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

and Applied Analysis 3 dη dτ ξ δ 2n 3 η ∞ ∑ k 2n 2 [( ξ2 1 2n 3 η2 ) Yk ( ξ, η ) − 2n 2 2n 3 ξηXk ( ξ, η ) ] × ( ξ2 η2 ) 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 /2 ( Az4 jz1−j Bz3z Cz2−jz j Dz ) , d 2m 1 ≥ 5, ż iz zz d−4 /2 ( Az3z Bz2z Cz ) , d 2m ≥ 4, ż λ i z zz d−3 /2 ( Az3 Bz2z Czz Dz ) , d 2m 1 ≥ 3, ż λ i z zz d−2 /2 ( Az2 Bzz Cz ) , 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 quarticlike 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 dx dt ( δx − y ( x2 y2 )λ X5 ( x, y )( x2 y2 )2 λ−1 − βy ( x2 y2 )3λ , dy dt ( x δy )( x2 y2 )λ Y5 ( x, y )( x2 y2 )2 λ−1 βx ( x2 y2 )3λ , 1.7 4 Abstract and Applied Analysis where


Introduction
In the qualitative theory of planar polynomial differential equations, one of open problems for planar polynomial differential systems dx dt P x, y , dy dt 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 socalled 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: By using their transformation 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 quarticlike 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 1.9 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.

Some Preliminary Results
In 36-38 , the authors defined complex center and complex isochronous center for the following complex system: Abstract and Applied Analysis Definition 2.3 see 37, 38 . Let μ 0 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: where c 00 1, for all c kk ∈ R, k 1, 2, . . ., and for any integer m, μ m is determined by the following recursive formulae:

2.12
Theorem 2.5 see 37 . For system 2.1 , one can derive uniquely the following formal series: 14 Abstract and Applied Analysis 7 and when k − j − 1 / 0, c kj and d kj are determined by the following recursive formulae:

2.15
and for any positive integer j, p j , and q j are determined by the following recursive formulae:
We introduce double parameter transformation groups where z, w are new variables, ρ, θ are complex parameters, and ρ / 0. Denote z x iy, w x − iy, z x i y, w x − i y. Transformation 2.17 can be turned into x ρ x cos θ − y sin θ , y ρ x sin θ y cos θ .

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 where ρ, θ are parameters, z, w, T are variables, and for all α ≥ 0, β ≥ 0 one has

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 Definition 2.6 see 38 . Suppose that there exist constants λ, σ, such that f ρ λ e iσθ f, 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.
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. 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

3.3
Abstract and Applied Analysis 9 Then, by using transformation system 3.2 can be transformed into the following system:

3.5
At last, by means of transformation 1.4 n 1 , system 3.5 is reduced to

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. where μ k 0, k / 5i, i ≤ 11, i ∈ N. In the above expression of μ k , we have already let μ 1 · · · μ k−1 0, k 2, 3, . . . , 45.
From Theorem 3.1, we get the following.

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.

3.19
there exists a transformation z u

3.22
Synthesizing all the above cases, we have the following.

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 a 21 b 21 r 21 , from the integrability conditions given in Section 4, we investigate the following three cases.

4.25
Because τ 10 a 12 b 12 / 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.

B.
The recursive formulae to compute the period constants of the origin of system 3.6 :