Abstract and Applied Analysis

Volume 2013 (2013), Article ID 762751, 8 pages

http://dx.doi.org/10.1155/2013/762751

Research Article

Limit Cycles and Isochronous Centers in a Class of Ninth Degree System

Li Hongwei,¹ Li Feng,¹ and Du Chaoxiong²

¹School of Science, Linyi University, Linyi, Shandong 276005, China

²School of Science, Shaoyang University, Shaoyang, Hunan 422000, China

Received 15 April 2013; Accepted 3 October 2013

Academic Editor: Ruediger Landes

Copyright © 2013 Li Hongwei et al. 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.

A class of ninth degree system is studied and the conditions ensuring that its five singular points can be centers and isochronous centers (or linearizable centers) at the same time by exact calculation and strict proof are obtained. What is more, the expressions of Lyapunov constants and periodic constants are simplified, and 21 limit circles could be bifurcated at least.

1. Introduction

As we all know, it is difficult for planar polynomial differential systems to characterize their centers and isochronous centers. Hence, up to now the sufficient and necessary condition determining an isochronous center can only be found by making some appropriate analytic changes of coordinates which let the original system be reduced to a linear system. This kind of appropriate analytic change is very difficult to obtain, so only a handful of isochronous systems have been investigated. Several classes of known isochronous systems studied are as follows: quadratic isochronous centers (see [1]); isochronous centers of a linear center perturbed by third, fourth and fifth degree homogeneous polynomials (see [2–4]); complex polynomial systems (see [5]); reversible systems (see [6, 7]); and isochronous centers of cubic systems with degenerate infinity (see [8]).

In [9], a class of ninth degree system with four isochronous centers was investigated. In this paper, we consider the following ninth degree system:

where

The paper is organized as follows. In Section 2, the preliminary methods to calculate the focal values (or Lyapunov constants) and period constants which are necessary in Sections 3 and 5 are presented. In Section 3, two appropriate transformations, changing system (1) to a class of Z4-equivariant fifth degree system in which the focal values are calculated, are provided. Based on it, the condition and proof that the infinity and four elementary singular points of (1) can be centers are gained. The bifurcations of 21 limit circles are the subject of Section 4. At last, in Section 5, the period constants with simpler expressions are given; moreover, the sufficient and necessary condition and proof are given to illustrate that the infinity and four elementary singular points of (1) become isochronous centers.

2. Some Preliminary Results

In [10, 11], the complex system of complex center is

where

and the authors gave two recursive algorithms to determine necessary conditions for a center. We now restate the definitions and algorithms.

Theorem 1 (see [11]). For system (3), we 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 (see [12]). For system (3), we 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, we have let , , and if or , let .

3. Singular Point Quantities and Center Conditions

It can be easily checked that system (1) have four elementary critical points , and the infinity. After introducing the method to calculate the focal values and period constants of system (1), we try to make some appropriate transformations so as to carry out our investigation.

By means of the Bendixson homeomorphous transformation

and time transformation

system (1) can be transformed into the following real system:

Then, the infinity and four elementary focuses , of (1) become the origin and , of system (14), respectively. Thus, we have the following theorems.

Theorem 3. System (14) is a class of Z4-equivariant differential polynomial system of fifth degree about point .

Proof. The system is invariant by the transformation of rotation

Then system (14) is a Z4-equivariant polynomial system about point .

Now, we will consider the of system (14). Let

System (14) can be transformed into the following complex system:

where

Theorem 4. The first two singular point quantities at the origin of system (17) are as follows:

Theorem 5. For system (17), the first two singular point quantities are zero if and only if the following condition holds:

Theorem 6. For system (14), all the singular point quantities at the origin are zero if and only if the first two singular point quantities are zero; that is, the condition in Theorem 5 holds. Correspondingly, the condition in Theorem 5 is the center condition of the origin.

Proof. If condition (20) holds, system (14) becomes

Remark (21) is symmetric with axis.

System (14) is a class of Z4-equivariant differential polynomial system, so that the four symmetric elementary focus points of system (14) have the same topological structure; then they have the same center conditions and isochronous center conditions and so on. Without loss of generality, we only need to consider the case of .

