This is a preprint of: “Limit cycles of a perturbation of a polynomial Hamiltonian systems of degree 4 symmetric with respect to the origin”, Jaume Llibre, Yohanna Paulina Martinez Mancilla, Claudio Vidal, Canad. Math. Bull., vol. 63(3), 547–561, 2020. DOI: [10.4153/S0008439519000626] LIMIT CYCLES OF A PERTURBATION OF A POLYNOMIAL HAMILTONIAN SYSTEMS OF DEGREE 4 SYMMETRIC WITH RESPECT TO THE ORIGIN JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL Abstract. We study the number of limit cycles bifurcating from the origin of a Hamiltonian system of degree 4. We prove using the averaging theory of order 7, that there are quartic polynomial systems close these Hamiltonian systems having 3 limit cycles. 1. Introduction and statement of the main result One of the main open problems in the qualitative theory of planar differential systems is the determination of limit cycles. Closely related to the Hilbert’s 16th problem is the study of the limit cycles from planar differential systems when we vary the parameters bifurcating from a center, or from its periodic solutions, and has been exhaustively studied in the last century. However there is no general method to solve completely this problem, the averaging theory as being largely studied in recent years in order to analyze the problem of the bifurcation of limit cycles, see for instance [4, 2, 7, 9, 8, 10, 13, 14, 17, 18, 15, 23]. For details about the averaging theory see the book of Sanders, Verhults and Murdock [21]. In this work we deal with polynomial differential systems in R2 of the form (1) ẋ = P (x, y), ẏ = Q(x, y), where the dot denotes derivative with respect to an independent real variable t, usually called the time. Assume that the origin O is an equilibrium point of system (1). When all the orbits of system (1) in a punctured neighborhood of the equilibrium point O are periodic, we say that the origin is a center. The study of the centers remain open in the present days and was started by Poincaré [20] and Dulac [6]. We focus on a polynomial differential system (1) having a center at the origin of linear type, i.e. after a linear change of variables and a scaling of the time variable, it can be written in the form: ẋ = −y + P2 (x, y), ẏ = x + Q2 (x, y), where P2 (x, y) and Q2 (x, y) are polynomials without constant and linear terms. This paper is a natural continuation of the work “Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis” [16] where we consider the Hamiltonian systems (2) ẋ = −y − x4 − 3bx2 y 2 − 5cy 4 , ẏ = x + 4x3 y + 2bxy 3 , of degree 4 with Hamiltonian function 1 2 (3) H(x, y) = (x + y 2 ) + x4 y + bx2 y 3 + cy 5 , 2 and are classify all the phase portraits of these Hamiltonian systems in the Poincaré disk, see Figure 1. 2010 Mathematics Subject Classification. Primary 34C07, Secondary 34C08. Key words and phrases. Hamiltonian systems, linear type centers, quartic polynomial, polynomial vector fields, phase portraits. 1 2 JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL In this work we perturb the Hamiltonian systems (2). Thus we consider these class of all polynomial differ- ential systems of degree 4, i.e. 7 X ẋ = −y − x4 − 3bx2 y 2 − 5cy 4 + εi pi (x, y), i=1 (4) 7 X 3 3 i ẏ = x + 4x y + 2bxy + ε qi (x, y), i=1 where pi (x, y) = ai0 + ai1 x + ai2 y + ai3 x2 + ai4 xy + ai5 y 2 + ai6 x3 + ai7 x2 y+ ai8 xy 2 + ai9 y 3 + ai10 x4 + ai11 x3 y + ai12 x2 y 2 + ai13 xy 3 + ai14 y 4 , (5) qi (x, y) = bi0 + bi1 x + bi2 y + bi3 x2 + bi4 xy + bi5 y 2 + bi6 x3 + bi7 x2 y+ bi8 xy 2 + bi9 y 3 + bi10 x4 + bi11 x3 y + bi12 x2 y 2 + bi13 xy 3 + bi14 y 4 . Our objective is to study the number of limit cycles bifurcating from the origin of system (4) using the averaging theory up to order 7. Our main result is the following one: Theorem 1. For ε > 0 sufficiently small the maximum number of small limit cycles of the differential system (4) bifurcating from the center (0, 0) obtained using the averaging theory of order (a) one and two is 0; (b) three and four is 1; (c) five and six is 2; (d) seven is 3. Theorem 1 is proved in Section 3. All the computations of this paper have been revised with the help of the algebraic manipulator Mathematica. Thus the two main objectives of this paper are: First to illustrate how to use the averaging theory up to order 7 for compute periodic solutions, and second how to use the averaging theory for studying the periodic solutions which are born in a Hopf bifurcation. We note that if the objective of this paper was to estimate the bound of the maximum number of periodic solutions of the differential system (4), this can be done using the techniques of the papers [11, 12]. In section 2 we provide the notations, basic definitions and results which will allow to do this study. 