This is a preprint of: “On the number of limit cycles in generalized Abel equations”, Jianfeng Huang, Joan Torregrosa, Jordi Villadelprat, SIAM J. Appl. Dyn. Syst., vol. 19(4), 2343–2370, 2020. DOI: [10.1137/20M1340083] ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT Abstract. Given p, q ∈ Z≥2 with p 6= q, we study generalized Abel differential equations dx = A(θ)xp + B(θ)xq , dθ where A and B are trigonometric polynomials of degrees n, m ≥ 1, respectively, and we are interested in the number of limit cycles (i.e., isolated periodic orbits) that can have. More concretely, in this context an open problem is to prove the existence of an integer, depending only on p, q, m, and n and that we denote by Hp,q (n, m), such that the above differential equation has at most Hp,q (n, m) limit cycles. In the present paper, by means of a second order analysis using Melnikov functions, we provide lower bounds of Hp,q (n, m) that, to the best of our knowledge, are larger than the previous ones appearing in the literature. In particular, for classical Abel differential equations (i.e., p = 3 and q = 2), we prove that H3,2 (n, m) ≥ 2(n + m) − 1. 1. Introduction and statements of main results The study of the existence of periodic orbits in ordinary differential equations has been an interesting problem for years in many areas of mathematics, particularly in qualitative theory of differential equations. In this area of interest, when we focus on planar polynomial vector fields, one of the most renowned classical problems arises: to know the number and location of isolated periodic orbits, the so-called limit cycles, in terms of its degree n. The study of this problem began at the end of the 19th century with the seminal works by Poincaré, but takes its name after Hilbert because of his famous list of unsolved problems published in 1900. From the original list of 23 problems, the 16th is still open, in particular its second part. More precisely (see [26] or [36] for details), the ‘existential’ Hilbert’s 16th problem is to prove that for any n ≥ 2 there exists a finite number H(n) such that any polynomial vector field of degree ≤ n has less than H(n) limit cycles. Motivated by 16th Hilbert’s problem, a very related line of research is to investigate the periodic solutions of scalar differential equations k dx X ẋ := = ai (θ)xi , dθ i=0 where ai are periodic analytic functions. In this context an isolated periodic solution is called limit cycle and it occurs that its number increases with k. (The reader is referred to [15] for an enlightening explanation of this fact.) Linear differential equations have at most 1 limit cycle, whereas the quadratic ones have at most 2. The latter are known as Riccati equations and the upper bound follows from the fact that the return map is a Möbius function. Nevertheless the situation is more intricate for degree three. The well-known trigonometric Abel differential equation writes as ẋ = A(θ)x3 + B(θ)x2 + C(θ)x, (1) being A, B, and C trigonometric polynomials. Pliss [33] proves using the Schwarzian derivative that if A does not change sign then the maximum number of limit cycles is 3. However it was Lins-Neto [31] the first to show that, in general, there is no upper bound for the number of limit cycles. Indeed, he proves that for every positive integer ` there exists an Abel differential 2010 Mathematics Subject Classification. Primary: 34C07. Secondary: 34C05 37C10. Key words and phrases. Generalized Abel equations, Melnikov theory, second order perturbation, limit cycles. 1 2 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT equation with periodic coefficients having ` limit cycles. He does it by studying the perturbation ẋ = εA(θ)x3 + B(θ)x2 , where both coeficients are trigonometric polynomials of degree `. In this paper we are interested in the number of limit cycles of generalized Abel equations in terms of the trigonometric polynomial degree of their coefficients. More concretely, we study the differential equation ẋ = A(θ)xp + B(θ)xq , (2) where A and B are trigonometric polynomials and p, q ∈ Z≥2 , with p 6= q. We say that a solution x = x(θ) of this differential equation is a periodic orbit if it satisfies x(−π) = x(π). As before, a periodic orbit is called limit cycle if it is isolated in the set of periodic orbits. For fixed exponents p and q, we define the Hilbert number H = Hp,q (n, m) as the maximum number of limit cycles that the differential equation (2) can have for any trigonometric polynomials A and B of degrees n and m, respectively. For the classical Abel equation, i.e., (p, q) = (3, 2), it is known as the Smale–Pugh problem, see [37]. So far it is even unknown whether H exists. Our main contribution in the present paper is to provide a lower bound of H that, to the best of our knowledge, improves the previous ones appearing in the literature. We shall restrict ourselves to the case n, m ≥ 1 because from the results in [16, 18, 31] the problem is completely solved when n = 0 or m = 0. It is proved in [18] that the upper bound for the number of limit cycles of the differential equation (1) is three provided that A or B does not change sign. The authors use this result in order to bound the number of hyperbolic limit cycles in some planar polynomial differential systems. (This idea is also used in many other papers, see [1, 2, 12, 20, 25] for example.) The natural extension of this result to the equation ẋ = A(θ)xp + B(θ)xq + C(θ)x is considered in [16] where, under the same hypothesis, it is proved that the upper bound is 5 (respectively, 4) when max(p, q) is odd (respectively, even). This result, particularised to C = 0, gives Hp,q (n, m) in case that nm = 0. Therefore, as we explain before, for the the problem that we tackle in the present paper is natural to assume that n, m ≥ 1. Certainly the problem is much more difficult when the coefficients A and B do change sign. This is the case studied in [4] where, under some other hypotheses on the coefficients, it is proved that only one limit cycle exists. Another upper bounds for the number of periodic solutions are given in [29] under some conditions on the number of zeros of B(θ). More generally, it Pis also to nbe referred the result in [3], where it is proved that the differential equation ẋ = m a i=0 i (θ)x i , with 1 ≤ n ≤ n and a periodic i i analytic functions, can have at most 3n − 1 limit cycles provided some transversal conditions are verified. The extension of the aforesaid Lins-Neto result to generalized Abel differential equations is also done in [16]. There are also some problems coming from planar polynomial differential systems that can be brought to a differential equation as in (1) or (2). Among others, the homogeneous nonlinear perturbations of the harmonic oscillator or the so-called rigid systems (θ̇ = 1), see [10] and [19], respectively. More recently, it is shown in [5] that Abel differential equations (2) have also limit cycles of alien type. They are not of small amplitude, like in a Hopf bifurcation, neither arising by the perturbation of an annulus foliated of periodic orbits. Among the long list of references to Abel differential equations (there are more than three hundred in the literature) there are some that reduce real problems to this type of differential equation. In [13] it is computed an approximation of an unstable limit cycle that appears in an Abel equation arising in a tracking control problem that affects an elementary, nonminimum phase, second order bilinear power converter. The authors in [23] study a second order differential equation that describes the relativistic evolution of a causal dissipative cosmological fluid in a conformally flat space-time. They reduce this evolution equation to an Abel differential equation. The same authors, in a more recent work [24], consider quasi-stationary (traveling wave type) solutions of a nonlinear reaction-diffusion equation, which describes the evolution of glioblastomas. These aggressive primary brain tumors are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth through traveling waves is reduced to a differential equation of Abel type. The relationship between the Einstein–Friedmann and Abel equations is studied in [38]. In that work the authors ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 3 demonstrate how the latter might be applied to the inflationary analysis in a