Linearizability Problem of Resonant Degenerate Singular Point for Polynomial Differential Systems

Abstract

The linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. A progressive way to find necessary conditions for linearizability is to compute period constants. In this paper, we are interested in the linearizability problem of p : −q resonant degenerate singular point for polynomial differential systems. Firstly, we transform degenerate singular point into the origin via a homeomorphism. Moreover, we establish a new recursive algorithm to compute the so-called generalized period constants for the origin of the transformed system. Finally, to illustrate the effectiveness of our algorithm, we discuss the linearizability problems of 1 : −1 resonant degenerate singular point for a septic system. We stress that similar results are hardly seen in published literatures up till now. Our work is completely new and extends existing ones.

1. Introduction

In the qualitative theory of planar polynomial differential equations, the problem of characterizing isochronous centers has stimulated a great deal of effort but which is also remarkably intractable. The authors of [1] pointed out that “a center of an analytic system is isochronous if and only if there exists an analytic change of variables such that the original system is reduced to a linear system,” thus an isochronous center is also called a linearizable center. A center is isochronous if the period of all period solutions in a neighborhood of it is constant.

The problem of determining isochronous centers for polynomial vector fields with linear part of center type𝑑𝑥𝑑𝑡=−𝑦+^∞_𝑘=2𝑋_𝑘𝑥,𝑦,𝑑𝑦𝑑𝑡=𝑥+^∞_𝑘=2𝑌_𝑘𝑥,𝑦, (1.1) is the subject of much work, where𝑋_𝑘𝑥,𝑦=_{𝛼+𝛽=𝑘}𝐴_𝛼𝛽𝑥^𝛼𝑦^𝛽,𝑌_𝑘𝑥,𝑦=_{𝛼+𝛽=𝑘}𝐵_𝛼𝛽𝑥^𝛼𝑦^𝛽. (1.2) The representative results are quadratic isochronous centers, see [2]; isochronous centers of a linear center perturbed by third, fourth, and fifth degree homogeneous polynomials, see [3–5]; complex polynomial systems, see [1]; reversible systems, see [6, 7]; and isochronous centers of cubic systems with degenerate infinity, see [8, 9].

By means of complex transformation𝑧=𝑥+𝑖𝑦,𝑤=𝑥−𝑖𝑦,𝑇=𝑖𝑡,𝑖=√−1, (1.3) system (1.1) becomes the following complex system:𝑑𝑧𝑑𝑇=𝑧+^∞_𝑘=2𝑍_𝑘(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝑤−^∞_𝑘=2𝑊_𝑘(𝑧,𝑤), (1.4) where𝑍_𝑘(𝑧,𝑤)=_{𝛼+𝛽=𝑘}𝑎_𝛼𝛽𝑧^𝛼𝑤^𝛽=𝑌_𝑘𝑧+𝑤2,𝑧−𝑤2𝑖−𝑖𝑋_𝑘𝑧+𝑤2,𝑧−𝑤,𝑊2𝑖_𝑘(𝑧,𝑤)=_{𝛼+𝛽=𝑘}𝑏_𝛼𝛽𝑤^𝛼𝑧^𝛽=𝑌_𝑘𝑧+𝑤2,𝑧−𝑤2𝑖+𝑖𝑋_𝑘𝑧+𝑤2,𝑧−𝑤.2𝑖 (1.5) Clearly, the coefficients of system (1.4) satisfy the conjugate condition, that is,𝑎_𝛼𝛽=𝑏_𝛼𝛽,𝛼≥0,𝛽≥0,𝛼+𝛽≥2. (1.6) As in [10], systems (1.1) and (1.4) are said to be concomitant. For isochronous center problem, we highlight the great contribution of Liu and Huang [11], which gave a new recursive algorithm to compute period constants at the origin of system (1.4).

At times, the problem is restricted to the following polynomial differential system with linear part of p : −q resonant saddle point type𝑑𝑥,𝑑𝑡=𝑝𝑥+𝑃𝑥,𝑦𝑑𝑦,𝑑𝑡=−𝑞𝑦+𝑄𝑥,𝑦 (1.7) where 𝑝,𝑞∈ℤ⁺,𝑥,𝑦,𝑡∈ℝ, 𝑃 and 𝑄 are polynomials. After a time scaling 𝑡→𝑝⁻¹𝑡, system (1.7) can be equivalently rewritten as𝑑𝑥,𝑑𝑡=𝑥+𝑃𝑥,𝑦𝑑𝑦,𝑑𝑡=−𝜆𝑦+𝑄𝑥,𝑦 (1.8) where 𝜆=𝑞/𝑝∈ℚ⁺. As to the problems of linearizability of system (1.7), systematic research has been done. In [12], the necessary and sufficient conditions (15 cases) for linearizable systems of the 1 : −2 resonance were given. For the Lotka-Volterra system𝑥̇=𝑥1+𝑎𝑥+𝑏𝑦,𝑦̇=𝑦−𝜆+𝑐𝑥+𝑑𝑦, (1.9) necessary and sufficient conditions for linearizability in the case 𝜆∈ℕ, that is the 1 : −n resonant cases, were already known in [12]. In [13], some sufficient conditions were given in the case of general 𝜆, and for the case 𝜆=𝑝/2or2/𝑝,𝑝∈ℕ⁺, necessary and sufficient conditions for linearizable systems were given. In [14], some new sufficient conditions for linearizable Lotka-Volterra system were presented. The linearizability was considered for 3 : −4 and 3 : −5 resonance in [15].

Wang and Liu [15] generalized the algorithm in [11]. Linearizability was investigated for the following general complex polynomial differential systems with a resonant singular point𝑑𝑧𝑑𝑇=𝑧+^∞_𝑘=2𝑍_𝑘(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝜆𝑤−^∞_𝑘=2𝑊_𝑘(𝑧,𝑤). (1.10)

Motivated by the above facts, in this paper, we concentrate on the linearizability problem for the polynomial differential system with a resonant degenerate singular point𝑑𝑧𝑑𝑇=𝑝𝑧^𝑛+1𝑤^𝑛+^∞_𝑘=2𝑛+2𝑍_𝑘(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝑞𝑤^𝑛+1𝑧^𝑛−^∞_𝑘=2𝑛+2𝑊_𝑘(𝑧,𝑤), (1.11) where 𝑛∈ℕ,𝑝,𝑞∈ℤ⁺,(𝑝,𝑞)=1,𝑧,𝑤,𝑇 are complex variables.

