# Fourteen Limit Cycles in a Seven-Degree Nilpotent System

Authors: Wentao Huang; Ting Chen; Tianlong Gu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.


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## Body

## 1. Introduction

In the qualitative theory of planar differential equations, the center-focus problem and bifurcation of limit cycles for nilpotent system(1)dxdt=y+∑k+j=2∞‍akjxkyj=X(x,y),dydt=∑k+j=2∞‍bkjxkyj=Y(x,y),
are known as a difficult problem. Some advance of this problem can be dated back to [1–3]. In recent years, due to the improvement of research method and development of computer symbolic computation, the problem has attracted more and more scholars’ attention and has received a lot of results. For instance, in [4, 5], the center conditions of the nilpotent critical points were obtained for several systems. In [6] the center conditions and the bifurcations of limit cycles were investigated for a quintic and a nine-degree nilpotent systems. The center and the limit cycles problems of a quintic nilpotent system were also solved in [7]. And in [8], the authors gave a recursive method to calculate quasi-Lyapunov constants of the nilpotent critical point. The nilpotent center problem and limit cycles bifurcations were performed also in [9]. It is interesting how many limit cycles can be bifurcated from the nilpotent critical point. Let N(n) be the maximum possible number of limit cycles bifurcated from a nilpotent critical point of system (1) when X and Y are of degree at most n. The known results of N(n) are:  Andreev et al. given have N(3)≥2, N(5)≥5, N(7)≥9, see [5]. Y. Liu and J. Li showed N(3)≥4, N(3)≥7, N(3)≥8, see [8, 10–12].  Li et al.  found N(7)≥12 in [13]. Recently,  Li et al.  [14] obtained N(7)≥13.In this paper, we study the bifurcation of limit cycles for a seven-degree nilpotent system with the following form:(2)dxdt=δx+y+a30x3+a12xy2+a32x3y2+a14xy4+a05y5+a06y6+a15xy5+a24x2y4+a33x3y3+a51x5y+a07y7+a16xy6+a25x2y5+a34x3y4+a43x4y3+a61x6y,dydt=2δy-2x3+xy2+b33x3y3+a51x4y2.
By the computation of the quasi-Lyapunov constants, we prove that its perturbed system has 14 small-amplitude limit cycles bifurcated from the origin, namely, N(7)≥14 which improves the result in [14].In Section2, we give some preliminary knowledge concerning the nilpotent critical point. In Section 3, we obtain the first 14 quasi-Lyapunov constants and derive the sufficient and necessary conditions of the origin to be a center and a 14th-order fine focus. At the end, it is proved that there exist 14 limit cycles in the neighborhood of the origin of the system.