spatially-flat Friedmann universe filled with a real-valued scalar field. They use an Abel equation to provide the necessary and sufficient conditions for both slow-rolling and inflation are estimated with respect to the initial value of the field. Coming back to the original Hilbert’s 16th problem, due to the difficulty to find uniform upper bounds for even subclasses of polynomial differential systems, some weak versions have appeared during last decades. One of them was proposed by Arnold [8] and it focus on the study of limit cycles bifurcating from the period annulus of Hamiltonian systems. Closely related to this, our approach in order to improve the lower bounds for Hp,q (n, m) when n, m ≥ 1 is to consider a second order perturbation of the generalized Abel differential equation ẋ = (sin θ + εP1 (θ) + ε2 P2 (θ))xp + (εQ1 (θ) + ε2 Q2 (θ))xq , (3) where the coefficients of the perturbation are trigonometric polynomials of degrees n and m, that is n X m X   Pi (θ) = bi0 + aik sin(kθ) + bik cos(kθ) and Qi (θ) = di0 + cik sin(kθ) + dik cos(kθ) k=1 k=1 for i = 1, 2. We note that the parameter space associated to (2) is R2(n+m+1) and that by taking (3) we study the perturbation of a specific point, say ξ0 ∈ R2(n+m+1) , corresponding to ẋ = sin θ xp . The coefficients of Pi and Qi , once determined, give a curve in R2(n+m+1) passing through ξ0 at ε = 0. In what follows, for the sake of convenience, we will treat these coefficients as parameters too, setting µ = (a, b, c, d) where a = (a1k , a2k ; k = 1, 2, . . . , n), c = (c1k , c2k ; k = 1, 2, . . . , m), (4) b = (b1k , b2k ; k = 0, 1, . . . , n), d = (d1k , d2k ; k = 0, 1, . . . , m). Thus µ ∈ R4(n+m+1) although as we already mentioned the ‘ambient’ parameter space is R2(n+m+1) . That being said we denote by x(θ, ρ; µ, ε) the solution of (3) with initial condi- tion x(−π, ρ; µ, ε) = ρ. One can readily prove (see Corollary 2.3) that the unperturbed system verifies x(π, ρ; µ, 0) = ρ for all ρ ∈ I := (κp , +∞), where ( −∞, if p is odd, κp := 1 − p−1 (5) −(2(p − 1)) , if p is even. Then, thanks to the analytic dependence of solutions with respect to initial conditions and parameters, we can write the Taylor series of the Poincaré transition map as ∞ X x(π, ρ; µ, ε) = ρ + εi Mi (ρ; µ), i=1 where Mi is an analytic function on I × Setting R+ = (0, +∞) and R− = (−∞, 0), R4(n+m+1) . in our first main result we study the case M1 = 0 and M2 6= 0 assuming that n, m ≥ 1. Theorem 1.1. The function M1 ( · ; µ) vanishes identically if, and only if, µ ∈ L = {µ ∈ R4(n+m+1) : b10 = d10 = · · · = d1m = 0}. For a fixed µ ∈ L , let K ± be the number of zeros of M2 ( · ; µ) on I ∩ R± taking multiplicities into account. Then the following properties hold: (a) If p and q are odd, then K ± ≤ n + m. Moreover, there exists µ0 ∈ L such that M2 ( · ; µ0 ) has 2(n + m) simple zeros in I \ {0}. (b) If p is odd and q is even, then K ± ≤ n + m and both equalities do not hold simultaneously. Moreover: (i) When p < q, there exists µ0 ∈ L such that M2 ( · ; µ0 ) has 2(n + m) − 1 simple zeros in I \ {0}. (ii) When p > q, there exists µ0 ∈ L such that M2 ( · ; µ0 ) has 2(n + m − 1) simple zeros in I \ {0}. 4 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT (c) If p is even, q is odd, and p < q, then K + + K − ≤ n + m. Moreover, there exists µ0 ∈ L such that M2 ( · ; µ0 ) has n + m simple zeros in I \ {0}. (d) If p is even and, either q is even or p > q, then K + + K − ≤ n + m + 1. Moreover, there exists µ0 ∈ L such that M2 ( · ; µ0 ) has n + m + 1 simple zeros in I \ {0}. Since x(π, 0; µ, ε) = 0 for all ε, note that Mi (0; µ) = 0 for all i ≥ 1 but we stress that the zero ρ = 0 is not counted in the previous result. That being said, in order to be consistent with the abovementioned papers on Abel differential equations, we do count this zero limit cycle in our next main result. Before giving its statement let us mention that in what follows we shall call Mi the Melnikov function of ith order for the perturbed differential equation (3). These functions are clearly analytic on I and, by applying the Weierstrass Preparation Theorem, the number of zeros (multiplicities taking into account) of the first non-identically zero Melnikov function gives an upper bound for the number of roots of x(π, ρ; µ, ε) = ρ for ε ≈ 0. In other words, it provides an upper bound for the number of limit cycles that bifurcate from the continuum of periodic orbits of the unperturbed differential equation. In its turn a lower bound is given by the number of simple zeros thanks to the Implicit Function Theorem. In short this is how the general lower bound of the Hilbert number that we give in our next result follows from Theorem 1.1. Theorem 1.2. The Hilbert number for the Abel differential equation (2) with p = 3 and q = 2 verifies H3,2 (n, m) ≥ 2(n + m) − 1, where n, m ≥ 1 are respectively the degrees of the trigonometric polynomials A and B. Moreover, H3,2 (1, 3) ≥ 8 and H3,2 (4, 1) ≥ 10. As we explain above, the general lower bound in the first assertion follows by Melnikov theory. (The Melnikov theory for planar autonomous differential equations is equivalent to the so-called averaging theory, see [9].) By contrast the second assertion, which improves the bound by one limit cycle in two particular cases, follows by using Lyapunov constants. In order to make this clear and to facilitate the reading of the paper, for reader’s convenience we prove the second assertion separately in an appendix, where we also introduce the basic notions on Lyapunov constants. There are two previous papers with results about the Hilbert number of Abel differential equations that should be referred. Recall that the general lower bound in Theorem 1.2 is obtained by a second order perturbation in ε. The authors in [6] give lower bounds for H3,2 (1, m) and H3,2 (n, 1) by a first order perturbation. On the other hand, the authors in [17] give a lower bound for Hp,q (n, 1) by a first order perturbation as well. The paper is organized in the following way. In Section 2 we study the perturbed differential equation ẋ = h(x)f (θ) + H(θ, x; ε) and we give the expression of its first non-identically zero Melnikov function (Theorem 2.1). This is a rather general result that, we believe, could be very useful in the development of further research on the issue. Next, in Section 3, we recall the notion of Chebyshev system and explain the related basic results. We also state a key result from [17] that turns out to be very important for our purposes (Theorem 3.4). Section 4 is devoted to show that the Melnikov function M2 for the perturbed differential equation (3) belongs to an appropriate Chebyshev system (Proposition 4.4). The proofs of our two main results are given in Section 5. Finally in the appendix we prove by using Lyapunov constants that H3,2 (1, 3) ≥ 8 and H3,2 (4, 1) ≥ 10 (Propositions A.2 and A.3, respectively), which improve in these particular cases the general lower bound that we obtain by using Melnikov theory of second order. These two lower bounds also improve some previous ones obtained using Lyapunov constants as well. Here we follow a new approach using first and second order developments of the Lyapunov constants at some specific parameters having a center. Finally we explain some numerical evidences in order to increase by using Lyapunov constants the lower bound given in Theorem 1.2 for H3,2 (1, 4) and H3,2 (2, 3). ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 5 2. Melnikov functions In this section we consider the perturbed differential equation dx = h(x)f (θ) + H(θ, x; ε), (6) dθ where • h is analytic on R, Rπ • f is a 2π-periodic analytic function with −π f (s)ds = 0, and • H is an analytic function on R × R × (−ε0 , ε0 ), for some ε0 > 0, such that θ 7→ H(θ, x; ε) is 2π-periodic and H(θ, x; 0) ≡ 0. Given ρ ∈ R, let x(θ, ρ; ε) denote the solution of (6) such that x(−π, ρ; ε) = ρ. (We shall use sometimes the more compact notation xε (θ, ρ) for the sake of shortness.) In this section we assume that the unperturbed differential equation, i.e., (6) with ε = 0, has a stripe of periodic orbits. More precisely, that there exists an open interval I of initial conditions such R πρ; 0) = ρ for all ρ ∈ I. Note that, due to H(θ, x; 0) ≡ 0, a necessary condition for this that x(π, is that −π f (s)ds = 0. Under this assumption, a sufficient condition for the existence of such an interval is that h vanishes at some point x0 ∈ R. This is precisely the setting that we have for the perturbed differential equation (3), for which x0 = 0 and, as we will see, I = (κp , +∞) where κp < 0 is given in (5). By the analytic dependence of solutions with respect to initial conditions and parameters, the solution x(θ, ρ; ε) is well defined and analytic for all (θ, ρ, ε) ∈ [−π, π] × U, where U is an open neighbourhood of I × {0} in R2 . We can thus consider the Taylor series of x(π, ρ; ε) at ε = 0, ∞ X x(π, ρ; ε) = ρ + Mi (ρ)εi , i=1 where each Mi is an analytic function on I. We aim to study the fixed points of ρ 7→ x(π, ρ; ε) that persist for small ε 6= 0 and to this end an explicit expression of the first Mi 6= 0 is needed. Our first result is addressed to this and in order to state it we introduce some more notation. We write the Taylor series of the perturbation at ε = 0 as ∞ X H(θ, x; ε) = `i (θ, x)εi . i=1 We also use the differential operator Θx := h(x)∂x (k and denote Θkx = Θx ◦ · · · ◦ Θx . Furthermore we consider the incomplete exponential Bell polynomials Bn,k (x1 , x2 , . . . , xn−k+1 ), which can be defined recursively by means X  n − 1 n−k+1 Bn,k = xi Bn−i,k−1 , i−1 i=1 setting B0,0 = 1, Bn,0 = 0 for n ≥ 1, and B0,k = 0 for k ≥ 1 (see for instance [14, 27]). 1 Theorem 2.1. If M1 = M2 = · · · = Mn−1 = 0 then Mn (ρ) = n! h(ρ)Ln (π, ρ) for all ρ ∈ I such that h(ρ) 6= 0, where the sequence {Ln }n∈Z≥1 is defined recursively by means n−1 X n! X i Z θ     `n−i (s, x) Ln (θ, ρ) = Θxk Bi,k L1 (s, ρ), L2 (s, ρ), . . . , Li−k+1 (s, ρ) ds. i! −π h(x) x=x0 (s,ρ) i=0 k=0 Proof. For each (θ, ρ) ∈ [−π, π] × I, let the Taylor series of x(θ, ρ; ε) at ε = 0 writes as ∞ X xε (θ, ρ) = x0 (θ, ρ) + Si (θ, ρ)εi . i=1 Notice in particular that, by definition, Mi (ρ) = Si (π, ρ). 6 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT Fix any ρ0 ∈ I such that h(ρ0 ) 6= 0. We claim that there exists δ > 0 small enough such 1 that Mn (ρ) = n! h(ρ)Ln (π, ρ) for all ρ ∈ I with |ρ − ρ0 | < δ. Clearly, due to the arbitrariness of ρ0 , the result will follow once we prove this claim. With this aim in view note first that if for a given ρ ∈ I there exists θ∗ ∈ [−π, π] such that h(x0 (θ∗ , ρ)) = 0 then x0 (θ, ρ) = ρ for all θ ∈ [−π, π]. Thus, if we denote by Iˆ the connect component of I \ {x ∈ R : h(x) = 0} that contains ρ0 , then h(x0 (θ, ρ)) 6= 0 for all (θ, ρ) ∈ [ − π, π] × I. ˆ In other words, setting ˜ ˆ I := {x0 (θ, ρ) : θ ∈ [−π, π], ρ ∈ I }, we have that 0 ∈ ˜ / h(I). Accordingly, if Z ρ du G(ρ) := ρ0 h(u) then G : I˜ → G(I) ˜ is a well-defined diffeomorphism. One can readily verify that the coordinate change y = G(x) brings the differential equation (6) to dy b y; ε) := H(θ, x; ε) b y; ε), with H(θ, = f (θ) + H(θ, . (7) dθ h(x) x=G−1 (y) ˆ we denote by yε (θ, ρ) the solution of (7) with initial condition yε (−π, ρ) = ρ. For each ρ ∈ G(I) −1 ˜ for each fixed ρ there exists ε > 0 small enough such that Due to x0 (θ, G (ρ)) ∈ I,  yε (θ, ρ) = G xε (θ, G−1 (ρ)) for all θ ∈ [−π, π]. Consequently, by continuity, there exists δ > 0 small enough such that if |ρ − ρ0 | + |ε| < δ then  xε (θ, ρ) = G−1 yε (θ, G(ρ)) for all θ ∈ [−π, π]. b is an analytic function in a neighbourhood of any (θ, y, ε) ∈ [−π, π] × G(I) Clearly H ˜ × {0} and so we can consider its Taylor series at ε = 0, ∞ X b y; ε) = `i (θ, x) H(θ, `ˆi (θ, y)εi where `ˆi (θ, y) = . (8) h(x) x=G−1 (y) i=1 Let us consider at this point the Taylor series of yε (θ, ρ) at ε = 0, say ∞ X yε (θ, ρ) = y0 (θ, ρ) + Sbi (θ, ρ)εi , (9) i=1 ci (ρ) := Sbi (π, ρ), which is well-defined for any ρ ∈ G(I) and set M ˆ such that |G−1 (ρ) − ρ0 | < δ. Then, taking the derivative with respect to θ on both sides of the above equality, from (7) and (8) we get that ∞ ! 1 X ∂θ Sbn (θ, ρ) = ∂ n ε `ˆi (θ, yε (θ, ρ)) i n! ε i=1 ε=0 ∞ X X n   k i   1 n d ε n−k ˆ = ∂ `i (θ, y ε (θ, ρ)) n! k dεk ε ε=0 i=1 k=0 n     1 X n = i! ∂εn−i `ˆi (θ, yε (θ, ρ)) . n! i ε=0 i=1 Accordingly n X 1   n−1 X 1 i ˆ  ∂θ Sbn (θ, ρ) = ∂εn−i `ˆi (θ, yε (θ, ρ)) = ∂ε `n−i (θ, yε (θ, ρ)) . (10) (n − i)! ε=0 i! ε=0 i=1 i=0 By applying Faà di Bruno’s formula for the chain rule (see [14, 27]) we can assert that   Xi     ∂εi ˆ `j (θ, yε (θ, ρ)) = ∂yk `ˆj (θ, yε (θ, ρ)) Bi,k ∂ε yε (θ, ρ), ∂ε2 yε (θ, ρ), . . . , ∂εi−k+1 yε (θ, ρ) . k=0 ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 7 Thus, on account of ∂εk yε (θ, ρ) ε=0 = k!Sbk (θ, ρ), we get   ∂εi `ˆj (θ, yε (θ, ρ)) = ε=0 i X     ∂yk `ˆj (θ, y0 (θ, ρ)) Bi,k Sb1 (θ, ρ), 2Ŝ2 (θ, ρ), . . . , (i − k + 1)!Sbi−k+1 (θ, ρ) . k=0 Therefore, since Sbn (−π, ρ) = 0, from (10) it follows that Sbn (θ, ρ) = (11) n−1 X i Z   1X θ ∂yk `ˆn−i (s, y0 (s, ρ)) Bi,k (Sb1 (s, ρ), 2Sb2 (s, ρ), . . . , (i − k + 1)!Sbi−k+1 (s, ρ)) ds. i! −π i=0 k=0  P Recall now that xε (θ, ρ) = G−1 yε (θ, G(ρ)) = x0 (θ, ρ) + ∞ i i=1 Si (θ, ρ)ε for all θ ∈ [−π, π] provided that |ρ − ρ0 | + |ε| < δ. Hence, since (G−1 )0 (y) = h(G−1 (y)) by definition, 1   Mn (ρ) = Sn (θ, ρ)|θ=π = ∂εn G−1 yε (θ, G(ρ)) n! ε=0,θ=π 1    = ∂εn−1 (G−1 )0 yε (θ, G(ρ)) ∂ε yε (θ, G(ρ)) n! ε=0,θ=π 1    = ∂εn−1 h xε (θ, ρ) ∂ε yε (θ, G(ρ)) n! ε=0,θ=π    n−1 1 X n−1    = ∂εn−1−k h xε (θ, ρ) ∂εk+1 yε (θ, G(ρ)) n! k ε=0,θ=π k=0 n−1    1 X n−1  = ∂εn−1−k h xε (θ, ρ) (k + 1)!M b k+1 (G(ρ)), n! k ε=0,θ=π k=0 b i (ρ) = Sbi (π, ρ). If M where in the last equality we use (11) and M b1 = M b2 = · · · = M b n−1 = 0 then only the term for k = n − 1 remains and b n (G(ρ)) = h(ρ)Sbn (π, G(ρ)) Mn (ρ) = h(ρ)M for all ρ ∈ (ρ0 − δ, ρ0 + δ). Note that by arguing recursively we get the same equality assuming M1 = M2 = · · · = Mn−1 = 0. It only remains to express Sbn (θ, ρ) in terms of the solution x0 (θ, ρ) of (6). To this end notice that, on account of (G−1 )0 (y) = h(G−1 (y)) once again, from (8) we get   kˆ k `i (θ, x) ∂y `i (θ, y) = Θx . h(x) x=G−1 (y)  Accordingly, due to y0 (θ, ρ) = G x0 (θ, G−1 (ρ)) , we can assert that     k ˆ k `i (s, x) ∂y `i (s, y0 (s, ρ)) = Θx . h(x) x=x0 (s,G−1 (ρ)) Taking this into account and setting Li (θ, ρ) := i!Sbi (θ, G(ρ)) for all i ∈ N, from (9) we obtain Ln (θ, ρ) = n!Sbn (θ, G(ρ)) = n−1 X n! X i Z θ     k `i (s, x) Θx Bi,k L1 (s, ρ), L2 (s, ρ), . . . , Li−k+1 (s, ρ) ds, i! −π h(x) x=x0 (s,ρ) i=0 k=0 and hence Mn (ρ) = h(ρ)Sbn (π, G(ρ)) = n! 1 h(ρ)Ln (π, ρ) for all ρ ∈ (ρ0 − δ, ρ0 + δ). This proves the claim and concludes the proof of the result.  8 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT Note that if we take ρ̂ ∈ I such that h(ρ̂) = 0 then x0 (θ, ρ̂) = ρ̂ for all θ. It happens then that the function Ln (θ, ρ) is not well-defined at ρ = ρ̂ due to the denominator h(x0 (s, ρ)) in its integrand. However, since Mn is continuous (in fact analytic) at ρ = ρ̂, the limit of h(ρ)Ln (π, ρ) as ρ tends to ρ̂ exists and is equal to n!Mn (ρ̂). Thus the singularity of h(ρ)Ln (π, ρ) at ρ = ρ̂ is removable. Our next result shows this for the perturbation associated to the differential equation (3), for which we have ρ̂ = 0. In its statement recall that κp is given in (5). Remark 2.2. For reader’s convenience we give the first terms in the recurrence of Theorem 2.1. Since B0,0 = 1, B1,0 = B2,0 = 0, B1,1 (x1 ) = x1 , B2,1 (x1 , x2 ) = x2 , and B2,2 (x1 ) = x21 , we get Z θ `1 (s, x) L1 (θ, ρ) = ds, −π h(x) x=x0 (s,ρ) Z θ     `2 (s, x) `1 (s, x) L2 (θ, ρ) = 2 + Θx L1 (s, ρ) ds, −π h(x) h(x) x=x0 (s,ρ) and Z θ    `3 (s, x) `2 (s, x) L3 (θ, ρ) = 3 2 + 2Θx L1 (s, ρ) −π h(x) h(x)      `1 (s, x) 2 `1 (s, x) 2 +Θx L2 (s, ρ) + Θx (L1 (s, ρ)) ds. h(x) h(x) x=x0 (s,ρ) Corollary 2.3. Following the previous notation, for the differential equation (3) we have ρ x(θ, ρ; 0) = 1/(p−1) , 1 + ρp−1 (p − 1)(1 + cos θ) so that x(π, ρ; 0) = ρ for all ρ ∈ I = (κp , +∞). Moreover Z π p  M1 (ρ) = ρ P1 (θ) + Q1 (θ)x0 (θ, ρ)q−p dθ −π and if M1 (ρ) ≡ 0 then Z π p  M2 (ρ) = ρ P2 (θ) + Q2 (θ)x0 (θ, ρ)q−p + (q − p)Q1 (θ)S(θ, ρ)x0 (θ, ρ)q−1 dθ, −π where Z θ  S(θ, ρ) = P1 (s) + Q1 (s)x0 (s, ρ)q−p ds. −π Proof. The statement concerning the unperturbed equation follows from an easy computation. The second part follows by Theorem 2.1 particularizing the expressions given in Remark 2.2 for the case f (θ) = sin θ, h(x) = xp , and `i (θ, x) = Pi (θ)xp + Qi (θ)xq , for i = 1, 2.  