Remark 1.1. (i) When 𝑝=𝑞, without loss of generality, taking 𝑝=𝑞=1, system (1.11) becomes 𝑑𝑧𝑑𝑇=𝑧^𝑛+1𝑤^𝑛+^∞_𝑘=2𝑛+2𝑍_𝑘(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝑤^𝑛+1𝑧^𝑛−^∞_𝑘=2𝑛+2𝑊_𝑘(𝑧,𝑤). (1.12)
(ii) When 𝑛=0, system (1.11) becomes 𝑑𝑧𝑑𝑇=𝑝𝑧+^∞_𝑘=2𝑍_𝑘(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝑞𝑤−^∞_𝑘=2𝑊_𝑘(𝑧,𝑤), (1.13) which is equivalent to system (1.10).
(iii) When 𝑝=𝑞=1,𝑛=0, system (1.11) becomes system (1.4).
Consequently, the proposed results are new and extend the existing ones with respect to systems (1.12) and (1.13).

The format of this paper is organized as follows. In Section 2, we recall some background definitions and lemmas which are very important in the proof of the main results in Section 3. In Section 3, we transfer degenerate singular point into the origin by a homeomorphic transformation and derive a new recursive algorithm for the calculation of generalized period constants at the origin of the transformed system. In Section 4, as an application of the new algorithm, we investigate the linearizability problem of 1 : −1 resonant degenerate singular point for a class of septic system. In the last section, we present the conclusions and remarks.

Most of the calculations in this paper have been done with the computer algebraic system-Mathematica.

2. Preliminaries

Lemma 2.1 (see [12, 13]). System (1.13) is normalizable at the origin if and only if there exists an analytic change of variables: 𝜉=𝑧+Φ(𝑧,𝑤)=𝑧+𝑜(|𝑧,𝑤|),𝜂=𝑤+Ψ(𝑧,𝑤)=𝑤+𝑜(|𝑧,𝑤|) (2.1) bringing the system to its normal form 𝑑𝜉𝑑𝑇=𝑝𝜉1+^∞_𝑖=1𝑝_𝑖𝑈^𝑖,𝑑𝜂𝑑𝑇=−𝑞𝜂1+^∞_𝑖=1𝑞_𝑖𝑈^𝑖, (2.2) where 𝑈=𝜉^𝑞𝜂^𝑝.

We write 𝜇₀=𝜏₀=0,𝜇_𝑘=𝑝_𝑘−𝑞_𝑘,𝜏_𝑘=𝑝_𝑘+𝑞_𝑘,𝑘=1,2,….

Definition 2.2 (see [15]). For any positive integer 𝑘, 𝜇_𝑘 is called the 𝑘th singular point quantity of the origin of system (1.13). If system (1.13) is real planar differential system, 𝜇_𝑘 is the 𝑘th saddle quantity. If system (1.13) is the concomitant system of (1.1), 𝜇_𝑘 is the 𝑘th focus quantity. Moreover, the origin of system (1.13) is called a generalized center if 𝜇_𝑘=0,𝑘=1,2,….

Definition 2.3 (see [15]). For any positive integer 𝑘, 𝜏_𝑘 is called the 𝑘th generalized period constant of the origin of system (1.13). And the origin of system (1.13) is called a generalized isochronous center if 𝜇_𝑘=𝜏_𝑘=0,𝑘=1,2,….

Then from Lemma 2.1, and similar to the analytical procedure in [16], we have the following.

Lemma 2.4. System (1.13) is integrable at the origin if and only if one can derive uniquely the following formal series: 𝜉=𝑧+^∞_𝑘+𝑗=2𝑐_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝜂=𝑤+^∞_𝑘+𝑗=2𝑑_𝑘𝑗𝑤^𝑘𝑧^𝑗, (2.3) where 𝑐_{𝑞𝑖+1,𝑝𝑖}=𝑑_{𝑝𝑖+1,𝑞𝑖}=0,𝑖=1,2,…, such that 𝜉̇=𝜉ℎ𝜉,𝜂=𝑝𝜉1+𝑂𝜉,𝜂,𝜂̇=−𝑞𝜂ℎ𝜉,𝜂=−𝑞𝜂1+𝑂𝜉,𝜂, (2.4) namely, 𝑝_𝑖=𝑞_𝑖(𝑖=1,2,…) in expression (2.2).

Lemma 2.5. System (1.13) is linearizable at the origin if and only if one can derive uniquely the following formal series (2.3), such that 𝜉̇=𝑝𝜉,𝜂̇=−𝑞𝜂, (2.5) namely, 𝑝_𝑖=𝑞_𝑖=0(𝑖=1,2,…) in expression (2.2).

Lemma 2.6. System (1.13) is linearizable at the origin if and only if one can derive uniquely the change of coordinates (2.3) which linearizes the system, that is, 𝜇_𝑘=𝜏_𝑘=0,𝑘=1,2,….

Obviously, system (1.13) is linearizable at the origin if and only if the origin is a generalized isochronous center.

3. Main Results

Making the transformation𝑧=𝑧₁𝑧₁𝑤₁^𝑛+1,𝑤=𝑤₁𝑧₁𝑤₁^𝑛+1,𝑑𝑇=(2𝑛+3)𝑧₁𝑤₁^{−𝑛(2𝑛+3)}𝑑𝑇₁, (3.1) and renaming (𝑧₁,𝑤₁,𝑇₁) by (𝑧,𝑤,𝑇), system (1.11) becomes𝑑𝑧𝑑𝑇=𝑝^∗𝑧+^∞_{𝛼+𝛽=2𝑛+2}(𝑛+2)𝑎_𝛼𝛽+(𝑛+1)𝑏_{𝛽+1,𝛼−1}𝑧^𝛼+1𝑤^𝛽+1(𝑧𝑤)^{(𝛼+𝛽−2𝑛−2)(𝑛+1)},𝑑𝑤𝑑𝑇=−𝑞^∗𝑤−^∞_{𝛼+𝛽=2𝑛+2}(𝑛+2)𝑏_𝛼𝛽+(𝑛+1)𝑎_{𝛽+1,𝛼−1}𝑤^𝛼+1𝑧^𝛽+1(𝑧𝑤)^{(𝛼+𝛽−2𝑛−2)(𝑛+1)}, (3.2) where𝑝^∗=(𝑛+1)𝑞+(𝑛+2)𝑝,𝑞^∗=(𝑛+1)𝑝+(𝑛+2)𝑞. (3.3) Transformation (3.1) enables us to reduce the problem of 𝑝∶−𝑞 resonant degenerate singular point to that of the 𝑝^∗∶−𝑞^∗ resonant elementary origin.

