This is a preprint of: “The centers and their cyclicity for a class of polynomial differential systems of degree 7”, Rebiha Benterki, Jaume Llibre, J. Comput. Appl. Math., vol. 368(112456), 1–16, 2020. DOI: [10.1016/j.cam.2019.112456] THE CENTERS AND THEIR CYCLICITY FOR A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 7 REBIHA BENTERKI1 AND JAUME LLIBRE2 Abstract. We classify the global phase portraits in the Poincaré disc of the generalized Kukles systems ẋ = −y, ẏ = x + axy 6 + bx3 y 4 + cx5 y 2 + dx7 , which are symmetric with respect to both axes of coordinates. Moreover using the averaging theory up to sixth order, we study the cyclicity of the center located at the origin of coordinates, i.e. how many limit cycles can bifurcate from the origin of coordinates of the previous differential system when we perturb it inside the class of all polynomial differential systems of degree 7. 1. Introduction and statement of the main results Two classical and difficult problems of the qualitative theory of planar polyno- mial differential systems are the characterization of their centers, and the study of their cyclicity, i.e. how many limit cycles can bifurcate from a center when we perturb it inside a given class of polynomial differential systems. Of course, this kind of bifurcation is called in the literature a Hopf bifurcation. In this work we deal with planar polynomial differential systems of the form (1) ẋ = −y, ẏ = x + Qn (x, y), having a center at the origin, being Qn (x, y) a homogeneous polynomial of degree n. As usual the dot in system (1) denotes derivative with respect an independent variable t usually called the time. Systems of this form were called by Giné [5] Kukles homogeneous systems. In 1999 Volokitin and Ivanov [16] conjectured that the systems (1) have a center at the origin if and only if they are symmetric with respect to one of the coordinate axes. For n = 2 and n = 3, the authors of the conjecture knew that it holds. Giné [5] in 2002 proved the conjecture for n = 4 and n = 5. Giné et al. [6, 7] proved the conjecture for all n under an additional assumption, that the authors believe that it is redundant. In this work we consider the class of polynomial differential systems (1) for n = 7 which are symmetric with respect to both coordinate axes, i.e. (2) ẋ = −y, ẏ = x + axy 6 + bx3 y 4 + cx5 y 2 + dx7 . 2010 Mathematics Subject Classification. Primary 34C15, 34C25. Key words and phrases. center, phase portrait, cyclicity, limit cycle, Hopf bifurcation, averaging method, Kukles. 1 http://www.gsd.uab.cat 2 R. BENTERKI AND J. LLIBRE We note that if we look for the systems (1) which are symmetric with respect to one of the coordinate axes, we obtain that they are symmetric with respect to both coordinate axes. The first main objective of this work is to classify the phase portraits of the polynomial differential systems (2) in the Poincaré disc. For more details on the Poincaré disc see for instance the Chapter 5 of [4]. The phase portraits of the centers of systems (1) with n = 2, are known because are known the phase portraits of all the center of quadratic polynomial differential systems, see Vulpe [17]. The phase portraits of cubic polynomial differential systems with a symmetry with respect to a straight line are also known and in particular those of system (1) with n = 3, see Buzzi et al. [3], see also Malkin [13]; Vulpe Sibirskii [18] and Żola̧dek [19, 20]. The phase portraits of systems (1) with n = 4 follows from Benterki and Llibre [2] and Llibre and Salhi [9]. In Llibre and Silva [10, 11] classified the phase portraits of the systems (1) with n = 5, 6. In order to present the classification of the phase portraits of systems (2) we write the homogeneous polynomial axy 6 + bx3 y 4 + cx5 y 2 + dx7 of degree 7 which appears in system (2) into the form p(x, y) = x(dx6 + cx4 y 2 + bx2 y 4 + ay 6 ) = x(dX 3 + cX 2 Y + bXY 2 + aY 3 ), where X = x2 and Y = y 2 . Then, according with the different kind of roots of the polynomial dX 3 + cX 2 Y + bXY 2 + aY 3 we can consider the following 62 cases for the polynomial p(x, y). For the next 14 cases the polynomial p(x, y) is ax(y 2 − r1 x2 )(y 2 − r2 x2 )(y 2 − r3 x2 ) with (1) a > 0 and 0 < r1 < r2 < r3 , (2) a > 0 and r1 = 0 < r2 < r3 , (3) a > 0 and r1 < 0 < r2 < r3 , (4) a > 0 and r1 < r2 = 0 < r3 , (5) a > 0 and r1 < r2 < 0 < r3 , (6) a > 0 and r1 < r2 < r3 = 0, (7) a > 0 and r1 < r2 < r3 < 0, (8) a < 0 and 0 < r1 < r2 < r3 , (9) a < 0 and r1 = 0 < r2 < r3 , (10) a < 0 and r1 < 0 < r2 < r3 , (11) a < 0 and r1 < r2 = 0 < r3 , (12) a < 0 and r1 < r2 < 0 < r3 , (13) a < 0 and r1 < r2 < r3 = 0, (14) a < 0 and r1 < r2 < r3 < 0. For the next 10 cases the polynomial p(x, y) is ax(y 2 − r1 x2 )(y 2 − r2 x2 )2 with (15) a > 0 and 0 < r1 < r2 , (16) a > 0 and r1 = 0 < r2 , (17) a > 0 and r1 < 0 < r2 , (18) a > 0 and r1 < r2 = 0, (19) a > 0 and r1 < r2 < 0, (20) a < 0 and 0 < r1 < r2 , (21) a < 0 and r1 = 0 < r2 , (22) a < 0 and r1 < 0 < r2 , (23) a < 0 and r1 < r2 = 0, CENTERS AND THEIR CYCLICITY 3 (24) a < 0 and r1 < r2 < 0. For the next 6 cases the polynomial p(x, y) is ax(y 2 − r1 x2 )3 with (25) a > 0 and 0 < r1 , (26) a > 0 and r1 = 0, (27) a > 0 and r1 < 0, (28) a < 0 and 0 < r1 , (29) a < 0 and r1 = 0, (30) a < 0 and r1 < 0. For the next 6 cases the polynomial p(x, y) is ax(y 2 −r1 x2 )(y 4 −2αx2 y 2 +(α2 +β 2 )x4 ) with (31) a > 0 and 0 < r1 , (32) a > 0 and r1 = 0, (33) a > 0 and r1 < 0, (34) a < 0 and 0 < r1 , (35) a < 0 and r1 = 0, (36) a < 0 and r1 < 0. For the next 10 cases the polynomial p(x, y) is bx3 (y 2 − r1 x2 )(y 2 − r2 x2 ) with (37) b > 0 and 0 < r1 < r2 , (38) b > 0 and r1 = 0 < r2 , (39) b > 0 and r1 < 0 < r2 , (40) b > 0 and r1 < r2 = 0, (41) b > 0 and r1 < r2 < 0, (42) b < 0 and 0 < r1 < r2 , (43) b < 0 and r1 = 0 < r2 , (44) b < 0 and r1 < 0 < r2 , (45) b < 0 and r1 < r2 = 0, (46) b < 0 and r1 < r2 < 0. For the next 6 cases the polynomial p(x, y) is bx3 (y 2 − r1 x2 )2 with (47) b > 0 and 0 < r1 , (48) b > 0 and r1 = 0, (49) b > 0 and r1 < 0, (50) b < 0 and 0 < r1 , (51) b < 0 and r1 = 0, (52) b < 0 and r1 < 0. For the next 2 cases the polynomial p(x, y) is bx3 (y 4 − 2αx2 y 2 + (α2 + β 2 )x4 ) with (53) b > 0, (54) b < 0. For the next 6 cases the polynomial p(x, y) is cx5 (y 2 − r1 x2 ) with (55) c > 0 and 0 < r1 , (56) c > 0 and r1 = 0, (57) c > 0 and r1 < 0, (58) c < 0 and 0 < r1 , (59) c < 0 and r1 = 0, (60) c < 0 and r1 < 0. For the next 2 cases the polynomial p(x, y) is dx7 with (61) d > 0, (62) d < 0. 