3. ECT-systems We begin this section by recalling some properties about Chebyshev systems, then we extend some results in [17]. All of them will be necessary in order to prove our main results. Definition 3.1. Let f0 , f1 , . . . , fn be analytic functions on an open interval I ⊂ R. The ordered set (f0 , f1 , . . . , fn ) is an Extended Complete Chebyshev system (for short, an ECT-system) on I if for each k = 0, 1, 2, . . . , n every nontrivial linear combination α0 f0 (x) + α1 f1 (x) + · · · + αk fk (x) has at most k isolated zeros on I counted with multiplicities. ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 9 Definition 3.2. Let f0 , f1 , . . . , fk be analytic functions on an open interval I ⊂ R. Then f0 (x) ··· fk (x) (i)  f00 (x) ··· fk0 (x) W [f0 , f1 , . . . , fk ](x) = det fj (x) 0≤i,j≤k = .. .. .. . . . (k) (k) f0 (x) · · · fk (x) is the Wronskian of (f0 , f1 , . . . , fk ) at x ∈ I. The following is a well-known result (see for instance [28, 32]) that enables to characterize Chebyshev systems in terms of Wronskians. Lemma 3.3. (f0 , f1 , . . . , fn ) is an ECT-system on an open interval I ⊂ R if and only if, for each k = 0, 1, 2, . . . , n, W [f0 , f1 , . . . , fk ](x) 6= 0 for all x ∈ I. To study the zeros of the Melnikov functions in Corollary 2.3 we will apply [17, Theorem A], that we state next for reader’s  convenience. In its statement g is an analytic function, Ig is the connected component of y ∈ R : 1 − yg(θ) > 0 for all θ ∈ [−π, π] containing the origin and, for each k ∈ Z≥0 and α ∈ R, Z π g k (θ) Tk,α (y) := α dθ for all y ∈ Ig . (12) −π 1 − yg(θ) Theorem 3.4. Consider α ∈ R and n ∈ Z≥0 . The following holds: / Z≤0 , then (T0,α , T1,α , . . . , Tn,α ) is an ECT-system on Ig . (a) If α ∈ (b) If α ∈ Z≤0 , then (T0,α , T1,α , . . . , Tn,α ) is an ECT-system on Ig if and only if n ≤ −α. The following technical lemmas extend some of the results in [17]. Lemma 3.5. If α ∈ R and k ∈ Z≥0 , then (y α Tk,α (y))0 = αy α−1 Tk,α+1 (y) for all y ∈ Ig ∩(0, +∞). Proof. This is an easy consequence of the following computation Z π Z π g(θ)k g(θ)k+1 (y α Tk,α (y))0 = αy α−1 α dθ + αy α α+1 dθ = αy α−1 Tk,α+1 (y). −π 1 − yg(θ) −π 1 − yg(θ)  Lemma 3.6. If α ∈ R \ Z≤0 and n ∈ Z≥0 then  (a) 1, y α T0,α (y), . . . , y α Tn,α (y) is an ECT-system on Ig ∩ (0, +∞), and (b) 1, (−y)α T0,α (y), . . . , (−y)α Tn,α (y) is an ECT-system on Ig ∩ (−∞, 0). Proof. For the sake of shortness let us prove the first assertion only (the other one follows exactly in the same way). We claim that, for each k = 0, 1, . . . , n, W [1, y α T0,α (y), . . . , y α Tk,α (y)] = (αy α−1 )k+1 W [T0,α+1 , T1,α+1 , . . . , Tk,α+1 ] (y) for all y ∈ Ig ∩ (0, +∞). Notice that, by applying Lemma 3.3 and Theorem 3.4, the result will follow once we show the claim. To this end a computation shows that   W [1, y α T0,α (y), . . . , y α Tk,α (y)] = W (y α T0,α (y))0 , . . . , (y α Tk,α (y))0   = W αy α−1 T0,α+1 (y), . . . , αy α−1 Tk,α+1 (y) = (αy α−1 )k+1 W [T0,α+1 , . . . , Tk,α+1 ] (y), where the second equality follows by applying Lemma 3.5 and the third one by the so-called Hesse–Christoffel’s identity (see [30, 34] for instance). This proves the validity of the claim and hence the result follows.  10 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT 4. Melnikov functions for the differential equation (3) Recall that Corollary 2.3 provides an expression of M2 assuming that M1 = 0. Our goal in this section is to write it as a linear combination of functions belonging to an ECT-system. This will be done in Proposition 4.4. With this aim in view we first particularise the integrals Tk,α defined in (12) with a specific choice of function g and parameter α that is very related with the solution of the unperturbed system. In order to stress this, and for reader’s convenience, we introduce the following additional notation Z π g k (θ) q−p Ik (y) := α dθ with g(θ) = −(p − 1)(1 + cos θ) and α = . (13) −π 1 − yg(θ) p−1 Related to this we also define Z π cos(kθ) Ck (y) :=  q−p dθ. (14) −π 1 + y(p − 1)(1 + cos θ) p−1 As it will be clear in a moment, these integrals constitute the building blocks for the Melnikov functions of the perturbed differential equation (3). In what follows recall that κp is given in (5).  Proposition 4.1. 1, ρq−p I0 (ρp−1 ), . . . , ρq−p In (ρp−1 ) is an ECT-system on (κp , 0) and (0, +∞) for every n ∈ Z≥0 . In addition the following equality between linear spans holds h1, ρq−p I0 (ρp−1 ), . . . , ρq−p In (ρp−1 )i = h1, ρq−p C0 (ρp−1 ), . . . , ρq−p Cn (ρp−1 )i. Proof. Note, see (12) and (13), that Ik (y) is Tk,α (y) particularised with g(θ) = −(p−1)(1+cos θ) q−p and α = p−1 . One can readily see that in this case the connected component of  y ∈ R : 1 − yg(θ) > 0 for all θ ∈ [−π, π]   −1 q−p containing the origin turns out to be Ig = 2(p−1) , +∞ . Moreover, since p−1 ∈ (−1, +∞) \ {0} due to p, q ∈ Z≥2 with p 6= q, we have that α ∈ R \ Z≤−1 . Consequently, by applying Lemma 3.6,  (i) 1, y α I0 (y), . . . , y α In (y) is an ECT-system on (0, +∞),  and   −1 (ii) 1, (−y)α I0 (y), . . . , (−y)α In (y) is an ECT-system on 2(p−1) ,0 . Next we make the substitution y = ρp−1 . This formally corresponds to compose each function with ρ 7→ ρp−1 , which restricted to (−∞, 0) and (0, +∞) is a diffeomorphism. In the first case we get that  1, ρq−p I0 (ρp−1 ), . . . , ρq−p In (ρp−1 ) (15) is an ECT-system on (0, +∞) for every p and that it is an ECT-system on (−∞, 0) in case that p is odd. In the second case we obtain that  1, (−ρ)q−p I0 (ρp−1 ), . . . , (−ρ)q−p In (ρp−1 ) 1 − is an ECT-system on (−(2(p − 1)) p−1 , 0) if p is even. On account of the fact that (−1)q−p is − 1 constant this implies that (15) is an ECT-system on (−(2(p−1)) p−1 , 0) if p is even. Accordingly, on account of the definition of κp given in (5), we have proved so far the validity of the first assertion in the statement. Finally, the assertion concerning the linear spans follows by noting that if we define Ek := h1, cos θ, . . . , cos(kθ)i then dim(Ek ) = k + 1 and Ek = h1, cos θ, . . . , cosk θi = h1, g(θ), . . . , g k (θ)i. To get the first equality one can use that cos(kθ) = Tk (cos θ), where Tk is the Chebyshev polynomial of the first kind and degree k. This proves the result.  Next result provides a more explicit expression of the Melnikov functions M1 and M2 for (3). In its statement we point out that aik , bik , cik , and dik are the coefficients of the trigonometric polynomials in the perturbation and Ck is the function defined in (14). ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 11 Proposition 4.2. The following holds for the perturbed differential equation (3). P  (a) The first Melnikov function is given by M1 (ρ) = 2πb10 ρp + m q k=0 d1k ρ Ck ρ p−1 . Moreover, M1 = 0 if, and only if, b10 = d10 = · · · = d1m = 0. (b) If M1 = 0, then the second Melnikov function is given by M2 (ρ) = M21 (ρ) + (q − p)ρp M22 (ρ), where m X M21 (ρ) := 2πb20 ρp + d2k ρq Ck (ρp−1 ) k=0 and X b1k c1l Z π M22 (ρ) := sin(kθ) sin(lθ)x0 (θ, ρ)q−1 dθ. k −π 1≤k≤n 1≤l≤m Proof. By applying Corollary 2.3, we get that the first Melnikov function writes as m X Z π m X p p q−p p  M1 (ρ) = 2πb10 ρ + ρ d1k cos(kθ)x0 (θ, ρ) dθ = 2πb10 ρ + d1k ρq Ck ρp−1 , k=0 −π k=0 where in the first equality we use that the solution x0 (θ, ρ) of the unperturbed equation is even in θ, whereas in the second one we take (14) into account. Then the second assertion in (a) follows by Proposition 4.1, which shows that the ordered set ρp , ρq C0 (ρp−1 ), . . . , ρq Cn (ρp−1 ) is an ECT-system on (κp , 0) and (0, +∞). Let us turn next to the proof of (b). If M1 = 0 then, by Corollary 2.3 again, the second Melnikov function writes as Z π  M2 (ρ) = ρp P2 (θ) + Q2 (θ)x0 (θ, ρ)q−p + (q − p)Q1 (θ)S(θ, ρ)x0 (θ, ρ)q−1 dθ, −π where Z θ  S(θ, ρ) = P1 (s) + Q1 (s)x0 (s, ρ)q−p ds. −π Rπ That the first summand in M2 writes as ρp −π (P2 (θ) + Q2 (θ)x0 (θ, ρ)q−p ) dθ = M21 (ρ) can be shown exactly as we did in (a). With regard to the second summand note that, on account of (a), n X m X P1 (θ) = (a1k sin(kθ) + b1k cos(kθ)) and Q1 (θ) = c1l sin(lθ). k=1 l=1 Rθ Thus, Q1 (θ)x0 (θ, ρ)q−1 and −π Q1 (s)x0 (s, ρ)q−p ds are odd and even functions in θ, respectively. Hence the second summand in M2 writes as Z π Z π Z θ  q−1 q−1 Q1 (θ)S(θ, ρ)x0 (θ, ρ) dθ = Q1 (θ)x0 (θ, ρ) P1 (s)ds dθ −π −π −π Z π n ! X b1k = Q1 (θ)x0 (θ, ρ)q−1 sin(kθ) dθ −π k k=1 X b1k c1l Z π = sin(kθ) sin(lθ)x0 (θ, ρ)q−1 dθ, k −π 1≤k≤n 1≤l≤m which is equal to M22 (ρ). This completes the proof.  As it is clear from the previous result, the function Skl that we introduce in next statement is a building block of the second Melnikov function. 