋯, 𝐶₁(𝑎,𝑏)=𝐵₁(𝑎,𝑏)𝑁^∗₁𝑓⁻_𝑛,𝑓⁺_𝑚,𝐶_𝑗(𝑎,𝑏)=𝐵_𝑗(𝑎,𝑏)𝑁^∗_𝑗𝑓⁻_𝑛,𝑓⁺_𝑚+𝑂𝐵₁,𝐵₂,…,𝐵_𝑗−1,𝑗≥2 (2.39) with 𝑁^∗_𝑗∈𝐶^∞, 𝑁_𝑗(0,0)>0 for 𝑗≥1 and 𝑑_𝑗(𝑗≥1) are 𝐶^∞ functions in 𝜌₀ satisfying 𝑑_𝑖(0)=0.

Proof. Denote 𝜌 by 𝜌(𝜃,𝜌₀,𝑎,𝑏)=𝜌(𝜋−𝜃,𝜌₀,𝑎,𝑏). Then 𝜌𝜋2,𝜌₀𝜋,𝑎,𝑏=𝜌2,𝜌₀,𝑎,𝑏,𝜌−𝜋2,𝜌₀,𝑎,𝑏=𝜌3𝜋2,𝜌₀,𝑎,𝑏=𝜌₂ (2.40) and it satisfies 𝑑𝜌.𝑑𝜃=−𝑅𝜋−𝜃,𝜌,𝑎,𝑏 (2.41) Let 𝑅(𝜃,𝜌,𝑎,𝑏)=−𝑅(𝜋−𝜃,𝜌,𝑎,𝑏). Then we can obtain that 𝜌(𝜃,𝜌₀,𝑎,𝑏) is a solution of the equation 𝑑𝜌=𝑑𝜃𝑅𝜃,𝜌,𝑎,𝑏 (2.42) satisfying 𝜌(𝜋/2,𝜌₀,𝑎,𝑏)=𝜌₀. Let 𝜃∈[−𝜋/2,𝜋/2]. Then 𝜋−𝜃∈[𝜋/2,3𝜋/2] and 𝑅𝜃,𝜌,𝑎,𝑏=𝑅₁(𝜃,𝑎,𝑏)𝜌^{((2𝑚/(𝑘+1))−1)𝛽+1}1+𝑂𝜌,𝑅𝜃,𝜌,𝑎,𝑏=𝑅₁(𝜃,𝑎,𝑏)𝜌^{((2𝑛/(𝑙+1))−1)𝛽+1}1+𝑂𝜌, (2.43) where 𝑅₁()=𝐾(0)𝑓𝜃,𝑎,𝑏^+𝑚𝛽(cos𝜃)^{(2𝑚/(𝑘+1))+1},𝑚>𝑘+12,𝐾(0)𝑓⁺_𝑚cos²𝜃𝛽1−sin𝜃cos𝜃𝐾(0)𝑓⁺_𝑚,𝑚=𝑘+12,𝑅₁()=−𝐾(0)𝑓𝜃,𝑎,𝑏^−𝑛𝛽(cos𝜃)^{(2𝑚/(𝑙+1))+1},𝑛>𝑙+12,−𝐾(0)𝑓⁻_𝑛cos²𝜃𝛽1−sin𝜃cos𝜃𝐾(0)𝑓⁻_𝑛,𝑛=𝑙+12. (2.44) Thus, by (2.43), (2.44) we have easily 𝜌𝜃,𝜌₀=𝜌,𝑎,𝑏⁰+𝑂𝜌₀^{((2𝑚/(𝑘+1))−1)𝛽+1},𝑚>𝑘+12,𝜌₀exp∫^𝜃_𝜋/2𝑅₁(𝑢,𝑎,𝑏)𝑑𝑢+𝑂𝜌²₀,𝑚=𝑘+12,𝑅𝜃,𝜌𝜃,𝜌₀,𝑎,𝑏,𝑎,𝑏−𝑅𝜃,𝜌𝜃,𝜌₀,𝑎,𝑏,𝑎,𝑏=𝜌𝜃,𝜌₀,𝑎,𝑏−𝜌𝜃,𝜌₀,𝑎,𝑏Δ𝜃,𝜌₀ (2.45)(2.46) by the mean value theorem, where Δ(𝜃,𝜌₀)=𝑂(𝜌₀^{((2𝑛/(𝑙+1))−1)𝛽}). Further by (2.34) and (2.42), we can obtain 𝑅𝜃,𝜌,𝑎,𝑏−𝑅=𝜌𝜃,𝜌,𝑎,𝑏−cos𝜃𝐾^𝛽sin𝜃𝜌^1−𝛽𝐹^∗−𝜌^𝛽cos𝜃,𝑎,𝑏−𝐹^∗𝜌^𝛽cos𝜃,𝑎,𝑏𝛽1−sin𝜃𝐾𝜌^𝛽sin𝜃𝜌^−𝛽𝐹^∗𝜌^𝛽cos𝜃,𝑎,𝑏1−sin𝜃𝐾𝜌^𝛽sin𝜃𝜌^−𝛽𝐹^∗−𝜌^𝛽cos𝜃,𝑎,𝑏 (2.47) which