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

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, when 𝜆=1, the system has been studied in Wu and Zhang (2010).

1. Introduction

In the qualitative theory of planar polynomial differential equations, one of open problems for planar polynomial differential systems 𝑑𝑥,𝑑𝑡=𝑃𝑥,𝑦𝑑𝑦,𝑑𝑡=𝑄𝑥,𝑦 (1.1) 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 be isochronous 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:𝑑𝑥=𝑥𝑑𝑡𝛿𝑥−𝑦²+𝑦^2𝑛+^∞_𝑘=2𝑛+2𝑋_𝑘𝑥,𝑦,𝑑𝑦=𝑥𝑑𝑡𝑥+𝛿𝑦²+𝑦^2𝑛+^∞_𝑘=2𝑛+2𝑌_𝑘𝑥,𝑦, (1.2) where𝑋_𝑘𝑥,𝑦=_{𝛼+𝛽=𝑘}𝐴_𝛼𝛽𝑥^𝛼𝑦^𝛽,𝑌_𝑘𝑥,𝑦=_{𝛼+𝛽=𝑘}𝐵_𝛼𝛽𝑥^𝛼𝑦^𝛽. (1.3) By using their transformation𝑥=𝜉𝜉²+𝜂^2𝑛+1,𝑦=𝜂𝜉²+𝜂^2𝑛+1,𝑑𝑡=𝜉²+𝜂^{2−𝑛(2𝑛+3)}𝑑𝜏, (1.4) system (1.2) becomes𝑑𝜉=𝛿𝑑𝜏2𝑛+3𝜉−𝜂+^∞_𝑘=2𝑛+21𝜉2𝑛+3²+𝜂²𝑋_𝑘𝜉,𝜂−2𝑛+22𝑛+3𝜉𝜂𝑌_𝑘×𝜉𝜉,𝜂₂+𝜂^{2(𝑘−2𝑛−2)(𝑛+1)},𝑑𝜂𝛿𝑑𝜏=𝜉+2𝑛+3𝜂+^∞_𝑘=2𝑛+2𝜉²+1𝜂2𝑛+3²𝑌_𝑘𝜉,𝜂−2𝑛+22𝑛+3𝜉𝜂𝑋_𝑘×𝜉𝜉,𝜂₂+𝜂^{2(𝑘−2𝑛−2)(𝑛+1)}. (1.5)

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:𝑧̇=(𝜆+𝑖)𝑧+(𝑧𝑧)^(𝑑−5)/2𝐴𝑧^4+𝑗𝑧^1−𝑗+𝐵𝑧³𝑧²+𝐶𝑧^2−𝑗𝑧^3+𝑗+𝐷𝑧⁵̇(𝑧,𝑑=2𝑚+1≥5,𝑧=𝑖𝑧+𝑧)^(𝑑−4)/2𝐴𝑧³𝑧+𝐵𝑧²𝑧²+𝐶𝑧⁴̇()(𝑧,𝑑=2𝑚≥4,𝑧=𝜆+𝑖𝑧+𝑧)^(𝑑−3)/2𝐴𝑧³+𝐵𝑧²𝑧+𝐶𝑧𝑧²+𝐷𝑧³̇()(𝑧,𝑑=2𝑚+1≥3,𝑧=𝜆+𝑖𝑧+𝑧)^(𝑑−2)/2𝐴𝑧²+𝐵𝑧𝑧+𝐶𝑧²,𝑑=2𝑚≥2 (1.6) 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 system𝑑𝑥=𝑥𝑑𝑡𝛿𝑥−𝑦²+𝑦^2𝜆+𝑋₅𝑥,𝑦𝑥²+𝑦^22(𝜆−1)−𝛽𝑦𝑥²+𝑦^23𝜆,𝑑𝑦=𝑥𝑑𝑡𝑥+𝛿𝑦²+𝑦^2𝜆+𝑌₅𝑥,𝑦𝑥²+𝑦^22(𝜆−1)+𝛽𝑥𝑥²+𝑦^23𝜆, (1.7) where𝑋₅𝑥,𝑦=_𝑘+𝑗=5𝐴_𝑘𝑗𝑥^𝑘𝑦^𝑗,𝑌₅𝑥,𝑦=_𝑘+𝑗=5𝐵_𝑘𝑗𝑥^𝑘𝑦^𝑗,𝐴₅₀=𝛽₀₃+𝛽₁₂+𝛽₂₁+𝛽₃₀,𝐴₄₁=−5𝛼₀₃−3𝛼₁₂−𝛼₂₁+𝛼₃₀,𝐴₃₂=−25𝛽₀₃+𝛽₁₂−𝛽₂₁−𝛽₃₀,𝐴₂₃=25𝛼₀₃−𝛼₁₂−𝛼₂₁+𝛼₃₀,𝐴₁₄=5𝛽₀₃−3𝛽₁₂+𝛽₂₁+𝛽₃₀,𝐴₀₅=−𝛼₀₃+𝛼₁₂−𝛼₂₁+𝛼₃₀,𝐵₅₀=𝛼₀₃+𝛼₁₂+𝛼₂₁+𝛼₃₀,𝐵₄₁=5𝛽₀₃+3𝛽₁₂+𝛽₂₁−𝛽₃₀,𝐵₃₂=−25𝛼₀₃+𝛼₁₂−𝛼₂₁−𝛼₃₀,𝐵₂₃=−25𝛽₀₃−𝛽₁₂−𝛽₂₁+𝛽₃₀,𝐵₁₄=5𝛼₀₃−3𝛼₁₂+𝛼₂₁+𝛼₃₀,𝐵₀₅=𝛽₀₃−𝛽₁₂+𝛽₂₁−𝛽₃₀,𝜆∈𝑅. (1.8)(1.9) When 𝜆=1, the system has been invested in [35].

The organization of this paper is as follows. In Section 2, 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:𝑑𝑧𝑑𝑇=𝑧+^∞_𝑘=2𝑍_𝑘(𝑧,𝑤)=𝑍(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝑤−^∞_𝑘=2𝑊_𝑘(𝑧,𝑤)=−𝑊(𝑧,𝑤), (2.1) where𝑍_𝑘(𝑧,𝑤)=_{𝛼+𝛽=𝑘}𝑎_𝛼𝛽𝑧^𝛼𝑤^𝛽,𝑊_𝑘(𝑧,𝑤)=_{𝛼+𝛽=𝑘}𝑏_𝛼𝛽𝑤^𝛼𝑧^𝛽, (2.2) 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 transformation𝑧=𝜌𝑒^𝑖𝜃,𝑤=𝜌𝑒^−𝑖𝜃,𝑇=𝑖𝑡,𝑖=√−1, (2.3) where 𝑟,𝜃 are complex numbers, system (2.1) can be transformed into𝑑𝜌=𝑑𝑡𝑖𝜌2^∞_{𝑘=1𝛼+𝛽=𝑘+2}𝑎_{𝛼,𝛽−1}−𝑏_{𝛽,𝛼−1}𝑒^{𝑖(𝛼−𝛽)𝜃}𝜌^𝑘,𝑑𝜃1𝑑𝑡=1+2^∞_{𝑘=1𝛼+𝛽=𝑘+2}𝑎_{𝛼,𝛽−1}+𝑏_{𝛽,𝛼−1}𝑒^{𝑖(𝛼−𝛽)𝜃}𝜌^𝑘. (2.4)

