# Limit Cycles and Analytic Centers for a Family of4n-1 Degree Systems with Generalized Nilpotent Singularities

Authors: Yusen Wu; Cui Zhang; Changjin Xu
Journal: Journal of Function Spaces
 (2015)

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## Abstract

With the aid of computer algebra systemMathematica  8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameter n controls the singular point type of the studied system. The main results generalize and improve the previously known results in Pan.


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## Body

## 1. Introduction

A famous problem for the plane analytic systems of differential equations is under what conditions the local phase portrait at a critical pointp is topologically equivalent to the local phase portrait of the linear part of the system at p. This problem has been solved by Poincaré and Bendixson for hyperbolic critical points and for elementary critical points, that is, for points having zero determinant and nonzero trace linear part. Another problem is to characterize the local phase portrait at an isolated critical point p.When the matrix of the linear part at the origin is not identically null but has its eigenvalues which are equal to zero, at this moment, the origin is a nilpotent critical point. An analytic system having an isolated nilpotent singularity at the origin, in some suitable coordinates, can be written as follows:(1)dxdt=y+∑i+j=2∞‍aijxiyj=y+X(x,y),dydt=∑i+j=2∞‍bijxiyj=Y(x,y).Suppose that the functiony=y(x) satisfies X(x,y)=0, y(0)=0. Amelkin et al. proved (see, for instance, [1]) that the origin of system (1) is a monodromic critical point (i.e., a center or a focus) if and only if (2)Yx,yx=αx2n-1+ox2n-1,α<0,∂X(x,y)∂x+∂Y(x,y)∂xy=y(x)=βxn-1+o(xn-1),β2+4nα<0,where n is a positive integer. Andreev [2] shows what the behavior of the solutions in a neighborhood of the nilpotent critical point is, except if it is a center or a focus (nilpotent center problem). This last result can not distinguish between a focus and a center. Takens [3] and Bogdanov [4] find easy formal normal forms for the nilpotent critical points. Moussu [5] has found the C∞ normal form for a nilpotent center and asks if there exists an analytic normal form for the nilpotent centers. Takens [3] proves that for any analytic nilpotent center there exists an analytic change of variables such that the new system can be written as a system of the form (1) and it is atime-reversible nilpotent center. However, the aforementioned result is difficult to implement because, in general, in order to decide if an analytic system has a nilpotent center, we must know explicitly the analytic change of variables which writes system (1) in the Berthier-Moussu normal form. Writing systems in a convenient normal form, Teixeira and Yang [6] study the relationship between time-reversibility and the center-focus problem for elementary singular points and nilpotent singular points. Giné [7] develops a method which provides necessary conditions for obtaining a local analytic integral in a neighborhood of a generalized nilpotent singular point. García and Giné [8] give a necessary condition to have local analytic integrability in an analytic nilpotent center. Giacomini et al. [9] studied the centers of planar analytic vector fields which are a limit of linear-type centers.In this paper, we consider the system of differential equations in the plane whose origin is a generalized nilpotent singular point(3)dxdt=y2n-1+yn-1(a30x3+a21x2yn+a12xy2n+a03y3n),dydt=-2x3+b02y2n+b12xy2n+b03y3n.The origin of system (3) is a third-order nilpotent singular point when n=1, while it is a total degenerate singular point when n>1.In Section2, we give some preliminary knowledge concerning the nilpotent critical point. In Section 3, we transform the origin into a third-order nilpotent singular point by a homeomorphism. Then, we compute the first several quasi-Lyapunov constants and derive the sufficient and necessary conditions for the origin to be an analytic center. In the last section, we prove that there exist seven small amplitude limit cycles in the neighborhood of the origin.


