Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems

Abstract

We study analytic properties of the Poincaré return map and generalized focal values of analytic planar systems with a nilpotent focus or center. We use the focal values and the map to study the number of limit cycles of this kind of systems and obtain some new results on the lower and upper bounds of the maximal number of limit cycles bifurcating from the nilpotent focus or center. The main results generalize the classical Hopf bifurcation theory and establish the new bifurcation theory for the nilpotent case.

1. Introduction and Main Result

Consider an analytic system of the form: 𝑥̇=𝑦+𝛼𝑥+𝑋𝑥,𝑦,𝛿,𝑦̇=−𝑥+𝛼𝑦+𝑌𝑥,𝑦,𝛿, (1.1) where 𝛼∈𝑅, 𝛿∈𝑅^𝑛, and 𝑋,𝑌=𝑂(|𝑥,𝑦|²) for (𝑥,𝑦) near the origin. A Poincaré map can be defined on a cross-section with an endpoint at the origin using positive orbits of the above system and can be written in the form: 𝑃(𝑟,𝛼,𝛿)=𝑟+_𝑗≥1𝑣_𝑗(𝛼,𝛿)𝑟^𝑗, (1.2) where the series converges for small 𝑟. A well-known fact is that for any 𝑘≥1𝑣_2𝑗−1(𝛼,𝛿)=0 for all 𝑗=1,…,𝑘 imply that 𝑣_2𝑘(𝛼,𝛿)=0; that is, only odd values of the expansion are important for determining the behavior of trajectories near the origin. The value 𝑣_2𝑘+1(𝛼,𝛿) is called the 𝑘th focal value or the 𝑘th Lyapunov constant. For quadratic systems, Bautin [1] proved that the Poincaré map can be written in the form: 𝑃(𝑟,𝛼,𝛿)=𝑟+⁴_𝑗=1𝑣_2𝑗−1(𝛼,𝛿)𝑟^2𝑗−11+𝑃_𝑗(𝑟,𝛼,𝛿), (1.3) where 𝑃_𝑗(𝑟,𝛼,𝛿)=𝑂(𝑟)∈𝐶^𝜔. This implies that there are at most 3 limit cycles near the origin.

Suppose now that the origin is a nilpotent singular point, so the system is written in the form: 𝑥̇=𝑦+𝑋𝑥,𝑦,𝑦̇=𝑌𝑥,𝑦, (1.4) where 𝑋, 𝑌=𝑂(|𝑥,𝑦|²) for (𝑥,𝑦) near the origin. The following criterion for the existence of a center or a focus at the origin of (1.4) has been established in [2–4].

Theorem 1.1 (see [2–4]). Let (1.4) have an isolated singular point at the origin. Let 𝑌(𝑥,𝐹(𝑥))=𝑎𝑥^2𝑛−1+𝑂𝑥^2𝑛,𝑎≠0,𝜕𝑋((𝑥+𝜕𝑥𝑥,𝐹))𝜕𝑌((𝑥𝜕𝑦𝑥,𝐹))=𝑏𝑥^𝑛−1+𝑂𝑥^𝑛, (1.5) where 𝑦=𝐹(𝑥) is the solution to the equation 𝑦+𝑋(𝑥,𝑦)=0 satisfying 𝐹(0)=0. Then the origin of (1.4) is a center or a focus if and only if 𝑎 is negative and 𝑏²+4𝑎𝑛<0.

Introducing the generalized polar coordinates: 𝑥=𝑟Cs(𝜃),𝑦=𝑟^𝑛Sn(𝜃), (1.6) where (Cs(𝑡),Sn(𝑡)) is the solution of the initial problem 𝑥̇=𝑦,𝑦̇=−𝑥^2𝑛−1,𝑥(0),𝑦(0)=(1,0), (1.7)

Liapunov [3, 4] proposed a method to solve the center-focus problem for (1.4). Sadovskiĭ [5] (see also [6]) and Moussu [7] investigated the problem using Lyapunov functions and normal forms, respectively. Other few approaches for computing focal values, Lyapunov constants or equivalent values and methods for studying bifurcations of local limit cycles were suggested by Chavarriga et al. [8], Giacomini et al. [9], Álvarez and Gasull [10, 11], and Liu and Li [12–15]. From [16] we know that (1.4) can be formally transformed into a formal normal form: 𝑥̇=𝑦,𝑦̇=−𝑔(𝑥)−𝑦𝑓(𝑥), (1.8) where 𝑔(𝑥)=𝑎𝑥^𝑚+𝑂(𝑥^𝑚+1), 𝑚≥2 (system (1.8) is a generalized Liénard system). Stróżyna and Żołądek [17] proved that this formal normal form can be achieved through an analytic change of variables. Thus, if (1.4) has a center or focus at the origin, then it can be changed into (1.8) with 𝑔(𝑥)=𝑥^2𝑛−1𝑎_2𝑛−1+𝑂(𝑥),𝑛≥2,𝑎_2𝑛−1>0. (1.9)

According to [11] under (1.9) by a change of variables 𝑥 and 𝑡 of the form: 𝑢=2𝑛^𝑥₀𝑔(𝑥)𝑑𝑥^1/2𝑛sgn𝑥≡𝑢(𝑥),𝑑𝑡𝑑𝑡₁=𝑢^2𝑛−1(𝑥)𝑔(𝑥) (1.10) system (1.8) is transformed into 𝑥̇=𝑦,𝑦̇=−𝑥^2𝑛−1−𝑦𝑓(𝑥), (1.11) where 𝑓(𝑥)=𝑥^2𝑛−1𝑓𝑢⁻¹(𝑥)𝑔𝑢⁻¹(𝑥)𝑢(𝑥)=,𝑛≥2,2𝑛^𝑥₀𝑔(𝑥)𝑑𝑥^1/2𝑛sgn𝑥=𝑎_2𝑛−1^1/2𝑛𝑥+𝑂𝑥². (1.12)

Then, by Theorem 1.1 system (1.11) has a center or a focus at the origin if and only if the function 𝑓 given in (1.12) satisfies 𝑓(𝑥)=_{𝑗≥𝑛−1}𝑏_𝑗𝑥^𝑗,𝑏²_𝑛−1−4𝑛<0. (1.13)

By Filippov's theorem (see, e.g., Ye et al. [18]) under (1.13) system (1.11) has a stable (unstable) focus at the origin if there exists an integer 𝑙 with 2𝑙≥𝑛−1 such that 𝑏_2𝑙>0(<0),𝑏_2𝑗=0for𝑗<𝑙, (1.14) and it has a center at the origin if 𝑏_2𝑗=0 for all 2𝑗≥𝑛−1.

Passing to the generalized polar coordinates (𝑥,𝑦)=(𝑟Cs(𝜃),𝑟^𝑛Sn(𝜃)) we obtain from (1.11) the following equation: 𝑑𝑟=∑𝑑𝜃^{𝑗≥𝑛−1}𝑏_𝑗(Sn(𝜃))²(Cs(𝜃))^𝑗𝑟^{2−𝑛+𝑗}1+∑_{𝑗≥𝑛−1}𝑏_𝑗Sn(𝜃)(Cs(𝜃))^𝑗+1𝑟^{1−𝑛+𝑗}. (1.15)

The function on the right hand side of (1.15) is periodic of the period 𝑇=2𝜋/𝑛Γ(1/2𝑛)/Γ((𝑛+1)/2𝑛). Let 𝑟(𝜃,𝑟₀) denote the solution of (1.15) with the initial value 𝑟(0)=𝑟₀. Then the Poincaré map of (1.15) has the form: 𝑟𝑇,𝑟₀=_𝑗≥1𝑉_𝑗𝑟^𝑗₀. (1.16)

Assuming that 𝑉₁=1,𝑉₂=⋯=𝑉_𝑘−1=0 Álvarez and Gasull [11] called the constant 𝑉_𝑘 the 𝑘th generalized Lyapunov constant of (1.15) (we will see that this definition is too rough since a half of the constants cannot be used to determine the stability of the origin of (1.8) or (1.11)). They also studied the normal form (1.11) and proved the following theorem.

Theorem 1.2 (see [11]). Let (1.13) and (1.14) be satisfied. Then(1)𝑉₁=exp(−2𝑏_𝑛−1𝜋/𝑛4𝑛−𝑏²_𝑛−1) if 2𝑙=𝑛−1;(2)𝑉₁=1, 𝑉_𝑗=0 for 1<𝑗<2−𝑛+2𝑙, and 𝑉_{2−𝑛+2𝑙}=−𝐾_𝑙𝑏_2𝑙 if either 𝑏_𝑛−1=0 or 𝑏_𝑛−1≠0 and 𝑛 is even, where 𝐾_𝑙 is a positive constant.

In the case 𝑛=2 Liu and Li [12] introduced different generalized polar coordinates of the form 𝑥=𝑟cos𝜃, 𝑦=𝑟²sin𝜃 to change (1.4) into the form: 𝑑𝑟(),𝑑𝑡=𝑅𝜃,𝑟𝑑𝜃(),𝑑𝑡=𝑄𝜃,𝑟 (1.17) where it is assumed that the origin is a center or a focus. Let 𝑟(̃𝜃,ℎ) denote the solution of the 2𝜋-periodic system 𝑑𝑟=𝑅()𝑑𝜃𝜃,𝑟𝑄(𝜃,𝑟) (1.18) satisfying 𝑟(̃0)=ℎ. Note that the initial value problem is well defined also for negative ℎ.

Let 𝐷 be a simply connected domain. Denote by 𝑅 the ring of analytic functions on 𝐷 and by ⟨𝜙₁,…,𝜙_𝑘⟩ the ideal in 𝑅 generated by the functions 𝜙₁,…,𝜙_𝑘 from 𝑅. Liu and Li [12] found the following facts.

Theorem 1.3 (see [12]). Consider system (1.4). Let the conditions of Theorem 1.1 be satisfied with 𝑛=2 (or 𝑚=3) so that the origin is a center or a focus. Then(1)𝑟(̃𝜃,−𝑟(̃𝜋,ℎ))=−𝑟(̃𝜋−𝜃,ℎ).(2)Δ(ℎ)=𝑟(̃−2𝜋,ℎ)−ℎ=∑_𝑘≥2𝑣_𝑘ℎ^𝑘, where 𝑣_2𝑘+1∈𝑣₂,𝑣₄,…,𝑣_2𝑘,𝑘≥1. (1.19) In particular, 𝑣_2𝑗=0, 𝑗=1,…,𝑘 imply that 𝑣_2𝑘+1=0.(3) The origin is a stable (unstable) focus if 𝑣_2𝑘<0(>0),𝑣_2𝑗=0for𝑗<𝑘. (1.20)
In the latter case the origin is called a 𝑘th order weak focus of (1.4).