2. Preliminary results We consider the center at the origin of system (2), the global phase portraits of this system was detailed studied and the results are summarize in the next theorem proved in [16]. Theorem 2. The phase portrait in the Poincaré disk of a linear type center of a polynomial Hamiltonian system with nonlinearities of degree 4 symmetric with respect to the y-axis is topologically equivalent to one of the 30 phase portraits of Figure 1. The averaging theory is fundamental to our study, so we introduce the main result for applying it, see [17]. Consider the system X (6) ẋ = εi Fi (t, x) + εk+1 R(t, x, ε), i=1 where Fi : R 7→ Rn for i = 1, 2, . . . , k and R : R × I × (−ε0 , ε0 ) 7→ Rn are continuous functions and T -periodic in the first variable, I being an open subset of Rn . LIMIT CYCLES OF QUARTIC POLYNOMIAL 3 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) Figure 1. Phase portraits of the Hamiltonian systems (2). The separatrices are in bold. 4 JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL For i = 1, . . . , k we define the averaging function fi : I 7→ Rn of order i as yi (T, z) (7) fi (z) = , i! where yi : R × I 7→ Rn , i = 1, . . . , k − 1 are defined recurrently by the following integral equation Z t yi (t, z) = i! [Fi (s, ϕ(s, z))+ 0  (8) X i X Yl 1 ∂ L Fi−l (s, ϕ(s, z)) yj (s, z)bj  ds b1 !b2 !2!b2 · · · bl !l!bl j=1 l=1 Sl where ∂ L G(u, v) denote the derivative of order L of a function G with respect to the variable u, and Sl is the set of all l-tuples of non-negative integers (b1 , b2 , . . . , bl ) satisfying b1 + 2b2 + · · · + lbl = l, and L = b1 + b2 + · · · + bl . The explicit expressions of the average functions for i = 1, . . . , 7 are given in the appendix A. Now we can enunciate the following result, proved in section 3 of [17]. Theorem 3. For the functions of (6) we assume the following conditions. (1) For each t ∈ R, Fi (t, ·) ∈ C k−i for i = 1, · · · k, ∂ k−i Fi is locally Lipschitz in the second variable for i = 1, · · · , k and R is a continuous function locally Lipschitz in the second variable. (2) Assume that fi = 0, i = 1, · · · , r − 1 and fr 6= 0, r ∈ {1, · · · , k}. Moreover suppose that for some a ∈ I with fr (a) = 0 there exists a neighborhood V ⊂ I of a such that fr (z) 6= 0, ∀z ∈ V̄ \ a and dB (fr (z), V, 0) 6= 0 (here dB (fr (z), V, 0) denotes the Brouwer degree of fr at a). Then for sufficiently small |ε| > 0 there exists a T -periodic solution x(·, ε) of (6) such that x(0, ε) → a when ε → 0. We note that when fr is C 1 then the Brouwer degree of fr at a is non-zero if the determinant of the Jacobian matrix Dfr (a) is non-zero. For more details see [3, 19]. Another important tool is the Descartes Theorem about the number of zeros of a real polynomial (see [1]). Theorem 4 (Descartes Theorem). Consider the real polynomial p(x) = ai1 xi1 + ai2 xi2 + · · · + air xir with 0 < i1 < i2 < · · · < ir and aij 6= 0 real constants for j ∈ {1, 2, · · · , r}. When aij aij+1 < 0, we say that aij and aij+1 have a variation of sign. If the number of variations of signs is m, then p(x) has at most m positive real roots. Moreover it is always possible to choose the coefficients of p(x) in such a way that p(x) has exactly r − 1 positive real roots. Gauss showed later on that the number of allowable positive roots is m, m − 2, m − 4, · · · given that the sign changes m times. 