## 2. Computation of Quasi-Lyapunov Constants and Determination of Analytic Centers

In this section, we first introduce some definitions, notations, and symbols in order to make the paper compact and clear, followed by an algorithm to obtain the necessary conditions for the third-order nilpotent critical point of system (1) to be an analytic center, and then we present several methods to prove the sufficiency. More details are due to [10, 11].If the origin of system (1) is a high-order critical point, it is called a (2n-1)th-order critical point when (2) is satisfied, and it could be broken into 2n-1 elementary critical points in the neighborhood of the origin in the complex plane. It is easy to show that the origin of system (1) is a third-order monodromic critical point if and only if b20=0 and (2a20-b11)2+8b30<0.Without loss of generality, we assume that(4)a20=μ,b20=0,b11=2μ,b30=-2.Otherwise, by letting (2a20-b11)2+8b30=-16λ2, 2a20+b11=4λμ, and taking the transformation ξ=λx, η=λy+(1/4)(2a20-b11)λx2, we obtain the desired form. Under (4), system (1) becomes the following real autonomous planar system: (5)dxdt=y+μx2+∑i+2j=3∞‍aijxiyj=X(x,y),dydt=-2x3+2μxy+∑i+2j=4∞‍bijxiyj=Y(x,y).In the following, we are going to classify the third-order nilpotent critical point.Theorem 1.
For system (5), there exists a series with nonzero convergence radius: (6)ux,y=x+∑α+β=2∞‍aαβ′xαyβ,vx,y=y+∑α+β=2∞‍bαβ′xαyβ,b20′=-μ,ζx,y=1+∑α+β=1∞‍cαβ′xαyβ,such that, by the transformation (7)u=u(x,y),v=v(x,y),dt=ζ(x,y)dτ,system (5) is reduced to the following Liénard equation: (8)dudτ=v+∑k=1∞‍Aku4k+∑k=0∞‍Bku4k+2+∑k=1∞‍Cku2k+1=U(u,v),dvdτ=-21+μ2u3=V(u,v),where B0=2μ. In addition, the origin of system (5) is a center if and only if Ck=0 for all k.The following two definitions are taken from [10].Definition 2.
(1) Ifμ≠0, then the origin of system (5) is called a third-order nilpotent critical point of zero-class.
(2) Ifμ=0 and there exists a positive integer s, such that B0=B1=⋯=Bs-1=0, but Bs≠0, then the origin of system (5) is called a third-order nilpotent critical point of s-class.
(3) Ifμ=0 and Bs=0 for all positive integer s, then the origin of system (5) is called a third-order nilpotent critical point of ∞-class.Definition 3.
Letfk, gk be continuous and bounded functions with respect to μ and aij, bij, k=1,2,…. If, for any integer m, there exist continuous and bounded functions of μ and all aij,bij: ξ1(m),ξ2(m),…,ξm-1(m), such that (9)fm=gm+ξ1(m)f1+ξ2(m)f2+⋯+ξm-1(m)fm-1,then we say that fm and gm are equivalent, denoted by fm~gm. If, for any integer m, we have fm~gm, we say that the sequences of functions {fm} and {gm} are equivalent, denoted by {fm}~{gm}.Remark 4.
It is easy to see from Definition3 that the following conclusions hold.(1)
The equivalence relationship of two sequences of functions is self-reciprocal, symmetric, and transmissible.(2)
If, for some integerm, fm~gm, then when f1=f2=⋯=fm-1=0, we have fm=gm.(3)
The relationshipf1~g1 implies that f1=g1.The following three theorems were proved in [10].Theorem 5.
If the origin of system (5) is s-class, then the origin of system (5) is a center if and only if there is an inverse integrating factor Ms+1.
If the origin of system (5) is ∞-class, then the origin of system (5) is a center if and only if for any natural number l there exists an inverse integrating factor Ml+1, where M is a power series given by (10)M=x4+y2+∑k+2j=5∞‍ckjxkyj.Theorem 6.
For any natural numbers and a given number sequence (11)c0β,β≥3,terms with the coefficients cαβ satisfying α≠0 of the formal series (10) can be constructed successively such that (12)∂∂xXMs+1+∂∂yYMs+1=1Ms+2∑m=6∞‍ωmxm;that is, (13)∂X∂x+∂Y∂yM-s+1∂M∂xX+∂M∂yY=∑m=6∞‍ωmxm,where sμ=0.Theorem 7.