4 R. BENTERKI AND J. LLIBRE For a definition of topological equivalence between two phase portraits in the Poincaré disc see subsections 2.1 and 2.2. Theorem 1. The polynomial differential systems (2) have 25 topologically non– equivalent phase portraits in the Poincaré disc. More precisely, the phase portrait in the Poincaré disc of Figure 1 is realizable by the case (1); 2 is realizable by the case (2); 3 is realizable by the cases (3) and (37); 4 is realizable by the case (4); 5 is realizable by the cases (5) and (39); 6 is realizable by the cases (6), (18), (26), (32) and (40); 7 is realizable by the cases (7), (19), (27), (33), (41), (49) and (53); 8 is realizable by the case (8); 9 is realizable by the case (9); 10 is realizable by the cases (10) and (42); 11 is realizable by the cases (11) and (43); 12 is realizable by the cases (12), (28), (34), (44), (50) and (58); 13 is realizable by the cases (13), (23), (29), (35), (45), (51) and (59); 14 is realizable by the cases (14), (24), (30), (36), (46), (52), (54), (60) and (62); 15 is realizable by the case (15); 16 is realizable by the case (16); 17 is realizable by the case (17); 18 is realizable by the case (20); 19 is realizable by the case (21); 20 is realizable by the case (22); 21 is realizable by the cases (25) and (31); 22 is realizable by the case (38); 23 is realizable by the cases (47) and (55); 24 is realizable by the cases (48) and (56); 25 is realizable by the cases (57) and (61). Figure 1 Figure 2 Figure 3 Theorem 1 is proved in section 3. The averaging theory described in subsection 2.3 allows to study analytically the existence of limit cycles of a non–autonomous differential system, by study- ing the simple zeros of the averaged function fk = fk (r). Here we shall use the CENTERS AND THEIR CYCLICITY 5 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 averaging theory up to sixth order for studying the number of limit cycles which can bifurcate from the center of system (2) when we perturb it inside the class of all polynomial differential systems of degree 7. More precisely, we deal with the 6 R. BENTERKI AND J. LLIBRE Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 polynomial differential systems 6   ẋ = −y + εs asij xi y j , s=1 0≤i+j≤7 (3) 6   ẏ = x + axy 6 + bx3 y 4 + cx5 y 2 + dx7 + εs bsij xi y j , s=1 0≤i+j≤7 where asij and bsij are real parameters, for 0 ≤ i, j ≤ 7 and 1 ≤ s ≤ 6. CENTERS AND THEIR CYCLICITY 7 Theorem 2. For |ε| = 0 sufficiently small the maximum number of small amplitude limit cycles of the differential system (3) bifurcating from the periodic solutions of the center (2) is (a) 0 if the first order average function f1 is non-zero, (b) 0 if f1 = 0 and the second order average function f2 is non-zero, (c) 1 if f1 = f2 = 0 and the third order average function f3 is non-zero, (d) 1 if f1 = f2 = f3 = 0 and the fourth order average function f4 is non-zero, (e) 2 if f1 = f2 = f3 = f4 = 0 and the fifth order average function f5 is non-zero, (f) 2 if f1 = f2 = f3 = f4 = f5 = 0 and the sixth order average function f6 is non-zero. Moreover, assume that fj = 0 for j = 1, . . . , k − 1 and fk = 0. Then if r is a simple zero of fk , the small amplitude limit cycle (x(t, ε), y(t, ε)) associated to this zero is of the form (x(t, ε), y(t, ε)) = ε(r cos t, r sin t) + O(ε2 ). Theorem 2 is proved in section 4. 2. Preliminaries and basic results In this section we present some basic results and notations which are necessary for proving our results. 2.1. Poincaré compactification. Let X = (−y, x+axy 6 +bx3 y 4 +cx5 y 2 +dy 7 ) be the planar polynomial vector field associated to system (2). We define the Poincaré compactified vector field p(X) associated to X as follows (see all the details for instance [11] or Chapter 5 of [4]). The Poincaré sphere is defined as S2 = {y = (y1 , y2 , y3 ) ∈ R3 : y12 + y22 + y32 = 1} and its tangent space at the point y ∈ S2 is denoted by Ty S2 . We identify the plane R2 where we have our vector field X with the plane T(0,0,1) S2 . We define the central projection f : T(0,0,1) S2 −→ S2 as follows: to each point q ∈ T(0,0,1) S2 the central projection associates the two intersection points of the straight line which connects the points q and (0, 0, 0) with the sphere S2 . This central projection gives two copies of X in S2 , one in each hemisphere. Let X  be the vector field Df ◦ X , which is defined in S2 minus its equator S1 = {y ∈ S2 : y3 = 0}. The equator S1 can be identified with the infinity of R2 . We extend the vector field X  on S2 \S1 to a vector field p(X) on S2 as follows: p(X) is the unique analytic extension of y37 X  to S2 . In summary, we have two symmetric copies of X on S2 \S1 , and studying the dynamics of p(X) near S1 , we have the dynamics of X at infinity. The Poincaré disc, denoted by D2 , is the closed northern hemisphere of {y ∈ S2 : y3 ≥ 0} projected on y3 = 0 under the projection (y1 , y2 , y3 ) → (y1 , y2 ). The infinity S1 is invariant under the flow of the Poincaré compactifcation p(X). Here two polynomial vector fields X and Y associated to systems (1) are topo- logically equivalent if there is a homeomorphism on S2 preserving the infinity S1 carrying orbits of the flow of p(X) into orbits of the flow of p(Y ), either reversing or preserving the sense of all orbits. 