We remark that the conclusions of Theorem 1.3 provide new and useful information on the property of the coefficients 𝑣_𝑘. Liu and Li [12] also gave some new methods to compute the focal values 𝑣₂,𝑣₄,…,𝑣_2𝑘, or equivalent values and studied the problem of limit cycle bifurcations near the origin (using the second conclusion of the above theorem). They found a new phenomenon: a node can generate a limit cycle when its stability changes.

In this paper we study the problem of limit cycle bifurcations near the origin of the analytic system 𝑥̇=𝑦+𝑋𝑥,𝑦,𝛿,𝑦̇=𝑌𝑥,𝑦,𝛿, (1.21) where 𝛿=(𝛿₁,…,𝛿_𝑚)∈𝐷⊂ℝ^𝑚, 𝐷 is a simply connected domain, and 𝑋,𝑌=𝑂(|𝑥,𝑦|²) for |𝑥| small and 𝛿∈𝐷.

To perform our analysis we first introduce a novel Poincaré map using a specific transversal section, study its analytical properties (Theorems 1.5 and 1.6), and then give a new definition of generalized focal values (or generalized Lyapunov constants) following [12]. Second, using the Poincaré map together with the generalized focal values we establish new bifurcation theory of limit cycles from a nilpotent focus or center and obtain conditions for finding a lower bound and an upper bound of the maximal number of limit cycles bifurcated directly from the nilpotent point (Theorems 1.7 and 1.8). Third, we provide a new method to compute the generalized focal values using the normal form (Theorems 1.10 and 1.11). Moreover, we prove that the normal form and the original system have the same generalized focal values when the higher-order term has a sufficiently high order (Theorem 1.12 and Corollary 1.13). For polynomial systems we prove that the maximal order of a nilpotent focus is uniformly bounded (Theorem 1.14). All these results directly generalize the classical Hopf bifurcation theory and establish the new bifurcation theory in the nilpotent case.

We now state our main results more precisely. Let 𝑦=𝐹(𝑥,𝛿) be the solution to the equation 𝑦+𝑋(𝑥,𝑦,𝛿)=0. We define the following two functions: 𝑔(𝑥,𝛿)=−𝑌(𝑥,𝐹(𝑥,𝛿),𝛿),𝑓(𝑥,𝛿)=−𝜕𝑋(())+𝜕𝑥𝑥,𝐹𝑥,𝛿,𝛿𝜕𝑌(().𝜕𝑦𝑥,𝐹𝑥,𝛿,𝛿) (1.22)

By Theorem 1.1, if 𝑔(𝑥,𝛿)=_{𝑗≥2𝑛−1}𝑎_𝑗(𝛿)𝑥^𝑗,𝑛≥2,𝑎_2𝑛−1(𝛿)𝑓()=>0,𝑥,𝛿_{𝑗≥𝑛−1}𝑏_𝑗(𝛿)𝑥^𝑗,𝑏²_𝑛−1(𝛿)−4𝑛𝑎_2𝑛−1(𝛿)<0, (1.23)(1.24) then the origin is a center or a focus of (1.21) for all 𝛿∈𝐷. For convenience, we introduce the following definition.

Definition 1.4. Let for all 𝛿∈𝐷 (1.23) and (1.24) be satisfied for some 𝑛≥2 so that the the origin is a center or a focus of (1.21). In this case we say that (1.21) has a singular point of multiplicity 𝑛 at the origin.
Next, let us define a Poincaré return map for the planar system (1.21). For each 𝛿∈𝐷 and 𝑥₀≠0 with |𝑥₀| small consider the solution (𝑥(𝑡,𝑥₀,𝛿),𝑦(𝑡,𝑥₀,𝛿)) of (1.21) with the initial condition (𝑥(0),𝑦(0))=(𝑥₀,𝐹(𝑥₀,𝛿)). Then there is a unique least positive number 𝜏=𝜏(𝑥₀,𝛿)>0 such that 𝑦(𝜏,𝑥₀,𝛿)=𝐹(𝑥(𝜏,𝑥₀,𝛿),𝛿) and 𝑥₀𝑥(𝜏,𝑥₀,𝛿)>0 (see Figure 1 for the case of small 𝑥₀>0).

Figure 1 

The Poincaré map of (1.21) with 𝑥0>0.

Assume that for all 𝛿∈𝐷 (1.23) and (1.24) are satisfied, so that the solution 𝑥(𝜏,𝑥₀,𝛿) exists for 0<|𝑥₀|<𝜀₀, where 𝜀₀=𝜀₀(𝐷) is a small positive constant. Define 𝑃𝑥₀=𝑥,𝛿𝜏,𝑥₀,𝛿,0<𝑥₀<𝜀₀,0,𝑥₀=0. (1.25)

The map 𝑃(𝑥₀,𝛿) is the Poincaré map we will use for the remainder of the paper.

Obviously, the function 𝑃(𝑥₀,𝛿) is continuous at 𝑥₀=0 if (1.23) and (1.24) hold. It is easily seen that (1.21) has a periodic orbit near the origin if and only if the map has two fixed points near zero: one positive and the other one negative. Moreover, we note that the function is uniquely defined since the Poincaré section is chosen to be on the curve 𝑦=𝐹(𝑥,𝛿). This enables us to obtain some nice analytical properties of this function at 𝑥₀=0, as stated in the following theorem.

Theorem 1.5. Let (1.21) satisfy (1.23) and (1.24) for all 𝛿∈𝐷. Then there is a unique analytic function 𝑃(𝑥₀,𝛿) in 𝑥₀ at 𝑥₀=0, satisfying (𝜕𝑃/𝜕𝑥₀)(0,𝛿)>0 and such that the displacement function 𝑑(𝑥₀,𝛿) has the expansion 𝑑𝑥₀,𝛿=𝑃𝑥₀,𝛿−𝑥₀=_𝑗≥1𝑣_𝑗(𝛿)𝑥^𝑗₀ (1.26) for |𝑥₀| sufficiently small, where(1)if 𝑛 is odd, then 𝑃(𝑥₀,𝛿)=𝑃(𝑥₀,𝛿) for all |𝑥₀| small;(2)if 𝑛 is even, then for all |𝑥₀| small 𝑃𝑥₀,𝛿=𝑃𝑥₀,𝛿for𝑥₀>0,𝑃⁻¹𝑥₀,𝛿for𝑥₀<0, (1.27)where 𝑃⁻¹ denotes the inverse of 𝑃 in 𝑥₀.

It follows from the theorem that system (1.21) has a periodic orbit near the origin if and only if the analytic function 𝑑 defined by (1.26) has two zeros in 𝑥₀ near 𝑥₀=0, among which one is positive and the other one is negative. The function 𝑑 is called the displacement function or the bifurcation function of (1.21).

The above theorem tells us that the function 𝑃(𝑥₀,𝛿) is analytic in 𝑥₀ at 𝑥₀=0 if 𝑛 is odd, and not analytic in 𝑥₀ at 𝑥₀=0 if 𝑛 is even unless the origin is a center (in this case 𝑃 is the identity). The theorem is a natural generalization of the case of elementary center or focus to the nilpotent case (𝑛>1), but it deals with two different cases (odd and even 𝑛), and the phenomenon in the case of 𝑛 even is new.

For the property of the coefficients 𝑣_𝑗 in (1.26) we have the following theorem which is more general than Theorem 1.3.

Theorem 1.6. Let (1.21) satisfy (1.23) and (1.24) for all 𝛿∈𝐷. Then(1)for 𝑛 odd we have 𝑣_2𝑘∈⟨𝑣₁,𝑣₃,…,𝑣_2𝑘−1⟩,𝑘≥1;(2)for 𝑛 even we have 𝑣₁=0, 𝑣_2𝑘+1∈⟨𝑣₂,𝑣₄,…,𝑣_2𝑘⟩,𝑘≥1.

Define 𝑝_𝑛=[1+(−1)^𝑛]/2. Then the conclusions of the above theorem can be written uniformly as 𝑣_{2𝑘+𝑝𝑛}∈𝑣_1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘−1+𝑝𝑛},𝑘≥1. (1.28)

From the proof of Theorem 1.6 we will see that 𝑣_{2𝑘+𝑝𝑛} depends on 𝑣_1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘−1+𝑝𝑛} smoothly. Using Theorem 1.6 we derive the following two statements on limit cycle bifurcations near the origin.

Theorem 1.7 (Bifurcations from the Focus). Let (1.21) satisfy (1.23) and (1.24) for all 𝛿∈𝐷. Denote that 𝑝_𝑛=[1+(−1)^𝑛]/2.(1) If there is an integer 𝑘≥1 such that ^𝑘+1_𝑗=1𝑣_{2𝑗−1+𝑝𝑛}(𝛿)>0,∀𝛿∈𝐷, (1.29) then there exists a neighborhood 𝑈 of the origin such that (1.21) has at most 𝑘 limit cycles in 𝑈 for all 𝛿∈𝐷, where 𝐷 is any compact subset of 𝐷.(2) If there is 𝛿₀∈𝐷 such that 𝑣_{2𝑘+1+𝑝𝑛}(𝛿₀)≠0, then for all 𝛿∈𝐷 near 𝛿₀ (1.21) has at most 𝑘 limit cycles in a neighborhood of the origin. If further, 𝑣_{2𝑗−1+𝑝𝑛}𝛿₀𝜕𝑣=0,𝑗=1,…,𝑘,rank^1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘−1+𝑝𝑛}𝜕𝛿₁,𝛿₂,…,𝛿_𝑚𝛿₀=𝑘, (1.30) then for an arbitrary sufficiently small neighborhood of the origin there are some 𝛿∈𝐷 near 𝛿₀ such that (1.21) has exactly 𝑘 limit cycles in the neighborhood.

Theorem 1.8 (Bifurcations from the Center). Let (1.21) satisfy (1.23) and (1.24) for all 𝛿∈𝐷. Assume(i) there exist 𝛿₀∈𝐷 and an integer 𝑘≥1 such that (1.30) is satisfied,(ii) the origin is a center of (1.21) if 𝑣_{2𝑗−1+𝑝𝑛}(𝛿₀)=0, 𝑗=1,…,𝑘, then there exists a neighborhood 𝑈 of the origin such that (1.21) has at most 𝑘−1 limit cycles in 𝑈 for all 𝛿∈𝐷 near 𝛿₀, and also, for an arbitrary sufficiently small neighborhood of the origin, there are some 𝛿∈𝐷 near 𝛿₀ such that (1.21) has exactly 𝑘−1 limit cycles in the neighborhood. Hence, the cyclicity of the system at the point 𝛿₀ is equal to 𝑘−1.

