This is a preprint of: “The local cyclicity problem. Melnikov method using Lyapunov constants”, Luiz Fernando da Silva Gouveia, Joan Torregrosa, Proc. Edinb. Math. Soc. (2), vol. 65(356-375), 2022. DOI: [10.1017/S0013091522000128] THE LOCAL CYCLICITY PROBLEM. MELNIKOV METHOD USING LYAPUNOV CONSTANTS LUIZ F. S. GOUVEIA AND JOAN TORREGROSA Abstract. In 1991, Chicone and Jacobs shown the equivalence between the compu- tation of the first-order Taylor developments of the Lyapunov constants and the devel- opments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small amplitude bifurcating from a quadratic center. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so M(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp (4) ≥ 43 and Mcp (5) ≥ 65. 1. Introduction In the last century, Hilbert presented a list of problems that almost all of them are solved. One problem that remains open consists in determining the maximal number H(n) of limit cycles, and their relative positions, of planar polynomial vector fields of degree n. This problem is known as the second part of the 16th Hilbert’s problem. In the year of 1977, Arnol’d in [4] proposed a weakened version, focused on the study of the number of limit cycles bifurcating from the period annulus of Hamiltonian systems. In this work, we are interested in another local version, that consists in providing the maximum number M(n) of small-amplitude limit cycles bifurcating from an elementary center or an elementary focus, clearly M(n) ≤ H(n). In other words, M(n) is an upper bound of the (local) cyclicity of such equilibrium points. For more details, see [49]. For n = 2, Bautin proved that M(2) = 3, see [5]. For n = 3, the family of cubic systems without quadratic terms was studied in [7, 51]. The proof of Mh (3) = 5 can be found in [54]. Żola̧dek in [55], shows the first evidence that M(3) ≥ 11. However, this problem has been recently revisited by himself in [57]. The first proof of this fact was obtained some years later by Christopher in [14], studying first-order perturbations of another cubic center also provided by Żola̧dek in [56]. In 2012, Giné conjectures that M(n) = n2 + 3n − 7, see [25, 26]. This suggests a higher value for M(n) for polynomial vector fields of low degree. Gouveia and Torregrosa in [30] show that M(5) ≥ 33, M(7) ≥ 61, M(8) ≥ 76, and M(9) ≥ 88. The first evidence that this conjecture fails is given in [53]. Recently, a complete proof that M(3) ≥ 12 is provided by Giné, Gouveia, and Torregrosa in [27]. Moreover, in the same paper, they show that M(4) ≥ 21. For n = 6, the best lower bound was given in [42], proving that M(6) ≥ 40. The first main result of this paper updates this value. Theorem 1.1. The local cyclicity of monodromic equilibrium points for polynomial vector fields of degree n = 6 is greater than or equal to 44. That is, M(6) ≥ 44. In this work, we are also interested in piecewise polynomial vector fields. Andronov, in [3] was the first studying such class of systems. In the last years, they have been widely studied, since many problems of engineering, physics, and biology can be modeled 2010 Mathematics Subject Classification. Primary 34C07; Secondary 34C23, 37C27. Key words and phrases. Melnikov theory, Lyapunov constants, local cyclicity. 1 2 LUIZ F. S. GOUVEIA AND J. TORREGROSA by them, see [1, 6]. One of the most studied situations in the plane is given by two vector fields defined in two half-planes separated by a straight line. As in the case of the classical qualitative theory of polynomial systems, the study of the number and location of the isolated periodic orbits, also called limit cycles, have received special attention, see for example [11, 17, 24, 36, 41, 46]. In particular, it can be seen as an extension of the 16th-Hilbert problem for planar piecewise polynomial vector fields, see [38]. Here, we study limit cycles bifurcating from the origin in the class of piecewise differ- ential equations of the form ( (ẋ, ẏ) = (P + (x, y, λ), Q+ (x, y, λ)), when y ≥ 0, (ẋ, ẏ) = (P − (x, y, λ), Q− (x, y, λ)), when y < 0, where P ± (x, y, λ) and Q± (x, y, λ) are polynomials in x and y. The straight line Σ = {y = 0} divides the plane in two half-planes, Σ± = {(x, y) : ±y > 0}, and the trajectories on Σ are defined following the Filippov convention, see [20]. Here we will consider only limit cycles of crossing type, that is, when both vector fields point out in the same direction on the separation line Σ. Then, we denote by Hpc (n) the number of limit cycles of crossing type for piecewise polynomial vector fields of degree n. Similarly, Mcp (n) counts the ones of small amplitude bifurcating from a monodromic equilibrium point. The upper index c means for crossing limit cycles and the subscript p for piecewise class. Clearly, M(n) ≤ H(n) and Mcp (n) ≤ Hpc (n). For n = 1, Huan and Yang in [37] show a numerical evidence that Hpc (1) ≥ 3. Llibre and Ponce in [46] provide an analytical proof of this property. Using the averaging technique this lower bound was reobtained by Buzzi, Pessoa, and Torregrosa in [11]. Freire, Ponce, and Torres