8 R. BENTERKI AND J. LLIBRE For computing the analytic expression of p(X) we use the fact that S2 is a differentiable manifold. Thus we take the six local charts Ui = {y2 ∈ S2 : yi > 0}, and Vi = {y2 ∈ S2 : yi < 0} for i = 1, 2, 3; and the associated diffeomorphisms Fi : Ui −→ R2 and Gi : Vi −→ R2 for i = 1, 2, 3 are respectively the inverses of the central projections from the planes tangent at the points (1, 0, 0); (−1, 0, 0); (0, 1, 0); (0, −1, 0); (0, 0, 1) and (0, 0, −1). The value of Fi (y) or Gi (y) for some i = 1, 2, 3 is denoted by z = (z1 , z2 ), consequently according to the local charts under consideration the same letter z represents different coordinates. After a rescaling in the independent variable in the local chart (U1 , F1 ) the expression for p(X) is        1 u 1 u 1 u u̇ = v n −uP , +Q , , v̇ = −v n+1 P , ; v v v v v v in the local chart (U2 , F2 ) the expression for p(X) is        u 1 u 1 u 1 u̇ = v n P , − uQ , v̇ = −v n+1 Q , ; v v v v v v and for the local chart (U3 , F3 ) the expression for p(X) is u̇ = P (u, v), v̇ = Q(u, v). In the chart (Vi , Gi ) the expression for p(X) is the same than in the chart (Ui , Fi ) multiplied by (−1)6 for i = 1, 2, 3. We note that the points at the infinity S1 in any chart have coordinates (u, v) = (u, 0). The equilibrium points of p(X) which come from the equilibrium points of X are called finite equilibrium points of X, and the equilibrium points of p(X) which are in S1 are called infinite equilibrium points of X. We observe that the unique infinite equilibrium points which cannot be contained in the charts U1 ∪ V1 are the origins of the local charts U2 and V2 . Therefore when we study the infinite equilibrium points on the charts U2 ∪ V2 , we only need to verify if the origin of these charts are equilibrium points. 2.2. Topological equivalence. Two polynomial vector fields X and Y on R2 are topologically equivalent if there is a homeomorphism on the Poincaré sphere S2 preserving the infinity S1 carrying trajectories of the flow of p(X) into trajectories of the flow of p(Y ), either preserving or reversing the sense of all trajectories. Here a separatrix of the Poincaré compactification p(X) is a trajectory which is either an equilibrium point, or a limit cycle, or a trajectory which belongs to the boundary of a hyperbolic sector at an equilibrium point, finite or infinity, or any trajectory contained at the infinity S1 . We denote by S(p(X)) the set formed by all separatrices of p(X). It is known that the set S(p(X)) is closed, see for instance Neumann [14]. A canonical region of p(X) is an open connected component of S2 \ S(p(X)). The union of S(p(X)) plus one trajectory chosen from each canonical region is the separatrix configuration of p(X). Two separatrix configurations S(p(X)) and S(p(Y )) are equivalent if there is a homeomorphism in S2 preserving the infinity S1 carrying trajectories of S(p(X)) into trajectories of S(p(Y )), either preserving or reversing the sense of all orbits. CENTERS AND THEIR CYCLICITY 9 Markus [12], Neumann [14] and Peixoto [15] characterized the topologically equivalence between two Poincaré compactified vector fields as follows. Theorem 3. Assume that two Poincaré compactified polynomial vector fields p(X) and p(Y ) have finitely many separatrices. Then their separatrix configurations are equivalent if and only if S(p(X)) and S(p(Y )) are topologically equivalent. Theorem 3 says that essentially we need to determine the α– and the ω–limit sets of all the separatrices of p(X) for obtaining the phase portrait of a Poincaré compactified polynomial vector field p(X) with finitely many separatrices. 2.3. The averaging theory up to order 6. In this subsection we present some results on the averaging theory that we shall need for studying the limit cycles which bifurcate from the center localized at the origin of coordinates of the systems (2), when they are perturbed inside the class of all polynomial differential systems of degree 7. We deal with a non–autonomous differential system k  (4) ẋ(t) = εi Fi (t, x) + εk+1 R(t, x, ε), i=0 where the functions Fi : R×D → R for i = 0, 1, · · · , k and R : R×D×(−ε0 , ε0 ) → R are T -periodic in the first variable and continuous functions, D is an open interval of R, and ε a small parameter. Following the results of [8] we define the functions yj (t, z) for j = 1, 2, 3, 4, 5 related to system (13) as  t y1 (t, z) = F1 (s, z)ds, 0 t  y2 (t, z) = 2F2 (s, z) + 2∂F1 (s, z)y1 (s, z) ds, 0  t y3 (t, z) = 6F3 (s, z) + 6∂F2 (s, z)y1 (t, z) 0 +3∂ 2 F1 (s, z)y1 (s, z)2 + 3∂F1 (s, z) y2 (s, z) ds,  t y4 (t, z) = 24F4 (s, z) + 24∂F3 (s, z)y1 (s, z) 0 +12∂ 2F2 (s, z)y1 (s, z)2 + 12∂F2 (s, z)y2 (s, z) +12∂ 2F1 (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,  t y5 (t, z) = 120F5 (s, z) + 120∂F4 (s, z)y1 (s, z) 0 +60∂ 2F3 (s, z)y1 (s, z)2 + 60∂F3 (s, z)y2 (s, z) +60∂ 2F2 (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∂ 2F1 (s, z)y2 (s, z)2 + 30∂ 3 F1 (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, 10 R. BENTERKI AND J. LLIBRE where the k–th partial derivative of the function F (s, z) with respect to the variable z has been denoted by ∂ k F (s, z). From [8] the functions averaged functions fj (z) for j = 1, . . . , 6 are  T f1 (z) = F1 (t, z)dt, 0  T  f2 (z) = F2 (t, z) + ∂F1 (t, z)y1 (t, z) dt, 0  T  f3 (z) = F3 (t, z) + ∂F2 (t, z)y1 (t, z) 0 1 1 + ∂ 2 F1 (t, z)y1 (t, z)2 + ∂F1 (t, z)y2 (t, z) dt, 2 2  T  f4 (z) = F4 (t, z) + ∂F3 (t, z)y1 (t, z) 0 1 1 + ∂ 2 F2 (t, z)y1 (t, z)2 + ∂F2 (t, z)y2 (t, z) 2 2 1 2 1 + ∂ F1 (t, z)y1 (t, z)y2 (t, z)dt + ∂ 3 F1 (t, z)y1 (t, z)3 2 6 1 + ∂F1 (t, z)y3 (t, z) dt, 6  T  1 f5 (z) = F5 (t, z) + ∂F4 (t, z)y1 (t, z) + ∂ 2 F3 (t, z)y1 (t, z)2 0 2 1 1 2 + ∂F3 (t, z)y2 (t, z) + ∂ F2 (t, z)y1 (t, z)y2 (t, z) 2 2 1 3 1 + ∂ F2 (t, z)y1 (t, z)3 + ∂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 1 + ∂ 3 F1 (t, z)y1 (t, z)2 y2 (t, z) + ∂ 4 F1 (t, z)y1 (t, z)4 4 24 1 + ∂F1 (t, z)y4 (t, z) dt, 24  T  1 f6 (z) = F6 (t, z) + ∂F5 (t, z)y1 (t, z) + ∂F4 (t, z)y2 (t, z) 0 2 1 2 2 1 + ∂ F4 (t, z)y1 (t, z) + ∂F3 (t, z)y3 (t, z) 2 6 1 2 1 + ∂ F3 (t, z)y1 (t, z)y2 (t, z) + ∂ 3 F3 (t, z)y1 (t, z)3 2 6 1 1 + ∂F2 (t, z)y4 (t, z) + ∂ 2 F2 (t, z)y1 (t, z)y3 (t, z) 24 6 1 3 1 + ∂ F2 (t, z)y1 (t, z)2 y2 (t, z) + ∂ 2 F2 (t, z)y2 (t, z)2 4 8 1 4 1 + ∂ F2 (t, z)y1 (t, z)4 + ∂F1 (t, z)y5 (t, z) 24 120 CENTERS AND THEIR CYCLICITY 11 1 2 1 + ∂ F1 (t, z)y1 (t, z)y4 (t, z) + ∂ 2 F1 (t, z)y2 (t, z)y3 (t, z) 24 12 1 1 + ∂ 3 F1 (t, z)y1 (t, z)2 y3 (t, z) + ∂ 4 F2 (t, z)y1 (t, z)3 y2 (t, z) 12 12 1 3 2 1 5 + ∂ F1 (t, z)y1 (t, z)y2 (t, z) + ∂ F1 (t, z)y1 (t, z)5 dt. 8 120 The averaging theory for studying the periodic solutions of a non–autonomous differential system (13) works as follows, see [8] for more details. Suppose that the average functions fj (z) = 0 for j = 1, . . . , k − 1 and fk (z) = 0 for some k ≥ 1, we assume that f0 (z) = 0. By [8] if z is a simple zero of fk (z), then there is a limit cycle r(θ, ε) of system (13) such that r(0, ε) = z + O(ε). 