This is a preprint of: “The center and cyclicity problems for quadratic linear-like reversible sys- tems”, Leonardo Pereira Costa da Cruz, Valery G. Romanovskii, Joan Torregrosa, Nonlinear Anal., vol. 190, 111593: 1–19, 2020. DOI: [10.1016/j.na.2019.111593] THE CENTER AND CYCLICITY PROBLEMS FOR QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS LEONARDO P. C. DA CRUZ, VALERY G. ROMANOVSKI, AND J. TORREGROSA Abstract. In this paper we study a family of quartic linear-like reversible polynomial systems having a nondegenerate center at the origin. This family has degree one with respect to one of the variables. We are interested in systems in this class having two extra nondegenerate centers outside the straight line of symmetry. The geometrical conguration of these centers is aligned or triangular. We solve the center problem in both situations and, in the second case, we study the limit cycles obtained from a simultaneous degenerate Hopf bifurcation in the quartic polynomials class. 1. Introduction Let us consider a planar analytic system of ordinary dierential equations dened in a neighborhood of the origin, (ẋ, ẏ) = (f (x, y), g(x, y)). We are interested in the local struc- ture of the solutions near an equilibrium point of nondegenerate center-focus type located at the origin. That is, a point with the Jacobian matrix having nonzero determinant and null trace. More specically, assuming f (0) = g(0) = 0, if we write the eigenvalues of the Jacobian matrix of (f, g) at 0 as α ± iβ, then α = 0 and β 6= 0. So, after a time rescaling, we can assume that β = 1, and we can write the corresponding system in the canonical form (ẋ, ẏ) = (−y + fn (x, y), x + gn (x, y)), (1) where fn and gn are polynomials of degree n, which do not contain neither constant nor linear terms. The problem of distinguishing whether the singular point at the origin of (1) is a center or a focus is known as the Poincaré center problem, the center-focus problem, or just the center problem. Even this problem was partially solved by Lyapunov, see [20], it has been studied for some xed values of the degree n during more than a century by many authors. The only family completely investigated is the quadratic one. The study of this family was started by Dulac in 1908 in [11], and also performed by Kapteyn some years later, see [18, 19]. Up to the work of Frommer ([14]), the coecient center conditions for existence of a center were not published. The problem was investigated for the quadratic system in the real form and the computations were rather dicult. The correct center conditions were published by Saharnikov ([28]) and later by Sibirski ([31, 32]). The center conditions are simpler and the center-focus problem is easier to solve if the system is written in complex coordinates, see [35]. For the complete cubic family (when in (1) n = 3), the problem remains unsolved. Only particular families has been studied, for example the linear plus cubic homogeneous in [33] or cubic with degenerate innity in [3]. Some recent works with other cubic families are [1, 7, 8, 29, 30]. For higher degree systems, none of the homogeneous nonlinearity cases n = 4 and n = 5 are completely solved, see [2, 13]. In last thirty years, there is a long list list of published papers studying many particular polynomial families (see, for example, the references in [13, 25]). 2010 Mathematics Subject Classication. Primary 34C07, 34C23, 37C27. Key words and phrases. Center-focus problem, linear-like reversible centers, quartic centers, center cyclicity. 1 2 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA In this work we are interested in the limit cycles bifurcating simultaneously from re- versible centers with two extra symmetric centers out of the symmetry line. All considered centers are nondegenerate. The aim is to do a similar study as the one done by Christopher in [6] or by Prohens and Torregrosa in [24]. In this last work, the classication problem for cubic systems having such properties was done. The natural continuation problem is the study of the cyclicity of reversible quartic systems having a center at the origin. These systems can be written as ẋ = −y + a11 xy + a21 x2 y + a03 y 3 + a31 x3 y + a13 xy 3 , (2) ẏ = x + a20 x2 + a02 y 2 + a30 x3 + a12 xy 2 + a40 x4 + a22 x2 y 2 + a04 y 4 . Before starting with the simultaneous bifurcation it is necessary to solve the center-focus problem for (2). This classication problem has been impossible to nish, as similar prob- lems having too many parameters. We remark that the above system has 12 parameters. In order to simplify and convert this problem to some tractable one, we propose a re- stricted family following the ideas in [22]. We consider the quartic linear-like reversible systems, that is, all polynomial systems with degree four, time reversible with respect to the x-axis, and having degree one also in the variable x. This family of systems writes as ẋ = −y + a11 xy + a03 y 3 + a13 xy 3 , (3) ẏ = x + a02 y 2 + a12 xy 2 + a04 y 4 . We recall that the above family is invariant with respect to the change of variables (x, y, t) → (x, −y, −t). In Section 3 we will prove that system (3), having two extra symmetric centers out of the symmetry line, can be written in two dierent canonical forms. The rst corresponds to the case with three aligned centers over x = 0 and the second with three centers in a triangular position. The following two results provide the answer to the center-focus classication problem. Theorem 1.1. The reversible quartic system (3) having three aligned singularities of center-focus type can be written as ẋ = −y − (2c + b)xy + y 3 + by 3 x, 1 (4) ẏ = x − cy 2 − (a2 + 4 c2 + 2)xy 2 + cy 4 , 2 with a > 0 and c ≥ 0 (see Figure 1 right). Moreover, the above system has three aligned centers at (0, 0) and (0, ±1) if and only if one of the following conditions holds: (A1 ) c = 0; (A2 ) b = a2 − 4c2 + 2 = 0. d 1 −1 −d −1 Figure 1. The two dierent congurations of centers of system (3) QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 3 Theorem 1.2. The reversible quartic system (3) having three singularities of center-focus type in a triangular position can be written as (c + d)xy (b + d)y 3 (c + b + d)xy 3 ẋ = −y − + + , d d3 d3 (5) (a2 d + b2 d + bcd + 2 c)y 2 (a2 d + b2 d − 2 c)xy 2 (bd + 2)y 4 ẏ = x + + − , 2cd2 2cd2 2d4 with a > 0, c 6= 0, and d > 0. See Figure 1 left. Moreover, the above system has three centers, one at the origin and two more at (−1, ±d), if and only if one of the following conditions holds: (T1 ) a2 d − 2c = b = 0; (T2 ) bd + 2 = a4 + a2 b2 − a2 c2 − 2a2 cd − a2 d2 + 4a2 + 4b2 + 4bc − 4 = 0; (T3 ) d2 − bd + cd − 4 = 2(a2 + b2 ) + bc − c2 − cd = 0; (T4 ) d − 2 = c + 1 = a2 + b2 − b = 0; (T5 ) cd + 4 = b + c + d = a2 + d2 − 4 = 0; (T6 ) cd + d2 + 2 = b + c + d = a2 − 2c2 + 2d2 + 10 = 0; (T7 ) cd + d2 − 2 = a2 + b2 + bc = 0; (T8 ) d − 2 = b − c = a2 + c2 − c = 0; (T9 ) 2c + 3d = 2b − d = d2 − 4a2 − 4 = 0; (T10 ) b + c + d = a2 d − c2 d − 2cd2 − d3 + 2c + 4d = 0; (T11 ) cd + d2 − 2 = 2b − d = d2 − 4a2 − 4 = 0; (T12 ) cd + d2 − 6 = 3b − c − d = 9a2 − 2c2 + 2d2 − 6 = 0; (T13 ) cd + d2 − 2 = c2 d − bd2 − d3 + 2b + 4d = bc + c2 − d2 + 4 = a2 + b2 − c2 + d2 − 4 = 0; (T14 ) cd − 2 = b + c − d = a2 + d2 − 4 = 0; (T15 ) c2 d2 +2cd3 +d4 −cd−2d2 −2 = b+c+d = c3 d−3cd3 −2d4 +a2 −2c2 −2cd+3d2 +8 = 0; (T16 ) cd − 2 = b = a − c = 0; (T17 ) cd + 2d2 − 8 = 2b − c = a2 + d2 − 4 = 0; (T18 ) a2 + cd + d2 = b + c + d = 0. The paper is structured as follows. In Section 2 we present some preliminary results on the center conditions computation, the Darboux integrability, and the degenerate Hopf bifurcation problem. Section 3 is devoted to deriving the canonical forms (4) and (5). The center classication is done in Section 4. Finally, the simultaneous limit cycles bifurcation in some systems with three centers listed above are studied in Section 5. 2. Preliminaries In this section we recall some classical concepts, necessary to state and prove the results of this paper. In Section 2.1 we provide the denition and computational algorithm to compute the center conditions which we use to study the center-focus problem. In Section 2.2 we explain how the simultaneous degenerate Hopf bifurcations are done for proving the results of Section 5. Finally, some results on the Darboux integrability are given in Section 2.3. 4 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA 2.1. The center conditions. The approach to characterize when system (1) has a center at the origin is based on the well known PoincaréLyapunov Theorem, see [21, 23]. Before stating it, we recall the denition of rst integral. We say that a nonconstant analytical function dened in a neighborhood Ω of the origin, Φ : Ω ⊂ R2 → R2 , is a rst integral of system (1) if it is constant along any solution γ or, equivalently, ∂Φ ∂Φ ẋ + ẏ ≡ 0. (6) ∂x ∂y γ Theorem 2.1. System (1) has a center at the origin if and only if it admits a local analytic rst integral of the form X Φ(x, y) = x2 + y 2 + qk,` xk y ` . (7) k+`≥3 Moreover, the existence of a formal rst integral Φ of the above form implies the existence of a local analytic rst integral of the same form. For a proof of this result we refer to [17, 25]. The necessary conditions for the existence of a rst integral (7) for system (1) are ob- tained looking for a formal series (7) satisfying (6). To start the computational procedure for nding the rst N conditions for integrability, we write down (7) up to order 2N + 2 2N X +2 e y) = x2 + y 2 + Φ(x, qk,` xk y ` . (8) k+`≥3 Then, for each i = 3, . . . , 2N + 2 we equate to zero the coecients of terms of degree i in the expression e ∂Φ e ∂Φ e ∂Φ e ∂Φ ẋ + ẏ = (−y + fn (x, y)) + (−x + gn (x, y)) . ∂x ∂y ∂x ∂y Starting at i = 3, we should solve in a recurrence way each linear system of i + 1 equations with i + 1 variables, qk,` such that k + ` = i. All linear systems corresponding to odd degrees, i = 2j + 1, have a unique solution in terms of previous values of qk,` . As the determinant of the linear system that corresponds to an even degree, i = 2j + 2, vanishes, we need to add an extra condition in order that the linear system has a unique solution. In fact, at this step, we have one equation more than the number of variables. We add suitable equations, for the terms x2j+2 for example, in order that the derivative over the associated vector eld writes as ∞ ∂Φ ∂Φ X ẋ + ẏ = Lj x2j+2 . (9) ∂x ∂y j=1 Then, when at least one Lj is dierent from zero Φ it is a Lyapunov function in a neigh- borhood of the origin. Then, system (1) has no local analytic rst integral and we say that the equilibrium point is a weak focus of order k if the rst nonzero coecient in (9) is Lk . The coecient Lj in (9) is called the j -th Lyapunov quantity. The stability of the origin is given by the sign of the rst nonzero Lj . It is obvious that Lj are polynomials in the parameters of system (1). Clearly, for N big enough, the above algorithm provides a necessary set of conditions, {Lj = 0 : j = 1, . . . , N }, for system (1) be a center. In other words, we can also say that the polynomials Lj represent obstacles for the existence of a rst integral. In particular, system (1) admits a rst integral of the form (7) if and only if Lj = 0, for all j ≥ 1. Thus, the simultaneous QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 5 vanishing of all focus quantities provides conditions which characterize when a system of the form (1) has a center at the origin. The next denition recalls the notion of Bautin ideal and the center variety. Denition 2.2. The ideal dened by the real focus quantities, BR = hL1 , L2 , . . .i ⊂ C[λ], where λ represents all the parameters of system (1), is called the real Bautin ideal. The ane variety VR = V(BR ) is called the real center variety of system (1). By the Hilbert Basis Theorem there exists a positive integer j such that B R = BjR = hL1 , . . . , Lj i. The main diculty is that there is no technique to get j a priori. Notice that the inclusion VR = V(B R ) ⊂ V(BjR ) holds for any j ≥ 1. The opposite inclusion, for a xed j , is veried nding the irreducible decomposition of V(BjR ), see [25], such that any point of each component of the decomposition corresponds to a system having a center at the origin. To nd the irreducible decomposition of V(BjR ), we perform the computations with the Computer Algebra System SINGULAR ([9, 10]). In Section 4 we explain this procedure in more detail. 