BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF PERTURBED KUKLES DIFFERENTIAL SYSTEMS

. We consider an integrable non-Hamiltonian system, which belongs to the quadratic Kukles diﬀerential systems. It has a center surrounded by a bounded period annulus. We study polynomial perturbations of such a Kukles system inside the Kukles family. We apply averaging theory to study the limit cycles that bifurcate from the period annulus and from the center of the unperturbed system. First, we show that the periodic orbits of the period an- nulus can be parametrized explicitly through the Lambert function. Later, we prove that at most one limit cycle bifurcates from the period annulus, under quadratic perturbations. Moreover, we give conditions for the non-existence, existence, and stability of the bifurcated limit cycles. Finally, by using aver- aging theory of seventh order, we prove that there are cubic systems, close to the unperturbed system, with 1 and 2 small limit cycles.

1. Introduction. In this paper we will focus on the limit cycles that can appear under polynomial perturbations of the following real quadratic systeṁ x = −y, y = x + x 2 + y 2 . (1) This system has two finite critical points: (0, 0) and (−1, 0). It is easy to see that the first one is a weak focus and the second is a saddle. Since the system is invariant under the transformation (x, y, t) → (x, −y, −t), (0, 0) is a center and the period annulus P surrounding the origin is bounded by a homoclinic loop Γ that joins the stable and the unstable manifolds of (−1, 0). Moreover, a simple computation shows that the analytic function is a first integral of (1) with integrating factor 2e 2x . Thus, (1) is an integrable non-Hamiltonian reversible system, which belongs to the family of Kukles systems. We recall that a Kukles system is a planar differential system of the forṁ where Q(x, y) is a real polynomial of degree at least two and without y as a divisor. The quadratic Kukles system (1) appears in [11] as a particular integrable system, which has a transversal to infinity invariant quadratic algebraic curve.
In this paper we will consider polynomial perturbations of system (1) inside the Kukles family, that is, systems of the forṁ x = −y, y = x + x 2 + y 2 + εQ(x, y, ε), where Q is a polynomial in the variables x and y, whose coefficients are analytic in the parameter ε which we will assume real and small enough. Thus (2) with ε = 0 is the original system (1). We ask about the maximum number of small and medium amplitude limit cycles of system (2) (with ε = 0), that is, the maximum number of limit cycles of (2) that bifurcate from the origin and from the period annulus P of (1), respectively.
The motivation in the Kukles family is because it is one of most important families related to the Hilbert 16th problem [4]. Moreover, some classes of Kukles systems appear in applied sciences: For example, they are used as predator-prey models [5]. Bifurcation of limit cycles in Kukles systems have been tackled by several authors and by using different approaches. See for example [1,6,12,13,15].
The Kukles systems studied in [1,6,13] are of arbitrary degree. However, they come essentially from polynomial perturbations of the harmonic oscillator. The paper [15] considers cubic perturbed Kukles systems whose unperturbed system is also cubic. The aim of this work is to apply averaging methods to find lower and upper estimations for the number of small and medium limit cycles of (2).
Concerning the medium limit cycles of (2) our main result is the following.
Theorem 1.1. The perturbed quadratic Kukles systeṁ where a i ∈ R, for i = 0, 1, . . . , 5, has at most one medium limit cycle γ ε . Moreover, a) if a 4 > a 2 > 0, then γ ε , if it exists, is asymptotically stable; b) if a 4 < a 2 < 0, then γ ε , if it exists, is asymptotically unstable; c) if 0 < a 2 and a 4 < a 2 , then the system does not have medium limit cycles; d) if a 2 < 0 and a 2 < a 4 , then the system does not have medium limit cycles.
Regarding the research of small limit cycles of system (2), we will consider special cubic perturbations of sixth order in ε of (1). Our result is the following. Theorem 1.2. Consider the cubic perturbed Kukles systeṁ where Q j (x, y), with j = 1, . . . , 6, is a polynomial of degree three vanishing at (0, 0). The maximum number of small limit cycles of this system, by using the averaging theory of order a) one and two, is 0; b) three and four, is 1; c) five, six, and seven, is 2. Moreover, we characterize the stability of the bifurcated limit cycles in terms of the coefficients of the perturbation and we show that there are suitable choices of Q 1 (x, y), . . . , Q 6 (x, y) such that the perturbed system has either one or two hyperbolic small limit cycles.
As far as we know, this is the first work where the theory of averaging up to seventh order is used to get limit cycles in the problem under study.
The organization of the paper is as follows. Section 2 is devoted to recall the main tools of the averaging theory that we will use along the paper. The transformation of system (2), in order to apply the averaging methods, as well as the parametrization of the solutions of the unperturbed system are developed in Section 3. Moreover, in such a section we will give some elementary but essential results for proving our first result. In Section 4 we will provide the proof of Theorem 1.1. Finally, in Section 5 we will prove Theorem 1.2.
To analyze the periodic orbits of (3) that bifurcate from the unperturbed differential equationẋ we will consider two different cases: F 0 (t, x) ≡ 0 and F 0 (t, x) ≡ 0. First, we assume that F 0 (t, x) ≡ 0 and that the unperturbed differential equation (4) has the following property: There exists a subset J ⊂ I such that for each p ∈ J its solution ϕ(t, p) with ϕ(0, p) = p is T -periodic in the variable t.
Let Y (t, p) be a solution of (5) such that Y (0, p) = 1. Then, the first order averaged function is defined as The following result is well-known from the averaging theory.
There are several versions of this result, in particular for higher dimensions. See for instance [7,8,9]. Here, we have stated a simplified version according to the purposes of this work. This version will be used in the proof of Theorem 1.1.
To prove Theorem 1.2 we will use averaging theory of higher order under the condition F 0 (t, x) ≡ 0. In such a case, the solution ϕ(t, p) of (4) with ϕ(0, p) = p is periodic, since ϕ(t, p) = p for all p ∈ J = I. The following definition and result come from [3,7,8] and they are adapted to the one-dimensional differential equation (3).
For i = 2, 3, . . . , k, it is defined the i−th order averaged function as