follows that by Lemma 2.3𝑅𝜃,𝜌,𝑎,𝑏−𝑅𝜃,𝜌,𝑎,𝑏=−cos𝜃𝜌^1−𝛽𝐹₀𝜌^𝛽cos𝜃,𝑎,𝑏𝑆𝜃,𝜌, (2.48) where 𝑆(𝜃,𝜌) is a 𝐶^∞ function in 𝜌 with 𝑆()=𝐾(0)𝜃,0𝛽,𝑚>𝑘+12,𝑛>𝑙+12,𝐾(0)𝛽1−sin𝜃cos𝜃𝐾(0)𝑓⁺_𝑚,𝑚=𝑘+12,𝑛>𝑙+12,𝐾(0)𝛽1−sin𝜃cos𝜃𝐾(0)𝑓⁻_𝑛,𝑚>𝑘+12,𝑛=𝑙+12,𝐾(0)𝛽1−sin𝜃cos𝜃𝐾(0)𝑓⁺_𝑚1−sin𝜃cos𝜃𝐾(0)𝑓⁻_𝑛,𝑚=𝑘+12,𝑛=𝑙+12, (2.49) which implies 𝑆(𝜃,0)>0 by the above discussion. Then it follows from (2.13), (2.45), and (2.48) that 𝑅𝜃,𝜌𝜃,𝜌₀,𝑎,𝑏,𝑎,𝑏−𝑅𝜃,𝜌𝜃,𝜌₀,𝑎,𝑏,𝑎,𝑏=−_𝑗≥1𝐴_𝑗(𝑎,𝑏)(cos𝜃)^{((2𝑘𝑗/(𝑘+1))+1)}𝜌^(2𝑘𝑗₀^{/(𝑘+1))𝛽+1−𝛽}𝑞_𝑗𝜃,𝜌₀, (2.50) where 𝑞_𝑗(𝑗≥1) are 𝐶^∞ functions in 𝜌₀ with 𝑞_𝑗()=𝑆()𝜃,0𝜃,0,𝑚>𝑘+12,𝑆(𝜃,0)exp2𝑘_𝑗𝑘+1𝛽∫^𝜃_𝜋/2𝑅₁(𝑢,𝑎,𝑏)𝑑𝑢,𝑚=𝑘+12, (2.51) which shows that 𝑞_𝑗(𝜃,0)>0 for all 𝑗≥1. Since 𝜌(𝜃,𝜌,𝑎,𝑏) and 𝜌(𝜃,𝜌,𝑎,𝑏) satisfy (2.42) and (2.34), respectively, using (2.46) we have 𝑑𝑑𝜃𝜌−𝜌=𝜌−𝜌Δ𝜃,𝜌₀+𝑅𝜃,𝜌,𝑎,𝑏−𝑅𝜃,𝜌,𝑎,𝑏, (2.52) where 𝜌=𝜌(𝜃,𝜌₀,𝑎,𝑏) and 𝜌=𝜌(𝜃,𝜌₀,𝑎,𝑏). Then noting (2.40), applying the formula of variation of constants to the above equation yields 𝜌𝜃,𝜌₀,𝑎,𝑏−𝜌𝜃,𝜌₀,𝑎,𝑏=^𝜃_𝜋/2𝑅𝛿,𝜌𝛿,𝜌,𝑎,𝑏,𝑎,𝑏−𝑅𝛿,𝜌𝛿,𝜌₀,𝑎,𝑏,𝑎,𝑏×exp^𝜃_𝛿Δ𝜏,𝜌₀𝑑𝜏𝑑𝛿. (2.53) Inserting (2.50) into the above formula and taking 𝜃=−𝜋/2 give by (2.37) that 𝑑𝜌₀,𝑎,𝑏=_𝑗≥1𝐴_𝑗(𝑎,𝑏)𝜌^(2𝑘𝑗₀^{/(𝑘+1))𝛽+1−𝛽𝜋/2}_−𝜋/2(cos𝜃)^{(2𝑘𝑗/(𝑘+1))+1}𝑞_𝑗(𝜃,0)×exp^𝜃_𝛿Δ(𝜏,0)𝑑𝜏1+𝑂𝜌₀𝑑𝜃, (2.54) which implies (2.38) by (2.33). Hence, The proof is completed.