2.2. Degenerate Hopf bifurcation. Roughly speaking, we can say that the cyclicity of an equilibrium point is the maximum number of isolated periodic orbits bifurcating from it. We are interested only in the limit cycles bifurcating from nondegenerate monodromic points, where the return map is well dened and analytic. Moreover, both unperturbed systems and perturbations are polynomials of degree four in the variables x, y . With the notation introduced above, the classical bifurcation known as the Hopf bifur- cation is the emergence of a limit cycle from a weak-focus of rst order varying the trace of the matrix of the linearized system from zero to a small enough but nonzero value. We will denote by L0 the trace of the perturbed system, clearly L0 = 0 for the unperturbed one. More concretely, the origin of the unperturbed system is stable (resp. unstable) when L1 < 0 (resp. L1 > 0). Then, the perturbed system, when L0 is a small enough positive (resp. negative) real number, a small stable (resp. unstable) limit cycle bifurcates from the origin. This is because the monodromic property remains but the local stability of the equilibrium point changes from stable to unstable. The degenerate Hopf bifurcation is the natural generalization of this bifurcation phenomenon when k small limit cycles appear from a weak-focus of order k . In general, the complete unfolding of k limit cycles near a weak focus of order k is only guarantied when the perturbation is analytic, see for example [27]. When the perturbation is restricted to be polynomial of some xed degree this property is not automatic. This is the case in our problem. Our unperturbed systems (3) are of degree four and we are perturbing them in the full quartic polynomial systems. This is the main reason why the problem of nding the cyclicity of a center, for systems like (3), is so dicult. A way to avoid this diculties is to study lower bounds for the cyclicity. This is the aim of the next result due to Christopher ([6]) that, as a direct application of the Implicit Function Theorem, provides necessary conditions to get lower bounds for the cyclicity of a center. In fact, there are similar previous results due to Chicone and Jacobs ([4, 5]). Also Han ([16]) applies them for Liénard families. Theorem 2.3. Suppose that c is a point on the center variety in the parameter space and that the rst L1 , . . . , Lk Lyapunov quantities have independent linear parts (with respect to the expansion of them with respect all perturbation parameters), then c lies on a component of the center variety of codimension at least k + 1, adding the trace parameter L0 , and there are bifurcations which produce k limit cycles locally from the center corresponding to the parameter value c. 6 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA In particular, the above result shows the existence of a curve of weak-foci of order k that unfold k hyperbolic limit cycles. The existence of this curve is obtained studying the Taylor developments of the varieties Lj = 0, for j = 0, . . . , k, that intersect transversally along it but Lk is nonvanishing. Finally, in [6] there are results about the existence of such transversal curves studying the homogeneous higher order terms when the previous ones vanish identically. They will be used in the future work to improve the results of this paper. In this work we are interested in the simultaneous bifurcation of limit cycles in the three center conguration that a system of type (3) has. We will explain how we use Theorem 2.3 for this purpose in Section 5. 2.3. Darboux integrability. More detailed versions of the next results can be found in [12, 25]. Here we only remind the statement on Darboux integrability that we will need for proving our classication results, together with some basic concepts and denitions. Let (ẋ, ẏ) = (P (x, y), Q(x, y)) be a C-polynomial dierential system of degree m. We say that an algebraic curve f = 0 is invariant if the vector eld associated to the polyno- mial dierential system is tangent along it. That is, ∂f ∂f P (x, y) + Q(x, y) = K(x, y)f (x, y). ∂x ∂y The polynomial K is known as the corresponding cofactor. Equivalently, exp(g/h) is called an exponential factor, with the associated cofactor K e , if it satises ∂ eg/h ∂ eg/h P (x, y) + Q(x, y) = K e (x, y) eg/h . ∂x ∂y Theorem 2.4. Suppose that a C-polynomial dierential system of degree m admits p irreducible invariant algebraic curves fi = 0 with cofactors Ki for i = 1, . . . , p and q ex- ponential factors exp(gj /hj ) with cofactorsPKje for j = 1, P.q. . , q. Then, there exist complex numbers λi and µj not all zero such that i=1 λi Ki + j=1 µj Kj = 0, if and only if the p e (multivalued) function   µ1   µq g1 gq f1λ1 · · · fpλp exp · · · exp h1 hq is a rst integral of the C-polynomial dierential system of degree m. 3. Canonical form As we have already mentioned, system (3) is time-reversible with respect to the x-axis, and it has a nondegenerate equilibrium point of center-focus type at the origin. Clearly, the reversibility condition and the linear part ensures that the origin will be always a center. As we have explained in the introduction, the aim of this paper is the study of system (3) having two extra nondegenerate centers out of the symmetry line. Then, as they can be located at (x0 , ±y0 ), there are two dierent possibilities, x0 = 0 and x0 6= 0. In the rest of the section we will prove the rst part of the statements of Theorems 1.1 and 1.2. In the rst case, x0 = 0, we can locate the equilibrium points at (0, ±1) after a rescaling. Then, imposing that the determinants of the Jacobian matrix at these points writes as a positive number, a2 , we get system (4). We notice that by the symmetry both determinants coincide. We can also assume the condition c ≥ 0 doing, if necessary, the QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 7 change (x, y) → (−x, −y). Hence, with the new variables (u, v) = (x, −1 + y), there exists a rst integral that writes H(u, v) = (au)2 + 4(cu − v)2 + · · · . (10) In the second case, x0 6= 0, we can locate the equilibrium points at (−1, ±d) after a rescaling, if necessary. The symmetry provides the condition d > 0. Assuming that the system has an equilibrium point with zero trace we get d4 a13 + d2 (a11 + 2a12 ) + 4 a02 = , 2d2 d2 a13 + a11 + 1 a03 = , d2 d4 a13 + d2 a11 + 2 a04 = − . 2d4 Then, adding the condition that it has a positive determinant, a2 , we obtain d6 a213 + 2d4 a11 a13 + d2 a211 + a2 + 2a11 + 2 a12 = − . 2d2 (a11 + 1) Clearly, it is not restrictive if we assume a > 0. The above denominator never vanishes because, when a11 = −1, the determinant, a2 = −d2 (a13 d2 − 1)2 , would be negative. Finally, we introduce two new parameters (b, c) performing the linear change c+d c+b+d a11 = − , a13 = . d d3 Hence, with the new variables (u, v) = (1 + x, −d + y), there exists a rst integral that writes H(u, v) = (adu)2 + (−bdu + 2cv)2 + · · · . (11) This expression can be easily transformed to (7). We remark that the condition a11 +1 6= 0 is equivalent to c 6= 0. From all the above transformations the system (3) changes to (5). We notice that, although the last parameter change does not seem necessary, it will help to obtain simpler expressions for the center conditions in the next section. 4. Centers classification This section is devoted to proving the center classication statements of Theorems 1.1 and 1.2. Because the proof of the second result is quite long we have written it separately in Propositions 4.1 and 4.2. The obtained rst integrals are globally dened and their level curves provide, among may be others, the three period annuli. 