3. Phase portraits in the Poincaré disc Now we shall study the phase portraits of the Poincaré compactified polynomial differential systems (2) with (a, b, c, d) = (0, 0, 0, 0). Remark 4. The polynomial differential systems (2) are reversible because they remain the same under the transformations (x, y, t) → (x, −y, −t) and (x, y, t) → (−x, y, −t). Therefore their phase portraits are symmetric with respect to the x–axis and y–axis. We shall study the phase portrait of a polynomial differential system (2) in the Poincaré disc as follows. First we shall determine the local phase portrait at all its finite and infinite equilibrium points. After with the help of the symmetries of its trajectories with respect to both coordinate axes, we shall determine its phase portrait in the Poincaré disc. Here we classify an equilibrium point p as hyperbolic when the eigenvalues of the linear part of system (2) at p have nonzero real part, as semi–hyperbolic when only one of these two eigenvalues is zero, as nilpotent when both eigenvalues are zero but the linear part of system (2) at p is not identically zero, and finally as linearly zero when the linear part of system (2) at p is identically zero. The local phase portraits of the hyperbolic, semi–hyperbolic and nilpotent equi- librium points can be determined using, for instance the Theorems 2.15, 2.19 and 3.5 of the book [4]. In order to determine the local phase portrait of a nilpotent equilibrium point at infinity it is not sufficient the mentioned Theorem 3.5, and we must studied it doing the changes of variables called blow–ups. These changes of variables are also necessary for analyze the local phase portraits of the linearly zero equilibrium points. For more information about the blow ups see Chapter 3 of [4] or [1]. 3.1. Finite equilibrium points. For a planar polynomial differential system (2) its finite equilibrium points are characterized in the following result. Proposition 5. Always the origin of the polynomial differential system (2) is a center. Furthermore if d < 0 then there are two additional equilibrium points, namely (±|d|−1/6 , 0), which are hyperbolic saddles. 12 R. BENTERKI AND J. LLIBRE Proof. Since the eigenvalues of the linear part of system (2) at the origin are ±i such equilibrium point is either a focus or a center, but due to the fact that the system is symmetric with respect to both coordinate axes, it is a center. −1/6 Clearly when d < 0 the system has the two equilibria √ (±|d| , 0), and the eigenvalues of the linear part at these equilibria are ± 6. So these equilibria are hyperbolic, and by Theorem 2.15 of [4] they are saddles.  3.2. Infinite equilibrium points. We shall use the notations and definitions given in subsection 2.2, for determining the local phase portraits at the infinite equilibrium points in the Poincaré disc. System (2) in the local chart U1 . The differential system (2) in the local chart U1 is (5) u̇ = d + cu2 + bu4 + au6 + v 6 + u2 v 6 , v̇ = uv 7 . An infinite equilibrium point of system (5) is a point (u0 , 0) such that u0 is a real root of the polynomial d + cu2 + bu4 + au6 . So the infinite equilibria of system √ (2) are (± rj , 0) when rj > 0, where the rj ’s are the ones which appear in the polynomials of the cases (1) to (62) described in section 1. √ The Jacobian matrix of system (5) evaluated at (± rj , 0) is  1/2 3/2 5/2  ± 2cr1 − 4br1 + 6ar1 0 . 0 0 Then all infinite equilibria of differential system (5) are semi–hyperbolic or linearly zero. In the next proposition we only provide the local phase portrait of the infinite √ equilibrium point ( rj , 0) with rj ≥ 0, because due to the symmetry (x, y, t) → (x, −y, −t) of system (2) with respect to x–axis the local phase portrait at the √ √ infinite equilibrium point (− rj , 0) is the same than at the equilibrium ( rj , 0) after reversing the sense of the trajectories. √ Proposition 6. The local phase portraits at the infinite equilibrium points ( rj , 0) with rj ≥ 0 of the local chart U1 ordered from the smallest value of rj to the biggest one are formed by (a) a semi–hyperbolic unstable node, a semi–hyperbolic saddle and a semi– hyperbolic unstable node in case (1); (b) a linearly zero at the origin of coordinates with two hyperbolic sectors, a semi–hyperbolic saddle and a semi–hyperbolic unstable node in case (2); (c) a semi–hyperbolic saddle and a semi–hyperbolic unstable node in cases (3) and (37); (d) a linearly zero with six hyperbolic sectors (the infinity line separates them in two groups of three sectors) at the origin of coordinates and a semi– hyperbolic unstable node in cases (4) and (38); (e) a semi–hyperbolic unstable node in cases (5), (39), (47) and (55); (f) a linearly zero at the origin of coordinates with two hyperbolic sectors in cases (6), (18), (26), (32), (40), (48) and (56); CENTERS AND THEIR CYCLICITY 13 (g) no infinite equilibria in the local chart U1 in cases (7), (14), (24), (27), (30), (33), (36), (41), (46), (49), (52)–(54), (57), (60)–(62); (h) a semi–hyperbolic saddle, a semi–hyperbolic unstable node and a semi– hyperbolic saddle in case (8); (i) a linearly zero with six hyperbolic sectors (the infinity line separates them in two groups of three sectors) at the origin of coordinates, a semi–hyperbolic unstable node and a semi–hyperbolic saddle in case (9); (j) a semi–hyperbolic unstable node and a semi–hyperbolic saddle in cases (10) and (42); (k) a linearly zero at the origin of coordinates with two hyperbolic sectors and a semi–hyperbolic saddle in cases (11) and (43); (l) a semi–hyperbolic saddle in cases (12), (28), (34), (44), (50) and (58); (m) a linearly zero at the origin of coordinates with six hyperbolic sectors (the infinity line separates them in two groups of three sectors) in cases (13), (23), (29), (35), (45), (51) and (59); (n) a semi–hyperbolic unstable node and a linearly zero with two hyperbolic sectors in case (15); (o) two linearly zero with two hyperbolic sectors, one of this equilibria is located at the origin of coordinates, in case (16); (p) a linearly zero