Abstract
By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.
1. Introduction
As we know that the second part of the 16 Hilbert problem ([1, 2]) wants to find a uniform upper bound for the number of limit cycles of all polynomial differential systems of a given degree, we refer the readers to see [3, 4]. The limit cycles problem and the center problem are fastened on specified classes of systems. For instance, we refer to Kukles systems (see, for example, [5–9]) and Liénard systems given bywhere is a polynomial in the variable of degree . For these systems, in 1977, Lins et al. [10] presented the conjecture that if has degree , then system (1) has at most limit cycles where [·] denotes the integer part function. They prove this conjecture for . The conjecture for has been proved recently by Chengzi and Llibre in [11].
Suppose that polynomials and are in the variable of degrees and , respectively; then, Llibre et al. [12] established the following generalized Liénard polynomial differential system:where is limit cycles.
Llibre and Makhlouf [13] studied the number of limit cycles of the following generalized Liénard polynomial differential system:where is a polynomial of degree , , and are positive integers, and is a small parameter.
They introduced the following theorem.
Theorem 1 (see [13]). Let be the degree of the polynomial , and ; then, the polynomial differential system (3) can have at least limit cycles.
Also, Jianyuan and Shuliang [14] investigated the maximum number of limit cycles of the following polynomial differential system:where is a small parameter, , , and is a polynomial of degree with .
In this manuscript, we discuss the maximum number of limit cycles of the following polynomial differential system:where is a polynomial of degree , is a small parameter, and . Clearly, system (5) with is a Hamiltonian system with Hamiltonian
More precisely, our main results are the following.
Theorem 2. For the sufficiently small , system (5) has at mostlimit cycles bifurcating from the periodic orbits of the center , , by using the averaging theory of first order.
The proof of Theorem 2 is given in Section 3.
Theorem 3. Consider system (5) with , where is a positive integer; then, for sufficiently small, 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 isThe proof of Theorem 3 is given in Section 4. Also, an example is given with its limit cycles (see Figure 1).
2. First-Order Averaging Method
Here, we state the basic outcomes from the averaging theory of first order, which will be used to prove the main outcomes.
Theorem 4. Consider the following two initial-value problems: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 , that is,
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:(a)On the time scale , we have(b)If is an equilibrium point of the averaged system (10) such that then system (9) has a -periodic solution as (c)If (13) is negative, the corresponding periodic solution of equation (9) according to is asymptotically stable for all sufficiently small; if (13) is positive, then it is unstable
For more information about the averaging theory, see, e. g., [15–17].
3. Proof of Theorem 2
The -trigonometrical functions were defined by Liapunov [18]. Let and be the solution of the following initial value problem:
Furthermore, the following properties are satisfied:(a)The functions and are –periodic with where is the gamma function.(b)For , we have and (c)(d)Let and be the -trigonometrical functions, when and are both even (see [19])
We shall need the first-order averaging theory to prove Theorem 2. We write system (5) in -polar coordinates , where and . In this way, system (5) will become written in the standard form for applying the averaging theory. If we write , then system (5) becomes
Treating as the independent variable, we get from system (17) the following:where
By using the notation which is introduced in Section 2, we getand we writewhere
It is known that
Hence,
For the simplicity of calculation, let ; therefore, (24) can be reduced to
The positive zeros number of , as we know, is equal to the following:and then, to find the real positive roots of , we must find the zeros of a polynomial in the variable :
Now, we stretch the polynomial (27) as follows:
So, the degree of is bounded by , and we conclude that has at most positive root . Hence, Theorem 2 is proved.
4. Proof of Theorem 3
Consider the polynomial differential system (5) withq = np. From equation (25), we obtain
The zeros positive number of is equal to the following:
We write (30) as follows:
Let us write (31) as
To find the number of positive roots of polynomials , we distinguish two cases.
Case 1. For , the number term in polynomial (32) isNow, the Descartes theorem of the Appendix will be applied, and the appropriate coefficients can be selected for the simple positive zeros number of as at mostHence, (a) of Theorem 3 is proved.
Case 2. For , the number term in polynomial (32) isby Descartes theorem of the Appendix, and we can choose the appropriate coefficients in order that the simple positive roots number of is at mostHence, (b) of Theorem 3 is proved.
Example 1. We consider system (5), where , andIn this case, and and are T-periodic functions with period . From equation (5), we obtainUpon using (16), we getSo,This polynomial has two positive real roots, and . According to statement (a) of Theorem 3, that system has exactly two limit cycles bifurcating from the periodic orbits of the center , , using the averaging theory of first order. Figure 1 shows the limit cycles for Example 1.
Appendix
We remember Descartes’ theorem regarding the real roots number of a real polynomial (for a proof, see, for example, [20]).
Descartes theorem: consider the following real polynomial:with and real constants for . Since , it can said that and admit a variation of sign. If the signs variations number of is , then admits at most positive real zeros. In aditios, it is always possible to pick out the coefficients of , where admits exactly positive real zero.
Data Availability
No data were used to support this study.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
The fifth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research group program under grant (R.G.P-2/1/42).