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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 482850, 7 pages
http://dx.doi.org/10.1155/2013/482850
Research Article

The Number of Limit Cycles of a Polynomial System on the Plane

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 23 April 2013; Accepted 16 June 2013

Academic Editor: Csaba Varga

Copyright © 2013 Chao Liu and Maoan Han. 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

We perturb the vector field , with a polynomial perturbation of degree , where , and study the number of limit cycles bifurcating from the period annulus surrounding the origin.

1. Introduction and Main Result

The main task in the qualitative theory of real plane differential systems is to determine the number of limit cycles, which is related to Hilbert's 16th problem as well as weakened Hilbert's 16th problem, posed by Arnold in [1].

Consider a planar system of the form

where , , and are real polynomials, , and is a small real parameter. It is well known that the number of zeros of the Abelian integral

where , controls the number of limit cycles of (1) that bifurcate from the periodic orbits of the unperturbed system (1) with ; see [2].

The problem of finding lower and upper bounds for the number of zeros of , when and are arbitrary polynomials of a given degree, say , and is a particular polynomial, has been faced in several recent papers. In the case of perturbing the linear center by arbitrary polynomials and of degree , that is, considering , , there are at most limit cycles up to first order in , see [3], where denotes the integer part function. Also it is known that perturbing the quadratic center , inside the polynomial systems of degree we can obtain at most limit cycles up to first order in (see [4]). The authors of [5] studied the perturbation of the cubic center , inside the polynomial differential systems of degree , and they obtained that is an upped bound for the number of limit cycles up to first order in . In [6], the authors studied the perturbations of , , and they obtained that if and, respectively, if , up to first order in , are upper bounds for the number of the limit cycles. In [7] the authors studied the maximum number of limit cycles which can bifurcate from the periodic orbits of the quartic center , with and by perturbing it inside the class of polynomial vector fields of degree . They proved that . In [8] the authors studied the bifurcation of limit cycles of the system , for sufficiently small, where and , are polynomials of degree . They obtained that up to first order in the upper bound for the number of limit cycles that bifurcate from the period annulus of the quintic center given by is if or if . More results can be found in [9, 10].

But few of these algebraic curves have a multiple factor. In [11], the authors took and proved that an upper bound for the number of zeros of on (0,1) is and that this bound is reached when . The approach of [11] is mainly based on the explicit computation of . In [12], the authors obtained the maximum of zeros of , taking into account their multiplicities, is when and when . In [12], the authors improve the upper bound given in [11] and provide the optimal upper bound for the zeros of . In this paper, motivated by [12] we take and obtain the following theorem.

Theorem 1. Consider (1) with . Let
where , , and and are polynomials of degree . Then the maximum number of zeros of , taking into account their multiplicities, is when and when . Moreover, when and , the corresponding maximum number, which is 1, can be reached by taking suitable and .

2. Preliminary Results

To study the property of , we need to make some preliminaries. First we introduce a function of the form

where and is an integer.

This section contains some preliminary computations to express the Abelian integral given in (3) in terms of polynomials.

Lemma 2. Let be a polynomial of degree in . Then for and it holds that
for some .

Proof. Note that, for ,
The result follows by applying the above formula to each term of

Lemma 3. Let be the functions introduced in (4). Then, for ,

Proof. Note that
Using integration by parts in the last integral, we obtain
Note that
Then
Substituting the formula above into (9), we find
Replacing by , we can obtain the conclusion.

Lemma 4. The functions in (4) satisfy
Moreover,
where denotes a polynomial of (exact) degree .

Proof. It is easy to check that equality (14) is true for . Now we prove that it is true for any by induction.
Suppose that it is true for and ; that is,
We need to prove that
By Lemma 3, we have
Hence,
Then it follows from assumption (16) that
By Lemma 3 again we have
Substituting it into (20) we obtain
which gives (17). Hence equality (14) holds for .
If , then . By applying (14) for , it is easy to see that (14) holds for all .
The first formula in (15) for the case of follows directly from (4). The second one follows from the first one together with (14). This completes the proof.

Some explicit expressions of are

Lemma 5. For any nonnegative integer numbers , , and , one has

Proof. Since the integrand is an odd function of , (24) follows. Further
Then (25) follows, and the proof is ended.

Lemma 6. Let be the Abelian integral given in (3). Then, there exist polynomials of degree , , such that, for ,

Proof. In polar coordinates, and , the integral writes as
Note that
Then by Lemma 5, we have
where denotes a homogeneous polynomial of degree in , and denotes a polynomial of degree in having the form
By the above formula, we get
where is a polynomial of degree in , .
Hence by (30) and Lemma 2,
This ends the proof.

Assume , . By Lemma 4, we have

where

where denotes a polynomial of degree .

Lemma 7. Let be the Abelian integral given in (3). Then for

Proof. We have the following cases.
Case  1 (). In this case we have , or .
By Lemmas 4 and 6, we have
Let . Then
where
which is a polynomial of degree    () in .
Hence,
Case  2 (). In this case we have or .

(i) When , we have

Thus,

where

Then

(ii) When , by Lemmas 4 and 6, we have

Thus,

This ends the proof.

The following lemma can be found in [5].

Lemma 8. Consider the family of functions
defined on , where and are polynomials of degrees and , respectively, and . Then each nontrivial function of the form has at most real zeros, taking into account their multiplicities. Moreover, there exist polynomials and such that the corresponding function has exactly this number of zeros on .

3. Proof of Theorem 1

Proof. By Lemma 7, when we have
As the numerator of the above expression is a polynomial in of degree , the maximum number of zeros of in [0,1) is .
When (), then
As before, the maximum number of zeros of in [0,1) is .
When , we have
Then, using Lemma 8, has at most zeros in .
Finally, for all and we know that . Then the maximum number of zeros of , taking into account their multiplicities, is when and when . The proof is completed.

4. Two Illustration Examples on the Maximum Number

Consider the system

where

and is a small real parameter.

Let

where , . Assuming , we have

where

Let , . Then

Obviously for if and only if . Thus for system (51) the function can have simple zero in .

Now we consider the system

where

and is a small real parameter.

Let

where , .

As before, let , . Then

It follows that for if and only if . Thus for system (57) the function can have simple zero in .

Acknowledgments

The project was supported by the National Natural Science Foundation of China (11271261), a grant from the Ministry of Education of China (20103127110001), and FP7-PEOPLE-2012-IRSES-316338.

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