with two hyperbolic sectors in case (17); (q) a semi–hyperbolic saddle and a linearly zero with two hyperbolic sectors and one parabolic sector, the straight line at infinity separates the two hyperbolic sectors and divides the parabolic one in case (20); (r) a linearly zero with six hyperbolic sectors (the infinity line separates them in two groups of three sectors) at the origin of coordinates, and a linearly zero with two adjacent hyperbolic sectors separated and with a parabolic sector, the infinite line separates the two hyperbolic sectors and divides the parabolic one in case (21); (s) a linearly zero saddle–node, the infinite line separates the two hyperbolic sectors and divides the parabolic one in case (22); (t) a linearly zero unstable node in cases (25) and (31); Proof. As we have mention near the beginning of this section the phase portraits of the semi–hyperbolic equilibrium can be determined using the Theorem 2.19 of [4], and the phase portraits of the linearly zero equilibrium points doing the blow–up changes of variables. Here we shall prove with all details the statements (a) and (b), the other statements are proved in a similar way. In statements (a) and (b) system (1) writes as (6) ẋ = −y; ẏ = x + ax(y 2 − r1 x2 )(y 2 − r2 x2 )(y 2 − r3 x2 ), with a > 0. This system in the local chart U1 becomes u̇ = −ar1 r2 r3 + a(r1 r2 + r1 r3 + r2 r3 )u2 (7) −a(r1 + r2 + r3 )u4 + au6 + v 6 + u2 v 6 , v̇ = uv 7 . Assume a > 0 and 0 < r1 < r2 < r3 . Then the eigenvalues at the infinite √ 1/2 equilibrium point ( rj , 0) for j = 1, 2, 3 are 0 and a 2rj (r1 r2 + r1 r3 + r2 r3 ) − 14 R. BENTERKI AND J. LLIBRE 3/2 5/2 √ √ 4rj (r1 + r2 + r3 )+ 6rj = h ( rj ) = 0, because the six roots ± rj for j = 1, 2, 3 of the polynomial h(u) = u̇|v=0 = −ar1 r2 r3 + a(r1 r2 + r1 r3 + r2 r3 )u2 − a(r1 + r2 + √ √ √ r3 )u4 + au6 are simple. Moreover h ( r1 ) > 0, h ( r2 ) < 0 and h ( r3 ) > 0. √ Hence the points ( rj , 0) for j = 1, 2, 3 are semi–hyperbolic equilibria. In what follows we shall apply Theorem 2.19 of [4] for determining the local √ phase portraits of the infinite equilibria ( rj , 0) for j = 1, 2, 3, and we shall use the √ notation of that theorem. First we translate the equilibrium point ( rj , 0) to the √ origin of coordinates doing the change (u, v) = (Y + rj , X). Thus we obtain the differential system √ √ Ẋ = A(X, Y ) = sign(h ( rj )) rj X 7 + X 7 Y , (8) √ √ Ẏ = sign(h ( rj )) h ( rj )Y + B(X, Y ) , where B(X, Y ) = a(r1 r2 + r1 r3 + r2 r3 − 6r1 rj − 6r2 rj − 6r3 rj + 15rj2 )Y 2 √ −4a(r1 + r2 + r3 − 5rj ) rj Y 3 − a(r1 + r2 + r3 − 15rj )Y 4 √ √ +6a rj Y 5 + aY 6 + (1 + rj )X 6 + 2 rj Y X 6 + Y 2 X 6 . √ If h ( rj ) < 0 we have changed the sign of the independent variable in the differen- tial system (8) in order that the coefficient of Y in the expression of Ẏ be positive as it is necessary in order to apply Theorem 2.19. Now the functions f (X) and g(X) of that theorem are 1 + rj f (X) = − √ X 6 + h.o.t., h ( rj ) √ √ g(X) = sign(h ( rj )) rj X 7 + h.o.t., where h.o.t. denotes higher order terms. So using the notation of Theorem 2.19 we √ √ have that am = sign(h ( rj )) rj and m = 7. Consequently, by Theorem 2.19 the √ √ equilibria ( rj , 0) for j = 1, 3 are unstable nodes, and the equilibrium ( r2 , 0) is a saddle. This completes the proof of statement (a). √ Assume a > 0 and r1 = 0 < r2 < r3 . The proof that the equilibria ( r2 , 0) √ and ( r3 , 0) are a semi–hyperbolic saddle and a semi–hyperbolic unstable node respectively, is identical to the proof of statement (a). So in order to complete the proof of statement (b) we shall show that the local phase portrait of the equilibrium √ ( r1 , 0) = (0, 0) is formed by two hyperbolic sectors. We do the blow–up change of variables v = wu to system (7) and it becomes u̇ = ar2 r3 u2 − a(r2 + r3 )u4 + au6 + u6 w6 + u8 w6 , (9) ẇ = −ar2 r3 uw + ar2 u3 w + ar3 u3 w − au5 w − u5 w7 . Now we remove the common factor u of u̇ and ẇ doing to system (9) the rescaling dτ = udt in the independent variable, we get the differential system u = ar2 r3 u − a(r2 + r3 )u3 + au5 + u5 w6 + u7 w6 , (10) w = −ar2 r3 w + ar2 u2 w + ar3 u2 w − au4 w − u4 w7 , where the prime denotes derivative with respect to the new independent variable τ . The unique equilibrium on the w–axis of system (10) is the origin of coordinates, CENTERS AND THEIR CYCLICITY 15 which is a saddle whose four separatrices are contained in the two axes. Going back through the two changes of variables, first the change dτ = udt and after the change v = wu, and taking into account that in system (7) we have that u̇|u=0 = v 6 , we obtain that the local phase portrait at the origin of system (7) is formed by two hyperbolic sectors. This completes the proof of statement (b).  System (2) in the local chart U2 . The differential system (2) in the local chart U1 is u̇ = −au2 − bu4 − cu6 − du8 − v 6 − u2 v 6 , v̇ = −auv − bu3 v − cu5 v − du7 v − uv 7 . Therefore the origin (0, 0) always is an infinite equilibrium point, which is linearly zero. Recall that from subsection 2.1 in this local chart the unique infinite equilib- rium point that we must study is its origin when it is an equilibrium point, as in the present case. Proposition 7. The local phase portrait at the origin of the local chart U2 is formed by a linearly zero equilibrium point with (a) two hyperbolic sectors separated by two parabolic sectors and the line of the infinity is contained into the two parabolic sectors in cases (1)–(7), (15)– (19), (25)–(27), (31)–(33), (37)–(41) and (53); (b) two elliptic sectors separated by the infinity in cases (8)–(13), (20)–(23), (28)–(29), (34)–(35), (42)–(45), (50), (51), (58) and (59); (c) two elliptic sectors separated by two parabolic sectors and the line of the infinity is contained into the two parabolic sectors in cases (14), (24), (30), (36), (46), (52), (54), (60) and (62). (d) two hyperbolic sectors in cases (47)–(49), (55)–(57) and (61). Proof. We shall prove statement (a) in the case (1). Statement (a) in the other cases, and the other statements are proved in a similar way. In the case (1) system (1) becomes system (6). This last system in the local chart U2 writes as u̇ = −au2 + a(r1 + r2 + r3 )u4 − a(r1 r2 + r1 r3 + r2 r3 )u6 +ar1 r2 r3 u8 − v 6 − u2 v 6 , (11) v̇ = −auv + a(r1 + r2 + r3 )u3 v − a(r1 r2 + r1 r3 + r2 r3 )u5 v +ar1 r2 r3 u7 v − uv 7 . This system doing the blow–up change of variables v = wu becomes u̇ = −au2 + a(r1 + r2 + r3 )u4 − a(r1 r2 + r1 r3 + r2 r3 )u6 (12) +ar1 r2 r3 u8 − u6 w6 − u8 w6 , ẇ = u5 w 7 . We eliminate the common factor u2 of u̇ and ẇ in system (12) doing the rescaling of the independent variable dτ = u2 dt, and we obtain the differential system u̇ = −a + a(r1 + r2 + r3 )u2 − a(r1 r2 + r1 r3 + r2 r3 )u4 +ar1 r2 r3 u6 − u4 w6 − u6 w6 , ẇ = u3 w7 . 