Forα≥1, α+β≥3 in (11) and (12), cαβ is uniquely determined by the recursive formula (14)cαβ=1s+1αAα-1,β+1+Bα-1,β+1.For m≥1, ωm(s,μ) is uniquely determined by the recursive formula (15)ωm=Am,0+Bm,0,where (16)Aαβ=∑k+j=2α+β-1‍k-s+1α-k+1akjcα-k+1,β-j,Bαβ=∑k+j=2α+β-1‍j-s+1β-j+1bkjcα-k,β-j+1.In (15), one sets (17)c00=c10=c01=0,c20=c11=0,c02=1,cαβ=0,ifα<0orβ<0.It follows from Theorems5–7 that if the origin of system (5) is a center with s-class or ∞-class, then, by choosing {c0β}, such that (18)ωk=0,k=6,7,….The following conclusion holds (see Theorem 3.4 of [10]).Theorem 8.
If the origin of system (5) is ∞-class, then when the origin of system (5) is a center, in a neighborhood of the origin, system (5) has an analytic inverse integrating factor (19)M∞=1+h.o.t.and an analytic first integral given by (20)Fx,y=x4+y2+∑k+2j=5∞‍Ckjxkyj.According to [11], we have the following lemma.Lemma 9.
For system (5), if there exists a first integral which is the power series (20) in the neighborhood of the origin, then the origin of system (5) is a center.By Theorem1, if the origin of system (5) is a center, there exist analytic transformations in the neighborhood of the origin such that system (5) can be transformed into the Liénard equations: (21)dudτ=v+∑k=1∞‍Aku4k+∑k=0∞‍Bku4k+2,dvdτ=-2(1+μ2)u3,B0=2μ.The vector field defined in (21) is symmetrical with respect to the v-axis. Further, by the transformation (22)w=u2,v=v,dτ=-dτ′2u,system (21) is reduced to (23)dwdτ′=-v-∑k=1∞‍Akw2k-∑k=0∞‍Bkw2k+1,dvdτ′=1+μ2w,B0=2μ.In the (w,v)-phase plane, the origin of system (23) is an elementary critical point (focus or center). Obviously, we have the following.Lemma 10.
If there exists a power seriesF=F(u,v) in u, v satisfying (24)dFdτ(21)=0,then F can be written as a power series in u2, v; namely, F(u,v)=G(u2,v).Then, we have the following theorem.Theorem 11.
The origin of system (5) is an analytic center if and only if the origin of system (5) is a center and ∞-class; namely, the origin of system (5) is a center and for any natural number k, Bk=0.Corollary 12.
Ifμ≠0, the origin of system (5) is not an analytic center.Theorem 13.
The origin of system (5) is an analytic center if and only if there exists an analytic first integral F(x,y) in the neighborhood of the origin, which is the power series (23).From Theorems8 and 13, we further have the following.Theorem 14.
The origin of system (5) is an analytic center if and only if in the neighborhood of the origin of system (5) there exists an analytic inverse integrating factor (25)M∞=1+∑k+j=1∞‍Ckj′xkyj.Similarly, by Theorems5 and 11, we have the following.Theorem 15.
The origin of system (5) is an analytic center, if and only if, for any natural number s, there exists an inverse integrating factor Ms+1, where M is the power series (10).Moreover, Theorems7 and 15 imply the following.Theorem 16.
The origin of system (5) is an analytic center, if and only if, for any natural number s, there exists a power series M satisfying ωm=0 in (12) for all m.Therefore, a new method of determining analytic nilpotent center for a given system has been obtained in Theorem16.Theorem 17.
If the origin of system (5) is a nilpotent center and system (5) is symmetric with respect to the origin, namely, (26)X(-x,-y)=-X(x,y),Y(-x,-y)=-Y(x,y),then the origin of system (5) is an analytic center.Theorem 18.
If system (5) is symmetric with respect to the x-axis, then the origin of system (5) is an analytic center.Remark 19.
If system (5) is symmetric with respect to the y-axis, then the origin of system (5) may not be an analytic center. For example, system (21) is symmetric with respect to the v-axis, but the origin is an analytic center if and only if Bk=0 for all k.Eventually, by Theorem11, we have the following.Theorem 20.
The origin of system (5) is an analytic center if and only if system (5) can be changed into (27)dudτ=v+∑k=1∞‍Aku4k,dvdτ=-2u3by the analytic transformation (7).