3. Proof of Theorem 1 We shall use the averaging theory up to order seven for studying the number of small limit cycles of system (4) which can bifurcate from the origin of system (4) with ε = 0 when this system is perturbed with ε 6= 0 and small. First we do the rescaling of the variables (x, y) → (X, Y ) with x = εX, y = εY , then the initial differential system (4) becomes a differential system of the form 7 X 7 X (9) Ẋ = −Y + εi ri (X.Y ) + O(ε8 ), Ẏ = X + εi si (X.Y ) + O(ε8 ). i=1 i=1 LIMIT CYCLES OF QUARTIC POLYNOMIAL 5 After we pass to polar coordinates (X, Y ) → (r, θ) given by X = r cos θ, Y = r sin θ, and the previous differential system writes 7 X 7 X (10) ṙ = εi Ri (r, θ) + O(ε8 ), θ̇ = 1 + εi Si (r, θ) + O(ε8 ). i=1 i=1 Finally we take as independent variable the variable θ and the differential system becomes the differential equation 7 dr X i (11) = ε Ti (r, θ) + O(ε8 ). dθ i=1 This differential equation is in the normal form for applying the averaging theory described in [14], which is summarized in section 2. Thus we apply the averaging theory from order 1 to 7 in ε and we shall obtain the number of periodic solutions indicated in Theorem 1 for the different orders. More precisely, let r(θ, ε) = r∗ + O(ε), be a periodic solution of the differential equation (11) obtained from the averaging theory of order k, being r∗ a simple zero of the averaged function of order k. Then this periodic solution provides the periodic solution (r(t, ε), θ(t, ε)) = (r∗ , 1) + O(ε), of the differential system (10). And this last periodic solution gives place to the periodic solution (X(t, ε), Y (t, ε)) = (r∗ cos t, r∗ sin t) + O(ε), of the differential system (9). Finally, we get the periodic solution (x(t, ε), y(t, ε)) = (εr∗ cos t, εr∗ sin t) + O(ε2 ), of system (1). In summary, all the periodic solutions r(θ, ε) obtained applying the averaging theory to the differential equation (4) provide periodic solutions (x(t, ε), y(t, ε)) of the differential system (1) which tends to the origin (i.e. to the center localized at the origin of coordinates) when ε → 0. Therefore are periodic solutions bifurcating from the origin in a Hopf bifurcation. By doing a Taylor expansion truncated at the 7th order in ε we obtain an expression in the form (6) for dr/dθ with k = 7. The explicit expansion is dr (12) = K1 ε + K2 ε2 + K3 ε3 + K4 ε4 + K5 ε5 + K6 ε6 + K7 ε7 + · · · , dθ where considering C = cos θ and S = sin θ the first three coefficients are: K1 = r(a11 C 2 + (a12 + b11 )CS + b12 S 2 ), K2 = r(a21 C 2 + a13 C 3 r + a22 CS + b21 CS + a14 C 2 rS + b13 C 2 rS + b22 S 2 + a15 CrS 2 + b14 CrS 2 + b15 rS 3 + (−b11 C 2 + S(a11 C − b12 C + a12 S))(a11 C 2 + S((a12 + b11 )C + b12 S))), and K3 = r(a31 C 2 + a23 C 3 r + a16 C 4 r2 − C 5 r3 + a32 CS + b31 CS + a24 C 2 rS + b23 C 2 rS + a17 C 3 r2 S + b16 C 3 r2 S + b32 S 2 + a25 CrS 2 + b24 CrS 2 + a18 C 2 r2 S 2 + b17 C 2 r2 S 2 + 4C 3 r3 S 2 + 3bC 3 r3 S 2 + b25 rS 3 + a19 Cr2 S 3 + b18 Cr2 S 3 + b19 r2 S 4 + 2bCr3 S 4 + 5cCr3 S 4 + (−b21 C 2 + (b11 )2 C 4 − b13 C 3 r + a21 CS − b22 CS + a13 C 2 rS − b14 C 2 rS + a22 S 2 + (a11 )2 C 2 S 2 − 2a11 b12 C 2 S 2 + (b12 )2 C 2 S 2 + a14 CrS 2 − b15 CrS 2 + 2a11 a12 CS 3 − 2a12 b12 CS 3 + a15 rS 3 + (a12 )2 S 4 − 2b11 C 2 S(a11 C − b12 C + a12 S))(a11 C 2 + S(a12 C + b11 C + b12 S)) + (−b11 C 2 + S(a11 C − b12 C + a12 S))(a21 C 2 + a13 C 3 r + S(a22 C + b21 C + a14 C 2 r + b13 C 2 r + b22 S + a15 CrS + b14 CrS + b15 rS 2 ))). 6 JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL The other coefficient are too longer and we do not provide them here. In this work we consider the center at the origin. Our system (4) is a polynomial differential system so the functions Fi and Ri are analytic, and the variable θ is 2π-periodic because appear through sinus and cosinus. Therefore we can apply Theorem 3 setting the interval I = {r : 0 < r̄}, for some r̄ > 0. Applying Theorem 3 we obtain the average function of first order f1 (r) = π(a11 + b12 )r. Clearly f1 (r) has no solution in I. Therefore the averaging method of first order does not provide any small limit cycle bifurcating from the origin. We set b12 = −a11 and obtain f1 (r) = 0. So we can apply the averaging theory of second order and we obtain the averaging function of order two: f2 (r) = π(a21 + b22 )r. Newly the averaging function has not solution in I. Thus, the statement (a) of Theorem 1 is proved. Doing b22 = −a21 we have f2 (r) = 0 and we can apply the averaging method of thirst order, the averaging function of order three is f3 (r) = r(A1 + A3 r2 ), where