also obtained the same number in [22]. In this last work, the three limit cycles are explained studying the full return map, two appear near the origin and the other one far from it. But the three limit cycles appear nested and surrounding one sliding segment. In fact, the two limit cycles of small amplitude appearing from an equilibrium point provide the lower bound Mcp (1) ≥ 2. This value can be proved with the results in [22] and we will recall it in the next section. Recently in [21], the three limit cycles have been obtained near infinity in a Hopf type bifurcation. For n = 2, using averaging theory of fifth-order, and perturbing the linear center, Llibre and Tang in [47] proved that Hpc (2) ≥ 8. Recently, da Cruz, Novaes, and Torregrosa in [18] improve this lower bound and, using local developments of the difference map, Gouveia and Torregrosa in [29] provide the first high lower bounds for piecewise polynomial classes of degrees three, four and five: Mcp (3) ≥ 26, Mcp (4) ≥ 40 and Mcp (5) ≥ 58. Here we update two of these lower bounds using averaging theory in our second main result. Theorem 1.2. The local cyclicity for piecewise polynomial vector fields of degree four and five is Mcp (4) ≥ 43 and Mcp (5) ≥ 65, respectively. In this paper, we are interesting also in differential equations containing a privileged ε that acts as small perturbation of a periodic behavior. Lagrange, in his study about the three-body problem, formulated the idea of averaging. During some years, this method was used in many fields without people bothering about proofs validity. In 1928, Fatou in [19] gave the first analytic proof. Nowadays, in planar differential systems, the averaging theory is also used for proving the existence of limit cycles which appear, after pertur- bation, from a fully periodic neighborhood. A similar mechanism, when the differential equation is nonautonomous, used for the same purpose is the Melnikov method. More- over, it is an excellent tool for studying global bifurcations that occur near homoclinic or heteroclinic loops in one-parameter families. MELNIKOV METHOD USING LYAPUNOV CONSTANTS 3 Let us consider the perturbation of a Hamiltonian system ( x0 = −Hy + ε P (x, y, ε, λ), (1) y0 = Hx + ε Q(x, y, ε, λ). Then, the first Melnikov function writes as Z M (h) = Q(x, y, 0, λ)dx − P (x, y, 0, λ)dy, (2) Γh where Γh = {H(x, y) = h2 } are closed ovals. This first-order analysis is based on the Implicit Function Theorem. In fact, for ε small enough, the simple zeros of M (h) corre- spond to limit cycles of the perturbed system (1). That is, if h∗ satisfies that M (h∗ ) = 0 and M 0 (h∗ ) 6= 0 then, there exists an hyperbolic limit cycle of (1) that goes to Γh∗ when h goes to h∗ . This result can be generalized also for the perturbation of centers, (ẋ, ẏ) = (Pc , Qc ), having an inverse integrating factor V (x, y). Then after a time rescaling the perturbed system ( ẋ = Pc (x, y) + ε P (x, y, ε, λ), (3) ẏ = Qc (x, y) + ε Q(x, y, ε, λ), can be written as    P (x, y, ε, λ)  x0 = −Hy + ε , V (x, y) (4)   Q(x, y, ε, λ)  y0 = Hx + ε . V (x, y) So, the corresponding generalized first Melnikov function of (4), or also of (3), is Z Q(x, y, 0, λ)dx − P (x, y, 0, λ)dy M (h) = . (5) Γh V (x, y) In this paper, we only consider the above Melnikov function for the study of perturba- tions of periodic orbits near planar autonomous differential systems. In this case, there are some works explaining the equivalence between Melnikov studies and the averaging theory, see [9, 32]. For more details in this theory we refer the reader to [10, 35, 50, 52]. This bifurcation mechanism can be used also in piecewise differential equations, see [43, 44, 45]. Although this problem can be treated in a more general way, in this work we will consider a simplified version of it. We will study that only the perturbation in (3) is defined by piecewise functions. This perturbation will be defined in two zones separated by a straight line, that is, the differential equation to consider can be written as ( (Pc (x, y), Qc (x, y)) + ε (P + (x, y, ε, λ), Q+ (x, y, ε, λ)), if y ≥ 0, (ẋ, ẏ) = (6) (Pc (x, y), Qc (x, y)) + ε (P − (x, y, ε, λ), Q− (x, y, ε, λ)), if y < 0. Therefore, the piecewise generalized first Melnikov function of (6) is Mp (h) = M + (h) + M − (h), (7) where Z ± Q± (x, y, 0, λ)dx − P ± (x, y, 0, λ)dy M (h) = Γ± h V (x, y) with Γ± h = Γh ∩ {±y > 0}. Clearly, in both piecewise regions we have the same center having V (x, y) as an inverse integrating factor. The main results of this work could be extended to perturbing piecewise centers but this is not the aim of this paper. 