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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 861329, 12 pages
http://dx.doi.org/10.1155/2013/861329
Research Article

Poincaré Bifurcations of Two Classes of Polynomial Systems

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China

Received 24 May 2013; Accepted 27 June 2013

Academic Editor: Luca Guerrini

Copyright © 2013 Jing Wang and Shuliang Shui. 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

Using bifurcation methods and the Abelian integral, we investigate the number of the limit cycles that bifurcate from the period annulus of the singular point when we perturb the planar ordinary differential equations of the form , with an arbitrary polynomial vector field, where or .

1. Introduction and Main Results

In the qualitative theory of real planar differential systems, one of the main problems is to determine the existence and number of the limit cycles of the polynomial differential system. In the general case, this is a very difficult task. Therefore, the researchers consider the weak Hilbert 16th problem. In addition, the existence of invariant algebraic curves in polynomial systems may influence the number of limit cycles. For example, the planar quadratic systems with one invariant line or conic curve or cubic curve can have at most one limit cycle [14]. In [3], the authors proved that the cubic systems with four invariant lines have at most one limit cycle. In [5, 6], the authors proved that a real polynomial system of degree with irreducible invariant algebraic curves has at most limit cycles if is even and limit cycles if is odd.

In this paper, we consider the weak Hilbert 16th problem that the unperturbed systems have a linear center and an invariant algebraic curve where and are polynomials of degree in , the algebraic curve satisfies , and is a sufficient small parameter.

It is obvious that, on the region , the system (1) is equivalent to the following form: When , system (2) is a Hamilton system with a family of ovals

Define the Abelian integral which is also called first-order Melnikov function of (2). According to the Poincaré-Pontryagin theorem [7], the number of isolated real zeros of controls the number of limit cycles of system (1) that bifurcate from the periodic annulus of the perturbed system (1) with . That is to say, when the does not vanish exactly, the maximum number of the isolated real zeros of is corresponding to the upper bound of the number of limit cycles which bifurcate from periodic annulus of unperturbed systems.

For the arbitrary polynomials , of given degree , the number of limit cycles of (2) depends on the different choices of . At present, several works have figured out this problem for the particular choices of . In [8], the authors studied the system (1) with and proved that the number of limit cycles that bifurcate from the period orbits is at most . The authors in [9, 10] studied the number of limit cycles which bifurcate from (1) when with and , respectively. The authors in [11] studied the number of limit cycles of system (1) with . In [12], the authors studied the number of limit cycles of system (1) with and obtain that the system can have at most limit cycles if and if , respectively. In [13], the authors studied the case the curves are three lines, two of them parallel and one perpendicular, and [14, 15] studied the case the curves are lines, and any two of them are parallel or perpendicular directions. The authors in [16] studied the case the curves are consistent by nonzero points. The authors in [17] considered system (1) with and proved that limit cycles can at most bifurcate from the periodic orbits of the unperturbed system. In [18], the authors proved that the system (1) with has at most limit cycles.

The aim of this paper is to investigate the upper bound of the number of limit cycles bifurcate from the periodic annulus of the center of the unperturbed system (1) with the perturbed polynomials , of given degree , and or .

Consider the planar differential system where is a sufficient small parameter. Applying the Abelian integral, we obtain the following two main theorems.

Theorem 1. If , the lower bound of the maximum number of limit cycles bifurcating from the period orbits of system (5) with is .

Theorem 2. If , the upper bound of the maximum number of limit cycles bifurcating from the period orbits of system (5) with is .

Our primary purpose is to calculate the concrete expression of ; then we can obtain the number of limit cycles of the perturbed system (5) by determining the isolated real zeros of Abelian integral . In Sections 2 and 3, we prove these two theorems with the different methods, respectively.

2. The proof of Theorem 1

Taking the change of variable , , we have where

Firstly, we have the following obvious result.

Lemma 3. If is odd, .

According to Lemma 3, we can rewrite the as follows: Denote and ; then Therefore, we have Let where if . Then, we have the following lemma.

Lemma 4. Abelian integral has an expansion in the form where

Proof. Firstly, we have Substituting the previous formulas into (10), we have
Using (12) and (13), (17) can be written as follows: Thus, we haveif , if , if , if , where and if . The proof is completed.

By Lemma 4, we regard , , , , , , , as free parameters, and denote vectors and Jacobian matrices where , . Then, is matrix, is matrix, and is matrix.

For matrices , and , we have Lemmas 5 and 6, respectively.

Lemma 5. For , and . That is, , .

Proof. According to Lemma 4 and (24), we can obtain that
Define matrix Then, we have Let ; then (). Denote where is matrix. Then, .
We add entries of the column which times to the th column and obtain
where . We can write as follows: then .
Summarizing above results, we have
therefore, .
According to Lemma 4 and (25), we have
By the above proof procedure, we can obtain in a similar way. That is . The proof is completed.

Lemma 6. For , .

Proof. Firstly, if , it is easy to know that .
If , we have Then, by Lemma 5, .
For , let , , , ; we have According to Lemma 4 and the definitions of and , simplifying it by elementary transformation of matrix, we obtain Therefore, . The proof is completed.

By Lemma 6, it is obvious that are independent. Now, we have the following lemma.

Lemma 7. For , one can write as follows: where , and are nonzero vectors.

Proof. According to Lemma 4, if , Substituting into the first equation, we obtain a linear equation where , . According to Lemma 5, , (41) has unique solution . That is, the first formula holds.
If , in a similar way, we can prove that the second formula holds.
If , we have substituting into the third equation, we obtain a linear equation where , . According to Lemma 5, , (43) has unique solution . That is, the third formula also holds. The proof is completed.

Now, we prove Theorem 1.

Proof. For , Abelian integral has an expansion of the following form: According to Lemma 6, for , are independent.
Let and ; then, by Lemma 7, . Thus (44) becomes , and if . Furthermore, we take , then still holds. Choosing proper such that , by Descartes’ rule of signs, (44) has a root on interval .
Let , and choose proper so that ; then (44) has the second root on interval (0, 1). In a similar way, we take proper in turn such that and . According to Descartes’ rule of signs, we can obtain zeros on interval .
Applying the Poincaré-Pontryagin theorem, the system (5) with can have at least limit cycles for suitable and . The proof of Theorem 1 is completed.

3. The proof of Theorem 2

In this section, we will prove Theorem 2. At first, all the primary computations to express the Abelian integral and some concerned lemmas are presented.

Taking the change of variable , , then by (4) we have where Denote

where ; then we have

Lemma 8. Let , , , ; then
where , , , . And

Proof. We use the residue theorem to compute the .
When , we have
Let ; then , . The previous formula becomes hence (49) holds. For the first formula of (50), and the others can be proved in a similar way. The proof is completed.

Lemma 9. If is odd, the integrands in and are odd functions with respect to the variable ; therefore, and .
Define then, according to Lemmas 8 and 9 and the definition of , one knows that .

Lemma 10. If is even, then

Proof. If is even, then Similarly, (56) also holds. The proof is completed.

Lemma 11. (i)  If is odd, .
(ii)  If is even, then

Proof. When ,
We use the residue theorem to compute the integrals and . Denote ; thus, , . We have Since is the first-order zero of the equation , the residue of at is For the residue at , we have the expansion of in the form the coefficient of is corresponding to the residue of at ; therefore, Substituting them into (60), we have We can compute in a similar way and obtain By the formulas (64) and (65), becomes from the previous formula, it is easy to know that, if is odd, , and if is even, formula (58) is obtained. The proof is completed.

In a similar way, we can prove the following lemma.

Lemma 12. (i) If is odd, .
(ii) If is even, then