A1 = π(a31 + b32 ), A3 = (3a16 + a18 + b17 + 3b19 )π/4. p Thus f3 (r) has one positive real root r∗ = −A1 /A3 in I if 0 < −A1 /A3 and in this case it holds that (df3 /dr)(r∗ ) 6= 0. Hence, applying the averaging theory of order three we can detect one limit cycle bifurcating from the center at the origin of system (4) with ε = 0. In order to apply the averaging method of fourth order we set b32 = −a31 and b19 = −a16 − a18 /3 − b17 /3. So the averaging function of fourth order is f4 (r) = r(B1 + B3 r2 ), where B1 = π(a41 + b42 ) and B3 = (−a13 a14 − a14 a15 + 3a12 a16 − 3a26 − 2a11 a17 − a28 + 3a16 b11 + 2a13 b13 + b13 b14 − 2a15 b15 + b14 b15 + a12 b17 + b11 b17 − b27 − 2a1 1b18 − 3b29 )π/4. Thus f4 (r) has one positive real zero in I if 0 < −B1 /B3 . Hence applying the averaging theory of order fourth we know that one small limit cycle bifurcates from the origin of system(4) with ε = 0. So statement (b) of Theorem 1 is proved. To apply the averaging method of fifth order we first set b42 = −a41 and b29 = (−a13 a14 − a14 a15 + 3a12 a16 − 3a26 − 2a11 a17 − a28 + 3a16 b11 + 2a13 b13 + b13 b14 − 2a15 b15 + b14 b15 + a12 b17 + b11 b17 − b27 − 2a11 b18 )/3 and then f4 (r) = 0. We continue applying the averaging method of fifth order where the averaging function is f5 (r) = r(C1 + C3 r2 + C5 r4 ), where C1 = (a51 + b52 )π, C3 = (2a11 (a13 )2 + a23 a14 + a11 (a14 )2 + a13 a24 + 2a11 a13 a15 + a12 a14 a15 + a24 a15 + a14 a25 − 3a22 a16 − 3a12 a26 + 3a36 + 2a21 a17 + 2a1 a7 + a38 − a13 a14 b11 + 3a12 a16 b11 − 3a26 b11 − 2a11 a17 b11 + 3a16 (b11 )2 − 3a16 b21 + 2a12 a13 b13 − 2a23 b13 − a11 a14 b13 + 4a13 b11 b13 − 2a13 b23 − 1 2 a11 a13 b14 + a11 a15 b14 + a12 b13 b14 + 2b11 b13 b14 − b23 b14 − a11 (b14 )2 − b13 b24 + a11 a14 b15 + 2a12 a15 b15 + 2a25 b15 − 2a11 b13 b15 + b11 b14 b15 − b24 b15 − 2a11 (b15 )2 + 2a15 b25 − b14 b25 − a22 b17 + a12 b11 b17 + (b11 )2 b17 − b21 b17 − a12 b27 − b11 b27 + b37 + 2a21 b18 − 2a11 b11 b18 + 2a11 b28 + 3b39 )π/4, and C5 = (−a14 + a14 b − 4bb13 − 2b15 − 10bb15 + 5a14 c + 10b15 c)π/8. LIMIT CYCLES OF QUARTIC POLYNOMIAL 7 The averaging function f5 (r) can have at most 2 positive real zeros in I because C1 , C3 and C5 are linearly independent since C1 depends only on a51 and b52 , C5 only depends on a14 , b13 and b15 and C3 depends of these coefficients and on other more, for example only C3 present the coefficient a36 , and by the Descartes Theorem 4 the averaging function f5 can present two change of sing so almost there is two positive simple roots for f5 . We prove through the averaging method of fifth order that at most 2 limit cycles can bifurcate from the origin of system (4) with ε = 0 using this averaging theory. At this moment we separate the study in three cases: b 6= 0, or b = 0 and c 6= 1/5, or b = 0 and c = 1/5. 3.1. Case b 6= 0. Setting b52 = −a51 , b13 = (−2b15 (1 + 5b − 5c) + a4 1(−1 + b + 5c))/(4b) (here we use that b 6= 0) and solving C3 for b39 we can apply the averaging method of sixth order. The sixth averaging function is f6 (r) = r(D1 + D3 r2 + D5 r4 ), where D1 = (a61 + b62 )π, D3 = π(2(a11 )2 a13 a14 − 2a22 a13 a14 − 2a12 a23 a14 + 2a33 a14 + a21 (a14 )2 + a11 a14 a24 + 8a21 (a13 )2 b + 16a11 a13 a23 b + 10(a11 )2 a13 a14 b + 2a2 a3 a4 b + 2a12 a23 a14 b + 2a33 a14 b + 3a21 (a14 )2 b + 4a11 a12 (a14 )2 b + 4a23 a24 b + 7a11 a14 a24 b + 4a13 a34 b + 8a21 a13 a15 b + 8a11 a12 a13 a15 b + 2 1 1 8a11 a23 a15 b + 4(a11 )2 a14 a15 b + 4(a12 )2 a14 a15 b + 4a22 a14 a15 b + 4a12 a24 a15 b + 4a34 a15 b + 8a11 a13 a25 b + 4a12 a14 a25 b + 4a24 a25 b + 4a14 a35 b − 12a32 a16 b − 12a22 a26 b − 12a12 a36 b + 12a46 b + 8a31 a17 b + 8a21 a27 b + 8a11 a37 b + 4a48 b + 4a12 a13 a14 b11 − 4a23 a14 b11 − 2a11 (a14 )2 b11 − 16a11 (a13 )2 bb11 − 4a12 a13 a14 bb11 − 2a11 (a14 )2 bb11 − 4a13 a24 bb11 − 8a11 a13 a15 bb11 + 12a22 a16 bb11 + 12a12 a26 bb11 − 12a36 bb11 − 8a21 a17 bb11 − 8a11 a27 bb11 +6a13 a14 (b11 )2 −2a13 a14 b(b11 )2 −12a12 a16 b(b11 )2 +12a26 b(b11 )2 +8a11 a17 b(b11 )2 −12a16 b(b11 )3 −4a13 a14 b21 +12a12 a16 bb21 − 12a26 bb21 − 8a11 a17 bb21 + 24a16 bb11 b21 − 12a16 bb31 + 8a12 a13 bb23 − 8a23 bb23 − 4a11 a14 bb23 + 16a13 bb11 b23 − 8a13 bb33 + (a11 )2 a14 b14 − a22 a14 b14 −4a21 a13 bb14 −4a11 a23 bb14 −(a11 )2 a14 