## 2. Focal Values and Quasi-Lyapunov Constants

In order to discuss limit cycles of the system, we state some preliminary results given by [8].According to [2], the origin of system is a 3th-order monodromic critical point and a center or a focus if and only if b20=0, (2a20-b11)2+8b30≤0. Without loss of generality, we assume that a20=μ, b20=0, b11=2μ, b30=-2, otherwise let (2a20-b11)2+8b30=-16λ2, 2a20+b11=4λμ.Under the substitutions(3)η=λy+14(2a20-b11)2λx2ξ=λx,
system (1) becomes
(4)dxdt=y+μx2+∑k+2j=3∞‍akjxkyj=X(x,y),dydt=-2x3+2μxy+∑k+2j=4∞‍bkjxkyj=Y(x,y).By the transformation of the generalized polar coordinates,(5)x=rcosθy=r2cosθ,
system (4) is transformed into
(6)drdθ=cosθR1(θ)Q1(θ)+o(r),
where
(7)R1(θ)=sinθ(1-2cos2θ)+μ(cos2θ+2sin2θ),Q1(θ)=-2(cos4θ+sin2θ)<0.For sufficiently smallh, let
(8)r=r~(θ,h)=∑k=1∞‍νk(θ)hk
be a solution of (6) satisfying the initial value condition r|θ=0=h, where
(9)ν1(θ)=(cos4θ+sin2θ)-1/4×exp((-μ2)arctan(sinθcos2θ)),ν1(kπ)=1,k=0,±1,±2….Because for all sufficiently smallr, there is dθ/dt<0, in a small neighborhood; we obtain the Poincaré return map of (6) in a small neighborhood of the origin as follows:
(10)Δ(h)=r~(-2π,h)-h=∑k=2∞‍νk(-2π)hk.Lemma 1.
For any positive integerm, ν2m+1(-2π) has the form
(11)ν2m+1(-2π)=∑k=1∞‍ζm(k)ν2k(-2π),
where ζm(k) is a polynomial of νi(π), νi(2π), νi(-2π), (i=2,3,…2m) with rational coefficients.Definition 2.
(
i
) For any positive integer m, ν2m(-2π) is called the mth-order focal value of system (4) at the origin; (ii) if ν2(-2π)≠0, the origin of system (4) is called an 1th-order weak focus; if there is an integer m>1 such that ν2(-2π)=ν4(-2π)=⋯=ν2m-2(-2π)=0, ν2m(-2π)≠0, then the origin of system (4) is called a mth-order weak focus; (iii) if for all positive integer m, we have ν2m(-2π)=0, the origin of system (4) is called a center.Lemma 3.
For system (4), one can derive successively the formal series
(12)M(x,y)=x4+y2+o(r4)
such that
(13)(∂X∂x+∂Y∂y)M-(s+1)(∂M∂xX+∂M∂yY)=∑m=1∞‍λm[(2m-4s-1)x2m+4+o(r2m+4)].Lemma 4.
If there exists a natural numbers and formal series
(14)M(x,y)=x4+y2+o(r4)
such that (13) holds, then
(15)v2m(-2π)~σmλm,m=1,2,3…,
where
(16)σm=12∫02π‍(1+sin2θ)cos2m+4θx×(exp((2m-12)μarctansinθcosθ)(sin4θ+sin2θ)2m+7/4x×exp((2m-12)μarctansinθcosθ))-1dθ>0.
In (15), ~ is the symbol of algebraic equivalence, meaning that there exists ξm(k)(k=1,2,…m-1), polynomial functions of the coefficients of system (4), such that
(17)ν2m+1(-2π)=σmλm+∑k=1m-1‍ξm(k)λk.Definition 5.
In Lemma4, λm is called the mth-order quasi-Lyapunov constant of the origin of system (4).Lemma 6.
For system (4), one can derive successively the formal series
(18)M(x,y)=y2+∑α+β=3∞‍cαβxαyβ
such that
(19)(∂X∂x+∂Y∂y)M-(s+1)(∂M∂xX+∂M∂yY)=∑m=3∞ωm(s,μ)xm,
where c00=c10=c01=c20=c11=0, c02=1. For α≥1, α+β≥3, cαβ, and ωm(s,μ) are determined by the following recursive formulas:
(20)cαβ=1(s+1)α(Aα-1,β+1+Bα-1,β+1),ωm(s,μ)=Am,0+Bm,0,
where
(21)Aαβ=∑k+j=2α+β-1‍[k-(s+1)(α-k+1)]akjcα-k+1,β-j,Bαβ=∑k+j=2α+β-1‍[j-(s+1)(β-j+1)]bkjcα-k,β-j+1.
By choosing {c0β} such that
(22)ω2k+1(s,μ)=0,k=1,2,…,
one has
(23)λm=ω2m+4(s,μ)2m-4s-1.One considers the perturbed system of system (4)
(24)dxdt=δx+y+μx2+∑k+2j=3∞‍akjxkyj,dydt=2δy-2x3+2μxy+∑k+2j=4∞‍bkjxkyj.For system (24)|δ=0, from Lemma 4, we know that the first nonvanishing quasi-Lyapunov constant λm is positive constant  times as much as the first nonvanishing focal value, so the former shows the same effect as the latter in the study of bifurcation of limit cycles. From [10, Theorem 4.7], we have the following.Theorem 7.
For the system (27)|δ=0, assume that the quasi-Lyapunov constants of the origin λi(i=1,2,…) have k independent parameters γ=(γ1,γ2,…,γk); that is, λi=λi(γ1,γ2,…,γk). If γ=γ0, the origin of the system (4) is an mth-order weak focus (m≤k),  and the Jacobian determinant
(25)∂(λ1,λ2,…,λm-1)∂(γ1,γ2,…,γm-1)|γ=γ0≠0,
then, the perturbed system (24) exists m small amplitude limit cycles bifurcated from the origin.