12 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT  Proposition 4.3. For each r ∈ Z≥0 , define Br := hρq−p C0 ρp−1 , . . . , ρq−p Cr ρp−1 )i. Then Z π Skl (ρ) := sin(kθ) sin(lθ)x0 (θ, ρ)q−1 dθ −π belongs to Bk+l−1 for all k, l ∈ Z≥1 . Furthermore,  Bk+l−1 = hρq−p C0 ρp−1 , . . . , ρq−p Ck ρp−1 ), Sk2 (ρ), . . . , Skl (ρ)i for all k ≥ 0 and l ≥ 2. Proof. In order to prove the first assertion we can suppose without loss of generality that k ≥ l. It is well-known that cos(iθ) = Ti (cos θ) and sin((i + 1)θ) = sin θUi (cos θ), where Ti and Ui are the ith degree Chebyshev polynomials of the first and second kind, respectively. (The reader is referred to [35] for the formulas relating the Chebyshev polynomials that we shall use hereunder.) Thus Z π Skl (ρ) = Uk−1 (cos θ)Ul−1 (cos θ) sin2 θ x0 (θ, ρ)q−1 dθ −π Z π (16) 1 0 = Uk−1 (cos θ)Ul−1 (cos θ) sin θ x0 (θ, ρ)q−p dθ, p − q −π where the second equality follows using that x0 (θ, ρ) is the solution of the unperturbed differential equation (3) to perform an integration by parts. Since l−1 X Uk−1 (x)Ul−1 (x) = Uk−l+2r (x) r=0 and, thanks to (x2 − 1)Ur0 (x) = (r + 1)Tr+1 (x) − xUr (x), 0 Ur (cos θ) sin θ = (r + 1)Tr+1 (cos θ) = (r + 1) cos((r + 1)θ), we get that k+l−1 X 0 Uk−1 (cos θ)Ul−1 (cos θ) sin θ = r cos(rθ). r=k−l+1 step 2 On account of (14), the substitution of this identity in (16) yields k+l−1 X ρq−p Skl (ρ) = r Cr (ρp−1 ). (17) p−q r=k−l+1 step 2 Accordingly the first assertion is true. With regard to the second one, it suffices to show that  ρq−p Ck+r ρp−1 ) ∈ hρq−p C0 ρp−1 , . . . , ρq−p Ck ρp−1 ), Sk2 (ρ), . . . , Skl (ρ)i for all r = 1, 2, . . . , l − 1, which can be proved by induction on r taking (17) into account again. This completes the proof of the result.  In what follows we will need to further emphasize the dependence of the Melnikov functions on the perturbative parameters. For this reason we use the notation µ = (a, b, c, d) as introduced in (4), so that µ ∈ R4(n+m+1) , and we denote the ith order Melnikov function associated to the perturbed differential equation given in (3) by Mi (ρ; µ). Proposition 4.4. Setting L = {µ ∈ R4(n+m+1) : M1 (ρ; µ) = 0 for all ρ ∈ I}, there exists a surjective map β : L −→ Rn+m+1 such that n+m−1 X p q M2 (ρ; µ) = β1 (µ)ρ + ρ βk+2 (µ)Ik (ρp−1 ). k=0 ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 13 Proof. We claim that there exists a surjective map β̂ : L −→ Rn+m+1 such that n+m−1 X M2 (ρ; µ) = β̂1 (µ)ρp + ρq β̂k+2 (µ)Ck (ρp−1 ). k=0 The result will follow once we prove this because, by Proposition 4.1, we know that the linear spans h1, ρq−p I0 (ρp−1 ), . . . , ρq−p Ik (ρp−1 )i and h1, ρq−p C0 (ρp−1 ), . . . , ρq−p Ck (ρp−1 )i are equal and have dimension k + 2 for all k ∈ Z≥0 . In order to prove the claim let us fix any µ = (a, b, c, d) ∈ L . Then, by applying Proposition 4.2, m X X b1k c1l M2 (ρ; µ) = 2πb20 ρp + d2k ρq Ck (ρp−1 ) + (q − p)ρp Skl (ρ). (18) k k=0 1≤k≤n 1≤l≤m By the first assertion in Proposition 4.3, for all k = 1, 2, . . . , n and l = 1, 2, . . . , m,  Skl (ρ) ∈ hρq−p C0 ρp−1 , . . . , ρq−p Cn+m−1 ρp−1 )i.  On account of (18), this shows that M2 (ρ; µ) ∈ hρp , ρq C0 ρp−1 , . . . , ρq Cn+m−1 ρp−1 )i, which has dimension n + m + 1 by Proposition 4.1. Hence there exists a unique β̂ = β̂(µ) ∈ Rn+m+1 such that n+m−1 X p q M2 (ρ; µ) = β̂1 ρ + ρ β̂k+2 Ck (ρp−1 ). k=0 It only remains to be proved that β̂ : L −→ Rn+m+1 is a surjective map. To this end it suffices to verify that Rn+m+1 = β̂(L ∩ P), where P := {µ ∈ R4(n+m+1) : b11 = 0, c1l = 0, for l = 1, . . . , m − 1, and c1m = 1}. Indeed, if µ ∈ L ∩ P then from (18) we get that m X n X p q p−1 p b1k M2 (ρ; µ) = 2πb20 ρ + d2k ρ Ck (ρ ) + (q − p)ρ Skm (ρ). k k=0 k=2 Hence the inclusion Rn+m+1 ⊂ β̂(L ∩ P) follows from the second assertion in Proposition 4.3. Since the other inclusion is clear, this proves the validity of the claim and so the result follows.  5. Proof of the main results This section is devoted to prove Theorems 1.1 and 1.2. Let us advance that in order to complete the proof of the latter we will need some additional results that for the sake of simplicity in the exposition we gather in the Appendix. Proof of Theorem 1.1. The fact that M1 ( · ; µ) ≡ 0 if, and only if, b10 = d10 = · · · = d1m = 0 follows from (a) in Proposition 4.2. By applying Proposition 4.4, there exists a surjective map β such that if µ ∈ L then n+m−1 X ∼ ∼ M2 (ρ; µ) = ρp M2 (ρ; µ) with M2 (ρ; µ) := β1 (µ) + ρq−p βk+2 (µ)Ik (ρp−1 ). (19) k=0 ∼ It is clear that M2 and the rescaled M2 have the same number of positive and negative zeros counted with multiplicities. For the sake of convenience we also define n+m−1 X q−p N (ρ; µ) := ρ βk+2 (µ)Ik (ρp−1 ). k=0 Recall, see (12) and (13), that Ik (y) is equal to Tk,α (y) particularised with g(θ) = −(p − 1)(1 + q−p cos θ) and α = p−1 . By abusing notation, in what follows we will write Ik (y) = Tk,α (y) for 14 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT simplicity. Taking this into account, by applying Lemma 3.5, we get that the derivative of N is n+m−1 X N 0 (ρ; µ) = (q − p)ρq−p−1 βk+2 (µ)Tk,α+1 (ρp−1 ). (20) k=0 In addition, the application of Theorem 3.4 to these Tk,α and Tk,α+1 shows easily the following: Claim 1: N has at most n + m − 1 zeros in (κp , 0) counted with multiplicities. The same is true in the interval (0, +∞). Claim 2: N 0 has at most n + m − 1 zeros in (κp , 0) counted with multiplicities. The same is true in the interval (0, +∞). Claim 3: In case that p is even, N has at most n + m − 1 zeros in (κp , +∞) \ {0} counted with multiplicities. The same is true for N 0 . Above we have omitted the dependence on µ for the sake of shortness. We will also do it in what follows when there is no risk of confusion. Let us prove next each one of the assertions in the statement of the result: (a) From (19), by applying Proposition 4.1, we get that K ± ≤ n + m. In addition, thanks to ∼ the surjectivity of β, we can assert that there exists µ0 ∈ R4(n+m+1) such that M2 (ρ; µ0 ) has ∼ exactly n + m positive simple zeros. Since M2 (ρ; µ0 ) is an even function in ρ in case that p and q are odd, we can conclude that it has the same number of negative simple zeros. (b) Exactly as before K ± ≤ n + m. We prove by contradiction that the equalities can not hold simultaneously. So suppose that M2 has n + m positive zeros, say 0 < ρ+ + 1 ≤ · · · ≤ ρn+m , and n + m negative zeros, say 0 > −ρ− − 1 ≥ · · · ≥ −ρn+m , counted with multiplicities. Note that N (ρ) is an odd function because q − p is odd and p − 1 is even. Therefore ∼ ∼ − − N (ρ+ + i ) = M2 (ρi ) − β1 = −β1 and N (ρj ) = −M2 (−ρj ) + β1 = β1 (21) for all i, j ∈ {1, 2, . . . , n + m}. This shows in particular that β1 6= 0 (otherwise we get a − contradiction with Claim 1 ), which in turn implies that ρ+ i 6= ρj for all i and j. ∼ On the other hand, N 0 = M20 has n + m − 1 positive zeros, say 0 < %+ + 1 ≤ · · · ≤ %n+m−1 , − − and n + m − 1 negative zeros, say 0 > −%1 ≥ · · · ≥ −%n+m−1 , counted with multiplicities, satisfying %± ± ± 0 i ∈ [ρi , ρi+1 ] for all i. Since N is even, taking Claim 2 into account we can − assert that %+ i = %i for all i. In particular − − [ρ+ + i , ρi+1 ] ∩ [ρi , ρi+1 ] 6= ∅ for all i. (22) − For each i = 1, 2, . . . , n + m, let Ji be the open interval with endpoints ρ+ i and ρi . Then, on account of (22), it is easy to show that J1 , J2 , . . . , Jn+m are pairwise non-intersecting. Since, due to (21), each Ji contains at least one zero of N , this contradicts Claim 1. Accordingly, either K + < n + m or K − < n + m.  (i) By Theorem 3.4, we have that ρq−p I0 (ρp−1 ), ρq−p I1 (ρp−1 ), . . . , ρq−p In+m−1 (ρp−1 ) is an ECT-system on (0, +∞). Thus there exist β2∗ , . . . , βn+m+1 ∗ ∈ R such that the func- tion n+m−1 X q−p ∗ ρ 7−→ ρ βk+2 Ik (ρp−1 ) k=0 has n + m − 1 positive simple zeros and n + m − 1 negative simple zeros. Here we use that the above function is odd due to the parity assumption on p and q. For the same reason, ρ = 0 is a zero with odd multiplicity. Hence, taking β1∗ ≈ 0, the function n+m−1 X ρ 7−→ β1∗ + ρq−p ∗ βk+2 Ik (ρp−1 ) k=0 has 2(n + m) − 1 simple zeros on (κp , +∞) \ {0}. We use at this point the surjectivity of µ 7→ β(µ) to choose some µ0 ∈ L such that β(µ0 ) = (β1∗ , β2∗ , . . . , βn+m+1 ∗ ). Then by construction, see (19), M2 (ρ; µ0 ) has 2(n + m) − 1 simple zeros on (κp , +∞) \ {0}. ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 15 (ii) Exactly the same argument as before, but choosing β1∗ = 0, guarantees the existence of µ0 ∈ L such that M2 (ρ; µ0 ) has 2(n + m − 1) simple zeros on (κp , +∞) \ {0}. ∼ − − (c) Let ρ+ + 1 ≤ · · · ≤ ρK + and ρ1 ≤ · · · ≤ ρK − be the zeros of M2 (ρ; µ) on (0, +∞) and (κp , 0), ∼ respectively. Then M20 = N 0 has at least K + − 1 zeros on [ρ+ 1 , +∞) and at least K − 1 − − zeros on (κp , ρ1 ], counted with multiplicities. Since p is even, it is clear, cf. (20), that n+m−1 X E(y; µ) := βk+2 (µ)Tk,α+1 (y) k=0    i − p−1 has at least K + − 1 zeros on (ρ+ 1 )p−1 , +∞ and at least K − − 1 zeros on −1 2(p−1) , (ρ1 ) , counted with multiplicities. (Here we use that κpp−1 = −1 2(p−1) .) By Theorem 3.4, (T0,α+1 , T1,α+1 , . . . , Tn+m−1,α+1 )   −1 is an ECT-system on 2(p−1) , +∞ . Hence E can have at most n + m − 1 zeros counted with multiplicities in this interval. If E(0) = 0 then E has at least K + + K − − 1 zeros counted with multiplicites in this interval. Consequently K + + K − ≤ n + m. If E(0) 6= 0 ∼ then the multiplicity of M20 = N 0 at ρ = 0 is exactly q − p − 1, which is an even number by assumption. On account of this, and the fact that M2 (ρ± − + 1 ) = 0 with ρ1 < 0 < ρ1 , one can ∼ − + easily conclude the existence of a zero of M20 = N 0 on (ρ1 , ρ1 ) \ {0}. Thus N 0 has at least n + m − 1 zeros on (κp , +∞) \ {0} counted with multiplicities. By Claim 3 we get the upper bound K + + K − ≤ n + m also in this case. Exactly as we did in the previous cases, Proposition 4.1 and the surjectivity of µ 7−→ β(µ) ensure the existence of some µ0 ∈ L such that M2 ( · ; µ0 ) has n + m simple zeros in, for instance, the interval (0, +∞). ∼ (d) The number of zeros of N 0 = M20 on (κp , +∞) \ {0} counted with multiplicities is at least K + + K − − 2. Then, by Claim 3 again, we get K + + K − ≤ n + m + 1. In order toprove that this upper bound   we will use that, by Theorem 3.4, I0 , I1 , . . . , In+m−1 is an is sharp −1 ECT-system on 2(p−1) , +∞ and we consider two cases: Case 1: q even and p < q. ∗ We take β2∗ , . . . , βn+m+1 ∈ R such that n+m−1 X ∗ F(ρ) := βk+2 Ik (ρp−1 ) (23) k=0 vanishes at 0 < ρ1 < ρ2 < · · · < ρn+m−1 with multiplicity one and satisfies F(0) 6= 0. Thus ρq−p F(ρ) has a zero at ρ = 0 of multiplicity q − p, which is an even number, and vanishes with multiplicity one at ρi , i = 1, 2, . . . , n + m − 1. Consequently we can choose β1∗ small enough such that β1∗ + ρq−p F(ρ) has n + m − 1 simple zeros near ρ1 < ρ2 < · · · < ρn+m−1 , together with one positive and one negative zero near ρ = 0, both being simple as well. Now, as we did before, we use the sujertivity of µ 7−→ β(µ) to choose some µ0 ∈ L such that β(µ0 ) = (β1∗ , β2∗ , . . . , βn+m+1 ∗ ). Then by construction, see (19), M2 (ρ; µ0 ) has n + m + 1 simple zeros on (κp , +∞) \ {0} as desired. Case 2: p > q. Pn+m−1 ∗ In this case we choose β2∗ , . . . , βn+m+1 ∗ ∈ R such that βk+2 Ik (y) has exactly   k=0 −1 n + m − 1 simple zeros in 2(p−1) , +∞ , one of them being y = 0 and the other ones positive. Thus the function F(ρ) in (23) has exactly n + m − 2 positive simple zeros, say ρ2 < ρ3 < · · · < ρn+m−1 , and there exists an analytic function η such that n+m−1 X ∗ F(ρ) = βk+2 Ik (ρp−1 ) = η(ρ)ρp−1 with η(0) 6= 0. k=0 16 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT Let us suppose, without loss of generality, that η(0) > 0. Then, since p is even, there exists ρ̂ ∈ (0, ρ2 ) such that ρF(ρ) > 0 for all ρ ∈ (−ρ̂, ρ̂) \ {0}. Let us define at this point b 2 (ρ; β1 , σ) := η(ρ)ρp−1 + β1 ρp−q + σI0 (ρp−1 ) M and split the proof into two subcases depending on the parity of q. Subcase 2a: q odd.  In this case M b 2 (ρ; β1 , 0) = ρp−q η(ρ)ρq−1 + β1 with p − q odd and q − 1 > 0 even. b 2 (ρ; β1∗ , 0) Hence, by continuity, there exist small enough β1∗ < 0 and ρ̃ ∈ (0, ρ̂) such that M has n + m − 2 positive simple zeros near ρ2 , . . . , ρn+m−1 and, moreover, b 2 (−ρ̂; β1∗ , 0) < 0, M M b 2 (−ρ̃; β1∗ , 0) > 0, M b 2 (ρ̃; β1∗ , 0) < 0, M b 2 (ρ̂; β1∗ , 0) > 0. Note also, see (13), that I0 (ρp−1 ) > 0 for all ρ. Thus, by continuity, we can take σ ∗ > 0 small enough, such that M b 2 (ρ; β1∗ , σ ∗ ) still has n + m − 2 positive simple zeros near ρ2 , . . . , ρn+m−1 b 2 (0; β1∗ , σ ∗ ) > 0, together with and also verifies M b 2 (−ρ̂; β1∗ , σ ∗ ) < 0, M M b 2 (−ρ̃; β1∗ , σ ∗ ) > 0, M b 2 (ρ̃; β1∗ , σ ∗ ) < 0, M b 2 (ρ̂; β1∗ , σ ∗ ) > 0. Therefore Mb 2 (ρ; β1∗ , σ ∗ ) vanishes at least once in each interval (−ρ̂, −ρ̃), (0, ρ̃) and (ρ̃, ρ̂). So the total number of zeros of M b 2 (ρ; β1∗ , σ ∗ ) on (κp , +∞) \ {0} is at least n + m + 1. On account of the surjectivity of β : L −→ Rn+m+1 , we can take µ0 ∈ L such that β(µ0 ) = (β1∗ , β2∗ + σ ∗ , β3∗ , . . . , βn+m−1 ∗ ). Hence, by construction, M2 (ρ; µ0 ) = ρq M b 2 (ρ; β1∗ , σ ∗ ) has at least n+m+1 zeros on (κp , +∞)\ {0}. Finally, since we have already proved that K + + K − ≤ n + m + 1, these zeros must be simple, as desired. Subcase 2b: q even.  In this case Mb 2 (ρ; β1 , 0) = ρp−q η(ρ)ρq−1 + β1 with p − q even and q − 1 > 0 odd. Then, by continuity, we can take small enough β1∗ < 0 and ρ̃ ∈ (0, ρ̂) such that M b 2 (ρ; β1∗ , 0) has n + m − 2 positive simple zeros near ρ2 , . . . , ρn+m−1 and b 2 (ρ̂; β1∗ , 0) > 0, M M b 2 (−ρ̂; β1∗ , 0) < 0, M b 2 (ρ̃; β1∗ , 0) < 0. Due to I0 (ρp−1 ) > 0 for all ρ, taking σ ∗ > 0 small enough, M b 2 (ρ; β1∗ , σ ∗ ) still has n + m − 2 positive simple zeros near ρ2 , . . . , ρn+m−1 and, additionally, b 2 (ρ̂; β1∗ , σ ∗ ) > 0, M M b 2 (−ρ̂; β1∗ , σ ∗ ) < 0, M b 2 (ρ̃; β1∗ , σ ∗ ) < 0, M b 2 (0; β1∗ , σ ∗ ) > 0. Thus M b 2 (ρ; β1∗ , σ ∗ ) vanishes at least once in each interval (−ρ̂, 0), (0, ρ̃) and (ρ̃, ρ̂). Exactly as we did in the previous subcase, there exists µ0 ∈ L such that M2 (ρ; µ0 ) = ρq M b 2 (ρ; β1∗ , σ ∗ ) has exactly n + m + 1 simple zeros on (κp , +∞) \ {0}.  Remark 5.1. If we only consider K + in Theorem 1.1, then by (19) and Proposition 4.1, K + ≤ n + m always holds and the upper bound can be achieved. Proof of Theorem 1.2. Taking p = 3 and q = 2, from point (ii) in assertion (b) of Theorem 1.1 we know that there exists µ0 ∈ R4(n+m+1) with M1 (ρ; µ0 ) = 0 for all ρ ∈ I and such that M2 (ρ; µ0 ) has 2(n + m − 1) simple zeros in I \ {0}. Hence, due to x(π, ρ; µ0 , ε) = ρ + ε2 M2 (ρ; µ0 ) + o(ε2 ), by applying the Implicit Function Theorem we can assert that ρ 7→ x(π, ρ; µ0 , ε) has at least 2(n + m − 1) fixed points in I \ {0} for all ε ≈ 0. Since x(π, 0; µ0 , ε) = 0 for all ε, the first assertion follows. With regard to the second assertion, the bound H3,2 (1, 3) ≥ 8 is proved in Proposition A.2, whereas H3,2 (4, 1) ≥ 10 is proved in Proposition A.3.  Appendix A. Improvements using Lyapunov constants Our goal in this appendix is to use Lyapunov constants in order to improve the general lower bound H3,2 (n, m) ≥ 2(n+m)−1 in case that (n, m) ∈ {(1, 3), (4, 1)}. At the end we shall discuss the difficulties we have found to tackle two other particular cases using the same approach. ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 17 In the last two decades, there have been several works about the Hilbert number H3,2 (n, m). Unfortunately the problem is far from being solved even for the case n = m = 1. We gather in the following theorem the main results obtained in [6]. Theorem A.1. For any nonnegative integers n and m, H3,2 (n, 0) = H3,2 (0, m) = 2, H3,2 (n, 1) ≥ n + 2, and H3,2 (1, m) ≥ 2m + 1. Moreover, H3,2 (3, 1) ≥ 7 and H3,2 (2, 2) ≥ 7. As it occurs with our proof of Theorem 1.2, the general lower bounds in the above result follow by using Melnikov functions, whereas the improvements for the particular cases follow by means of Lyapunov constants, that enable to study those limit cycles bifurcating from ρ = 0. With this aim let us consider the differential equation ẋ = A(θ)x3 + B(θ)x2 and write its trigonometric polynomials as Xn Xm   A(θ) = b0 + ak sin(kθ) + bk cos(kθ) and B(θ) = d0 + ck sin(kθ) + dk cos(kθ) . (24) k=1 k=1 If x(θ, ρ) denotes the solution with initial condition x(0, ρ) = ρ then the hyperbolic limit cycles near ρ = 0 can be viewed as simple zeros of the displacement map ∞ x(2π, ρ) − ρ X ∆(ρ) := = Xj ρj . (25) 2π j=2 The coefficients Xj of the Taylor development of ∆ at ρ = 0 are polynomial on ak , bk , ck , dk and the first Xj which is not identically zero is called the jth order Lyapunov constant of the corresponding Abel equation. Usually the Lyapunov constants appear in the context of planar polynomial vector fields when studying the stability of equilibrium points of monodromic