bb14 +a22 a14 bb14 +4a21 a15 bb14 +4a11 a12 a15 bb14 +4a11 a25 bb14 +2a12 a14 b11 b14 +8a11 a13 bb11 b14 − 2a12 a14 bb11 b14 −4a11 a15 bb11 b14 +3a14 (b11 )2 b14 −3a14 b(b11 )2 b14 −2a14 b21 b14 +2a14 bb21 b14 +4a12 bb23 b14 +8bb11 b23 b14 −4bb33 b14 −4a21 b(b14 )2 + 8a11 bb11 (b14 )2 − a12 a14 b24 − 4a11 a13 bb24 + a12 a14 bb24 + 4a11 a15 bb24 − 2a14 b11 b24 + 2a14 bb11 b24 − 4bb23 b24 − 8a11 bb14 b24 + a14 b34 − a14 bb34 + 4(a11 )2 a13 b15 − 4a22 a13 b15 − 4a12 a23 b15 + 4a33 b15 + 4a21 a14 b15 + 2a11 a24 b15 + 20(a11 )2 a13 bb15 − 20a22 a13 bb15 − 20a12 a23 bb15 + 20a33 bb15 + 12a21 a14 bb15 + 4a11 a12 a14 bb15 + 14a11 a24 bb15 + 8(a11 )2 a15 bb15 + 8(a12 )2 a15 bb15 + 8a22 a15 bb15 + 8a12 a25 bb15 + 8a35 bb15 + 8a12 a13 b11 b15 − 8a23 b11 b15 − 8a11 a14 b11 b15 + 40a12 a13 bb11 b15 − 40a23 bb11 b15 − 20a11 a14 bb11 b15 + 12a13 (b11 )2 b15 + 60a13 b(b11 )2 b15 − 8a13 b21 b15 − 40a13 bb21 b15 − 8a11 bb23 b15 + 2(a11 )2 b14 b15 − 2a22 b14 b15 − 2(a11 )2 bb14 b15 − 10a22 bb14 b15 + 4a12 b11 b14 b15 + 20a12 bb11 b14 b15 + 6(b11 )2 b14 b15 + 26b(b11 )2 b14 b15 − 4b21 b14 b15 −16bb21 b14 b15 −2a12 b24 b15 −10a12 bb24 b15 −4b11 b24 b15 −16bb11 b24 b15 +2b34 b15 +6bb34 b15 +4a21 (b15 )2 +12a21 b(b15 )2 −8a11 a12 b(b15 )2 − 8a11 b11 (b15 )2 −32a11 bb11 (b15 )2 +2a11 a14 b25 +2a11 a14 bb25 +8a12 a15 bb25 +8a25 bb25 +4bb11 b14 b25 −4bb24 b25 +4a11 b15 b25 +4a11 bb15 b25 +8a15 bb35 − 4bb14 b35 −4a32 bb17 +4a22 bb11 b17 −4a12 b(b11 )2 b17 −4b(b11 )3 b17 +4a12 bb21 b17 +8bb11 b21 b17 −4bb31 b17 −4a22 bb27 +4a12 bb11 b27 +4b(b11 )2 b27 − 4bb21 b27 − 4a12 bb37 − 4bb11 b37 + 4bb47 + 8a31 bb18 − 8a21 bb11 b18 + 8a11 b(b11 )2 b18 − 8a11 bb21 b18 + 8a21 bb28 − 8a11 bb11 b28 + 8a11 bb38 + 12bb49 − 10(a11 )2 a13 a14 c+10a22 a13 a14 c+10a12 a23 a14 c−10a33 a14 c−5a21 (a14 )2 c−5a11 a14 a24 c−20a12 a13 a14 b11 c+20a23 a14 b11 c+10a11 (a14 )2 b11 c− 30a13 a14 (b11 )2 c+20a13 a14 b21 c−5(a11 )2 a14 b14 c+5a22 a14 b14 c−10a12 a14 b11 b14 c−15a14 (b11 )2 b14 c+10a14 b21 b14 c+5a12 a14 b24 c+10a14 b11 b24 c− 5a14 b34 c − 20(a11 )2 a13 b15 c + 20a22 a13 b15 c + 20a12 a23 b15 c − 20a33 b15 c − 20a21 a14 b15 c − 10a11 a24 b15 c − 40a12 a13 b11 b15 c + 40a23 b11 b15 c + 40a11 a14 b11 b15 c − 60a13 (b11 )2 b15 c + 40a13 b21 b15 c − 10(a11 )2 b14 b15 c + 10a22 b14 b15 c − 20a12 b11 b14 b15 c − 30(b11 )2 b14 b15 c + 20b21 b14 b15 c + 10a12 b24 b15 c + 20b11 b24 b15 c − 10b34 b15 c − 20a21 (b15 )2 c + 40a11 b11 (b15 )2 c − 10a11 a14 b25 c − 20a11 b15 b25 c)/(16b), and D5 = π(20a110 a14 +2a112 a14 −120a11 a13 b+36a111 a13 b+28a113 a13 b−40a110 a14 b+2a112 a14 b+12a114 a14 b−12a24 b+12a111 a15 b+ 20a113 a15 b + 12a16 a17 b + 8a17 a18 b − 36a16 a19 b + 312a11 a13 b2 + 12a12 a14 b2 + 12a24 b2 + 240a11 a15 b2 − 24a13 bb110 + 12a14 b2 b11 + 5a14 b111 − 13a14 bb111 + 16a13 bb112 + 8a15 bb112 + 3a14 b113 − 3a14 bb113 + 88a13 bb114 + 80a15 bb114 − 48b2 b23 − 60a11 bb14 + 8a113 bb14 − 60a11 b2 b14 − 12bb110 b14 − 4bb112 b14 + 20bb114 b14 + 40a110 b15 + 4a112 b15 + 112a110 bb15 + 4a112 bb15 + 24a114 bb15 − 120a12 b2 b15 − 120b2 b11 b15 + 10b111 b15 + 22bb111 b15 + 6b113 b15 − 6bb113 b15 − 24bb25 − 120b2 b25 − 36a16 bb16 − 4a17 bb17 − 12a19 bb17 − 12bb16 b17 + 12a16 bb18 + 8a18 bb18 − 4bb17 b18 − 100a110 a14 c − 10a112 a14 c + 120a11 a13 bc + 120a12 a14 bc + 60a24 bc + 120a14 bb11 c − 25a14 b111 c − 15a14 b113 c + 60a11 bb14 c − 200a110 b15 c − 20a112 b15 c + 240a12 bb15 c + 240bb11 b15 c − 50b111 b15 c − 30b113 b15 c + 120bb25 c)/(96b). 8 JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL Therefore f6 (r) can have two positive real zeros in I following the arguments used for f5 . Note that D1 , D3 and D5 are linearly independent functions. In fact D1 only present the coefficients a61 and b62 , only D3 has the coefficients a22 , a23 , a28 and b21 , and D5 is the only with the coefficients b112 , b113 and b114 . So applying the averaging theory of order 6 we can detect that at most two small limit cycles bifurcating from the center at the origin of system (4) with ε = 0 and this number can be reached. Thus, the statement (c) of Theorem 1 is proved in the case b 6= 0. We do b62 = −a61 , and solving D3 for b49 and D5 for b25 we can apply the averaging theory of seven order. The averaging function of order 