LIMIT CYCLES OF PERTURBED GLOBAL ISOCHRO-NOUS CENTER

We apply the averaging method of first order to study the maximum number of limit cycles of the ordinary differential systems of the form


INTRODUCTION AND STATEMENT OF THE MAIN RESULT
At the Paris International Congress of Mathematics in 1900, Hilbert presented twenty-three problems in mathematics. Some problems are still unsolved so far, they were a challenge for all mathematicians of that era. The second part of the well-known Hilbert's 16th problem is to find the maximum number of limit cycles and their position for an ordinary differential planar system of degree n of the form ẋ = ψ(x, y), where n is a positive integer, the dot above the variables represents the first derivative with respect to the variable t, ψ(x, y) and η(x, y) are real polynomials, see for instance [13,14,17]. This problem has so far remained unresolved, for n ≥ 2. Let us denote by H(n) the maximum number of limit cycles of differential system (1) which is usually called a Hilbert number. For example, Chen and Wang in [3], Shi in [22] gave the best result up to now about the lower bounds of H (2), which is H(2) ≥ 4. Li, Liu, and Yang in [15] proved H(3) ≥ 13, Li and Li in [16] proved H(3) ≥ 11, also Han, Wu and Bi in [10] and Han, Zhang and Zang in [12] proved H(3) ≥ 11. For more results about the Hilbert number, see, for example, the paper [9] and the references therein.
There are many papers that studied number of limit cycles using several methods, including the Poincaré-Melnikov integrals, see for instance [11]; the Poincaré return map, see for example [1]; the Abelian integrals, see for example [4]; the averaging method, see [5,6]; the inverse integrating factor, see for instance [8].
In [19], Llibre and Teixeira used the averaging method of first-order for study the existence of limit cycles of the system of second-order differential equations ẍ where f (x, y) and g (x, y) are real cubic polynomials and ε is a small parameter.
In this paper, we apply the averaging method of first-order for study the existence of limit cycles of system of second-order differential equations ẍ where f 1 (x, y) and g 1 (x, y) are real cubic polynomials, f 2 (x, y) and g 2 (x, y) are real quadratic polynomials such that f i (0, 0) = g i (0, 0) = 0, for i = 1, 2, and ε is a small parameter. These polynomials are expressions of the form The system of second order differential equations (2) can be expressed as the following system of first-order differential equations in the usual way: Note that system (3) when ε = 0, is We notice that the system (4) have a global isochronous center at the the origin, i.e. all orbits different from the origin are 2π-periodic, for more detail see [18].
The main result of our work is the following.
Theorem 1. Using the first-order averaging method, system of second-order differential equations (3), where f 1 (x, y) and g 1 (x, y) are real cubic polynomials; f 2 (x, y) and g 2 (x, y) are real quadratic polynomials, has at most four limit cycles bifurcating from the periodic orbits of the linear centeṙ Here ε is a small parameter. Moreover if a 4 = 0 and a 3 = 0, the system (3) has at most two periodic solutions.
The first-order averaging method theory, that we summarize in the sequel, can be found in a more extended way in [2]. Similar works where the perturbations via polynomials play an important role are for instance [7] and [20].

THE FIRST-ORDER AVERAGING METHOD FOR COMPUTING PERIODIC ORBITS
Theorem 2. We consider the following two problemṡ where t ∈ [0, +∞), x and y in some open D of R n and ε ∈ (−ε 0 , ε 0 ) is a small parameter. Moreover, we suppose that the vector functions F (t, x) and R(t, x, ε) are T −periodic in the first variable and we consider the first-order averaging function Then, the statements (I) and (II) satisfied: (II) If y = α ( the equilibrium point ) of (6) is hyperbolic. Then, for ε > 0 sufficiently small, the corresponding periodic solution of system (5) is unique, hyperbolic and of the same stability type as α.
The proof of this theorem can be seen in [21,23].