Theorem 3.1. For system (3.2), one can derive uniquely the following formal series: 𝑓(𝑧,𝑤)=𝑧+^∞_{𝑘+𝑗=2𝑛+4}𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝑔(𝑧,𝑤)=𝑤+^∞_{𝑘+𝑗=2𝑛+4}𝑑^′_𝑘𝑗𝑤^𝑘𝑧^𝑗, (3.4) where 𝑐^′_{𝑞∗𝑖+1,𝑝∗𝑖}=𝑑^′_{𝑝∗𝑖+1,𝑞∗𝑖}=0,𝑖=1,2,…, such that 𝑑𝑓𝑑𝑇=𝑝^∗𝑓(𝑧,𝑤)+^∞_𝑖=𝑛+2𝑝^∗𝑝^′_𝑖𝑧^{𝑞∗𝑖+1}𝑤^𝑝∗𝑖,𝑑𝑔𝑑𝑇=−𝑞^∗𝑔(𝑧,𝑤)−^∞_𝑖=𝑛+2𝑞^∗𝑞^′_𝑖𝑤^{𝑝∗𝑖+1}𝑧^𝑞∗𝑖, (3.5) and when 𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗≠0,(𝑖.𝑒.,|𝑘−𝑞^∗𝑖−1|+|𝑗−𝑝^∗𝑖|≠0,𝑖=0,1,2,…),𝑐^′_𝑘𝑗 is determined by the following recursive formula: 𝑐^′_𝑘𝑗=1𝑞^∗𝑗−𝑝^∗()×𝑘−1_{∞𝛼+𝛽=2𝑛+3}(𝑛+2)(𝑘−𝛼)−(𝑛+1)𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑗−𝛽−(𝑛+1)(𝑘−𝛼)−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}×𝑐^′_{))𝑘−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1}, (3.6) and when 𝑞^∗𝑘−𝑝^∗𝑗−𝑞^∗≠0,(𝑖.𝑒.,|𝑘−𝑝^∗𝑖−1|+|𝑗−𝑞^∗𝑖|≠0,𝑖=0,1,2,…),𝑑^′_𝑘𝑗 is determined by the following recursive formula: 𝑑^′_𝑘𝑗=1𝑝^∗𝑗−𝑞^∗()×𝑘−1_{∞𝛼+𝛽=2𝑛+3}(𝑛+2)𝑘−𝛽−(𝑛+1)𝑗−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}−(𝑛+2)𝑗−𝛼−(𝑛+1)𝑘−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}×𝑑^′_{))𝑘−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1,𝑗−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1}=1𝑝^∗𝑗−𝑞^∗()×𝑘−1_{∞𝛼+𝛽=2𝑛+3}(𝑛+2)(𝑘−𝛼)−(𝑛+1)𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛼,𝛽−1}−(𝑛+2)𝑗−𝛽−(𝑛+1)(𝑘−𝛼)−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛽,𝛼−1}×𝑑^′_{))𝑘−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1}, (3.7) and for any positive integer 𝑖,𝑝^′_𝑖 and 𝑞^′_𝑖 are determined by the following recursive formulae: 𝑝^′_𝑖=1𝑝^∗∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑞^∗𝑖+1−𝛼−(𝑛+1)𝑝^∗𝑖−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑝^∗𝑖−𝛽−(𝑛+1)𝑞^∗𝑖+1−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}×𝑐^′_{𝑞∗)𝑖+1−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑝∗)𝑖−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1},𝑞^′_𝑖=1𝑞^∗∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑝^∗𝑖+1−𝛽−(𝑛+1)𝑞^∗𝑖−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}−(𝑛+2)𝑞^∗𝑖−𝛼−(𝑛+1)𝑝^∗𝑖+1−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}×𝑑^′_{𝑝∗)𝑖+1−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1,𝑞∗)𝑖−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1}=1𝑞^∗∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑝^∗𝑖+1−𝛼−(𝑛+1)𝑞^∗𝑖−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛼,𝛽−1}−(𝑛+2)𝑞^∗𝑖−𝛽−(𝑛+1)𝑝^∗𝑖+1−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛽,𝛼−1}×𝑑^′_{𝑝∗)𝑖+1−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑞∗)𝑖−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1}. (3.8)(3.9) In expressions (3.6)–(3.9), for 1≤𝑘+𝑗≤2𝑛+3, one has defined 𝑐^′₁₀𝑐=1,_′𝑘𝑗,𝑑=0,forother𝑘,𝑗_′10𝑑=1,_′𝑘𝑗=0,forother𝑘,𝑗, (3.10) and if 𝑘<0 or 𝑗<0, let 𝑎_𝑘𝑗=𝑏_𝑘𝑗=𝑐^′_𝑘𝑗=𝑑^′_𝑘𝑗=0.

