The Upper Bound Estimation of Abelian Integral for a Class of Quadratic Reversible System under Small Perturbations

In this article, by using the Riccati equation method, we investigate the maximal values of the isolated zeros of the Abelian integral for a set of quadratic reversible systems (r11) that belong to genus one, while experiencing varying 3rd, 2nd, and 1st-polynomial perturbations. Specifically, we aimed to find the upper bound for the maximal zeros of the system’s limit cycle (a special dynamic behavior in a stable state, characterized by the existence of specific periodic orbits). We know that the Abelian integral is a function of h, so when studying the maximal zeros of the function related to h, we not only consider the highest degree of the relevant function but also take into account the parity of the function and the range of values of h. Then through variable substitution, a smaller upper bound can be obtained: our findings show that the maximal values of the isolated zeros count under varying 3rd, 2nd, and 1st-polynomial perturbations is 12, improving upon previous results where the upper bound was 34 for the 3rd polynomial perturbation and 22 for the 2nd and 1st polynomial perturbations. This study represents an improvement upon previous research.


Introduction and Main Conclusion
Consider the quadratic reversible system ( , ) ( , ) ( , ), ( , ) ( , ) ( , ) where  is a small parameter, and ( , ) x H x y G x y are quadratic polynomials in , xy, while ( , ) u x y , ( , )  v x y are polynomials of degree ( 1, 2,3)   nn in , xy .The system given by ( 1) is a quadratic reversible system with a center that is integrable when the value of  is zero.In this case, the function ( , )  H x y serves as the first integral of the system and is accompanied by an integrating factor ( , )  Gxy, which permits the construction of a continuous periodic domain can be defined as follows is the largest open interval.The primary focus of this article is to determine how many limit cycles can arise from the periodic domain   h  of system (1) for any given decimal value of  .It is well-established that the quantity of isolated zero points in the Abelian integral () Ihas follows which sets a ceiling on the maximum number of limit cycles of system (1) that can exist in any compact region of the periodic orbits [1][2][3][4].
Reference [5] was the first to use the Riccati equation method to study the maximum number of zeros of the Abelian integral of system (1).Reference [6] divided quadratic reversible systems of genus one into 22 classes, specifically (   1)  r . Using the Riccati equation method, the maximal number of zeros that can be obtained in Hamiltonian systems (   1)  r and ( 2) when n is small was explored in reference [7]; Reference [8] studied systems ( 3) r -(

6) r
; References [9][10][11][12] with the integrating factor and nearly all of its trajectories are sixth-degree curves.Reference [9] provides the following theorem on the upper bound of Abelian integral for system ( 11) r .Theorem 1.1 [9] For arbitrary n -degree polynomials ( , ) u x y and ( , ) v x y , the maximum number of the isolated zeros of the Abelian integral ()

Ih
of system (   11)  r linearly depends on n .Specifically, the maximum number is 21 29 and the maximum number is 22  when The central finding of this paper can be summarized in the following theorem.Theorem 1.2.For arbitrary n -degree polynomials ( , ) u x y and ( , ) ,the maximum number of the isolated zeros of the Abelian integral ()

Ih
of system (   11)  r is: the maximum number is 12 when 1, 2,3 n  .are any polynomials, then by equation (2) , we can obtain that Theorem 1.2 states the Abelian integral ()  Ih which has the following expression

Simple Expression for the Abelian Integral I(h)
To simplify the derivation, the integral function presented here will be used throughout the following calculations:  Since the periodic orbit h  possesses axisymmetry about the x -axis, it can be concluded that , ( ) 0 rs Ih  when s is even.Therefore, only the odd values of s need to be taken into account.Then we have where . 9 The formulas (21) and ( 22) in document [9]   .

Riccati Equation
In document [9], the following lemma's conclusion is established for the system ( 11)  r .Lemma 3.1 [9] If 1 n… , then () Ih obeys the given Raccita equation as follows where

( ) ( ) ( ) ( ), B h E h W h A W h D h W h G h
where  .In order to fully demonstrate the validity of Theorem 1.2, we will present the following lemma.

( ) G h B h E h F h B h E h F h
Lemma 4.1 [7] If the functions . Finally, we will utilize the technique of Riccati equations to prove Theorem 1.2.Proof.We can derive the following conclusion by using Lemma 3.  .Therefore, from equation (21), we can obtain ( ) 2 0 1 3 6 2 12.

Conclusion
For system ( 11)  r , this paper applies the Riccati equation method to study the upper bound of the number of isolated zeros of the Abelian integral under arbitrary 3rd, 2nd, and 1st-polynomial perturbations.When studying the maximal zeros of the function related to h , we not only consider the highest degree of the relevant function but also take into account the parity of the function and the range of values of h .Then we obtained the results are as follows: when 1, 2,3 n  , the upper bound is 12.These results are improvements upon the original ones.

4 .
The Upper Bound Estimation of Abelian IntegralIn this paper, better results are obtained by considering only the degrees of the functions()   h, while also taking into account the range of values for h and the parity of the functions.The purpose of this section is to utilize the Riccati equation method in order to demonstrate Theorem 1.2.Let () Ih ♯ represent the count of zeros of the Abelian integral that lie within the interval 

Table 1 .
Comparison between the new results and the original ones.