4 LUIZ F. S. GOUVEIA AND J. TORREGROSA From the mechanism described above, the functions (2), (5), and (7) provide the num- ber of limit cycles bifurcating from the period annulus up to first-order analysis. Another tool for obtaining limit cycles bifurcating from the origin comes from the study of the Taylor development of the return map, the so-called Lyapunov constants. In this context the perturbation terms of the non-degenerate linear center, (ẋ, ẏ) = (−y, x), usually have degree at least two because in this case, the Lyapunov constants are polynomials in the perturbation parameters, [15]. Another small limit cycle bifurcates from the origin using the classical Hopf bifurcation when the linear coefficients are added in the perturbation terms and the trace parameter change sign adequately. This bifurcation phenomenon is also known as the degenerated Hopf bifurcation. More details on it can be seen in [2]. In piecewise differential equations, a similar bifurcation can also be introduced, but adding two limit cycles instead of only one. The linear terms are responsible for one of them and the other is due to the existence of what is known as the sliding segment that appears when the constant terms are added. The latter is known as the pseudo-Hopf bifurcation and they are summarized in the next result. Proposition 1.3 ([28]). Consider the perturbed system    P ∞  P ∞  2  ẋ = −(1 + c )y + + k ` ak` x y ,   ẋ = −y + a− k ` k` x y , k+`=2 k+`=2  P ∞  P∞   ẏ = x + 2cy + b+ k ` k` x y ,   ẏ = d + x + b− k ` k` x y , k+`=2 k+`=2 − − − for y ≥ 0 and y < 0, respectively. If a+ + + 11 − a11 + 2b02 − 2b02 + b20 − b20 6= 0 then there exist c and d small enough such that two crossing limit cycles of small amplitude bifurcate from the origin. − − − We notice that when c = d = 0, the expression a+ + + 11 − a11 + 2b02 − 2b02 + b20 − b20 defines the stability of the origin. So, we could define it as the first Lyapunov constant for the piecewise perturbation of the linear center. More details on the pseudo-Hopf bifurcation are collected in [12, 22, 23]. In [14, 27, 29, 30], we can see that a good and simple tool for studying the number of limit cycles of small amplitude bifurcating from the origin is the first-order development of the Lyapunov constants. We notice that, in this case, the Jacobian matrix of the vector field at the origin must be of nondegenerate monodromic type. Then, a natural question arises. Are there any relation between both mechanisms? Chicone and Jacobs in 1991 provided a positive answer for quadratic families of vector fields, see [13]. But, their proof is also valid for polynomial vector fields of any degree. Hence, we can say that the next result comes from their original work. These ideas appear also in the works of Han and Yu, see [33, 35], where the Melnikov function in the Hopf bifurcation is studied. Theorem 1.4 ([13]). Let (ẋ, ẏ) = (Pc (x, y), Qc (x, y)) be a polynomial vector field of degree n, with a non degenerated center at the origin. Consider the perturbed systems, in the class of polynomial vector fields of degree n,  Xn n X  (ẋ, ẏ) = Pc (x, y) + akl xk y l , Qc (x, y) + bkl xk y l (8) k+l=2 k+l=2 and  n X n X  k l 2 (ẋ, ẏ) = Pc (x, y) + ε akl x y + O(ε ), Qc (x, y) + ε bkl xk y l + O(ε2 ) . (9) k+l=2 k+l=2 MELNIKOV METHOD USING LYAPUNOV CONSTANTS 5 [1] If we denote by Lk the first-order expansion, with respect to the parameters akl , bkl , for the Taylor series of the Lyapunov constant Lk associated to (8) then, for ρ small, the first Melnikov function of (9) is N X  X∞  [1] M (ρ) = Lk 1 + αkj0 ρ ρ2k+2 , j (10) k=1 j=1 with the Bautin ideal hL1 , . . . , LN , . . .i = hL1 , . . . , LN i. We remark that the perturbations in the above result are taken without linear terms. This restriction is due to the fact that the elements in the Bautin ideal should be poly- nomials. A direct consequence is the first part of the next corollary. For the second part the classical Hopf bifurcation is necessarily to be used. As usual, O(ε2 ) denotes the terms of degree at least 2 in the perturbative parameter ε. [1] [1] Corollary 1.5. Let Am be the matrix corresponding to (L1 , . . . , Lm ) with respect to the 2 [1] parameters (a20 , a11 , a02 , . . . , b20 , b11 , b02 , . . .) ∈ Rn +3n−4 , where each Lk is the linear k- Lyapunov constant of system (8) and m ≤ N . Then, if rank Am = ` and for ε small enough, system (9) has ` − 1 hyperbolic limit cycles of small amplitude bifurcating from the origin. Additionally, adding the trace parameter, there are polynomial perturbations of the center (ẋ, ẏ) = (Pc (x, y), Qc (x, y)), exhibiting ` hyperbolic limit cycles of small amplitude in a small enough neighborhood of the origin. Recently, see [27, 29, 30], a detailed study of the return map of a differential equation near a non-degenerated monodromic point located at the origin has been done. The authors focus their attention not only on first-order Taylor developments, with respect to parameters, of the coefficients of the return map with respect to the initial condition. These papers are a continuation of the work given by Christopher in [14]. The degenerated [1] Hopf bifurcation technique applies when Lk = uk for k = 1, . . . , N, and all uk are new independent parameters. Then, the existence of N limit cycles of small amplitude bifurcating from the origin is guaranteed. This approach fails when the rank does not [1] increase in each step. That is when at least one Lk is a linear combination of the previous ones. The main advantage of Theorem 1.4 is that we can compute easier