Proof. Denote system (3.2) as 𝑑𝑧𝑑𝑇=𝑝^∗𝑧+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}𝑧^{𝛼+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)},𝑑𝑤𝑑𝑇=−𝑞^∗𝑤−^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑏_{𝛽,𝛼−1}+(𝑛+1)𝑎_{𝛼,𝛽−1}𝑧^{𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}. (3.11) Differentiating 𝑓(𝑧,𝑤) with respect to 𝑇 along the trajectories of system (3.2) yields 𝑑𝑓=𝑑𝑇𝑑𝑧+𝑑𝑇^∞_{𝑘+𝑗=2𝑛+4}𝑐^′_𝑘𝑗𝑘𝑧^𝑘−1𝑤^𝑗𝑑𝑧𝑑𝑇+𝑗𝑧^𝑘𝑤^𝑗−1𝑑𝑤𝑑𝑇=𝑝^∗𝑧+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}𝑧^{𝛼+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}+^∞_{𝑘+𝑗=2𝑛+4}𝑐^′_𝑘𝑗𝑘𝑧^𝑘−1𝑤^𝑗𝑝^∗𝑧+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}×𝑧^{𝛼+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}−𝑗𝑧^𝑘𝑤^𝑗−1𝑞^∗𝑤+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑏_{𝛽,𝛼−1}+(𝑛+1)𝑎_{𝛼,𝛽−1}×𝑧^{𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}=𝑝^∗𝑧+𝑝^∗∞_{𝑘+𝑗=2𝑛+4}𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗+^∞_{𝑘+𝑗=2𝑛+4}𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}𝑧^{𝛼+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}+^∞∞_{𝑘+𝑗=2𝑛+4𝛼+𝛽=2𝑛+3}𝑐^′_𝑘𝑗𝑘(𝑛+2)𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}−𝑗(𝑛+2)𝑏_{𝛽,𝛼−1}+(𝑛+1)𝑎_{𝛼,𝛽−1}×𝑧^{𝑘+𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝑗+𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}. (3.12) For 1≤𝑘+𝑗≤2𝑛+3, let 𝑐^′₁₀𝑐=1,_′𝑘𝑗=0,forother𝑘,𝑗, (3.13) then one arrives at (𝑛+2)^∞_{𝛼+𝛽=2𝑛+3}𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}𝑧^{𝛼+1+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}=^2𝑛+3∞_{𝑘+𝑗=1𝛼+𝛽=2𝑛+3}𝑐^′_𝑘𝑗𝑘(𝑛+2)𝑎_{𝛼,𝛽−1}+(𝑛+1)𝑏_{𝛽,𝛼−1}−𝑗(𝑛+2)𝑏_{𝛽,𝛼−1}+(𝑛+1)𝑎_{𝛼,𝛽−1}×𝑧^{𝑘+𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝑗+𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}, (3.14) thus 𝑑𝑓𝑑𝑇=𝑝^∗𝑓(𝑧,𝑤)+^∞_{𝑘+𝑗=2𝑛+4}𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗+^∞∞_{𝑘+𝑗=1𝛼+𝛽=2𝑛+3}(𝑛+2)𝑘−(𝑛+1)𝑗𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑗−(𝑛+1)𝑘𝑏_{𝛽,𝛼−1}×𝑐^′_𝑘𝑗𝑧^{𝑘+𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝑗+𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}. (3.15) Subscript transformation 𝑘′=𝑘+𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1),𝑗′=𝑗+𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1) implies 𝑘′+𝑗′=𝑘+𝑗+(𝛼+𝛽−2𝑛−2)(2𝑛+3)≥2𝑛+4. Still denoting (𝑘^′,𝑗^′) with (𝑘,𝑗), then one derives (𝑛+2)^∞∞_{𝑘+𝑗=1𝛼+𝛽=2𝑛+3}𝑘−(𝑛+1)𝑗𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑗−(𝑛+1)𝑘𝑏_{𝛽,𝛼−1}×𝑐^′_𝑘𝑗𝑧^{𝑘+𝛼+(𝛼+𝛽−2𝑛−3)(𝑛+1)}𝑤^{𝑗+𝛽+(𝛼+𝛽−2𝑛−3)(𝑛+1)}=^∞∞_{𝑘+𝑗=2𝑛+4𝛼+𝛽=2𝑛+3}(𝑛+2)(𝑘−𝛼)−(𝑛+1)𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑗−𝛽−(𝑛+1)(𝑘−𝛼)−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}×𝑐^′_{))𝑘−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1}𝑧^𝑘𝑤^𝑗. (3.16) Therefore, 𝑑𝑓𝑑𝑇=𝑝^∗𝑓()+𝑧,𝑤_{∞𝑘+𝑗=2𝑛+4}𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗𝑐^′_𝑘𝑗+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)(𝑘−𝛼)−(𝑛+1)𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑗−𝛽−(𝑛+1)(𝑘−𝛼)−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}×𝑐^′_{))𝑘−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1}𝑧^𝑘𝑤^𝑗. (3.17) Denote that ℎ_𝑘𝑗=𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗𝑐^′_𝑘𝑗+^∞_{𝛼+𝛽=2𝑛+3}(𝑛+2)(𝑘−𝛼)−(𝑛+1)𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1)𝑎_{𝛼,𝛽−1}−(𝑛+2)𝑗−𝛽−(𝑛+1)(𝑘−𝛼)−𝛼+𝛽−2𝑛−3(𝑛+1)𝑏_{𝛽,𝛼−1}×𝑐^′_{))𝑘−𝛼−𝛼+𝛽−2𝑛−3(𝑛+1,𝑗−𝛽−𝛼+𝛽−2𝑛−3(𝑛+1}. (3.18) When 𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗≠0, take ℎ_𝑘𝑗=0, else take ℎ_𝑘𝑗=𝑝^∗𝑝^′_𝑖(𝑘=𝑞^∗𝑖+1,𝑗=𝑝^∗𝑖,𝑖=1,2,…), then one obtains 𝑑𝑓𝑑𝑇=𝑝^∗𝑓(𝑧,𝑤)+^∞_𝑖=𝑛+2𝑝^∗𝑝^′_𝑖𝑧^{𝑞∗𝑖+1}𝑤^𝑝∗𝑖. (3.19) When 𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗≠0, from ℎ_𝑘𝑗=0, one gets (3.6); and when 𝑝^∗𝑘−𝑞^∗𝑗−𝑝^∗=0, from ℎ_𝑘𝑗=ℎ_{𝑞∗𝑖+1,𝑝∗𝑖}=𝑝^′_𝑖, one gets (3.8).
For 𝑔(𝑧,𝑤), in the same way as 𝑓(𝑧,𝑤) was done, one can get (3.7) and (3.9).

The relations between 𝑝_𝑖,𝑞_𝑖 and 𝑝^′_𝑖,𝑞^′_𝑖(𝑖=1,2,…) are as follows.

Theorem 3.2. Let 𝑝₀=𝑞₀=𝑝^′₀=𝑞^′₀=0. If there exists a positive integer 𝑙, such that 𝑝₀=𝑞₀=𝑝₁=𝑞₁=⋯=𝑝_𝑙−1=𝑞_𝑙−1=0, (3.20) then 𝑝^′₀=𝑞^′₀=𝑝^′₁=𝑞^′₁=𝑝^′_𝑙−1=𝑞^′_𝑙−1=0,𝑝_𝑙=𝑝^′_𝑙,𝑞_𝑙=𝑞^′_𝑙 (3.21) per contra holds as well.