In the following part, we verify Theorems 1.6 and 1.7 using the above lemmas.

Proof of Theorem 1.6. One can see that the conclusion (i) is true by Lemmas 2.1 and 2.3. Now, we prove the conclusion (ii). Obviously, it suffices to prove the following two points.(1)There are at most 𝑠 limit cycles near the origin for all (𝑎,𝑏) near (𝑎₀,𝑏₀).(2)There can appear 𝑠 limit cycles in any given neighborhood of the origin for some (𝑎,𝑏) arbitrarily close to (𝑎₀,𝑏₀).
In fact, noting that 𝐶_𝑠+1𝑎₀,𝑏₀=𝐵_𝑠+1𝑎₀,𝑏₀𝑁^∗_𝑠+1≠0. (2.55) By our assumption, we have from (2.38) and (2.55) 𝑑𝜌₀,𝑎,𝑏=^𝑠+1_𝑗=1𝐶_𝑗(𝑎,𝑏)𝜌^𝑟𝑗₀1+𝑑_𝑗𝜌₀+𝜌^𝑟𝑠+2₀𝑄𝜌₀=,𝑎,𝑏_𝑠+1𝑗=1𝐶_𝑗(𝑎,𝑏)𝜌^𝑟𝑗₀1+𝑑_𝑗𝜌₀≡𝐷_𝑠, (2.56) where 𝑄(𝜌₀,𝑎,𝑏)=∑_{𝑗≥𝑠+2}𝐶_𝑗(𝑎,𝑏)𝜌^{𝑟𝑗−𝑟𝑠+1}₀, 𝑑_𝑗=𝑑_𝑗,𝑗=1,…,𝑠,𝑑_𝑠+1=𝑑_𝑠+1+𝑄𝜌₀,𝑎,𝑏𝐶_𝑠+1()𝜌𝑎,𝑏^{𝑟𝑠+2−𝑟𝑠+10}, (2.57) which implies 𝑑_𝑗 in (2.56) are 𝐶^∞ in 𝜌₀ satisfying 𝑑_𝑗(0)=0 for (𝑎,𝑏) sufficiently close to (𝑎₀,𝑏₀).
We claim that 𝐷_𝑠 in (2.56) has at most 𝑠 zeros in 𝜌₀>0 small for |𝑎−𝑎₀|+|𝑏−𝑏₀|≪1. We can proceed with the proof by induction on 𝑠.
First, when 𝑠=1, from (2.56), we can obtain 𝐷₁=1+𝑑₁𝜌₀𝜌^𝑟1₀𝐶₁(𝑎,𝑏)+𝐶₂(𝑎,𝑏)𝜌^{𝑟2−𝑟1}₀1+𝑑₂𝜌₀=1+𝑑₁𝜌₀𝜌^𝑟1₀𝑑₁𝜌₀,𝑎,𝑏. (2.58) Note that 𝑑𝑑₁𝜌₀,𝑎,𝑏𝑑𝜌₀=𝐶₂(𝑎,𝑏)𝜌^{𝑟2−𝑟1}₀⁻¹𝑟₂−𝑟₁1+𝑑₂𝜌₀+𝜌₀𝑑^′₂𝜌₀, (2.59) which implies 𝑑𝑑₁𝜌₀,𝑎,𝑏𝑑𝜌₀≠0 (2.60) for 0<𝜌₀≪1 and (𝑎,𝑏) near (𝑎₀,𝑏₀) since 𝐶₂(𝑎₀,𝑏₀)=𝐵₂(𝑎₀,𝑏₀)𝑁^∗₂≠0 by (2.55). Hence, by Rolle’s theorem, the conclusion (1) is true for 𝑠=1.