the [1] expressions of Lk than the coefficients of the series expansion of M. In [34] the study of the multiplicity of zeros is also taken into account and moreover it provides upper bounds for the number of limit cycles up to this perturbation order. This paper is structured as follows. In Section 2, for completeness, we prove Theo- rem 1.1 recovering the original proof for quadratic vector fields obtained in [13]. As a natural application of it, we also prove Corollary 1.5. Then, we can get limit cycles of small amplitude bifurcating from a center equilibrium point. Finally, we extend Theo- rem 1.4 and Corollary 1.5 to the class of piecewise polynomial vector fields. In Section 3 we show, with a simple example of a polynomial vector field of degree 6, that the max- imal computed rank of the linear developments does not coincide with the subscript of the corresponding Lyapunov constant. Then, the result using only linear parts given by Christopher in [14] does not apply. But using Theorem 1.4 we provide more limit cycles of small amplitude. In Section 4 we prove our first main result, Theorem 1.1, and we study the local cyclicity problem for some vector fields of degrees 7, 8, and 10, proving that M(7) ≥ 60, M(8) ≥ 70, and M(10) ≥ 97. Finally, Section 5 is devoted to piecewise polynomial vector fields of degrees 3, 4, and 5, proving our second main result Theo- rem 1.2. We also provide a new proof of Mcp (3) ≥ 26, different from the one published in [29], using now only first-order developments. 6 LUIZ F. S. GOUVEIA AND J. TORREGROSA 2. The proof of Chicone–Jacobs’ result This section is devoted to the proof of Theorem 1.4 and Corollary 1.5. The proof of Theorem 1.4 for quadratic vector fields can be found in [13]. You can see it partially also in [48]. Here we reproduce it but extended for every polynomial vector field of degree n. Proof of Theorem 1.4. To simplify notation, we rewrite the perturbed parameters in (8) and (9) as (a20 , a11 , a02 , . . . , b20 , b11 , b02 , . . .) = (λ1 , . . . , λm ) ∈ Rm , with m = n2 + 3n − 4. After a linear change if necessary, equations (8) and (9), in the usual polar coordinates, write as dr r2 R3 (θ) + · · · + rn Rn+1 (θ) = (11) dθ 1 + rΘ3 (θ) + · · · + rn−1 Θn+1 (θ) where Ri and Θi are homogeneous trigonometric polynomials in sin θ, cos θ of degree i. We define r(θ; ρ) the solution of the initial value problem defined by (11) with initial condition r(0; ρ) = ρ. Then, from the first return map r(2π; ρ), we can define the displacement function associated to (11) as ∆(ρ) = r(2π; ρ) − ρ. Clearly, the zeros of this function provide periodic orbits of the corresponding system in Cartesian coordinates. The Taylor series in ρ of the displacement function ∆1 associated to (8) is N X  ∞ X  2k+1 j ∆1 (ρ, λ) = Lk (λ)ρ 1+ βkj (λ)ρ . (12) k=1 j=1 with βkj polynomials vanishing at zero. Similarly, the Taylor series in ε of the displace- ment function ∆2 associated to (9) is X∞ X∞   k 1 ∂ k d(ρ, ε) ∆2 (ρ, ε) = dk (ρ)ε = k εk . (13) k=1 k=1 k! ∂ε ε=0 We notice that both Taylor series representation are only local, but with the Global Bifurcation Lemma, see [13], the coefficients dk (ρ) are defined and being analytic on the ρ-domain corresponding to the portion of the x−axis cut by the periodic trajectories surrounding the center at the origin of the system. We observe that the idea of Melnikov theory is determining dk (ρ) under the condition dj (ρ) = 0 for all j < k. Writing the parameters P λ as their Taylor series expansion in terms of a privileged parameter ε, λl (ε) = ∞ j j=1 jl ε , in (12), we have for each k and i, λ ∞ X ∞ X (j) Lk (λ(ε)) = Lk (λ(ε))εj and βki (λ(ε)) = βkij εj . j=1 j=1 Rearranging the Taylor series (12), for ε and ρ small enough, it follows that XN X ∞  ∞ X X ∞  (j) j ∆1 (ρ, λ(ε)) = Lk ε 1 + βkij ρ ε ρ2k+1 . j i (14) k=1 j=1 i=0 j=1 As the first-orders in ε of equation (8), changing λ by ελ, and equation (9) coincide, the coefficients of degree one in ε for (14) and (13) are equal. So, XN  X∞  [1] d1 (ρ) = Lk 1 + βkj0 ρ ρ2k+1 . j k=1 j=1 MELNIKOV METHOD USING LYAPUNOV CONSTANTS 7 It is well known that in the above expression the odd exponents in ρ corresponds to the property that the first non vanishing Lyapunov constant has always an odd index, see [2]. For a proof that the Lyapunov constants of even indices are in the ideal of the odd ones, we refer the reader to [16, 31]. Applying the ideas in [13, Sect. 4] to differential system (4) we can write 1 d1 (ρ) = M (ρ), Hx (ρ, 0) where the function M is defined in (5). The statement follows because, in our case, H(x, y) = (x2 + y 2 )/2 + · · · .  Proof of Corollary 1.5. We study first the case when the trace parameter is zero. In this case, it is well-known that the Lyapunov constants are polynomials in the perturbation parameters, see [15]. Then we can compute the linear terms of the Lyapunov constants with, for example, the classical mechanism described in [42]. Consequently, Theorem 1.4 gets the expression of the first Melnikov function (10). Hence, if the rank of the computed expressions is `, we have `−1 simple zeros near ρ = 0 and there exists a perturbed system exhibiting ` − 1 limit cycles of small amplitude. Finally, adding the trace parameter as in the classical Hopf bifurcation (see [40]), we get the total ` limit cycles as it is stated in the statement.  