4.1. Proving the center classication in the aligned case. The reversibility prop- erty and the canonical form (4) ensures that the origin is a center. Then, by the sym- metry, we only need to study the equilibrium point (0, 1). First we check that the two families of the statement satisfy the center conditions, and second we compute the ex- plicit expressions of the rst integrals, which are of Darboux type, H1 = f1λ1 f2λ2 (eg )µ and H2 = f1λ1 (eg )µ . In both cases µ = 1. We will denote by Kfi , i = 1, 2, and Kg the corresponding cofactors. The change given by expression (10) allows us to use the algorithm described in Sec- tion 2.1 to compute the rst center conditions, that is, the rst Lyapunov quantities. Although they are polynomials in the parameters, the change (10) introduces some de- nominators. Straightforward computations show that the denominators of L2 and L3 before the usual simplications are a4 (a2 + 4c2 )2 and a8 (a2 + 4c2 )5 , respectively, which 8 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA never vanish because a > 0. Hence, we can consider only the numerators, modulo positive constants, L1 =c(a2 b2 + 2a2 bc + 4b2 c2 + 8bc3 − 2a2 + 8c2 − 4), L2 = − c(a2 + 4c2 + 2)(a2 + 8bc − 4c2 + 2), L3 =c5 (a2 + 4c2 + 2)(a2 − 4c2 + 2)(6220800a2 c14 + 24883200c16 + 8121600a2 c12 + 65664000c14 + 3124800a2 c10 + 51206400c12 + 290800a2 c8 + 13476800c10 − 11200a2 c6 + 620800c8 + 600a2 c4 − 36000c6 − 52a2 c2 + 1840c4 − a2 + 60c2 ). These values for Li are dened when the previous are zero. Moreover, L4 , L5 also vanish if L1 = L2 = L3 = 0. The necessary part of the statement follows easily checking that the real solutions of {L1 = L2 = L3 = 0} are the families A1 and A2 . For the sucient conditions we only list the functions to get the Darboux rst integrals for both families. ◦ Family A1 satises c = 0, then f1 = −2 + (a2 + 2)y 2 , f2 = bx + 1, g = −b(a2 + 2)((a2 + 2)x + y 2 b), Kf1 = −(a2 + 2)xy, Kf2 = b(y 2 − 1)y, Kg = b(a2 + 2)(a2 bx − (a2 + 2)y 2 + a2 + 2)y, with λ1 = a2 b2 and λ2 = (a2 + 2)2 . √ ◦ Family A2 satises b = 0 and c = a2 + 2, then f1 = −1 − 2σx + 2(a2 + 2)y 2 + 2σ(a2 + 2)xy 2 − (a2 + 2)y 4 , g = −2σx, Kf1 = −2σy(1 + σx − y 2 ), Kg = 2σy(1 + σx − y 2 ), √ with σ = a2 + 2 and λ1 = 1. This, together with the canonical forms derived in Section 3, nishes the proof of Theorem 1.1. 4.2.Proving the center classication in the triangular case. Proposition 4.1. If the equilibrium point (−1, ±d) of the quartic system (5) is a center then the parameters (a, b, c, d) satisfy one of the conditions given in the statement of Theorem 1.2. Proof. As the system (5) is reversible with respect to the x-axis, the center conditions for the point (−1, −d) are the same as for (−1, d). Thus, we only study this point. From the canonical form computations done in Section 3, (−1, d) is a nondegenerate equilibrium point of center-focus type, that is, the trace and the determinant of the Jacobian matrix are zero and positive, respectively. The proof follows computing a few rst Lyapunov quantities and then solving the obtained system to check that all the families are the ones described in the statement. First, we translate the point to the origin, we do an ane change of coordinates, see (11), and rescaling time. Then, we follow the approach described in Section 2.1 for QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 9 the computation of the center conditions Li in (9). From the ane change, they are rational functions with denominators of the form ck d` . As usual, the four parameters, (a, b, c, d), appearing in system (5) indicate that only the rst four Lyapunov quantities will be necessary to be computed, but as we will see, we use the rst ve. Instead of the complete expression, for simplicity, we denote by Li only the numerators of the center conditions, that are polynomials with integer coecients in (a, b, c, d). Because of the size of them, we only show the rst one, L1 =(cd2 + d3 − 4d)b4 + (−8 + d4 + 2cd3 + (c2 − 2)d2 − 4cd)b3 + (a2 cd2 + a2 d3 − 2a2 d − 12c)b2 + (a2 d4 + 2a2 cd3 + a2 (c2 − 2)d2 − 8a2 − 4c2 )b + 2a4 d − 4a2 c. The other four numerators are polynomials of degrees 19, 33, 49, and 67 with 317,1524, 4835, and 12152 monomials, respectively. For the second step, we need to solve the algebraic system of equations S = {L1 = L2 = L3 = L4 = L5 = 0}. Although this system has only four variables and ve equations the usual mechanisms for solving it fails. Instead of nding the solutions directly, we will use the GianniTragerZacharias algorithm, see [15], to determine the irreducible components of the variety V = V(L1 , L2 , L3 , L4 , L5 ). The main function used is minAssGTZ, it is implemented in the library primdec.lib included on the algebraic computational system SINGULAR, see [9, 10]. However using the function directly we were not able to compute the decomposition neither over the eld of rational numbers nor in the nite eld Zp , with p = 32003. Hence, the irreducible components will be computed considering some auxiliary polynomials obtained from some crossed resultants with respect to one of the parameters, because the new polynomials have less variables (but higher degree) and contain the solutions that we are interested. Thus, we compute the four resultants with respect to the parameter c, Res(L1 , L2 , c), Res(L1 , L3 , c), Res(L1 , L4 , c), and Res(L1 , L5 , c). We denote by Res12 , Res13 , Res14 , and Res15 , the corresponding factorized expressions, leaving only one term, when they have multiplicity bigger than one, and removing the nonvanishing terms, that is the powers of a, d, and a2 + b2 . Then, we get Res12 = R · R12 , Res13 = R · R13 , Res14 = R · R14 , Res15 = R · R15 , with the common factor R = R1 R2 · · · R8 , where R1 = b, R2 = bd + 2, R3 = a2 d2 + b2 d2 − 2bd + 2d2 − 8, R4 = a2 d + b2 d − bd2 + 2b, R5 = a2 d − b2 d − 2b + 2d, R6 = a2 d2 + b2 d2 − 4, R7 = a2 − bd, R8 = a2 d2 + (bd + 2)2 + 2d2 . We have not written, because of their size, the polynomials R12 , R13 , R14 , R15 . They have degrees 6, 32, 62, 92 in (a, b, d) and 9, 293, 1642, and 4803 monomials, respectively. With the above discussion, and taking into account that the surface R8 = 0 has no real points because a, d > 0, the solution of S can be studied solving the eight algebraic systems S0 = {L1 = L2 = L3 = L4 = L5 = R12 = R13 = R14 = R15 = 0} and Si = {L1 = L2 = L3 = L4 = Ri = 0} for i = 1, . . . , 7. 