Now, different from [11–15], we give the following definition.

Definition 1.9. We call 𝑣_{2𝑘+1+𝑝𝑛}(𝛿) the generalized focal value of order 𝑘 of (1.21) at the origin and call the origin a focus of order 𝑘 if 𝑣_{2𝑘+1+𝑝𝑛}(𝛿)≠0 and 𝑣_{2𝑗−1+𝑝𝑛}(𝛿)=0 for 𝑗=1,…,𝑘.
The above definition is very reasonable and natural, since by Theorem 1.7, we see that a nilpotent focus of order 𝑘 generates at most 𝑘 limit cycles under perturbations which satisfy (1.23) and (1.24).
By the above definition, condition (ii) of Theorem 1.8 means that the origin is a focus of (1.21) of order at most 𝑘−1. This condition alone is not enough to ensure the conclusion of the theorem. For example, using Theorem 1.10 stated below one can prove that the system 𝑥̇=𝑦−𝑎₁𝑥³−𝑎²₂𝑥⁵+𝑎₂𝑥⁷,𝑦̇=−𝑥³ (1.31) has exactly two limit cycles near the origin for 0<𝑎₁≪𝑎₂≪1. But, the focus at the origin has the order at most 1 for (𝑎₁,𝑎₂)≠0 (the origin is a center for 𝑎₁=𝑎₂=0).
The generalized focal values 𝑣_1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘+1+𝑝𝑛},… can be calculated using the normal form of system (1.21). We will give a different method to compute them. From a result of Stróżyna and Żołądek [17] we know that (1.21) has the analytic normal form: 𝑥̇=𝑦,𝑦̇=−𝑔(𝑥,𝛿)−𝑦𝑓(𝑥,𝛿). (1.32)
Note that 𝑓 and 𝑔 in (1.32) may be different from the ones given by (1.22). As before, let 𝛿∈𝐷⊂ℝ^𝑚 where 𝐷 is a domain. Also, suppose that for small |𝑥| the function 𝑔(𝑥,𝛿) satisfies (1.23). Define 𝐹(𝑥,𝛿)=^𝑥₀𝑓(𝑥,𝛿)𝑑𝑥,𝐺(𝑥,𝛿)=^𝑥₀𝑔(𝑥,𝛿)𝑑𝑥. (1.33)
It is easy to see that for 𝑥𝑦<0 the equation 𝐺(𝑥,𝛿)=𝐺(𝑦,𝛿) defines a unique analytic function 𝑦=𝛼(𝑥,𝛿)=−𝑥+𝑂(𝑥²). Set 𝐹(𝛼(𝑥,𝛿),𝛿)−𝐹(𝑥,𝛿)=_𝑗≥1𝐵_𝑗(𝛿)𝑥^𝑗. (1.34)
By Theorem 1.1, if (1.32) satisfies (1.23) and (1.24), then it has a center or focus at the origin. Thus, under (1.23) and (1.24) the Poincaré map for (1.32) is well defined near the origin.

Theorem 1.10. Let (1.32) satisfy (1.23) and (1.24) for all 𝛿∈𝐷. Then, for 𝑥₀>0 small, the Poincaré map 𝑃(𝑥₀,𝛿) has the form: 𝑃𝑥₀,𝛿−𝑥₀=_𝑗≥0𝑣_{2𝑗+1+𝑝𝑛}(𝛿)𝑥^{2𝑗+1+𝑝𝑛}₀1+𝑃^∗_𝑗𝑥₀,𝛿, (1.35) where 𝑃^∗_𝑗(𝑥₀,𝛿)=𝑂(𝑥₀), 𝑣_1+𝑝𝑛(𝛿)=𝐾^∗_𝑙𝐵_2𝑙+1(𝛿)+1−𝑝_𝑛𝑂𝐵²_2𝑙+1,𝑣_{2𝑗+1+𝑝𝑛}(𝛿)=𝐾^∗_𝑙+𝑗𝐵_{2𝑙+2𝑗+1}(𝛿)+𝐵_{2𝑙+2𝑗+1}(𝛿),𝑗≥1, (1.36)𝑙=[𝑛/2], 𝐾^∗_𝑙+𝑗, 𝑗≥0 are positive constants and 𝐵_{2𝑙+2𝑗+1}∈⟨𝐵_2𝑙+1,𝐵_2𝑙+3,…,𝐵_{2𝑙+2𝑗−1}⟩. Thus, Theorems 1.7 and 1.8 hold if 𝑣_{2𝑗+1+𝑝𝑛} is replaced by 𝐵_{2𝑙+2𝑗+1}, 𝑗≥0.

Let 𝑓(𝑥,𝛿)=_𝑗≥0𝑏_𝑗(𝛿)𝑥^𝑗. (1.37)

For system (1.32) we have the following result.

Theorem 1.11. Let (1.32) satisfy (1.23), (1.34), and (1.37) for all 𝛿∈𝐷. Assume there exist 𝛿₀∈𝐷 and 𝑘≥[𝑛/2] such that 𝐵_2𝑘+1𝛿₀<0(>0),𝐵_2𝑗−1𝛿₀=0,𝑗=1,…,𝑘. (1.38)
Let one of the following conditions be satisfied:(a)𝑛=2, and 𝑏₀𝛿₀=0,𝑏²₁𝛿₀−8𝑎₃𝛿₀<0; (1.39)(b)𝑛>2, 𝑔(−𝑥,𝛿)=−𝑔(𝑥,𝛿), 𝑓(−𝑥,𝛿)=𝑓(𝑥,𝛿), and 𝑏_𝑗𝛿₀=0for𝑗=0,…,𝑛−2,𝑏²_𝑛−1𝛿₀−4𝑛𝑎_2𝑛−1𝛿₀<0. (1.40)
Then(1)for 𝛿=𝛿₀ (1.32) has a stable (unstable) focus at the origin.(2)If further 𝜕𝐵rank¹,𝐵₃,…,𝐵_2𝑘−1𝜕𝛿₁,𝛿₂,…,𝛿_𝑚𝛿₀=𝑘, (1.41)then for an arbitrary sufficiently small neighborhood of the origin there are some 𝛿∈𝐷 near 𝛿₀ such that (1.32) has at least 𝑘 limit cycles in the neighborhood.

From Theorems 1.5–1.10, it seems that under (1.23) and (1.24) we have solved the problem of limit cycle bifurcation for generic systems. Theoretically it is, but in practice it is not. The reason is that in general we do not know what is the transformation from (1.21) to its normal form (1.32). Here we give a method to solve the problem completely both theoretically and in practice. It includes three main steps described below.

Under (1.23) and (1.24) by the normal form theory (see, e.g., [16]) for any integer 𝑚>2𝑛−1 there is a change of variables of the form: 𝑥𝑦=𝑢𝑣+𝐻_𝑚(𝑢,𝑣,𝛿), (1.42) where 𝐻_𝑚(𝑢,𝑣,𝛿)=𝑂(|𝑢,𝑣|²) is a polynomial in 𝑢,𝑣 of degree at most 𝑚, such that it transforms (1.21) into (1.43) (called the normal form of order 𝑚 of (1.21), or the Takens normal form; we still use (𝑥,𝑦) for the new variables 𝑢,𝑣) 𝑥̇=𝑦+𝑋_𝑚+1𝑥,𝑦,𝛿,𝑦̇=−𝑔_𝑚(𝑥,𝛿)−𝑦𝑓_𝑚−1(𝑥,𝛿)+𝑌_𝑚+1𝑥,𝑦,𝛿, (1.43) where 𝑔_𝑚(𝑥,𝛿)=^𝑚_{𝑗=2𝑛−1}𝑎_𝑗(𝛿)𝑥^𝑗,𝑓_𝑚−1(𝑥,𝛿)=^𝑚−1_{𝑗=𝑛−1}𝑏_𝑗(𝛿)𝑥^𝑗 (1.44) with 𝑎_2𝑛−1(𝛿)>0 and 𝑏²_𝑛−1(𝛿)−4𝑛𝑎_2𝑛−1(𝛿)<0, and 𝑋_𝑚+1(𝑥,𝑦,𝛿), 𝑌_𝑚+1(𝑥,𝑦,𝛿) being analytic functions satisfying 𝑋_𝑚+1,𝑌_𝑚+1=𝑂(|𝑥,𝑦|^𝑚+1). Here, we should mention that the functions 𝑔_𝑚 and 𝑓_𝑚−1 depend only on the terms of degree at most 𝑚 of the expansions of the functions 𝑋 and 𝑌 in (1.21) at the origin.

The Poincaré maps of (1.21) and (1.43) are essentially the same. Denote the Poincaré map of (1.43) by 𝑃(𝑥₀,𝛿), and then the displacement function has the expansion 𝑃𝑥₀,𝛿−𝑥₀=_𝑗≥1𝑣_𝑗(𝛿)𝑥^𝑗₀, (1.45) where the series converge for small |𝑥₀|.

Truncating the series (1.43) at terms of order 𝑚 we obtain the polynomial system 𝑥̇=𝑦,𝑦̇=−𝑔_𝑚(𝑥,𝛿)−𝑦𝑓_𝑚−1(𝑥,𝛿). (1.46)

In practice, for given system (1.21) it is not difficult to find the corresponding system (1.46). For (1.46) we can use Theorem 1.10 to find its focal values at the origin up to any large order. Let 𝑃_𝑚(𝑥₀,𝛿) denote the Poincaré map of (1.46). We write the expansion of the displacement function as 𝑃_𝑚𝑥₀,𝛿−𝑥₀=_𝑗≥1𝑣_𝑗(𝛿)𝑥^𝑗₀, (1.47) where the series converge for small |𝑥₀|.

We intend to use 𝑣_𝑗(𝛿) instead of 𝑣_𝑗(𝛿). To this end, we have to solve the following problem. For any given 𝑘>1 find 𝑚>2𝑛−1 such that 𝑣_𝑗(𝛿)=𝑣_𝑗(𝛿) for 1≤𝑗≤𝑘. The following theorem gives a solution.

Theorem 1.12. Consider (1.43) and (1.46). Then, for any integer 𝑘≥1, if 𝑚≥(𝑘+2)𝑛−2, then 𝑣_𝑗(𝛿)=𝑣_𝑗(𝛿)for1≤𝑗≤𝑘𝑛. (1.48)

Therefore, we have the following.

Corollary 1.13. Under (1.23) and (1.24) for any integer 𝑘≥1 for (1.21) the coefficients 𝑣₁,𝑣₂,…,𝑣_𝑘𝑛 in (1.26) depend only on the terms of degree at most (𝑘+2)𝑛−2 of the expansions of the functions 𝑋 and 𝑌 at the origin.

In the case of elementary center or focus, the above conclusion is well-known.

