Poincaré Bifurcations of Two Classes of Polynomial Systems

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 ̇𝑥 = −𝑦𝐶(𝑥,𝑦) , ̇𝑦 = 𝑥𝐶(𝑥,𝑦) with an arbitrary polynomial vector field, where 𝐶(𝑥,𝑦) = 1 − 𝑥 3 or 𝐶(𝑥,𝑦) = 1 − 𝑥 4 .


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 [1][2][3][4]. 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 1 + ( − 1)( − 2)/2 limit cycles if is even and ( − 1)( − 2)/2 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 curvė = − ( , ) + ( , ) , where ( , ) and ( , ) are polynomials of degree in R 2 , the algebraic curve ( , ) satisfies (0, 0) ̸ = 0, and ∈ R is a sufficient small parameter.
It is obvious that, on the region Ω = {( , ) | ( , ) ̸ = 0}, the system (1) is equivalent to the following form: (2) When = 0, system (2) is a Hamilton system with a family of ovals ℎ = {( , ) ∈ R 2 | ( , ) = 2 + 2 = ℎ, ℎ > 0} . (3) 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 Φ(ℎ) controls the number of limit cycles of system (1) that bifurcate from the periodic annulus of the perturbed system (1) with = 0. That is to say, when the Φ(ℎ) does not vanish exactly, the maximum number of the isolated real zeros of Φ(ℎ) 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 2 Abstract and Applied Analysis 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 ( , ) = 1 + 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 = 0 with ( , ) = 1 + + 2 and ( , ) = 1 + + + ( 2 + 2 ), respectively. The authors in [11] studied the number of limit cycles of system (1) with ( , ) = 2 + 2 + + . In [12], the authors studied the number of limit cycles of system (1) with ( , ) = ( + )( + ) and obtain that the system can have at most 3[( − 1)/2] + 2 limit cycles if ̸ = and 2[( − 1)/2] + 1 if = , respectively. In [13], the authors studied the case the curves ( , ) = 0 are three lines, two of them parallel and one perpendicular, and [14,15] studied the case the curves are ( > 3) 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 ( , ) = 1 + 4 and proved that 3[( + 1)/2] − 2 limit cycles can at most bifurcate from the periodic orbits of the unperturbed system. In [18], the authors proved that the system (1) with ( , ) = (1 − ) has at most + − 1 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) ( = 0) with the perturbed polynomials ( , ), ( , ) of given degree , and ( , ) = 1 − 3 or ( , ) = 1 − 4 .
Consider the planar differential systeṁ where ∈ R is a sufficient small parameter. Applying the Abelian integral, we obtain the following two main theorems.  Our primary purpose is to calculate the concrete expression of Φ(ℎ); then we can obtain the number of limit cycles of the perturbed system (5) by determining the isolated real zeros of Abelian integral Φ(ℎ). In Sections 2 and 3, we prove these two theorems with the different methods, respectively.

The proof of Theorem 1
Taking the change of variable = √ ℎ cos , = √ ℎ sin (0 < ℎ < 1), we have where Firstly, we have the following obvious result.
For matrices , and , we have Lemmas 5 and 6, respectively.

(34)
By the above proof procedure, we can obtain rank( −1 ) = − 1 in a similar way. That is rank( ) = . The proof is completed.
it is easy to know that rank( 1 ) = 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 Φ(ℎ) and some concerned lemmas are presented.

Lemma 10. If is even, then
Proof. If is even, then Similarly, (56) also holds. The proof is completed.

(58)
Proof. When > 1, We use the residue theorem to compute the integrals 1 and 2 . Denote = ; thus, cos = ( 2 +1)/2 , d = d / . We have Since 1 = (1 − √ 1 − ℎ)/ √ ℎ is the first-order zero of the equation √ ℎ 2 − 2 + √ ℎ = 0, the residue of 1 at 1 is For the residue at = 0, we have the expansion of 1 in the form the coefficient of −1 is corresponding to the residue of 1 at = 0; therefore, Substituting them into (60), we have We can compute 2 in a similar way and obtain from the previous formula, it is easy to know that, if is odd, Φ 1 ,0 = 0, and if is even, formula (58) is obtained. The proof is completed.
In a similar way, we can prove the following lemma.
To prove Lemma 13, we need the following lemma and a known principle, the Derivation-division algorithm.
By Lemmas 11 and 12, the previous formula becomes ] . (74) According to Lemma 10, we have that, if is even, if is odd, where = 1, 2.
According to (75) and (76), we can obtain that, if is odd, is even, and if is even, is odd, where is odd, is even.
Applying the Poincaré-Pontryagin theorem, the upper bound of number of limit cycles for the system (5) with ( , ) = 1 − 4 is 3 − 2. That is, the maximum number of limit cycles bifurcating from the period orbits of system (5) with = 0 is 3[( + 1)/2] − 2. The proof of Theorem 2 is completed.