# Integrability and Pseudo-Linearizable Conditions in a Quasi-Analytic System

Authors: Feng Li; Yusen Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with the problems of integrability and linearizable conditions at
degenerate singular point in a class of quasianalytic septic polynomial differential
system. We solve the problems by an indirect method, that is, we transform the quasianalytic
system into an analytic system firstly, and the degenerate singular point into
an elementary singular point. Then we calculate the singular values at the origin of
the analytic system by the known classical methods. We obtain the center conditions
and isochronous center conditions. Accordingly, integrability and pseudolinearizable
conditions at degenerate singular point in the quasianalytic system are obtained. Especially, whenλ=1, the system has been studied in Wu and Zhang (2010).


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## Body

## 1. Introduction

In the qualitative theory of planar polynomial differential equations, one of open problems for planar polynomial differential systems(1.1)dxdt=P(x,y),dydt=Q(x,y),
is how to characterize their centers and isochronous centers. The characterization of centers for concrete families of differential equations is solved theoretically by computing the so-called Lyapunov constants. In most cases the procedure to study all centers is as follows: compute several Lyapunov constants and when you get one significant, that is, zero, try to prove that the system obtained indeed has a center. Nevertheless, to completely solve this problem, there are two main obstacles. How can you be sure that you have computed enough Lyapunov constants? How do you prove that some system candidate to have a center actually has a center? As far as the case of the center is concerned, a lot of work has been done. Here we will not give an exhaustive bibliography.In the case of a center, it makes sense to locally define a period function associated with a center, whose value at any point is the minimum period of the periodic orbit through the point. A center is said to beisochronous if the associated period function is constant. It is well known that isochronous centers are nondegenerate and systems with an isochronous center can be locally linearized by an analytic change of coordinates in a neighborhood of the center. The problem of characterizing isochronous centers of the origin has attracted the attention of several authors, and many good results have been published. The characterization of isochronous centers has been treated by several authors. However, there is a low number of families of polynomial systems for which there is a complete classification of their isochronous centers. For example, quadratic isochronous center [1]; isochronous centers of a linear center perturbed by third, fourth, and fifth degree homogeneous polynomials [2–4]; the cubic system of Kukles [5, 6]; the class of systems which in complex variable  z=x+iy  writes as dz/dt=iP(z)=iz+o(z) (all of which have an isochronous center at the origin) and the cubic time-reversible systems with dφ/dt=1, see [7]; some isochronous cubic systems with four invariant lines, see [8]; isochronous centers of cubic systems with degenerate infinity [9, 10]; isochronous center conditions of infinity for rational systems [11–13]; and so forth. For more details about centers and isochronous centers, we refer the reader to the [14, 15].Theory of center focus for a class of higher-degree critical points was established in [16], the authors there considered the following polynomial differential system:(1.2)dxdt=(δx-y)(x2+y2)n+∑k=2n+2∞Xk(x,y),dydt=(x+δy)(x2+y2)n+∑k=2n+2∞Yk(x,y),
where(1.3)Xk(x,y)=∑α+β=kAαβxαyβ,Yk(x,y)=∑α+β=kBαβxαyβ.
By using their transformation(1.4)x=ξ(ξ2+η2)n+1,y=η(ξ2+η2)n+1,dt=(ξ2+η2)-n(2n+3)dτ,
system (1.2) becomes(1.5)dξdτ=δ2n+3ξ-η+∑k=2n+2∞[(12n+3ξ2+η2)Xk(ξ,η)-2n+22n+3ξηYk(ξ,η)]×(ξ2+η2)(k-2n-2)(n+1),dηdτ=ξ+δ2n+3η+∑k=2n+2∞[(ξ2+12n+3η2)Yk(ξ,η)-2n+22n+3ξηXk(ξ,η)]×(ξ2+η2)(k-2n-2)(n+1).Furthermore, Liu in [16] gave the definition of singular value and pseudo-isochronous center at a degenerate singular point.Definition 1.1.