We notice that the above two results can be easily generalized for piecewise vector fields. Let us consider the piecewise version of the perturbed differential systems (8) and (9) given by  n X n X  ± k l ± k l (ẋ, ẏ) = Pc (x, y) + akl x y , Qc (x, y) + bkl x y (15) k+l=2 k+l=2 and  n X n X  (ẋ, ẏ) = Pc (x, y) + ε a± kl x k l y + O(ε 2 ), Qc (x, y) + ε b ± k l kl x y + O(ε 2 ) . (16) k+l=2 k+l=2 − − They are defined for the parameters a+ + kl , bkl and akl , bkl , in the regions y ≥ 0 and y < 0, respectively. Here, as previously, the unperturbed system (ẋ, ẏ) = (Pc (x, y), Qc (x, y)) has a non-degenerated center at the origin. The next corollary provides the piecewise extension of Theorem 1.4. So, its proof follows closely to the proof of Theorem 1.4. In this case, the return map is also analytic in ρ and all the steps are similar. The main difference is that the Taylor series of d1 in ρ have all powers starting in 2, because now we have not the property of symmetry that vanishes all the even terms in the developments. We remark that the perturbation in (16) has no constant nor linear terms. [1] Corollary 2.1. Let Lk be the linear terms of the Lyapunov constants of a polynomial piecewise vector field (15) near a monodromic equilibrium point. Then the corresponding first Melnikov function (5) of (16) writes as N X  ∞ X  [1] M (ρ) = Lk 1+ αkj0 ρ ρk+1 , j k=2 j=1 being the Bautin ideal hL2 , . . . , LN , . . .i = hL2 , . . . , LN i. Finally, similarly as for analytic perturbations here also the rank of the matrix provides a lower bound for the number of limit cycles bifurcating from a monodromic equilibrium 8 LUIZ F. S. GOUVEIA AND J. TORREGROSA point. But, using piecewise perturbations, we have two extra limit cycles of small ampli- tude as we have shown in Proposition 1.3. Then we have at least as many limit cycles as the number of linearly independent linear parts plus one. [1] [1] Corollary 2.2. Let Am be the matrix corresponding to (L2 , . . . , Lm ) with respect to the 2n2 +6n−8 [1] parameters (a± ± ± ± ± ± 20 , a11 , a02 , . . . , b20 , b11 , b02 , . . .) ∈ R , where each Lk is the linear part of the k-Lyapunov constant of system (15). Then, if rank Am = `, for ε small enough, there exist perturbation parameters, (a± , b± ), such that system (16) has ` − 1 limit cycles of small amplitude bifurcating from the origin. Additionally, adding the trace parameter, there are piecewise polynomial perturbations of (ẋ, ẏ) = (Pc (x, y), Qc (x, y)) exhibiting `+1 hyperbolic limit cycles of small amplitude in a small enough neighborhood of the origin. We remark that the definition of the matrix Am for Corollaries 1.5 and 2.2 does not coincide, because of the different definitions of the subscripts in the nonvanishing coeffi- cients of the respectively Taylor developments in ρ. But the important property for the number of the limit cycles of small amplitude is which is the value of its rank. 3. A first but not trivial example In this section we show the explicit linear developments of the Lyapunov constants in a simple example provided by [39]. In the proof of the next result, we can see how Corollary 1.5 is better than the direct use of the Lyapunov constants. We remark that, most probably, the number of limit cycles presented will be the maximum using only first-order analysis. We have chosen a vector field of a high degree to see better the effectiveness of our results. Proposition 3.1. Let us consider the system (ẋ, ẏ) = (−y + x5 y, x + x4 y 2 ). (17) Then, there exist polynomial perturbations of degree 6 such that from the origin of the center (17) bifurcate 16 limit cycles of small amplitude. Proof. System (17) has a time-reversible center at the origin. Because it is invariant by the change (x, y, t) → (x, −y, −t). Additionally, it has also a rational first integral obtained from its corresponding inverse integrating factor. They are, respectively, (x2 + y 2 )5 x10 − 2x5 + 1 H(x, y) = and V (x, y) = . (1 − x5 )2 (x2 + y 2 )3/2 The next step is the computation of the linear terms of the Taylor developments of [1] the Lyapunov constants. We have computed Lk for k = 1, . . . , 40 with, for example, the algorithm described in [30]. As in the previous section, let An be the matrix corresponding [1] [1] to (L1 , . . . , Ln ) with respect to the parameters (a20 , a11 , a02 , . . . , b20 , b11 , b02 , . . .) ∈ R50 . Then, the proof of the statement follows using Corollary 1.5 because rank A20 = 16. We point out that we have checked that rank Ak = k for k = 1, . . . , 12, but rank A13 = 12. Additionally, that rank A14 = 13, rank Ak = 14, for k = 15, . . . , 18, rank A19 = 15, and rank Ak = 16, for k = 20, . . . , 40. From these values we can say that, up to k = 40 there are no more limit cycles than the ones in the statement. Moreover, without Corollary 1.5, that is with the