type. They are polynomial in the coefficients of the vector field when we restrict the analysis to the trace zero class and typically they are computed writing the planar differential equation in polar coordinates, see [11]. In fact, equation (1) is the third degree truncation of this type of equations and the zero trace class here is reduced to C = 0, which is precisely the equation that we are analyzing. Hence, the standard Lyapunov scheme applies, so that the Lyapunov constants that we obtain are polynomials in the coefficients of A and B. As in the standard scheme here we consider the transition map from θ = 0 to θ = 2π. Only a constant translation θ 7→ θ − π is needed to obtain the transition from θ = −π to θ = π. Clearly, the number of limit cycles does not depend on this initial angle. Moreover, the Lyapunov constants are always defined modulus the vanishing of all the previous ones, i.e., Xj := Xj |X1 =···=Xj−1 =0 . For simplicity, by abusing notation we keep the same symbol to denote them. As it is usual, we say that ρ = 0 is a weak focus of order k if Xk 6= 0 and Xj = 0, for j = 2, . . . , k − 1. Moreover, we say that ρ = 0 is a center if all the solutions in a neighborhood of ρ = 0 are periodic, i.e. ∆(ρ) ≡ 0. This theory was initially developed for planar ordinary differential systems having an equilibrium point of center-focus type (see [7]) but writing in polar coordinates both are equivalent. We point out that here, contrary to what happens in the standard planar scheme, the coefficients with an even subindex do not vanish identically. The bifurcation phenomenon is known as the degenerated Hopf bifurcation. In our context, since we have that X1 = 0, one limit cycle is always missing when we perturb a weak focus or a center. Note that if bk and dk in (24) are all zero then the equation ẋ = A(θ)x3 + B(θ)x2 is invariant by the change θ 7→ −θ and, consequently, it has a reversible center at the origin, so that all Xj vanish. Therefore, when b = (b0 , . . . , bn ) and d = (d0 , . . . , dm ) are nonzero parameters, we can write the truncated first order Taylor series of Xj at (b, d) = (0, 0) as n X m X Xj1 := fj,k (a, c)bk + gj,k (a, c)dk , for all j ≥ 2, (26) k=0 k=0 where a = (a1 , . . . , an ) and c = (c1 , . . . , cm ). We remark that the above expression has no constant term because the return map is identically zero when b = d = 0 for every a and c. Here we will use the ideas developed in [21] to work only with these linear developments in order to increase the number of limit cycles of small amplitude for families of centers. In [22] it is proved 18 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT that these linear developments can also be obtained by computing the Taylor series at ρ = 0 of the first Melnikov function M1 as introduced in Section 2. In the next two propositions, since we treat the cases n = 1 and m = 1, we can follow similarly as with the perturbed problem (3), taking the coefficients of sin θ and cos θ to be 1 and 0, respectively, since we can rescale x and do a translation in θ if necessary. In the proofs we will see that we are using all the other free parameters. Showing also that the weak focus order is maximal in the considered families. Proposition A.2. Let us consider the Abel differential equation ẋ = A(θ)x3 + B(θ)x2 taking the trigonometric polynomials A and B as introduced in (24) with (n, m) = (1, 3) and (a1 , b1 ) = (1, 0). Then there exist parameters (b0 , c, d) such that the origin is a weak focus of order 9 unfolding 7 nonzero limit cycles of small amplitude. Consequently H3,2 (1, 3) ≥ 8. Proof. In this case it turns out that all the Lyapunov constants that we need in order to prove the result are linear with respect to (b0 , d0 , d1 , d2 , d3 ), so that Xj = Xj1 . The proof follows by finding a transversal intersection point on the zero level set of X2 , . . . , X8 in which X9 is nonvanishing. Then 7 nonzero limit cycles of small amplitude bifurcate from the origin and ρ = 0 is still an hyperbolic solution that remains. The complete expressions are quite large and we only show the first ones: X2 = d0 , X3 = b0 , X4 = 2−1 d1 , X5 = 12−1 (−3c1 + c3 )d2 − 12−1 c2 d3 , (27) X6 = 16−1 (3c1 c2 − 2)d2 + 32−1 (4c21 + c22 )d3 , X7 = 432−1 (−1674c31 − 1620c21 c2 − 504c21 c3 − 432c1 c22 + 162c1 c2 c3 + 150c1 c23 + 225c22 c3 + 234c2 c23 + 64c33 − 864c1 − 162c2 + 72c3 )d2 + 864−1 (648c31 − 828c21 c2 + 216c21 c3 − 1242c1 c22 − 720c1 c2 c3 − 369c32 − 414c22 c3 − 128c2 c23 + 144c1 − 360c2 )d3 . Clearly X2 , X3 , X4 , X5 have degree one with respect to b0 , d0 , d1 , d2 . Then, when 3c1 − c3 6= 0 we can write Xj = uj for j = 2, . . . , 5 and we get fj−5 (c1 , c2 , c3 ) Xj |u2 =···=u5 =0 = Cj d3 , for j = 6, 7, 8, 9, 3c1 − c3 with some nonvanishing rational numbers Cj and f1 = 12c31 − 4c21 c3 − 3c1 c22 − c22 c3 + 4c2 , f2 = 216c41 + 96c31 c2 − 54c21 c22 − 36c21 c2 c3 − 24c21 c23 − 27c1 c32 − 36c1 c22 c3 + 4c1 c2 c23 − 9c32 c3 − 6c22 c23 + 48c21 + 72c1 c2 − 16c1 c3 + 36c22 + 24c2 c3 , f3 = 18792c51 + 16128c41 c2 + 5832c41 c3 − 828c31 c22 − 672c31 c2 c3 − 1824c31 c23 − 4536c21 c32 − 6156c21 c22 c3 − 1344c21 c2 c23 − 736c21 c33 − 1197c1 c42 − 3024c1 c32 c3 − 1224c1 c22 c23 + 224c1 c2 c33 − 399c42 c3 − 504c32 c23 − 184c22 c33 + 13248c31 + 10068c21 c2 − 1728c21 c3 + 4752c1 c22 + 2688c1 c2 c3 − 896c1 c23 + 1608c32 + 1584c22 c3 + 780c2 c23 + 432c1 + 1728c2 − 144c3 , f4 = 80352c61 + 99252c51 c2 + 48384c51 c3 + 26352c41 c22 + 26388c41 c2 c3 + 1440c41 c23 − 20034c31 c32 − 28800c31 c22 c3 − 7944c31 c2 c23 − 5376c31 c33 − 14364c21 c42 − 30618c21 c32 c3 − 15120c21 c22 c23 − 1728c21 c2 c33 − 1152c21 c43 − 2646c1 c52 − 9576c1 c42 c3 − 8235c1 c32 c23 − 1728c1 c22 c33 + 480c1 c2 c43 − 882c52 c3 − 1596c42 c23 − 1104c32 c33 − 288c22 c43 + 110538c41 + 101808c31 c2 + 16470c31 c3 + 33678c21 c22 + 13344c21 c2 c3 − 11934c21 c23 + 11196c1 c32 + 9369c1 c22 c3 + 4992c1 c2 c23 − 1946c1 c33 + 3600c42 + 3732c32 c3 + 2925c22 c23 ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 19 + 1328c2 c33 + 17280c21 + 22698c1 c2 − 4032c1 c3 + 10800c22 + 6156c2 c3 − 576c23 . For d3 6= 0, the solutions of the system of equations defined by {f1 = f2 = f3 = 0} write as   ∗ ∗ ∗ ∗ β 3 p2 (β) β p3 (β) c = (c1 , c2 , c3 ) = β, − , 126976 31744 where β a simple real zero of p(x) = 50625x16 − 207900x12 + 112644x8 − 26880x4 + 4096, that one can verify it has exactly 4 simple real zeros near ±0.7796202641 and ±1.369217569, and p2 (x) = 1771875x12 − 7456500x8 + 4643340x4 − 842624, p3 (x) = 16875x12 − 177300x8 + 398508x4 − 18304. The proof will follow once we check that f4 , the denominator 3c1 − c3 , and the determinant of the Jacobian matrix of (f1 , f2 , f3 ) with respect to c = (c1 , c2 , c3 ) are all different from zero evaluated at c = c∗ . Straightforward computations show that f4 (c∗ ) = (151875β 12 + 3649500β 8 − 2148180β 4 + 184448)/1984 =: p4 (β), det Jac(f1 ,f2 ,f3 ) (c∗ ) = β(172800β 12 + 843264β 8 − 516096β 4 + 65536) =: p5 (β), 3c1 − c3 |c∗ = −β(16875β 12 − 177300β 8 + 398508β 4 − 113536)/31744 =: p6 (β). For i = 4, 5, 6, one can verify that the resultant between pi (x) and p(x) is different from zero. Consequently this implies that pi (β) 6= 0 for i = 4, 5, 6, as desired. This concludes the proof of the result.  