16 R. BENTERKI AND J. LLIBRE This last system has no equilibria on the w–axis and on it we have that u̇|u=0 = −a < 0. Then going back through the two changes of variables, first dτ = u2 dt and second v = wu, and taking into account that in system (11) we have u̇|u=0 = −v 6 , we obtain that the local phase portrait at the origin of system (11) is formed by two hyperbolic sectors whose separatrices are tangent to the w–axis, one in v > 0 and the other in v < 0, separated by two parabolic sectors which contain locally the invariant u–axis, i.e. the infinite line. This completes the proof of case (1) of statement (a).  We note that in the proof of statement (b) doing blow–ups changes of variables we only obtain that the local phase portrait is formed by two elliptic sectors sepa- rated by two parabolic sectors and the line of the infinity is contained into the two parabolic sectors, but that when we shall consider the global phase portrait in the Poincaré disc these two local parabolic sectors are contained in two global elliptic sectors, for this reason in statement (b) we only said two elliptic sectors instead of two elliptic sectors separated by two parabolic sectors and the line of the infinity is contained into the two parabolic sectors. Proof of Theorem 1. This proof follows taking into account the local phase portraits of the finite and infinite equilibria described in Propositions 5, 6 and 7, and taking into account first Theorem 3 (i.e. that for obtaining the global phase portrait in the Poincaré disc we essentially must determine the α– and ω–limit of all the separatrices of the system) and after that the global phase portrait of a system (2) is symmetric with respect both coordinate axes.  4. Proof of Theorem 2 For studying the limit cycles which bifurcate in a Hopf bifurcation from the center of the differential system (2) when it is perturbed inside the class of all polynomial differential systems of degree 7, see (3), we work as follows. First doing the scaling x = εX, y = εY we introduce a small parameter ε. Thus we obtain the differential system (Ẋ, Ẏ ). Now performing the polar change of coordinates X = r cos θ, Y = r sin θ, the differential system (Ẋ, Ẏ ) written in polar coordinates becomes a differential system (ṙ, θ̇). Taking as independent variable the angle θ the differential system (ṙ, θ̇) produces the differential equation dr/dθ. Finally doing a Taylor expansion in the variable r at r = 0 and truncating at 6–th order in ε we obtain the differential equation 6 dr  i (13) r = = ε Fi (θ, r) + O(ε7 ). dθ i=0 The functions Fi (θ, r) i = 1, . . . , 6 of the differential system (13) are analytic, and since the independent variable θ appears through the sinus and cosinus of θ, they are 2π–periodic. Hence the assumptions for applying the averaging theory described in subsection 2.3 are satisfied. Now we shall study the limit cycles bifurcating from the center of system (2) when it is perturbed as in (3) following the steps described in subsection 2.3. We give only the expressions of functions F1 (r, θ) and F2 (r, θ). The explicit expressions CENTERS AND THEIR CYCLICITY 17 of Fi (r, θ) for i = 3, . . . , 6 are very long, therefore we shall omit them here. Thus we have 1 F1 (r, θ) = a200 cos θ + b200 sin θ + r(b101 + a110 − b101 cos 2θ + a110 cos 2θ 2 +b110 sin 2θ + a101 sin 2θ). and 1 2 F2 (r, θ) = (a cos θ + a110 r cos2 θ + b200 sin θ + b110 r cos θ sin θ + a101 r cos θ sin θ r 00 +b101 r sin2 θ)(−b200 cos θ − b110 r cos2 θ + a200 sin θ − b101 r cos θ sin θ +a101 r sin2 θ) + (a300 cos θ + a210 r cos2 θ + a120 r2 cos3 θ + b300 sin θ +a201 r cos θ sin θ + b120 r2 cos2 θ sin θ + a111 r2 cos2 θ sin θ + b201 r sin2 θ +b111 r2 cos θ sin2 θ + a102 r2 cos θ sin2 θ + b102 r2 sin3 θ + a110 r cos θ sin θ +b210 r cos θ sin θ). Using the formulas given in subsection 2.3 the averaged function of first order is f1 (r) = (b101 + a110 )r. Since the polynomial f1 (r) = 0 has no positive roots, the first average function does not give any information on the limit cycles that bifurcate from the center when we perturb it as in system (3). This proves statement (a). Taking b101 = −a110 , we obtain f1 (r) ≡ 0. Therefore we can apply the averaging theory of second order. Thus the averaged function of second order is f2 (r) = (b201 + a210 )r. Again the second averaged function does not provide information on the bifurcating limit cycles. This completes the proof of statement (b). Now taking b201 = −a210 we get f2 (r) ≡ 0, and applying the averaging theory of third order, we obtain the third averaged function 1 f3 (r) = −(b111 b200 −b301 +2b200 a120 −2b102a200 −a111 a200 −a310 )r+ (3b103 +b121 +a112 +3a130 )r3 . 4 Since the polynomial f3 (r) can have at most one positive real root, statement (c) of the theorem is proved. For applying the averaging theory of fourth order we must have f3 (r) ≡ 0, so we take a310 = b111 b200 − b301 + 2b200 a120 − 2b102 a200 − a111 a200 , a112 = −3b103 − b121 − 3a130 . 