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

Abstract

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.

1. 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 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, [3–5].

In [6], Llibre and Mereu studied the maximum number of limit cycles using the averaging theory as follows:where, for every , the polynomials , and have degree , and , respectively. is a real number and is a small parameter.

Also, Makhlouf and Menaceur [7] studied the maximum number for the more generalized polynomial Kukles differential systems in the form

The number of limit cycles bifurcating from the center and , where 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 , and are positive integers, the polynomials , and have degree , and , respectively, and is a small positive parameter. Clearly, system (5) with is an Hamiltonian system with

Our main theorems are given as follows.

Theorem 1. For the sufficiently small , system (5), using averaging theory of first order, has at mostThe limit cycles bifurcating from the periodic orbits of the center are and , where denotes the integer part function.

The proof of Theorem 1 is given in Section 3.

Theorem 2. Consider system (5) with , is a positive integer, and sufficiently small; let denote the maximum number of limit cycles of the polynomial differential system (5) bifurcating from the periodic orbits of the center using the averaging theory of first order; then,

The proof of Theorem 2 is given Section 4.

2. First-Order Averaging Method

The averaging theory is an interesting method to research the limit cycles. Here, some specific function, associated to the initial system, is stated.

Theorem 3. The two initial value problems are as follows:where and which is an open domain of , , , and are periodic functions with their period with its variable , and is the average function of with respect to , i.e.,Assume that(i), , , , and are well defined, continuous, and bounded by a constant independent by in .(ii) is a constant independent of .(iii) belongs to on the time scale . Then, the following statements hold:(i)On the time scale , we have(ii)If is an equilibrium point of the averaged system (10), such that then system (9) has a -periodic solution as .(iii)If (11) is a negative, therefore, the corresponding periodic solution of equation (9) according to is asymptotically stable, for all sufficiently small; if (11) is a positive, then it is unstable.

For more information about the averaging theory, see [10–12].

3. Proof of Theorem 1

Here, we need to transform system (5) to the canonical from (9). Doing the change of -polar coordinates and (see Appendix) and taking as an independent variable, then system (5) can be written as

If we writethen system (14) becomeswhere is the independent variable we get from system (16). Fromwhere

According to the notation introduced in Section 2, we haveand we writewhere

It is known that

Hence,we obtain

For the simplicity of calculation, let and ; therefore, (24) can be reduced to

As we all know, the number of positive roots of is equal to that of

Then, to find the real positive roots of , we must find the zeros of a polynomial in the variable :

So, the degree of is bounded by , we conclude that has at most positive root . Hence, Theorem 1 is proved.

4. Proof of Theorem 2

Consider the polynomial differential system (5) with ; from equation (25) we obtain

As we all know, the number of positive roots of is equal to that of

To find the number of positive roots of polynomials , we distinguish 3 cases.where and . Using (A.3) of the Appendix, we obtain

Case 1. For , the number terms in polynomial (29) is . Now, we shall apply the Descartes theorem of the Appendix, we can choose the appropriate coefficients and so that the simple positive roots’ number of is at most . Hence, (a) of Theorem 2 is proved.

Case 2. For , the number terms in polynomial (29) isBy Descartes Theorem, we can choose the appropriate coefficients and so that the simple positive roots’ number of is at most . Hence, (b) of Theorem 2 is proved.

Case 3. For , the number terms in polynomial (29) is ; by Descartes Theorem, we can choose the appropriate coefficients and so that the simple positive roots number of is at most . Hence, (c) of Theorem 2 is proved.

Example 1. We consider system (5), with , andwhereIn this case, and are T-periodic function with period . From equation (28), we obtain

So,

This polynomial has four positive real roots: , and . According to statement (a) of Theorem 2, the system has exactly 5 limit cycles bifurcating from the periodic orbits of the center and , using the averaging theory of first order.

5. 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 condition with a new method will be used to prove our main result in this study.

Appendix

1--Polar Coordinates

Following Lyapunov [13], we introduce the -trigonometric functions and as the solution of the following initial value problem:

Moreover, they satisfy the following properties:(i)The functions and are -periodic with where is the gamma function.(ii)For , we have and .(iii).(iv)Let and be the -trigonometrical functions, for and are both even (see [1]):

2-Descartes Theorem

The purpose of the Descartes theorem is to provide an insight on how many real roots a polynomial may have.

Theorem A.1 (see [14]). Consider the real polynomialwith and real constants, for . When , we say that and have a variation of sign. If the number of variations of signs is , then has at most positive real roots. Moreover, it is always possible to choose the coefficients of in such a way that has exactly positive real roots.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The third author extends appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project, under Grant no. R.G.P-2/53/42.