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Bifurcation of limit cycles at degenerate singular point and infinity in a septic system
Advances in Difference Equations volume 2013, Article number: 5 (2013)
Abstract
In this paper, the problem of bifurcation of limit cycles from degenerate singular point and infinity in a class of septic polynomial differential systems is investigated. Using the computer algebra system Mathematica, the limit cycle configurations of and are obtained under synchronous perturbation at degenerate singular point and infinity. To our knowledge, up to now, this is the first time that the problem of limit cycles bifurcated from degenerate singular point and infinity under synchronous perturbed conditions in a septic system has been investigated.
MSC:34C05, 34C07.
1 Introduction
In the qualitative theory of planar dynamical systems, bifurcation of limit cycles for planar polynomial differential system
which belongs to the second part of the Hilbert 16th problem, is known as a hot but intractable issue. In the case of limit cycles bifurcated from the origin, a lot of work has been done (see [1, 2]). Let be the maximum possible number of limit cycles in the neighborhood of the infinity (large amplitude limit cycles) of system (1.1) when , are of degree at most n. As far as the number of limit cycles bifurcated from infinity is concerned, representative results are as follows: cubic systems, [3] got , [4] got , [5] got , [6, 7] got ; quintic systems, [8] got , [9] got , [10] got , [11] got ; septic systems, [12] got , [13] got , [14] got .
Definition 1.1 [4]
We call the configuration the configuration of a vector field with ‘small’ limit cycles and ‘large’ ones.
Few papers are concerned with bifurcation of limit cycles from the origin and infinity under synchronous perturbation: [4] obtained the limit cycle configurations of and in a cubic polynomial differential system; [15] and [16] respectively got the limit cycle configurations of , and , in two certain quintic systems. However, for a higher degree system, there is not any result.
In this paper, we deal with limit cycles bifurcated from degenerate singular point and infinity under synchronous perturbation in the following real septic polynomial differential system:
where and
This paper is organized as follows. In Section 2, some preliminaries are given and system (1.2) is transferred to its complex concomitant system. In Sections 3 and 4, the first nine singular point quantities at degenerate singular point and the first seven singular point quantities at infinity are deduced. At the same time, their center conditions are obtained. In Section 5, the sufficient conditions of limit cycle configurations and are presented.
2 Conversion of the system
Obviously, the origin of system (1.2) is either a center or a focus, and it is also a degenerate singular point. System (1.2) contains no real singular point on the equator of the Poincaré compactification on the sphere. is called infinity on the Gauss sphere or the equator of system (1.2).
By means of polar coordinate transformation
system (1.2) takes the form
For a sufficiently small h, let
be the solution of Eq. (2.2) satisfying the initial value condition , where
The 0th focal value at the origin of system (1.2) is , i.e., , the k th one is ,  .
By means of generalized polar coordinate transformation
system (1.2) takes the form
For a sufficiently small h, let
be the solution of Eq. (2.6) satisfying the initial value condition , where
The 0th focal value at infinity of system (1.2) is , i.e., , the k th one is ,  .
By means of transformation
system (1.2) can be transformed into the following complex system:
where
It is obvious that the coefficients of system (2.10) satisfy the conjugate condition, i.e.,
We call that systems (1.2) and (2.10) are associated.
3 Singular point quantities and center conditions at degenerate singular point
From Theorem 3.2 in [3], we have
Lemma 3.1 For system , we can determine uniquely an extended formal power series
such that
where , ,  , , and for any positive integer m, the mth singular point quantity at the origin can be determined by the following recursion formulas:
Using the recursion formulas to do symbolic computation, we have
Theorem 3.1 The first nine singular point quantities at the origin of system are as follows:
In the above expression of , we have already let , .
From Theorem 3.1, it is easy to get
Theorem 3.2 For system , the first nine singular point quantities at the origin are zero if and only if one of the following conditions holds:
Proof Putting condition (I) or (II) into expression (3.5) can easily complete the proof of sufficiency.