results in [14], only 12 limit cycles of small amplitude can bifurcate from the origin. The explicit expression of the necessary Lyapunov constants to get the statement are: [1] 2 L1 = (3a30 + a12 + b21 + 3b03 ), 3 MELNIKOV METHOD USING LYAPUNOV CONSTANTS 9 [1] 2 L2 = (b41 + a32 + b23 + 5b05 + a14 + 5a50 ), 5 [1] 2 L3 = (33a20 + 7a02 − b11 ), 35 [1] 2 L4 = (161a40 − 21b31 − 9b13 + 29a22 + 21a04 ), 315 [1] 2 L5 = (21a60 − 7b33 − 5b15 + 5a24 + 7a06 + 7a42 − 21b51 ), 231 [1] 2 L6 = − (2661b03 + 873b21 + 901a12 + 2773a30 ), 715 [1] 2 L7 = − (42775b05 + 8639a14 + 8755a32 + 8447b23 + 8051b41 + 44259a50 ), 10725 [1] 2 L8 = − (−265929b11 + 1931573a02 + 9055937a20 ), 2127125 [1] 2 L9 = (−40561101b31 − 18052569b13 + 42963501a04 + 320497961a40 121246125 + 58729949a22 ), [1] 2 L10 = − (12806957a60 + 4471569a42 + 4628869a06 − 12806957b51 24249225 − 4471569b33 + 3270585a24 − 3270585b15 ), [1] 2 L11 = (9725859717a12 + 29642231001a30 + 28891397801b03 663966875 + 9538151417b21 ), [1] 2 L12 = (770680333485b05 + 789987106585a50 + 155267074197a14 16599171875 + 156791990497a32 + 147654869097b41 + 152709627597b23 ), [1] 2 L13 = (158564044415887a20 + 33790154009423a02 3137243484375 − 4680685675639b11 ), [1] 2 L14 = (44434290786711a04 + 332135401856251a40 10108895671875 + 60777619220839a22 − 18692855217819b13 − 42136878469511b31 ), [1] 2 L15 = (23966540375301a60 + 8360451650717a42 3418644718125 + 6115824649905a24 − 23966540375301b51 − 8360451650717b33 − 6115824649905b15 + 8657859935137a06 ), [1] 2 L16 =− (3245412818709921797b03 + 1075093752985900449b21 3357597491015625 + 1088751590673845749a12 + 3300044169461702997a30 ), [1] 2 L17 =− (441743420639301995b05 + 85703797233564799b41 430461216796875 + 87765192673441699b23 + 89435736465275399a32 + 88812279725145499a14 + 449630929414297095a50 ), [1] 2 L18 =− (63431621172437564093a20 56468685076171875 − 1883311935706466021b11 + 13503765373476308497a02 ), 10 LUIZ F. S. GOUVEIA AND J. TORREGROSA [1] 2 L19 = − (464272204405895182111a40 621155535837890625 + 84747082997571253579a22 − 59162027050841642971b31 − 26101104368457380559b13 + 61867863575213428971a04 ), [1] 2 L20 =− (5933622583442761308861a60 35654327757094921875 + 1503042242167592202705a24 − 2059684414443886787537b33 + 2125467391558746447037a06 − 1503042242167592202705b15 − 5933622583442761308861b51 + 2059684414443886787537a42 ).  4. Local cyclicity for systems of degrees 6, 7, 8, and 10 In this section we use Corollary 1.5 to study lower bounds for the cyclicity in some centers of low degree, n = 6, 7, 8, and 10. Next result proves Theorem 1.1. The other propositions do not provide better lower bounds for the local cyclicity in the indicated degrees. But we have decided to be added here because the centers and their perturbation studies are new. From the proofs, it seems that the number of limit cycles that bifurcate from the origin will be the maximum values for first-order studies. We have computed some more terms than the necessary to prove the statements. In all cases, we show that only studying first-order developments and using the technique described in [14] the results on lower bounds for the cyclicity are worst. We have not studied degrees n = 3, 4, 5, 9 because we do not have new centers having this special property. Unfortunately, we have not improved the lower bounds previously obtained in [27, 30]. Proposition 4.1. There are polynomial perturbations of degree 6 such that from the origin of the center  128 6 128 5 416 4 2 448 3 3 256 2 4 256 5 8 6   ẋ = −y + x − x y− xy + xy − xy + xy + y , 15 15 45 45 15 45 9 (18)   ẏ = 2x − 896 x5 y − 1664 x4 y 2 + 96 x3 y 3 − 512 x2 y 4 + 112 xy 5 + 32 y 6 , 45 45 5 45 45 15 bifurcate at least 44 limit cycles of small amplitude. Proof. The vector field in the statement and the proof that it has a center at the origin is given by Giné in [25]. The proof of the statement follows from Corollary 1.5 computing [1] Lk , for k = 1, . . . , 80, and checking that rank Ak = 44, for k = 52, . . . , 80. We notice that rank Ak = k, for k = 1, . . . , 32. Because of the size, we only show the linear developments of the first three Lyapunov constants: [1] 4 L1 = (2a12 + b21 ) + 4(a30 + b03 ), 3 [1] 4 L2 = (2a32 + 4a14 + b41 + 2b23 ) + 4(a50 + 4b05 ), 5 [1] 512 L3 = (488a02 − 86a11 + 236a20 − 118b02 + 172b11 − 61b20 ). 1575  Remark 4.2. We notice that for the center (18) we have also computed 44 Lyapunov constants up to fourth-order and, using [30], we can prove only the existence of 41 limit cycles of small amplitude bifurcating from the origin. MELNIKOV METHOD USING LYAPUNOV CONSTANTS 11 Proposition 4.3. There are polynomial perturbations of degree 7 such that from the origin of the center   2527 6 2968 5 2 4186 4 3 2800 3 4   ẋ = − x y− xy − xy − x y − 553x2 y 5 + 56xy 6   3 3 3 3     184 3 88 + x y + x2 y 2 + 48xy 3 − y, 3 3   2219 5 2 5684 4 3 742 3 4 1148 2 5   ẏ = 672x + 1484x6 y + 7 xy + xy − xy + xy   3 3 3 3   104 2 2 44 3  −315xy 6 − 28y 7 − 58x4 − 44x3 y − x y − xy + 10y 4 + x, 3 3 bifurcate at least 60 limit cycles of small amplitude. Proof. The system in the statement has a center at the origin