Finally, we consider the following polynomial system of degree 𝑘𝑥̇=𝑦+_{2≤𝑖+𝑗≤𝑘}𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗,𝑦̇=_{2≤𝑖+𝑗≤𝑘}𝑏_𝑖𝑗𝑥^𝑖𝑦^𝑗. (1.49)

Theorem 1.14. For any 𝑘≥3 there is an integer 𝑁_𝑘>0 such that an arbitrary polynomial system of the form (1.49) has a singular point of multiplicity at most 𝑁_𝑘 at the origin (i.e., one must have 2≤𝑛≤𝑁_𝑘 if (1.49) satisfies (1.23) and (1.24), see Definition 1.4). Further, for each 2≤𝑛≤𝑁_𝑘, if (1.49) satisfies (1.23) and (1.24), then there exists an integer 𝐾_𝑛(𝑘)>0 such that for (1.49) the origin is a focus of order at most 𝐾_𝑛(𝑘). Hence, the origin as a nilpotent focus of (1.49) generates at most 𝐾_𝑛(𝑘) limit cycles.

We organize the paper as follows. In Section 2 we give preliminary lemmas. In Section 3 we prove our main results. In Section 4 few examples are provided.

2. Preliminaries

Consider system (1.21). In this section we will always suppose that (1.23) and (1.24) are satisfied. Introducing the new variable 𝑣=𝑦−𝐹(𝑥,𝛿) we obtain from (1.21) (reusing 𝑦 for 𝑣) 𝑥̇=𝑦1+𝑍₁,̇()()𝑥,𝑦,𝛿𝑦=−𝑔𝑥,𝛿−𝑦𝑓𝑥,𝛿+𝑦²𝑍₂𝑥,𝑦,𝛿, (2.1) where the functions 𝑓 and 𝑔 are given by (1.22), 𝑍₁ and 𝑍₂ are analytic functions near the origin with 𝑍₁(𝑥,𝑦,𝛿)=𝑂(|𝑥,𝑦|). In the discussion below for convenience we will often omit 𝛿. As suggested by Liu and Li in [15] we pass in (2.1) to the generalized polar coordinates 𝑥=𝑟cos𝜃,𝑦=𝑟^𝑛sin𝜃,𝑟>0. (2.2)

Lemma 2.1. Let (1.23) and (1.24) be satisfied. Then the substitution (2.2) transforms (2.1) into the system 𝜃̇=𝑆(𝜃,𝑟)=𝑟^𝑛−1𝐻(𝜃)𝑆₀(𝜃)(𝑟,̇()=𝑟+𝑂)𝑟=𝑅𝜃,𝑟^𝑛𝐻(𝜃)𝑅₀(𝜃)+𝑂(𝑟), (2.3) where 𝑆 and 𝑅 are 2𝜋-periodic in 𝜃 and have the properties 𝑆𝜋+(−1)^𝑛−1𝜃,−𝑟=(−1)^𝑛−1𝑆(𝜃,𝑟),𝑅𝜋+(−1)^𝑛−1()𝐻(𝜃)𝜃,−𝑟=−𝑅𝜃,𝑟,=cos²𝜃+𝑛sin²𝑆𝜃>0,₀(𝜃)=−𝑛sin²𝜃+𝑏_𝑛−1cos^𝑛𝜃sin𝜃+𝑎_2𝑛−1cos^2𝑛𝜃𝑅<0,₀(𝜃)=cos𝜃sin𝜃1−𝑎_2𝑛−1cos^2𝑛−2𝜃−𝑏_𝑛−1sin𝜃cos^𝑛−2𝜃. (2.4)(2.5)

Proof. From (2.2) we have 𝑥̇=cos𝜃𝑟̇−𝑟sin𝜃𝜃̇,𝑦̇=𝑛𝑟^𝑛−1sin𝜃𝑟̇+𝑟^𝑛cos𝜃𝜃̇. (2.6)
We solve the above equations for 𝜃̇ and 𝑟̇ and obtain (2.3) with 𝑆()=̇𝜃,𝑟cos𝜃𝑦−𝑛𝑟^𝑛−1sin𝜃𝑥̇𝑟^𝑛cos²𝜃+𝑛sin²𝜃,𝑅()=̇𝜃,𝑟sin𝜃𝑦+𝑟^𝑛−1cos𝜃𝑥̇𝑟^𝑛−1cos²𝜃+𝑛sin²𝜃. (2.7)
Then noting that cos(𝜋±𝜃)=−cos𝜃,sin(𝜋±𝜃)=∓sin𝜃 (2.8) and that (2.2) is invariant as (𝜃,𝑟) is replaced by (𝜋+(−1)^𝑛−1𝜃,−𝑟) one can easily prove (2.4). The other conclusions are direct. This ends the proof.

By (2.3) and (2.4) we obtain the following analytic 2𝜋-periodic equation 𝑑𝑟=𝑑𝜃𝑅(𝜃,𝑟), (2.9) where 𝑅()̇𝜃,𝑟=𝑟sin𝜃𝑦+𝑟^𝑛−1cos𝜃𝑥̇cos𝜃𝑦̇−𝑛𝑟^𝑛−1𝑥̇𝑅sin𝜃=𝑟⁰(𝜃)𝑆₀(𝜃)+𝑂(𝑟),𝑅𝜋+(−1)^𝑛−1𝜃,−𝑟=(−1)^𝑛𝑅(𝜃,𝑟). (2.10)

Let 𝑟(𝜃,ℎ) denote the solution of (2.9) with the initial value 𝑟(0)=ℎ. For properties of the solution we have the following.

Lemma 2.2. The solution 𝑟(𝜃,ℎ)=𝑂(ℎ) is analytic in (𝜃,ℎ) for |ℎ| small and satisfies the following:(1)𝑟(𝜃,−𝑟(𝜋,ℎ))=−𝑟(𝜋+(−1)^𝑛−1𝜃,ℎ);(2)𝑟(𝜃±2𝜋,ℎ)=𝑟(𝜃,𝑟(±2𝜋,ℎ)).

Proof. Let 𝑟(̃𝜃)=−𝑟(𝜋+(−1)^𝑛−1𝜃,ℎ). Then by (2.9) and (2.10) we have 𝑑𝑟̃=()𝑑𝜃−1^𝑛𝑅𝜋+(−1)^𝑛−1𝜃,𝑟𝜋+(−1)^𝑛−1=()𝜃,ℎ−1^𝑛𝑅𝜋+(−1)^𝑛−1𝑟̃(𝜃)=𝜃,−𝑅(𝜃,𝑟̃(𝜃)). (2.11)
This means that 𝑟(̃𝜃) is also a solution to (2.9). Then the first conclusion follows from the uniqueness of the solution to the initial problem. The second one follows in the same way. This completes the proof.

Further we have the following.

Lemma 2.3. Let 𝑃(𝑥₀,𝛿) be the Poincaré return map of (1.21) defined in Section 1. Then for |𝑥₀|>0 small we have 𝑃(𝑥₀,𝛿)=𝑟(−2𝜋,𝑥₀) for 𝑥₀>0, and 𝑃(𝑥₀,𝛿)=𝑟((−1)^𝑛2𝜋,𝑥₀) for 𝑥₀<0.

Proof. First, it is easy to see that (1.21) and (2.1) have the same Poincaré map 𝑃(𝑥₀,𝛿). Then, noting from (2.3) that 𝜃̇<0 for small 𝑟>0, by the definition of 𝑃 and (2.2), we see that 𝑃𝑥₀,𝛿=𝑥𝜏,𝑥₀=𝑟−2𝜋,𝑥₀ (2.12) for 𝑥₀>0 small. Now consider the case of 𝑥₀<0. Let 𝑟^∗(𝜃,ℎ) denote the solution of (2.9) satisfying 𝑟^∗(𝜋)=ℎ. Similarly as above we have 𝑃𝑥₀,𝛿=𝑥𝜏,𝑥₀=−𝑟^∗−𝜋,−𝑥₀, (2.13) since under (2.2) the points (𝑥₀,0) and (𝑃(𝑥₀,𝛿),0) on the (𝑥,𝑦) plane correspond to the points (𝜋,−𝑥₀) and (−𝜋,−𝑃(𝑥₀,𝛿)) on the (𝜃,𝑟) plane, respectively.
Further, by Lemma 2.2(1), we have 𝑟^∗()()𝑟𝜃,−ℎ=−𝑟𝜋−𝜃,ℎfor𝑛even,_∗(𝜃,−𝑟(2𝜋,ℎ))=−𝑟(𝜋+𝜃,ℎ)for𝑛odd. (2.14)(2.15)
Noting that, by Lemma 2.2(2), 𝑥₀=𝑟(2𝜋,ℎ) if and only if ℎ=𝑟(−2𝜋,𝑥₀), we see that (2.15) becomes 𝑟^∗𝜃,−𝑥₀=−𝑟𝜋+𝜃,𝑟−2𝜋,𝑥₀for𝑛odd. (2.16)
Therefore, by (2.14) and (2.16) we have for 𝑥₀<0𝑃𝑥₀=𝑟,𝛿2𝜋,𝑥₀𝑟for𝑛even,−2𝜋,𝑥₀for𝑛odd. (2.17)
This ends the proof.

Lemma 2.4. Let 𝑑(𝑥₀,𝛿)=𝑃(𝑥₀,𝛿)−𝑥₀. Then there exists an analytic function 𝐾(ℎ,𝛿) convergent for small |ℎ| with 𝐾(0,𝛿)=(𝜕𝑟/𝜕𝑥₀)(𝜋,0)>0 such that 𝑑𝑥₀,𝛿=−𝐾𝑥₀,𝛿𝑑𝑥₀,𝛿 (2.18) for 𝑥₀>0 small, where 𝑥₀=−𝑟(𝜋,𝑥₀).

Proof. By Lemma 2.2, we have 𝑟(−1)^𝑛2𝜋,𝑥₀=−𝑟−𝜋,𝑥₀=−𝑟𝜋,𝑟−2𝜋,𝑥₀. (2.19)
Hence, by Lemma 2.3 for 𝑥₀>0𝑑𝑥₀,𝛿=𝑟(−1)^𝑛2𝜋,𝑥₀−𝑥₀=−𝑟𝜋,𝑟−2𝜋,𝑥₀+𝑟𝜋,𝑥₀=−𝐾𝑥₀,𝛿𝑟−2𝜋,𝑥₀−𝑥₀=−𝐾𝑥₀,𝛿𝑑𝑥₀,𝛿, (2.20) where 𝐾𝑥₀,𝛿=¹₀𝜕𝑟𝜕𝑥₀𝜋,𝑥₀+𝑠𝑟−2𝜋,𝑥₀−𝑥₀𝑑𝑠. (2.21)
It is obvious that 𝐾 is analytic for |𝑥₀| small and 𝐾(0,𝛿)=(𝜕𝑟/𝜕𝑥₀)(𝜋,0)>0. This completes the proof.