The degenerate singular point of system (1.2)δ=0 is called a pseudo-isochronous center if the origin of system (1.5)δ=0 is an isochronous center.The problems of center conditions and pseudo-isochronous center conditions for degenerate singular point are poorly understood in the qualitative theory of ordinary differential equations. There are only a few papers concerning centers of degenerate singular points [17–24].Recently, the following systems:(1.6)ż=(λ+i)z+(zz̅)(d-5)/2(Az4+jz̅1-j+Bz3z̅2+Cz2-jz̅3+j+Dz̅5),d=2m+1≥5,ż=iz+(zz̅)(d-4)/2(Az3z̅+Bz2z̅2+Cz̅4),d=2m≥4,ż=(λ+i)z+(zz̅)(d-3)/2(Az3+Bz2z̅+Czz̅2+Dz̅3),d=2m+1≥3,ż=(λ+i)z+(zz̅)(d-2)/2(Az2+Bzz̅+Cz̅2),d=2m≥2
were investigated by Llibre and Valls, see [25–28]. The conditions of centers and isochronous centers were obtained. But the d is restricted in order to assure the system is polynomial system. In [29], centers and isochronous centers for two classes of generalized seventh and ninth systems were investigated. In [30], linearizable conditions of a time-reversible quartic-like system were obtaied.For the case of nonanalytic, being difficult, there are very few results. As far as integrability at origin are concerned, several special systems have been studied, see [31–34].In this paper, we investigate integrability and linearizable conditions at degenerate singular point for a class of quasanalytic polynomial differential system(1.7)dxdt=(δx-y)(x2+y2)λ+X5(x,y)(x2+y2)2(λ-1)-βy(x2+y2)3λ,dydt=(x+δy)(x2+y2)λ+Y5(x,y)(x2+y2)2(λ-1)+βx(x2+y2)3λ,
where(1.8)X5(x,y)=∑k+j=5Akjxkyj,Y5(x,y)=∑k+j=5Bkjxkyj,(1.9)A50=β03+β12+β21+β30,A41=-5α03-3α12-α21+α30,A32=-2(5β03+β12-β21-β30),A23=2(5α03-α12-α21+α30),A14=5β03-3β12+β21+β30,A05=-α03+α12-α21+α30,B50=α03+α12+α21+α30,B41=5β03+3β12+β21-β30,B32=-2(5α03+α12-α21-α30),B23=-2(5β03-β12-β21+β30),B14=5α03-3α12+α21+α30,B05=β03-β12+β21-β30,λ∈R.
When λ=1, the system has been invested in [35].The organization of this paper is as follows. In Section2, we introduce some preliminary results which are useful throughout this paper. In Section 3, we make two appropriate transformations which let research on the degenerate singular point of system (1.7) be reduced to research on the elementary singular point of a twenty-one degree system. Furthermore, we compute the singular point quantities and derive the center conditions of the origin for the transformed system. Accordingly, the conditions of integrability at the degenerate singular point are obtained. In Section 4, we compute the period constants and discuss isochronous center conditions at the origin of the twenty-one degree system, meanwhile, the pseudolinearizable conditions at degenerate singular point are classified.All calculations in this paper have been done with the computer algebra system: MATHEMATICA.


## 2. Some Preliminary Results

In [36–38], the authors defined complex center and complex isochronous center for the following complex system:(2.1)dzdT=z+∑k=2∞Zk(z,w)=Z(z,w),dwdT=-w-∑k=2∞Wk(z,w)=-W(z,w),
where(2.2)Zk(z,w)=∑α+β=kaαβzαwβ,Wk(z,w)=∑α+β=kbαβwαzβ,
and gave two recursive algorithms to determine necessary conditions for a center and an isochronous center. We now restate the definitions and algorithms.By means of transformation(2.3)z=ρeiθ,w=ρe-iθ,T=it,i=-1,
where r,θ are complex numbers, system (2.1) can be transformed into(2.4)dρdt=iρ2∑k=1∞∑α+β=k+2(aα,β-1-bβ,α-1)ei(α-β)θρk,dθdt=1+12∑k=1∞∑α+β=k+2(aα,β-1+bβ,α-1)ei(α-β)θρk.For the complex constanth, |h|≪1, we write the solution of system (2.4) satisfying the initial condition ρ∣θ=0=h as(2.5)r=ρ̃(θ,h)=h+∑k=2∞νk(θ)hk,
which could be thought of as the first Poincaré displacement map and denote the period function by(2.6)τ(φ,h)=∫0φdtdθdθ=∫0φ[1+12∑k=1∞∑α+β=k+2(aα,β-1+bβ,α-1)ei(α-β)θr̃k(θ,h)]-1dθ.Definition 2.1.