because it has a rational first integral of the form H(x(x2 + y 2 ), y(x2 + y 2 )), where (42x − 7y − 1)3 ψ(x, y) H(x, y) = (19) (448x2 + 336xy + 63y 2 − 44x − 12y + 1)3 (1183x2 − 68x + 1) with ψ(x, y) = 10752x3 + 29568x2 y + 17640xy 2 + 3024y 3 − 1600x2 − 2760xy − 576y 2 + 74x + 57y − 1. The original cubic polynomial vector field having a center at the origin is given in [8]. [1] The proof follows also from Corollary 1.5 computing Lk , for k = 1, . . . , 80, and checking that rank A63 = 60. We notice that rank Ak = k, for k = 1, . . . , 55. We show only the first three Lyapunov constants: [1] 1 L1 = (a12 + b21 ) + a30 + b03 , 3 [1] 1 L2 = (1664a02 + 352a11 + 9a14 + 3952a20 + 9a32 + 45a50 + 968b02 + 45b05 45 + 1064b11 + 616b20 + 9b23 + 9b41 ), [1] 2192 L3 = (a04 + 176a13 + 15a16 + 1392a22 + 528a31 + 9a34 + 6800a40 + 15a52 105 + 105a70 + 880b04 + 105b07 + 912b13 + 528b22 + 15b25 + 1072b31 + 880b40 + 9b43 + 15b61 ).  Proposition 4.4. There are polynomial perturbations of degree 8 such that from the origin of the center   54 8 18 7 54 6 2 894 5 3   ẋ = x + x y− xy + x y − 2x4 y 4   175 35 175 175   66 26 342 7 16 8    + x3 y 5 − x2 y 6 − xy + y − y, 25 35 175 25   198 1254 586 258 4 4   ẏ = − x7 y − x6 y 2 − x5 y 3 − xy   175 175 175 35     22 18 382 7 162 8 − x3 y 5 + x2 y 6 − xy + y + x, 5 25 175 175 bifurcate at least 70 limit cycles of small amplitude. Proof. We consider the center with quartic homogeneous nonlinearities given in [25] but written in polar coordinates, (x, y) = (r cos θ, r sin θ). Then, with the change R = r3/5 and recovering again new Cartesian coordinates, we obtain the system in the statement of 12 LUIZ F. S. GOUVEIA AND J. TORREGROSA degree 8. Then it has also a center at the origin. The proof of the statement follows again [1] from Corollary 1.5 computing Lk , for k = 1, . . . , 130, and checking that rank A87 = 70. We notice that rank Ak = k, for k = 1, . . . , 45. The first three Lyapunov constants are [1] 1 L1 = (a12 + b21 ) + a30 + b03 , 3 [1] 1 L2 = (a14 + a32 + 5a50 + 5b05 + b23 + b41 ), 5 [1] 1 L3 = (5a16 + 3a34 + 5a52 + 35a70 + 35b07 + 5b25 + 3b43 + 5b61 ). 35  Proposition 4.5. There are polynomial perturbations of degree 10 such that from the origin of the center   6 10 2 9 8 152 7 3 28 6 4 44 5 5 8 4 6   ẋ = x + x y + x8 y 2 + xy − xy + xy − xy   25 5 25 25 25 5 5   24 6 54 16    + x3 y 7 + x2 y 8 − xy 9 + y 10 − y, 25 25 25 25   ẏ = − 6 x9 y − 182 x8 y 2 − 104 x7 y 3 − 352 x6 y 4 − 164 x5 y 5 − 28 x4 y 6     5 25 25 25 25 5     136 48 46 18 − x3 y 7 + x2 y 8 − xy 9 + y 10 + x, 25 25 25 25 bifurcate at least 97 limit cycles of small amplitude. Proof. The center in the statement is obtained following the same procedure than in the proof of Proposition 4.4 but with the change R = r3/9 . The proof of the existence [1] of such limit cycles of small amplitude follows again from Corollary 1.5 computing Lk , for k = 1, . . . , 130, and checking that rank A126 = 97. We notice that rank Ak = k, for k = 1, . . . , 58. Curiously, the expressions of the first three Lyapunov constants coincide with the ones of the previous proposition.  5. Perturbing piecewise systems of degrees 3, 4, and 5 This section is devoted to proving Theorem 1.2. We also show that there are other cubic centers having cyclicity also higher or equal than 26 as it was proved in [29] but using second-order developments. The following result only uses first-order analysis. The best lower bound for the local cyclicity, up to our knowledge, for piecewise quadratic vector fields is [18]. As we have no new nor better results for n = 2 we have concentrated our efforts to high degrees. But, the computational difficulties arise for n ≥ 6 even using the parallelization algorithms. This is why we have stopped in degree five polynomial piecewise differential equations. So we do not write them here again. As in the previous section, we have computed some more Lyapunov constants than the strictly necessary to get the proofs. With the aim to convince ourselves that we have obtained the optimal results for first-order studies. From the proofs, it can be seen also that the classical study using only the return map and the Lyapunov constants provide worse results. See again [29]. The general perturbed system considered in this section is     Pn Pn   (ẋ, ẏ) = Pc (x, y) + + k ` ak` x y , Qc (x, y) + + k ` bk` x y for y ≥ 0, k+`=0 k+`=0  Pn Pn  (20)   − k ` − k `  (ẋ, ẏ) = Pc (x, y) + ak` x y , Qc (x, y) + bk` x y for y < 0. k+`=0 k+`=0 MELNIKOV METHOD USING LYAPUNOV CONSTANTS 13 Proposition 5.1. There exist polynomial piecewise perturbations of degree n = 3 as (20) such that 26 crossing limit cycles of small amplitude bifurcate from the origin of system   168 2 8252 2968 2  ẋ = −y +  x + xy − y   125 125 125   44436 3 533631 2 592508 2 69552 3    − x − x y+ xy + y , 625 625 625 625   4974 2 9164 2874 2   ẏ = x − x + xy + y   125 125 125     232848 3 910392 2 385231 2 407064 3 + x − x y+ xy + y . 625 625 625 625 Proof. The system in the statement is the Bondar–Sadovskiı̆ cubic center system given by the rational first integral (19), see again [8], but rotated with the matrix ! 