10 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA System S0 is studied computing the two by two crossed resultants, rst with respect to d, then with respect to b. In each step we remove, as previously, all the multiple factors and also the powers of the nonvanishing variables or factors (d a, a2 + b2 , a2 + 2). Finally, the last two polynomials have only one common factor a2 − 1. Next, as a > 0, we consider the system {R12 = R√ 13 = R14√ = R15 = a − 1 = 0}, that has, since d > 0, only one solution √ {a = 1, √ b = 1/√ 3, d = 3}. Finally,√S0 has only√two real √ solutions {a = 1, b = 1/ 3, c = 2/ 3, d = 3} and {a = 1, b = 1/ 3, c = −4/ 3, d = 3}. They, satisfy the conditions of cases T17 and T5 of the statement, respectively. Although some systems can be easily solved, as S1 that writes equivalently as the rst family, {b = a2 d − 2c = 0}, most of them need an accurate procedure. We will follow step by step the algorithm described in [26]. The 18 families in the statement are obtained solving systems Si , for i = 1, . . . , 7. For each i, except for i = 4, we use the mentioned GianniTragerZacharias algorithm and we apply the routine minAssGTZ to the ideal Bi = hRi , L1 , L2 , L3 , L4 i, for simplicity we have not used L5 . That is, we obtain the necessary conditions to have a center nding the irreducible decomposition of the variety of each ideal Bi . We illustrate the procedure only for one family, S7 , the other follow similarly. Working in Zp [a, b, c, d], with p = 32003, instead of Q[a, b, c, d], the minimal associate prime ideals of B7 , provided by SINGULAR, are ha, bi, ha, d, b − 16001ci, hcd + 2d2 − 8, b + 16001c, a2 + d2 − 4i, hcd − d2 − 2, b + c, a2 + d2 + 2i, (12) hd3 + c − 2d, b + d, a2 + d2 i, ha2 + cd + d2 , b + c + di. Next, we use the rational reconstruction algorithm provided by [34] to get a candidate to be the minimal associate prime ideal but with rational coecients. In fact, we should apply the next function to each coecient of the above polynomials. Rational_Reconstruction(x,p) u=[1,0,p] v=[0,1,x] while sqrt(p/2)<= v[3] do { q=floor(u[3]/v[3]) r=u-qv u=v v=r } if abs(v[2])>= sqrt(p/2) then error() return(v[3]/v[2]) We recall that the oor function, bxc, gives the greatest integer less than or equal to x. Given an integer x and a prime p, the rational reconstruction function dened above, pro- vides a rational number y such thatpy = x (mod p) and, in absolute value, the numerator and denominator of y are less than p/2. In the primary decomposition ideals of B7 given in (12) all the rational reconstructed values are the same integers except 16001, which is −1/2. Consequently, the candidate QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 11 to be the minimal associate prime ideals of B7 are Pe1 = ha, bi, Pe2 = ha, d, b + (1/2)ci, Pe3 = hcd + 2d2 − 8, b − (1/2)c, a2 + d2 − 4i, Pe4 = hcd − d2 − 2, b + c, a2 + d2 + 2i, Pe5 = hd3 + c − 2d, b + d, a2 + d2 i, Pe6 = ha2 + cd + d2 , b + c + di. √ √ T The next √step is to show that P7 = B7 , where P7 = 6k=1 Pek in Q[a, b, c, d]. We have denoted by P the radical of the ideal P . In general, it is simpler to verify the double inclusion instead of computing the radicals. Adding a new articial parameter w, this property can be seen checking that {1} is the Gröbner basis of the next list of ideals, h1 − wLk , P7 i, for k = 1, . . . , 4 and h1 − wp, B7 i, for every p ∈ P7 . In the last step we check which Pek , for k = 1, . . . , 6, will appear in the statement. As a, d, a2 + d2 , a2 + d2 + 2 are nonvanishing, only Pe3 and Pe6 are necessary conditions for system (5) be a center. In fact, they are cases T17 and T18 , respectively. Finally, we study the family S4 = {R4 , L1 , L2 , L3 , L4 } with another procedure. Because, for √ the rational √ reconstructed candidate to be the primary decomposition ideals we have P4 6= B4 . As the condition b = 0 has been studied before, here we will assume that b 6= 0. We start again computing the four resultants but with respect to the parameter a, Res(R4 , L1 , a), Res(R4 , L2 , a), Res(R4 , L3 , a), Res(R4 , L4 , a). d 1 , Res We denote by Res d 2 , Res d 3 , and Res d 4 , the corresponding factorized expressions. But leaving only one term, when they have multiplicity bigger than one, and removing the nonvanishing terms, that are powers of b, d. Then, we get d1 = R Res b·R b 1, d2 = R Res b·R b 2, d3 = R Res b·R b 3, d4 = R Res b·R b 4, with the common factor R b = cd + d2 − 2. Hence, the solution {L1 = L2 = L3 = L4 = Rb4 = R b = 0} is case (7). As we have reduced the set of variables to (b, c, d), we can follow as before solving the system {R b1 = R b2 = R b3 = R b 4 = 0} by computing the crossed resultants Res(R b 1, R b 2 , b), Res(R b 1, R b 3 , b), and Res(R b 1, R b 4 , b). Now,leaving also only one term, when they have mul- tiplicity bigger than one, and removing the nonvanishing terms, that are powers of c, d, d2 + 2, d + 2, and cd + d2 − 2, we get b 12 = R R e·R e 12 , b 13 = R R e·R e 13 , b 14 = R R e·R e 14 , where R e = (d−2)(d2 −2)(2c+3d)(cd+2d2 −4)(d2 −6). As R e 12 = d2 +6 6= 0, we have to check that, for each factor D in R,e the solution of SD = {L1 = L2 = L3 = L4 = R4 = D = 0} is in one of the cases listed in the statement. For example, for d = 2, we obtain the two solutions {a2 + b2 − b = c + 1 = d − 2 = 0} and {a2 + b2 − b = c − b = d − 2 = 0}. The rst is (4) and the second is (8). The other factors are found similarly.  Proposition 4.2. For each family in the statement of Theorem 1.2, the quartic sys- tem (5), which is reversible with respect to the x-axis, has a center at the origin and two centers at the points (−1, ±d). 12 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA Proof. The origin of (5) is a nondegenerate center-focus point then, by the symmetry property, it is a center. The points (−1, ±d) are also nondegenerate equilibria of center- focus type. The proof follows straightforward doing a case by case study. We compute the rst integrals, which are of Darboux type, and check that all of them are well dened in a neighborhood of such equilibrium points. In particular, there are only four types of rst integrals: H1 = f1 , H2 = f1λ1 f2λ2 , H3 = f1λ1 (eg )µ , and H4 = f1λ1 f2λ2 (eg )µ . In the rest of the proof we only show the explicit expressions of the polynomials f1 , f2 , g and the real numbers λ1 , λ2 , because µ = 1. We also list the associated cofactors Kf1 , Kf2 , and Kg , when their expressions are not so big. We recall that f1 = 0 and f2 = 0 are invariant algebraic curves and eg is an exponential factor. The cases are listed grouped by the expression of the corresponding rst integral. ◦ Cases corresponding to the Hamiltonian systems with the rst integral H1 = f1 : (T3 ) In this case, we have a2 = −(d2 c2 + 2d(d2 − 5)c + (d2 − 4) √ ) and b = (cd + d − 4)/d. 2 2 Then, √ as a should be nonnegative, 2 √ we have that (−d + 5 − −2d + 9)/d < c < (−d + 2 2 2 5 + −2d2 + 9)/d, and 0 < d < 3 2/2. We remark that system (5) depends quadratically in a. With