Proof. Let 𝑙 be any a positive integer, and 𝜑(𝑧,𝑤)=𝑧+^{(𝑝∗+𝑞∗)𝑙+1}_{𝑘+𝑗=2𝑛+4}𝑐_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝜓(𝑧,𝑤)=𝑤+^{(𝑝∗+𝑞∗)𝑙+1}_{𝑘+𝑗=2𝑛+4}𝑑_𝑘𝑗𝑤^𝑘𝑧^𝑗,𝑓(𝑧,𝑤)=𝑧+^{(𝑝∗+𝑞∗)𝑙+1}_{𝑘+𝑗=2𝑛+4}𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝑔(𝑧,𝑤)=𝑤+^{(𝑝∗+𝑞∗)𝑙+1}_{𝑘+𝑗=2𝑛+4}𝑑^′_𝑘𝑗𝑤^𝑘𝑧^𝑗. (3.22) When expression (3.20) holds, from (2.3) we have 𝑑𝜑𝑑𝑇=𝑝^∗𝜑+𝑝^∗𝑝_𝑙𝑧^{𝑞∗𝑙+1}𝑤^𝑝∗𝑙𝑑𝜓+h.o.t.,𝑑𝑇=−𝑞^∗𝜓−𝑞^∗𝑞_𝑙𝑤^{𝑝∗𝑙+1}𝑧^𝑞∗𝑙+h.o.t., (3.23)
where h.o.t. stands for higher order term.
From expression (3.5), we get 𝑑𝑓𝑑𝑇=𝑝^∗𝑓+^𝑙_𝑖=𝑛+2𝑝^∗𝑝^′_𝑖𝑧^{𝑞∗𝑖+1}𝑤^𝑝∗𝑖+h.o.t.,𝑑𝑔𝑑𝑇=−𝑞^∗𝑔−^𝑙_𝑖=𝑛+2𝑞^∗𝑞^′_𝑖𝑤^{𝑝∗𝑖+1}𝑧^𝑞∗𝑖+h.o.t. (3.24)Considering the uniqueness in Lemma 2.5 and Theorem 3.1, from expressions (3.23) and (3.24), it is easy to get expression (3.21) and 𝜑=𝑓,𝜓=𝑔 with mathematical induction.

Theorems 3.1 and 3.2 give a new algorithm to compute the generalized period constants at the origin of system (3.2) by applying the coefficients of system (1.11). For any a positive integer 𝑙, in order to compute 𝜏_𝑙, we only need to force addition, subtraction, multiplication, and division to the coefficients of system (1.11). The algorithm is recursive and then avoids complex integration operations and solving equations. It is symbolic and easy to realize with computer algebraic system such as Mathematica or Maple.

Remark 3.3. We cannot use Theorems 3.1 and 3.2 to compute singular point quantities 𝜇_𝑙=𝑝_𝑙−𝑞_𝑙(𝑙=1,2,…), because the one condition of Theorem 3.2 is expression (3.20), while the computation of 𝜇_𝑙 is only under the condition 𝜇₀=𝜇₁=⋯=𝜇_𝑙−1=0. However, we can apply directly the two theorems to find necessary conditions for linearizability, needless to solve firstly the problem of integrability.

4. Application

In this section, we investigate the linearizability problem at degenerate singular point for the following septic system:𝑑𝑧𝑑𝑇=𝑧²𝑤+𝑏₀₃𝑧⁵+𝑏₁₂𝑧⁴𝑤+𝑏₂₁𝑧³𝑤²+𝑏₃₀𝑧²𝑤³+𝜆𝑧⁴𝑤³,𝑑𝑤𝑑𝑇=−𝑤²𝑧−𝑎₀₃𝑤⁵−𝑎₁₂𝑤⁴𝑧−𝑎₂₁𝑤³𝑧²−𝑎₃₀𝑤²𝑧³−𝜆𝑤⁴𝑧³, (4.1) where𝑏_𝑘𝑗=𝑎_𝑘𝑗,𝑘,𝑗∈{(3,0),(2,1),(1,2),(0,3)}. (4.2)

Factually, the pseudo-isochronous center problem at the degenerate singular point of system (4.1) has already been solved in [17, 18]. Here the aim that we chose this system as an illustrative example is just to demonstrate the effectiveness of this theory.

Firstly, let us see the computational method of singular point quantities at 1 : −1 resonant degenerate singular point.

When 𝑝=𝑞=1, by the time scaling 𝑇→1/(2𝑛+3)𝑇, system (3.2) can be simplified as𝑑𝑧1𝑑𝑇=𝑧+2𝑛+3^∞_{𝛼+𝛽=2𝑛+2}(𝑛+2)𝑎_𝛼𝛽+(𝑛+1)𝑏_{𝛽+1,𝛼−1}𝑧^𝛼+1𝑤^𝛽+1×(𝑧𝑤)^{(𝛼+𝛽−2𝑛−2)(𝑛+1)}=𝑍(𝑧,𝑤),𝑑𝑤1𝑑𝑇=−𝑤−2𝑛+3^∞_{𝛼+𝛽=2𝑛+2}(𝑛+2)𝑏_𝛼𝛽+(𝑛+1)𝑎_{𝛽+1,𝛼−1}𝑤^𝛼+1𝑧^𝛽+1×(𝑧𝑤)^{(𝛼+𝛽−2𝑛−2)(𝑛+1)}=−𝑊(𝑧,𝑤). (4.3)

Definition 4.1 (see [19]). (i) For any positive integer 𝑘, 𝜇_𝑘=(2𝑛+3)𝜇^′_{(2𝑛+3)𝑘} is called the 𝑘th singular point quantity at the origin of system (1.12), where 𝜇^′_{(2𝑛+3)𝑘} is the (2𝑛+3)𝑘th singular point quantity at the origin of system (4.3).
(ii) If 𝜇₀=𝜇₁=⋯=𝜇_𝑘−1=0,𝜇_𝑘≠0, the origin is called the 𝑘th fine singular point of system (1.12).
(iii) If for all 𝑘, 𝜇_𝑘=0, then the origin is called a complex center of system (1.12).

Definition 4.2. Degenerate singular point of system (1.12) or the origin of system (3.2)𝑝=𝑞=1 is called a complex pseudo-isochronous center if the origin of system (4.3) is a complex isochronous center.

Lemma 4.3 (see [19]). For system (4.3), one can derive successively the terms of the following formal series: 𝑀(𝑧,𝑤)=1+^∞_𝑘=1𝑓_{(2𝑛+3)𝑘}(𝑧,𝑤), (4.4) where 𝑓_{(2𝑛+3)𝑘}(𝑧,𝑤)=_{()𝑘𝛼+𝛽=2𝑛+3}𝑐_𝛼𝛽𝑧^𝛼𝑤^𝛽, (4.5) such that 𝜕𝑀𝜕𝑧𝑍−𝜕𝑀𝜕𝑤𝑊+𝜕𝑍−𝜕𝑧𝜕𝑊𝜕𝑤𝑀=^∞_𝑚=12𝑚𝑛+3𝑚+1𝜆2𝑛+3^𝑚(𝑧𝑤)^{(2𝑛+3)𝑚+1},𝜆_𝑚∼𝜇_𝑚,𝑚=1,2,…. (4.6)