3/5 −4/5 . 4/5 3/5 [1] The proof of the statement follows from Corollary 2.2 computing Lk , for k = 2, . . . , 32, and checking that rank A28 = 25. We notice that rank Ak = k − 1, for k = 2, . . . , 24. Because of the size, we only show the linear developments of the first three Lyapunov constants: [1] 2 − − − L2 = (a+ + + 11 − a11 + b20 − b20 + 2b02 − 2b02 ), 3 [1] 1 L3 = (126000πa− − − − − 02 − 25200πa11 + 1125πa12 + 163800πa20 + 3375πa30 9000 + 126000πa+ + + + + 02 − 25200πa11 + 1125πa12 + 163800πa20 + 3375πa30 − 135900πb− − − − − + 02 + 3375πb03 + 18900πb11 − 85500πb20 + 1125πb21 − 135900πb02 − + 3375πb+ + + + + 03 + 18900πb11 − 85500πb20 + 1125πb21 − 577664a11 + 577664a11 − 1155328b− − + + 02 − 577664b20 + 1155328b02 + 577664b20 ), [1] 1 L4 = (4738650000πa− − − 02 − 947730000πa11 + 42309375πa12 2812500 + 6160245000πa− − + + 20 + 126928125πa30 + 4738650000πa02 − 947730000πa11 − + 42309375πa+ + + 12 + 6160245000πa20 + 126928125πa30 − 5110972500πb02 + 126928125πb− − − − 03 + 710797500πb11 − 3215512500πb20 + 42309375πb21 − 5110972500πb+ + + + 02 + 126928125πb03 + 710797500πb11 − 3215512500πb20 − − − + 42309375πb+ 21 + 1547032000a02 − 84000000a03 − 14072769976a11 + 3736000a− − − − 12 + 2604141664a20 − 37356000a21 + 41772000a30 − 1547032000a+ + + + 02 + 84000000a03 + 14072769976a11 − 3736000a12 − − 2604141664a+ + + 20 + 37356000a21 − 41772000a30 − 27939203952b02 + 68208000b− − − − 03 + 628004832b11 + 4644000b12 − 14413096320b20 + 42424000b− − + + 21 + 9390000b30 + 27939203952b02 − 68208000b03 − 628004832b+ + + + + 11 − 4644000b12 + 14413096320b20 − 42424000b21 − 9390000b30 ).  We remark that, in the above result, we have rotated because, without it, the local cyclicity takes a lower value, up to first-order analysis. We have not proved that this 14 LUIZ F. S. GOUVEIA AND J. TORREGROSA rotation plays any special role. Except that the entries of the matrix are small rational numbers. With other rotation matrices, the result will be generically the same. This example shows how the separation straight line change the local cyclicity of the origin. Proposition 5.2. There exist polynomial piecewise perturbations of degree n = 4 as (20) such that 43 crossing limit cycles of small amplitude bifurcate from the origin of system ( ẋ = x4 − 6x2 y 2 + y 4 + x3 − 3xy 2 + x2 − y 2 − y, ẏ = 4x3 y − 4xy 3 + 3x2 y − y 3 + 2xy + x. Proof. The above vector field has a center at the origin because it is the holomorphic system of degree four ż = i z+z 2 +z 3 +z 4 , but written in Cartesian coordinates, z = x+i y. [1] The proof follows also from Corollary 2.2 computing Lk , for k = 2, . . . , 60, and checking that rank A60 = 42. We notice that rank Ak = k − 1, for k = 2, . . . , 38. Because of the size, we only show the linear developments of the first three Lyapunov constants: [1] 2 L2 = (a+ − a− + − 11 + b20 − b20 ), 3 11 [1] 1 − − − − L3 = π(a+ + + + 02 + a02 + a12 + a12 + 3a30 + 3a30 + 3b03 + 3b03 8 − − − 4b+ + 20 − 4b20 + b21 + b21 ), [1] − − − − L4 = 8b+ + + + + 20 − 8b20 + 4a02 − 4a02 − 6b21 + 6b21 + 4b22 − 4b22 − 2a11 + 2a− + − + − + − + 11 + 4a12 − 4a12 + 4a13 − 4a13 − 16a20 + 16a20 + 6a30 − 6a− + − + − + − + − 30 + 6a31 − 6a31 − 4b03 + 4b03 + 16b04 − 16b04 − 14b11 + 14b11 .  Proposition 5.3. There exist polynomial piecewise perturbations of degree n = 5 as (20) such that 65 crossing limit cycles of small amplitude bifurcate from the origin of system ( ẋ = x5 − 10x3 y 2 + 5xy 4 + x4 − 6x2 y 2 + y 4 + x3 − 3xy 2 + x2 − y 2 − y, ẏ = 5x4 y − 10x2 y 3 + y 5 + 4x3 y − 4xy 3 + 3x2 y − y 3 + 2xy + x. Proof. The vector field in the statement is, written in Cartesian coordinates (z = x + i y), the holomorphic system of degree five ż = i z + z 2 + z 3 + z 4 + z 5 . The proof follows again [1] from Corollary 2.2 computing Lk , for k = 2, . . . , 80, and checking that rank A72 = 64. We notice that rank Ak = k − 1, for k = 2, . . . , 58. Because of the size, we only show the linear developments of the first three Lyapunov constants: [1] 2 − − − L2 = (a+ + + 11 − a11 + 2(b02 − 2b02 ) + b20 − b20 ), 3 [1] 1 − − − − L3 = π(a+ + + + 12 + a12 + 3(a30 + a30 ) − b02 − b02 + 3(b03 + b03 ) 8 − − − 4(b+ + 20 + b20 ) + b21 + b21 ), [1] 2 − − − − − L4 = (3b+ + + + + 40 − 3b40 + 3b21 − 3b21 + 4b20 − 4b20 + 2b03 − 2b03 − 7b11 + 7b11 15 − 3a− + − + + − + − + 30 + 3a30 − 8b04 + 8b04 − 8a20 + 8a20 + 3a31 − 3a31 + 2a13 − 2a13 − − 6b− + + − + − + − + − 02 + 6b02 + 2a12 − 2a12 + 2b22 − 2b22 − a11 + a11 − 12a02 + 12a02 ).  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In Mathematical Sciences with Multidisciplinary Applications, pages 595–625, Cham, 2016. Springer International Publishing. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellater- ra, Barcelona (Spain); Departamento de Matemática, Universidade Estadual Paulista, 15054-000 São José do Rio Preto (Brazil) Email address: fernandosg@mat.uab.cat, fernando.gouveia@unesp.br Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellater- ra, Barcelona (Spain); Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Barcelona (Spain) Email address: torre@mat.uab.cat