these conditions, we get f1 = 2d4 x2 + 2d4 y 2 + 2d3 (c + d)xy 2 − (cd + 2d2 − 4)y 4 − 2(cd + d2 − 2)xy 4 . (T8 ) Here a2 = c − c2 , b = c, and d = 2. Hence, when 0 < c < 1, system (5) is well dened. Then, f1 = 16x2 + 16y 2 + 8(c + 2)xy 2 − (c + 2)y 4 − 2(c + 1)xy 4 . (T12 ) In this case, we have a2 = 2(4 − d2 )/d2 , b = 2/d and c = (6 − d2 )/d. Thus, when 0 < d < 2, system (5) is well dened and we get f1 = 2d4 x2 + 2d4 y 2 + 12d2 xy 2 − (d2 + 2)y 4 − 8xy 4 . (T14 ) Here, we have a2 = 4 − d2 , b = (d2 − 2)/d and c = 2/d. So, the condition 0 < d < 2 is necessary and we have f1 = d4 x2 + d4 y 2 + (d2 + 2)d2 xy 2 − (d2 − 1)y 4 − d2 xy 4 . (T17 ) In this case, we have a2 = 4−d2 , b = (4−d2 )/d, and c = 2(4−d2 )/d. Here, 0 < d < 2 and f1 = d4 x2 + d4 y 2 − (d2 − 8)d2 xy 2 − 2y 4 + (d2 − 6)xy 4 . ◦ Cases corresponding to H2 = f1λ1 f2λ2 : (T1 ) We can write b = 0 and c = a2 d/2, for obtaining f1 = 2d6 − 8d4 x − (2a2 d2 + 4d2 + 16)d2 y 2 + (a2 d2 + 2d2 + 8)y 4 , f2 = 2 + (a2 + 2)x, 4 Kf1 = 4 (d2 − y 2 )y, d (a2 + 2) 2 K f2 = − (d − y 2 )y, 2d2 with λ1 = d2 (a2 + 2), λ2 = 8. QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 13 (T16 ) Here we have a = 2/d, b = 0, and c = 2/d. Then, we get λ1 = 1, λ2 = 4/(d2 + 2), and f1 = d6 − 4d4 x − 2(d2 + 6)d2 y 2 + (d2 + 6)y 4 , f2 = d2 + (d2 + 2)x, 4(d2 − y 2 )y K f1 = , d4 (d2 + 2)(d2 − y 2 )y K f2 =− . d4 ◦ Cases corresponding to H3 = f1λ1 (eg )µ , with µ = 1 : (T2 ) In this case we have b = −2/d and −a2 d2 c2 − 2 a2 cd4 − 8 cd2 + a4 d2 − a2 d4 + 4 a2 d2 + 4 a2 − 4 d2 + 16 = 0. As a > 0, the discriminant σ = (a2 + 2)2 (a2 d2 + 4) is positive for all a, d. Adding an articial variable, e, the curve σ − e2 = (a2 + 2)2 a2 d2 − e2 + (2a2 + 4)2 = 0 is a hyperbola with respect to the variables (d, e). Then, we can nd a rational parameterization 4(a2 + 2)2 − t2 d= 2at(a2 + 2) such that σ is a perfect square. Hence, we can isolate b, c in terms of a, t: (t2 + 2(a2 + 2)2 t + 4(a2 + 2)2 )(t2 + 4(a2 + 2)2 ) c(a, t) = − , 2at(4(a2 + 2)2 − t2 )(a2 + 2) 4at(a2 + 2) b(a, t) = 2 . t − 4(a2 + 2)2 Then, we get λ1 = −1 and f1 =4(a2 + 2)5 t3 (2a2 + t + 4)2 a2 xy 2 − 2(a2 + 2)3 a2 t2 (t4 − 8(a2 + 2)2 (a2 + 1)t2 − 16(a2 + 2)4 )y 2 − 2(a2 + 2)2 t(4(a2 + 2)2 − t2 )2 (t2 + 2(a2 + 2)2 t + 4(a2 + 2)2 )x + (2a2 − t + 4)4 (t2 + 2(a2 + 2)2 t + 4(a2 + 2)2 )2 , 4(a2 + 2)3 t2 y 2 2(a2 + 2)2 (2a2 + t + 4)2 tx g= − . (2a2 − t + 4)4 (2a2 − t + 4)2 (t2 + 2(a2 + 2)2 t + 4(a2 + 2)2 ) We remark that without the parameterization the expressions for f1 and g are more dicult to be found. (T4 ) Here a2 = b(1 − b), c = −1 and d = 2. So, system (5) is well dened only when 0 < b < 1. Moreover, λ1 = 4b and f1 = −4 + (b + 1)y 2 , g = −(b + 1)((2b + 2)x2 + (b + 2)y 2 + (b + 1)xy 2 ), (b + 1)(y 2 + 4x)y K f1 = − , 8 b(b + 1)(y 2 + 4x)y Kg = . 2 14 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA (T6 ) For this case we have a2 = 2(4 − d2 )/d2 , b = 2/d, and c = −(d2 + 2)/d. So, when 0 < d < 2 system (5) is well dened. Additionally, f1 = d4 (2 + d2 )2 − 8d4 (d2 + 2)x − 32d4 y 2 + 64d2 xy 2 + 16(d2 + 2)y 4 , 8x g= 2 , d +2 8(d4 − 2d2 x − (d2 + 2)y 2 )y K f1 = , (d2 + 2)d4 8(d4 − 2d2 x − (d2 + 2)y 2 )y Kg = − . (d2 + 2)d4 and λ1 = 1. (T7 ) Here, a2 = b(d2 − bd − 2)/d and c = −(d√ − 2)/d. There √ 2 are two cases such that system (5) is √ d < (b − b + 8)/2 or (b + b + 8)/2 < d, and (ii) √ well dened: (i) b > 0, 2 2 b < 0, (b − b2 + 8)/2 < d < (b + b2 + 8)/2. Independently, we have f1 = −2d2 + (bd + 2)y 2 , g = −(bd + 2)((bd3 + 2d2 )x2 + (2bd + 2d2 )y 2 + (2bd + 4)xy 2 ), (bd + 2)(y 2 + d2 x)y K f1 = − , d4 2b(d2 − 2)(bd + 2)(y 2 + d2 x)y Kg = , d with λ1 = 2bd3 (d2 − 2). (T9 ) In this case, a2 = (d2 − 4)/4, b = d/2, and c = −3d/2. Hence, when d > 2 system (5) is well dened. Here we get f1 = 18d6 − (12d6 + 48d4 )x − (4d6 + 32d4 + 64d2 )y 2 + (2d6 + 16d4 + 32d2 )xy 2 + (3d4 + 24d2 + 48)y 4 , g1 = 2x(d2 + 4)/3d2 , (d2 + 4)(3y 3 + d2 x − 2d2 )y K f1 = , d4 (d2 + 4)(3y 3 + d2 x − 2d2 )y Kg = − , d4 with λ1 = 1. (T10 ) The discriminant with respect to c of the second condition writes as σ = 4(a2 d2 + 2d2 + 1). Similarly, as for family T2 , adding an articial variable, e, from the rational parameterization of the hyperbola σ − e2 = 0, we know that −2d2 + t2 − 1 a= . 2dt Then, the second condition allow us to write 2d2 + t2 − 2t + 1 b= , 2dt 2d2 t + 2d2 + t2 − 2t + 1 c=− . 2dt QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 15 In fact, the other solution is equivalent to changing t by −t, moreover system (5) depends on a2 . Hence, f1 = 2d4 t(2d2 (t + 1) + (t − 1)2 )2 − 4d4 t(2d2 + (t + 1)2 )(2d2 (t + 1) + (t − 1)2 )x − 4d4 t(2d2 + (t + 1)2 )2 y 2 + 2d2 (2d2 + (t − 1)2 )(2d2 + (t + 1)2 )2 xy 2 + (2d2 (t + 1) + (t − 1)2 )(2d2 + (t + 1)2 )2 y 4 , 2(2d2 + (t + 1)2 )x g= 2 , 2d (t + 1) + (t − 1)2 (2d2 + (t + 1)2 )(2d4 t − d2 (2d2 + (t − 1)2 )x − (2d2 (ty 2 + 1) + (t − 1)2 )y 2 )y K f1 = , td4 (2d2 (t + 1) + (t − 1)2 ) (2d2 + (t + 1)2 )(2d4 t − d2 (2d2 + (t − 1)2 )x − (2d2 (ty 2 + 1) + (t − 1)2 )y 2 )y Kg = − . td4 (2d2 (t + 1) + (t − 1)2 ) with λ1 = 1. √ √ (T√11 ) In this case, as d > 0, we have b = a2 + 1, c = −(2a2 + 1)/ a2 + 1, and d = 2 a2 + 1. Then, f1 = −4(1 + a2 ) + (a2 + 2)y 2 , g = (a2 + 2)((2a4 + 6a2 + 4)x2 + (a2 + 2)xy 2 + (3a2 + 3)y 2 ), (a2 + 2)((4a2 + 4)x + y 2 )y Kf 1 = − , 8(a2 + 1)2 (2a2 + 1)(a2 + 2)((4a2 + 4)x + y 2 )y Kg = − , 2 with λ1 = −4(2a2 + 1)(a2 + 1)2 . (T13 ) Here we have b = 4/d(d2 − 2), c = −(d2 − 2)/d, and (d2 − 2)2 a2 − 4(d2 − 4) = 0. Last condition can be written equivalently as the rational parameterization   8(2t − 1)(2t + 1)t 2(4t2 + 1) (a, d) = ,− . 16t4 + 24t2 + 1 (2t − 1)(2t + 1) This expression simplies the writing of f1 = −32t4 − 48t2 − 2 + (16t4 − 8t2 + 1)y 2 , g = −(512t8 + 512t6 + 192t4 + 32t2 + 2)x2 − (384t8 + 384t6 + 272t4 + 24t2 + 3/2)y 2 − (256t8 − 32t4 + 1)xy 2 , (4t2 − 1)2 ((64t4 + 32t2 + 4)xy + (16t4 − 8t2 + 1)y 3 ) K f1 = − , 4(16t4 + 24t2 + 1)(4t2 + 1)2 (4t2 − 1)2 (16t4 + 24t2 + 1)((64t4 + 32t2 + 4)xy + (16t4 − 8t2 + 1)y 3 ) Kg = , 4(4t2 + 1)2 with λ1 = (16t4 + 24t2 + 1)2 . (T15 ) In this case, we have two solutions but, as a2 > 0, only one corresponds to real values: √ √ √ 2 −2d2 + 4d2 + 9 + 3 1 − 4d2 + 9 −2d2 + 1 − 4d2 + 9 a = , b=− , c= . 