Lemma 4.4 (see [19]). For all pairs (𝛼,𝛽) with 𝛼≠𝛽, when 𝛼+𝛽=(2𝑛+3)𝑁, 𝑐_𝛼𝛽 is given by 𝑐_𝛼𝛽=1(2𝑛+3)𝛽−𝛼^2𝑛+2+𝑁_{𝑘+𝑗=2𝑛+3}(𝑛+2)𝛼−(𝑛+1)𝛽+1𝑎_{𝑘,𝑗−1}−(𝑛+2)𝛽−(𝑛+1)𝛼+1𝑏_{𝑗,𝑘−1}×𝑐_{𝛼−(𝑛+2)𝑘−(𝑛+1)𝑗+(𝑛+1)(2𝑛+3),𝛽−(𝑛+2)𝑗−(𝑛+1)𝑘+(𝑛+1)(2𝑛+3)}, (4.7) where, in (4.5), one take 𝑐₀₀=1 and 𝑐_𝑘𝑘 as an arbitrary constant (𝑘=1,2,…).
For any positive integer 𝑚, 𝜆_𝑚 is given by 𝜆_𝑚=^2n+2+2𝑚_𝑘+𝑗=1𝑎_{𝑘,𝑗−1}−𝑏_{𝑗,𝑘−1}𝑐_{(𝑚+𝑛+1)(2𝑛+3)−(𝑛+2)𝑘−(𝑛+1)𝑗,(𝑚+𝑛+1)(2𝑛+3)−(𝑛+2)𝑗−(𝑛+1)𝑘}, (4.8) where for all pairs (𝛼,𝛽), when 𝛼<0 or 𝛽<0, one take 𝑎_𝛼𝛽=𝑏_𝛼𝛽=𝑐_𝛼𝛽=0.

Performing the transformation (3.1)_{(n = 1)} and renaming (𝑧₁,𝑤₁,𝑇₁) by (𝑧,𝑤,𝑇), system (4.1) is reduced to𝑑𝑧𝑑𝑇=5𝑧+3𝑏₀₃𝑧⁸𝑤³+2𝑎₃₀+3𝑏₁₂𝑧⁷𝑤⁴+2𝑎₂₁+3𝑏₂₁𝑧⁶𝑤⁵+2𝑎₁₂+3𝑏₃₀𝑧⁵𝑤⁶+2𝑎₀₃𝑧⁴𝑤⁷+5𝜆𝑧¹¹𝑤¹⁰,𝑑𝑤𝑑𝑇=−5𝑤−3𝑎₀₃𝑤⁸𝑧³−2𝑏₃₀+3𝑎₁₂𝑤⁷𝑧⁴−2𝑏₂₁+3𝑎₂₁𝑤⁶𝑧⁵−2𝑏₁₂+3𝑎₃₀𝑤⁵𝑧⁶−2𝑏₀₃𝑤⁴𝑧⁷−5𝜆𝑤¹¹𝑧¹⁰. (4.9) By the time scaling 𝑇→(1/5)𝑇, system (4.9) can be interpreted to𝑑𝑧3𝑑𝑇=𝑧+5𝑏₀₃𝑧⁸𝑤³+152𝑎₃₀+3𝑏₁₂𝑧⁷𝑤⁴+152𝑎₂₁+3𝑏₂₁𝑧⁶𝑤⁵+152𝑎₁₂+3𝑏₃₀𝑧⁵𝑤⁶+25𝑎₀₃𝑧⁴𝑤⁷+𝜆𝑧¹¹𝑤¹⁰,𝑑𝑤3𝑑𝑇=−𝑤−5𝑎₀₃𝑤⁸𝑧³−152𝑏₃₀+3𝑎₁₂𝑤⁷𝑧⁴−152𝑏₂₁+3𝑎₂₁𝑤⁶𝑧⁵−152𝑏₁₂+3𝑎₃₀𝑤⁵𝑧⁶−25𝑏₀₃𝑤⁴𝑧⁷−𝜆𝑤¹¹𝑧¹⁰. (4.10)

According to Definition 4.1, Lemmas 4.3 and 4.4, we have the following.

Theorem 4.5. The first 9 singular point quantities at degenerate singular point of system (4.1) are as follows: 𝜇₁=−𝑎₂₁+𝑏₂₁,𝜇₂=𝑎₁₂𝑎₃₀−𝑏₁₂𝑏₃₀,𝜇₃=18−9𝑎₀₃𝑎²₃₀−𝑎²₁₂𝑏₀₃+𝑎₀₃𝑏²₁₂+9𝑏₀₃𝑏²₃₀,𝜇₄1=−𝑎12²¹+𝑏₂₁𝑎²₁₂𝑏₀₃+3𝑎₀₃𝑎₃₀𝑏₁₂−𝑎₀₃𝑏²₁₂−3𝑎₁₂𝑏₀₃𝑏₃₀,𝜇₅1=−𝑎648²¹²𝑏₀₃+3𝑎₀₃𝑎₃₀𝑏₁₂−𝑎₀₃𝑏²₁₂−3𝑎₁₂𝑏₀₃𝑏₃₀−27𝑎₀₃𝑏₀₃+16𝑎₁₂𝑏₁₂,𝜇+216𝜆₆𝜇=0,₇1=−466560405𝑎²₀₃𝑏²₀₃−8928𝑎₀₃𝑎₁₂𝑏₀₃𝑏₁₂+6400𝑎²₁₂𝑏²₁₂×−𝑎²₁₂𝑏₀₃−3𝑎₀₃𝑎₃₀𝑏₁₂+𝑎₀₃𝑏²₁₂+3𝑎₁₂𝑏₀₃𝑏₃₀,𝜇₈=76998445𝑎₀₃𝑏₀₃−32𝑎₁₂𝑏₁₂𝑎²₁₂𝑏₀₃+𝑎₀₃𝑏²₁₂−𝑎²₁₂𝑏₀₃−3𝑎₀₃𝑎₃₀𝑏₁₂+𝑎₀₃𝑏²₁₂+3𝑎₁₂𝑏₀₃𝑏₃₀,𝜇₉=11𝑎13778100²¹²𝑏²₁₂−224181𝑎₀₃𝑏₀₃+164000𝑎₁₂𝑏₁₂𝑎²₁₂𝑏₀₃+3𝑎₀₃𝑎₃₀𝑏₁₂−𝑎₀₃𝑏²₁₂−3𝑎₁₂𝑏₀₃𝑏₃₀. (4.11) In the above expression of 𝜇_𝑘, one has already let 𝜇₁=⋯=𝜇_𝑘−1=0,𝑘=2,3,…,9.

From Theorem 4.5, we get the following.