## 3. Analytic Center Conditions

In this section we will derive conditions for the origin of system (3) to be an analytic center.After the change(28)x1=x,y1=1nyn,dt1=nyn-1dt,and renaming (x1,y1,t1) with (x,y,t), system (3) takes the form (29)dxdt=y+1na30x3+a21x2y+na12xy2+na03y3,dydt=-2x3+nb02y2+nb12xy2+n3/2b03y3.According to Theorem7, we have the following.Lemma 21.
Assume thats is a natural number. One can derive a power series (10) for system (29) under which (12) is satisfied, where (30)c00=0,c10=0,c01=0,c20=0,c11=0,c02=1.In addition, for any natural numbers α, β, cαβ is given by the following recursive formula: (31)cαβ=2(1+s)(2+β)c-4+α,2+β+1na30(3-(1+s)(-3+α))c-3+α,1+β+a21(2-(1+s)(-2+α))c-2+α,β+b12n(2-(1+s)β)c-2+α,β+a12n(1-(1+s)(-1+α))c-1+α,-1+β+b03n3/23-1+s-1+βc-1+α,-1+β+b02n2-1+sβc-1+α,β-a03n1+sαcα,-2+β×s+1α-1,and, for any natural number m, ωm is given by the following recursive formula: (32)ωm=2(1+s)c-3+m,1+1na30(3-(-2+m)(1+s))c-2+m,0+b12n(3+s)c-1+m,-1+a212--1+m1+sc-1+m,-1+b03n3/23+21+scm,-2+a12n1-m1+scm,-2+b02n3+scm,-1-a031+mn1+sc1+m,-3.Applying Lemma21 and computing with Mathematica, we have (33)ω6=-4s-1na30,ω7~3(s+1)c03,ω8~-25n(4s-3)(a12+3nb03),ω9~-4n5/2s-1b02b03.Fromω9=0, the analytic center problem of system (29) can be broken down into three cases: (1) b02=0, (2) b03=0, and (3) s=1.Case 1.
Considerb02=0.
In this case, further calculation gives the following.Theorem 22.
For system (29), the first three quasi-Lyapunov constants at the origin are given by (34)λ1=1na30,λ2~25n(a12+3nb03),λ3~47n3/2b03(a21+nb12).In the above expression of λk, one has already let λ1=λ2=⋯=λk-1=0, k=2,3.From Theorem22, we obtain the following assertion.Theorem 23.
For system (29), all the quasi-Lyapunov constants at the origin are zero if and only if the first three quasi-Lyapunov constants at the origin are zero; that is, one of the following two conditions holds: (35)a30=a12=b02=b03=0;(36)a30=b02=0,a12+3nb03=0,a21+nb12=0.Relevantly, both conditions are the analytic center conditions of the origin.Proof.
When condition (35) is satisfied, system (29) can be brought to (37)dxdt=y1+a21x2+na03y2,dydt=-x2x2-nb12y2,whose vector field is symmetric with respect to the origin.
When condition (36) is satisfied, system (29) can be brought to (38)dxdt=y(1-nb12x2-3n3/2b03xy+na03y2),dydt=-2x3+nb12xy2+n3/2b03y3,which is Hamiltonian and possesses the analytic first integral (39)F1(x,y)=x4+y2-nb12x2y2-2n3/2b03xy3+12na03y4.Case 2.
Considerb03=0.
In this case,λ1=λ2=0 yields that (40)a30=a12=b03=0.Under this condition, system (29) becomes (41)dxdt=y(1+a21x2+na03y2),dydt=-2x3+nb02y2+nb12xy2,whose vector field is symmetric with respect to axis x. Thus, system (29) has an analytic center at the origin when condition (40) is satisfied.
Obviously, condition (35) is a special case of condition (40).Case 3.
Consders=1.
In this case, we have the following.Proposition 24.
For system (29), one can determine successively the terms of the formal series M(x,y)=x4+y2+o(r4), such that (42)∂X∂x+∂Y∂yM-2∂M∂xX+∂M∂yY=∑m=17‍λm2m-5x2m+4+or18,where λm is the mth quasi-Lyapunov constant at the origin of system (29), m=1,2,…,7.Further calculation gives the following.Theorem 25.
For system (29), the first seven quasi-Lyapunov constants at the origin are given by (43)λ1=1na30,λ2~25n(a12+3nb03),λ3~435n3/2b03(5a21+5nb12+11n2b022),λ4~-47875n9/2b022b03(-1475b12+744nb022),λ5~-82512846875n9/2b022b03×(589594375a03+395195814n3b024),λ6~-244352606774140625n13/2b022b03×-1131416865765625b032+529893701720802n3b026,λ7~-1057311533261767691721638365073074138357421875n23/2b0210b03.In the above expression of λk, we have already let λ1=λ2=⋯=λk-1=0, k=2,3,4,5,6,7.λ
i, i=1,2,3,4,5,6,7, in expression (43) vanish if and only if one of conditions (36) and (40) holds.Therefore, we see from Cases1–3 the following.Corollary 26.
The origin of system (29) ((3)) is an analytic center if and only if condition (36) or (40) holds.Actually, when condition (36) holds, system (3) goes over to (44)dxdt=y2n-1(1-nb12x2-3nb03xyn+a03y2n),dydt=-2x3+b12xy2n+b03y3n,which is Hamiltonian and possesses the analytic first integral (45)F2(x,y)=x4+1ny2n-b12x2y2n-2b03xy3n+12na03y4n.When condition (40) holds, system (3) goes over to (46)dxdt=y2n-1(1+a21x2+a03y2n),dydt=-2x3+b02y2n+b12xy2n,whose vector field is symmetric with respect to axis x.