3. Proof of the Main Results

In this section we prove our main results presented in Theorems 1.5–1.12.

Proof of Theorem 1.5. We take 𝑃(𝑥₀,𝛿)=𝑟(−2𝜋,𝑥₀) for |𝑥₀| small. Then by Lemma 2.2  𝑃 is analytic. Note that by Lemma 2.2, 𝑟(2𝜋,𝑥₀) is the inverse of 𝑟(−2𝜋,𝑥₀) in 𝑥₀. Then Theorem 1.5 follows directly from Lemma 2.3. The proof is complete.

Proof of Theorem 1.6. There are two cases to consider separately.
Case A (𝑛 odd). By (1.26) and Theorem 1.5(1), we have 𝑑𝑥₀,𝛿=𝑑𝑥₀,𝛿=_𝑗≥1𝑣_𝑗(𝛿)𝑥^𝑗₀ (3.1) for all |𝑥₀| small.
By Lemma 2.4, we can suppose that 𝐾𝑥₀,𝛿=_𝑗≥0𝑘_𝑗𝑥^𝑗₀,𝑥₀=−𝑟𝜋,𝑥₀=_𝑗≥1𝑙_𝑗𝑥^𝑗₀, (3.2) where 𝑘₀>0, 𝑙₁=−𝑘₀. Substituting (3.1) and (3.2) into (2.18), we obtain _𝑗≥1𝑣_𝑗𝑖≥1𝑙_𝑖𝑥^𝑖₀^𝑗=−_{𝑖≥0,𝑗≥1}𝑘_𝑖𝑣_𝑗𝑥₀^𝑖+𝑗. (3.3)
Comparing the coefficients of the terms 𝑥²₀, 𝑥⁴₀ and 𝑥₀^2𝑗 on both sides yields 𝑣₂𝑙²₁+𝑣₁𝑙₂=−𝑘₀𝑣₂+𝑘₁𝑣₁,𝑣₄𝑙⁴₁+3𝑣₃𝑙²₁𝑙₂+𝑣₂𝑙²₂+2𝑙₁𝑙₃+𝑣₁𝑙₄=−𝑘₀𝑣₄+𝑘₁𝑣₃+𝑘₂𝑣₂+𝑘₃𝑣₁,⋯𝑣_2𝑗𝑙₁^2𝑗+𝑣_2𝑗−1𝐿_1,𝑗𝑙₁,𝑙₂+⋯+𝑣₂𝐿_{2𝑗−2,𝑗}𝑙₁,𝑙₂,…,𝑙_2𝑗−1+𝑣₁𝑙_2𝑗=−^2𝑗−1_𝑖=0𝑘_𝑖𝑣_2𝑗−𝑖,⋯, (3.4) where 𝐿_𝑖,𝑗(𝑙₁,𝑙₂,…,𝑙_𝑖+1), 𝑖=1,2,…,2𝑗−2, are polynomials. Thus, from the above equations we obtain 𝑣_2𝑘∈𝑣₁,𝑣₃,…,𝑣_2𝑘−1,𝑘≥1. (3.5)
Case B (𝑛 even). By (1.26) and 𝑥₀=𝑃⁻¹(𝑃(𝑥₀,𝛿),𝛿) we find that 𝑃⁻¹𝑥₀,𝛿=𝑣₁𝑥₀+𝑣₂𝑥²₀+𝑣₃𝑥³₀+⋯, (3.6) where 𝑣₁=𝑣₁+1⁻¹,𝑣₂=−𝑣₂𝑣₁+1⁻³,⋯𝑣_𝑗=−𝑣_𝑗𝑣₁+1^−(𝑗+1)+𝐿_𝑗𝑣₂,𝑣₃,…,𝑣_𝑗−1,⋯ (3.7) and each 𝐿_𝑗 is a polynomial of degree at least 2. Now we suppose that 𝑥₀>0. Then (3.1) holds by Theorem 1.5. Further, noting that 𝑥₀<0 by Theorem 1.5 again 𝑑𝑥₀,𝛿=𝑃𝑥₀,𝛿−𝑥₀=𝑃⁻¹𝑥₀,𝛿−𝑥₀. (3.8)
Then, inserting (3.1), (3.2), (3.6), and (3.8) into (2.18) we obtain 𝑣₁−1𝑙₁=−𝑘₀𝑣₁,𝑣₁−1𝑙₂+𝑣₂𝑙²₁=−𝑘₀𝑣₂+𝑘₁𝑣₁,⋯𝑣₁−1𝑙_𝑗+𝐿_𝑗𝑣₂,𝑣₃,…,𝑣_𝑗−1+𝑣_𝑗𝑙^𝑗₁=−𝑘₀𝑣_𝑗+𝑘₁𝑣_𝑗−1+⋯+𝑘_𝑗−1𝑣₁,⋯, (3.9) where 𝐿_𝑗𝑣₂,𝑣₃,…,𝑣_𝑗−1∈𝑣₂,𝑣₃,…,𝑣_𝑗−1. (3.10)
Finally, noting that 𝑙₁=−𝑘₀ and substituting (3.7) into (3.9) we easily see that 𝑣_2𝑗+1∈𝑣₂,𝑣₄,…,𝑣_2𝑗,𝑗≥1. (3.11)
This ends the proof.

Proof of Theorem 1.7. For the first part, suppose that the conclusion is not true. Then there exists a sequence {𝛿_𝑚} in 𝐷 such that for 𝛿=𝛿_𝑚 (1.21) has 𝑘+1 limit cycles 𝐿_𝑚,1,𝐿_𝑚,2,…,𝐿_𝑚,𝑘+1 which approach the origin as 𝑚→∞. Then by Theorem 1.6, the function 𝑑(𝑥₀,𝛿_𝑚) has 2𝑘+2 nonzero roots in 𝑥₀ which approach zero as 𝑚→∞.
Since 𝐷 is compact, we can assume 𝛿_𝑚→𝛿₀∈𝐷 as 𝑚→∞. By our assumption, ∑^𝑘+1_𝑗=1|𝑣_{2𝑗−1+𝑝𝑛}(𝛿₀)|>0. Thus, for some 1≤𝑙≤𝑘+1, 𝑣_{2𝑙−1+𝑝𝑛}𝛿₀≠0,𝑣_{2𝑗−1+𝑝𝑛}𝛿₀=0for1≤𝑗≤𝑙−1. (3.12)
Therefore, by (1.26) and Theorem 1.6, we have 𝑑𝑥₀,𝛿₀=𝑣_{2𝑙−1+𝑝𝑛}𝛿₀𝑥^{2𝑙−1+𝑝𝑛}₀+𝑂𝑥^{2𝑙+𝑝𝑛}₀. (3.13)
Note that 𝑑(0,𝛿)=0. It follows from Rolle's theorem (see [1]) that for some 𝜀₀>0 the function 𝑑(𝑥₀,𝛿) has at most 2𝑙−2+𝑝_𝑛 nonzero roots in (−𝜀₀,𝜀₀) for all |𝛿−𝛿₀|<𝜀₀. We have proved that the function 𝑑(𝑥₀,𝛿_𝑚) has 2𝑘+2 nonzero roots which approach zero as 𝑚→∞. It then follows that 2𝑘+2≤2𝑙−2+𝑝_𝑛, contradicting to 2𝑙−2+𝑝_𝑛≤2𝑘+𝑝_𝑛≤2𝑘+1. The first conclusion follows.
For the second one, from the above proof one can see that for all 𝛿∈𝐷 near 𝛿₀ (1.21) has at most 𝑘 limit cycles in a neighborhood of the origin. Then, by Theorem 1.6, the displacement function 𝑑 can be written as 𝑑𝑥₀,𝛿=_𝑗≥1𝑣_{2𝑗−1+𝑝𝑛}(𝛿)𝑥^{2𝑗−1+𝑝𝑛}₀1+𝑃_𝑗𝑥₀,𝛿, (3.14) where 𝑃_𝑗(0,𝛿)=0. Like in [19] one can show that 𝑃_𝑗 are series convergent in a neighborhood of 𝛿₀ (see also, e.g., [20, 21]). Further, by (1.30), we can take 𝑣_1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘−1+𝑝𝑛} as free parameters, varying near zero. Precisely, if we change them such that 0<𝑣_1+𝑝𝑛≪𝑣_3+𝑝𝑛≪⋯≪𝑣_{2𝑘−1+𝑝𝑛}≪1,𝑣_1+𝑝𝑛𝑣_3+𝑝𝑛𝑣<0,…,_{2𝑘−1+𝑝𝑛}𝑣_{2𝑘+1+𝑝𝑛}<0, (3.15) then by (3.14) the function 𝑑 has exactly 𝑘 positive zeros in 𝑥₀ near 𝑥₀=0, which give 𝑘 limit cycles. This finishes the proof.

By Theorem 1.5 and (3.14) we immediately have the following.

Corollary 3.1. Let (1.21) satisfy (1.23) and (1.24) for a fixed 𝛿∈𝐷. Then, if 𝑣_{2𝑘+1+𝑝𝑛}(𝛿)<0(>0),𝑣_{2𝑗−1+𝑝𝑛}(𝛿)=0for𝑗=1,…,𝑘, (3.16) then the origin is a stable (unstable) focus of order 𝑘 of (1.21). If 𝑣_{2𝑗−1+𝑝𝑛}(𝛿)=0∀𝑗≥1, (3.17) the origin is a center of (1.21).

Proof of Theorem 1.8. Under (1.30) the values 𝑣_1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘−1+𝑝𝑛} can be taken as free parameters. Further, by our assumption, the origin is a center of (1.21) as 𝑣_{2𝑗−1+𝑝𝑛}(𝛿)=0, 𝑗=1,…,𝑘. It then follows that 𝑣_{2𝑗−1+𝑝𝑛}(𝛿)∈𝑣_1+𝑝𝑛,𝑣_3+𝑝𝑛,…,𝑣_{2𝑘−1+𝑝𝑛}∀𝑗≥𝑘+1. (3.18)
Therefore, (3.14) can be further written in the form: 𝑑𝑥₀,𝛿=^𝑘_𝑗=1𝑣_{2𝑗−1+𝑝𝑛}(𝛿)𝑥^{2𝑗−1+𝑝𝑛}₀1+𝑃_𝑗𝑥₀,𝛿, (3.19) where 𝑃_𝑗(0,𝛿)=0 and 𝑃_𝑗 are series convergent in a neighborhood of 𝛿₀ [19]. Using the reasoning of Bautin [1] (see also e.g., [20–22]) one can easily see that the conclusion of the theorem holds. The proof is completed.