For the complex constant ℎ, |ℎ|≪1, we write the solution of system (2.4) satisfying the initial condition 𝜌∣_𝜃=0=ℎ as𝑟=𝜌(𝜃,ℎ)=ℎ+^∞_𝑘=2𝜈_𝑘(𝜃)ℎ^𝑘, (2.5) which could be thought of as the first Poincaré displacement map and denote the period function by𝜏𝜑,ℎ=^𝜑₀𝑑𝑡=𝑑𝜃𝑑𝜃_𝜑011+2^∞_{𝑘=1𝛼+𝛽=𝑘+2}𝑎_{𝛼,𝛽−1}+𝑏_{𝛽,𝛼−1}𝑒^{𝑖(𝛼−𝛽)𝜃}𝑟^𝑘̃(𝜃,ℎ)⁻¹𝑑𝜃. (2.6)

Definition 2.1. For a sufficiently small complex constant ℎ, the origin of system (2.1) is called a complex center if 𝜌(2𝜋,ℎ)≡ℎ, and it is called a complex isochronous center if 𝜌(2𝜋,ℎ)≡ℎ,𝜏(2𝜋,ℎ)≡2𝜋. (2.7)

Lemma 2.2. For system (2.1), one can derive uniquely the following formal series: 𝜉=𝑧+^∞_𝑘+𝑗=2𝑐_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝜂=𝑤+^∞_𝑘+𝑗=2𝑑_𝑘𝑗𝑤^𝑘𝑧^𝑗, (2.8) where 𝑐_𝑘+1,𝑘=𝑑_𝑘+1,𝑘=0, 𝑘=1,2,…, such that 𝑑𝜉𝑑𝑇=𝜉+^∞_𝑗=1𝑝_𝑗𝜉^𝑗+1𝜂^𝑗,𝑑𝜂𝑑𝑇=−𝜂−^∞_𝑗=1𝑞_𝑗𝜂^𝑗+1𝜉^𝑗. (2.9)

Definition 2.3 (see [37, 38]). Let 𝜇₀=0,  𝜇_𝑘=𝑝_𝑘−𝑞_𝑘,  𝜏_𝑘=𝑝_𝑘+𝑞_𝑘,  𝑘=1,2,….𝜇_𝑘 be called the 𝑘_th singular point quantity of the origin of system (2.1) and 𝜏_𝑘 be called the 𝑘_th 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: 𝑀(𝑧,𝑤)=^∞_𝛼+𝛽=0𝑐_𝛼𝛽𝑧^𝛼𝑤^𝛽, (2.10) such that 𝜕(𝑀𝑍)−𝜕()𝜕𝑧𝑀𝑊=𝜕𝑤^∞_𝑚=1(𝑚+1)𝜇_𝑚(𝑧𝑤)^𝑚, (2.11) where 𝑐₀₀=1, for all 𝑐_𝑘𝑘∈𝑅, 𝑘=1,2,…, and for any integer 𝑚, 𝜇_𝑚 is determined by the following recursive formulae: 𝑐₀₀=1,when𝛼=𝛽>0or𝛼<0,or𝛽<0,𝑐_𝛼𝛽𝑐=0,else_𝛼𝛽=1𝛽−𝛼^𝛼+𝛽+2_𝑘+𝑗=3(𝛼+1)𝑎_{𝑘,𝑗−1}−𝛽+1𝑏_{𝑗,𝑘−1}𝑐_{𝛼−𝑘+1,𝛽−𝑗+1},𝜇_𝑚=^2𝑚+2_𝑘+𝑗=3𝑎_{𝑘,𝑗−1}−𝑏_{𝑗,𝑘−1}𝑐_{𝑚−𝑘+1,𝑚−𝑗+1}. (2.12)

Theorem 2.5 (see [37]). For system (2.1), one can derive uniquely the following formal series: 𝑓(𝑧,𝑤)=𝑧+^∞_𝑘+𝑗=2𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝑔(𝑧,𝑤)=𝑤+^∞_𝑘+𝑗=2𝑑^′_𝑘𝑗𝑤^𝑘𝑧^𝑗, (2.13) where 𝑐^′_𝑘+1,𝑘=𝑑^′_𝑘+1,𝑘=0, 𝑘=1,2,…, such that 𝑑𝑓()+𝑑𝑇=𝑓𝑧,𝑤^∞_𝑗=1𝑝^′_𝑗𝑧^𝑗+1𝑤^𝑗,𝑑𝑔()−𝑑𝑇=−𝑔𝑧,𝑤^∞_𝑗=1𝑞^′_𝑗𝑤^𝑗+1𝑧^𝑗, (2.14) and when 𝑘−𝑗−1≠0, 𝑐^′_𝑘𝑗 and 𝑑^′_𝑘𝑗 are determined by the following recursive formulae: 𝑐^′_𝑘𝑗=1𝑗+1−𝑘^𝑘+𝑗+1_𝛼+𝛽=3(𝑘−𝛼+1)𝑎_{𝛼,𝛽−1}−𝑗−𝛽+1𝑏_{𝛽,𝛼−1}𝑐^′_{𝑘−𝛼+1,𝑗−𝛽+1},𝑑^′_𝑘𝑗=1𝑗+1−𝑘^𝑘+𝑗+1_𝛼+𝛽=3(𝑘−𝛼+1)𝑏_{𝛼,𝛽−1}−𝑗−𝛽+1𝑎_{𝛽,𝛼−1}𝑑^′_{𝑘−𝛼+1,𝑗−𝛽+1}, (2.15) and for any positive integer 𝑗,  𝑝^′_𝑗, and 𝑞^′_𝑗 are determined by the following recursive formulae: 𝑝^′_𝑗=^2𝑗+2_𝛼+𝛽=3𝑗−𝛼+2𝑎_{𝛼,𝛽−1}−𝑗−𝛽+1𝑏_{𝛽,𝛼−1}𝑐^′_{𝑗−𝛼+2,𝑗−𝛽+1},𝑞^′_𝑗=^2𝑗+2_𝛼+𝛽=3𝑗−𝛼+2𝑏_{𝛼,𝛽−1}−𝑗−𝛽+1𝑎_{𝛽,𝛼−1}𝑑^′_{𝑗−𝛼+2,𝑗−𝛽+1}. (2.16) In the above expression, one has let 𝑐^′₁₀=𝑑^′₁₀=1,  𝑐^′₀₁=𝑑^′₀₁=0, and if 𝛼<0 or 𝛽<0, let 𝑎_𝛼𝛽=𝑏_𝛼𝛽=𝑐^′_𝛼𝛽=𝑑^′_𝛼𝛽=0.
We introduce double parameter transformation groups 𝑧=𝜌𝑒^𝑖𝜃𝑧,𝑤=𝜌𝑒^−𝑖𝜃𝑤, (2.17) where 𝑧,𝑤 are new variables, 𝜌,𝜃 are complex parameters, and 𝜌≠0. Denote 𝑧=𝑥+𝑖𝑦,  𝑤=𝑥−𝑖𝑦,  𝑧=𝑥+𝑖𝑦,  𝑤=𝑥−𝑖𝑦. Transformation (2.17) can be turned into 𝑥=𝜌𝑥cos𝜃−𝑦sin𝜃,𝑦=𝜌𝑥sin𝜃+𝑦cos𝜃. (2.18) 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 𝑑𝑧=𝑑𝑇𝑧+^∞_𝛼+𝛽=2𝑎_𝛼𝛽𝑧^𝛼𝑤^𝛽,𝑑𝑤𝑑𝑇=−𝑤−^∞_𝛼+𝛽=2𝑏̃_𝛼𝛽𝑤^𝛼𝑧^𝛽, (2.19) where 𝜌, 𝜃 are parameters, 𝑧, 𝑤, 𝑇 are variables, and for all 𝛼≥0, 𝛽≥0 one has 𝑎_𝛼𝛽=𝑎_𝛼𝛽𝜌^{𝛼+𝛽−1}𝑒^{𝑖(𝛼−𝛽−1)𝜃},𝑏̃_𝛼𝛽=𝑏_𝛼𝛽𝜌^{𝛼+𝛽−1}𝑒^{−𝑖(𝛼−𝛽−1)𝜃}. (2.20)
Under the transformation (2.17), suppose that 𝑓=𝑓(𝑎_𝛼𝛽,𝑏_𝛼𝛽) is a polynomial of 𝑎_𝛼𝛽, 𝑏_𝛼𝛽 with complex coefficients, and denote 𝑓=𝑓𝑎_𝛼𝛽,𝑏̃_𝛼𝛽,𝑓^∗=𝑓𝑎^∗_𝛼𝛽,𝑏^∗_𝛼𝛽, (2.21) where 𝑎^∗_𝛼𝛽=𝑏_𝛼𝛽,  𝑏^∗_𝛼𝛽=𝑎_𝛼𝛽,  𝛼≥0,  𝛽≥0,  𝛼+𝛽≥2.

