On Behavior of the Periodic Orbits of a Hamiltonian System of Bifurcation of Limit Cycles

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.


Introduction
Among the many interesting problems in the qualitative theory of planar polynomial differential systems is the study of their limit cycles (see [1,2]). In particular, concerning Kukles differential system of the form, has a long history, where f(x, y) is a polynomial with real coefficients of degree n. Since it was first introduced in Kukles 1944, many researchers have concentrated on its maximum number of limit cycles and their location. See, for example, [3][4][5].
In [6], Llibre and Mereu studied the maximum number of limit cycles using the averaging theory as follows: where, for every k, the polynomials f k (x), g k (x), and h k (x) have degree n 1 , n 2 , and n 3 , respectively. d k 0 ≠ 0 is a real number and ε is a small parameter.
Also, Makhlouf and Menaceur [7] studied the maximum number for the more generalized polynomial Kukles differential systems in the form e number of limit cycles bifurcating from the center _ x � − y 2p− 1 and _ y � x 2q− 1 , where p, q are positive integers, for the following two kinds of polynomial differential systems, were investigated in the works [8,9], respectively. In the current study, we discuss the maximum number of limit cycles of the following differential system: where p, q, and n are positive integers, the polynomials f(x), g(x), h(x), and l(x) have degree n 1 , n 2 , n 3 , and n 4 , respectively, and ε is a small positive parameter. Clearly, system (5) with ϵ � 0 is an Hamiltonian system with Our main theorems are given as follows.

Theorem 1.
For the sufficiently small |ε|, system (5), using averaging theory of first order, has at most max n 2 2 p, n 4 2 p + q , e limit cycles bifurcating from the periodic orbits of the center are _ x � − y 2p− 1 and _ y � x 2q− 1 , where [.] denotes the integer part function. e proof of eorem 1 is given in Section 3. Theorem 2. Consider system (5) with q � lp, l is a positive integer, and |ε| sufficiently small; let H(n i , l) denote the maximum number of limit cycles of the polynomial differential system (5) bifurcating from the periodic orbits of the center _ x � − y 2p− 1 and _ y � x 2lp− 1 using the averaging theory of first order; then, e proof of eorem 2 is given Section 4.

First-Order Averaging Method
e averaging theory is an interesting method to research the limit cycles. Here, some specific function, associated to the initial system, is stated.
e two initial value problems are as follows: where x, y and , R and G are 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, (iii) y(t) belongs to D on the time scale 1/ε. en, the following statements hold: (i) On the time scale 1/ε, we have (ii) If p is an equilibrium point of the averaged system (10), such that then system (9) has a T-periodic solution (11) is a negative, therefore, the corresponding periodic solution ϕ(t, ε) of equation (9)  For more information about the averaging theory, see [10][11][12].

Proof of Theorem 1
Here, we need to transform system (5) to the canonical from (9). Doing the change of (p, q)-polar coordinates x � r p Csθ and y � r q Snθ (see Appendix) and taking θ as an independent variable, then system (5) can be written as 2 Mathematical Problems in Engineering If we write then system (14) becomes where θ is the independent variable we get from system (16). From where According to the notation introduced in Section 2, we have Mathematical Problems in Engineering and we write where It is known that Hence, f 0 (r) � − r q(2n− p− 1)+p+1 T k�0 k even n 2 b k I k,2(p+n− 1) r pk + k�0 k even we obtain For the simplicity of calculation, let B s � b 2s I 2s,2(p+n− 1) and D s � d 2s I 2s,2(p+n) ; therefore, (24) can be reduced to As we all know, the number of positive roots of f 0 (r) is equal to that of en, to find the real positive roots of N(r), we must find the zeros of a polynomial in the variable ρ � r 2 : So, the degree of M(ρ) is bounded by μ � max [n 2 /2]p, [n 4 /2]p + q , we conclude that f 0 (r) has at most μ positive root r. Hence, eorem 1 is proved.

Proof of Theorem 2
Consider the polynomial differential system (5) with q � lp; from equation (25) we obtain 4 Mathematical Problems in Engineering As we all know, the number of positive roots of f 0 (r) is equal to that of ) . (29) To find the number of positive roots of polynomials G(r), we distinguish 3 cases.  (29) is By Descartes eorem, we can choose the appropriate coefficients B i and D j so that the simple positive roots' number of G(r) is at most [n 4 /2] + l. Hence, (b) of eorem 2 is proved. Example 1. We consider system (5), with p � 1, q � 3, n � 1, and where a 0 � 1, is polynomial has four positive real roots: r 1 � 0.6, r 2 � 0.8, r 3 � 1.1, r 4 � 1.3, and r 5 � 2. According to statement (a) of eorem 2, the system has exactly 5 limit cycles bifurcating from the periodic orbits of the center _ x � − y and _ y � x 5 , using the averaging theory of first order.

Conclusion
In this work, by using averaging theory of the first order, we have proved upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of the Hamiltonian system. In addition, in the next work, a new Mathematical Problems in Engineering condition with a new method will be used to prove our main result in this study.