Proposition A.3. Let us consider the Abel differential equation ẋ = A(θ)x3 + B(θ)x2 taking the trigonometric polynomials A and B as introduced in (24) with (n, m) = (4, 1) and (c1 , d1 ) = (1, 0). Then there exist parameters (a, b, d0 ) such that the origin is a weak focus of order 11 unfolding 9 nonzero limit cycles of small amplitude. Consequently H3,2 (4, 1) ≥ 10. Proof. The proof follows similarly as in the previous result, but here we must use the linear Taylor developments of some Xj as we have previously explained. This is so because in this case not all Lyapunov constants are of degree 1 in the parameters (b0 , b1 , b2 , b3 , b4 , d0 ). For simplicity we will write a = (a1 , a2 , a3 , a4 ). Being the expressions of the Lyapunov constants very large, for the sake of shortness we only show the first ones. X2 = d0 , X3 = b0 , X4 = −2−1 b1 , X5 = 4−1 b2 , X6 = 48−1 ((2a2 − a4 − 6)b3 + a3 b4 ), X7 = 24−1 (−4a1 + 6a2 − 3a4 − 18)b3 + 16−1 (−a2 + 2a3 + 1)b4 , X8 = 960−1 ((−60a21 + 240a1 a2 − 120a1 a4 + 135a22 + 80a2 a3 − 15a2 a4 − 40a3 a4 − 30a24 − 1840a1 + 575a2 − 240a3 − 643a4 − 2610)b3 + (−45a1 a2 + 120a1 a3 + 75a2 a3 + 40a23 + 30a3 a4 + 145a1 − 420a2 + 463a3 + 420)b4 ). The next necessary Xk , for k = 9, 10, 11, have degree 3 in a3 , a4 and we linearize them with respect to a3 , a4 . As above, we start simplifying with the first that are linear, X2 , . . . , X5 , and X6 , writing, when a3 6= 0, Xk = uk for k = 2, . . . , 6. Then, naming b3 = u7 we can write Cj Xj |u2 =···=u6 =0 = Xj1 + O2 (b3 ) = fj−6 (a) u7 + O2 (u7 ), for j = 7, 8, 9, 10, 11, (28) a3 with some Cj nonvanishing rational numbers and f1 = 8a1 a3 − 6a22 + 3a2 a4 + 24a2 − 3a4 − 18, f2 = 60a21 a3 − 90a1 a22 + 45a1 a2 a4 + 15a22 a3 + 560a1 a2 + 1120a1 a3 − 145a1 a4 − 840a22 20 JIANFENG HUANG, JOAN TORREGROSA, AND JORDI VILLADELPRAT − 99a2 a3 + 420a2 a4 − 870a1 + 3360a2 − 168a3 − 420a4 − 2520, f3 = 750a21 a2 + 4800a21 a3 − 375a21 a4 − 4680a1 a22 + 1653a1 a2 a3 + 2340a1 a2 a4 + 1120a1 a23 + 840a1 a3 a4 − 1212a32 − 480a22 a3 − 24a22 a4 − 98a2 a23 + 420a2 a3 a4 + 315a2 a24 − 49a23 a4 − 2250a21 + 23520a1 a2 + 13281a1 a3 − 4740a1 a4 − 6012a22 + 984a2 a3 + 7344a2 a4 + 294a23 − 420a3 a4 − 315a24 − 28440a1 + 39384a2 − 6552a3 − 7110a4 − 31320, f4 = 3600a31 a2 + 23040a31 a3 − 1800a31 a4 − 34560a21 a22 + 11520a21 a2 a3 + 17280a21 a2 a4 + 7680a21 a23 + 5760a21 a3 a4 − 17280a1 a32 − 5760a1 a22 a3 − 1200a1 a2 a23 + 5760a1 a2 a3 a4 + 4320a1 a2 a24 − 600a1 a23 a4 + 3780a32 a3 + 1920a22 a23 + 1440a22 a3 a4 − 225a2 a3 a24 − 10800a31 + 323040a21 a2 + 579120a21 a3 − 109680a21 a4 − 426816a1 a22 + 235856a1 a2 a3 + 293088a1 a2 a4 + 144720a1 a23 + 88064a1 a3 a4 − 13920a1 a24 − 147648a32 − 101144a22 a3 − 6144a22 a4 − 26784a2 a23 + 43416a2 a3 a4 + 39984a2 a24 + 816a33 − 7056a23 a4 + 1523a3 a24 − 658080a21 + 2311584a1 a2 + 446352a1 a3 − 521328a1 a4 − 5568a22 + 221676a2 a3 + 544128a2 a4 + 20832a23 − 69048a3 a4 − 39984a24 − 2626848a1 + 1881216a2 − 625176a3 − 507744a4 − 1607040, f5 = 720000a31 a2 + 2386800a31 a3 − 360000a31 a4 − 2438640a21 a22 + 1869360a21 a2 a3 + 1354320a21 a2 a4 + 1147200a21 a23 + 772080a21 a3 a4 − 67500a21 a24 − 1621440a1 a32 − 526920a1 a22 a3 − 28800a1 a22 a4 + 266160a1 a2 a23 + 855540a1 a2 a3 a4 + 419760a1 a2 a24 + 147600a1 a33 + 136320a1 a23 a4 + 92040a1 a3 a24 − 221400a42 − 121080a32 a3 − 107460a32 a4 − 60240a22 a23 + 58560a22 a3 a4 + 46890a22 a24 − 23520a2 a33 + 20340a2 a23 a4 + 64350a2 a3 a24 + 31095a2 a34 − 11760a33 a4 − 8820a23 a24 − 2160000a31 + 16558560a21 a2 + 12839120a21 a3 − 5026320a21 a4 − 4939840a1 a22 + 7959200a1 a2 a3 + 9117760a1 a2 a4 + 3502960a1 a23 − 377600a2 a23 + 1230980a1 a3 a4 − 848560a1 a24 − 2428520a32 − 1888240a22 a3 + 489400a22 a4 + 1309488a2 a3 a4 + 1108880a2 a24 + 111360a33 − 191740a23 a4 + 550a3 a24 − 33295a34 − 27727920a21 + 47743680a1 a2 − 2597880a1 a3 − 14256960a1 a4 + 3526680a22 + 5056920a2 a3 + 9864060a2 a4 − 199440a23 − 2398224a3 a4 − 1126770a24 − 54993600a1 + 24634440a2 − 13051440a3 − 9252000a4 − 22140000. We remark that the terms O2 are polynomial in u7 and rational in a. We claim that there exists at least a transversal intersection point a∗ of the zero level sets of f1 , . . . , f4 where f5 is nonvanishing. Once we prove the claim the result will follow because near this point a∗ , thanks to the Implicit Function Theorem, we can write (28) as Xj = u7 vj−6 , being v1 , . . . , v4 new independent variables. Hence, the existence of a weak focus of maximal order 11 is clear and also its unfolding providing only 9 nonzero limit cycles because the displacement map (25) starts with degree 2 terms. The extra limit cycle for proving the last statement follows from the fact that ρ = 0 is an isolated solution. Let us prove finally that the claim is true. To this end we note that the system of equations {f1 = f2 = f3 = f4 = 0} has solutions that write as a∗ = (a∗1 , a∗2 , a∗3 , a∗4 ) = (β, α, βp3 (α), p4 (α)), where β 2 = p2 (α), with p2 , p3 and p4 some polynomials of degree 13 with rational coefficients, and α is a simple real solution of the polynomial p1 (x) = 14352187500x14 − 657776700000x13 + 11284736929875x12 − 42406416759825x11 − 1391899076716315x10 + 23023762786511909x9 ON THE NUMBER OF LIMIT CYCLES IN GENERALIZED ABEL EQUATIONS 21 − 123393106663586826x8 − 218236503571470586x7 + 5756360884347363494x6 − 27272611466754481126x5 + 55780715677026807263x4 − 48094920597945273157x3 + 9193962961957763353x2 + 5105496738368043633x − 1609769302079739192 verifying p2 (α) > 0. Straightforward computations show that there exist p5 and p6 , polynomials of degree 13 with rational coefficients such that f5 (a∗ ) = p5 (α) and the determinant of the Jacobian matrix det Jac(f1 ,f2 ,f3 ,f4 ) (a∗ ) = p6 (α). Moreover, the polynomials pk , for k = 2, . . . , 6 do not vanish at α, because the respective resultants with p1 , with respect to α, are nonzero rational numbers. Only remains to prove that there exists α such that also p2 (α) > 0. This follows just computing the real zeros of the polynomials p1 and p2 , ordering them in the real line and comparing their plots. From the 8 simple real zeros of p1 only 4 satisfy the condition p2 > 0. They are located near −12.079846278, −6.6037190290, 1.81965668348, 1.84169431112. This concludes the proof of the result.  We remark that working in the last proof with the complete Lyapunov constants instead of the linear developments we have not obtained more limit cycles and the computations to provide the weak focus of maximum order are even worse. Moreover, although from the proof it seems that we are computing only linear developments, from the final writing Xj = u7 vj−6 it is clear that a second order bifurcation mechanism is used, as we have shown in all the present paper. We finish the appendix by making some considerations about numerical simulations regarding other values of m and n. Theorem 1.2 improves the general lower bounds of H3,2 (n, m) that appear in the literature, in particular the ones in [6]. The new general lower bound is 2(n+m)−1 and, as we have commented before, it is very close to the total number of parameters 2(n+m)+2 in the system. Since we can rescale x and do a translation in θ, two of these parameters can be removed and only 2(n + m) remain. Hence it is reasonable to conjecture that H3,2 (n, m) = 2(n + m). Nevertheless it can be checked that with the degenerated Hopf bifurcation explained above we can not get such number of limit cycles when n + m ≤ 4 except for (n, m) = (1, 3), see Proposition A.2. Some of these computations where done in [6] by studying the maximum weak focus order. With regard to the segment n + m = 5, Proposition A.3 gets this value of limit cycles for (n, m) = (4, 1) whereas, numerically, we can get (also using the technique explained in the last section) that H3,2 (2, 3) ≥ 10∗ and H3,2 (1, 4) ≥ 10∗ . But we have not been able to improve, not even numerically, the lower bound H3,2 (3, 2) ≥ 9. (Here and below the superscript ∗ means that we have not an analytic proof but only numerical evidences.) The main difficulty is not in finding the system of equations to solve but in solving it. The numerical solutions that we have found seem to be values to have a center at the origin and not a weak focus of the order that we look for. For (n, m) = (1, 4), as we have done in Proposition A.2, we fix a1 = 1 and b1 = 0. The next step is the computation of the first linearized Lyapunov constants in the form (26) that are similar as (27). Then, again with the Implicit Function Theorem, we write Xk = uk for k = 2, . . . , 6 and we obtain for X7 , . . . , X11 the functions f1 , . . . , f5 similarly as the ones in (28), depending only on c = (c1 , c2 , c3 , c4 ). Here the polynomials fk have degree k + 5 for k = 1, . . . , 5. Hence, an equivalent Proposition A.3 could be conjectured for this case because, working with enough precision up to see the stabilization of the digits, we have found numerically a weak focus of order 11 at c∗ = (0.8339985012, 3.0122982805, 1.9052668985, −5.4437166429), that unfolds 9 nonzero limit cycles of small amplitude and yields to H3,2 (1, 4) ≥ 10∗ . For (n, m) = (2, 3) we can get a similar numerical result for H3,2 (2, 3) ≥ 10∗ . The main difference with the latter case is that the functions fk depend on (a1 , a2 , c1 , c2 ) and have de- gree 3k + 7, for k = 1, . . . , 5. 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China Email address: thuangjf@jnu.edu.cn Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barce- lona, Spain Email address: torre@mat.uab.cat Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain Email address: jordi.villadelprat@urv.cat