Abstract
In this work, 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 obtain at most fifteenth limit cycles by using the averaging theory of first order.
1. 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. This 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 [3–7].
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 is a conic, , and 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 is a conic and , by using the first order of the averaging theory method.
In this work, we perturb the cubic systems equation (1). Thus, we consider these classes of all polynomial differential systems of degree , i.e.,where is a conic, , and are the real polynomials of degree , and is a small parameter. Main result of this study is the following Theorem 1.
Theorem 1. For the sufficiently small and the polynomials and having degree 6, suppose that , 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).
2. The Averaging Theory of First Order
Theorem 2. Consider the following two initial value problems:andwhere , and is an open domain of , , , and are the 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.(a)On the time scale , we have(b)If is an equilibrium point of the averaged system equation (5), such that Then, system equation (4) has a -periodic solution as .(c)If equation (8) is a negative, the corresponding periodic solution of equation (4) according to is asymptotically stable for all sufficiently small, and if equation (8) is a positive, then it is unstable.
For more details on the averaging method, see [10, 11].
3. Proof of Theorem 1
For , the cubic system equation (1) has a unique period annulus:
According to Figures 1 and 2, this proof is based on the first order of the averaging theory method, in polar coordinates , where , , and . We take
Equation (3) can be written as follows:whereand
Therefore, we have
The averaged function of equation (14) is
For , we getwhere
According to Theorem 2, every simple zero of the average function provides a limit cycle of system equation (3). Now, we prove Theorem 1; in the first step, we compute the integral , and in the second step, the number of its simple zeros is studied.
Lemma 1. From the above, we havewherewith
Proof. Assume that and is the circle ; we getwhose poles areBy applying the residue theorem, for , we obtain , encloses the two singular points of the integrand, soTherefore, we haveTherefore, we getThis completes the proof.
Lemma 2. Under the previous notations, we have(a)(b), Where (c)(d), .
Proof. Putting , , we get for (a) and (b)Thus,Therefore,andwhere ([12]).
Thus,thenThus,This completes the proof.
Remark 1. .
By Lemmas 1 and 2, we haveThen,and we also haveandIn addition, we haveUsing equation (15), we getwhereandwith the coefficients , and the polynomials in the coefficients of , and .
In fact, there are only ten independent parameters between , , and with respect to , and . In order to bound the zeros number of numerator of , it is sufficient to bound the zeros number ofSinceandwe haveFinally, in order to bound the zeros number of the above expression, we should bound the zeros of the following polynomial:We haveTherefore, we getwhere are the polynomials in , and . We conclude that has at most 15 simple zeros. Hence, Theorem 1 is proved.
4. 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
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project (R.G.P-2/53/42).