For a sufficiently small complex constanth,the origin of system (2.1) is called a complex center if ρ̃(2π,h)≡h,and it is called a complex isochronous center if
(2.7)ρ̃(2π,h)≡h,τ(2π,h)≡2π.Lemma 2.2.
For system (2.1), one can derive uniquely the following formal series:
(2.8)ξ=z+∑k+j=2∞ckjzkwj,η=w+∑k+j=2∞dkjwkzj,
where ck+1,k=dk+1,k=0, k=1,2,…, such that
(2.9)dξdT=ξ+∑j=1∞pjξj+1ηj,dηdT=-η-∑j=1∞qjηj+1ξj.Definition 2.3 (see [37, 38]).
Letμ0=0,  μk=pk-qk,  τk=pk+qk,  k=1,2,….μk be called the kth singular point quantity of the origin of system (2.1) and τk be called the kth period constant of the origin of system (2.1).Reeb’s criterion (see for instance [39]) says that system (2.1) has a center if and only if there is a nonzero analytic integrating factor (or integral factor) in a neighborhood of the origin. In [16], it is developed an algorithm to compute the focal values through the analytic integrating factor that must exist when we have a center, namely, the following theorem.Theorem 2.4 (see [16]).
For system (2.1), one can derive successively the terms of the following formal series:
(2.10)M(z,w)=∑α+β=0∞cαβzαwβ,
such that
(2.11)∂(MZ)∂z-∂(MW)∂w=∑m=1∞(m+1)μm(zw)m,
where c00=1, for all ckk∈R, k=1,2,…, and for any integer m, μm is determined by the following recursive formulae:
(2.12)c00=1,when(α=β>0)orα<0,orβ<0,cαβ=0,elsecαβ=1β-α∑k+j=3α+β+2[(α+1)ak,j-1-(β+1)bj,k-1]cα-k+1,β-j+1,μm=∑k+j=32m+2(ak,j-1-bj,k-1)cm-k+1,m-j+1.Theorem 2.5 (see [37]).
For system (2.1), one can derive uniquely the following formal series:
(2.13)f(z,w)=z+∑k+j=2∞ckj′zkwj,g(z,w)=w+∑k+j=2∞dkj′wkzj,
where ck+1,k′=dk+1,k′=0, k=1,2,…, such that
(2.14)dfdT=f(z,w)+∑j=1∞pj′zj+1wj,dgdT=-g(z,w)-∑j=1∞qj′wj+1zj,
and when k-j-1≠0, ckj′ and dkj′ are determined by the following recursive formulae:
(2.15)ckj′=1j+1-k∑α+β=3k+j+1[(k-α+1)aα,β-1-(j-β+1)bβ,α-1]ck-α+1,j-β+1′,dkj′=1j+1-k∑α+β=3k+j+1[(k-α+1)bα,β-1-(j-β+1)aβ,α-1]dk-α+1,j-β+1′,
and for any positive integer j,  pj′, and qj′ are determined by the following recursive formulae:
(2.16)pj′=∑α+β=32j+2[(j-α+2)aα,β-1-(j-β+1)bβ,α-1]cj-α+2,j-β+1′,qj′=∑α+β=32j+2[(j-α+2)bα,β-1-(j-β+1)aβ,α-1]dj-α+2,j-β+1′.
In the above expression, one has let c10′=d10′=1,  c01′=d01′=0, and if α<0 or β<0, let aαβ=bαβ=cαβ′=dαβ′=0.
We introduce double parameter transformation groups(2.17)z=ρ̅eiθz̃,w=ρ̅e-iθw̃,
where z̃,w̃ are new variables, ρ̅,θ are complex parameters, and ρ̅≠0. Denote z=x+iy,  w=x-iy,  z̃=x̃+iỹ,  w̃=x̃-iỹ. Transformation (2.17) can be turned into
(2.18)x=ρ̅(x̃cos⁡θ-ỹsin⁡θ),y=ρ̅(x̃sin⁡θ+ỹcos⁡θ).