18 R. BENTERKI AND J. LLIBRE The averaged function of fourth order is 1 f4 (r) = − r3 (b102 b111 + b111 b120 + b110 b121 − 3b203 − b221 + b121 a101 − 2b102 a102 4 −2b112 a110 − a102 a111 + 2b120 a120 − a111 a120 − 2a110 a121 + 3b110 a130 +3a101 a130 − a212 − 3a230 ) − r(−b110 b111 b200 + b200 b211 + b111 b300 −b401 + 2b102 b200 a110 + b200 a110 a111 − 2b110 b200 a120 + 2b300 a120 −2b102 a101 a200 − b111 a110 a200 − a101 a111 a200 − 2a110 a120 a200 − a200 a211 −2b202 a200 + 2b200 a220 − 2b102 a300 − a111 a300 − a410 ). From the expression of the polynomial f4 (r) we get that it has at most one positive real root. Hence statement (d) of the theorem is proved. Doing a410 = −b110 b111 b200 + b200 b211 + b111 b300 − b401 + 2b102 b200 a110 + b200 a110 a111 +2b300 a120 − 2b202 a200 − 2b102 a101 a200 − b111 a110 a200 − a101 a111 a200 −a200 a211 + 2b200 a220 − 2b102 a300 − a111 a300 − 2b110 b200 a120 − 2a110 a120 a200 . a212 = b102 b111 + b111 b120 + b110 b121 − 3b203 − b221 + b121 a101 − 2b102 a102 − 3a230 −2b112 a110 − a102 a111 + 2b120 a120 − a111 a120 − 2a110 a121 + 3a101 a130 +3b110 a130 , CENTERS AND THEIR CYCLICITY 19 we have that f4 (r) ≡ 0, and we can apply the averaging theory of order five, and the fifth averaged function is f5 (r) = K3 r5 + K2 r3 + K1 r, where K1 = −((b110 )2 b111 b200 − b121 (b200 )2 − b111 b200 b210 − b110 b200 b211 − b110 b111 b300 + b211 b300 +b200 b311 + b111 b400 − 2b102 b110 b200 a110 + 2b200 b202 a110 + 2b102 b300 a110 +2b102 b200 a101 a110 + b111 b200 (a110 )2 − b110 b200 a110 a111 + b300 a110 a111 +b200 a101 a110 a111 + 2(b110 )2 b200 a120 − 2b200 b210 a120 − 2b110 b300 a120 + 2b400 a120 +2b200 (a110 )2 a120 − 3(b200 )2 a130 + 2b112 b200 a200 − 2b302 a200 − 2b202 a101 a200 −2b102 (a101 )2 a200 + b110 b111 a110 a200 − b211 a110 a200 − b111 a101 a110 a200 − a211 a300 −(a101 )2 a111 a200 − (a110 )2 a111 a200 + 2b110 a110 a120 a200 − 2a101 a110 a120 a200 +b121 (a200 )2 + 3a130 (a200 )2 − 2b102 a200 a201 − a111 a200 a201 + 2b102 b200 a210 −b111 a200 a210 − 2a120 a200 a210 + b200 a110 a211 − a101 a200 a211 − 2b110 b200 a220 +2b300 a220 − 2a110 a200 a220 − 2b202 a300 − 2b102 a101 a300 − b111 a110 a300 − a101 a111 a300 −a200 a311 + 2b200 a320 − a510 − 2b102 a400 − a111 a400 − b501 + b200 a111 a210 −2b102 (a110 )2 a200 − 2a110 a120 a300 + 2b200 a121 a200 ), −1 K2 = (−b102 b110 b111 − 2b110 b111 b120 − (b110 )2 b121 + b111 b202 + b121 b210 + b102 b211 4 +b120 b211 + b111 b220 + b110 b221 − 3b303 − b321 − b111 b120 a101 − b110 b121 a101 −2b202 a102 − 2b102 a101 a102 + 2(b102 )2 a110 + (b111 )2 a110 + 2b110 b112 a110 −2b212 a110 − b111 a102 a110 − a101 a102 a111 − b102 a110 a111 + b120 a110 a111 −4b110 b120 a120 + 2b220 a120 − 2b120 a101 a120 + b111 a110 a120 − 2a102 a110 a120 +b110 a111 a120 − 2a110 (a120 )2 + 2b110 a110 a121 + 2b200 a122 − 3(b110 )2 a130 +3b210 a130 − 3b110 a101 a130 + 12b200 a140 − 12b104 a200 − 2b122 a200 + b121 a201 3a130 a201 − 2b102 a202 − a111 a202 − 2b112 a210 − 2a121 a210 − a102 a211 −a120 a211 + 2b120 a220 − a111 a220 − 2a110 a221 + 3b110 a230 − a110 (a111 )2 3a101 a230 − a312 − 3a330 + b221 a101 + 2b102 b120 a110 ), 1 1 K3 = (5b + b123 + a132 + 5a150 ). 8 05 The rank of the Jacobian matrix of the function K = (K1 , K2 , K3 ) with respect to the coefficient asij and bsij which appear in their expressions is maximal, i.e. it is 3. Then the coefficients Ki for i = 1, 2, 3 which appear in th expression of f5 (r) are linearly independent. By the roots of a quadratic polynomial in the variable r2 it follows that f5 (r) can have at most two positive real roots. Therefore statement (e) of the theorem is proved. Imposing that K1 = 0, K2 = 0 and K3 = 0 we obtain that f5 (r) ≡ 0, and applying the averaging theory of order six we obtain the sixth average function 20 R. BENTERKI AND J. LLIBRE f6 (r) = r(C1 + C2 r2 + C3 r4 ), where 1 C1 = (2(b110 )3 b111 b200 − 4b102 b111 (b200 )2 − 2b111 b120 (b200 )2 − 4b110 b121 (b200 )2 − 4b110 b111 b200 b210 2 −2(b110 )2 b200 b211 + 2b200 b210 b211 + 2(b200 )2 b221 − 2(b110 )2 b111 b300 + 4b121 b200 b300 +2b110 b211 b300 + 4b102 b200 b301 + 2b111 b200 b310 + 2b110 b200 b311 − 2b300 b311 + 2b110 b111 b400 −2b200 e11 + 2b601 − 4b102 (b110 )2 b200 a110 + 4(b112 )2 a110 + 4b110 b200 b202 a110 + 2b111 b210 b300 +4b102 b200 b210 a110 + 4b102 b110 b300 a110 − 4b202 b300 a110 − 4b200 b302 a110 − 4b102 b400 a110 +4b102 b110 b200 a101 a110 − 4b200 b202 a101 a110 − 4b102 b300 a101 a110 − 4b102 b200 (a101 )2 a110 +4b110 b111 b200 (a110 )2 − 2b200 b211 (a110 )2 − 2b111 b300 (a110 )2 − 2b111 b200 a101 (a110 )2 − 2b211 b400 −4b102 b200 (a110 )3 − 2b111 (b200 )2 a111 + 2b200 b301 a111 − 2(b110 )2 b200 a110 a111 + 2b200 b210 a110 a111 +2b110 b300 a110 a111 − 2b400 a110 a111 + 2b110 b200 a101 a110 a111 − 2b300 a101 a110 a111 − 4b120 (b200 )2 a120 −2b200 (a101 )2 a110 a111 − 2b200 (a110 )3 a111 + 4(b110 )3 b200 a120 − 4b102 (b200 )2 a120 + 4(b200 )2 a110 a121 −8b110 b200 b210 a120 − 4(b110 )2 b300 a120 + 4b210 b300 a120 + 4b200 b310 a120 + 4b110 b400 a120 +8b110 b200 (a110 )2 a120 − 4b300 (a110 )2 a120 − 4b200 a101 (a110 )2 a120 − 2(b200 )2 a111 a120 −12b110 (b200 )2 a130 + 12b200 b300 a130 + 8(b102 )2 b200 a200 + 2(b111 )2 b200 a200 + 4b110 b112 b200 a200 −4b200 b212 a200 − 4b112 b300 a200 − 2b111 b301 a200 + 4b402 a200 − 4b112 b200 a101 a200 + 4b302 a101 a200 +4b202 (a101 )2 a200 + 4b102 (a101 )3 a200 + 2(b110 )2 b111 a110 a200 − 2b111 b210 a110 a200 −2b110 b211 a110 a200 + 2b311 a110 a200 − 2b110 b111 a101 a110 a200 + 2b211 a101 a110 a200 +2b111 (a101 )2 a110 a200 − 4b102 b110 (a110 )2 a200 + 4b202 (a110 )2 a200 + 8b102 a101 (a110 )2 a200 +2b111 (a110 )3 a200 + 4b102 b200 a111 a200 + 2(a101 )3 a111 a200 − 2b110 (a110 )2 a111 a200 +4b111 b200 a120 a200 − 4b301 a120 a200 + 4(b110 )2 a110 a120 a200 − 4b210 a110 a120 a200 −4b110 a101 a110 a120 a200 + 4(a101 )2 a110 a120 a200 + 4(a110 )3 a120 a200 + 4b110 b200 a121 a200 −4b300 a121 a200 − 4b200 a101 a121 a200 + 2b111 (b120 a200 )2 + 2b110 b121 (a200 )2 − 2b221 (a200 )2 −2b121 a101 (a200 )2 − 4b102 a120 (a200 )2 + 4b120 a120 (a200 )2 − 2a111 a120 (a200 )2 + 6b110 a130 (a200 )2 −6a101 a130 (a200 )2 − 4b102 b200 a110 a201 − 2b200 a110 a111 a201 + 4b202 a200 a201 + 8b102 a101 a200 a201 +2b111 a110 a200 a201 + 4a101 a111 a200 a201 + 4a110 a120 a200 a201 + 4b102 b110 b200 a210 + 4b110 b300 a220 −4b200 b202 a210 − 4b102 b300 a210 − 4b102 b200 a101 a210 − 4b111 b200 a110 a210 + 4b102 (a101 )2 a300 +2b110 b200 a111 a210 − 2b300 a111 a210 − 2b200 a101 a111 a210 − 8b200 a110 a120 a210 − 2b110 b111 a200 a210 +2b211 a200 a210 + 2b111 a101 a200 a210 + 8b102 a110 a200 a210 + 4a110 a111 a200 a210 − 4b110 a120 a200 a210 +4a101 a120 a200 a210 + 2b110 b200 a110 a211 − 2b300 a110 a211 − 2b200 a101 a110 a211 + 2(a101 )2 a200 a211 +2(a110 )2 a200 a211 + 2a200 a201 a211 − 2b200 a210 a211 − 4(b110 )2 b200 a220 + 4b200 b210 a220 −4b400 a220 − 4b200 (a110 )2 a220 − 4b110 a110 a200 a220 + 4a101 a110 a200 a220 + 4a200 a210 a220 −4b200 a200 a221 + 6(b200 )2 a230 − 6(a200 )2 a230 − 4b112 b200 a300 + 4b302 a300 + 4b202 a101 a300 −2b110 b111 a110 a300 + 2b211 a110 a300 + 2b111 a101 a110 a300 + 4b102 (a110 )2 a300 + 2(a101 )2 a111 a300 CENTERS AND THEIR CYCLICITY 21 +2(a110 )2 a111 a300 − 4b110 a110 a120 a300 + 4a101 a110 a120 a300 − 4b200 a121 a300 − 4b121 a200 a300 −12a130 a200 a300 + 4b102 a201 a300 + 2a111 a201 a300 + 2b111 a210 a300 + 4a120 a210 a300 +2a101 a211 a300 + 4a110 a220 a300 + 4b102 a200 a301 + 2a111 a200 a301 − 2b200 a110 a311 + 2a111 a500 +2a101 a200 a311 + 2a300 a311 + 4b110 b200 a320 − 4b300 a320 + 4a110 a200 a320 + 4b102 a500 +2a610 + 4b200 2a400 + 4b102 a101 a400 + 2b111 a110 a400 + 2a101 a111 a400 + 4a110 a120 a400 +2a200 v11 − 4b200 v20 − 2b111 b500 − 4a120 b500 + 4a101 (a110 )2 a111 a200 + 2a211 a400 ). 