Assume that the conclusion (1) is true for 𝑠=𝑖. That is by (2.56) 𝐷_𝑖=^𝑖+1_𝑗=1𝐶_𝑗(𝑎,𝑏)𝜌^𝑟𝑗₀1+𝑑_𝑗𝜌₀ (2.61) has at most 𝑖 zeros for 𝜌₀>0 and |𝑎−𝑎₀|+|𝑏−𝑏₀| sufficiently small with 𝐶_𝑖+1(𝑎₀,𝑏₀)≠0. Let us prove that the conclusion (1) is also true for 𝑠=𝑖+1. Then from (2.56), we can obtain 𝐷_𝑖+1=^𝑖+2_𝑗=1𝐶_𝑗(𝑎,𝑏)𝜌^𝑟𝑗₀1+𝑑_𝑗𝜌₀=𝜌^𝑟1₀1+𝑑₁𝜌₀^𝑖+2_𝑗=1𝐶_𝑗(𝑎,𝑏)𝜌^{𝑟𝑗−𝑟1}₀𝑑^∗_𝑗≡𝜌^𝑟1₀1+𝑑₁𝜌₀𝐷_𝑖+1, (2.62) where 𝑑^∗₁=1, 𝑑^∗_𝑗=(1+𝑑_𝑗(𝜌₀))/(1+𝑑₁(𝜌₀))=1+𝑂(𝜌₀), 𝑗=2,3,…,𝑖+2. Then, 𝑑𝐷_𝑖+1𝑑𝜌₀=^𝑖+2_𝑗=2𝑟_𝑗−𝑟₁𝐶_𝑗(𝑎,𝑏)𝜌^{𝑟𝑗−𝑟1}₀⁻¹𝑑^∗_𝑗+𝜌₀𝑟_𝑗−𝑟₁𝑑𝑑^∗_𝑗𝑑𝜌₀=^𝑖+1_𝑗=1𝐶_𝑗()𝜌𝑟̃𝑎,𝑏^𝑗₀1+𝑑_𝑗𝜌₀, (2.63) where 𝐶_𝑗(𝑎,𝑏)=(𝑟_𝑗+1−𝑟₁)𝐶_𝑗+1(𝑎,𝑏), 𝑟_𝑗̃=𝑟_𝑗+1−𝑟₁−1, 𝑗=1,…,𝑖+1 satisfying 1≤𝑟₁̃<⋯<𝑟̃_𝑖+1 and 𝑑_𝑗(𝜌₀) is 𝐶^∞ in 𝜌₀ with 𝑑_𝑗(0)=0, which implies that 𝑑𝐷_𝑖+1/𝑑𝜌₀ has at most 𝑖 zeros in 𝜌₀>0 small for (𝑎,𝑏) near (𝑎₀,𝑏₀) by induction assumption since 𝐶_𝑖+1(𝑎₀,𝑏₀)=(𝑟_𝑖+2−𝑟₁)𝐶_𝑖+2(𝑎₀,𝑏₀)≠0. Hence, by Rolle’s theorem, it follows that 𝐷_𝑖+1 in (2.62) has at most 𝑖+1 zeros in 0<𝜌₀≪1 for (𝑎,𝑏) near (𝑎₀,𝑏₀). This ends the proof of conclusion (1).
Now, we verify the conclusion (2). For simplicity, we can assume (𝑎,𝑏)=𝑎=(𝑎₁,𝑎₂,…,𝑎_𝑛1). Further, by (1.22), without loss of generality, suppose 𝜕𝐵det¹,…,𝐵_𝑠𝜕𝑎₁,…,𝑎_𝑠≠0. (2.64) Then we can fix 𝑎_𝑗=𝑎_𝑗0 for 𝑗=𝑠+1,…,𝑛₁ since 𝑛₁≥𝑠. Consider the equations 𝐵_𝑗=𝐵_𝑗(𝑎),𝑗=1,…,𝑠. (2.65) By the implicit function theorem, the equations have a unique set of solutions 𝑎_𝑗=𝑎^∗_𝑗𝐵₁,…,𝐵_𝑠,𝑎_𝑠+1,…,𝑎_𝑛1,𝑗=1,2,…,𝑠 (2.66) for 𝑎 near 𝑎₀, which means 𝐵_𝑗(𝑗=1,…,𝑠) can be taken as free parameters. Hence, by changing the sign of 𝐵_𝑠,𝐵_𝑠−1,…,𝐵₁ in turn such that 𝐵_𝑗−1𝐵_𝑗<0,𝑗=𝑠+1,…,1,0<𝐵₁≪𝐵₂≪⋯=𝐵_𝑠≪1, (2.67) which follows that 𝐶_𝑗 in (2.56) satisfy 𝐶_𝑗−1𝐶_𝑗<0,𝑗=𝑠+1,…,1,0<𝐶₁≪𝐶₂≪⋯≪𝐶_𝑠≪1. (2.68) This ensures that the bifurcation function 𝑑 has 𝑠 zeros in 𝜌₀ for some (𝑎,𝑏) near (𝑎₀,𝑏₀), which implies the conclusion (2). This ends the proof.