In the case of real variables and real parameters, (2.18) is a transformation of similar rotation. With (2.17) being used, system (2.1) can be transformed into
(2.19)dz̃dT=z̃+∑α+β=2∞ãαβz̃αw̃β,dw̃dT=-w̃-∑α+β=2∞b̃αβw̃αz̃β,
where ρ̅, θ are parameters, z̃, w̃, T are variables, and for all α≥0, β≥0 one has
(2.20)ãαβ=aαβρ̅α+β-1ei(α-β-1)θ,b̃αβ=bαβρ̅α+β-1e-i(α-β-1)θ.
Under the transformation (2.17), suppose that f=f(aαβ,bαβ) is a polynomial of aαβ, bαβ with complex coefficients, and denote
(2.21)f̃=f(ãαβ,b̃αβ),f*=f(aαβ*,bαβ*),
where aαβ*=bαβ,  bαβ*=aαβ,  α≥0,  β≥0,  α+β≥2.Definition 2.6 (see [38]).
Suppose that there exist constantsλ,σ, such that f̃=ρ̅λeiσθf, we say that λ is a similar exponent and σ a rotation exponent of system (2.1) under the transformation (2.17), which are denoted by Is(f)=λ,Iρ(f)=σ.Definition 2.7 (see [38]).
(i) A polynomialf=f(aαβ,bαβ) is called a Lie invariant of order k, if f̃=ρ̅2kf.
(ii) An invariantf is called a monomial Lie invariant, if f is both of a Lie invariant and a monomial of aαβ,bαβ.
(iii) A monomial Lie invariantf is called an elementary Lie invariant, if it can not be expressed as a product of two monomial Lie invariants.Definition 2.8 (see [38]).
A polynomialf=f(aαβ,bαβ) is called self-symmetry if f*=f. It is called self-antisymmetry if f*=-f.Theorem 2.9 (see the extended symmetric principle in [38]).
Letg denote  an elementary Lie invariant of system (2.1). If for all g the symmetric condition g=g* is satisfied, then the origin of system (2.1) is a complex center. Namely, all singular point quantities of the origin are zero.


## 3. Integrability at the Origin of (1.7)

In this section, the integrability at the origin of (1.7) is discussed by an indirect method. By means of transformation(3.1)u=x+iy,v=x-iy,T=it,i=-1,
system (1.7)δ=0 becomes its concomitant complex system(3.2)dudT=u(uv)λ+(uv)2(λ-1)(a03u5+a12vu4+a21u3v2+a30u2v3)-βu(uv)3λ,dvdT=-v(uv)λ-(uv)2(λ-1)(b03v5+b12v4u+b21u2v3+b30u3v2)+βv(uv)3λ,
where(3.3)a30=α30+iβ30,a21=α21+iβ21,a12=α12+iβ12,a03=α03+iβ03,b30=α30-iβ30,b21=α21-iβ21,b12=α12-iβ12,b03=α03-iβ03.Then, by using transformation(3.4)ξ=u(λ+1)/2v(λ-1)/2,η=v(λ+1)/2u(λ-1)/2,
system (3.2) can be transformed into the following system:(3.5)dξdT=ξ2η+12a03(1+λ)ξ5+12(a12+b30+a12λ-b30λ)ξ4η+12(a21+b21+a21λ-b21λ)ξ3η2+12(a30+b12+a30λ-b12λ)η3ξ2+12b03(-1+λ)ξη4+βξ4η3,dηdT=-η2ξ-12b03(1+λ)η5-12(b12+a30+b12λ-a30λ)η4ξ-12(b21+a21+b21λ-a21λ)η3ξ2-12(b30+a12+b30λ-a12λ)η2ξ3-12a03(-1+λ)ηξ4+βξ3η4.At last, by means of transformation (1.4)n=1, system (3.5) is reduced to(3.6)dzdτ=z+110w3z4(-b03(-5+λ)w4+(-b12(-5+λ)+a30(5+λ))w3z+(-b21(-5+λ)+a21(5+λ))w2z2+(-b30(-5+λ)+a12(5+λ))wz3+a03(5+λ)z4)+βw10z11,dwdτ=-w-110w4z3(b03(-5+λ)w4+(b12(5+λ)-a30(-5+λ))w3z+(b21(-5+λ)-a21(-5+λ))w2z2+(b30(5+λ)-a12(-5+λ))wz3-a03(-5+λ)z4)+βz10w11.By those transformations, we transform the quasanalytic system into an analytic system firstly, and the degenerate singular point into an elementary singular point. Under the conjugate condition (3.6): it is obvious that the origin of system (3.5) to be integrability (linearizable) is equivalent to the degenerate singular point of system (1.7) to be integrability (pseudolinearizable).Using the recursive formulae of Theorem2.4 to compute the singular point quantities at the origin of system (3.6) (for detailed recursive formulae, see Appendix A) and simplify them with the constructive theorem of singular point quantities, we get the following.Theorem 3.1.