1 C2 = (−2b102 (b110 )2 b111 − 6(b110 )2 b111 b120 − 2(b110 )3 b121 + 12b102 b103 b200 + 6b111 b112 b200 8 +8b102 b121 b200 + 8b120 b121 b200 + 6b111 b130 b200 + 2b110 b111 b200 2 + 2b102 b111 b210 +4b111 b120 b210 + 4b110 b121 b210 + 2b102 b110 b211 + 4b110 b120 b211 − 2b200 2b211 +4b110 b111 b220 − 2b211 b220 + 2(b110 )2 b221 − 2b210 b221 − 4b112 b301 − 2b111 b302 −2b121 b310 − 2b102 b311 − 2b120 b311 − 2b111 b320 − 2b110 b321 + 6b403 + 2b421 −4b110 b111 b120 a101 − 2(b110 )2 b121 a101 + 2b121 b210 a101 + 2b120 b211 a101 + 4b302 a102 +4b102 a101 a102 + 4b102 (a101 )2 a102 + 4(b102 )2 b110 a110 + 4b110 (b111 )2 a110 + 2a412 +4(b110 )2 b112 a110 + 8b102 b110 b120 a110 − 24b104 b200 a110 − 12b122 b200 a110 − 2b321 a101 −8b102 b102 a110 − 4b120 b102 a110 − 4b112 b210 a110 − 4b111 b211 a110 − 4b110 b212 a110 −4b102 b220 a110 + 4b312 a110 − 4(b102 )2 a101 a110 − 2b110 b111 a102 a110 + 2b111 b220 a101 +2b211 a102 a110 + 2b111 a101 a102 a110 − 6b102 b111 (a110 )2 − 2b111 b120 (a110 )2 − 4b112 b200 a102 +4b102 a102 (a110 )2 + 6b103 b200 a111 + 4b121 b200 a111 + 2(a101 )2 a102 a111 + 2b110 b221 a101 −2b102 b110 a110 a111 + 4b110 b120 a110 a111 + 2b102 a110 a111 − 2b220 a110 a111 + 2b102 a101 a110 a111 +2a102 (a110 )2 a111 − 2b110 a110 (a111 )2 + 2a101 a110 (a111 )2 − 12(b110 )2 b120 a120 +12b130 b200 a120 + 8b120 b210 a120 + 8b110 b220 a120 − 4b320 a120 − 8b110 b120 a101 a120 +4b220 a101 a120 + 4b110 b111 a110 a120 − 2b211 a110 a120 − 4b110 a102 a110 a120 − 4b110 b122 a200 +4a101 a102 a110 a120 − 4b120 (a110 )2 a120 + 2(b110 )2 a111 a120 − 2b210 a111 a120 +6(a110 )2 a111 a120 − 8b110 a110 (a120 )2 + 4b111 b200 a121 − 4b301 a121 − 4b200 a102 a121 +4(b110 )2 a110 a121 − 4b210 a110 a121 − 4b200 a120 a121 + 4b110 b200 a122 − 4b300 a122 −6(b110 )3 a130 + 12b102 b200 a130 + 24b120 b200 a130 + 12b110 b210 a130 − 6b310 a130 −6(b110 )2 a101 a130 + 6b210 a101 a130 + 6b200 a111 a130 + 48b110 b200 a140 − 24b300 a140 +24b200 a101 a140 − 6b103 b111 a200 − 12b102 b112 a200 − 4b112 b120 a200 − 4b111 b121 a200 +24b200 4a200 + 4c22a200 + 24b104 a101 a200 − 8b121 a102 a200 + 12b102 a103 a200 +6a103 a111 a200 − 12b103 a120 a200 − 8b121 a120 a200 − 8b102 a121 a200 − 4b120 a121 a200 +2a111 a121 a200 + 12a110 a122 a200 − 6b111 a130 a200 − 24a102 a130 a200 − 12a120 a130 a200 +24a110 a140 a200 + 2b111 b120 a201 + 2b110 b121 a201 − 2b221 a201 + 4b102 a102 a201 +2a102 a111 a201 + 4b120 a120 a201 + 6b110 a130 a201 + 4b200 2a202 + 4b102 a101 a202 +2b111 a110 a202 + 2a101 a111 a202 + 4a110 a120 a202 − 4(b102 )2 a210 − 2(b111 )2 a210 22 R. BENTERKI AND J. LLIBRE −4b110 b112 a210 − 4b102 b120 a210 + 4b212 a210 + 2b111 a102 a210 + 2b102 a111 a210 − 6b110 a330 −2b120 a111 a210 + 2(a111 )2 a210 − 2b111 a120 a210 + 4a102 a120 a210 + 4(a120 )2 a210 −4b110 a121 a210 + 2a101 a102 a211 + 2b102 a110 a211 − 2b120 a110 a211 + 4a110 a111 a211 −2b110 a120 a211 + 2a202 a211 + 8b110 b120 a220 − 4b220 a220 + 4b120 a101 a220 − 6a101 a330 −2b111 a110 a220 + 4a102 a110 a220 − 2b110 a111 a220 + 8a110 a120 a220 + 2a211 a220 + 2a111 a320 −4b110 a110 a221 + 4a210 a221 − 4b200 a222 + 6(b110 )2 a230 − 6b210 a230 + 6a430 +6b110 a101 a230 − 6a201 a230 − 24b200 a240 + 24b104 a300 + 4b122 a300 − 2b121 a301 −6a130 a301 + 4b102 a302 + 2a111 a302 + 2a102 a311 + 2a120 a311 − 4b120 a320 + 4a110 a321 ). 1 C3 = (−6b103 b112 − 3b112 b121 − b111 b122 − 3b121 b130 − 3b111 b140 + 15b105 − 15b110 a150 24 +3b223 + 15b105 (b110 + a101 ) + 2b122 a102 − 3b121 a103 + 15a250 − 5a111 a140 +6b102 a104 + 6b132 a110 + 3a104 a111 + 4b122 a120 − 6b140 a120 + 3a232 +b104 (5b111 + 20a102 + 22a120 ) − 6b103 a121 − 3b121 a121 − 4b102 a122 −2b120 a122 + a111 a122 + 6a110 a123 − 3b112 a130 − 9b130 a130 − 15a101 a150 −9a103 a130 − 3a121 a130 − 22b102 a140 − 20b120 a140 ). Since the rank of the Jacobian matrix of the function C = (C1 , C2 , C3 ) with respect to the coefficients asij and bsij which appear in their expressions is 3, the three coefficients Ci for i = 1, 2, 3 of the polynomial f6 (r) are linearly independent. Therefore il follows that the polynomial f6 (r) can have at most 2 positive real roots. Consequently, by subsection 2.3 for |ε| > 0 sufficiently small the differential equation (ṙ, θ̇) has at most 2 limit cycles surrounding the origin, going back through the changes of variables we obtain that there are at most 2 limit cycles bifurcating from the center of system (2) when we perturb it as in system (3) if fk = 0 for k = 1, 2, 3, 4, 5 and f6 = 0. So statement (e) of the theorem is proved for the averaged function of order 6. Acknowledgments The second author is partially supported by a FEDER-MINECO grant MTM2016- 77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568. References [1] V. I. Arnold and Y. S. 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Nauk SSSR 301 (1988), 1297–1301; translation in Soviet Math. Dokl. 38 (1989), 198–201. [19] H. Żola̧dek, The classification of reversible cubic systems with center, Topol. Methods Non- linear Anal. 4 (1994), 79–136. [20] H. Żola̧dek, Remarks on: “The classification of reversible cubic systems with center” [Topol. Methods Nonlinear Anal. 4 (1994), 79–136 ; Topol. Methods Nonlinear Anal. 8 (1996), 335– 342. 1 Département de Mathématiques, Université Bachir El Ibrahimi, Bordj Bou Arréridj, Bordj Bou Arréridj 34265, El Anasser, Algeria E-mail address: r benterki@yahoo.fr 2 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat