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

Authors: Maoan Han; Valery G. Romanovski
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction and Main Result

Consider an analytic system of the form:(1.1)x˙=y+αx+X(x,y,δ),y˙=-x+αy+Y(x,y,δ),
where α∈R, δ∈Rn, 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:
(1.2)P(r,α,δ)=r+∑j≥1vj(α,δ)rj,
where the series converges for small r. A well-known fact is that for any k≥1v2j-1(α,δ)=0 for all j=1,…,k imply that v2k(α,δ)=0; that is, only odd values of the expansion are important for determining the behavior of trajectories near the origin. The value v2k+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:
(1.3)P(r,α,δ)=r+∑j=14v2j-1(α,δ)r2j-1(1+Pj(r,α,δ)),
where Pj(r,α,δ)=O(r)∈Cω. 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:(1.4)x˙=y+X(x,y),y˙=Y(x,y),
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
(1.5)Y(x,F(x))=ax2n-1+O(x2n),a≠0,∂X∂x(x,F(x))+∂Y∂y(x,F(x))=bxn-1+O(xn),
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:(1.6)x=rCs(θ),y=rnSn(θ),
where (Cs(t),Sn(t)) is the solution of the initial problem
(1.7)x˙=y,y˙=-x2n-1,(x(0),y(0))=(1,0),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:
(1.8)x˙=y,y˙=-g(x)-yf(x),
where g(x)=axm+O(xm+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
(1.9)g(x)=x2n-1(a2n-1+O(x)),n≥2,a2n-1>0.According to [11] under (1.9) by a change of variables x and t of the form:
(1.10)u=[2n∫0xg(x)dx]1/2n(sgnx)≡u(x),dtdt1=u2n-1(x)g(x)
system (1.8) is transformed into
(1.11)x˙=y,y˙=-x2n-1-yf-(x),
where
(1.12)f-(x)=x2n-1f(u-1(x))g(u-1(x)),n≥2,u(x)=[2n∫0x‍g(x)dx]1/2n(sgnx)=(a2n-1)1/2n(x+O(x2)).Then, by Theorem1.1 system (1.11) has a center or a focus at the origin if and only if the function f- given in (1.12) satisfies
(1.13)f-(x)=∑j≥n-1bjxj,bn-12-4n<0.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
(1.14)b2l>0(<0),b2j=0forj<l,
and it has a center at the origin if b2j=0 for all 2j≥n-1.Passing to the generalized polar coordinates(x,y)=(rCs(θ),rnSn(θ)) we obtain from (1.11) the following equation:
(1.15)drdθ=∑j≥n-1bj(Sn(θ))2(Cs(θ))jr2-n+j1+∑j≥n-1bjSn(θ)(Cs(θ))j+1r1-n+j.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:
(1.16)r(T,r0)=∑j≥1Vjr0j.Assuming thatV1=1,V2=⋯=Vk-1=0 Álvarez and Gasull [11] called the constant Vk 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(-2bn-1π/n4n-bn-12) if 2l=n-1;(2)
V1=1, Vj=0 for 1<j<2-n+2l, and V2-n+2l=-Klb2l if either bn-1=0 or bn-1≠0 and n is even, where Kl is a positive constant.In the casen=2 Liu and Li [12] introduced different generalized polar coordinates of the form x=rcosθ, y=r2sinθ to change (1.4) into the form:
(1.17)drdt=R(θ,r),dθdt=Q(θ,r),
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
(1.18)drdθ=R(θ,r)Q(θ,r)
satisfying r~(0)=h. Note that the initial value problem is well defined also for negative h.LetD be a simply connected domain. Denote by R the ring of analytic functions on D and by 〈ϕ1,…,ϕk〉 the ideal in R generated by the functions ϕ1,…,ϕk 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≥2vkhk, where
(1.19)v2k+1∈〈v2,v4,…,v2k〉,k≥1.In particular,v2j=0,  j=1,…,k imply that v2k+1=0.(3)
The origin is a stable (unstable) focus if(1.20)v2k<0(>0),v2j=0forj<k.
In the latter case the origin is called akth order weak focus of (1.4).We remark that the conclusions of Theorem1.3 provide new and useful information on the property of the coefficients vk. Liu and Li [12] also gave some new methods to compute the focal values v2,v4,…,v2k, 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(1.21)x˙=y+X(x,y,δ),y˙=Y(x,y,δ),
where δ=(δ1,…,δm)∈D⊂ℝm, 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 (Theorems1.5 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. Lety=F(x,δ) be the solution to the equation y+X(x,y,δ)=0. We define the following two functions:
(1.22)g(x,δ)=-Y(x,F(x,δ),δ),f(x,δ)=-[∂X∂x(x,F(x,δ),δ)+∂Y∂y(x,F(x,δ),δ)].By Theorem1.1, if
(1.23)g(x,δ)=∑j≥2n-1aj(δ)xj,n≥2,a2n-1(δ)>0,(1.24)f(x,δ)=∑j≥n-1bj(δ)xj,bn-12(δ)-4na2n-1(δ)<0,
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 allδ∈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 allδ∈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
(1.25)P(x0,δ)={x(τ,x0,δ),0<|x0|<ε0,0,x0=0.The mapP(x0,δ) is the Poincaré map we will use for the remainder of the paper.Obviously, the functionP(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
(1.26)d-(x0,δ)=P-(x0,δ)-x0=∑j≥1vj(δ)x0j
for |x0| sufficiently small, where(1)
ifn is odd, then P(x0,δ)=P-(x0,δ) for all |x0| small;(2)
ifn is even, then for all |x0| small
(1.27)P(x0,δ)={P-(x0,δ)forx0>0,P--1(x0,δ)forx0<0,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 functionP(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 coefficientsvj 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)
forn odd we have v2k∈〈v1,v3,…,v2k-1〉,k≥1;(2)
forn even we have v1=0, v2k+1∈〈v2,v4,…,v2k〉,k≥1.Definepn=[1+(-1)n]/2. Then the conclusions of the above theorem can be written uniformly as
(1.28)v2k+pn∈〈v1+pn,v3+pn,…,v2k-1+pn〉,k≥1.From the proof of Theorem1.6 we will see that v2k+pn depends on v1+pn,v3+pn,…,v2k-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 pn=[1+(-1)n]/2.(1)
If there is an integerk≥1 such that
(1.29)∑j=1k+1|v2j-1+pn(δ)|>0,∀δ∈D,then there exists a neighborhoodU 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 isδ0∈D such that v2k+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,
(1.30)v2j-1+pn(δ0)=0,j=1,…,k,rank∂(v1+pn,v3+pn,…,v2k-1+pn)∂(δ1,δ2,…,δm)(δ0)=k,then for an arbitrary sufficiently small neighborhood of the origin there are someδ∈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 existδ0∈D and an integer k≥1 such that (1.30) is satisfied,(ii)
the origin is a center of (1.21) if v2j-1+pn(δ0)=0, j=1,…,k,then there exists a neighborhoodU 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 callv2k+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 v2k+1+pn(δ)≠0 and v2j-1+pn(δ)=0 for j=1,…,k.
The above definition is very reasonable and natural, since by Theorem1.7, 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 Theorem1.8 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
(1.31)x˙=y-(a1x3-a22x5+a2x7),y˙=-x3
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 valuesv1+pn,v3+pn,…,v2k+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:
(1.32)x˙=y,y˙=-g(x,δ)-yf(x,δ).
Note thatf and g in (1.32) may be different from the ones given by (1.22). As before, let δ∈D⊂ℝm where D is a domain. Also, suppose that for small |x| the function g(x,δ) satisfies (1.23). Define
(1.33)F(x,δ)=∫0xf(x,δ)dx,G(x,δ)=∫0xg(x,δ)dx.
It is easy to see that forxy<0 the equation G(x,δ)=G(y,δ) defines a unique analytic function y=α(x,δ)=-x+O(x2). Set
(1.34)F(α(x,δ),δ)-F(x,δ)=∑j≥1Bj(δ)xj.
By Theorem1.1, 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:
(1.35)P(x0,δ)-x0=∑j≥0v2j+1+pn(δ)x02j+1+pn(1+Pj*(x0,δ)),
where Pj*(x0,δ)=O(x0),
(1.36)v1+pn(δ)=Kl*B2l+1(δ)+(1-pn)O(B2l+12),v2j+1+pn(δ)=Kl+j*B2l+2j+1(δ)+B~2l+2j+1(δ),j≥1,l=[n/2], Kl+j*, j≥0 are positive constants and B~2l+2j+1∈〈B2l+1,B2l+3,…,B2l+2j-1〉. Thus, Theorems 1.7 and 1.8 hold if v2j+1+pn is replaced by B2l+2j+1, j≥0.Let(1.37)f(x,δ)=∑j≥0bj(δ)xj.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
(1.38)B2k+1(δ0)<0(>0),B2j-1(δ0)=0,j=1,…,k.