Let

Then, system (14) can be changed into

By noting that

system (23) can be transformed into its complex system:

Through computing, we have the following theorem.

Theorem 7. The first five singular point quantities at the origin of system (25) are

where

Theorem 8. For system (25), the first five singular point quantities are zero if and only if one of the following conditions holds:

Then we can get the following theorem.

Theorem 9. For system (25), all the singular point quantities at the origin are zero if and only if the first five singular point quantities are zero; that is, one of the conditions in Theorem 8 holds. Correspondingly, the conditions in Theorem 8 are the center conditions of the origin.

Proof. Under condition (28), the system (25) becomes

System (30) has an integrating factor

Under condition (29), system (25) becomes

where (32) is symmetric with axis.

So we get the following theorem.

Theorem 10. The system (14) has five centers , , when . Namely, the system (1) has five centers the infinity and , when .

The system (14) has four centers , when , , . Namely, the system (1) has four centers , when , , and .

4. Bifurcation of Limit Circles

Theorem 11. For system (14), the is a fifth order weak focus if and only if

Proof. When , we could find suitable and to satisfy

So the is a fifth order weak focus.

Theorem 12. Assume that the conditions in Theorem 11 hold. Then the 5 limit cycles can bifurcate from the of System (14), which lie in the neighborhood of the origin.

Proof. If the conditions in Theorem 11 hold, we can obtain that the Jacobian of the with respect to

So . Applying the theory in [13], the result is completed.

Theorem 13. Assume that the conditions in Theorem 11 hold. Then 1 limit cycle can bifurcate from the of System (14), which lie in the neighborhood of the origin.

Since System (14) is Z4-equivariant, taking into account Theorems 12 and 13, we conclude the following.

Theorem 14. Assume that the conditions in Theorem 11 hold. Then 5 limit cycles can bifurcate from each of the four fine focus points , of System (14), and one limit circle could be created from the origin at the same time; namely, 21 limit cycles can bifurcate from the System (14) in all, five of which are located in the neighborhood of each of the four fine focus points , and one of which is located in the neighborhood of origin.

In fact, there are 21 limit cycles for system (1); 5 limit cycles can bifurcate from each of the four fine focus points , of System (1), and one limit circle could be created from the infinity at the same time; namely, 21 limit cycles can bifurcate from the system (1) in all. It is a very interesting result.

5. Period Constants and Isochronous Center Conditions

In this section, we focus on the calculation of period constants and isochronous center conditions of the origin and for system (17) and (25). According to Theorems 2, we begin by computing period constants through recursive formulae. Under the center conditions given in Section 3, we discuss three different relevant cases: (a) about the origin, case condition (20) holds.

Theorem 15. Under condition (20), the origin of system (17) is a complex isochronous center if and only if the following conditions holds

Proof. If condition (20) holds,substituting into the recursive formulae in Theorem 2, we can easily get the first four period constants

Clearly, implies that condition (37) holds.

On the other hand, case condition (37) holds, and system (17) is changed into

By means of transformation

system (39) is reduced to a linear system.

Thus, the origin of system (17) is a complex isochronous center.

(b) about of system (25).

When condition (28) holds, substituting , , into the recursive formulae in Theorem 2, we easily obtain the first two period constants

Thus could not be an isochronous center when condition (28) holds.

Theorem 16. of system (25) is a complex isochronous center if and only if condition (37) holds.

Proof. When condition (29) holds, substituting into the recursive formulae in Theorem 2, we easily obtain the first four period constants:

Denoting by

where is the resultant of with respect to , we have . From , we could get that . When , deduce , , .

Besides, when condition (37) holds, system (25) is changed into

Letting that

system (44) is reduced to a linear system.

From Theorems 15 and 16, we can conclude the following theorem.

Theorem 17. The infinity and four elementary singular points , are five isochronous centers of (1) if and only if condition (37) holds; namely , , .

Acknowledgments

This research is partially supported by the National Nature Science Foundation of China (11201211, 111101126, and 1371373) and Nature Science Foundation of Shandong Province (ZR2012AL04) and Applied Mathematics Enhancement Program of Linyi University.

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