Definition 2.6 (see [38]). Suppose that there exist constants 𝜆,𝜎, such that 𝑓=𝜌^𝜆𝑒^𝑖𝜎𝜃𝑓, 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 𝐼_𝑠(𝑓)=𝜆,𝐼_𝜌(𝑓)=𝜎.

Definition 2.7 (see [38]). (i) A polynomial 𝑓=𝑓(𝑎_𝛼𝛽,𝑏_𝛼𝛽) is called a Lie invariant of order 𝑘, if 𝑓=𝜌^2𝑘𝑓.
(ii) An invariant 𝑓 is called a monomial Lie invariant, if 𝑓 is both of a Lie invariant and a monomial of 𝑎_𝛼𝛽,𝑏_𝛼𝛽.
(iii) A monomial Lie invariant 𝑓 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 polynomial 𝑓=𝑓(𝑎_𝛼𝛽,𝑏_𝛼𝛽) 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]). Let 𝑔 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 transformation𝑢=𝑥+𝑖𝑦,𝑣=𝑥−𝑖𝑦,𝑇=𝑖𝑡,𝑖=√−1, (3.1) system (1.7)_δ=0 becomes its concomitant complex system𝑑𝑢()𝑑𝑇=𝑢𝑢𝑣^𝜆+(𝑢𝑣)^2(𝜆−1)𝑎₀₃𝑢⁵+𝑎₁₂𝑣𝑢⁴+𝑎₂₁𝑢³𝑣²+𝑎₃₀𝑢²𝑣³−𝛽𝑢(𝑢𝑣)^3𝜆,𝑑𝑣()𝑑𝑇=−𝑣𝑢𝑣^𝜆−(𝑢𝑣)^2(𝜆−1)𝑏₀₃𝑣⁵+𝑏₁₂𝑣⁴𝑢+𝑏₂₁𝑢²𝑣³+𝑏₃₀𝑢³𝑣²+𝛽𝑣(𝑢𝑣)^3𝜆, (3.2) where𝑎₃₀=𝛼₃₀+𝑖𝛽₃₀,𝑎₂₁=𝛼₂₁+𝑖𝛽₂₁,𝑎₁₂=𝛼₁₂+𝑖𝛽₁₂,𝑎₀₃=𝛼₀₃+𝑖𝛽₀₃,𝑏₃₀=𝛼₃₀−𝑖𝛽₃₀,𝑏₂₁=𝛼₂₁−𝑖𝛽₂₁,𝑏₁₂=𝛼₁₂−𝑖𝛽₁₂,𝑏₀₃=𝛼₀₃−𝑖𝛽₀₃. (3.3)

Then, by using transformation𝜉=𝑢^(𝜆+1)/2𝑣^(𝜆−1)/2,𝜂=𝑣^(𝜆+1)/2𝑢^(𝜆−1)/2, (3.4) system (3.2) can be transformed into the following system:𝑑𝜉𝑑𝑇=𝜉²1𝜂+2𝑎₀₃(1+𝜆)𝜉⁵+12𝑎₁₂+𝑏₃₀+𝑎₁₂𝜆−𝑏₃₀𝜆𝜉⁴1𝜂+2𝑎₂₁+𝑏₂₁+𝑎₂₁𝜆−𝑏₂₁𝜆𝜉³𝜂²+12𝑎₃₀+𝑏₁₂+𝑎₃₀𝜆−𝑏₁₂𝜆𝜂³𝜉²+12𝑏₀₃(−1+𝜆)𝜉𝜂⁴+𝛽𝜉⁴𝜂³,𝑑𝜂𝑑𝑇=−𝜂²1𝜉−2𝑏₀₃(1+𝜆)𝜂⁵−12𝑏₁₂+𝑎₃₀+𝑏₁₂𝜆−𝑎₃₀𝜆𝜂⁴1𝜉−2𝑏₂₁+𝑎₂₁+𝑏₂₁𝜆−𝑎₂₁𝜆𝜂³𝜉²−12𝑏₃₀+𝑎₁₂+𝑏₃₀𝜆−𝑎₁₂𝜆𝜂²𝜉³−12𝑎₀₃(−1+𝜆)𝜂𝜉⁴+𝛽𝜉³𝜂⁴. (3.5)

At last, by means of transformation (1.4)𝑛=1, system (3.5) is reduced to𝑑𝑧1𝑑𝜏=𝑧+𝑤10³𝑧⁴−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,𝑑𝑤1𝑑𝜏=−𝑤−𝑤10⁴𝑧³𝑏₀₃(−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. (3.6)