Let one of the following conditions be satisfied:(a)
n=2, and
(1.39)b0(δ0)=0,b12(δ0)-8a3(δ0)<0;(b)
n>2, g(-x,δ)=-g(x,δ), f(-x,δ)=f(x,δ), and
(1.40)bj(δ0)=0forj=0,…,n-2,bn-12(δ0)-4na2n-1(δ0)<0.
Then(1)
forδ=δ0 (1.32) has a stable (unstable) focus at the origin.(2)
If further(1.41)rank∂(B1,B3,…,B2k-1)∂(δ1,δ2,…,δm)(δ0)=k,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 Theorems1.5–1.10, 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:
(1.42)(xy)=(uv)+Hm(u,v,δ),
where Hm(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)
(1.43)x˙=y+Xm+1(x,y,δ),y˙=-gm(x,δ)-yfm-1(x,δ)+Ym+1(x,y,δ),
where
(1.44)gm(x,δ)=∑j=2n-1maj(δ)xj,fm-1(x,δ)=∑j=n-1m-1bj(δ)xj
with a2n-1(δ)>0 and bn-12(δ)-4na2n-1(δ)<0, and Xm+1(x,y,δ), Ym+1(x,y,δ) being analytic functions satisfying Xm+1,Ym+1=O(|x,y|m+1). Here, we should mention that the functions gm and fm-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
(1.45)P(x0,δ)-x0=∑j≥1vj(δ)x0j,
where the series converge for small |x0|.Truncating the series (1.43) at terms of order m we obtain the polynomial system
(1.46)x˙=y,y˙=-gm(x,δ)-yfm-1(x,δ).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 Pm(x0,δ) denote the Poincaré map of (1.46). We write the expansion of the displacement function as
(1.47)Pm(x0,δ)-x0=∑j≥1v-j(δ)x0j,
where the series converge for small |x0|.We intend to usev-j(δ) instead of vj(δ). To this end, we have to solve the following problem. For any given k>1 find m>2n-1 such that vj(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
(1.48)vj(δ)=v-j(δ)for1≤j≤kn.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,…,vkn 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 degreek(1.49)x˙=y+∑2≤i+j≤kaijxiyj,y˙=∑2≤i+j≤kbijxiyj.Theorem 1.14.
For anyk≥3 there is an integer Nk>0 such that an arbitrary polynomial system of the form (1.49) has a singular point of multiplicity at most Nk at the origin (i.e., one must have 2≤n≤Nk if (1.49) satisfies (1.23) and (1.24), see Definition 1.4). Further, for each 2≤n≤Nk, if (1.49) satisfies (1.23) and (1.24), then there exists an integer Kn(k)>0 such that for (1.49) the origin is a focus of order at most Kn(k). Hence, the origin as a nilpotent focus of (1.49) generates at most Kn(k) limit cycles.We organize the paper as follows. In Section2 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)
(2.1)x˙=y(1+Z1(x,y,δ)),y˙=-g(x,δ)-yf(x,δ)+y2Z2(x,y,δ),
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
(2.2)x=rcosθ,y=rnsinθ,r>0.Lemma 2.1.
Let (1.23) and (1.24) be satisfied. Then the substitution (2.2) transforms (2.1) into the system
(2.3)θ˙=S(θ,r)=rn-1H(θ)[S0(θ)+O(r)],r˙=R(θ,r)=rnH(θ)[R0(θ)+O(r)],
where S and R are 2π-periodic in θ and have the properties
(2.4)S(π+(-1)n-1θ,-r)=(-1)n-1S(θ,r),R(π+(-1)n-1θ,-r)=-R(θ,r),(2.5)H(θ)=cos2θ+nsin2θ>0,S0(θ)=-[nsin2θ+bn-1cosnθsinθ+a2n-1cos2nθ]<0,R0(θ)=cosθsinθ(1-a2n-1cos2n-2θ-bn-1sinθcosn-2θ).Proof.
From (2.2) we have
(2.6)x˙=cosθr˙-rsinθθ˙,y˙=nrn-1sinθr˙+rncosθθ˙.
We solve the above equations forθ˙ and r˙ and obtain (2.3) with
(2.7)S(θ,r)=cosθy˙-nrn-1sinθx˙rn(cos2θ+nsin2θ),R(θ,r)=sinθy˙+rn-1cosθx˙rn-1(cos2θ+nsin2θ).