## 4. Multiple Bifurcation of Limit Cycles

In this section, we are going to establish the conditions forO(0,0) to be at most a weak focus of order seven, and, more, we will prove that the perturbed system of (29) can generate seven limit cycles enclosing an elementary node at the origin.From the factλ1=λ2=λ3=λ4=λ5=λ6=0, λ7≠0, we obtain the following.Theorem 27.
For system (29), the origin is a weak focus of order seven if and only if (47)a30=0,a12=-3nb03,a21=-15n(5b12+11nb022),b12=7441475nb022,a03=-395195814589594375n3b024,b032=5298937017208021131416865765625n3b026,b02b03≠0.Consider the perturbed system of (29): (48)dxdt=δεx+y+1na30(ε)x3+a21(ε)x2y+na12εxy2+na03(ε)y3,dydt=2δ(ε)y-2x3+nb02y2+nb12(ε)xy2+n3/2b03(ε)y3.In order to get seven limit cycles, we only need to show that, when condition (47) holds, the Jacobian of the function group (λ1,λ2,λ3,λ4,λ5,λ6) with respect to the parameter group (a30,a12,a21,b12,a03,b03) does not equal zero. A direct calculation gives that (49)∂λ1,λ2,λ3,λ4,λ5,λ6∂a30,a12,a21,b12,a03,b03(47)=-127789436073926783638623356338176145260521069842113268585205078125n23b0218b03≠0.The above discussions indicate the following.Theorem 28.
If the origin of system (29) ((3)) is a weak focus of order seven, for 0<δ≪1, making a small perturbation to the coefficient group (a30,a12,a21,b12,a03,b03), then, for system (48), in a small neighborhood of the origin, there exist exactly seven small amplitude limit cycles enclosing the origin O(0,0), which is an elementary node.Example 29.
Take(50)δ(ε)=ε56,a30(ε)=-ε42,a12ε=-958877077968978/208016897375n5/2c3sign(c)-3nε2+ε30,a21(ε)=-39891475n2c2-nε12-ε20,b12(ε)=7441475nc2+ε12,a03(ε)=-395195814589594375n3c4+ε6,b03(ε)=358877077968978/208016897375n3/2c3sign(c)+ε2,b02=csign(c),where c is an arbitrary nonzero real constant.
Straightforward computations by using Theorem25 give the first seven quasi-Lyapunov constants at the origin of system (48): (51)λ1=-1nε42+oε42,λ2~25nε30+o(ε30)=0.4nε30+o(ε30),λ3~-1258877077968978/2080168951625n3c3sign(c)ε20+o(ε20)≈-0.391061n3c3sign(c)ε20+o(ε20),λ4~419625692656326/624050674375n6c5sign(c)ε12+o(ε12)≈0.512725n6c5signcε12+oε12,λ5~-815955688129593038/767592839375n6c5sign(c)ε6+o(ε6)≈-1.28458n6c5sign(c)ε6+o(ε6),λ6~254348976825984964352606774140625n19/2c8ε2+o(ε2)≈5.8436n19/2c8ε2+oε2,λ7~-3171934599785303075164858877077968978/20801689282942413921770385986328125×n13c13sign(c)+o(1)≈-0.188604n13c13sign(c)+o(1).Then, for 0<ε≪1, system (48) has seven limit cycles Γk:r=r~(θ,hk(ε)) in a small neighborhood of the origin, where hk(ε)=O(εk), k=1,2,3,4,5,6,7.



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