Theorem 4.6. For system (4.1), the first 9 singular point quantities are zero if and only if one of the following conditions holds:(I)𝑎₂₁=𝑏₂₁,3𝑎₃₀−𝑏₁₂=0,3𝑏₃₀−𝑎₁₂=0,𝑎₁₂𝑏₁₂≠0; (II)𝑎₂₁=𝑏₂₁,𝑎₃₀𝑎₁₂=𝑏₃₀𝑏₁₂,𝑎²₁₂𝑏₀₃=𝑏²₁₂𝑎₀₃,𝑏₃₀𝑎₁₂𝑏₀₃=𝑎₃₀𝑏₁₂𝑎₀₃,𝑏²₃₀𝑏₀₃=𝑎²₃₀𝑎₀₃.

In order to obtain the center conditions of degenerate singular point, we have to find out all the elementary Lie invariants of system (4.1); following the technique used in [10], we have the following.

Lemma 4.7. All the elementary Lie invariants of system (4.1) are as follows: 𝜆,𝑎₂₁,𝑏₂₁,𝑎₃₀𝑏₃₀,𝑎₁₂𝑏₁₂,𝑎₀₃𝑏₀₃,𝑎₃₀𝑎₁₂,𝑏₃₀𝑏₁₂,𝑎²₃₀𝑎₀₃,𝑎₃₀𝑏₁₂𝑎₀₃,𝑏²₁₂𝑎₀₃,𝑏²₃₀𝑏₀₃,𝑏₃₀𝑎₁₂𝑏₀₃,𝑎²₁₂𝑏₀₃. (4.12)

Theorem 4.8. For system (4.1), all the singular point quantities at degenerate singular point are zero if and only if the first 9 singular point quantities are zero, that is, one of the conditions in Theorem 4.6 holds. Correspondingly, the conditions in Theorem 4.6 are the center conditions of degenerate singular point.

Proof. If condition (I) is satisfied, system (4.1) has the integrating factor (𝑧𝑤)⁻⁵; if condition (II) is satisfied, system (4.1) satisfies the conditions of the extended symmetric principle (Theorem  2.6 in [10]).

We next discuss the linearizability problem of the origin of system (4.9), according to Theorems 3.1 and 3.2; we can get the recursive formulae to compute period constants at the origin. Since transformation (3.1)_{(n = 1)} is a homeomorphism, the center conditions of the origin of system (4.9) are identical with those of degenerate singular point of system (4.1). From the center conditions in Theorem 4.6, we investigate the following two cases.

Case 1 (center condition (I) holds.). Substituting center condition (I) into the recursive formulae in Theorem 3.1 we obtain the first 8 period constants 𝜏₁=2𝑟₂₁,𝜏₂=118−9𝑎₀₃𝑏₀₃−16𝑎₁₂𝑏₁₂,𝜏+36𝜆₃𝜏=0,₄=12592−81𝑎²₀₃𝑏²₀₃+1152𝑎₀₃𝑎₁₂𝑏₀₃𝑏₁₂−256𝑎²₁₂𝑏²₁₂,𝜏₅=71944−27𝑎₀₃𝑏₀₃+32𝑎₁₂𝑏₁₂𝑎²₁₂𝑏₀₃+𝑎₀₃𝑏²₁₂,𝜏₆𝜏=0,₇=14228𝑎10935²¹²𝑏²₁₂𝑎²₁₂𝑏₀₃+𝑎₀₃𝑏²₁₂,𝜏₈=−65𝑎59049³¹²𝑏³₁₂−5571𝑎₀₃𝑏₀₃+1280𝑎₁₂𝑏₁₂. (4.13)
In the above expression of 𝜏_𝑘, we have already let 𝜏₁=⋯=𝜏_𝑘−1=0,𝑘=2,3,…,8.
From 𝜏₇=0, there exists a complex constant ℎ, such that 𝑎₀₃=ℎ𝑎²₁₂,𝑏₀₃=−ℎ𝑏²₁₂, (4.14) putting (4.14) into the expression of 𝜏₄, we have 𝜏₄1=−𝑎2592²¹²𝑏²₁₂256+1152𝑎₁₂𝑏₁₂ℎ²+81𝑎²₁₂𝑏²₁₂ℎ⁴. (4.15) By virtue of 𝑎₁₂𝑏₁₂≠0, 𝜏₄=0 yields 4ℎ=3−4±√15𝑎₁₂𝑏₁₂, (4.16) this, together with (4.14), leads to 𝜏₈=1040√590492396∓61915𝑎⁴₁₂𝑏⁴₁₂≠0. (4.17) Therefore, under center condition (I), the origin is not a complex isochronous center.

Case 2 (center condition (II) holds). When center condition (II) holds, the right hand of system (4.9) satisfies the extended symmetric principle (Theorem  2.6 in [10]). Write that 3𝑎₃₀−𝑏₁₂=𝐵,3𝑏₃₀−𝑎₁₂=𝐴. Then when 𝐴=𝐵=0, center condition (I) holds. We next assume that 𝐴𝐵≠0. Hence, we have from center condition (II) that 𝑎₃₀𝐴−𝑏₃₀𝐵=0,𝑎₀₃𝐵²−𝑏₀₃𝐴²=0. (4.18) It follows that there exist 𝑟₂₁,𝑟,𝑠 such that 𝑎₂₁=𝑏₂₁=𝑟₂₁,𝑎₃₀=𝑟𝐵,𝑏₃₀=𝑟𝐴,𝑎₀₃=𝑠𝐴²,𝑏₀₃=𝑠𝐵², (4.19) then center condition (II) is equivalent to 𝑎₂₁=𝑏₂₁=𝑟₂₁,𝑎₃₀=𝑟𝐵,𝑏₃₀=𝑟𝐴,𝑎₁₂=(3𝑟−1)𝐴,𝑏₁₂=()𝑎3𝑟−1𝐵,₀₃=𝑠𝐴²,𝑏₀₃=𝑠𝐵². (4.20) System (4.9) is brought to 𝑑𝑧𝑑𝑇=5𝑧+3𝐵²𝑠𝑧⁸𝑤³+𝐵(11𝑟−3)𝑧⁷𝑤⁴+5𝑟₂₁𝑧⁶𝑤⁵+𝐴(9𝑟−2)𝑧⁵𝑤⁶+2𝐴²𝑠𝑧⁴𝑤⁷+5𝜆𝑧¹¹𝑤¹⁰,𝑑𝑤𝑑𝑇=−5𝑤−3𝐴²𝑠𝑤⁸𝑧³−𝐴(11𝑟−3)𝑤⁷𝑧⁴−5𝑟₂₁𝑤⁶𝑧⁵−𝐵(9𝑟−2)𝑤⁵𝑧⁶−2𝐵²𝑠𝑤⁴𝑧⁷−5𝜆𝑤¹¹𝑧¹⁰. (4.21)
Putting expression (4.20) into the recursive formulae in Theorem 3.1 and calculating in Mathematica, we have the following.