Then noting that(2.8)cos(π±θ)=-cosθ,sin(π±θ)=∓sinθ
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
(2.9)drdθ=R-(θ,r),
where
(2.10)R-(θ,r)=rsinθy˙+rn-1cosθx˙cosθy˙-nrn-1sinθx˙=r[R0(θ)S0(θ)+O(r)],R-(π+(-1)n-1θ,-r)=(-1)nR(θ,r).Letr(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 solutionr(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.
Letr~(θ)=-r(π+(-1)n-1θ,h). Then by (2.9) and (2.10) we have
(2.11)dr~dθ=(-1)nR-(π+(-1)n-1θ,r(π+(-1)n-1θ,h))=(-1)nR-(π+(-1)n-1θ,-r~(θ))=R-(θ,r~(θ)).
This means thatr~(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.
LetP(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 θ˙<0 for small r>0, by the definition of P and (2.2), we see that
(2.12)P(x0,δ)=x(τ,x0)=r(-2π,x0)
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
(2.13)P(x0,δ)=x(τ,x0)=-r*(-π,-x0),
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 Lemma2.2(1), we have
(2.14)r*(θ,-h)=-r(π-θ,h)forneven,(2.15)r*(θ,-r(2π,h))=-r(π+θ,h)fornodd.
Noting that, by Lemma2.2(2), x0=r(2π,h) if and only if h=r(-2π,x0), we see that (2.15) becomes
(2.16)r*(θ,-x0)=-r(π+θ,r(-2π,x0))fornodd.
Therefore, by (2.14) and (2.16) we have for x0<0(2.17)P(x0,δ)={r(2π,x0)forneven,r(-2π,x0)fornodd.
This ends the proof.Lemma 2.4.
Letd(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
(2.18)d(x~0,δ)=-K(x0,δ)d(x0,δ)
for x0>0 small, where x~0=-r(π,x0).Proof.
By Lemma2.2, we have
(2.19)r((-1)n2π,x~0)=-r(-π,x0)=-r(π,r(-2π,x0)).
Hence, by Lemma2.3 for x0>0(2.20)d(x~0,δ)=r((-1)n2π,x~0)-x~0=-r(π,r(-2π,x0))+r(π,x0)=-K(x0,δ)[r(-2π,x0)-x0]=-K(x0,δ)d(x0,δ),
where
(2.21)K(x0,δ)=∫01∂r∂x0(π,x0+s(r(-2π,x0)-x0))ds.
It is obvious thatK 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 Theorems1.5–1.12.Proof of Theorem1.5.
We takeP-(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 Theorem1.6.
There are two cases to consider separately.
Case A (n odd). By (1.26) and Theorem 1.5(1), we have
(3.1)d(x0,δ)=d-(x0,δ)=∑j≥1vj(δ)x0j
for all |x0| small.
By Lemma2.4, we can suppose that
(3.2)K(x0,δ)=∑j≥0kjx0j,x~0=-r(π,x0)=∑j≥1ljx0j,
where k0>0, l1=-k0. Substituting (3.1) and (3.2) into (2.18), we obtain
(3.3)∑j≥1vj(∑i≥1lix0i)j=-∑i≥0,j≥1kivjx0i+j.
Comparing the coefficients of the termsx02, x04 and x02j on both sides yields
(3.4)v2l12+v1l2=-(k0v2+k1v1),v4l14+3v3l12l2+v2(l22+2l1l3)+v1l4=-(k0v4+k1v3+k2v2+k3v1),⋯v2jl12j+v2j-1L1,j(l1,l2)+⋯+v2L2j-2,j(l1,l2,…,l2j-1)+v1l2j=-∑i=02j-1kiv2j-i,⋯,
where Li,j(l1,l2,…,li+1), i=1,2,…,2j-2, are polynomials. Thus, from the above equations we obtain
(3.5)v2k∈〈v1,v3,…,v2k-1〉,k≥1.
Case B (n even). By (1.26) and x0=P--1(P-(x0,δ),δ) we find that
(3.6)P--1(x0,δ)=v~1x0+v~2x02+v~3x03+⋯,
where
(3.7)v~1=(v1+1)-1,v~2=-v2(v1+1)-3,⋯v~j=-vj(v1+1)-(j+1)+Lj(v2,v3,…,vj-1),⋯
and each Lj 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 x~0<0 by Theorem 1.5 again
(3.8)d(x~0,δ)=P(x~0,δ)-x~0=P--1(x~0,δ)-x~0.