PROOF OF THEOREM 1
In this work, we consider ρ > 0, s > 0 and writing differential system (3) in the new variables (θ, ρ, s, ω) given by The previous differential system in the new independent variable θ becomes as follows We observe that this system is into the normal form of the averaging method (5), with t = θ and x = (ρ, s, ω) that the all assumptions of the Theorem 2 are satisfied for the system (7). We compute the average functions of the first-order associated with the system (7) f i (ρ, s, ω) = 1 2π If (ρ 0 , s 0 , ω 0 ) is a zero of the system such that det then Theorem 2 assures that the system (3) has a periodic solutions. So, in particular a zero of (8) must be isolated in the set of all zeros of (8). We note that the zeros of (8) having sin(ω) = 0 are non-isolated, so we cannot apply to them the averaging theory for obtaining limit cycles. Moreover, since the differential system (7) is only well defined when s > 0 and ρ > 0, in the rest of this section we will assume that ρ > 0, s > 0 and sin(ω) ̸ = 0, and consequently we can restrict to look for the zeros of        ξ 1 (ρ, s, ω) = 0, The rest of the proof of Theorem 1 is divided into the following cases and subcases.
Case 1 If a 4 ̸ = 0. Then, by solving the first equation ξ 1 = 0 with respect to cos (ω) we get Substituting the expression of cos (ω) in the second equation, ξ 2 = 0, we obtain Substituting the expressions of ρ and cos(ω) in the third equation, ξ 3 = 0, we obtain an equation of the form where A, B, C are constants. From now onwards, we are going to denote by A, B, C this kind of generic constants. As A + Bs 2 + Cs 4 is a quadratic polynomial in s 2 , which can have at most two positive solutions for s, in this subcase we get two values for ρ, s and cos(ω). Observe that each value of (ρ, s) provides at most two solutions for ω. Hence, assuming that in these four solutions the determinant (9) is not zero, by Theorem 2, it follows that in this subcase we have at most four periodic solutions of system (3).

Example 1.
Let us consider the system of differential equations: , respectively. So, this system has two periodic solutions coming from periodic orbits of the center (4) . 4 ), and we need to consider the following subcases. (11) we obtain that We must consider that −B 1 a 4 + b 4 A 2 ̸ = 0, otherwise s = 0 and we cannot get periodic solutions. If we substitute the expressions of cos(ω) and s in ξ 3 = 0, we get an equation of the form ρ(A + Bρ 2 + Cρ 4 ) = 0. Since ρ must be positive, again in this subcase we get two values for ρ, s and cos(ω); and consequently at most four periodic solutions of system (3). Case 2 a 4 = 0. Again we need to consider the following subcases. Since the first factor cannot be zero, as in the previous subcases, we can get at most four periodic solutions of system (3).  Subcase 2.2 a 3 = 0. Looking at equation ξ 1 = 0, we see that a 5 cannot be zero, otherwise ξ 1 = 0 reduces to A 2 = 0, and either we do not have solutions or we have a continuum of solutions. Then, from ξ 1 = 0 we get Substituting the expression of s in the second equation ξ 2 = 0, we must consider the subcases: Substituting the expressions of cos(ω) and s in ξ 3 = 0, we obtain an equation of the form ρ(A + Bρ 2 ) = 0. Hence, as in previous subcases, system (3) has at most two periodic solutions.
Then, the second equation ξ 2 = 0 is of the form A + Bρ 2 = 0, so, there is at most one positive solution for ρ. Hence, by substituting the value of s and ρ in ξ 3 = 0, we obtain an equation of the form A + B cos(ω) = 0. Therefore, we get at most one solution for ρ and cos(ω). In short, putting aside those subcases where we obtain a continuum of solutions, there is at most one solution for s, ρ and cos(ω) and so, there are at most two periodic solutions for system (3).
In the previous case, we gave a particular solution obtained from the subcase 1.1, that is the most general one. Now we are going to see the general solution of subcase 2.2.2, characterized by a 3 = a 4 = b 4 = 0.
From the first equation, ξ 1 = 0, we obtain that s 2 = − 4A 2 3a 5 , and, by substituting this value in the second equation, ξ 2 = 0, it follows that ρ 2 = − 4B 1 +b 5 s 2 This three equalities provides all the possible solutions of this subcase.

Example 2.
In the previous corollary, if we take A 1 = B 2 , we have that ω = π 2 or ω = 3π 2 , and if we take the values a 5 = −1, b 3 = −1, b 5 = 1, A 2 = 9, B 1 = 1, we obtain the following two solutions: Observe that this subcase does not depend on the constants not listed in this example, so, we can choose any value for them.

CONCLUSIONS
This paper shows that the application of averaging method of first-order it is useful for study the existence of limit cycles of perturbated system of second-order differential equations.