The first 55 singular point quantities at the origin of system (3.6) are as follows:
(3.7)μ5=15(a21-b21)λ,μ10=-15(a30a12-b30b12)λ.
Case 1.
a12b12≠0, then there exist k to make a30=kb12,  b30=ka12,
(3.8)μ15=λ40(a03b122-b03a122)(-1+3k)(2+2k-λ+kλ),μ20=λ210(λ+2)b21(a03b122-b03a122)(3k-1),μ25=λ2240(λ+2)3(3k-1)(a03b122-b03a122)×(-32a03b03-28a03b03λ+32a12b12λ2+5a03b03λ3+a03b03λ4+192β+192λβ+48λ2β),μ30=0,μ35=-λ219200(λ+2)5(3k-1)(a03b122-b03a122)×(1024a032b032+1920a032b032λ-13824a12a03b12b03λ2+224a032b032λ2-11392a12a03b12b03λ3-1584a032b032λ3+12800a122b122λ4+2432a12a03b12b03λ4-1056a032b032λ4+4064a12a03b12b03λ5+864a12a03b12b03λ6+206a032b032λ6+69a032b032λ7+7a032b032λ8),μ40=-7λ22880(λ+2)5(3k-1)(a03b122-b03a122)(a03b122+b03a122)(λ+1)×(-32a03b03-24a03b03λ+32a12b12λ2+4a03b03λ2+6a03b03λ3+a03b03λ4).
If a12b12=-a03b03(-2+λ)(2+λ)2(4+λ)/32λ2,
(3.9)μ45=2413167184640(3k-1)a033b033(a03b122-b03a122).
If a03b122+b03a122=0, then there exist m to make a03=ma122,  b03=-mb122,
(3.10)μ45=7λ24838400000(λ+2)3(3k-1)(λ+1)a12b12a032b032(a03b122-b03a122)×(-23257088a12b12m2+6577280a12b12m2λ-23257088λ2+26650064a12b12m2λ2+75164416λ3-2764244a12b12m2λ3+8304896λ4-10922212a12b12m2λ4-18884608λ5-916685a12b12m2λ5+1341691a12b12m2λ6+255854a12b12m2λ7),μ50=0,μ55=-1099511627776λ198859375m8(λ+2)17(3k-1)(a03b122-b03a122)(λ+1).Case 2.
a12=b12=0,
(3.11)μ15=3λ40(a03a302-b03b302)(λ+2),μ20=310(a03a302-b03b302)b21,μ25=-140(a03a302-b03b302)(4a30b30-3a03b03+24β),μ30=0,μ35=-1400(a03a302-b03b302)(11a032b032-19a03b03a30b30-50a302b302),μ40=7480(a03a302-b03b302)(a03a302+b03b302)(a03b03-a30b30),μ45=-11344000a303b303(a03a302-b03b302)×(102334400+28122192a30b30m2-26626536a302b302m4+2330697a303b303m6),
where μk=0, k≠5i,  i≤11,  i∈N. In the above expression of μk, we have already let μ1=⋯=μk-1=0, k=2,3,…,45.From Theorem3.1, we get the following.Theorem 3.2.
For system (3.6), the first 55 singular point quantities are zero if and only if one of the following conditions holds:
(3.12)a21=b21,a12=b12=0,a302a03=b03b302,(3.13)a21=b21,a30=13b12,b30=13a12,a12b12≠0,(3.14)a21=b21,a30a12=b30b12,a122b03=b122a03,a12b12≠0,(3.15)λ=-1,β=0,a21=b21=0,a30=-3b12,b30=-3a12,a03b03=4a12b12,a12b12≠0.In order to obtain the integrability conditions of the origin, we have to find out all the elementary Lie invariants of system (3.6). According to Definitions 2.6, 2.7 and 2.8, we have the following.Lemma 3.3.