Now, let us prove the necessity. If , then , . Therefore, () yields
If , notice that , , there exist and p such that
Substituting (3.7) into expression (3.5), we get
and do not simultaneously hold, thus from , we have (i) or (ii) , . At this moment, . Condition (i) or (ii) combining with implies the following conditions, respectively:
Condition (3.6) plus (3.10) is equivalent to condition (II). The proof is completed. □
From the definition of elementary Lie invariants in [17], we can obtain
Lemma 3.2 All the elementary Lie invariants of system are as follows:
Theorem 3.3 For system , all the singular point quantities at the origin are zero if and only if the first nine singular point quantities are zero, i.e., one of the two conditions in Theorem 3.2 holds. Correspondingly, the two conditions in Theorem 3.2 are the center conditions of the origin.
Proof When condition (I) holds, system has the integrating factor ; when condition (II) holds, system satisfies the conditions of the extended symmetric principle (Theorem 2.6 in [17]). □
4 Singular point quantities and center conditions at infinity
Lemma 4.1 [18]
For the system
where
to any integer , , we can derive successively the terms of the following formal series:
such that
and for any integer m,
where is the mth singular point quantity at infinity, ‘∼’ is the symbol of algebraic equivalence, , can be taken arbitrarily,  . When , is determined by the following recursion formulas:
Take , , , applying Lemma 4.1 to system and executing calculation in Mathematica, we have
Theorem 4.1 The first seven singular point quantities at infinity of system are as follows:
In the above expression of , we have already let , .
By the similar method used in the proof of Theorem 3.1, we can obtain the following result.
Theorem 4.2 For system , the first seven singular point quantities at infinity are zero if and only if condition (I) or (II) is satisfied.
The following result is immediate from Theorems 3.3 and 4.2.
Theorem 4.3 The degenerate singular point and infinity of system have the same center conditions.
5 Bifurcation of limit cycles under synchronous perturbation
From Theorems 3.1 and 4.1, we have
Theorem 5.1 The degenerate singular point of system is a fine focus of order nine if and only if the coefficients of its associated complex system satisfy
Theorem 5.2 The infinity of system is a fine focus of order seven if and only if the coefficients of its associated complex system satisfy
Actually, when the degenerate singular point of system is a fine focus of order nine, its infinity is at most a fine focus of order three; when the infinity of system is a fine focus of order seven, its degenerate singular point is at most a fine focus of order three. Therefore, we have
Theorem 5.3 By synchronous perturbation at degenerate singular point and infinity, system (1.2) has the limit cycle configurations of and .
For the sake of simplicity, we will prove the first case of Theorem 5.3. The other case can be proved with the same principle.
Theorem 5.4 Let the coefficients in (2.10) satisfy
(accordingly, the coefficients of system (1.2) are determined by (1.4), (2.11) and (5.3)), where , are small parameters with
Then for system (1.2), there are eight limit cycles bifurcating from the degenerate singular point, and at the same time, there are three limit cycles bifurcating from infinity.
Proof For the degenerate singular point of system (1.2), from , , Theorem 3.1 and , after computing carefully, we have the following:
where are analytic at and , .
Simultaneously, for the infinity of system (1.2), we have
According to the classical theory of Bautin, there exist eight limit cycles in a small enough neighborhood of the degenerate singular point and three limit cycles in a small enough neighborhood of the infinity. □
6 Concluding remarks
We have presented the solution to the problem of bifurcation of limit cycles at degenerate singular point and infinity for a class of seventh degree polynomial vector field. There are two important points to make here. Firstly, by computing the first several singular point quantities with Mathematica and vanishing them, the same center conditions at degenerate singular point (Theorem 3.2) and at infinity (Theorem 4.2) are obtained. Secondly, by synchronous perturbation of degenerate singular point and infinity, the limit cycle configurations of and are determined. We hope that our work could make some contributions in this direction.
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Acknowledgements
Acknowledgments are due to the referees for their useful suggestions and valuable comments. This work is supported in part by the National Nature Science Foundation of China (No. 11101126, 11126114 and 11201211), the Fund of Henan Education Commission (No. 2011B110014), and the Scientific Research Foundation for Doctoral Scholars of Haust (09001524).
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The study was carried out in collaboration between both of the authors. YW completed the main part of this paper, FL corrected the main theorems and polished the manuscript. All authors read and approved the final manuscript.
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Wu, Y., Li, F. Bifurcation of limit cycles at degenerate singular point and infinity in a septic system. Adv Differ Equ 2013, 5 (2013). https://doi.org/10.1186/1687-1847-2013-5
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DOI: https://doi.org/10.1186/1687-1847-2013-5