Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

Laboratory of Analysis and Control of Differential Equations ACED, Department of Mathematics, University of Guelma, P.O. Box 401, Guelma 24000, Algeria Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Buraydah, Saudi Arabia College of Industrial Engineering, King Khalid University, Abha, Saudi Arabia Department of Mathematics, College of Sciences, Juba University, Juba, Sudan Department of Mathematics, Faculty of Science, SVU, Qena 83523, Egypt


Introduction and Statement of the Main Result
Hilbert in 1900 was interested in the maximum number of the limit cycles that a polynomial differential system of a given degree can have. is problem is the well-known 16th Hilbert problem, which together with the Riemann conjecture are the two problems of the famous list of 23 problems of Hilbert which remain open. See for more details [1,2]. A classical way to produce limit cycles is by perturbing a system which has a center, in such a way that limit cycles bifurcate in the perturbed system from some of the periodic orbits of the period annulus of the center of the unperturbed system [3][4][5][6][7].
In [8], the authors improved the result of the maximum number of limit cycles for a class of polynomial differential systems which bifurcate from the period annulus surrounding the origin of the system: where (u − y + a)(u + v + a) � 1 is a conic, a 2 ≠ 1, and |a| ≤ � 2 √ by using the first order of the averaging theory method.
In [9], the authors improved the result of the maximum number of limit cycles of sixth polynomial differential systems which bifurcate from the period annulus surrounding the origin of the system: where u − v 2 − a � 0 is a conic and a ≠ 0, by using the first order of the averaging theory method.
In this work, we perturb the cubic systems equation (1). us, we consider these classes of all polynomial differential systems of degree n, i.e., where , P(u, v) and Q (u, v) are the real polynomials of degree n ≥ 3, and ε is a small parameter. Main result of this study is the following eorem 1.

Theorem 1.
For the sufficiently small |ε| and the polynomials P(u, v) and Q(u, v) having degree 6, suppose that |a| > � 2 √ , system equation (3) has at most 15 limit cycles bifurcating from the period annulus surrounding the origin of cubic polynomial differential system equation (1) using averaging theory of first order (Figures 1 and 2).

The Averaging Theory of First Order
Theorem 2. Consider the following two initial value problems: and where x, y, and x 0 ∈ D is an open domain of R, t ∈ [0, ∞), ε ∈ (0, ε 0 ], R and G are the periodic functions with their period T with its variable t, and f 0 (y) is the average function of R(t, y) with respect to t, i.e., Assume that (i) R, zR/zx, z 2 R/zx 2 , G, and zG/zx are well defined, continuous, and bounded by a constant independent by ε ∈ (0, ε 0 ] in [0, ∞) × D (ii) T is a constant independent of ε (iii) y(t) belongs to D on the time scale 1/ε. en, the following statements hold.
(a) On the time scale 1/ε, we have (b) If p is an equilibrium point of the averaged system equation (5), such that en, system equation (4) has a T-periodic solution ϕ(t, ε) ⟶ p as ε ⟶ 0. (c) If equation (8) is a negative, the corresponding periodic solution ϕ(t, ε) of equation (4) according to (t, x) is asymptotically stable for all ε sufficiently small, and if equation (8) is a positive, then it is unstable.
For more details on the averaging method, see [10,11].
e averaged function of equation (14) is 2 Discrete Dynamics in Nature and Society For n � 6, we get where According to eorem 2, every simple zero of the average function f 0 (r) provides a limit cycle of system equation (3). Now, we prove eorem 1; in the first step, we compute the integral f 0 (r), and in the second step, the number of its simple zeros is studied.

Lemma 1. From the above, we have
where Figure 1: Phase portrait of the cubic system equation (1) with a � 2. Discrete Dynamics in Nature and Society Proof. Assume that z � e iθ and C is the circle |z| � 1; we get whose poles are By applying the residue theorem, for |a| > � 2 √ , we obtain |z 1 | < 1, |z 3 < 1, C encloses the two singular points of the integrand, so erefore, we have 4 Discrete Dynamics in Nature and Society erefore, we get is completes the proof.

us,
is completes the proof.
□ Remark 1. A p,2k+1 � 0. By Lemmas 1 and 2, we have Discrete Dynamics in Nature and Society en, and we also have and Discrete Dynamics in Nature and Society In addition, we have

Discrete Dynamics in Nature and Society
Using equation (15), we get where X � x 4 r 4 + x 2 r 2 + x 0 , Y � y 6 r 6 + y 4 r 4 + y 2 r 2 + y 0 , and with the coefficients x i , y i , and z i the polynomials in the coefficients of a, p i,j , and q i,j . In fact, there are only ten independent parameters between x i , y i , and z i with respect to p ij , q ij , and a. In order to bound the zeros number of numerator of f 0 (r), it is sufficient to bound the zeros number of Since and we have Finally, in order to bound the zeros number of the above expression, we should bound the zeros of the following polynomial: We have (48) erefore, we get H(r) � S 2 d 30 r 30 + · · · + d 2 r 2 + d 0 , where d i are the polynomials in a, x i , y i , and z i . We conclude that f 0 (r) has at most 15 simple zeros. Hence, eorem 1 is proved.

Conclusion
As we know, the limit cycles and a polynomial differential system is the well-known 16th Hilbert problem, which together with the Riemann conjecture are the two problems of the famous list of 23 problems of Hilbert which remain open. In addition, a classical way to produce limit cycles is by perturbing a system which has a center, in such a way that limit cycles bifurcate in the perturbed system from some of the periodic orbits of the period annulus of the center of the unperturbed system; in this work, by using the averaging theory of first order, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we have obtained at most 15th limit cycles for this kind of the problem; in the next study, we will try to extend the same tools but for higher degrees.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.