All the elementary Lie invariants of system (3.6) are as follows:
(3.16)β,a21,b21,a30b30,a12b12,a03b03,a30a12,b30b12,a302a03,a30b12a03,b122a03,b302b03,b30a12b03,a122b03.The following result holds.Theorem 3.4.
For system (3.6), all the singular point quantities at the origin are zero if and only if the first 55 singular point quantities are zero, that is, one of the conditions in Theorem 3.2 holds. Correspondingly, the conditions in Theorem 3.2 are the integrability conditions of the origin.Proof.
If condition (3.12) or (3.14) holds, system (3.5)δ=0 satisfies the conditions of Theorem 2.9. If condition (3.13) holds, system (3.6) has the first integral
(3.17)zwe3a03z7w3+4a12z6w4+6b21z5w5+4b12z4w6+3b03z3w7+3z10w10β-3,λ=-2,(zw)5(λ+2)/λf1,λ≠-2,
where
(3.18)f1=(-12a03+3a03λ2)z7w3+(4a41λ2-16a12)z6w4+(6b21λ2-24b21)z5w5+(4b12λ2+16b12)z4w6+(3b03λ2-12b03)z3w7+(24β-12rβ)z10w10-24+12λ.
If condition (3.15) holds, system (3.6) becomes
(3.19)dzdT=110(10+4a03z7w3-14a12z6w4-6b12z4w6+6b03z3w7),dwdT=-110(10+4b03z3w7-14b12w6z4-6a12w4z6+6a03w3z7),
there exists a transformation
(3.20)z=u(uv)2/5,w=v(uv)2/5,
system (3.19) is changed into
(3.21)dudT=u+b03v3+b12v2u-3a12u3=U,dvdT=-(v+a03u3+a12u2v-3b12v3)=V,
system (3.21) has the integral factor f2-5/6, where
(3.22)f2=1-6(b12u2+a12v2)+3(3b122u4-2a12b03u3v+2a12b12u2v2-2b12a03v3u+3a124v4)+12(2a12u-a03v)(2b12v-b03u)(b03u4-2b12u3v-2a12v3u+a03v4),df2dt=-12(b12u2-a12v2)f2=65(∂U∂u-∂V∂v)f2.Synthesizing all the above cases, we have the following.Theorem 3.5.
The system (1.7) is integrability at the origin if and only if one of conditions in Theorem 3.2 holds.


## 4. Linearizable Conditions at the Origin of (1.7)

In this section we classify the pseudolinearizable conditions at the origin of (1.7). We discuss the linearizable conditions for system (3.6) firstly. According to Theorem 2.5, we get the recursive formulae to compute period constants (detailed recursive formulae, see Appendix B). Denote a21=b21=r21, from the integrability conditions given in Section 4, we investigate the following three cases.Case 1.
Integrability condition (3.12) holds.
Ifa30=b30=0, we easily obtain the first 30 period constants
(4.1)τ5=2r21,τ10=12((λ-2)a03b03+4β),τ15=0,τ20=-132a032b032(-2+λ)2(-2+3λ),τ25=0,τ30=125600a032b032(-2+λ)2(50+365λ-1962λ2+2153λ3).
Ifa30b30≠0, from condition (3.12), there exists an arbitrary complex constant s, such that
(4.2)a03=sb302,b03=sa302,
then we get the first 30 period constants
(4.3)τ5=2r21,τ10=12(-2a30b30-2a302b302s2-2a30b30λ+a302b302s2λ+4β),τ15=34a302b302s(λ+2),τ20=14a302b302(1+λ)2(3λ+1),τ25=0,τ30=-1800a303b303(1+λ)2(-25+35λ+981λ2+2153λ3).
In expressions (4.1) and (4.3), τk=0, k≠5i, i≤4, i∈N, and we have already let τ1=⋯=τk-1=0, k=2,3,…,30.From expressions (4.1) and (4.3), we have the following.Theorem 4.1.
The first 30 period constants at the origin of system (3.6) are zero if and only if one of the following conditions holds:
(4.4)β=a21=b21=a12=b12=a30=b30=0,λ=2.(4.5)β=a21=b21=a12=b12=a03=b03=0,λ=-1,a30b30≠0.Theorem 4.2.
Under integrability condition (3.12), the origin of system (3.6) is a complex isochronous center if and only if one of the conditions in Theorem 4.1 holds.Proof.