7 is f7 (r) = r(E1 + E3 r2 + E5 r4 + E7 r6 ), where E1 = (a71 + b72 )π, E7 = (−15a111 − 3a113 − 180a11 b + 9a111 b + 13a113 b + 36a11 b2 − 12bb110 − 10b112 − 6bb112 − 12b114 − 8bb114 + 15a111 c + 35a113 c + 420a11 bc + 10b112 c + 140b114 c)π/64, and we do not provided the explicit expressions of E3 and E5 because they are huge. The averaging function f7 (r) can have at most 3 positive real zeros in I. This because E1 , E3 , E5 and E7 are linearly independent, since E1 depends on a71 and b72 , only E3 has the coefficients (for example) a33 , b34 and b44 , the coefficient a213 only appear in E5 and E7 depends of a11 , a111 , a113 , b110 , b112 and b114 , and using Descartes Theorem 4 we can affirm that f7 can has three changes of sing so the averaging function of seven order can has three different positive real zeros. Therefore we can detect through the averaging method of order seven that at most 3 limit cycles can bifurcate from the origin of system (4) with ε = 0. Hence statement (d) of Theorem 1 is proved for averaging function of order 7 in the case b 6= 0. 3.2. Case b = 0 and c 6= 1/5. Under these conditions system (4) becomes 7 X 7 X (13) ẋ = −y − x4 − 5cy 4 + εi pi (x, y), ẏ = x + 4x3 y + εi qi (x, y). i=1 i=1 The averaging function until order four are the same that before. So following the previous elections of coefficient, we continue applying the averaging method of fifth order where the averaging function is f˜5 (r) = r(C1 + C3 r2 + C̃5 r4 ), where C1 and C3 are the same that before and C̃5 = C5 |b=0 = (a14 + 2b15 )(−1 + 5c)π/8. Since C1 , C3 and C̃5 are linearly independent because C1 depends on a51 and b52 , C̃5 only depends on a14 and b13 and C3 depends of these coefficients and other more, for example only C3 present the coefficients b37 , b39 and a36 and using Descartes Theorem 4 we have that the averaging function f5 (r) can have at most 2 positive real zeros in I. We can detect through the averaging method of fifth order that at most 2 limit cycles can bifurcate from the origin of (2). Note that if c = 1/5 the coefficient C̃5 vanish, in this case the averaging function f5 (r) can have at most one positive real root in I. This situation will we studied in detail in the next case. Setting b52 = −a51 , b15 = a14 /2 and solving C3 for b39 we can apply the averaging method of sixth order. The averaging function of order six when b = 0 is f˜6 (r) = r(D1 + D̃3 r2 + D̃5 r4 ), LIMIT CYCLES OF QUARTIC POLYNOMIAL 9 where D̃3 = (4a21 (a13 )2 + 8a11 a13 a23 + 6(a11 )2 a13 a14 + 2a33 a14 + 2a23 a24 + 3a11 a14 a24 + 2a13 a34 + 4a21 a13 a15 + 4a11 a12 a13 a15 + 4a11 a23 a15 + 2a2 a4 a5 + 2a34 a15 + 4a11 a13 a25 + 2a24 a25 − 6a32 a16 − 6a22 a26 − 6a12 a36 + 6a46 + 4a31 a17 + 4a21 a27 + 4a11 a37 + 2a48 − 8a11 (a13 )2 b11 − 1 2 1 2a23 a14 b11 −2a13 a24 b11 −4a11 a13 a15 b11 +6a22 a16 b11 +6a12 a26 b11 −6a36 b11 −4a21 a17 b11 −4a11 a27 b11 +2a13 a14 (b11 )2 −6a12 a16 (b11 )2 +6a26 (b11 )2 + 4a11 a17 (b11 )2 −6a16 (b11 )3 −2a13 a14 b21 +6a12 a16 b21 −6a26 b21 −4a11 a17 b21 +12a16 b11 b21 −6a16 b31 −4(a11 )2 a13 b13 +4a22 a13 b13 +4a12 a23 b13 − 4a33 b13 − 2a11 a24 b13 − 8a12 a13 b11 b13 + 8a23 b11 b13 − 12a13 (b11 )2 b13 + 8a13 b21 b13 + 4a12 a13 b23 − 4a23 b23 + 8a13 b11 b23 − 4a13 b33 − 2a21 a13 b14 − 2a11 a23 b14 + 3(a11 )2 a14 b14 + 2a21 a15 b14 + 2a11 a12 a15 b14 + 2a11 a25 b14 + 4a11 a13 b11 b14 − 2a11 a15 b11 b14 + a14 (b11 )2 b14 − a14 b21 b14 − 2(a11 )2 b13 b14 + 2a22 b13 b14 −4a12 b11 b13 b14 −6(b11 )2 b13 b14 +4b21 b13 b14 +2a12 b23 b14 +4b11 b23 b14 −2b33 b14 −2a21 (b14 )2 +4a11 b11 (b14 )2 −2a11 a13 b24 +2a11 a15 b24 − a14 b11 b24 + 2a12 b13 b24 + 4b11 b13 b24 − 2b23 b24 − 4a11 b14 b24 + a14 b34 − 2b13 b34 + 6a11 a14 b25 + 4a12 a15 b25 + 4a25 b25 − 4a11 b13 b25 + 2b11 b14 b25 − 2b24 b25 + 4a15 b35 − 2b14 b35 − 2a32 b17 + 2a22 b11 b17 − 2a12 (b11 )2 b17 − 2(b11 )3 b17 + 2a12 b21 b17 + 4b11 b21 b17 − 2b31 b17 − 2a22 b27 + 2a12 b11 b27 + 2(b11 )2 b27 − 2b21 b27 − 2a12 b37 − 2b11 b37 + 2b47 + 4a31 b18 − 4a21 b11 b18 + 4a11 (b11 )2 b18 − 4a11 b21 b18 + 4a21 b28 − 4a11 b11 b28 + 4a11 b38 + 6b49 )π/8, and D̃5 = (−60a11 a13 + 18a111 a13 + 14a113 a13 + 12a110 a14 + 6a112 a14 − 6a24 + 6a111 a15 + 10a113 a15 + 6a16 a17 + 4a17 a18 − 18a16 a19 − 12a13 b110 + 3a14 