Proof of Theorem 1.10. Now we consider (1.32), where 𝑔 satisfies (1.23). Let 𝐹(𝑥,𝛿)=^𝑥₀𝑓(𝑥,𝛿)𝑑𝑥,𝐺(𝑥,𝛿)=^𝑥₀𝑔(𝑥,𝛿)𝑑𝑥. (3.20)
If 𝑓 satisfies (1.24), then the origin is a center or focus of (1.32), and 𝐹(𝛼(𝑥,𝛿),𝛿)−𝐹(𝑥,𝛿)=_𝑗≥𝑛𝐵_𝑗(𝛿)𝑥^𝑗=_𝑗≥𝑛1𝐵_𝑗(𝛿)𝑥^𝑗, (3.21) where 𝐵_𝑛=(−1)^𝑛−1𝑛𝑏_𝑛−1,𝑛₁𝑛=2𝑙+1,𝑙=2, (3.22) and 𝛼(𝑥,𝛿)=−𝑥+𝑂(𝑥²) satisfies 𝐺(𝛼(𝑥,𝛿),𝛿)=𝐺(𝑥,𝛿) for |𝑥| small. Note that (1.32) is equivalent to the following system: 𝑥̇=𝑦−𝐹(𝑥,𝛿),𝑦̇=−𝑔(𝑥,𝛿) (3.23) which has the same Poincaré return map 𝑃(𝑥₀,𝛿) as (1.32). Introducing the change of variables 𝑥 and 𝑡𝑢=[2𝑛𝐺(𝑥,𝛿)]^1/2𝑛sgn𝑥=𝑎_2𝑛−1^1/2𝑛𝑥+𝑂𝑥²≡𝜑(𝑥),𝑑𝑡𝑑𝑡₁=𝜑^2𝑛−1(𝑥)𝑔(𝑥,𝛿) (3.24) system (3.23) becomes 𝑢̇=𝑦−𝐹(𝑢,𝛿),𝑦̇=−𝑢^2𝑛−1, (3.25) which is equivalent to 𝑢̇=𝑦,𝑦̇=−𝑢^2𝑛−1−𝑦𝑓(𝑢,𝛿), (3.26) where 𝐹(𝑢,𝛿)=𝐹𝜑⁻¹(𝑢),𝛿,𝑓()=𝜕𝑢,𝛿𝐹().𝜕𝑢𝑢,𝛿 (3.27)
The systems (3.25) and (3.26) have the same Poincaré return map, denoted by 𝑃₁(𝑢₀,𝛿). One can see that the maps 𝑃 and 𝑃₁ have the relation 𝑃₁∘𝜑=𝜑∘𝑃. Hence, 𝑃𝑥₀,𝛿−𝑥₀=𝐾𝑢₀𝑃₁𝑢₀,𝛿−𝑢₀, (3.28) where 𝐾(𝑢₀)=(𝑎_2𝑛−1)^−1/2𝑛+𝑂(𝑢₀) is analytic. By (1.26) and (3.14), for 𝑢₀>0 small we have 𝑃₁𝑢₀,𝛿−𝑢₀=_𝑗≥1𝑣_𝑗(𝛿)𝑢^𝑗₀=_𝑗≥1𝑣_{2𝑗−1+𝑝𝑛}(𝛿)𝑢^{2𝑗−1+𝑝𝑛}₀1+𝑃_𝑗𝑢₀,𝛿. (3.29)
Hence, 𝑃𝑥₀,𝛿−𝑥₀=_𝑗≥1𝑣_{2𝑗−1+𝑝𝑛}(𝛿)𝑎_2𝑛−1^−1/2𝑛𝑢^{2𝑗−1+𝑝𝑛}₀1+𝑃_𝑗𝑢₀,𝛿=_𝑗≥1𝑣_{2𝑗−1+𝑝𝑛}(𝛿)𝑎_2𝑛−1^{(2𝑗−2+𝑝𝑛)/2𝑛}𝑥^{2𝑗−1+𝑝𝑛}₀1+𝑃^∗_𝑗𝑥₀,𝛿, (3.30) where 𝑃_𝑗(𝑢₀,𝛿)=𝑂(𝑢₀), 𝑃^∗_𝑗(𝑥₀,𝛿)=𝑂(𝑥₀).
Since 𝛼 satisfies 𝐺(𝛼(𝑥,𝛿),𝛿)=𝐺(𝑥,𝛿) and 𝑥𝛼<0 for |𝑥| small, we have 𝜑(𝛼)=−𝜑(𝑥) or 𝛼=𝜑⁻¹(−𝜑(𝑥))=𝜑⁻¹(−𝑢), where 𝑢=𝜑(𝑥). Thus, we have 𝐹(𝛼(𝑥,𝛿),𝛿)−𝐹(𝑥,𝛿)=𝐹𝜑⁻¹(−𝑢),𝛿−𝐹𝜑⁻¹(𝑢),𝛿=𝐹(−𝑢,𝛿)−𝐹(𝑢,𝛿). (3.31)
Let 𝑓(𝑢,𝛿)=_{𝑗≥𝑛−1}𝑏_𝑗(𝛿)𝑢^𝑗. (3.32)
Then 𝐹(𝑢,𝛿)=_𝑗≥𝑛𝑏_𝑗−1(𝛿)𝑗𝑢^𝑗. (3.33)
Thus, by (3.31) we have 𝐹(𝛼(𝑥,𝛿),𝛿)−𝐹(𝑥,𝛿)=−2_{[]𝑗≥𝑛/2}𝑏_2𝑗(𝛿)𝑢2𝑗+1^2𝑗+1. (3.34)
Substituting 𝑢=𝜑(𝑥)=(𝑎_2𝑛−1)^1/2𝑛(𝑥+𝑂(𝑥²)) into the above equality and comparing with (3.21) we obtain 𝐵_2𝑙+1=−𝐾_𝑙𝑏_2𝑙,𝐵_2𝑙+2=𝑂𝑏_2𝑙,𝐵_{2𝑙+2𝑗+1}=−𝐾_𝑙+𝑗𝑏_2𝑙+2𝑗+𝐵_{2𝑙+2𝑗+1},𝐵_{2𝑙+2𝑗+2}∈𝑏_2𝑙,𝑏_2𝑙+2,…,𝑏_2𝑙+2𝑗,𝑗≥1, (3.35) where 𝐾_𝑙, 𝐾_𝑙+1, … are positive constants and 𝐵_{2𝑙+2𝑗+1}∈⟨𝑏_2𝑙,𝑏_2𝑙+2,…,𝑏_{2𝑙+2𝑗−2}⟩.
Then by Theorem 1.5 for 𝑢₀>0 small we clearly have 𝑃₁𝑢₀,𝛿=𝑢₀+_𝑗≥1𝑣_𝑗(𝛿)𝑢^𝑗₀=_𝑗≥1𝑉_𝑗(𝛿)𝑢^𝑗₀, (3.36) where 𝑉_𝑗 are introduced before Theorem 1.2. Thus, by Theorem 1.2, we have 𝑣₁=−𝐾_𝑙𝑏_2𝑙+𝑂𝑏²_2𝑙,𝑣_{2𝑗+1𝑣1=⋯=𝑣2𝑗−1=0}=−𝐾_𝑙+𝑗𝑏_2𝑙+2𝑗,𝑗≥1 (3.37) for 𝑛=2𝑙+1 odd, and 𝑣₂=−𝐾_𝑙𝑏_2𝑙,𝑣_{2𝑗+2𝑣2=⋯=𝑣2𝑗=0}=−𝐾_𝑙+𝑗𝑏_2𝑙+2𝑗,𝑗≥1 (3.38) for 𝑛=2𝑙 even, where 𝐾_𝑙+𝑗, 𝑗≥0 are positive constants. Hence, 𝑣_1+𝑝𝑛=−𝐾_𝑙𝑏_2𝑙+1−𝑝_𝑛𝑂𝑏²_2𝑙,𝑣_{2𝑗+1+𝑝𝑛}=−𝐾_𝑙+𝑗𝑏_2𝑙+2𝑗+𝜑𝑏_2𝑙,𝑏_2𝑙+2,…,𝑏_{2𝑙+2𝑗−2},𝑗≥1, (3.39) where 𝜑(0,0,…,0)=0. Note that (1.32) is analytic in each 𝑏_𝑗. It follows from Theorem 1.5 that 𝜑 is analytic in (𝑏_2𝑙,𝑏_2𝑙+2,…,𝑏_{2𝑙+2𝑗−2}), which yields 𝜑∈⟨𝑏_2𝑙,𝑏_2𝑙+2,…,𝑏_{2𝑙+2𝑗−2}⟩. Then (3.35) and (3.39) together give 𝑣_1+𝑝𝑛=𝐾_𝑙𝐾_𝑙𝐵_2𝑙+1+1−𝑝_𝑛𝑂𝐵²_2𝑙+1,𝑣_{2𝑗+1+𝑝𝑛}=𝐾_𝑙+𝑗𝐾_𝑙+𝑗𝐵_{2𝑙+2𝑗+1}+𝐵_{2𝑙+2𝑗+1},𝑗≥1, (3.40) where 𝐵_{2𝑙+2𝑗+1}∈⟨𝐵_2𝑙+1,𝐵_2𝑙+3,…,𝐵_{2𝑙+2𝑗−1}⟩.
Then (1.36) follows from (3.30) and (3.40). This finishes the proof.

Proof of Theorem 1.11. Let |𝛿−𝛿₀| be small. For 𝑛=2 we have 𝑝_𝑛=1. Then the first conclusion follows directly from Corollary 3.1 and (1.36)–(1.39). In fact, we have by Theorem 1.10𝑣_2𝑘𝛿₀=𝐾^∗_𝑘𝐵_2𝑘+1𝛿₀,𝐾^∗_𝑘>0,𝑣_2𝑗𝛿₀=0for𝑗=1,…,𝑘−1. (3.41)
For the second conclusion, we first keep 𝐵₁(𝛿)=0 and vary 𝐵₃(𝛿),…,𝐵_2𝑘−1(𝛿) near zero to obtain exactly 𝑘−1 simple limit cycles near the origin. These limit cycles are bifurcated by changing the stability of the focus at the origin 𝑘−1 times. Then we vary 𝐵₁ such that 0<|𝐵₁|≪|𝐵₃|, and 𝐵₁𝐵₃<0. This step produces one more limit cycle bifurcated from the origin by changing the stability of the origin which is a node now by [23]. The theorem is proved for the case 𝑛=2.
For 𝑛>2 since 𝑔(−𝑥,𝛿)=−𝑔(𝑥,𝛿), 𝑓(−𝑥,𝛿)=𝑓(𝑥,𝛿) we have 𝑏_𝑗(𝛿)=0for𝑗=0,…,𝑛−2,𝑏²_𝑛−1(𝛿)−4𝑛𝑎_2𝑛−1(𝛿)<0 (3.42) if 𝐵_2𝑙+1(𝛿)<0(>0),𝐵_2𝑗−1(𝛿)=0,𝑗=1,…,𝑙 (3.43) for some [𝑛/2]≤𝑙≤𝑘. In this case the origin is a stable (unstable) focus of (3.23) by Theorem 1.10. If (3.43) holds for some 0≤𝑙<[𝑛/2], then by [23] again the origin is a stable (unstable) node of (3.23). Thus, we can proceed similarly as above. The proof is complete.