Proof of Theorem 1.7. From the proof of Theorem 1.6, it is easy to see 𝐵_𝑗(𝑗=1,…,𝑠) can be taken as free parameters by (1.22), which implies 𝐶_𝑗(𝑗=1,…,𝑠) can also be taken as free parameters by the definition of 𝐶_𝑗 in Lemma 2.4. From (1.23) and Lemma 2.4, we have 𝑑𝜌₀,𝑎,𝑏=0when𝐶_𝑗=0,𝑗=1,…,𝑠 (2.69) for |𝑎−𝑎₀|+|𝑏−𝑏₀| sufficiently small since 𝐵_𝑗=0 if and only if 𝐶_𝑗=0, which means that the coefficients 𝐶_𝑗(𝑗≥𝑠+1) in (2.38) satisfy 𝐶_𝑗(𝑎,𝑏)=0,𝑗=𝑠+1,𝑠+2,…aslongas𝐶₁=⋯=𝐶_𝑠=0. (2.70) Since 𝐶_𝑗 can be taken as free parameters, we can write 𝐶_𝑗(𝑎,𝑏)=^𝑠_𝑖=1𝐶_𝑖𝑄_𝑖𝑗(𝑎,𝑏), (2.71) where 𝑄_𝑖𝑗∈𝐶^∞,𝑗≥𝑠+1. Inserting (2.71) into (2.38) gives that 𝑑𝜌₀,𝑎,𝑏=^𝑠_𝑗=1𝐶_𝑗𝜌^𝑟𝑗₀1+𝑒_𝑗𝜌₀, (2.72) where 𝑒_𝑗𝜌₀=1+𝑑_𝑗𝜌₀+_{𝑖≥𝑠+1}𝑄_𝑗𝑖𝜌^{𝑟𝑖−𝑟𝑗}₀1+𝑑_𝑖𝜌₀for𝑗=1,…,𝑠. (2.73) We can easily prove 𝑑 in (2.72) has at most 𝑠−1 zeros for (𝑎,𝑏) near (𝑎₀,𝑏₀) by induction 𝑠 in a similar way to the above proof. Further, we can choose 𝐶_𝑗 satisfying 𝐶_𝑗−1𝐶_𝑗<0,𝑗=𝑠,…,1,0<𝐶₁≪𝐶₂≪⋯≪𝐶_𝑠≪1, (2.74) which ensures that 𝑑 in (2.72) has 𝑠−1 zeros for (𝑎,𝑏) near (𝑎₀,𝑏₀). The proof is finished.

From the proof of Theorems 1.6 and 1.7, we have immediately the following corollaries.

Corollary 2.5. Let (1.15), (1.16), (H₁), and (H₂) hold. If there exist a point (𝑎₀,𝑏₀) and an integer 𝑠≥1such that 𝐵_𝑗𝑎₀,𝑏₀=0,𝑗=1,2,…,𝑠,𝐵_𝑠+1𝑎₀,𝑏₀<0resp.,>0, (2.75) then the origin is a stable (resp., unstable) focus of system (1.13) for (𝑎,𝑏)=(𝑎₀,𝑏₀) and there has at most 𝑠 limit cycles at the origin for all (𝑎,𝑏) near (𝑎₀,𝑏₀).

Corollary 2.6. Let (1.15), (1.16), (H₁), and (H₂) hold. If there exists 𝑠≥1 such that 𝐵_𝑗=𝑂𝐵₁,𝐵₂,…,𝐵_𝑠 (2.76) for 𝑗≥𝑠+1, then for any given 𝑁>0 there exists a neighborhood 𝑈 of the origin such that for all |𝐵_𝑗|<𝑁, 𝑗=1,…,𝑠 system (1.13) has at most 𝑠−1 limits cycles in 𝑈.

Further, about system (1.13), we have the below two remarks in order.

Remark 2.7. If the function ℎ(𝑦) in (1.13) has the form ℎ(𝑦)=ℎ₁𝑦+𝑂(𝑦²) with ℎ₁>0, we also can obtain the same conclusions since this system is equivalent to the following: ̇ℎ𝑦𝑥=ℎ₁−𝐹(𝑥)ℎ₁,̇𝑔(𝑥)𝑦=−ℎ₁. (2.77)

Remark 2.8. When vector parameter 𝑎 or 𝑏 is constant in (1.13), Theorems 1.6 and 1.7 remain true in this case.

3. Applications

In this section, we present some applications of our main results to systems of the form ̇(),̇(),𝑥=𝑦−𝐹𝑥,𝑎𝑦=−𝑔𝑥,𝑏 (3.1) where 𝐹,𝑔 satisfy (1.15) and (1.16).