Then, inserting (3.1), (3.2), (3.6), and (3.8) into (2.18) we obtain
(3.9)(v~1-1)l1=-k0v1,(v~1-1)l2+v~2l12=-(k0v2+k1v1),⋯(v~1-1)lj+Lj(v~2,v~3,…,v~j-1)+v~jl1j=-(k0vj+k1vj-1+⋯+kj-1v1),⋯,
where
(3.10)Lj(v~2,v~3,…,v~j-1)∈〈v~2,v~3,…,v~j-1〉.
Finally, noting thatl1=-k0 and substituting (3.7) into (3.9) we easily see that
(3.11)v2j+1∈〈v2,v4,…,v2j〉,j≥1.
This ends the proof.Proof of Theorem1.7.
For the first part, suppose that the conclusion is not true. Then there exists a sequence{δm} in D- such that for δ=δm (1.21) has k+1 limit cycles Lm,1,Lm,2,…,Lm,k+1 which approach the origin as m→∞. Then by Theorem 1.6, the function d-(x0,δm) has 2k+2 nonzero roots in x0 which approach zero as m→∞.
SinceD- is compact, we can assume δm→δ0∈D- as m→∞. By our assumption, ∑j=1k+1|v2j-1+pn(δ0)|>0. Thus, for some 1≤l≤k+1,
(3.12)v2l-1+pn(δ0)≠0,v2j-1+pn(δ0)=0for1≤j≤l-1.
Therefore, by (1.26) and Theorem 1.6, we have
(3.13)d-(x0,δ0)=v2l-1+pn(δ0)x02l-1+pn+O(x02l+pn).
Note thatd-(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+pn nonzero roots in (-ε0,ε0) for all |δ-δ0|<ε0. We have proved that the function d-(x0,δm) has 2k+2 nonzero roots which approach zero as m→∞. It then follows that 2k+2≤2l-2+pn, contradicting to 2l-2+pn≤2k+pn≤2k+1. The first conclusion follows.
For the second one, from the above proof one can see that for allδ∈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
(3.14)d-(x0,δ)=∑j≥1v2j-1+pn(δ)x02j-1+pn(1+Pj(x0,δ)),
where Pj(0,δ)=0. Like in [19] one can show that Pj are series convergent in a neighborhood of δ0 (see also, e.g., [20, 21]). Further, by (1.30), we can take v1+pn,v3+pn,…,v2k-1+pn as free parameters, varying near zero. Precisely, if we change them such that
(3.15)0<|v1+pn|≪|v3+pn|≪⋯≪|v2k-1+pn|≪1,v1+pnv3+pn<0,…,v2k-1+pnv2k+1+pn<0,
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 Theorem1.5 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
(3.16)v2k+1+pn(δ)<0(>0),v2j-1+pn(δ)=0forj=1,…,k,
then the origin is a stable (unstable) focus of order k of (1.21). If
(3.17)v2j-1+pn(δ)=0∀j≥1,
the origin is a center of (1.21).Proof of Theorem1.8.
Under (1.30) the values v1+pn,v3+pn,…,v2k-1+pn can be taken as free parameters. Further, by our assumption, the origin is a center of (1.21) as v2j-1+pn(δ)=0, j=1,…,k. It then follows that
(3.18)v2j-1+pn(δ)∈〈v1+pn,v3+pn,…,v2k-1+pn〉∀j≥k+1.
Therefore, (3.14) can be further written in the form:
(3.19)d-(x0,δ)=∑j=1kv2j-1+pn(δ)x02j-1+pn(1+P-j(x0,δ)),
where P-j(0,δ)=0 and Pj 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 Theorem1.10.
Now we consider (1.32), where g satisfies (1.23). Let
(3.20)F(x,δ)=∫0xf(x,δ)dx,G(x,δ)=∫0xg(x,δ)dx.