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 Theorem 2.4 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: 𝜇₅=15𝑎₂₁−𝑏₂₁𝜇𝜆,₁₀1=−5𝑎₃₀𝑎₁₂−𝑏₃₀𝑏₁₂𝜆. (3.7)
Case 1. 𝑎₁₂𝑏₁₂≠0, then there exist 𝑘 to make 𝑎₃₀=𝑘𝑏₁₂,  𝑏₃₀=𝑘𝑎₁₂, 𝜇₁₅=𝜆𝑎40⁰³𝑏²₁₂−𝑏₀₃𝑎²₁₂(),𝜇−1+3𝑘)(2+2𝑘−𝜆+𝑘𝜆₂₀=𝜆²()𝑏10𝜆+2²¹𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂(),𝜇3𝑘−1₂₅=𝜆²240(𝜆+2)³(3𝑘−1)𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂×−32𝑎₀₃𝑏₀₃−28𝑎₀₃𝑏₀₃𝜆+32𝑎₁₂𝑏₁₂𝜆²+5𝑎₀₃𝑏₀₃𝜆³+𝑎₀₃𝑏₀₃𝜆⁴+192𝛽+192𝜆𝛽+48𝜆²𝛽,𝜇₃₀𝜇=0,₃₅𝜆=−²19200(𝜆+2)⁵(3𝑘−1)𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂×1024𝑎²₀₃𝑏²₀₃+1920𝑎²₀₃𝑏²₀₃𝜆−13824𝑎₁₂𝑎₀₃𝑏₁₂𝑏₀₃𝜆²+224𝑎²₀₃𝑏²₀₃𝜆²−11392𝑎₁₂𝑎₀₃𝑏₁₂𝑏₀₃𝜆³−1584𝑎²₀₃𝑏²₀₃𝜆³+12800𝑎²₁₂𝑏²₁₂𝜆⁴+2432𝑎₁₂𝑎₀₃𝑏₁₂𝑏₀₃𝜆⁴−1056𝑎²₀₃𝑏²₀₃𝜆⁴+4064𝑎₁₂𝑎₀₃𝑏₁₂𝑏₀₃𝜆⁵+864𝑎₁₂𝑎₀₃𝑏₁₂𝑏₀₃𝜆⁶+206𝑎²₀₃𝑏²₀₃𝜆⁶+69𝑎²₀₃𝑏²₀₃𝜆⁷+7𝑎²₀₃𝑏²₀₃𝜆⁸,𝜇₄₀=−7𝜆²2880(𝜆+2)⁵(3𝑘−1)𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂𝑎₀₃𝑏²₁₂+𝑏₀₃𝑎²₁₂()×𝜆+1−32𝑎₀₃𝑏₀₃−24𝑎₀₃𝑏₀₃𝜆+32𝑎₁₂𝑏₁₂𝜆²+4𝑎₀₃𝑏₀₃𝜆²+6𝑎₀₃𝑏₀₃𝜆³+𝑎₀₃𝑏₀₃𝜆⁴. (3.8) If 𝑎₁₂𝑏₁₂=−𝑎₀₃𝑏₀₃(−2+𝜆)(2+𝜆)²(4+𝜆)/32𝜆², 𝜇₄₅=24131()𝑎671846403𝑘−1³₀₃𝑏³₀₃𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂. (3.9) If 𝑎₀₃𝑏²₁₂+𝑏₀₃𝑎²₁₂=0, then there exist 𝑚 to make 𝑎₀₃=𝑚𝑎²₁₂,  𝑏₀₃=−𝑚𝑏²₁₂, 𝜇₄₅=7𝜆²4838400000(𝜆+2)³(3𝑘−1)(𝜆+1)𝑎₁₂𝑏₁₂𝑎²₀₃𝑏²₀₃𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂×−23257088𝑎₁₂𝑏₁₂𝑚²+6577280𝑎₁₂𝑏₁₂𝑚²𝜆−23257088𝜆²[phone]𝑎₁₂𝑏₁₂𝑚²𝜆²[phone]𝜆³−2764244𝑎₁₂𝑏₁₂𝑚²𝜆³+8304896𝜆⁴−10922212𝑎₁₂𝑏₁₂𝑚²𝜆⁴−18884608𝜆⁵−916685𝑎₁₂𝑏₁₂𝑚²𝜆⁵+1341691𝑎₁₂𝑏₁₂𝑚²𝜆⁶+255854𝑎₁₂𝑏₁₂𝑚²𝜆⁷,𝜇₅₀𝜇=0,₅₅=−1099511627776𝜆¹⁹8859375𝑚⁸(𝜆+2)¹⁷(3𝑘−1)𝑎₀₃𝑏²₁₂−𝑏₀₃𝑎²₁₂(𝜆+1). (3.10)Case 2. 𝑎₁₂=𝑏₁₂=0, 𝜇₁₅=3𝜆𝑎40⁰³𝑎²₃₀−𝑏₀₃𝑏²₃₀(),𝜇𝜆+2₂₀=3𝑎10⁰³𝑎²₃₀−𝑏₀₃𝑏²₃₀𝑏₂₁,𝜇₂₅1=−𝑎40⁰³𝑎²₃₀−𝑏₀₃𝑏²₃₀4𝑎₃₀𝑏₃₀−3𝑎₀₃𝑏₀₃,𝜇+24𝛽₃₀𝜇=0,₃₅1=−𝑎400⁰³𝑎²₃₀−𝑏₀₃𝑏²₃₀11𝑎²₀₃𝑏²₀₃−19𝑎₀₃𝑏₀₃𝑎₃₀𝑏₃₀−50𝑎²₃₀𝑏²₃₀,𝜇₄₀=7𝑎480⁰³𝑎²₃₀−𝑏₀₃𝑏²₃₀𝑎₀₃𝑎²₃₀+𝑏₀₃𝑏²₃₀𝑎₀₃𝑏₀₃−𝑎₃₀𝑏₃₀,𝜇₄₅1=−𝑎1344000³³⁰𝑏³₃₀𝑎₀₃𝑎²₃₀−𝑏₀₃𝑏²₃₀×102334400+28122192𝑎₃₀𝑏₃₀𝑚²−26626536𝑎²₃₀𝑏²₃₀𝑚⁴+2330697𝑎³₃₀𝑏³₃₀𝑚⁶, (3.11) where 𝜇_𝑘=0, 𝑘≠5𝑖,  𝑖≤11,  𝑖∈𝑁. In the above expression of 𝜇_𝑘, we have already let 𝜇₁=⋯=𝜇_𝑘−1=0, 𝑘=2,3,…,45.

From Theorem 3.1, 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: 𝑎₂₁=𝑏₂₁,𝑎₁₂=𝑏₁₂=0,𝑎²₃₀𝑎₀₃=𝑏₀₃𝑏²₃₀,𝑎₂₁=𝑏₂₁,𝑎₃₀=13𝑏₁₂,𝑏₃₀=13𝑎₁₂,𝑎₁₂𝑏₁₂𝑎≠0,₂₁=𝑏₂₁,𝑎₃₀𝑎₁₂=𝑏₃₀𝑏₁₂,𝑎²₁₂𝑏₀₃=𝑏²₁₂𝑎₀₃,𝑎₁₂𝑏₁₂≠0,𝜆=−1,𝛽=0,𝑎₂₁=𝑏₂₁=0,𝑎₃₀=−3𝑏₁₂,𝑏₃₀=−3𝑎₁₂,𝑎₀₃𝑏₀₃=4𝑎₁₂𝑏₁₂,𝑎₁₂𝑏₁₂≠0. (3.12)(3.13)(3.14)(3.15)

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: 𝛽,𝑎₂₁,𝑏₂₁,𝑎₃₀𝑏₃₀,𝑎₁₂𝑏₁₂,𝑎₀₃𝑏₀₃,𝑎₃₀𝑎₁₂,𝑏₃₀𝑏₁₂,𝑎²₃₀𝑎₀₃,𝑎₃₀𝑏₁₂𝑎₀₃,𝑏²₁₂𝑎₀₃,𝑏²₃₀𝑏₀₃,𝑏₃₀𝑎₁₂𝑏₀₃,𝑎²₁₂𝑏₀₃. (3.16)