b111 + 8a13 b112 + 4a15 b112 + 9a14 b113 + 44a13 b114 + 40a15 b114 − 40a110 b13 − 4a112 b13 − 10b111 b13 − 6b113 b13 − 30a11 b14 + 4a113 b14 − 6b110 b14 − 2b112 b14 + 10b114 b14 − 12b25 − 18a16 b16 − 2a17 b17 − 6a19 b17 − 6b16 b17 + 6a16 b18 + 4a18 b18 − 2b17 b18 + 60a11 a13 c + 30a24 c + 30a11 b14 c + 60b25 c)π/48. As before (in the case b 6= 0) f6 (r) can have two positive real zeros in I, because the coefficients of the averaging function of order 6, D1 , D̃3 and D̃5 , are linearly independent functions, and then we can apply the Descartes Theorem 4. So applying the averaging theory of order 6 we can detect at most two small limit cycles bifurcating from the center at the origin and this number can be reached. Solving D1 for b62 , D̃3 for b49 and D̃5 for b25 , we can apply the averaging theory of seventh order and the averaging function of order seven is f˜7 (r) = r(E1 + Ẽ3 r2 + Ẽ5 r4 + Ẽ7 r6 ), where Ẽ7 = (−15a111 − 3a113 − 10b112 − 12b114 + 15a111 c + 35a113 c + 10b112 c + 140b114 c)π/64. again we do not provide the explicit expressions of Ẽ3 and Ẽ5 because are very big. Under the hypothesis b = 0 and c 6= 1/5 the averaging function of order 7 associated to system (13) has at most three positive real zeros, becausethe coefficients of f˜7 (r) are linearly independent and we can apply the Descartes Theorem. Thus we can detect through the averaging method of order seven that at most 3 limit cycles can bifurcate from the origin of (4) with ε = 0. S o the statement (d) is proved when b = 0 and c 6= 1/5. 3.3. Case b = 0, c = 1/5. In this case the averaging function of fifth order f5 (r) is fˆ5 (r) = r(C1 + C3 r2 ). The averaging function fˆ5 (r) can have only at most 1 positive real zero in I. We detect through the averaging method of fifth order that at most 1 limit cycle can bifurcate from the origin of system (2) with ε = 0. Solving C1 for b52 and C3 for b39 . We can apply the averaging method of sixth order. The averaging function of order six is fˆ6 (r) = r(D1 + D̂3 r2 + D̂5 r4 ), where D̂3 = D̃3 + (a14 + 2b15 )(2(a12 )2 a15 + 2a22 a15 + 2a12 a25 + 2a35 + (a11 )2 (2a15 − 3b14 ) − (b11 )2 b14 + b21 b14 + b11 b24 − b34 + 2a21 (a14 − b13 − b15 ) + a11 (2a12 a14 + a24 − 2a14 b11 + 4b11 b13 − 2b23 − 2a12 b15 + 2b11 b15 − 4b25 ))π/8, and 10 JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL D̂5 = −(24a11 a13 − 9a111 a13 − 7a113 a13 + 5a110 a14 − a112 a14 − 3a114 a14 − 6a12 a14 − 3a111 a15 − 5a113 a15 − 3a16 a17 − 2a17 a18 + 9a6 a9 + 6a13 b110 − 6a14 b11 + 2a14 b111 − 4a13 b112 − 2a15 b112 − 22a13 b114 − 20a15 b114 + 20a110 b13 + 2a112 b13 + 5b111 b13 + 3b113 b13 + 1 1 12a11 b14 − 2a113 b14 + 3b110 b14 + b112 b14 − 5b114 b14 + 22a110 b15 + 4a112 b15 − 6a114 b15 − 12a12 b15 − 12b11 b15 + 7b111 b15 + 9b113 b15 + 9a16 b16 + a17 b17 + 3a19 b17 + 3b16 b17 − 3a16 b18 − 2a18 b18 + b17 b18 )π/24. As for the previous cases (b 6= 0, and b = 0 with c 6= 1/5) the averaging function of sixth order f6 (r) can have two real positive zeros in I, because D1 , D̂3 and D̂5 are linearly independent functions and we can apply Descartes Theorem 4. Therefore, through the averaging theory of order 6 we can detect that at most two small limit cycles bifurcating from the center at the origin and this number can be reached. So statement (c) is proved in the case b = 0 and c = 1/5. Solving D1 for b62 , D̂3 for b49 and D̂5 for b25 , we can apply the averaging theory of seven order and the averaging function is fˆ7 (r) = r(E1 + Ê3 r2 + Ê5 r4 + Ê7 r6 ), where Ê1 = (a71 + b72 )π, Ê7 = −(1/16)(3a111 − a113 + 2b112 − 4b114 )π. we do not provide the explicit expressions of Ê3 and Ê5 because they are very long. Thus fˆ7 (r) can have three positive real roots in I since the coefficients of fˆ7 (r) are linearly independent and we can apply Descartes Theorem. So, applying the averaging theory of order 7 we can detect that at most three small limit cycles can bifurcate from the center at the origin and this number can be reached. This prove the statement (d) in Theorem 1. In summary, for the averaging theory of order 1 and 2 we can not detect the existence of small limit cycles bifurcating from the center at the origin. For the averaging theory of order 3 and 4 we can detect that at most one small