Iff satisfies (1.24), then the origin is a center or focus of (1.32), and
(3.21)F(α(x,δ),δ)-F(x,δ)=∑j≥nBj(δ)xj=∑j≥n1Bj(δ)xj,
where
(3.22)Bn=(-1)n-1nbn-1,n1=2l+1,l=[n2],
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:
(3.23)x˙=y-F(x,δ),y˙=-g(x,δ)
which has the same Poincaré return map P(x0,δ) as (1.32). Introducing the change of variables x and t(3.24)u=[2nG(x,δ)]1/2n(sgnx)=(a2n-1)1/2n(x+O(x2))≡φ(x),dtdt1=φ2n-1(x)g(x,δ)
system (3.23) becomes
(3.25)u˙=y-F-(u,δ),y˙=-u2n-1,
which is equivalent to
(3.26)u˙=y,y˙=-u2n-1-yf-(u,δ),
where
(3.27)F-(u,δ)=F(φ-1(u),δ),f-(u,δ)=∂F-∂u(u,δ).
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,
(3.28)P(x0,δ)-x0=K(u0)(P1(u0,δ)-u0),
where K(u0)=(a2n-1)-1/2n+O(u0) is analytic. By (1.26) and (3.14), for u0>0 small we have
(3.29)P1(u0,δ)-u0=∑j≥1vj(δ)u0j=∑j≥1v2j-1+pn(δ)u02j-1+pn(1+Pj(u0,δ)).
Hence,(3.30)P(x0,δ)-x0=∑j≥1v2j-1+pn(δ)(a2n-1)-1/2nu02j-1+pn(1+P~j(u0,δ))=∑j≥1v2j-1+pn(δ)(a2n-1)(2j-2+pn)/2nx02j-1+pn(1+Pj*(x0,δ)),
where P~j(u0,δ)=O(u0), Pj*(x0,δ)=O(x0).
Sinceα 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
(3.31)F(α(x,δ),δ)-F(x,δ)=F(φ-1(-u),δ)-F(φ-1(u),δ)=F-(-u,δ)-F-(u,δ).
Let(3.32)f-(u,δ)=∑j≥n-1b-j(δ)uj.
Then(3.33)F-(u,δ)=∑j≥nb-j-1(δ)juj.
Thus, by (3.31) we have
(3.34)F(α(x,δ),δ)-F(x,δ)=-2∑j≥[n/2]b-2j(δ)2j+1u2j+1.
Substitutingu=φ(x)=(a2n-1)1/2n(x+O(x2)) into the above equality and comparing with (3.21) we obtain
(3.35)B2l+1=-K-lb-2l,B2l+2=O(b-2l),B2l+2j+1=-K-l+jb-2l+2j+B~2l+2j+1,B2l+2j+2∈〈b-2l,b-2l+2,…,b-2l+2j〉,j≥1,
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 Theorem1.5 for u0>0 small we clearly have
(3.36)P1(u0,δ)=u0+∑j≥1vj(δ)u0j=∑j≥1Vj(δ)u0j,
where Vj are introduced before Theorem 1.2. Thus, by Theorem 1.2, we have
(3.37)v1=-Klb-2l+O(b-2l2),v2j+1|v1=⋯=v2j-1=0=-Kl+jb-2l+2j,j≥1
for n=2l+1 odd, and
(3.38)v2=-Klb-2l,v2j+2|v2=⋯=v2j=0=-Kl+jb-2l+2j,j≥1
for n=2l even, where Kl+j, j≥0 are positive constants. Hence,
(3.39)v1+pn=-Klb-2l+(1-pn)O(b-2l2),v2j+1+pn=-Kl+jb-2l+2j+φ-(b-2l,b-2l+2,…,b-2l+2j-2),j≥1,
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
(3.40)v1+pn=KlK-lB2l+1+(1-pn)O(B2l+12),v2j+1+pn=Kl+jK-l+jB2l+2j+1+B~2l+2j+1,j≥1,
where B~2l+2j+1∈〈B2l+1,B2l+3,…,B2l+2j-1〉.
Then (1.36) follows from (3.30) and (3.40). This finishes the proof.Proof of Theorem1.11.
Let|δ-δ0| be small. For n=2 we have pn=1. Then the first conclusion follows directly from Corollary 3.1 and (1.36)–(1.39). In fact, we have by Theorem 1.10(3.41)v2k(δ0)=Kk*B2k+1(δ0),Kk*>0,v2j(δ0)=0forj=1,…,k-1.
For the second conclusion, we first keepB1(δ)=0 and vary B3(δ),…,B2k-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.
Forn>2 since g(-x,δ)=-g(x,δ), f(-x,δ)=f(x,δ) we have
(3.42)bj(δ)=0forj=0,…,n-2,bn-12(δ)-4na2n-1(δ)<0
if
(3.43)B2l+1(δ)<0(>0),B2j-1(δ)=0,j=1,…,l
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 ifg(-x,δ)=-g(x,δ), then α(x,δ)=-x.Proof of Theorem1.12.