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 𝑧𝑤𝑒^{3𝑎03𝑧7𝑤3+4𝑎12𝑧6𝑤4+6𝑏21𝑧5𝑤5+4𝑏12𝑧4𝑤6+3𝑏03𝑧3𝑤7+3𝑧10𝑤10𝛽−3}(),𝜆=−2,𝑧𝑤^{5(𝜆+2)/𝜆}𝑓₁,𝜆≠−2, (3.17) where 𝑓₁=−12𝑎₀₃+3𝑎₀₃𝜆²𝑧⁷𝑤³+4𝑎₄₁𝜆²−16𝑎₁₂𝑧⁶𝑤⁴+6𝑏₂₁𝜆²−24𝑏₂₁𝑧⁵𝑤⁵+4𝑏₁₂𝜆²+16𝑏₁₂𝑧⁴𝑤⁶+3𝑏₀₃𝜆²−12𝑏₀₃𝑧³𝑤⁷+24𝛽−12𝑟𝛽𝑧¹⁰𝑤¹⁰−24+12𝜆. (3.18) If condition (3.15) holds, system (3.6) becomes 𝑑𝑧=1𝑑𝑇1010+4𝑎₀₃𝑧⁷𝑤³−14𝑎₁₂𝑧⁶𝑤⁴−6𝑏₁₂𝑧⁴𝑤⁶+6𝑏₀₃𝑧³𝑤⁷,𝑑𝑤1𝑑𝑇=−1010+4𝑏₀₃𝑧³𝑤⁷−14𝑏₁₂𝑤⁶𝑧⁴−6𝑎₁₂𝑤⁴𝑧⁶+6𝑎₀₃𝑤³𝑧⁷, (3.19) there exists a transformation 𝑢𝑧=(𝑢𝑣)^2/5𝑣,𝑤=(𝑢𝑣)^2/5, (3.20) system (3.19) is changed into 𝑑𝑢𝑑𝑇=𝑢+𝑏₀₃𝑣³+𝑏₁₂𝑣²𝑢−3𝑎₁₂𝑢³=𝑈,𝑑𝑣𝑑𝑇=−𝑣+𝑎₀₃𝑢³+𝑎₁₂𝑢²𝑣−3𝑏₁₂𝑣³=𝑉, (3.21) system (3.21) has the integral factor 𝑓₂^−5/6, where 𝑓₂=1−6𝑏₁₂𝑢²+𝑎₁₂𝑣²+33𝑏²₁₂𝑢⁴−2𝑎₁₂𝑏₀₃𝑢³𝑣+2𝑎₁₂𝑏₁₂𝑢²𝑣²−2𝑏₁₂𝑎₀₃𝑣³𝑢+3𝑎⁴₁₂𝑣⁴+122𝑎₁₂𝑢−𝑎₀₃𝑣2𝑏₁₂𝑣−𝑏₀₃𝑢𝑏₀₃𝑢⁴−2𝑏₁₂𝑢³𝑣−2𝑎₁₂𝑣³𝑢+𝑎₀₃𝑣⁴,𝑑𝑓₂𝑏𝑑𝑡=−12₁₂𝑢²−𝑎₁₂𝑣²𝑓₂=65𝜕𝑈−𝜕𝑢𝜕𝑉𝑓𝜕𝑣². (3.22)

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.
If 𝑎₃₀=𝑏₃₀=0, we easily obtain the first 30 period constants 𝜏₅=2𝑟₂₁,𝜏₁₀=12(𝜆−2)𝑎₀₃𝑏₀₃,𝜏+4𝛽₁₅𝜏=0,₂₀1=−𝑎32²⁰³𝑏²₀₃(−2+𝜆)²(),𝜏−2+3𝜆₂₅𝜏=0,₃₀=1𝑎25600²⁰³𝑏²₀₃(−2+𝜆)²50+365𝜆−1962𝜆²+2153𝜆³. (4.1)
If 𝑎₃₀𝑏₃₀≠0, from condition (3.12), there exists an arbitrary complex constant 𝑠, such that 𝑎₀₃=𝑠𝑏²₃₀,𝑏₀₃=𝑠𝑎²₃₀, (4.2) then we get the first 30 period constants 𝜏₅=2𝑟₂₁,𝜏₁₀=12−2𝑎₃₀𝑏₃₀−2𝑎²₃₀𝑏²₃₀𝑠²−2𝑎₃₀𝑏₃₀𝜆+𝑎²₃₀𝑏²₃₀𝑠²,𝜏𝜆+4𝛽₁₅=34𝑎²₃₀𝑏²₃₀𝑠(),𝜏𝜆+2₂₀=14𝑎²₃₀𝑏²₃₀(1+𝜆)²(),𝜏3𝜆+1₂₅𝜏=0,₃₀1=−𝑎800³³⁰𝑏³₃₀(1+𝜆)²−25+35𝜆+981𝜆²+2153𝜆³. (4.3) In expressions (4.1) and (4.3), 𝜏_𝑘=0, 𝑘≠5𝑖, 𝑖≤4, 𝑖∈𝑁, and we have already let 𝜏₁=⋯=𝜏_𝑘−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: 𝛽=𝑎₂₁=𝑏₂₁=𝑎₁₂=𝑏₁₂=𝑎₃₀=𝑏₃₀=0,𝜆=2.𝛽=𝑎₂₁=𝑏₂₁=𝑎₁₂=𝑏₁₂=𝑎₀₃=𝑏₀₃=0,𝜆=−1,𝑎₃₀𝑏₃₀≠0. (4.4)(4.5)

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 𝑑𝑧=1𝑑𝑇𝑧1010+7𝑎₀₃𝑧⁷𝑤³+3𝑏₀₃𝑧³𝑤⁷,𝑑𝑤1𝑑𝑇=−𝑤1010+3𝑎₀₃𝑧⁷𝑤³+7𝑏₀₃𝑧³𝑤⁷. (4.6) There exists a transformation 𝑢=𝑧(1+𝑏₀₃𝑧³𝑤⁷)^3/401+𝑎₀₃𝑧⁷𝑤^37/40𝑤,𝑣=1+𝑎₀₃𝑧⁷𝑤^33/401+𝑏₀₃𝑧³𝑤^77/40, (4.7) such that system (4.6) is reduced to a linear system.
When condition (4.5) is satisfied, system (3.6) becomes 𝑑𝑧=1𝑑𝑇𝑧1010+6𝑏₃₀𝑧⁶𝑤⁴+4𝑎₃₀𝑧⁴𝑤⁶,𝑑𝑤1𝑑𝑇=−𝑤1010+4𝑏₃₀𝑧⁴𝑤⁶+6𝑎₃₀𝑧⁶𝑤⁴. (4.8) There exists a transformation 𝑧𝑢=1+𝑎₃₀𝑧⁴𝑤^61/51+𝑏₃₀𝑧⁶𝑤^43/10𝑤,𝑣=1+𝑏₃₀𝑧⁶𝑤^41/51+𝑎₃₀𝑧⁴𝑤^63/10, (4.9) 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 constants𝜏₅=2𝑟₂₁,𝜏₁₀=118−18𝑎₀₃𝑏₀₃−32𝑎₁₂𝑏₁₂+9𝑎₀₃𝑏₀₃𝜆+16𝑎₁₂𝑏₁₂,𝜏𝜆+36𝛽₁₅=13𝑎²₁₂𝑏₀₃+𝑎₀₃𝑏²₁₂(),𝜏−4+𝜆)(−1+𝜆₂₀1=−𝑎2592²¹²𝑏²₁₂(𝜆−2)(3𝜆+2)−512−1152𝑎₁₂𝑏₁₂𝑠²−162𝑎²₁₂𝑏²₁₂𝑠⁴+256𝜆+81𝑎²₁₂𝑏²₁₂𝑠⁴𝜆,𝜏₂₅𝜏=0,₃₀=19256+81𝑎²₁₂𝑏²₁₂𝑠⁴⁵2048𝑎⁵₁₂𝑏⁵₁₂𝑠⁴256+720𝑎₁₂𝑏₁₂𝑠²+81𝑎²₁₂𝑏²₁₂𝑠⁴×256+1152𝑎₁₂𝑏₁₂𝑠²+81𝑎²₁₂𝑏²₁₂𝑠⁴−262144−2801664𝑎₁₂𝑏₁₂𝑠²+4147200𝑎²₁₂𝑏²₁₂𝑠⁴−2192832𝑎³₁₂𝑏³₁₂𝑠⁶+183708𝑎⁴₁₂𝑏⁴₁₂𝑠⁸+59049𝑎⁵₁₂𝑏⁵₁₂𝑠¹⁰,𝜏₃₅𝜏=0,₄₀=12025256+81𝑎²₁₂𝑏²₁₂𝑠⁴⁷64𝑎⁶₁₂𝑏⁶₁₂𝑠⁴×109239312561498750976+2393515709059622240256𝑎₁₂𝑏₁₂𝑠²+16627534144531656081408𝑎²₁₂𝑏²₁₂𝑠⁴+45808666111836080308224𝑎³₁₂𝑏³₁₂𝑠⁶+34487314499860709769216𝑎⁴₁₂𝑏⁴₁₂𝑠⁸−43671898596853816492032𝑎⁵₁₂𝑏⁵₁₂𝑠¹⁰−31656165840700764585984𝑎⁶₁₂𝑏⁶₁₂𝑠¹²+49193367282534498435072𝑎⁷₁₂𝑏⁷₁₂𝑠¹⁴−2655678216972439388160𝑎⁸₁₂𝑏⁸₁₂𝑠¹⁶−8653122543235415408640𝑎⁹₁₂𝑏⁹₁₂𝑠¹⁸−539890165260393578496𝑎¹⁰₁₂𝑏¹⁰₁₂𝑠²⁰+408937711778677748736𝑎¹¹₁₂𝑏¹¹₁₂𝑠²²+82987106695640213760𝑎¹²₁₂𝑏¹²₁₂𝑠²⁴+5972974608886179588𝑎¹³₁₂𝑏¹³₁₂𝑠²⁶+151421489259386859𝑎¹⁴₁₂𝑏¹⁴₁₂𝑠²⁸, (4.10) where 𝜏_𝑘=0, 𝑘≠5𝑖, 𝑖≤8, 𝑖∈𝑁. In the above expression of 𝜏_𝑘, we have already let 𝜏₁=⋯=𝜏_𝑘−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: 𝜆=2,𝛽=𝑎₂₁=𝑏₂₁=𝑎₀₃=𝑏₀₃=0,𝑎₃₀=13𝑏₁₂,𝑏₃₀=13𝑎₁₂. (4.11)