2d2 2d 2d 16 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA Then, f1 = 2d6 − (σ + 3)d4 + 8(σ − 2)d2 − 12σ + 36 − 2d2 (σd2 − d2 − 2σ + 6)x − 4d4 y 2 + 2d2 (σ − 1)xy 2 + (2d2 + σ − 1)y 4 , ((σ + 1)d2 − 2σ − 6)x g= , d4 − 2d2 − 2 (d4 (σd2 + d2 − 2σ − 6) + 2d2 (−d4 + σ + 3)x − (d4 − 2d2 − 2)(σ + 3)y 2 )y K f1 = , (d4 − 2d2 − 2)d4 (d4 (σd2 + d2 − 2σ − 6) + 2d2 (−d4 + σ + 3)x − (d4 − 2d2 − 2)(σ + 3)y 2 )y Kg = − , (d4 − 2d2 − 2)d4 √ with σ = 4d2 + 9 and λ1 = 1. ◦ Cases corresponding to H4 = f1λ1 f2λ2 (eg )µ , with µ = 1 : (T5 ) Here, the parameters satisfy a2 = 4 − d2 , b = (4 − d2 )/d, and c = −4/d. Thus, the condition 0 < d < 2 is necessary in order that system (5) is well dened. Then, f1 = d4 + d2 (d2 − 6)x + (d2 − 6)y 2 , f2 = 2d2 + (d2 − 6)y 2 , g = (d2 − 6)x, (d2 − 6)(d2 x + y 2 )y K f1 = , d4 (d2 − 6)(d2 − y 2 )y K f2 = − , d4 (d2 − 6)(d4 + (d2 − 4)d2 x − 4y 2 ) Kg = − , d4 with λ1 = −d2 , λ2 = d2 − 4. (T18 ) In this case, we have b = a2 /d and c = −(a2 + d2 )/d. Then, f1 = −d4 + d2 (a2 + 2)x + (a2 + 2)y 2 , f2 = −2d2 + (a2 + 2)y 2 , g = (a2 + 2)x, (a2 + 2)(d2 − y 2 )y K f1 = , d4 (a2 + 2)(d2 x + y 2 )y K f2 = − , d4 (a2 + 2)(−d4 + a2 d2 x + (a2 + d2 )y 2 )y Kg = , d4 with λ1 = d2 , λ2 = a2 .  5. Simultaneous Cyclicity This section is devoted to studying the simultaneous degenerate Hopf bifurcation of system (5) for dierent values of the parameters (a, b, c, d). Before stating and proving the main result of this section, we recall in few words the bifurcation scheme presented in [6] for rst order perturbations and degree four systems. The unperturbed system (5) has three centers located in the vertex of a triangle. As two of them are symmetric with respect to the x-axis, we study only two simultaneous degenerated Hopf bifurcations. The bifurcation will be done in two steps. First, we consider a perturbation which is symmetric QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 17 with respect to the x-axis to get n symmetric hyperbolic small limit cycles surrounding the symmetric centers using Theorem 2.3. With this rst perturbation, the origin remains a center. Second, we consider a general perturbation to get m small hyperbolic limit cycles surrounding the origin, also by Theorem 2.3. We obtain in total 2n + m small limit cycles in conguration hn, m, ni. Recently, the same procedure has been used in [24], but with a higher order develop- ment, to some systems with degrees up to ten, improving the best global Hilbert numbers known up to now for polynomial systems from degree four to ten. Here, as in [6], we only study rst order developments of the Lyapunov quantities introduced in Section 2.2. The study of higher order perturbations will be done in a near future. Proposition 5.1. There exist polynomial perturbations of degree four such that at least 13 small limit cycles bifurcate from the centers of system (5). They present at least two dierent congurations varying the parameters (a, b, c, d): h4, 5, 4i and h3, 7, 3i. Moreover the phase portraits in the Poincaré disk for the unperturbed systems are also dierent. See Figure 2. Proof. The proof follows the bifurcation procedure described in the beginning of this section. We will describe only the statement to get the 13 small limit cycles in con- guration h4, 5, 4i, xing the values of the parameters of the unperturbed center as (a, b, c, d) = (1/2, 1/2, −1, 2). It corresponds to family T4 . The same result and con- guration can be found in family T7 , xing the parameters to (1/2, −1/2, 1, 1). The phase portraits of the unperturbed systems are drawn in the two left pictures in Fig- ure 2. In the same way, we can also obtain the 13 small limit cycles bifurcating √ from the centers of√families T√9 and T √15 , xing now the parameter values to ( 5/2, 3/2, −9/2, 3) and (1, 3 7/7, −13 7/14, 7/2), respectively. The conguration of limit cycles is now h3, 7, 3i. The respective phase portraits are drawn in the two right pictures in Figure 2. First we take a perturbation of system (5), for the xed values in T4 mentioned above, keeping the reversible symmetry at the origin, X 4 1 5 3 ẋ = −y − xy + y 3 + xy 3 + fk,2`+1 xk y 2`+1 , 2 16 16 k+2`+1=1 4 (13) 1 2 3 2 3 4 X ẏ = x + y − xy − y + fk,2` xk y 2` . 4 8 32 k+2`=1 It is important to mention that it is not restrictive to assume that the origin remains as an equilibrium point of center type. Then, we do a translation in (13) in order that the center (−1, 2) moves to the origin. Moreover, after the translation, the perturbation should start with terms of degree 2. Because the last limit cycle appears moving the value of the trace at (−1, 2) in a usual Hopf bifurcation. This fact restrict the values of the perturbation parameters fk,` . Now, we can compute and check that the linear developments of the rst four Lyapunov quantities are linearly independent. Hence, we have found at least four small limit cycles surrounding (−1, 2) and (−1, −2), respectively. Second, it is enough to consider only a perturbation that does not respect the symmetry around the x-axis, X4 1 5 3 3 3 ẋ = −y − xy + y + xy + ek,2` xk y 2` , 2 16 16 k+2`=2 4 (14) 1 2 3 2 3 4 X ẏ = x + y − xy − y + ek,2`+1 xk y 2`+1 . 4 8 32 k+2`+1=2 18 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA Here we compute and check, as above, that the linear developments of the rst ve Lyapunov quantities are linearly independent. Then, we get ve small limit cycles sur- rounding the origin, via a degenerated Hopf bifurcation. We remark that the last limit cycle also appear moving the trace at the origin from zero to a nonzero value, in a usual Hopf bifurcation. We remark that the parameters that control both bifurcations, (13) and (14), are in- dependent. Considering all together we have at least 13 limit cycles in conguration h4, 5, 4i.  We remark that, in the last result, we have not obtained more limit cycles from a rst order analysis because using more Lyapunov quantities we do not have a higher rank to increase the number of limit cycles. Moreover, we have only stated the highest value for the total cyclicity found, studying all the cases listed in Theorem 1.2. We have not done a complete study and we have performed only up to a rst order analysis. We have picked a point (a, b, c, d) in each of the 18 families and we have obtained which are the ones that provide the highest cyclicity value. The cases with 13 limit cycles, in conguration h4, 5, 4i, corresponds to families T4 , T7 , T11 , and T13 and, in conguration h3, 7, 3i, to families T1 , T2 , T6 , T9 , T15 , and T18 . Other families provide less limit cycles. The families T5 , T10 , and T16 exhibit 11 limit cycles in conguration h2, 7, 2i. Also 11 limit cycles appear in conguration h3, 5, 3i for families T8 , T14 , and T17 . Finally, family T12 is the one with less limit cycles, only 9 in conguration h2, 5, 2i. Figure 2. Dierent phase portraits in the Poincaré disc of system (5) The equilibrium points in Figure 2 are drawn following the local phase portraits: The centers points in black, the stable nodes in blue, the unstable nodes in red, the saddles in green and the degenerated ones in white. In these last equilibrium points some blowups have been necessary to be done to get the local phase