Consider (1.43). Without loss of generality, we can assume Xm+1=0 in (1.43) (otherwise, introduce the change of variables v=y+Xm+1(x,y)). In this case, we can write (1.43) into the form:
(3.44)x˙=y,y˙=-g(x)-f(x)y+y2∑j≥0φj(x)yj,
where
(3.45)g(x)=gm(x)+O(|x|m+1),f(x)=fm-1(x)+O(|x|m),φj(x)=O(|x|m-1-j).
For the sake of convenience we rewrite the functionsg, f, and φj as follows:
(3.46)g(x)=x2n-1[g0(x)+xng1(x)+x2ng2(x)+⋯],f(x)=xn-1[f0(x)+xnf1(x)+x2nf2(x)+⋯],φj(x)=φj0(x)+xn-1φj1(x)+x2n-1φj2(x)+⋯,
where fj, gj, and φjl, j≥0, l≥1, are polynomials in x of degree at most n-1, and φj0, j≥0, are polynomials in x with degree at most n-2.
Now we change (3.44) using (2.2) to the system
(3.47)x˙=rnsinθ,y˙=r2n-1∑j≥0Vj(θ,r)rjn,
where by (3.46)
(3.48)V0(θ,r)=-cos2n-1θg0(rcosθ)-sinnθf0(rcosθ)+rsin2θφ00(rcosθ),Vj(θ,r)=-cos2n-1+jnθgj(rcosθ)-sinnθcosjnθfj(rcosθ)+rsin2+jθφj0(rcosθ)+∑k=0j-1sin2+kθcos(j-k)n-1θφk,j-k(rcosθ),j≥1.
This yields the equation(3.49)drdθ=r∑j≥0Rj(θ,r)rjn∑j≥0Sj(θ,r)rjn,
where
(3.50)S0(θ,r)=-nsin2θ+cosθV0(θ,r),R0(θ,r)=sinθcosθ+sinθV0(θ,r),Sj(θ,r)=cosθVj(θ,r),Rj(θ,r)=sinθVj(θ,r),j≥1.
By (3.48) and (3.50) we can further expand Sj and Rj in r to obtain for j≥0(3.51)Sj(θ,r)=∑l=0n-1S-l+jn(θ)rl,Rj(θ,r)=∑l=0n-1R-l+jn(θ)rl,
so that the above differential equation can be written as
(3.52)drdθ=r∑j≥0R-j(θ)rj∑j≥0S-j(θ)rj.
Further, letting(3.53)1∑j≥0S-j(θ)rj=∑j≥0S~j(θ)rj,(3.54)R~j(θ)=∑k+l=jR-k(θ)S~l(θ),j≥0
we obtain
(3.55)drdθ=r∑j≥0R~j(θ)rj.
Note that for anyj≥0, S~j depends only on S-k with 0≤k≤j. Then by (3.54) one can see that,
(3.56)foranyj≥0,R~jdependsonlyonR-kandS-kwith0≤k≤j.
Letr(θ,r0) denote the solution of (3.55) with the initial value r0. The for r0 small we have
(3.57)r(θ,r0)=∑j≥1rj(θ)r0j,
where r1,r2,r3,… satisfy r1(0)=1, r2(0)=r3(0)=⋯=0, and
(3.58)r1′=R~0r1,r2′=R~0r2+R~1r12,r3′=R~0r3+2R~1r1r2+R~2r13,⋯
which implies that, for any j≥1, the function rj 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. 							(3.59)Foranyj≥1,vj(δ)dependsonlyonR-kandS-kwith0≤k≤j-1.
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, vj depends only on the coefficients of degree at most j-1 of the polynomials g0,f0, and xφ00 in x.
Similarly, forj≥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 gj, fj, xφj0 and φi,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 gj,fj, xφj0, and φi,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
(3.60)N[0,u-1]=⋃i=0j-1N[in,(i+1)n-1]⋃N[jn,u-1].
Thus, for allk∈N[in,(i+1)n-1], S-k and R-k depend only on gi, fi, xφi0, and φl,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 gj, fj, xφj0, and φl,j-l with l=0,…,j-1 in x.
Therefore, by (3.59) for jn+1≤u≤(j+1)n, vu(δ) depends only on the functions gi, fi, xφi0 and φl,i-l with l=0,…,i-1, i=0,…,j-1 and the coefficients of degree at most u-1-jn of the polynomials gj, fj, xφj0, and φl,j-l with l=0,…,j-1 in x.