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 𝑑𝑧=1𝑑𝑇𝑧1010+8𝑎₁₂𝑧⁶𝑤⁴+163𝑏₁₂𝑧⁴𝑤⁶,𝑑𝑤1𝑑𝑇=−𝑤1010+8𝑏₁₂𝑧⁴𝑤⁶+163𝑎₁₂𝑧⁶𝑤⁴. (4.12) There exists a transformation 𝑧𝑢=3+4𝑏₁₂𝑧⁴𝑤^61/53+4𝑎₁₂𝑧⁶𝑤^43/10𝑤,𝑣=3+4𝑎₁₂𝑧⁶𝑤^41/53+4𝑏₁₂𝑧⁴𝑤^63/10, (4.13) such that system (4.12) is reduced to a linear system.

Case 3. Integrability condition (3.14) holds.
Because 𝑎₁₂𝑏₁₂≠0, we can let 𝑎₃₀=𝑘𝑏₁₂,𝑏₃₀=𝑘𝑎₁₂,𝑎₀₃=𝑚𝑎²₁₂,𝑏₀₃=𝑚𝑏²₁₂, (4.14) where 𝑘,𝑚 are arbitrary complex constants. Substituting (4.14) into the recursive formulae in Appendix B, we obtain the first 30 period constants 𝜏₅=2𝑟₂₁,𝜏₁₀=12−2𝑎₁₂𝑏₁₂−4𝑎₁₂𝑏₁₂𝑘−2𝑎₁₂𝑏₁₂𝑘²−2𝑎²₁₂𝑏²₁₂𝑚²+2𝑎₁₂𝑏₁₂𝜆−2𝑎₁₂𝑏₁₂𝑘²𝜆+𝑎²₁₂𝑏²₁₂𝑚²,𝜏𝜆+4𝛽₁₅=14𝑎²₁₂𝑏²₁₂𝑚(3+3𝑘−4𝜆)(2+2𝑘−𝜆+𝑘𝜆). (4.15) If 𝑚=0, 𝜏₂₀=14𝑎²₁₂𝑏²₁₂(1+𝑘)(1+𝑘−𝜆+𝑘𝜆)²(),𝜏1+𝑘−3𝜆+3𝑘𝜆₂₅𝜏=0,₃₀=1(1+3𝜆)³2𝑎³₁₂𝑏³₁₂(1+𝑘)𝜆³(1+𝑘−𝜆+𝑘𝜆)². (4.16) If 𝑘=(1/3)(−3+4𝜆), 𝜏₂₀1=−𝑎2592²¹²𝑏²₁₂−648𝑎²₁₂𝑏²₁₂𝑚⁴+1620𝑎²₁₂𝑏²₁₂𝑚⁴𝜇−1152𝑎₁₂𝑏₁₂𝑚²𝜆²−1134𝑎²₁₂𝑏²₁₂𝑚⁴𝜆²+4608𝑎₁₂𝑏₁₂𝑚²𝜆³+243𝑎²₁₂𝑏²₁₂𝑚⁴𝜆³+1792𝑟⁴−4608𝑎₁₂𝑏₁₂𝑚²𝜆⁴−8704𝜆⁵+13312𝜆⁶−6144𝜆⁷,𝜏₂₅1=−𝑎1944³¹²𝑏³₁₂𝑚𝜆²−864𝑎₁₂𝑏₁₂𝑚²−459𝑎₁₂𝑏₁₂𝑚²𝜆+576𝜆²+6075𝑎₁₂𝑏₁₂𝑚²𝜆²−256𝜆³−5670𝑎₁₂𝑏₁₂𝑚²𝜆³−5888𝜆⁴+1296𝑎₁₂𝑏₁₂𝑚²𝜆⁴+11264𝜆⁵−6144𝜆⁶,𝜏₃₀1=−492075−32−17𝜆+225𝜆²−210𝜆³+48𝜆⁴³4𝑎³₁₂𝑏³₁₂𝜆⁶(1+𝜆)(−1+2𝜆)²×16223998464−25771071744𝜆−266392474620𝜆²[phone]𝜆³[phone]𝜆⁴−4814334558957𝜆⁵[phone]𝜆⁶−2358342484708𝜆⁷−5235799647872𝜆⁸[phone]𝜆⁹−3998287419904𝜆¹⁰[phone]𝜆¹¹[phone]𝜆¹²−135353438208𝜆¹³[phone]𝜆¹⁴. (4.17) If 𝑘=(𝜆−2)/(𝜆+2), 𝜏₂₀1=−96(𝜆+2)⁴𝑎²₁₂𝑏²₁₂−384𝑎²₁₂𝑏²₁₂𝑚⁴+192𝑎²₁₂𝑏²₁₂𝑚⁴𝜆−1536𝑎₁₂𝑏₁₂𝑚²𝜆²+672𝑎²₁₂𝑏²₁₂𝑚⁴𝜆²−1280𝑎₁₂𝑏₁₂𝑚²𝜆³+48𝑎²₁₂𝑏²₁₂𝑚⁴𝜆³+1920𝜆⁴+384𝑎₁₂𝑏₁₂𝑚²𝜆⁴−264𝑎²₁₂𝑏²₁₂𝑚⁴𝜆⁴+576𝑎₁₂𝑏₁₂𝑚²𝜆⁵−60𝑎²₁₂𝑏²₁₂𝑚⁴𝜆⁵+128𝑎₁₂𝑏₁₂𝑚²𝜆⁶+30𝑎²₁₂𝑏²₁₂𝑚⁴𝜆⁶+9𝑎²₁₂𝑏²₁₂𝑚⁴𝜆⁷,𝜏₂₅1=−12(2+𝜆)⁴𝑎³₁₂𝑏³₁₂𝑚𝜆²96𝑎₁₂𝑏₁₂𝑚²+76𝑎₁₂𝑏₁₂𝑚²𝜆−96𝜆²−40𝑎₁₂𝑏₁₂𝑚²𝜆²−32𝜆³−33𝑎₁₂𝑏₁₂𝑚²𝜆³+5𝑎₁₂𝑏₁₂𝑚²𝜆⁴+4𝑎₁₂𝑏₁₂𝑚²𝜆⁵,𝜏₃₀=2𝑎³₁₂𝑏³₁₂𝜆⁷75(2+𝜆)⁶24−5𝜆−11𝜆²+4𝜆³³−8200224−114773490𝜆+250179993𝜆²−32032585𝜆³−178032915𝜆⁴[phone]𝜆⁵[phone]𝜆⁶−24125520𝜆⁷+4769504𝜆⁸, (4.18) where 𝜏_𝑘=0,  𝑘≠5𝑖,  𝑖≤6,  𝑖∈𝑁. In the above expression of 𝜏_𝑘, we have already let 𝜏₁=⋯=𝜏_𝑘−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: 𝛽=𝑎₂₁=𝑏₂₁=𝑎₀₃=𝑏₀₃=0,𝑎₃₀=−𝑏₁₂,𝑏₃₀=−𝑎₁₂,𝛽=𝑎₂₁=𝑏₂₁=𝑎₀₃=𝑏₀₃=0,𝑎₃₀=−1+𝜆𝑏1+𝜆¹²,𝑏₃₀=−1+𝜆𝑎1+𝜆¹². (4.19)(4.20)