## 3. Criterion of Center Focus and Bifurcation of Limit Cycles

Applying the recursive formulas in Lemma6, we compute the quasi-Lyapunov constants of the origin of system (2)|δ=0 with the computer algebra system Mathematica and obtain the following result.Theorem 8.
For system (2)|δ=0, the first 14 quasi-Lyapunov constants are as follows:
(26)λ1=a30,λ2=25a12,λ3=27a32,λ4=415a14,λ5=1277a34,λ6=2195(20a16+3a51b33),λ7=1385b33(35a51-8a33),λ8=713260b33(128a15-355a51),λ9=333440b33a51(1385+64a61),λ10=1278460b33a51×(-192495+12320a05+1904a43),λ11=91184444800b33a51×(317763455+1688064a43+1158080a512),λ12=1505504614521088000b33a51×(424870735079675775-8480461063976518a512xxx-164955456258816b332),λ13=12497759223828804812800×b33a51(a5121154557205782671354192175-25287050037965301847744a512),λ14=-11926846314779614102444810240000b33a51×(a5121913839774991447312487020909964625-38616043776955260227746202006848a512+457974511144735287048192000a514).
Here, every λk (k=1,2,…,14) was computed under the assumption λ1=λ2=⋯=λk-1=0.
It is easy to obtain the following Theorem.Theorem 9.
For system (2)|δ=0, the first 14 quasi-Lyapunov constants at the origin are all zero if and only if the following condition is satisfied:
(27)a30=a12=a32=a14=a34=a51=a33=a15=a16=0.
Ifδ=0 and the condition (27) holds, system (2) becomes
(28)dxdt=y+a05y5+a06y6+a24x2y4+a07y7+a25x2y5+a43x4y3+a61x6y,dydt=-2x3+xy2+b33x3y3,
which is symmetric with respect to the y-axis, one has the following.Theorem 10.
The origin of system (2) is a center if and only if δ=0 and (27) holds.
Byλ1=λ2=⋯=λ13=0, λ14≠0, one has the following.Theorem 11.
The origin of system (2) is a 14th-order weak focus if and only if
(29)δ=a30=a12=a32=a14=a34=0,a61=-138564,a05=30075794600575314214479775606889200911167244345856,a43=-66625696625444520068811785303444600455583622172928,b332=109139947163472258472470037254779252457175442049223616,a512=115455720578267135419217525287050037965301847744,a16=-320a51b33,a33=358a51,a15=355128a51.By computing carefully, we obtain that the Jacobian determinant(30)∂(λ1,λ2,λ3,λ4,λ5,λ6,λ7,λ8,λ9,λ10,λ11,λ12,λ13)∂(a30,a12,a32,a14,a34,a16,a33,a15,a61,a05,a43,a51,b33)|(29)=-11259131158497337756164795883686035195310097999201613627491381814272a514b336110636634525265639383282317978327920684865639296808353136757452754003615234375≈-2526.4563514134≠0.From (30) and Theorem 7, one has the following.Theorem 12.
For system (2), under the condition (29), by small perturbations of the parameter group (δ,a30,a12,a32,a14,a34,a16,a33,a15,a61,a05,a43,a51,b33), then there are 14 small amplitude limit cycles bifurcated from the origin.



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