We claim that ifj≥0, m≥(j+1)n, then, for jn+1≤u≤(j+1)n, vu(δ) depends only on the functions gi, fi, with i=0,…,j-1 and the coefficients of degree at most u-1-jn of the polynomials gj, fj in x.
In fact, by the above discussion, we need only to proveφ00=0 in the case j=0 and φls=0 for l+s≤j and 0≤l≤j-1 in the case j>0. This can be shown easily since
(3.61)φj0=O(|x|m-1-j),φjs=O(|x|m-j-sn)fors≥1,degφj0≤n-2,degφjs≤n-1fors≥1
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, vu(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, vu(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
(3.62)2n+u-2≤m,n+u-2≤m-1foru≤(k+1)n.
In this case, for all1≤u≤(k+1)n, vu(δ) depends only on gm and fm-1 in (3.45). Then the conclusion of Theorem 1.12 follows.Proof of Theorem1.14.
Let(3.63)X(x,y,δ)=∑2≤i+j≤kaijxiyj,Y(x,y,δ)=∑2≤i+j≤kbijxiyj.
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
(3.64)g(x,δ)=∑j≥2ajxj,
where all aj are polynomials in aij and bij. Thus, by the well-known Hilbert basis theorem there exists Nk>0 such that aj=0 for all j≥2Nk if a2=⋯=a2Nk-1=0. That is, a2=⋯=a2Nk-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 Nk as a nilpotent focus or center.
Further, for any2≤n≤Nk 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 aj in gm and bj in fm-1 are all polynomials in aij and bij. Moreover, a2n-1>0 and bn-12-4na2n-1<0. Let
(3.65)F(x)=∫0xfm-1dx,G(x)=∫0xgmdx,
and α(x)=-x+∑j≥2αjxj satisfies G(α)=G(x). Then, noting that
(3.66)G1/2n=[a2n-1(2n)]1/2n|x|(1+∑j≥1βjxj),
where all βj are polynomials in aij, bij, and c≡a2n-1-1, one can see that all αj are polynomials in aij, bij, and c. Therefore, if we let
(3.67)F(α(x))-F(x)=∑j≥nBjmxj=∑j≥2l+1Bjmxj,
where l=[n/2], then all Bjm are polynomials in aij, bij, and c. By Theorems 1.10 and 1.12, for any integer k0>0 there exists an integer m0>0 such that Bjm, j=2l+1,…,k0 are independent of m for all m≥m0, denoting Bjm by Bj. Then, we have defined a series of coefficients Bj for all j≥2l+1, which are all polynomials in aij, bij, and c. Again, by the Hilbert basis theorem there exists Kn(k)>0 such that
(3.68)B2l+1=B2l+2=B2l+3=⋯=B2l+2Kn(k)+1=0⇒B2l+j+1=0,j≥2Kn(k)+1.
Now, we takeK- and m such that 2Kn(k)+1+pn≤K-n, m>(K-+2n)-2. Then by Theorems 1.10 and 1.12 again, for (1.49) we have
(3.69)v1+pn=v3+pn=⋯=v2Kn(k)+1+pn=0
yielding vj+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
(4.1)x˙=y,y˙=-(a11xy+a02y2+a30x3+a21x2y+a12xy2+a03y3).The authors of [11] proved that if a30>0 and a112-8a30<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 Theorem1.7 we have immediately the following.Proposition 4.1.
Leta11, a02, a30, a21, a12, and a03 be bounded parameters satisfying
(4.2)a30>0,a112-8a30<0,|a21|+|a03|+|a11a02|>0.
Then there exists a neighborhoodV of the origin such that the system (4.1) has at most 3 limit cycles in V.Consider now the system(4.3)x˙=-y+Ax2+Bxy+Cy2,y˙=x3+xy2+y3.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.
LetA,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(4.4)x˙=y,y˙=-(x3+x5)-∑j=0kb2jx2jy,
where k≥2. By Theorems 1.8–1.11, we obtain the following.Proposition 4.3.
Letb2j be bounded parameters. Then(1)
ifb0=0, the system (4.4) has at most k-1 limit cycles near the origin, and k-1 limit cycles can appear.(2)
Ifb0≠0, there are systems inside (4.4) which have at least k limit cycles near the origin.



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*Source: 720830-2012-11-28.xml*