Example 3.1. First consider system (3.1) with 𝑔(𝑥,𝑏)=^𝑘_𝑗∑₌₀𝑏_𝑗𝑥^𝑗𝐹()=,𝑥≥0,−1,𝑥<0,𝑥,𝑎^𝑛_𝑗∑₌₁𝑎⁺_𝑗𝑥^𝑗,𝑥≥0,_𝑚𝑗∑₌₁𝑎⁻_𝑗𝑥^𝑗,𝑥<0, (3.2) where 𝑏₀>0 and 𝑎=(𝑎⁺₁,…,𝑎⁺_𝑛,𝑎⁻₁,…,𝑎⁻_𝑚), 𝑏=(𝑏₀,𝑏₁,…,𝑏_𝑘). By Lemma 2.1, it is clear that 𝛼(𝑥,𝑏)=−^𝑘_𝑗=0𝑏_𝑗𝑥𝑗+1^𝑗+1, (3.3) for 𝑥≥0 small. For any given 𝑁>0, denote 𝐿(𝑘,𝑛,𝑚) the cyclicity of system (3.1) near the origin for |𝑎|+|𝑏|≤𝑁. We study the cyclicity of system (3.1) under (3.2) near the origin with the following three cases.

Case 1. 𝑚=1, 𝑘,𝑚,𝑛≥1. In this case, we claim that 𝐿(𝑘,𝑛,1)=max{𝑘+1,𝑛}−1. In fact, by (3.2) and (3.3), we have 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=−^𝑘_𝑗=0𝑎⁻₁𝑏_𝑗𝑥𝑗+1^𝑗+1−^𝑛_𝑗=1𝑎⁺_𝑗𝑥^𝑗=^{max{𝑘+1,𝑛}}_𝑗=1𝐵_𝑗(𝑎,𝑏)𝑥^𝑗 (3.4) for 0≤𝑥≪1. From (3.4) it is easy to see that 𝐹⁻(𝛼(𝑥,𝑏),𝑎)≡𝐹⁺(𝑥,𝑎) for 0≤𝑥≪1 if and only if 𝐵_𝑗=0 for 𝑗=1,…,max{𝑘+1,𝑛}, which shows that 𝐿(𝑘,𝑛,1)≤max{𝑛,𝑘+1}−1 by Corollary 2.6. Thus we only need to prove that 𝐿(𝑘,𝑛,1)≥max{𝑛,𝑘+1}−1. For 𝑛<𝑘+1, by (3.4), we can obtain 𝐵_𝑗()=𝑎𝑎,𝑏⁻¹𝑏_𝑗−1𝑗−𝑎⁺_𝑗,𝑗=1,…,𝑛,𝐵_𝑗()=𝑎𝑎,𝑏⁻¹𝑏_𝑗−1𝑗,𝑗=𝑛+1,…,𝑘+1. (3.5) Take (𝑎,𝑏)=(𝑎₀,𝑏̃₀) such that 𝑎⁺_𝑗0=(𝑎⁻₁₀𝑏_𝑗−1,0)/𝑗, 𝑗=1,…,𝑛, 𝑏_𝑗0=0, 𝑗=𝑛,…,𝑘 and 𝑎⁻₁₀≠0. Then by (3.5), we have 𝐵_𝑗𝑎₀,𝑏₀𝜕𝐵=0,𝑗=1,…,𝑘+1,rank¹,…,𝐵_𝑛,𝐵_𝑛+1,…,𝐵_𝑘+1𝜕𝑎⁺₁,…,𝑎⁺_𝑛,𝑏_𝑛,…,𝑏_𝑘|_(𝑎0,̃𝑏₀₎=𝑘+1 (3.6) which implies 𝑘 limit cycles can appear near the origin for (𝑎,𝑏) near (𝑎₀,𝑏̃₀). For 𝑛≥𝑘+1, we can discuss similarly.

Case 2. 𝑘=2, 𝑚=2, 𝑛≥1. We will see that 𝐿(2,2,1)=3, 𝐿(2,2,2)=𝐿(2,2,3)=4, 𝐿(2,2,4)=𝐿(2,2,5)=5, and 𝐿(2,2,𝑛)=𝑛−1 for 𝑛≥6.
In fact, combining (3.2) and (3.3) gives that 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=𝑎⁻₁−𝑏₀𝑏𝑥−¹2𝑥²−𝑏₂3𝑥³+𝑎⁻₂−𝑏₀𝑏𝑥−¹2𝑥²−𝑏₂3𝑥³²−^𝑛_𝑗=1𝑎⁺_𝑗𝑥^𝑗=^max{6,𝑛}_𝑗=1𝐵_𝑗(𝑎,𝑏)𝑥^𝑗. (3.7) As before, it is easy to get that 𝐿(2,2,𝑛)=𝑛−1 for 𝑛≥6. For 𝑛≤5, we can discuss system (3.1) case by case with respect to 𝑛.
For 𝑛=1, by (3.7) we can obtain 𝐵₁(𝑎,𝑏)=−𝑎⁻₁𝑏₀−𝑎⁺₁,𝐵₂()1𝑎,𝑏=−2𝑎⁻₁𝑏₁+𝑎⁻₂𝑏²₀,𝐵₃()1𝑎,𝑏=−3𝑎⁻₁𝑏₂+𝑎⁻₂𝑏₀𝑏₁,𝐵₄(𝑎,𝑏)=𝑎⁻₂23𝑏₀𝑏₂+14𝑏²₁,𝐵₅()=1𝑎,𝑏3𝑎⁻₂𝑏₁𝑏₂,𝐵₆()=1𝑎,𝑏9𝑎⁻₂𝑏²₂. (3.8) From (3.8), it is easy to see that 𝐵₅=𝑂(|𝐵₂,𝐵₃,𝐵₄|), 𝐵₆=𝑂(|𝐵₂,𝐵₃,𝐵₄|). By Corollary 2.6, one can see that 𝐿(2,2,1)≤3. Further, from (3.8) we can have 𝑎₀=(𝑎⁺₁₀,𝑎⁻₁₀,𝑎⁻₂₀) with 𝑎⁺₁₀=−𝑎⁻₁₀𝑏₀,𝑎⁻₂₀=𝑏₁2𝑏²₀𝑎⁻₁₀,𝑎⁻₁₀≠0 (3.9) such that 𝐵₁𝑎₀,𝑏=𝐵₂𝑎₀,𝑏=0,𝐵₃𝑎₀,𝑏=𝑎⁻₁₀Δ₁(𝑏),𝐵₄𝑎₀,𝑏=𝑎⁻₁₀Δ₂(𝑏),𝐵₅𝑎₀,𝑏=𝑎⁻₁₀Δ₃(𝑏),𝐵₆𝑎₀,𝑏=𝑎⁻₁₀Δ₄(𝑏), (3.10) where Δ₁(𝑏)𝑏=−²3+𝑏²₁2𝑏₀,Δ₂(𝑏)=𝑏₁𝑏²₀13𝑏₀𝑏₂+18𝑏²₁,Δ₃(𝑏)=𝑏²₁𝑏₂6𝑏²₀,Δ₄(𝑏)=𝑏₁𝑏²₂18𝑏²₀. (3.11) Take 𝑏=𝑏̃₀=(𝑏₀₀,𝑏₁₀,(3𝑏²₁₀/2𝑏₀₀)) with 𝑏₀₀≠0, 𝑏₁₀≠0. Then by (3.11), we have Δ₁𝑏₀=0,Δ₂𝑏₀=5𝑏³₁₀8𝑏²₀₀≠0. (3.12) Note that 𝜕𝐵det¹,𝐵₂𝜕𝑎⁺₁,𝑎⁻₂=−100𝑏²₀=−𝑏²₀≠0,𝜕Δ₁𝜕𝑏₂1=−3. (3.13) Then by Theorem 1.8, we know that system (3.1) has cyclicity 3 for (𝑎,𝑏) sufficiently close to (𝑎₀,𝑏̃₀). In other words, 𝐿(2,2,1)=3. Similarly, we have 𝐿(2,2,2)=𝐿(2,2,3)=4, 𝐿(2,2,4)=𝐿(2,2,5)=5.