portrait. We remark that the two left centers in Figure 2 have three period annulus while the two right ones have four. Acknowledgements This work has been realized thanks to the Brazilian CAPES Agency (grant BEX 13473/13-1), the Catalan AGAUR Agency (grant 2017 SGR 1617), the Slovenian Re- search Agency (program P1-0306, project N1-0063), and the Spanish Ministerio de Cién- cia, Innovación y Universidades (FEDER grant MTM2016-77278-P). References [1] Y. L. Bondar and A. P. Sadovskii. Solution of the center and focus problem for a cubic system that reduces to the Liénard system. Dier. Uravn., 42(1):1122, 141, 2006. [2] J. Chavarriga and J. Giné. Integrability of a linear center perturbed by homogeneous polynomial. In Proceedings of the 2nd Catalan Days on Applied Mathematics (Odeillo, 1995), Collect. Études, pages 6175. Presses Univ. Perpignan, Perpignan, 1995. QUARTIC LINEAR-LIKE REVERSIBLE SYSTEMS 19 [3] J. Chavarriga and J. Giné. Integrability of cubic systems with degenerate innity. In XIV CEDYA/IV Congress of Applied Mathematics (Spanish)(Vic, 1995), page 12. Univ. Barcelona, Barcelona, 1996. [4] C. Chicone and M. Jacobs. Bifurcation of critical periods for plane vector elds. Trans. Amer. Math. Soc., 312(2):433486, 1989. [5] C. Chicone and M. Jacobs. Bifurcation of limit cycles from quadratic isochrones. J. Dierential Equations, 91(2):268326, 1991. [6] C. Christopher. Estimating limit cycle bifurcations from centers. In Dierential equations with sym- bolic computation, Trends Math., pages 2335. Birkhäuser, Basel, 2005. [7] D. Cozma. The problem of the center for cubic systems with two parallel invariant straight lines and one invariant cubic. ROMAI J., 11(2):6375, 2015. [8] D. Cozma and A. “ub . The solution of the problem of center for cubic dierential systems with four invariant straight lines. An. “tiinµ. Univ. Al. I. Cuza Ia³i. Mat. (N.S.), 44(suppl.):517530 (2000), 1998. Mathematical analysis and applications (Ia³i, 1997). [9] W. Decker, G.-M. Greuel, G. Pster, and H. Schönemann. Singular 4-1-1  A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2018. [10] W. Decker, G. Pster, H. Schönemann, and S. Laplagne. primdec.lib a singular 4-1-1 library for computing the primary decomposition and radical ideals, 2018. [11] H. Dulac. Détermination et intégration d'une certaine class d'équations diérentialles ayant pour point singulier un centre. Bull. Sci. Math., 32(2):230252, 1908. [12] F. Dumortier, J. Llibre, and J. C. Artés. Qualitative theory of planar dierential systems. Universi- text. Springer-Verlag, Berlin, 2006. [13] B. Fer£ec, J. Giné, V. G. Romanovski, and V. F. Edneral. Integrability of complex planar systems with homogeneous nonlinearities. J. Math. Anal. Appl., 434(1):894914, 2016. [14] M. Frommer. über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkur- ven) in der Umgebung rationaler Unbestimmtheitsstellen. Math. Ann., 109(1):395424, 1934. [15] P. Gianni, B. Trager, and G. Zacharias. Gröbner bases and primary decomposition of polynomial ideals. J. Symbolic Comput., 6(2-3):149167, 1988. Computational aspects of commutative algebra. [16] M. Han. Liapunov constants and Hopf cyclicity of Liénard systems. Ann. Dierential Equations, 15(2):113126, 1999. [17] Y. Ilyashenko and S. Yakovenko. Lectures on analytic dierential equations, volume 86 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008. [18] W. Kaptyen. On the midpoints of integral curves of dierntial equations of the rst degree. Nederl. Adak. Wetecnsch. Verslag. Afd. Natuurk. Konikl., pages 14461457, 1911. [19] W. Kaptyen. New investigations on the midpoints of integral of dierential equations of the rst degree. Nederl. Adak. Wetecnsch. Verslag. Afd. Natuurk. Konikl., 20:13541365, 1912. [20] A. M. Liapunov. Investigation of one of the special cases of the stability of motion. Mat. Sb., 17:253 333, 1893. [21] A. M. Liapunov. Problème Général de la Stabilité du Mouvement. Annals of Mathematics Studies, no. 17. Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1947. [22] N. G. Lloyd, C. J. Christopher, J. Devlin, J. M. Pearson, and N. Yasmin. Quadratic-like cubic systems. Dierential Equations Dynam. Systems, 5(3-4):329345, 1997. Planar nonlinear dynamical systems (Delft, 1995). [23] H. Poincaré. Mémoire sur les courbes dénies par une équation diérentielle. Sér.3)7(1881), 375422; Sér.(3)8(1882),251296; Sér.(4)1(1885),167244; Sér. (4)2(1886), 151217. [24] R. Prohens and J. Torregrosa. New lower bounds for the Hilbert numbers using reversible centers. Nonlinearity, 32(331), 2019. [25] V. G. Romanovski and D. S. Shafer. The center and cyclicity problems: a computational algebra approach. Birkhäuser Boston, Inc., Boston, MA, 2009. [26] V. G. Romanovski and D. S. Shafer. Centers and limit cycles in polynomial systems of ordinary dierential equations. In School on real and complex singularities in São Carlos, 2012, volume 68 of Adv. Stud. Pure Math., pages 267373. Math. Soc. Japan, [Tokyo], 2016. [27] R. Roussarie. Bifurcation of planar vector elds and Hilbert's sixteenth problem, volume 164 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1998. [28] N. A. Saharnikov. On Frommer's conditions for the existence of a center. Akad. Nauk SSSR. Prikl. Mat. Meh., 12:669670, 1948. [29] B. Sang. Center-focus problems for two classes of cubic dierential systems. J. Nanjing Norm. Univ. Nat. Sci. Ed., 35(2):1621, 2012. 20 L. P. C. DA CRUZ, VALERY ROMANOVSKI, AND JOAN TORREGROSA [30] B. Sang and C. Niu. Solution of center-focus problem for a class of cubic systems. Chin. Ann. Math. Ser. B, 37(1):149160, 2016. [31] K. S. Sibirski. On conditions for the presence of a center and a focus. Ki²inev. Gos. Univ. U£. Zap., 11:115117, 1954. [32] K. S. Sibirski. The principle of symmetry and the problem of the center. Ki²inev. Gos. Univ. U£. Zap., 17:2734, 1955. [33] K. S. Sibirski. On the number of limit cycles in the neighborhood of a singular point. Dierencialnye Uravnenija, 1:5366, 1965. [34] P. S. Wang, M. J. T. Guy, and J. H. Davenport. P-adic reconstruction of rational numbers. SIGSAM Bull., 16(2):23, 1982. [35] H. ›oª¡dek. Quadratic systems with center and their perturbations. J. Dierential Equations, 109(2):223273, 1994. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address : leonardo@mat.uab.cat Faculty of Electrical Engineering and Computer Science, University of Maribor, Koro²ka cesta 46, SI-2000, Maribor, Slovenia Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000, Maribor, Slovenia Faculty of Natural Science and Mathematics, University of Maribor, Koro²ka cesta 160, SI-2000, Maribor, Slovenia Email address : valerij.romanovskij@um.si Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address : torre@mat.uab.cat