This is a preprint of: “Limit cycles bifurcating from a family of reversible quadratic centers via averaging theory”, Jaume Llibre, Arefeh Nabavi, Marzieh Mousavi, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 30(4), 2050051:1–8, 2020. DOI: [10.1142/S0218127420500510] LIMIT CYCLES BIFURCATING FROM A FAMILY OF REVERSIBLE QUADRATIC CENTERS VIA AVERAGING THEORY JAUME LLIBRE1 , AREFEH NABAVI2 AND MARZIEH MOUSAVI2 Abstract. Consider the class of reversible quadratic systems ẋ = y, ẏ = −x + x2 + y 2 − r2 , with r > 0. These quadratic √ polynomial differential systems have a center at the point ((1 − 1 + 4r2 )/2, 0) and the circle x2 + y 2 = r2 is one of the periodic orbits surrounding this center. These systems can be written into the form ẋ = y + (4 + A)x2 − Ay 2 , ẏ = −x, with A ∈ (−2, 0). We prove for all A ∈ R that the averaging theory up to seven order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles (respectively one limit cycle) bifurcating from the periodic orbits surrounding the center (0, 0) of that system. Up to now this result was only know for A = −2 (see [22, 23]). 1. Introduction and statement of the main results One of the important problems in the qualitative theory of differ- ential equations is the study of their limit cycles. The second part of the well-known Hilbert’s 16th problem [26] asks about the maximal number and the possible relative positions of limit cycles in the planar polynomial differential systems of degree n ≥ 2. This problem is still open, even for the case n = 2. An easier problem than the Hilbert’s 16th problem is the study of the number of limit cycles which can bifurcate from the periodic orbits surrounding a center of a polynomial differential system. Many authors these last years have studied this last problem restricted to the centers of the quadratic polynomial differential systems, see for instance the 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. limit cycles, quadratic reversible centers, averaging theory. 1 2 J. LLIBRE, A. NABAVI AND M. MOUSAVI a Figure 1. The phase portrait in the Poincaré disc of system (1). book of Christopher and Li [6] and the hundreds of references quoted therein. The tools for studying the limit cycles bifurcating from the peri- odic orbits surrounding a center are the Poincaré return map (see for instance [3, 20]), the Poincaré-Melnikov integrals (see for exam- ple [13, 14]), the Abelian integrals (see [1, 6, 28]), the averaging theory (see for instance [2, 7]), and the inverse integrating factor (see [11]). While the first three methods only provide the number of bifurcated limit cycles, the averaging method and method which use the inverse integrating factor can also give the shape of bifurcated limit cycles. For a general overview and more details about the mentioned methods see reference [12]. According with Iliev [17] and using complex notation the centers of the quadratic polynomial differential systems are classified into the following five families: ż = −iz − z 2 + 2|z|2 + (b̄ + ic̄)z̄ 2 , Hamiltonian (QH 3 ), ż = −iz + az 2 + 2|z|2 + bz̄ 2 , Reversible (QR 3 ), ż = −iz + 4z 2 + 2|z|2 + (b + ic)z̄ 2 , |b + ic| = 2, codimension four (Q4 ), ż = −iz + z 2 + (b + ic)z̄ 2 , Generalized Lotka-Voltera (QLV 3 ), ż = −iz + z̄ 2 , Hamiltonian triangle, where a, b, c are arbitrary real constants. The major part of papers studying the bifurcation of limit cycles from the periodic orbits surrounding a center consider centers of poly- nomial Hamiltonian systems. Thus this problem for the centers of the quadratic polynomial Hamiltonian systems has been solved completely, see [5, 10, 15, 16, 19, 21]. CYCLICITY OF QUADRATIC REVARSIBLE CENTERS 3 Here for r > 0 we consider the following family of quadratic polyno- mial differential systems (1) ẋ = y, ẏ = −x + x2 + y 2 − r 2 , 2 2 2 which have the√ circle x + y = r as a periodic orbit surrounding the center (1 − 1 + 4r 2 )/2, 0 . The phase portraits in the Poincaré disc of these quadratic polynomial differential systems have been studied in [25], see Figure 1. It is easy to check that system (1) has the non- algebraic first integral H(x, y) = e−2x (x2 + y 2 − r 2 ). Taking z = x + iy we obtain that any quadratic reversible center QR 3 can be written in the following real form (2) ẋ = y + (a + b + 2)x2 − (a + b − 2)y 2, ẏ = −x [1 − 2(a − b)y] . There are some papers studying the limit cycles which bifurcate from the periodic orbits surrounding the centers of (2) with a 6= b, see for instance [4, 18]. The study of the case a = b is more difficult because the first integral contains an exponential function. The following lemma shows that system (1) is a reversible system (2) with a = b. Lemma 1. Systems (1) are inside the class of QR 3 because they can be written as (3) ẋ = y + (4 + A)x2 − Ay 2, ẏ = −x, A where a = b = 1 + and A ∈ (−2, 0). 2 This lemma is proved in section 3. As far as we know the first work on the bifurcation of limit cycles from the periodic orbits surrounding a center of a system (3) under quadratic perturbations was done by Li [22] for A = −2 proving that at most 2 limit cycles bifurcate from that center. Note that for A = −2 system (3) with the change of variables (x, y, t) → (x/2, y/2, −t) becomes the system ẋ = −y − x2 − y 2 , ẏ = x, studied by Li. Later on Liu [23] provided a shorter proof of the result of Li. In this paper we study this problem for all values of A. From statement (ii) of Theorem 1 in Iliev [17] the essential quadratic perturbations for finding the upper bound on the number of limit cycles which can bifurcate from the periodic orbits surrounding the center 4 J. LLIBRE, A. NABAVI AND M. MOUSAVI (0, 0) of the reversible quadratic polynomial differential