Case 3. 𝑘=𝑚=𝑛, 𝑛=1,2,3. We claim that 𝐿(𝑛,𝑛,𝑛)=3𝑛−2 for 𝑛=1,2,3. In fact, for 𝑘=𝑚=𝑛, we have 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=^𝑛_𝑗=1𝑎⁻_𝑗−^𝑛_𝑖=0𝑏_𝑖𝑥𝑖+1^𝑖+1𝑗−^𝑛_𝑗=1𝑎⁺_𝑗𝑥^𝑗=^𝑛(𝑛+1)_𝑗=1𝐵_𝑗(𝑎,𝑏)𝑥^𝑗. (3.14) From Cases 1 and 2, it is easy to see that 𝐿(1,1,1)=1, 𝐿(2,2,2)=4. For 𝑘=𝑚=𝑛=3, 𝐵_𝑗(𝑎,𝑏) in (3.14) can be written as 𝐵₁(𝑎,𝑏)=−𝑎⁻₁𝑏₀−𝑎⁺₁,𝐵₂()1𝑎,𝑏=−2𝑎⁻₁𝑏₁+𝑎⁻₂𝑏²₀−𝑎⁺₂,𝐵₃()1𝑎,𝑏=−3𝑎⁻₁𝑏₂+𝑎⁻₂𝑏₀𝑏₁−𝑎⁻₃𝑏³₀−𝑎⁺₃,𝐵₄()1𝑎,𝑏=−4𝑎⁻₁𝑏₃+𝑎⁻₂23𝑏₀𝑏₂+14𝑏²₁−32𝑎⁻₃𝑏²₀𝑏₁,𝐵₅(𝑎,𝑏)=𝑎⁻₂12𝑏₀𝑏₃+13𝑏₁𝑏₂−𝑎⁻₃𝑏²₀𝑏₂+34𝑏₀𝑏²₁,𝐵₆(𝑎,𝑏)=𝑎⁻₂14𝑏₁𝑏₃+19𝑏²₂−𝑎⁻₃𝑏₀𝑏₁𝑏₂+34𝑏²₀𝑏₃+18𝑏³₁,𝐵₇()=1𝑎,𝑏6𝑎⁻₂𝑏₂𝑏₃−𝑎⁻₃34𝑏₀𝑏₁𝑏₃+13𝑏₀𝑏²₂+14𝑏²₁𝑏₂,𝐵₈()=1𝑎,𝑏𝑎16⁻²𝑏²₃−𝑎⁻₃12𝑏₀𝑏₂𝑏₃+3𝑏16²¹𝑏₃+16𝑏₁𝑏²₂,𝐵₉(𝑎,𝑏)=−𝑎⁻₃14𝑏₁𝑏₂𝑏₃+3𝑏16⁰𝑏²₃+1𝑏27³²,𝐵₁₀(𝑎,𝑏)=−𝑎⁻₃3𝑏32¹𝑏²₃+1𝑏12²²𝑏₃,𝐵₁₁()1𝑎,𝑏=−𝑎15⁻³𝑏₂𝑏²₃,𝐵₁₂1=−𝑎64⁻³𝑏³₃. (3.15) From (3.15), it is easy to prove that 𝐵_𝑖=𝑂(|𝐵₄,…,𝐵₈|), 𝑖=9,…,12. Then by Corollary 2.6, we have 𝐿(3,3,3)≤7. In order to prove 𝐿(3,3,3)=7, let for convenience 𝑍(𝑏)=36𝑏²₀𝑏₁𝑏₃−12𝑏₀𝑏²₁𝑏₂−32𝑏²₀𝑏²₂−9𝑏⁴₁. (3.16) Then we can find (𝑎,𝑏)=(𝑎₀,𝑏̃₀) with 𝑍(𝑏̃₀)≠0 and 𝑎⁺₁₀=−𝑎⁻₁₀𝑏₀,𝑎⁺₂₀=12𝑎⁻₁₀𝑏₁−𝑎⁻₂₀𝑏²₀,𝑎⁺₃₀1=−3𝑎⁻₁₀𝑏₂+𝑎⁻₂𝑏₀𝑏₁−𝑎⁻₃₀𝑏³₀,𝑎⁻₁₀≠0,𝑎⁻₂₀=3𝑎⁻₁₀𝑏₃4𝑏₀𝑏₂+3𝑏²₁𝑍(𝑏),𝑎⁻₃₀=2𝑎⁻₁₀𝑏₃3𝑏₀𝑏₃+2𝑏₁𝑏₂𝑏₀𝑍(𝑏) (3.17) for 𝑏=𝑏̃₀ such that 𝐵₁𝑎₀,𝑏̃₀=⋯=𝐵₅𝑎₀,𝑏̃₀=0,𝐵₆𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₁𝑏̃₀,𝐵₇𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₂𝑏̃₀,𝐵₈𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₃𝑏̃₀,𝐵₉𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₄𝑏̃₀,𝐵₁₀𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₅𝑏̃₀,𝐵₁₁𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₆𝑏̃₀,𝐵₁₂𝑎₀,𝑏̃₀=𝑎⁻₁₀Δ₇𝑏̃₀, (3.18) where Δ₁(𝑏)=𝑍₁−92𝑏³₀𝑏²₃−6𝑏²₀𝑏₁𝑏₂𝑏₃+43𝑏²₀𝑏³₂+32𝑏₀𝑏²₁𝑏₃−3𝑏₀𝑏²₁𝑏²₂−12𝑏⁴₁𝑏₂,Δ₂(𝑏)=𝑍₁−92𝑏²₀𝑏₁𝑏²₃−3𝑏₀𝑏²₁𝑏₂𝑏²₃−43𝑏₀𝑏₁𝑏³₂−𝑏³₁𝑏²₂,Δ₃(𝑏)=𝑍₁−94𝑏²₀𝑏₂𝑏²₃−9𝑏16⁰𝑏²₁𝑏²₃−3𝑏₀𝑏₁𝑏²₂𝑏₃−34𝑏²₁𝑏₂𝑏₃−23𝑏²₁𝑏³₂,Δ₄(𝑏)=𝑍₁−98𝑏²₀𝑏²₃−94𝑏₀𝑏₁𝑏₂𝑏²₃−29𝑏₀𝑏³₂𝑏₃−𝑏²₁𝑏²₂𝑏₃−4𝑏27¹𝑏⁴₂,Δ₅(𝑏)=𝑍₁−9𝑏16⁰𝑏₁𝑏²₃−12𝑏₀𝑏²₂𝑏²₃−38𝑏²₁𝑏₂𝑏²₃−13𝑏₁𝑏³₂𝑏₃,Δ₆(𝑏)=𝑍₁−38𝑏₂𝑏³₃−14𝑏₁𝑏²₂𝑏²₃,Δ₇(𝑏)=𝑍₁−3𝑏32⁰𝑏⁴₃−1𝑏16¹𝑏₂𝑏²₃, (3.19) with 𝑍₁=𝑏₃/𝑏₀𝑍. Let’s choose 𝑏̃₀=(𝑏₀₀,0,(3/2)𝑏^1/3₀₀𝑏^2/3₃₀,𝑏₃₀) with 𝑏₀₀>0, 𝑏₃₀≠0. Obviously, 𝑍(𝑏̃₀)≠0. Then by (3.19), we have Δ₁𝑏̃₀=Δ₂𝑏̃₀=0,Δ₃𝑏̃₀=3𝑏^7/3₃₀64𝑏^4/3₀₀≠0. (3.20) Note that 𝜕𝐵det¹,𝐵₂,𝐵₃,𝐵₄,𝐵₅𝜕𝑎⁺₁,𝑎⁺₂,𝑎⁺₃,𝑎⁻₂,𝑎⁻₃∣_(𝑎0,̃𝑏₀₎=98𝑏^8/3₀₀𝑏^4/3₃₀𝜕Δ≠0,det¹,Δ₂𝜕𝑏₁,𝑏₂∣̃𝑏₀=812𝑏^14/3₀₀𝑏^10/3₃₀≠0. (3.21) Then by Theorem 1.8, it is clear that 7 limit cycles can appear near the origin for (𝑎,𝑏) sufficiently close (𝑎₀,𝑏̃₀). Then we have 𝐿(3,3,3)=7.