limit cycle bifurcating from the origin of system (4) with ε = 0. For the averaging theory of order 5 and 6 we can detect at most two small limit cycles bifurcating from the center at the origin and this number can be reached. Finally for order seven we detect trough the averaging theory at most three limit cycles. This complete the proof of Theorem 1. Acknowledgements The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA- RISE-2017-777911. Y. Paulina Martı́nez was supported by a CONICYT fellowship (Chile). C. Vidal was partially supported by CONICYT (Chile) through FONDECYT project 1130644. This paper is part of Y. Paulina Martı́nez Ph.D. thesis in the Program Doctorado en Matemática Aplicada, Universidad del Bı́o-Bı́o (Chile). LIMIT CYCLES OF QUARTIC POLYNOMIAL 11 Appendix A. Averaging functions We present explicitly the averaging function until order 7. Rt y1 (t, z) = F1 (s, ϕ(s, z))ds, R0t   y2 (t, z) = 0 2F2 (s, ϕ(s, z)) + 2∂F1 (s, ϕ(s, z))y1 (s, z) ds, Rt y3 (t, z) = 0 6F3 (s, ϕ(s, z)) + 6∂F2 (s, ϕ(s, z))y1 (s, z)  +3∂ 2 F1 (s, ϕ(s, z))y1 (s, z)2 + 3∂F1 (s, ϕ(s, z)) y2 (s, z) ds, Rt y4 (t, z) = 0 24F4 (s, ϕ(s, z)) + 24∂F3 (s, ϕ(s, z))y1 (s, z) +12∂ 2 F2 (s, ϕ(s, z))y1 (s, z)2 + 12∂F2 (s, ϕ(s, z))y2 (s, z) +12∂ 2 F1 (s, ϕ(s, z))y1 (s, z)y2 (s, z)  +4∂ 3 F1 (s, ϕ(s, z))y1 (s, z)3 + 4∂F1 (s, ϕ(s, z))y3 (s, z) ds, Rt y5 (t, z) = 0 120F5 (s, ϕ(s, z)) + 120∂F4 (s, ϕ(s, z))y1 (s, z) +60∂ 2 F3 (s, ϕ(s, z))y1 (s, z)2 + 60∂F3 (s, ϕ(s, z))y2 (s, z) +60∂ 2 F2 (s, ϕ(s, z))y1 (s, z)y2 (s, z) + 20∂ 3 F2 (s, ϕ(s, z))y1 (s, z)3 +20∂F2 (s, ϕ(s, z))y3 (s, z) + 20∂ 2 F1 (s, ϕ(s, z))y1 (s, z)y3 (s, z) +15∂ 2 F1 (s, ϕ(s, z))y2 (s, z)2 + 30∂ 3 F1 (s, ϕ(s, z))y1 (s,z)2 y2 (s, z) +5∂ 4 F1 (s, ϕ(s, z))y1 (s, z)4 + 5∂F1 (s, ϕ(s, z))y4 (s, z) ds, Rt y6 (t, z) = 0 720F6 (s, ϕ(s, z)) + 720∂F5 (s, ϕ(s, z))y1 (s, z) +360∂ 2 F4 (s, ϕ(s, z))y1 (s, z)2 + 360∂F4 (s, ϕ(s, z))y2 (s, z) +120∂ 3 F3 (s, ϕ(s, z))y1 (s, z)3 + 360∂ 2 F3 (s, ϕ(s, z))y1 (s, z)y2 (s, z) +120∂F3 (s, ϕ(s, z))y3 (s, z) + 30∂ 4 F2 (s, ϕ(s, z))y1 (s, z)4 +180∂ 3 F2 (s, ϕ(s, z))y1 (s, z)2 y2 (s, z) + 120∂ 2 F2 (s, ϕ(s, z))y1 (s, z)y3 (s, z) +90∂ 2 F2 (s, ϕ(s, z))y2 (s, z)2 + 30∂F2 (s, ϕ(s, z))y4 (s, z) +60∂ 4 F1 (s, ϕ(s, z))y1 (s, z)3 y2 (s, z) + 60∂ 3 F1 (s, ϕ(s, z))y1 (s, z)2 y3 (s, z) +90∂ 3 F1 (s, ϕ(s, z))y1 (s, z)y2 (s, z)2 + 30∂ 2 F1 (s, ϕ(s, z))y1 (s, z)y4 (s, z) +60∂ 2 F1 (s, ϕ(s, z))y2 (s,  z)y3 (s, z) + 6∂ 5 F1 (s, ϕ(s, z))y1 (s, z)5 +6∂F1 (s, ϕ(s, z))y5 (s, z) ds, Rt y7 (t, z) = 0 (5040F7 (s, ϕ(s, z)) + 5040∂F6 (s, ϕ(s, z))y1 (s, z) +2520∂ 2 F5 (s, ϕ(s, z))y1 (s, z)2 + 2520∂F5 (s, ϕ(s, z))y2 (s, z) +2520∂ 2 F4 (s, ϕ(s, z))y1 (s, z)y2 (s, z) + 840∂ 3 F4 (s, ϕ(s, z))y1 (s, z)3 +840∂F4 (s, ϕ(s, z))y3 (s, z) + 840∂ 2 F3 (s, ϕ(s, z))y1 (s, z)y3 (s, z) +630∂ 2 F3 (s, ϕ(s, z))y2 (s, z)2 + 1260∂ 3 F3 (s, ϕ(s, z))y1 (s, z)2 y2 (s, z) +210∂ 4 F3 (s, ϕ(s, z))y1 (s, z)4 + 210∂F3 (s, ϕ(s, z))y4 (s, z) +210∂ 2 F2 (s, ϕ(s, z))y1 (s, z)y4 (s, z) + 420∂ 3 F2 (s, ϕ(s, z))y1 (s, z)2 y3 (s, z) +420∂ 4 F2 (s, ϕ(s, z))y1 (s, z)3 y2 (s, z) + 630∂ 3 F2 (s, ϕ(s, z))y2 (s, z)2 y1 (s, z) +42∂ 5 F2 (s, ϕ(s, z))y1 (s, z)5 + 420∂ 2 F2 (s, ϕ(s, z))y2 (s, z)y3 (s, z) +42∂F2 (s, ϕ(s, z))y5 (s, z) + 630∂ 3 F2 (s, ϕ(s, z))y2 (s, z)2 y1 (s, z) +7∂ 6 F1 (s, ϕ(s, z))y1 (s, z)6 + 105∂ 5 F1 (s, ϕ(s, z))y1 (s, z)4 y2 (s, z) +140∂ 4 F1 (s, ϕ(s, z))y1 (s, z)3 y3 (s, z) + 630∂ 4 F1 (s, ϕ(s, z))y1 (s, z)2 y2 (s, z)2 +105∂ 3 F1 (s, ϕ(s, z))y1 (s, z)2 y4 (s, z) + 42∂ 2 F1 (s, ϕ(s, z))y1 (s, z)y5 (s, z) +420∂ 3 F1 (s, ϕ(s, z))y1 (s, z)y2 (s, z)y3 (s, z) +105∂ 3 F1 (s, ϕ(s, z))y2 (s, z)3 + 105∂ 2 F1 (s, ϕ(s, z))y2 (s, z)y4 (s, z) +70∂ 2 F1 (s, ϕ(s, z))y3 (s, z)2 + 7∂F1 (s, ϕ(s, z))y6 (s, z) ds. 12 JAUME LLIBRE, PAULINA MARTÍNEZ, AND CLAUDIO VIDAL References [1] I.S. 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(J Llibre) Departament de Matemàtiques, Facultat de Ciències Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat (Y.P. Martı́nez) Departamento de Matemática, Facultad de Ciencias, Universidad de Bı́o-Bı́o, Casilla 5–C, Con- cepción, VIII–Región, Chile Email address: yohanna.martinez@uab.cat (C. Vidal) Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad de Bı́o-Bı́o, Casilla 5–C, Concepción, VIII–región, Chile Email address: clvidal@ubiobio.cl