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 𝑑𝑧=1𝑑𝑇𝑧1010+2𝑎₁₂𝜆𝑧⁶𝑤⁴−2𝑏₁₂𝜆𝑧⁴𝑤⁶,𝑑𝑤1𝑑𝑇=−5𝑤10+2𝑏₁₂𝜆𝑧⁴𝑤⁶−2𝑎₁₂𝜆𝑧⁶𝑤⁴, (4.21) we have for system (4.21) that 𝑑𝜃=1𝑑𝑡21𝑧𝑑𝑧−1𝑑𝑇𝑤𝑑𝑤𝑑𝑇=1. (4.22)
When condition (4.20) is satisfied, system (3.6) becomes 𝑑𝑧=1𝑑𝑇5()𝑧5()1+𝜆1+𝜆+4𝑎₁₂𝜆𝑧⁶𝑤⁴+6𝑏₁₂𝜆𝑧⁴𝑤⁶,𝑑𝑤1𝑑𝑇=−5()𝑤5()1+𝜆1+𝜆+4𝑏₁₂𝜆𝑧⁴𝑤⁶+6𝑎₁₂𝜆𝑧⁶𝑤⁴. (4.23) There exists a transformation 𝑧𝑢=1+𝜆+2𝑏₁₂𝜆𝑧⁴𝑤^61/51+𝜆+2𝑎₁₂𝜆𝑧⁶𝑤^43/10𝑤,𝑣=1+𝜆+2𝑎₁₂𝜆𝑧⁶𝑤^41/51+𝜆+2𝑏₁₂𝜆𝑧⁴𝑤^63/10, (4.24) 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 𝜏₅=2𝑟₂₁,𝜏₁₀=−2𝑎₁₂𝑏₁₂. (4.25) 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.

Appendices

A.

The recursive formulae to compute the singular point quantities at the origin of system (3.6):𝑐[0,0]=1, when (𝑘=𝑗>0) or 𝑘<0, or 𝑗<0,𝑐[𝑘,𝑗]=0. Else𝑐𝑘,𝑗=−110𝑗−𝑘10𝑗𝛽𝑐−10+𝑘,−10+𝑗−10𝑘𝛽𝑐−10+𝑘,−10+𝑗+5𝑎₀₃𝑗𝑐−7+𝑘,−3+𝑗−5𝑎₀₃𝑘𝑐−7+𝑘,−3+𝑗−2𝑎₀₃𝜆𝑐−7+𝑘,−3+𝑗−𝑎₀₃𝑗𝜆𝑐−7+𝑘,−3+𝑗−𝑎₀₃𝑘𝜆𝑐−7+𝑘,−3+𝑗+5𝑎₁₂𝑗𝑐−6+𝑘,−4+𝑗+5𝑏₃₀𝑗𝑐−6+𝑘,−4+𝑗−5𝑎₁₂𝑘𝑐−6+𝑘,−4+𝑗−5𝑏₃₀𝑘𝑐−6+𝑘,−4+𝑗−2𝑎₁₂𝜆𝑐−6+𝑘,−4+𝑗+2𝑏₃₀𝜆𝑐−6+𝑘,−4+𝑗−𝑎₁₂𝑗𝜆𝑐−6+𝑘,−4+𝑗+𝑏₃₀𝑗𝜆𝑐−6+𝑘,−4+𝑗−𝑎₁₂𝑘𝜆𝑐−6+𝑘,−4+𝑗+𝑏₃₀𝑘𝜆𝑐−6+𝑘,−4+𝑗+5𝑎₂₁𝑗𝑐−5+𝑘,−5+𝑗+5𝑏₂₁𝑗𝑐−5+𝑘,−5+𝑗−5𝑎₂₁𝑘𝑐−5+𝑘,−5+𝑗−5𝑏₂₁𝑘𝑐−5+𝑘,−5+𝑗−2𝑎₂₁𝜆𝑐−5+𝑘,−5+𝑗+2𝑏₂₁𝜆𝑐−5+𝑘,−5+𝑗−𝑎₂₁𝑗𝜆𝑐−5+𝑘,−5+𝑗+𝑏₂₁𝑗𝜆𝑐−5+𝑘,−5+𝑗−𝑎₂₁𝑘𝜆𝑐−5+𝑘,−5+𝑗+𝑏₂₁𝑘𝜆𝑐−5+𝑘,−5+𝑗+5𝑎₃₀𝑗𝑐−4+𝑘,−6+𝑗+5𝑏₁₂𝑗𝑐−4+𝑘,−6+𝑗−5𝑎₃₀𝑘𝑐−4+𝑘,−6+𝑗−5𝑏₁₂𝑘𝑐−4+𝑘,−6+𝑗−2𝑎₃₀𝜆𝑐−4+𝑘,−6+𝑗+2𝑏₁₂𝜆𝑐−4+𝑘,−6+𝑗−𝑎₃₀𝑗𝜆𝑐−4+𝑘,−6+𝑗+𝑏₁₂𝑗𝜆𝑐−4+𝑘,−6+𝑗−𝑎₃₀𝑘𝜆𝑐−4+𝑘,−6+𝑗+𝑏₁₂𝑘𝜆𝑐−4+𝑘,−6+𝑗+5𝑏₀₃𝑗𝑐−3+𝑘,−7+𝑗−5𝑏₀₃𝑘𝑐−3+𝑘,−7+𝑗+2𝑏₀₃𝜆𝑐−3+𝑘,−7+𝑗+𝑏₀₃𝑗𝜆𝑐−3+𝑘,−7+𝑗+𝑏₀₃,𝜇𝑘𝜆𝑐−3+𝑘,−7+𝑗_𝑘=15𝜆𝑎₀₃𝑐[−7+𝑘,−3+𝑘]+𝑎₁₂𝑐[−6+𝑘,−4+𝑘]−𝑏₃₀𝑐[−6+𝑘,−4+𝑘]+𝑎₂₁𝑐[−5+𝑘,−5+𝑘]−𝑏₂₁𝑐[−5+𝑘,−5+𝑘]+𝑎₃₀𝑐[−4+𝑘,−6+𝑘]−𝑏₁₂𝑐[−4+𝑘,−6+𝑘]−𝑏₀₃𝑐[−3+𝑘,−7+𝑘]. (A.1)

B.