Example 3.2. Consider system (3.1) with 𝑔()=𝑥𝑥,𝑏²,𝑥≥0,−1+𝑏₁𝑥+𝑏₂𝑥²𝐹()=,𝑥<0,𝑥,𝑎^𝑛_𝑗∑₌₂𝑎⁺_𝑗𝑥^𝑗,𝑥≥0,_𝑚𝑗∑₌₁𝑎⁻_𝑗𝑥^𝑗,𝑥<0. (3.22) Clearly, the origin is a center or focus by Theorem 1.2. Denote 𝐻(𝑚,𝑛) the cyclicity of system (3.1) near the origin. By (3.22), we can obtain 𝐺()=𝑥𝑥,𝑏3³𝑏,𝑥≥0,−𝑥+¹2𝑥²+𝑏₂3𝑥³,𝑥<0. (3.23) Using Lemma 2.1 and (3.23), for 𝑥≥0 small, we can solve from the equation 𝐺⁻(𝛼(𝑥,𝑏),𝑏)=𝐺⁺(𝑥,𝑏)𝛼(𝑥,𝑏)=_𝑗≥1𝛼_𝑗𝑥^3𝑗1=−3𝑥³+1𝑏18¹𝑥⁶−𝑏²₁+𝑏54²𝑥81⁹+5𝑏648³¹+5𝑏486¹𝑏₂𝑥¹²−7𝑏1944⁴¹+7𝑏972²¹𝑏₂+1𝑏729²²𝑥¹⁵+7𝑏3888⁵¹+7𝑏1458³¹𝑏₂+14𝑏6561¹𝑏²₂𝑥¹⁸+𝑂𝑥²¹. (3.24) Take 𝑚=4, 𝑛=5. Combining (3.22) and (3.24), we can obtain 𝐹⁻(𝛼(𝑥,𝑏),𝑎)−𝐹⁺(𝑥,𝑎)=⁵_𝑗=1𝐵_𝑗(𝑎,𝑏)𝑥^𝑗+1+_𝑗≥6𝐵_𝑗(𝑎,𝑏)𝑥^3(𝑗−3), (3.25) where 𝐵₁(𝑎,𝑏)=−𝑎⁺₂,𝐵₂(𝑎,𝑏)=−𝑎⁺₃−13𝑎⁻₁,𝐵₃(𝑎,𝑏)=−𝑎⁺₄,𝐵₄=−𝑎⁺₅,𝐵₅()=1𝑎,𝑏𝑎18⁻¹𝑏₁+19𝑎⁻₂,𝐵₆(𝑎,𝑏)=𝑎⁻₁−1𝑏54²¹−1𝑏81²−1𝑎27⁻²𝑏₁−1𝑎27⁻³,𝐵₇(𝑎,𝑏)=𝑎⁻₁5𝑏648³¹+5𝑏486¹𝑏₂+𝑎⁻₂5𝑏324²¹+2𝑏243²+1𝑎54⁻³𝑏₁+1𝑎81⁻⁴,𝐵₈(𝑎,𝑏)=𝑎⁻₁−1𝑏1944⁴¹−1𝑏972²¹𝑏₂−1𝑏729²²+𝑎⁻₂−7𝑏972³¹−2𝑏243¹𝑏₂−𝑎⁻₃1𝑏108²¹+1𝑏243²−2𝑎243⁻⁴𝑏₁,𝐵₉(𝑎,𝑏)=𝑎⁻₁7𝑏3888⁵¹+7𝑏1458³¹𝑏₂+14𝑏6561¹𝑏²₂+𝑎⁻₂1𝑏648⁴¹+5𝑏2187²¹𝑏₂+7𝑏6561²²+𝑎⁻₃7𝑏1458³¹+7𝑏1458¹𝑏₂+𝑎⁻₄7𝑏1458²¹+4𝑏2187², (3.26) which implies that 𝐵_𝑗=𝑂(|𝐵₁,…,𝐵₉|), 𝑗≥10. Then we can obtain 𝐻(4,5)≤8 by Corollary 2.6. Now we prove that 𝐻(4,5)=8. Suppose 𝑎=(𝑎⁺₂,…,𝑎⁺₅,𝑎⁻₁,…,𝑎⁻₄). Then we can find 𝑎₀=𝑎⁺₂₀,…,𝑎⁺₅₀,𝑎⁻₁₀,…,𝑎⁻₄₀, (3.27) where 𝑎⁺₂₀=𝑎⁺₄₀=𝑎⁺₅₀=𝑎⁻₄₀=0,𝑎⁺₃₀1=−3𝑎⁻₁₀,𝑎⁻₂₀1=−2𝑎⁻₁₀𝑏₁,𝑎⁻₃₀1=−3𝑎⁻₁₀𝑏₂,𝑎⁻₁₀≠0, (3.28) such that 𝐵₁𝑎₀,𝑏=𝐵₂𝑎₀,𝑏=𝐵₃𝑎₀,𝑏=𝐵₄𝑎₀,𝑏=𝐵₅𝑎₀,𝑏=𝐵₆𝑎₀,𝑏=𝐵₇𝑎₀𝐵,𝑏=0,₈𝑎₀=1,𝑏𝑎162⁻¹Δ₀,𝐵₉𝑎₀=𝑎,𝑏⁻¹Δ162¹, (3.29) where Δ₀=𝑏²₁12𝑏²₁+𝑏₂,Δ₁=𝑏³₁5185𝑏31096²¹+13𝑏₂. (3.30) Solving the equation Δ₀=0 gives 𝑏=(𝑏₁₀,−(1/2)𝑏²₁₀)≡𝑏₀, 𝑏₁₀≠0. Inserting 𝑏₀ into (3.30) gives Δ₁𝑏₀=5185𝑏186576⁵¹⁰≠0. (3.31) Further, by (3.26) and (3.30), we have 𝜕𝐵₁,𝐵₂,𝐵₃,𝐵₄,𝐵₅,𝐵₆,𝐵₇𝜕𝑎⁺₂,𝑎⁺₃,𝑎⁺₄,𝑎⁺₅,𝑎⁻₂,𝑎⁻₃,𝑎⁻₄=1−10000000−10000000−10000000−10000000−9𝑏000000−¹−12702700005𝑏²₁+3242𝑏₂𝑏243¹154,81𝜕Δ₀𝜕𝑏₂𝑏₀=𝑏²₁₀≠0. (3.32) Hence, by the conclusion (i) of Theorem 1.8, it is easy to see that there exists cyclicity 8 when (𝑎,𝑏) near (𝑎₀,𝑏₀), which implies 𝐻(4,5)=8.

Remark 3.3. In a similar way, for 1≤𝑚≤4, 2≤𝑛≤5, we can obtain 𝐻(𝑚,𝑛)=𝑚+𝑛−1 for 𝑛≠2, 𝐻(𝑚,𝑛)=𝑚+𝑛−2 for 𝑛=2, 𝑚≠2 and 𝐻(2,2)=3.