When condition (4.4) is satisfied, system (3.6) becomes
(4.6)dzdT=110z(10+7a03z7w3+3b03z3w7),dwdT=-110w(10+3a03z7w3+7b03z3w7).
There exists a transformation
(4.7)u=z(1+b03z3w7)3/40(1+a03z7w3)7/40,v=w(1+a03z7w3)3/40(1+b03z3w7)7/40,
such that system (4.6) is reduced to a linear system.
When condition (4.5) is satisfied, system (3.6) becomes
(4.8)dzdT=110z(10+6b30z6w4+4a30z4w6),dwdT=-110w(10+4b30z4w6+6a30z6w4).
There exists a transformation
(4.9)u=z(1+a30z4w6)1/5(1+b30z6w4)3/10,v=w(1+b30z6w4)1/5(1+a30z4w6)3/10,
such that system (4.8) is reduced to a linear system.Case 2.
Integrability condition (3.13) holds.Substituting condition (3.13) into the recursive formulae in Appendix B, we obtain the first 40 period constants(4.10)τ5=2r21,τ10=118(-18a03b03-32a12b12+9a03b03λ+16a12b12λ+36β),τ15=13(a122b03+a03b122)(-4+λ)(-1+λ),τ20=-12592a122b122(λ-2)(3λ+2)(-512-1152a12b12s2-162a122b122s4+256λ+81a122b122s4λ),τ25=0,τ30=19(256+81a122b122s4)52048a125b125s4(256+720a12b12s2+81a122b122s4)×(256+1152a12b12s2+81a122b122s4)(-262144-2801664a12b12s2+4147200a122b122s4-2192832a123b123s6+183708a124b124s8+59049a125b125s10),τ35=0,τ40=12025(256+81a122b122s4)764a126b126s4×(b1214s28109239312561498750976+2393515709059622240256a12b12s2+16627534144531656081408a122b122s4+45808666111836080308224a123b123s6+34487314499860709769216a124b124s8-43671898596853816492032a125b125s10-31656165840700764585984a126b126s12+49193367282534498435072a127b127s14-2655678216972439388160a128b128s16-8653122543235415408640a129b129s18-539890165260393578496a1210b1210s20+408937711778677748736a1211b1211s22+82987106695640213760a1212b1212s24+5972974608886179588a1213b1213s26+151421489259386859a1214b1214s28),
where τk=0, k≠5i, i≤8, i∈N. In the above expression of τk, we have already let τ1=⋯=τk-1=0,  k=2,3,…,40.From expressions (4.10), we have the following.Theorem 4.3.
The first 40 period constants at the origin of system (3.6) are zero if and only if one of the following conditions holds:
(4.11)λ=2,β=a21=b21=a03=b03=0,a30=13b12,b30=13a12.Theorem 4.4.
Under integrability condition (3.13), the origin of system (3.6) is a complex isochronous center if and only if the condition in Theorem 4.3 holds.Proof.
When condition (4.11) is satisfied, system (3.6) becomes
(4.12)dzdT=110z(10+8a12z6w4+163b12z4w6),dwdT=-110w(10+8b12z4w6+163a12z6w4).
There exists a transformation
(4.13)u=z(3+4b12z4w6)1/5(3+4a12z6w4)3/10,v=w(3+4a12z6w4)1/5(3+4b12z4w6)3/10,
such that system (4.12) is reduced to a linear system.Case 3.
Integrability condition (3.14) holds.
Becausea12b12≠0, we can let
(4.14)a30=kb12,b30=ka12,a03=ma122,b03=mb122,
where k,m are arbitrary complex constants. Substituting (4.14) into the recursive formulae in Appendix B, we obtain the first 30 period constants
(4.15)τ5=2r21,τ10=12(-2a12b12-4a12b12k-2a12b12k2-2a122b122m2+2a12b12λ-2a12b12k2λ+a122b122m2λ+4β),τ15=14a122b122m(3+3k-4λ)(2+2k-λ+kλ).
If m=0,
(4.16)τ20=14a122b122(1+k)(1+k-λ+kλ)2(1+k-3λ+3kλ),τ25=0,τ30=1(1+3λ)32a123b123(1+k)λ3(1+k-λ+kλ)2.