We remark that if 𝑔(−𝑥,𝛿)=−𝑔(𝑥,𝛿), then 𝛼(𝑥,𝛿)=−𝑥.

Proof of Theorem 1.12. Consider (1.43). Without loss of generality, we can assume 𝑋_𝑚+1=0 in (1.43) (otherwise, introduce the change of variables 𝑣=𝑦+𝑋_𝑚+1(𝑥,𝑦)). In this case, we can write (1.43) into the form: 𝑥̇=𝑦,𝑦̇=−𝑔(𝑥)−𝑓(𝑥)𝑦+𝑦²_𝑗≥0𝜑_𝑗(𝑥)𝑦^𝑗, (3.44) where 𝑔(𝑥)=𝑔_𝑚(𝑥)+𝑂|𝑥|^𝑚+1,𝑓(𝑥)=𝑓_𝑚−1(𝑥)+𝑂|𝑥|^𝑚,𝜑_𝑗(𝑥)=𝑂|𝑥|^{𝑚−1−𝑗}. (3.45)
For the sake of convenience we rewrite the functions 𝑔, 𝑓, and 𝜑_𝑗 as follows: 𝑔(𝑥)=𝑥^2𝑛−1𝑔₀(𝑥)+𝑥^𝑛𝑔₁(𝑥)+𝑥^2𝑛𝑔₂(𝑥),𝑓(𝑥)+⋯=𝑥^𝑛−1𝑓₀(𝑥)+𝑥^𝑛𝑓₁(𝑥)+𝑥^2𝑛𝑓₂(𝑥),𝜑+⋯_𝑗(𝑥)=𝜑_𝑗0(𝑥)+𝑥^𝑛−1𝜑_𝑗1(𝑥)+𝑥^2𝑛−1𝜑_𝑗2(𝑥)+⋯, (3.46) where 𝑓_𝑗, 𝑔_𝑗, and 𝜑_𝑗𝑙, 𝑗≥0, 𝑙≥1, are polynomials in 𝑥 of degree at most 𝑛−1, and 𝜑_𝑗0, 𝑗≥0, are polynomials in 𝑥 with degree at most 𝑛−2.
Now we change (3.44) using (2.2) to the system 𝑥̇=𝑟^𝑛sin𝜃,𝑦̇=𝑟^2𝑛−1_𝑗≥0𝑉_𝑗(𝜃,𝑟)𝑟^𝑗𝑛, (3.47) where by (3.46) 𝑉₀(𝜃,𝑟)=−cos^2𝑛−1𝜃𝑔₀(𝑟cos𝜃)−sin^𝑛𝜃𝑓₀(𝑟cos𝜃)+𝑟sin²𝜃𝜑₀₀(),𝑉𝑟cos𝜃_𝑗(𝜃,𝑟)=−cos^{2𝑛−1+𝑗𝑛}𝜃𝑔_𝑗(𝑟cos𝜃)−sin^𝑛𝜃cos^𝑗𝑛𝜃𝑓_𝑗(𝑟cos𝜃)+𝑟sin^2+𝑗𝜃𝜑_𝑗0()+𝑟cos𝜃_{𝑗−1𝑘=0}sin^2+𝑘𝜃cos^{(𝑗−𝑘)𝑛−1}𝜃𝜑_{𝑘,𝑗−𝑘}(𝑟cos𝜃),𝑗≥1. (3.48)
This yields the equation 𝑑𝑟∑𝑑𝜃=𝑟^𝑗≥0𝑅_𝑗(𝜃,𝑟)𝑟^𝑗𝑛∑_𝑗≥0𝑆_𝑗(𝜃,𝑟)𝑟^𝑗𝑛, (3.49) where 𝑆₀(𝜃,𝑟)=−𝑛sin²𝜃+cos𝜃𝑉₀(𝜃,𝑟),𝑅₀(𝜃,𝑟)=sin𝜃cos𝜃+sin𝜃𝑉₀(),𝑆𝜃,𝑟_𝑗(𝜃,𝑟)=cos𝜃𝑉_𝑗(𝜃,𝑟),𝑅_𝑗(𝜃,𝑟)=sin𝜃𝑉_𝑗(𝜃,𝑟),𝑗≥1. (3.50)
By (3.48) and (3.50) we can further expand 𝑆_𝑗 and 𝑅_𝑗 in 𝑟 to obtain for 𝑗≥0𝑆_𝑗(𝜃,𝑟)=^𝑛−1_𝑙=0𝑆_{𝑙+𝑗𝑛}(𝜃)𝑟^𝑙,𝑅_𝑗(𝜃,𝑟)=^𝑛−1_𝑙=0𝑅_{𝑙+𝑗𝑛}(𝜃)𝑟^𝑙, (3.51) so that the above differential equation can be written as 𝑑𝑟∑𝑑𝜃=𝑟^𝑗≥0𝑅_𝑗(𝜃)𝑟^𝑗∑_𝑗≥0𝑆_𝑗(𝜃)𝑟^𝑗. (3.52)
Further, letting 1∑_𝑗≥0𝑆_𝑗(𝜃)𝑟^𝑗=_𝑗≥0𝑆_𝑗(𝜃)𝑟^𝑗,𝑅_𝑗(𝜃)=_{𝑘+𝑙=𝑗}𝑅_𝑘(𝜃)𝑆_𝑙(𝜃),𝑗≥0 (3.53)(3.54) we obtain 𝑑𝑟𝑑𝜃=𝑟_𝑗≥0𝑅_𝑗(𝜃)𝑟^𝑗. (3.55)
Note that for any 𝑗≥0, 𝑆_𝑗 depends only on 𝑆_𝑘 with 0≤𝑘≤𝑗. Then by (3.54) one can see that, forany𝑗≥0,𝑅_𝑗dependsonlyon𝑅_𝑘and𝑆_𝑘with0≤𝑘≤𝑗. (3.56)
Let 𝑟(𝜃,𝑟₀) denote the solution of (3.55) with the initial value 𝑟₀. The for 𝑟₀ small we have 𝑟𝜃,𝑟₀=_𝑗≥1𝑟_𝑗(𝜃)𝑟^𝑗₀, (3.57) where 𝑟₁,𝑟₂,𝑟₃,… satisfy 𝑟₁(0)=1, 𝑟₂(0)=𝑟₃(0)=⋯=0, and 𝑟^′₁=𝑅₀𝑟₁,𝑟^′₂=𝑅₀𝑟₂+𝑅₁𝑟²₁,𝑟^′₃=𝑅₀𝑟₃+2𝑅₁𝑟₁𝑟₂+𝑅₂𝑟³₁,⋯ (3.58) which implies that, for any 𝑗≥1, the function 𝑟_𝑗 depends only on 𝑅_𝑘 with 0≤𝑘≤𝑗−1. Hence, by Lemma 2.3, (1.45), and (3.56) we come to the following conclusion. Forany𝑗≥1,𝑣_𝑗(𝛿)dependsonlyon𝑅_𝑘and𝑆_𝑘with0≤𝑘≤𝑗−1. (3.59)
Further, by (3.48)–(3.51), one can observe that, for 0≤𝑙≤𝑛−1, 𝑆_𝑙 and 𝑅_𝑙 depend only on the coefficients of degree 𝑙 of the polynomials 𝑔₀, 𝑓₀, and 𝑥𝜑₀₀ in 𝑥. Hence, by (3.59) we see that for 1≤𝑗≤𝑛, 𝑣_𝑗 depends only on the coefficients of degree at most 𝑗−1 of the polynomials 𝑔₀,𝑓₀, and 𝑥𝜑₀₀ in 𝑥.
Similarly, for 𝑗≥1 and 0≤𝑙≤𝑛−1, or 𝑗𝑛≤𝑙+𝑗𝑛≤(𝑗+1)𝑛−1, 𝑆_{𝑙+𝑗𝑛} and 𝑅_{𝑙+𝑗𝑛} depend only on the coefficients of degree 𝑙 of the polynomials 𝑔_𝑗, 𝑓_𝑗, 𝑥𝜑_𝑗0 and 𝜑_{𝑖,𝑗−𝑖} with 𝑖=0,…,𝑗−1 in 𝑥. In other words, for 𝑗𝑛+1≤𝑢≤(𝑗+1)𝑛, 𝑆_𝑢−1, and 𝑅_𝑢−1 depend only on the coefficients of degree 𝑢−1−𝑗𝑛 of the polynomials 𝑔_𝑗,𝑓_𝑗, 𝑥𝜑_𝑗0, and 𝜑_{𝑖,𝑗−𝑖} with 𝑖=0,…,𝑗−1 in 𝑥. Let 𝑁_[𝑎,𝑏] denote the set of integers in the interval [𝑎,𝑏]. Then, for 𝑗𝑛+1≤𝑢≤(𝑗+1)𝑛, we have 𝑁_[0,𝑢−1]=^𝑗−1_𝑖=0𝑁_{[𝑖𝑛,(𝑖+1)𝑛−1]}𝑁_{[𝑗𝑛,𝑢−1]}. (3.60)
Thus, for all 𝑘∈𝑁_{[𝑖𝑛,(𝑖+1)𝑛−1]}, 𝑆_𝑘 and 𝑅_𝑘 depend only on 𝑔_𝑖, 𝑓_𝑖, 𝑥𝜑_𝑖0, and 𝜑_{𝑙,𝑖−𝑙} with 𝑙=0,…,𝑖−1. And for 𝑘∈𝑁_{[𝑗𝑛,𝑢−1]}, 𝑆_𝑘 and 𝑅_𝑘 depend only on the coefficients of degree 𝑘−𝑗𝑛 of the polynomials 𝑔_𝑗, 𝑓_𝑗, 𝑥𝜑_𝑗0, and 𝜑_{𝑙,𝑗−𝑙} with 𝑙=0,…,𝑗−1 in 𝑥.
Therefore, by (3.59) for 𝑗𝑛+1≤𝑢≤(𝑗+1)𝑛, 𝑣_𝑢(𝛿) depends only on the functions 𝑔_𝑖, 𝑓_𝑖, 𝑥𝜑_𝑖0 and 𝜑_{𝑙,𝑖−𝑙} with 𝑙=0,…,𝑖−1, 𝑖=0,…,𝑗−1 and the coefficients of degree at most 𝑢−1−𝑗𝑛 of the polynomials 𝑔_𝑗, 𝑓_𝑗, 𝑥𝜑_𝑗0, and 𝜑_{𝑙,𝑗−𝑙} with 𝑙=0,…,𝑗−1 in 𝑥.
We claim that if 𝑗≥0, 𝑚≥(𝑗+1)𝑛, then, for 𝑗𝑛+1≤𝑢≤(𝑗+1)𝑛, 𝑣_𝑢(𝛿) depends only on the functions 𝑔_𝑖, 𝑓_𝑖, with 𝑖=0,…,𝑗−1 and the coefficients of degree at most 𝑢−1−𝑗𝑛 of the polynomials 𝑔_𝑗, 𝑓_𝑗 in 𝑥.
In fact, by the above discussion, we need only to prove 𝜑₀₀=0 in the case 𝑗=0 and 𝜑_𝑙𝑠=0 for 𝑙+𝑠≤𝑗 and 0≤𝑙≤𝑗−1 in the case 𝑗>0. This can be shown easily since 𝜑_𝑗0=𝑂|𝑥|^{𝑚−1−𝑗},𝜑_𝑗𝑠=𝑂|𝑥|^{𝑚−𝑗−𝑠𝑛}for𝑠≥1,deg𝜑_𝑗0≤𝑛−2,deg𝜑_𝑗𝑠≤𝑛−1for𝑠≥1 (3.61) by (3.45) and (3.46).
By (3.46), the above claim can be restated as the claim that if 𝑗≥0, 𝑚≥(𝑗+1)𝑛, then for 𝑗𝑛+1≤𝑢≤(𝑗+1)𝑛, 𝑣_𝑢(𝛿) depends only on the coefficients of degree at most 2𝑛+𝑢−2 of 𝑔 and the coefficients of degree at most 𝑛+𝑢−2 of 𝑓 in 𝑥. Thus, for any integers 𝑘 and 𝑚 satisfying 𝑘≥1 and 𝑚≥(𝑘+1)𝑛, taking 𝑗=0,…,𝑘 we see that, for all 1≤𝑢≤(𝑘+1)𝑛, 𝑣_𝑢(𝛿) depends only on the coefficients of degree at most 2𝑛+𝑢−2 of 𝑔 and the coefficients of degree at most 𝑛+𝑢−2 of 𝑓 in 𝑥.
Finally, by (3.45), if 𝑚≥(𝑘+3)𝑛−2, then 2𝑛+𝑢−2≤𝑚,𝑛+𝑢−2≤𝑚−1for𝑢≤(𝑘+1)𝑛. (3.62)
In this case, for all 1≤𝑢≤(𝑘+1)𝑛, 𝑣_𝑢(𝛿) depends only on 𝑔_𝑚 and 𝑓_𝑚−1 in (3.45). Then the conclusion of Theorem 1.12 follows.