Theorem 4.9. The first 6 period constants of the origin of system (4.21) are given by 𝜏₁=2𝑟₂₁,𝜏₂=124𝐴𝐵𝑟−16𝐴𝐵𝑟²−𝐴²𝐵²𝑠²,𝜏+4𝜆₃=14𝐴²𝐵²()𝜏−1+6𝑟𝑠,₄=−2𝐴²𝐵²(−1+𝑟)𝑟²()−1−1+4𝑟𝐴288³𝐵³𝑠²−16+9𝐴𝐵𝑠²,𝜏₅1=−𝐴972³𝐵³𝑟−8+27𝐴𝐵𝑠²,𝜏₆=1𝐴25³𝐵³𝑟²(−1+4𝑟)−154+715𝑟−1067𝑟²+531𝑟³−5053𝐴77760⁴𝐵⁴𝑠². (4.22) In the above expression of 𝜏_𝑘, one has already let 𝜏₁=⋯=𝜏_𝑘−1=0,𝑘=2,3,…,6.

Theorem 4.10. The first 6 period constants at the origin of system (4.21) are zero if and only if one of the following conditions holds: 𝜆=𝑟₂₁=𝑟=𝑠=0,𝐴𝐵≠0,𝜆=𝑟₂₁1=𝑠=0,𝑟=4,𝐴𝐵≠0. (4.23)(4.24)

Proof. Being 𝐴𝐵≠0, 𝜏₃=0 implies 𝑟=1/6 or 𝑠=0.
When 𝑟=1/6, 𝜏₄=−(1/2592)𝐴²𝐵²(40−144𝐴𝐵𝑠²+81𝐴²𝐵²𝑠⁴). From 𝜏₄=0, we have 𝑠=±(1/3)2(4±√6)/𝐴𝐵, then 𝜏₅=(∓1/729)(1/2)(4±√6)(±8+3√6)√𝐴⁵𝐵⁵≠0.
When 𝑠=0, 𝜏₄=−2𝐴²𝐵²(−1+𝑟)𝑟²(−1+4𝑟),𝜏₆=(1/25)𝐴³𝐵³𝑟²(−1+4𝑟)(−154+715𝑟−1067𝑟²+531𝑟³). From 𝜏₄=𝜏₆=0, we get 𝑟=0 or 𝑟=1/4.

Theorem 4.11. Under center condition (II), the origin of system (4.21) is a complex pseudo-isochronous center if and only if one of the conditions in Theorem 4.10 holds.

Proof. When condition (4.23) is satisfied, after the time scaling 𝑇→(1/5)𝑇, system (4.21) becomes 𝑑𝑧=1𝑑𝑇5𝑧5−3𝐵𝑧⁶𝑤⁴−2𝐴𝑧⁴𝑤⁶,𝑑𝑤1𝑑𝑇=−5𝑤5−3𝐴𝑤⁶𝑧⁴−2𝐵𝑤⁴𝑧⁶. (4.25) There exists a transformation 𝑧𝑢=1−𝐴𝑧⁴𝑤^61/51−𝐵𝑧⁶𝑤^43/10𝑤,𝑣=1−𝐵𝑤⁴𝑧^61/51−𝐴𝑤⁶𝑧^43/10, (4.26) such that system (4.25) is reduced to 𝑑𝑢/𝑑𝑇=𝑢,𝑑𝑣/𝑑𝑇=−𝑣.
When condition (4.24) is satisfied, after the time scaling 𝑇→(1/5)𝑇, system (4.21) becomes 𝑑𝑧=1𝑑𝑇𝑧2020−𝐵𝑧⁶𝑤⁴+𝐴𝑧⁴𝑤⁶,𝑑𝑤1𝑑𝑇=−𝑤2020−𝐴𝑤⁶𝑧⁴+𝐵𝑤⁴𝑧⁶, (4.27) from the polar coordinate transformation 𝑧=𝜌𝑒^𝑖𝜃,𝑤=𝜌𝑒^−𝑖𝜃,𝑇=𝑖𝑡,𝑖=√−1, (4.28) and we have for system (4.27) that 𝑑𝜃=1𝑑𝑡21𝑧𝑑𝑧−1𝑑𝑇𝑤𝑑𝑤𝑑𝑇=1. (4.29) From the above discussion and Definition 4.2, the conclusion of this theorem follows.

We have from Definition 4.2, expression (4.20), Theorems 4.10 and 4.11 the following.

Theorem 4.12. Degenerate singular point of system (4.1) is a complex pseudo-isochronous center if and only if one of the following conditions is satisfied: (i)𝜆=𝑎₃₀=𝑏₃₀=𝑎₂₁=𝑏₂₁=𝑎₀₃=𝑏₀₃=0,𝑎₁₂𝑏₁₂≠0, (ii)𝜆=𝑎₂₁=𝑏₂₁=𝑎₀₃=𝑏₀₃=0,𝑎₃₀+𝑏₁₂=0,𝑏₃₀+𝑎₁₂=0,𝑎₁₂𝑏₁₂≠0.

5. Conclusions and Remarks

In the qualitative theory of ODE, the linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. Up to now, most of the existing literatures are concerned with elementary singular point. A progressive way to find necessary conditions for linearizability is to compute period constants.

The linearizability problem of degenerate singular point is much more difficult. As far as the degenerate case is concerned, the classical methods are invalid. As a result, the literatures about linearizability problem of degenerate singular point are limited. One reason is that the problem itself is very complicated; another is that the problem is subject to limited methods. Therefore, it is completely natural to develop a "new" theory to analyze this dynamical behavior for the planar dynamical systems with degenerate singular points.

In our paper, firstly, the degenerate singular point is taken to the elementary origin by a blow-up; besides, we establish a new recursive algorithm to compute the so-called generalized period constants, so we can deal with the linearizability problem of degenerate singular point via the classical theory developed for elementary singular point. Synthesizing the above, we give an indirect approach regarding to the linearizability problem of the degenerate singular point.

Acknowledgments

The authors are grateful to the anonymous referees for their careful comments and valuable suggestions. Especially, the authors greatly appreciated the help of the referees in posing important questions, which are sure to enhance the readability and interest of the paper and suggest some neat ideas for future studies. This work is supported in part by the National Nature Science Foundation of China (NSFC 11101126 and 11101127) and Scientific Research Foundation for Doctoral Scholars of HAUST (09001524).