The recursive formulae to compute the period constants of the origin of system (3.6):𝑐^′[1,0]=𝑑^′[1,0]=1;𝑐^′[0,1]=𝑑^′[0,1]=0, if 𝑘<0 or 𝑗<0 or (𝑗>0 and 𝑘=𝑗+1) then 𝑐^′[𝑘,𝑗]=0, 𝑑^′[𝑘,𝑗]=0.Else𝑐^′=1𝑘,𝑗−()𝛽𝑐+1𝑗+1−𝑘−10+𝑗𝛽+−10+𝑘−10+𝑘,−10+𝑗𝑎10⁰³()+1−3+𝑗−5+𝜆𝑎10⁰³()𝑐+1−7+𝑘)(5+𝜆−7+𝑘,−3+𝑗()10−6+𝑘5𝑎₁₂+5𝑏₃₀+𝑎₁₂𝜆−𝑏₃₀𝜆−1×10−4+𝑗5𝑎₁₂+5𝑏₃₀−𝑎₁₂𝜆+𝑏₃₀𝜆𝑐+1−6+𝑘,−4+𝑗(10−5+𝑘)5𝑎₂₁+5𝑏₂₁+𝑎₂₁𝜆−𝑏₂₁𝜆−1×10−5+𝑗5𝑎₂₁+5𝑏₂₁−𝑎₂₁𝜆+𝑏₂₁𝜆𝑐+1−5+𝑘,−5+𝑗()10−4+𝑘5𝑎₃₀+5𝑏₁₂+𝑎₃₀𝜆−𝑏₁₂𝜆−1×10−6+𝑗5𝑎₃₀+5𝑏₁₂−𝑎₃₀𝜆+𝑏₁₂𝜆𝑐+−1−4+𝑘,−6+𝑗𝑏10⁰³()−1−3+𝑘)(−5+𝜆𝑏10⁰³(),𝑑−7+𝑗5+𝜆×𝑐−3+𝑘,−7+𝑗_′=1𝑘,𝑗−()𝛽𝑑+1𝑗+1−𝑘−10+𝑗𝛽+−10+𝑘−10+𝑘,−10+𝑗𝑏10⁰³()+1−3+𝑗−5+𝜆𝑏10⁰³()𝑑+−1−7+𝑘)(5+𝜆−7+𝑘,−3+𝑗10−4+𝑗5𝑎₃₀+5𝑏₁₂+𝑎₃₀𝜆−𝑏₁₂𝜆+1()×10−6+𝑘5𝑎₃₀+5𝑏₁₂−𝑎₃₀𝜆+𝑏₁₂𝜆𝑑+−1−6+𝑘,−4+𝑗10−5+𝑗5𝑎₂₁+5𝑏₂₁+𝑎₂₁𝜆−𝑏₂₁𝜆+1()×10−5+𝑘5𝑎₂₁+5𝑏₂₁−𝑎₂₁𝜆+𝑏₂₁𝜆𝑑+−1−5+𝑘,−5+𝑗10−6+𝑗5𝑎₁₂+5𝑏₃₀+𝑎₁₂𝜆−𝑏₃₀𝜆+1()×10−4+𝑘5𝑎₁₂+5𝑏₃₀−𝑎₁₂𝜆+𝑏₃₀𝜆𝑑+−1−4+𝑘,−6+𝑗𝑎10⁰³()−1−3+𝑘)(−5+𝜆𝑎10⁰³(,𝜏[𝑚]=−𝛽𝑐+1−7+𝑗5+𝜆)×𝑑−3+𝑘,−7+𝑗−10+𝑗𝛽+−9+𝑗−9+𝑗,−10+𝑗𝑎10⁰³()+1−3+𝑗−5+𝜆𝑎10⁰³()𝑐+1−6+𝑗5+𝜆−6+𝑗,−3+𝑗10−5+𝑗5𝑎₁₂+5𝑏₃₀+𝑎₁₂𝜆−𝑏₃₀𝜆−1×10−4+𝑗5𝑎₁₂+5𝑏₃₀−𝑎₁₂𝜆+𝑏₃₀𝜆𝑐+1−5+𝑗,−4+𝑗10−4+𝑗5𝑎₂₁+5𝑏₂₁+𝑎₂₁𝜆−𝑏₂₁𝜆−1×10−5+𝑗5𝑎₂₁+5𝑏₂₁−𝑎₂₁𝜆+𝑏₂₁𝜆𝑐+1−4+𝑗,−5+𝑗10−3+𝑗5𝑎₃₀+5𝑏₁₂+𝑎₃₀𝜆−𝑏₁₂𝜆−1×10−6+𝑗5𝑎₃₀+5𝑏₁₂−𝑎₃₀𝜆+𝑏₁₂𝜆𝑐+−1−3+𝑗,−6+𝑗𝑏10⁰³()−1−2+𝑗−5+𝜆𝑏10⁰³(+−𝛽𝑑+1−7+𝑗5+𝜆)×𝑐−2+𝑗,−7+𝑗−10+𝑗𝛽+−9+𝑗−9+𝑗,−10+𝑗𝑏10⁰³()+1−3+𝑗−5+𝜆𝑏10⁰³()𝑑+−1−6+𝑗5+𝜆−6+𝑗,−3+𝑗10−4+𝑗5𝑎₃₀+5𝑏₁₂+𝑎₃₀𝜆−𝑏₁₂𝜆+1×10−5+𝑗5𝑎₃₀+5𝑏₁₂−𝑎₃₀𝜆+𝑏₁₂𝜆𝑑+−1−5+𝑗,−4+𝑗10−5+𝑗5𝑎₂₁+5𝑏₂₁+𝑎₂₁𝜆−𝑏₂₁𝜆+1×10−4+𝑗5𝑎₂₁+5𝑏₂₁−𝑎₂₁𝜆+𝑏₂₁𝜆𝑑+−1−4+𝑗,−5+𝑗10−6+𝑗5𝑎₁₂+5𝑏₃₀+𝑎₁₂𝜆−𝑏₃₀𝜆+1×10−3+𝑗5𝑎₁₂+5𝑏₃₀−𝑎₁₂𝜆+𝑏₃₀𝜆𝑑−1−3+𝑗,−6+𝑗𝑎10⁰³()+1−2+𝑗−5+𝜆𝑎10⁰³().−7+𝑗5+𝜆×𝑑−2+𝑗,−7+𝑗 (B.1)

Acknowledgments

Many thanks are due to the anonymous referee for his/her valuable comments and suggestions that helped improve the presentation of the work. This research is supported by the National Nature Science Foundation of China (11071222, 11101126).