Proof of Theorem 1.14. Let 𝑋𝑥,𝑦,𝛿=_{2≤𝑖+𝑗≤𝑘}𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗,𝑌𝑥,𝑦,𝛿=_{2≤𝑖+𝑗≤𝑘}𝑏_𝑖𝑗𝑥^𝑖𝑦^𝑗. (3.63)
Then (1.49) is equivalent to an analytic system of the form (2.1), where the functions 𝑓 and 𝑔 are given by (1.22). We can write 𝑔(𝑥,𝛿)=_𝑗≥2𝑎_𝑗𝑥^𝑗, (3.64) where all 𝑎_𝑗 are polynomials in 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗. Thus, by the well-known Hilbert basis theorem there exists 𝑁_𝑘>0 such that 𝑎_𝑗=0 for all 𝑗≥2𝑁_𝑘 if 𝑎₂=⋯=𝑎_{2𝑁𝑘−1}=0. That is, 𝑎₂=⋯=𝑎_{2𝑁𝑘−1}=0 imply that 𝑔=0. In this case, all points on the line 𝑦=0 and near the origin are singular for (2.1). Thus, the origin has a multiplicity at most 𝑁_𝑘 as a nilpotent focus or center.
Further, for any 2≤𝑛≤𝑁_𝑘 let (1.49) have a nilpotent focus or center at the origin, having multiplicity 𝑛. Then for any sufficiently large 𝑚, (1.49) is equivalent to a system of the form (1.43), where the coefficients 𝑎_𝑗 in 𝑔_𝑚 and 𝑏_𝑗 in 𝑓_𝑚−1 are all polynomials in 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗. Moreover, 𝑎_2𝑛−1>0 and 𝑏²_𝑛−1−4𝑛𝑎_2𝑛−1<0. Let 𝐹(𝑥)=^𝑥₀𝑓_𝑚−1𝑑𝑥,𝐺(𝑥)=^𝑥₀𝑔_𝑚𝑑𝑥, (3.65) and 𝛼(𝑥)=−𝑥+∑_𝑗≥2𝛼_𝑗𝑥^𝑗 satisfies 𝐺(𝛼)=𝐺(𝑥). Then, noting that 𝐺^1/2𝑛=𝑎_2𝑛−1(2𝑛)^1/2𝑛|𝑥|1+_𝑗≥1𝛽_𝑗𝑥^𝑗, (3.66) where all 𝛽_𝑗 are polynomials in 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, and 𝑐≡𝑎⁻¹_2𝑛−1, one can see that all 𝛼_𝑗 are polynomials in 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, and 𝑐. Therefore, if we let 𝐹(𝛼(𝑥))−𝐹(𝑥)=_𝑗≥𝑛𝐵^𝑚_𝑗𝑥^𝑗=_{𝑗≥2𝑙+1}𝐵^𝑚_𝑗𝑥^𝑗, (3.67) where 𝑙=[𝑛/2], then all 𝐵^𝑚_𝑗 are polynomials in 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, and 𝑐. By Theorems 1.10 and 1.12, for any integer 𝑘₀>0 there exists an integer 𝑚₀>0 such that 𝐵^𝑚_𝑗, 𝑗=2𝑙+1,…,𝑘₀ are independent of 𝑚 for all 𝑚≥𝑚₀, denoting 𝐵^𝑚_𝑗 by 𝐵_𝑗. Then, we have defined a series of coefficients 𝐵_𝑗 for all 𝑗≥2𝑙+1, which are all polynomials in 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, and 𝑐. Again, by the Hilbert basis theorem there exists 𝐾_𝑛(𝑘)>0 such that 𝐵_2𝑙+1=𝐵_2𝑙+2=𝐵_2𝑙+3=⋯=𝐵_{2𝑙+2𝐾𝑛(𝑘)+1}=0⇒𝐵_2𝑙+𝑗+1=0,𝑗≥2𝐾_𝑛(𝑘)+1. (3.68)
Now, we take 𝐾 and 𝑚 such that 2𝐾_𝑛(𝑘)+1+𝑝_𝑛≤𝐾𝑛, 𝑚>(𝐾+2𝑛)−2. Then by Theorems 1.10 and 1.12 again, for (1.49) we have 𝑣_1+𝑝𝑛=𝑣_3+𝑝𝑛=⋯=𝑣_{2𝐾𝑛(𝑘)+1+𝑝𝑛}=0 (3.69) yielding 𝑣_{𝑗+𝑝𝑛}=0, 𝑗≥1. Then the conclusion follows.

4. Application Examples

In this section we provide some application examples based on the ones given in [11]. Consider a Kukles type system of the form 𝑥̇=𝑦,𝑦̇=−𝑎₁₁𝑥𝑦+𝑎₀₂𝑦²+𝑎₃₀𝑥³+𝑎₂₁𝑥²𝑦+𝑎₁₂𝑥𝑦²+𝑎₀₃𝑦³. (4.1)

The authors of [11] proved that if 𝑎₃₀>0 and 𝑎²₁₁−8𝑎₃₀<0, then for (4.1) 𝑣₂=𝑣₄=𝑣₆=𝑣₈=0 if and only if 𝑎₂₁=𝑎₀₃=𝑎₁₁𝑎₀₂=0, which implies that the origin is a center. Moreover, there can be 3 limit cycles near the origin. See Theorem 4.1 in [11] and its proof.

From this conclusion and Theorem 1.7 we have immediately the following.

Proposition 4.1. Let 𝑎₁₁, 𝑎₀₂, 𝑎₃₀, 𝑎₂₁, 𝑎₁₂, and 𝑎₀₃ be bounded parameters satisfying 𝑎₃₀>0,𝑎²₁₁−8𝑎₃₀<0,𝑎₂₁+𝑎₀₃+𝑎₁₁𝑎₀₂>0. (4.2)
Then there exists a neighborhood 𝑉 of the origin such that the system (4.1) has at most 3 limit cycles in 𝑉.

Consider now the system 𝑥̇=−𝑦+𝐴𝑥²+𝐵𝑥𝑦+𝐶𝑦²,𝑦̇=𝑥³+𝑥𝑦²+𝑦³. (4.3)

By Theorem 4.2 in [11] and its proof if 𝐴²<2, then the origin of (4.3) is always a focus with |𝑣₂|+|𝑣₄|+|𝑣₆|+|𝑣₈|>0. Moreover, there are systems inside (4.3) with at least 3 limit cycles around the origin. Then by Theorem 1.7 again we have the following.

Proposition 4.2. Let 𝐴,𝐵, and 𝐶 be bounded parameters with 𝐴²<2. Then there exists a neighborhood 𝑉 of the origin such that the system (4.3) has at most 3 limit cycles in 𝑉.

Finally, consider the system 𝑥̇=𝑦,𝑦̇=−𝑥³+𝑥⁵−^𝑘_𝑗=0𝑏_2𝑗𝑥^2𝑗𝑦, (4.4) where 𝑘≥2. By Theorems 1.8–1.11, we obtain the following.

Proposition 4.3. Let 𝑏_2𝑗 be bounded parameters. Then(1)if 𝑏₀=0, the system (4.4) has at most 𝑘−1 limit cycles near the origin, and 𝑘−1 limit cycles can appear.(2)If 𝑏₀≠0, there are systems inside (4.4) which have at least 𝑘 limit cycles near the origin.

Acknowledgments

The project supported by the National Natural Science Foundation of China (10971139), the Slovenian Research Agency, and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Programme, FP7-PEOPLE-2012-IRSES-316338. The authors thank the referees for their valuable remarks and helpful suggestions.