If k=(1/3)(-3+4λ),
(4.17)τ20=-12592a122b122(-648a122b122m4+1620a122b122m4μ-1152a12b12m2λ2-1134a122b122m4λ2+4608a12b12m2λ3+243a122b122m4λ3+1792r4-4608a12b12m2λ4-8704λ5+13312λ6-6144λ7),τ25=-11944a123b123mλ2(-864a12b12m2-459a12b12m2λ+576λ2+6075a12b12m2λ2-256λ3-5670a12b12m2λ3-5888λ4+1296a12b12m2λ4+11264λ5-6144λ6),τ30=-1492075(-32-17λ+225λ2-210λ3+48λ4)34a123b123λ6(1+λ)(-1+2λ)2×(λ1416223998464-25771071744λ-266392474620λ2+779771224771λ3+502091493918λ4-4814334558957λ5+7235802457920λ6-2358342484708λ7-5235799647872λ8+7279836287888λ9-3998287419904λ10+800816397376λ11+199732916736λ12-135353438208λ13+20141015040λ14).
If k=(λ-2)/(λ+2),
(4.18)τ20=-196(λ+2)4a122b122(-384a122b122m4+192a122b122m4λ-1536a12b12m2λ2+672a122b122m4λ2-1280a12b12m2λ3+48a122b122m4λ3+1920λ4+384a12b12m2λ4-264a122b122m4λ4+576a12b12m2λ5-60a122b122m4λ5+128a12b12m2λ6+30a122b122m4λ6+9a122b122m4λ7),τ25=-112(2+λ)4a123b123mλ2(96a12b12m2+76a12b12m2λ-96λ2-40a12b12m2λ2-32λ3-33a12b12m2λ3+5a12b12m2λ4+4a12b12m2λ5),τ30=2a123b123λ775(2+λ)6(24-5λ-11λ2+4λ3)3(-8200224-114773490λ+250179993λ2-32032585λ3-178032915λ4+94932055λ5+18512302λ6-24125520λ7+4769504λ8),
where τk=0,  k≠5i,  i≤6,  i∈N. In the above expression of τk, we have already let τ1=⋯=τk-1=0,  k=2,3,…,30.From expressions (4.15), (4.16), (4.17) and (4.18), we have the following.Theorem 4.5.
The first 30 period constants at the origin of system (3.6) are zero if and only if one of the following conditions holds:
(4.19)β=a21=b21=a03=b03=0,a30=-b12,b30=-a12,(4.20)β=a21=b21=a03=b03=0,a30=-1+λ1+λb12,b30=-1+λ1+λa12.Theorem 4.6.
Under integrability condition (3.14), the origin of system (3.6) is a complex isochronous center if and only if one of the conditions in Theorem 4.5 holds.Proof.
When condition (4.19) is satisfied, system (3.6) becomes
(4.21)dzdT=110z(10+2a12λz6w4-2b12λz4w6),dwdT=-15w(10+2b12λz4w6-2a12λz6w4),
we have for system (4.21) that
(4.22)dθdt=12(1zdzdT-1wdwdT)=1.
When condition (4.20) is satisfied, system (3.6) becomes
(4.23)dzdT=15(1+λ)z(5(1+λ)+4a12λz6w4+6b12λz4w6),dwdT=-15(1+λ)w(5(1+λ)+4b12λz4w6+6a12λz6w4).
There exists a transformation
(4.24)u=z(1+λ+2b12λz4w6)1/5(1+λ+2a12λz6w4)3/10,v=w(1+λ+2a12λz6w4)1/5(1+λ+2b12λz4w6)3/10,
such that system (4.23) is reduced to a linear system.Case 4.
Integrability condition (3.15) holds.
Substituting condition (3.15) into the recursive formulae in Appendix B, we obtain the first 10 period constants
(4.25)τ5=2r21,τ10=-2a12b12.
Because τ10=a12b12≠0, under integrability condition (3.15), the origin of system (3.5)δ=0 is not a complex isochronous center.Synthesizing all the above cases, we get the main result of this paper.Theorem 4.7.
The degenerate singular point (origin) of system system (1.7)δ=0 ((3.5)δ=0) is pseudo-linearizable (linearizable) if and only if one of conditions (4.4), (4.5), (4.11), (4.19), (4.20) holds.



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*Source: 783546-2012-03-22.xml*