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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 783546, 24 pages
http://dx.doi.org/10.1155/2012/783546
Research Article

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

Feng Li1Β and Yusen Wu2

1School of Mathematical Science and Computing Technology, Central South University, Hunan, Changsha 410075, China
2School of Mathematics and Statistics, Henan University of Science and Technology, Henan, Luoyang 471003, China

Received 30 October 2011; Revised 27 November 2011; Accepted 13 January 2012

Academic Editor: P. J. Y.Β Wong

Copyright Β© 2012 Feng Li and Yusen Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , 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 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    writes as (all of which have an isochronous center at the origin) and the cubic time-reversible systems with , 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: where By using their transformation system (1.2) becomes

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) is called a pseudo-isochronous center if the origin of system (1.5) 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: were investigated by Llibre and Valls, see [25–28]. The conditions of centers and isochronous centers were obtained. But the 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 where When , 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: where 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 where are complex numbers, system (2.1) can be transformed into

For the complex constant , , we write the solution of system (2.4) satisfying the initial condition as which could be thought of as the first PoincarΓ© displacement map and denote the period function by

Definition 2.1. For a sufficiently small complex constant , the origin of system (2.1) is called a complex center if , and it is called a complex isochronous center if

Lemma 2.2. For system (2.1), one can derive uniquely the following formal series: where , , such that

Definition 2.3 (see [37, 38]). Let ,  ,  ,   be called the singular point quantity of the origin of system (2.1) and be called the 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: such that where , for all , , and for any integer , is determined by the following recursive formulae:

Theorem 2.5 (see [37]). For system (2.1), one can derive uniquely the following formal series: where , , such that and when , and are determined by the following recursive formulae: and for any positive integer ,  , and are determined by the following recursive formulae: In the above expression, one has let ,  , and if or , let .
We introduce double parameter transformation groups where are new variables, are complex parameters, and . Denote ,  ,  ,  . Transformation (2.17) can be turned into 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 , are parameters, , , are variables, and for all , one has
Under the transformation (2.17), suppose that is a polynomial of , with complex coefficients, and denote where ,  ,  ,  ,  .

Definition 2.6 (see [38]). Suppose that there exist constants , such that , we say that is a similar exponent and a rotation exponent of system (2.1) under the transformation (2.17), which are denoted by .

Definition 2.7 (see [38]). (i) A polynomial is called a Lie invariant of order , if .
(ii) An invariant is called a monomial Lie invariant, if is both of a Lie invariant and a monomial of .
(iii) A monomial Lie invariant 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 is called self-symmetry if . It is called self-antisymmetry if .

Theorem 2.9 (see the extended symmetric principle in [38]). Let denote an elementary Lie invariant of system (2.1). If for all the symmetric condition 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 system (1.7)Ξ΄=0 becomes its concomitant complex system where

Then, by using transformation system (3.2) can be transformed into the following system:

At last, by means of transformation (1.4), system (3.5) is reduced to

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:
Case 1. , then there exist to make ,  , If , If , then there exist to make ,  , Case 2. , where , ,  ,  . In the above expression of , we have already let , .

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:

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:

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) satisfies the conditions of Theorem 2.9. If condition (3.13) holds, system (3.6) has the first integral where If condition (3.15) holds, system (3.6) becomes there exists a transformation system (3.19) is changed into system (3.21) has the integral factor , where

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 , from the integrability conditions given in Section 4, we investigate the following three cases.

Case 1. Integrability condition (3.12) holds.
If , we easily obtain the first 30 period constants
If , from condition (3.12), there exists an arbitrary complex constant , such that then we get the first 30 period constants In expressions (4.1) and (4.3), , , , , and we have already let , .

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:

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 There exists a transformation such that system (4.6) is reduced to a linear system.
When condition (4.5) is satisfied, system (3.6) becomes There exists a transformation 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 where , , , . In the above expression of , we have already let ,  .

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:

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 There exists a transformation such that system (4.12) is reduced to a linear system.

Case 3. Integrability condition (3.14) holds.
Because , we can let where are arbitrary complex constants. Substituting (4.14) into the recursive formulae in Appendix B, we obtain the first 30 period constants If , If , If , where ,  ,  ,  . In the above expression of , we have already let ,  .

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:

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 we have for system (4.21) that
When condition (4.20) is satisfied, system (3.6) becomes There exists a transformation 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 Because , under integrability condition (3.15), the origin of system (3.5)Ξ΄=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) ((3.5)) 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):, when or , or ,. Else<