SALOMÓN REBOLLO-PERDOMO AND CLAUDIO VIDAL
where y i : R × J → R, for i = 1, . . . , k − 1, are defined recurrently by the integral equation: where ∂ L F (s, u) denotes the derivative of order L of F with respect to the variable u, S l is the set of all l-tuples of non-negative integers (b 1 , b 2 , . . . , b l ) that satisfy b 1 + 2b 2 + · · · + lb l = l, and L = b 1 + b 2 + · · · + b l .
3. Preliminary results. The period annulus P of (1) is contained in the region bounded by the unit circle with center at the origin. Indeed, we have then the intersection points of Γ with the x-axis are (−1, 0) and (x 0 , 0), where x 0 is positive and satisfies the equation As e −(2+2x) is a strictly decreasing function we have e −(2+2x) ≤ 1 for −1 ≤ x ≤ x 0 . Thus, x 2 + y 2 ≤ 1. This implies that Γ is contained in the unitary disc D 1 with center at the origin. Hence P D 1 . By using polar coordinates x = r cos θ, y = r sin θ, the system (2) becomeṡ r = r 2 sin θ + ε sin θ Q(r cos θ, r sin θ, ε), θ = 1 + r cos θ + ε cos θ Q(r cos θ, r sin θ, ε)/r.
Moreover we know that Then, by changing the independent variable t of system (8) by the variable θ, we obtain the equivalent one-dimensional differential equation where prime denotes differentiation with respect to the variable θ, and Q 0 (θ, r) := Q 0 (r cos θ, r sin θ). Thus, each medium limit cycle of (2) corresponds to an isolated periodic orbit of (9) that bifurcates from the unperturbed differential equation r = F 0 (θ, r). The first integral for (9) with ε = 0 is H(θ, r) = r 2 e 2r cos θ .
We now multiply (10) by cos θ to get where X = r(θ; r 0 ) cos θ. Moreover, as we are interested in the orbits of the unperturbed system living in P D 1 , then we can assume −1 < X < 1.
A simple computation shows that the real function se s is strictly increasing for s > −1, therefore for s > −1 there exists the inverse: if se s = u with s > −1, then s = W (u), where W (·) is the so-called Lambert W -function (see [2], [10] or [14] for details and properties of the Lambert W -function). Hence, applying this property to equation (11) we get r(θ; r 0 ) cos θ = W (r 0 e r0 cos θ), whence r(θ; r 0 ) = W (r 0 e r0 cos θ) cos θ .
A straightforward computation gives us that r 0 ∈ [0, W (1/e)) and that r(θ; r 0 ) is an analytic function for all θ ∈ R. Moreover, it is an even and 2π-periodic function. This proves that equation (9) with ε = 0 has a submanifold foliated by 2π-periodic orbits. To study the limit cycles bifurcating from the periodic orbits of this submanifold we will apply methods of averaging theory.

SALOMÓN REBOLLO-PERDOMO AND CLAUDIO VIDAL
which, by means of the change of variable z = r 0 e r0 cos θ, can be written as The first term in the right-hand side of the previous equation is the derivative of W (z) and the second one can be integrated by using the change of variable v = W (z) + 1. Thus, we have Finally, from the condition y(0; r 0 ) = 1 we obtain the particular solution Hence, the first order averaged function for our system (8) is where Y −1 (θ; r 0 ) = W (r 0 e r0 ) (W (r 0 e r0 cos θ) + 1) cos θ (W (r 0 e r0 ) + 1) W (r 0 e r0 cos θ) and The proof of Theorem 1.1 will be based in the research of the number of zeros of the function F 1 (r 0 ). For such investigation it will be useful the following two technical lemmas.
Lemma 3.1. The function g(x) := x − W (x) defined in (−1/e, 1/e) is non negative and has a global minimum at the origin. In particular, for Proof. It is clear that g(0) = 0. Easy computations show that g (0) = 0 and g (0) = 2. Thus, g has a minimum at the origin. Moreover, it is the unique critical point. Therefore, g(x) ≥ 0 in (−1/e, 1/e).
The right-hand side of the previous inequality can be written as whose denominator is positive, and W (x)W (−x) is negative. Moreover, from Lemma 3.1, we have that W (x) + W (−x) is negative, so by using this property and (15) we have that π/2 0 [S(θ; r 0 ) + S(π − θ; r 0 )] dθ < 0.