systems (3) under general quadratic perturbations are (4) ż = (λ1 (ε) − i)z + (a + iλ3 (ε))z 2 + 2|z|2 + (b + iλ5 (ε))z̄ 2 , where we can take the λk (ε) for k = 1, 3, 5 as real analytical functions in ε sufficiently small with λk (0) = 0. That is λk (ε) = Σ∞ j j=1 λkj ε . In fact according with Iliev it would be sufficient to take λk (ε) = λk ε, but since we shall use averaging theory up to order seven for studying the limit cycles of system (4) it is convenient to take λk (ε) = Σ∞ j j=1 λkj ε . In short, for studying the maximum number of limit cycles which can bifurcate from the periodic orbits surrounding the center (0, 0) of system (3) perturbed by a general quadratic polynomial, it is enough to study for ε 6= 0 sufficiently small the limit cycles of system (4), or equivalently the limit cycles of system (4) written in the real variables (x, y) where z = x + iy, i.e. X∞ 2 ẋ = y + (4 + A)x − Ay + 2 εi (λ1i x + 2(λ5i − λ3i )xy) , i=1 (5) ∞ X  ẏ = −x + εi λ1i y + (λ3i + λ5i )(x2 − y 2 ) . i=1 Our main result is the following one. Theorem 2. For |ε| = 6 0 sufficiently small and for all A ∈ R the maximum number of limit cycles of system (5) obtained using averaging theory up to seven order is two. Theorem 2 is proved in section 3. As far as we know in this paper it is the first time that it is used in an application the average functions until order seven. We note, as it is shown in the proof of Theorem 2, that the two limit cycles can be obtained already from the fifth average function when A ∈ R \ {−4, 6}, and only from the seventh average function when A ∈ {−4, 6}. 2. Preliminary results In this section we recall the results on the averaging theory that we need for proving Theorem 2. We consider a differential system of the form k X (6) ẋ = εi Fi (t, x) + εk+1 G(t, x, ε), i=1 CYCLICITY OF QUADRATIC REVARSIBLE CENTERS 5 where Fi : R × D → Rn for i = 1, ..., k, and G : R × D × (−ε0 , ε0 ) → Rn are continuous functions and T-periodic in the first variable, D is an open subset of Rn and ε a small parameter. From [27] (see also (1.103), (1.104) and section 1.3.4 of [24]) we define the functions yi (t, z) and the average functions fi (z) for i = 1, ..., 7 associated to system (6). In these functions we denote by ∂ k Fm (s, z) the k-th partial derivative of the function Fm (s, z) with respect to the variable z. Thus we have Z t y1 (t, z) = F1 (s, z)ds, 0 Z t y2 (t, z) = (2F2 (s, z) + 2∂F1 (s, z)y1 (s, z)) ds, 0 Z t y3 (t, z) = 6F3 (s, z) + 6∂F2 (s, z)y1 (s, z) + 3∂ 2 F1 (s, z)y1 (s, z)2 0 +3∂F1 (s, z)y2 (s, z)) ds, Z t y4 (t, z) = 24F4 (s, z) + 24∂F3 (s, z)y1 (s, z) + 12∂ 2 F2 (s, z)y1 (s, z)2 0 + 12∂F2 (s, z)y2 (s, z) + 12∂ 2 F1 (s, z)y1 (s, z)y2 (s, z)  +4∂ 3 F1 (s, z)y1 (s, z)3 + 4∂F1 (s, z)y3 (s, z) ds, Z t y5 (t, z) = 120F5 (s, z) + 120∂F4 (s, z)y1 (s, z) + 60∂ 2 F3 (s, z)y1 (s, z)2 0 + 60∂F3 (s, z)y2 (s, z) + 60∂ 2 F2 (s, z)y1 (s, z)y2 (s, z)  +20∂ 3 F2 (s, z)y1 (s, z)3 + 20∂F2 (s, z)y3 (s, z) + 20∂ 2 F1 (s, z)y1 (s, z)y3 (s, z) + 15∂ 2 F1 (s, z)y2 (s, z)2 + 30F1 (s, z)y1 (s, z)2 y2 (s, z) + 5∂ 4 F1 (s, z)y1 (s, z)4 +5∂F1 (s, z)y4 (s, z)) ds. Z t y6 (t, z) = (720F6 (s, z) + 720∂F5 (s, z)y1 (s, z) + 360∂F4 (s, z)y2 (s, z) 0 + 360∂ 2 F4 (s, z)y1 (s, z)2 + 120∂F3 (s, z)y3 (s, z) + 360∂ 2 F3 (s, z)y1 (s, z)y2 (s, z) + 120∂ 3 F3 (s, z)y1 (s, z)3 + 30∂F2 (s, z)y4 (s, z) + 120∂ 2 F2 (s, z)y1 (s, z)y3 (s, z) + 30∂ 4 F2 (s, z)y1 (s, z)4 + 90∂ 2 F2 (s, z)y2 (s, z)2 + 180∂ 3 F2 y1 (s, z)2 y2 (s, z) + 6∂F1 (s, z)y5 (s, z) 6 J. LLIBRE, A. NABAVI AND M. MOUSAVI + 30∂ 2 F1 (s, z)y1 (s, z)y4 (s, z) + 60∂ 2 F1 (s, z)y2 (s, z)y3 (s, z) + 60∂ 3 F1 (s, z)y1 (s, z)2 y3 (s, z) + 60∂ 4 F1 (s, z)y1 (s, z)3 y2 (s, z)  +90∂ 3 F1 (s, z)y1 (s, z)y2 (s, z)2 + 6∂ 5 F1 (s, z)y1 (s, z)5 ds. Moreover we have Z 1 T f1 (z) = F1 (t, z)dt, T Z0 1 T f2 (z) = (F2 (t, z) + ∂F1 (t, z)y1 (t, z)) dt, T Z0  1 T 1 f3 (z) = F3 (t, z) + ∂F2 (t, z)y1 (t, z) + ∂ 2 F1 (t, z)y1 (t, z)2 + T 0  2 1 + ∂F1 (t, z)y2 (t, z) dt, 2Z  1 T 1 f4 (z) = F4 (t, z) + ∂F3 (t, z)y1 (t, z) + ∂ 2 F2 (t, z)y1 (t, z)2 T 0 2 1 1 2 + ∂F2 (t, z)y2 (t, z) + ∂ F1 (t, z)y1 (t, z)y2 (t, z) 2 2  1 3 3 1 + ∂ F1 (t, z)y1 (t, z) + ∂F1 (t, z)y3 (t, z) dt, 6Z  6 1 T 1 f5 (z) = F5 (t, z) + ∂F4 (t, z)y1 (t, z) + ∂ 2 F3 (t, z)y1 (t, z)2 T 0 2 1 1 2 + ∂F3 (t, z)y2 (t, z) + ∂ F2 (t, z)y1 (t, z)y2 (t, z) 2 2 1 3 3 1 + ∂ F2 (t, z)y1 (t, z) + ∂F2 (t, z)y3 (t, z) 6 6 1 2 1 + ∂ F1 (t, z)y1 (t, z)y3 (t, z) + ∂ 2 F1 (t, z)y2 (t, z)2 6 8 1 3 1 + ∂ F1 (t, z)y1 (t, z) y2 (t, z) + ∂ 4 F1 (t, z)y1 (t, z)4 2 4  24 1 + ∂F1 (t, z)y4 (t, z) ds. 24 Z  1 T 1 f6 (z) = F6 (t, z) + ∂F5 (t, z)y1 (t, z) + ∂F4 (t, z)y2 (t, z) T 0 2 1 2 1 + ∂ F4 (t, z)y1 (t, z)2 + ∂F3 (t, z)y3 (t, z) 2 6 1 1 + ∂ 2 F3 (t, z)y1 (t, z)y2 (t, z) + ∂ 3 F3 (t, z)y1 (t, z)3 2 6 1 1 2 + ∂F2 (t, z)y4 (t, z) + ∂ F2 (t, z)y1 (t, z)y3 (t, z) 24 6 1 3 1 + ∂ F2 (t, z)y1 (t, z) y2 (t, z) + ∂ 2 F2 (t, z)y2 (t, z)2 2 4 8 CYCLICITY OF QUADRATIC REVARSIBLE CENTERS 7 1 4 1 + ∂ F2 (t, z)y1 (t, z)4 + ∂F1 (t, z)y5 (t, z) 24 120 1 1 + ∂ 2 F1 (t, z)y1 (t, z)y4 (t, z) + ∂ 2 F1 (t, z)y2 (t, z)y3 (t, z) 24 12 1 3 1 + ∂ F1 (t, z)y1 (t, z) y3 (t, z) + ∂ 4 F1 (t, z)y1 (t, z)3 y2 (t, z) 2 12 12  1 3 2 1 5 5 + ∂ F1 (t, z)y1 (t, z)y2 (t, z) + ∂ F1 (t, z)y1 (t, z) dt. 8Z 120 1 T f7 (z) = (F7 (s, z) + ∂F6 (s, z)y1 (s, z) T 0 1 1 + ∂F5 (s, z)y2 (s, z) + ∂ 2 F5 (s, z)y1 (s, z)2 2 2 1 1 + ∂F4 (s, z)y3 (s, z) + ∂ 2 F4 (s, z)y1 (s, z)y2 (s, z) 6 2 1 3 3 1 + ∂ F4 (s, z)y1 (s, z) + ∂F3 (s, z)y4 (s, z) 6 24 1 2 1 + ∂ F3 (s, z)y1 (s, z)y3 (s, z) + ∂ 3 F3 (s, z)y1 (s, z)2 y2 (s, z) 6 4 1 1 + ∂ 4 F3 (s, z)y1 (s, z)4 + ∂ 2 F3 (s, z)y2 (s, z)2 24 8 1 1 2 + ∂F2 (s, z)y5 (s, z) + ∂ F2 (s, z)y1 (s, z)y4 (s, z) 120 24 1 3 1 + ∂ F2 (s, z)y1 (s, z)y2 (s, z)2 + ∂ 3 F2 (s, z)y1 (s, z)2 y3 (s, z) 8 12 1 4 3 1 5 + ∂ F2 (s, z)y1 (s, z) y2 (s, z) + ∂ F2 (s, z)y1 (s, z)5 12 120 1 1 + ∂ 2 F2 (s, z)y2 (s, z)y3 (s, z) + ∂F1 (s, z)y6 (s, z) 12 720 1 2 1 + ∂ F1 (s, z)y1 (s, z)y5 (s, z) + ∂ 3 F1 (s, z)y1 (s, z)y2 (s, z)y3 (s, z) 120 12 1 1 + ∂ 3 F1 (s, z)y1 (s, z)2 y4 (s, z) + ∂ 4 F1 (s, z)y1 (s, z)2 y2 (s, z)2 48 16 1 4 1 + ∂ F1 (s, z)y1 (s, z) y3 (s, z) + ∂ 5 F1 (s, z)y1 (s, z)4 y2 (s, z) 3 36 48 1 6 6 1 2 + ∂ F1 (s, z)y1 (s, z) + ∂ F1 (s, z)y2 (s, z)y4 (s, z) 120 48  1 3 3 1 2 2 + ∂ F1 (s, z)y2 (s, z) + ∂ F1 (s, z)y3 (s, z) ds. 48 72 If we denote the solution of the differential system (6) with x(t, z, ε) such that x(0, z, ε) = z, then we have 7 X x(T, z, ε) − z = εfi (z) + . . . i=1 Therefore by using the Implicit Function Theorem we conclude that if f1 (z) = f2 (z) = ... = fk−1 (z) ≡ 0 and fk (z) 6≡ 0, then the simple zeros 8 J. LLIBRE, A. NABAVI AND M. MOUSAVI of fk (z) provide limit cycles of the differential system (6). For more details see Theorem A of [27]. 3. Proof of Theorem 2 Before proving Theorem 2 we prove Lemma 1. √ Proof of Lemma 1. First we translate the center ((1 − 1 + 4r 2)/2, 0) of the differential system (1) to the origin of coordinates and we obtain the differential system √ (7) ẋ = y, ẏ = − 1 + 4r 2 x + x2 + y 2 . p Doing the linear change (x, y) = (v/ 4 (1 + 4r 2 ), u) we write the linear part of the differential system (7) into its canonical real Jordan normal form, and system (7) becomes p v2 u̇ = − 4 (1 + 4r 2 ) v + u2 + √ , (8) p 1 + 4r 2 v̇ = 4 (1 + 4r 2 ) u. Performing the transformation u = αX, v = βY with √ ! p 1 + 4r 2−1 α = −β = 4 (1 + 4r 2 )3 , r2 and changing (X, Y ) by√ (x, y) the differential system (8) becomes sys- 1 − 1 + 4r 2 tem (3) with A = . Moreover system (3) is into the form r2 A (2) of reversible quadratic centers taking a = b = 1 + .  2 Proof of Theorem 1. The aim of this section is to study the limit cycles of the quadratic system (5) when ε 6= 0 is sufficiently small by using the averaging theory of six order. We see that system (5) has a center at the origin and the linear part of the system is in its canonical real Jordan normal form when ε = 0. We consider the change to polar coordinates x = r cos θ, y = r sin θ and we get the differential system (5) in polar coordinates (ṙ, θ̇). After rescaling the variable r as r = εR and taking the angle θ as the new independent variable, we obtain the differential equation 7 dR X i (9) = ε Fi (θ, R) + O(ε7), dθ i=1 CYCLICITY OF QUADRATIC REVARSIBLE CENTERS 9 Now we can apply the averaging theory described in section 2 to this differential system, because system (9) is equivalent to system (6) tak- ing x = R, t = θ and T = 2π. For the differential equation (9) we have that Fi = Fi (θ, R) for i = 1, 2, 3, 4 are F1 = −λ11 R + − cos θ((A + 2) cos(2θ) + 2)R2 , F2 = −λ12 R + sin θ(cos(2θ)((A + 2)λ11 − 2λ51 ) + 2λ11 + λ31 − λ51 )R2 1 + sin(2θ)((A + 2) cos(2θ) + 2)2 R3 , 2 F3 = −λ13 R + (sin θ(cos(2θ)((A + 2)λ12 − 2λ52 ) + 2λ12 + λ32 − λ52 ) 1 −λ11 (λ31 cos θ + λ51 cos(3θ)))R2 + ((A + 2) cos(2θ) + 2) 4 (cos(4θ)((A + 2)λ11 − 4λ51 ) − 2Aλ11 cos(2θ) +(A − 2)λ11 − 4λ31 )R3 − sin2 θ cos θ((A + 2) cos(2θ) + 2)3 R4 , F4 = −λ14 R + (sin θ(cos(2θ)((A + 2)λ13 − 2λ53 ) + 2λ13 + λ33 − λ53 ) + cos θ(−λ11 λ32 − λ12 λ31 ) − cos(3θ)(λ11 λ52 + λ12 λ51 ))R2 1 + (−2 cos(4θ) ((A2 − 4) λ12 + 8λ52 ) − 2 ((A2 + 4) λ12 + 8λ32 ) 8 +4 sin(2θ)((A − 2)λ11 λ51 + 4λ11 λ31 + λ31 (λ31 − 2λ51 )) +4λ11 sin(4θ)((A + 2)λ31 − (A − 2)λ51 ) + 4λ51 sin(6θ) ((A + 2)λ11 − λ51 ) + cos(2θ)((A − 2)(3A + 2)λ12 −4(A + 2)(2λ32 + λ52 )) + (A + 2) cos(6θ)((A + 2)λ12 − 4λ52 )) R3 1 + sin θ((A + 2) cos(2θ) + 2)2 (2 cos(2θ)(Aλ11 + λ31 + λ51 ) 4 − cos(4θ)((A + 2)λ11 − 6λ51 ) − Aλ11 + 2λ11 + 6λ31 )R4 + sin3 θ cos θ((A + 2) cos(2θ) + 2)4 R5 . We do not provide the functions Fi (θ, R) for i = 5, 6, 7 because their expressions are huge, and they are easy to compute with the help of an algebraic manipulator as mathematica or mapple. From the formulas of section 2 we can compute again by using an algebraic manipulator as mathematica or mapple the average functions. Thus the average function of first order is f1 (R) = −λ11 R. It is clear that f1 (R) = 0 has no positive roots. Therefore we get no information about the bifurcated limit cycles of system (5) from the 10 J. LLIBRE, A. NABAVI AND M. MOUSAVI average function of first order. By setting λ11 = 0, we have f1 (R) ≡ 0. So computing the average function of second order we obtain f2 (R) = −λ12 R. Again f2 (R) does not provide any information about the limit cycles. Taking λ12 = 0 we compute the third order average function f3 (R) = −λ13 R − 2λ31 R3 . Clearly equation f3 (R) = 0 has at most one simple positive root, show- ing that the third average function can provide one limit cycle for the differential equation (9). If λ13 = λ31 = 0, then f3 (R) ≡ 0 and we compute the fourth order average function f4 (R) = −λ14 R − 2λ32 R3 . Therefore as in the case of the third average function the fourth one can provide at most one limit cycle for the differential equation (9). By setting λ14 = λ32 = 0 we compute 1 f5 (R) = −λ15 R − 2λ33 R3 − (A − 6)(A + 4)λ51 R5 . 3 It follows immediately that equation f5 (R) = 0 can have at most two simple positive roots if A ∈/ {−4, 6}, and consequently the fifth average function can provide at most two limit cycles for the differential equa- tion (9) if A ∈/ {−4, 6}. When A ∈ {−4, 6} as for the third average function the fifth average function can provide at most one limit cycle for the differential equation (9). Now we must distinguish three cases. Case 1: λ15 = λ33 = λ51 = 0. Then we compute the sixth order averaging function and we obtain 1 f6 (R) = −λ16 R − 2λ34 R3 − (A − 6)(A + 4)λ52 R5 . 3 So the same conclusions than for the fifth average function, and we must consider three subcases. Subcase 1.1: λ16 = λ34 = λ52 = 0 and we obtain the seventh order averaging function 1 f7 (R) = −λ17 R − 2λ35 R3 − (A − 6)(A + 4)λ53 R5 . 3 Again the same conclusions than for the fifth average function. Subcase 1.2: λ16 = λ34 = 0, A = −4. Then the seventh order averaging function is (10) f7 (R) = −λ17 R − 2λ35 R3 . CYCLICITY OF QUADRATIC REVARSIBLE CENTERS 11 So equation f7 (R) = 0 can have at most one simple positive root, and consequently the seventh average function can provide at most one limit cycle for the differential equation (9). Subcase 1.3: λ16 = λ34 = 0, A = 6. Therefore the seventh order averaging function is (10). So the same conclusion than in the subcase 1.2. Case 2: λ15 = λ33 = 0, A = −4 and we obtain the sixth order averaging function (11) f6 (R) = −λ16 R − 2λ34 R3 . Therefore in this case we can obtain at most one limit cycle for the differential equation (9). Taking λ16 = λ34 = 0 we get that the seventh order averaging function is 11011 f7 (R) = −λ17 R − 2λ35 R3 + λ51 R7 . 1296 Hence equation f7 (R) = 0 can have at most two simple positive roots, and consequently the seventh average function can provide at most two limit cycles for the differential equation (9). In short, when A = −4 the two limit cycles appear in order seven and not in order five as it is the case for A ∈ R \ {−4, 6}. Case 3: λ15 = λ33 = 0, A = 6 and we obtain the sixth order average function is (11). So at most one limit cycle at order sixth. Taking λ16 = λ34 = 0 we get that the seventh order averaging function is 193594 f7 (R) = −λ17 R − 2λ35 R3 − λ51 R7 . 81 Again as in the case A = −4 for A = 6 the two limit cycles do not appear until order seven. This completes the proof of Theorem 2.  Acknowledgements This work is supported by the Ministerio de Economı̀a, Industria y competitividad, Agencia Estatal de Investigación grant MTM2016- 77278-P (FEDER), the Agència de Gestió d’Ajusts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics- H2020-MSCA-RISE-2017-777911. The second and third authors are supported by Isfahan University of Technology (IUT). 12 J. LLIBRE, A. NABAVI AND M. MOUSAVI References [1] A. Atabaigi, N. Nyamoradi and H. Zangeneh, The number of limit cycles of a quintic polynomial system with a center, Nonlinear Anal. 7 (2009), 3008– 3017. [2] R. Benterki and J. Llibre, Centers and limit cycles of polynomial differ- ential systems of degree 4 via averaging theory, J. Comput. Appl. Math. 407 (2013), 16–22. [3] T. Blows and L. Perko, Bifurcation of limit cycles from centers and sepa- ratrix cycles of planar analytic systems, SIAM Rev. 36 (1994), 341–376. [4] G. Chen, C. Li, C. Liu and J. 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E-mail address: a.nabavi@math.iut.ac.ir E-mail address: marzieh.mousavi@math.iut.ac.ir