4.
Proof of Theorem 1.1. Recall that Theorem 1.1 is related to the perturbed differential systeṁ with a i ∈ R, for i = 0, 1, . . . , 5. In this section we will give the proof of Theorem 1.1, whose assertion is that this quadratic system has at most one medium amplitude limit cycle.
Proof of Theorem 1.1. By using polar coordinates, system (16) can be transformed in the form (8), with Q(x, y, ε) = Q 0 (x, y) = a 0 + a 1 x + a 2 y + a 3 x 2 + a 4 xy + a 5 y 2 . Moreover, the transformed system can be written in the form (9) by changing the independent variable t by the variable θ. Since r(θ; r 0 ), given in (12), is the periodic solution of the unperturbed system, the first order averaged function (13) becomes where with x = r 0 e r0 cos θ.
Hence, the first order averaged function reduces to Furthermore, it is not difficult to see that the derivative of a 2 I 2 (r 0 ) + a 4 I 4 (r 0 ) is which has a defined sign if a 4 = a 2 , according to Lemma 3.2, or it vanishes identically if a 4 = a 2 . Hence, in the case a 4 = a 2 , the function a 2 I 2 (r 0 ) + a 4 I 4 (r 0 ) is strictly monotone, which implies that F 1 (r 0 ) has at most one zero in the interval (0, W (1/e)) because G(r 0 ) > 0 on such interval. For the case a 4 = a 2 , we have to consider two possibilities: a 2 = 0 and a 2 = 0. In the former, the function a 2 (I 2 (r 0 ) + I 4 (r 0 )) is a non-zero constant, because a 2 (I 2 (0) + I 4 (0)) = a 2 π = 0, whence F 1 (r 0 ) does not have any zero in (0, W (1/e)). In the later, the function a 2 (I 2 (r 0 ) + I 4 (r 0 )) vanishes identically, which implies that F 1 (r 0 ) vanishes identically on (0, W (1/e)). In addition, if a 2 = a 4 = 0, then the perturbed differential system is reversible under the transformation (x, y, t) → (x, −y, −t). Thus, in such a case and for ε small enough, the perturbed system has no limit cycles bifurcating from P.
Therefore, either F 1 (r 0 ) ≡ 0 and has at most one zero in (0, W (1/e)) or the perturbed system has no medium limit cycles. Thus, by applying Theorem 2.1 we complete the proof of the main part of the theorem.
Finally, we will prove the four items of the theorem. If a 4 > a 2 > 0, then from (19) it follows that the derivative of a 2 I 2 (r 0 ) + a 4 I 4 (r 0 ) is negative, whence the function a 2 I 2 (r 0 ) + a 4 I 4 (r 0 ) is strictly monotone decreasing on (0, W (1/e)). Thus, if F 1 (r 0 ) has a zero r * in (0, W (1/e)), then F 1 (r * ) < 0. Hence, Theorem 2.3 implies that the medium limit cycle γ ε bifurcating from r(θ; r * ) is asymptotically stable. This proves the first item. The second item follows from the same idea.
If 0 < a 2 and a 4 < a 2 , then from (19) it follows that the derivative of a 2 I 2 (r 0 ) + a 4 I 4 (r 0 ) is positive. Moreover, since a 2 I 2 (0)+a 4 I 4 (0) = πa 2 > 0, a 2 I 2 (r 0 )+a 4 I 4 (r 0 ) is positive. Hence, F 1 (r 0 ) has no zeros in (0, W (1/e)). Thus the system does not have medium limit cycles. This proves the third item. The proof of the fourth one is analogous.
We finish this section by providing an example and a remark of our Theorem 1.1.
First, we will apply the averaging theory of first order. By Theorem 2.2 we must study the simple positive zeros of the first order averaged function F 1 (r 0 ), which in our case has the form Thus, if a 2 = 0, then F 1 (r 0 ) has not positive zeros, which implies that (23) has no periodic orbits bifurcating from the unperturbed equation.

SALOMÓN REBOLLO-PERDOMO AND CLAUDIO VIDAL
and by computing the third averaged function according (6), we arrive to Then, under the condition c 2 (2a 4 − a 7 − 3a 9 ) > 0, the function F 3 (r) has a simple positive zero at because F 3 (r * 1 ) = −2c 2 π. Hence, Theorem 2.2 ensures that system (22) has a medium limit cycle, which is stable if c 2 > 0 and it is unstable if c 2 < 0 according to Theorem 2.3. This implies that the Kukles system (21), for ε small enough, has a small limit cycle.