This is a preprint of: “Asymptotic development of an integral operator and boundedness of the criticality of potential centers”, David Rojas, J. Dyn. Diff. Equat., 1–40, 2019. DOI: [10.1007/s10884-019-09753-2] Asymptotic development of an integral operator and boundedness of the criticality of potential centers David Rojas Departament d’Informàtica, Matemàtica Aplicada i Estadı́stica, Universitat de Girona, 17003 Girona, Spain. Abstract. We study the asymptotic development at infinity of an integral operator. We use this development to give sufficient conditions to upper bound the number of critical periodic orbits that bifurcate from the outer boundary of the period function of planar potential centers. We apply the main results to two different families: the power-like potential family ẍ = xp − xq , p, q ∈ R, p > q; and the family of dehomogenized Loud’s centers. 1 Introduction Consider a continuous family of planar potential systems ẋ = −y, ẏ = Vµ0 (x), (1) where µ ∈ Λ is a parameter, Λ is an open subset of Rd , d > 1, and Vµ is an analytic function defined in an open interval Iµ ⊂ R containing x = 0. In the case when Vµ (0) = Vµ0 (0) = 0 and Vµ00 (0) > 0 equation (1) has a non-degenerate center at the origin for each value of the parameter and so the point (0, 0) has a punctured neighbourhood that is entirely foliated by periodic orbits surrounding it. The largest neighbourhood with this property is the period annulus of the center and we shall denote it by Pµ . If we consider the embedding of Pµ into RP2 , its boundary, namely ∂Pµ , is divided in two connected components: the origin itself, which is called the inner boundary of the period annulus, and the outer boundary of the period annulus defined by Πµ := ∂Pµ \ {(0, 0)}. When the center is a potential oscillator the natural parametrization of the closed orbits inside the period annulus is given by the energy level of the Hamiltonian H(x, y; µ) = 21 y 2 + Vµ (x). Since Vµ (0) = 0 by convention, we have that H(Pµ ) = (0, h0 (µ)), where h0 (µ) ∈ R+ ∪ {+∞} denotes the energy level of the outer boundary Πµ . The object under study in this paper is the period function of the center. The minimal period Tµ (h) of the periodic orbit γh,µ inside the energy level {H(x, y; µ) = h} can be written as the Abelian integral Z dx Tµ (h) = . γh,µ y This function is analytic on (0, h0 (µ)) for each value of the parameter and it can be extended analytically to h = 0 due to the non-degeneracy of the center. The derivative Tµ0 (h) can also be written as an Abelian integral and its zeros correspond to critical periodic orbits of the system. This paper is concerned with the bifurcation of such critical periodic orbits from the outer boundary Πµ . That is, for a fixed µ0 ∈ Λ, we aim to control the number of critical periodic orbits of system (1) that may emerge or disappear from Πµ0 as we move slightly the parameter µ ≈ µ0 . This number is called the criticality of the outer boundary. Definition 1.1. Consider a continuous family {Xµ }µ∈Λ of planar analytic vector fields with a center and fix some µ0 ∈ Λ. Suppose that the outer boundary of the period annulus varies continuously at µ0 ∈ Λ, 2010 Mathematics Subject Classification. 34C07; 34C23; 34C25. Key words and phrases: Center; Period function; Critical periodic orbit; Bifurcation; Criticality. 1 meaning that for any ε > 0 there exists δ > 0 such that dH (Πµ , Πµ0 ) 6 ε for all µ ∈ Λ with kµ − µ0 k 6 δ. Then, setting N (δ, ε) := sup{#critical periodic orbits γ of Xµ in Pµ with dH (γ, Πµ0 ) 6 ε and kµ − µ0 k 6 δ},  the criticality of (Πµ0 , Xµ0 ) with respect to the deformation Xµ is Crit (Πµ0 , Xµ0 ), Xµ := inf δ,ε N (δ, ε).  In the previous definition dH stands for the Hausdorff distance between compact sets of RP2 . Notice that according with this definition the criticality may be infinite but, in the case it is not, it gives the maximal number of critical periodic orbits of Xµ tending to the outer boundary Πµ0 in the Hausdorff sense as the parameter µ approaches µ0 . The requirement of the continuity of Pµ with respect to the parameters of the system ensures that the possible changes of Pµ do not occur abruptly. We refer to [8] for details illustrating the necessity of this extra assumption. Definition 1.2. A parameter µ0 ∈ Λ is called a localregular value of the period function at the outer boundary of the period annulus if Crit (Πµ0 , Xµ0 ), Xµ = 0. Otherwise the parameter is called a local bifurcation value at the outer boundary.  The present paper is a contribution that follows the spirit of the series of works [5,6,15]. In these papers, we develop analytical tools which allow to give an upper bound of the criticality at the outer boundary of the period annulus of families of planar potential systems (1). The key idea is to find a collection of functions φiµ (h), i = 1, 2, . . . , n, verifying that there exist δ, ε > 0 such that (φ1µ , φ2µ , . . . , φnµ , Tµ0 ) form an Extended Complete Chebyshev system (ECT-system for short, see Definition 2.10) on the interval (h0 (µ) − ε, h0 (µ)) and kµ − µ0 k 6 δ. This fact implies that Tµ0 (h) has at most n zeros in (h0 (µ) − ε, h0 (µ)), counting  multiplicities, uniformly on the parameters µ ≈ µ0 . In particular, Crit (Πµ0 , Xµ0 ), Xµ 6 n. According with Lemma 2.12, to give an upper bound of the criticality is reduced to guarantee that the Wronskian (see Definition 2.11) W [φ1µ , φ2µ , . . . , φnµ , Tµ0 ](h) does not vanish for all (h, µ) ≈ (h0 (µ0 ), µ0 ). The tools of the previous works, and also the ones we present here, allow to tackle this problem in the following two situations: either h0 (µ) = +∞ or h0 (µ) < +∞ for all µ ≈ µ0 . That is, the case in which there exist µ1 and µ2 in any neighbourhood of µ0 with h0 (µ1 ) = +∞ and h0 (µ2 ) < +∞ is not considered. Roughly speaking, the results in the previous papers relate the first term in the asymptotic develop- ment of the potential Vµ at the endpoints of Iµ with the first term of the asymptotic development of W [φ1µ , φ2µ , . . . , φnµ , Tµ0 ](h) at h = h0 (µ). This allows to control the sign of the Wronskian under consideration for (h, µ) ≈ (h0 (µ0 ), µ0 ); that is, uniformly on the parameters µ ≈ µ0 . However, there are some situations where these first terms of Vµ are not enough to compute the first term of the Wronskian at h = h0 (µ) and so more terms in the asymptotic development must be employed. Theorem D and E in Section 3 aim to generalize the results in [5, 6, 15] in this direction. To accomplish the desired results, we will employ a generalization of [6, Proposition 2.16] and [15, Theorem D] (see Theorem C in Section 2.) As an illustration of these generalizations we recover the study of two different families of planar centers. The first application is on the two-parametric family of potential differential system given by  ẋ = −y, (2) ẏ = (x + 1)p − (x + 1)q , which has a non-degenerate center at the origin for all µ := (q, p) varying in Λ := {(q, p) ∈ R2 : p > q}. As far as we know, the period function of this center was originally studied by Miyamoto and Yagasaki [14], giving a monotonicity result for q = 1 and p ∈ N. This result was improved later by Yagasaki [18] showing that the period function of (2) is monotonous for q = 1 and any p > 1 real. Motivated by those results, we considered the whole family (2) with p > q and performed an exhaustive study of the period function in [7]. Concerning the criticality at the outer boundary, the family (2) became our testing ground for the techniques mentioned before. In these  works, see Figure 1a, we proved that parameters µ0 ∈ Λ \ ΓB ∪ {q + 1 = 0} ∪ {(− 21 , p1 )} ∪ {(− 13 , p2 )} , with p1 ≈ 1.20175 and p2 ≈ 1.15685,  are local regular values of the period function at the outer boundary. In addition, Crit (Πµ0 , Xµ0 ), Xµ > 1 if µ0 ∈ ΓB and it is exactly one for parameters µ0 = (q0 , p0 ) satisfying either q0 = 0 and p0 ∈ (0, +∞) \ {1}, p0 = 1 and q0 < −3, or p0 + 2q0 + 1 = 0 and q0 ∈ (− 53 , − 13 ) \ {− 21 }. Using the tools in the present paper, the bifurcation diagram in Figure 1a is improved by the following result. (See Figure 1b.) 2 (a) Previous bifurcation diagram (b) New bifurcation diagram Figure 1: On the left, bifurcation diagram of the period function of the family (2) at the outer boundary of the period annulus according with [5, 6, 15]. On the right, improvement of the bifurcation diagram according with Theorem A. In both figures, ΓB stands for the union of the bold lines. In black the parameters with criticality at least one. In blue the parameters with criticality exactly one. Black squares are parameters that correspond to isochronous centers. Crosses are parameters where techniques do not apply. The q-axis do not preserve the scale for the sake of space. Theorem A. Let {Xµ }µ∈Λ be the family of analytic potential systems (2) and consider the period function  p0 = 1 and q0 ∈ (−3, −1) \ {−2}, or p0 + 2q0 + 1 = 0 of the center at the origin. If µ0 = (q0 , p0 ) with either and q0 ∈ (−1, − 21 ) \ {− 23 } then Crit (Πµ0 , Xµ0 ), Xµ = 1. The proof of this result is presented in Section 4.1.1 (see Proposition 4.2) and Section 4.1.2 (see Propo- sition 4.6). The result finishes the bifurcation diagram of the period function at the outer boundary of the family (2) except for the line {q + 1 = 0} and four points in the parameter space. (See Figure 1b.) We point out that the line corresponds to parameters such that the energy at the outer boundary h0 (µ) changes from infinite (q < −1) to finite (q > −1). The points correspond to parameters that do not satisfy the technical hypothesis to apply the analytic tools. The bifurcation diagram in Figure 1b agrees with the global bifurcation diagram conjectured in [7]. The second application is on the family of quadratic polynomial planar centers. The literature clas- sify quadratic centers in four families: Hamiltonian, reversible QR 3 , codimension four Q4 , and generalized Lotka-Volterra QLV3 . Chicone [2] conjectured that reversible centers have at most two critical periodic or- bits whereas the centers of the other three families have monotonic period function. Regarding quadratic reversible centers, by an affine transformation and a constant rescaling of time, they can be brought to the Loud normal form  ẋ = −y + Bxy, ẏ = x + Dx2 + F y 2 . In [3] the authors show that if B = 0 the period of the center at the origin is globally monotone. When B 6= 0 one can reduce the system, by means of a rescaling, to B = 1. That is,  ẋ = −y + xy, (3) ẏ = x + Dx2 + F y 2 . This family is known as dehomogenized Loud’s centers and it has a center at the origin for all parameters µ := (D, F ) ∈ R2 . The bifurcation of critical periodic orbits from the outer boundary of the period annulus of system (3) has been extensively studied in the recent years. (See Figure 2.) We refer to the series of papers [8, 10–12, 16, 17] and references therein. Our contribution to the bifurcation diagram at the outer boundary of the dehomogenized Loud’s centers is to show that almost all parameters in the bifurcation curve D = G(F ) have criticality exactly one. Up to 3 Figure 2: Bifurcation diagram of the period function at the polycycle of system (3), where µ? = (−F? , F? ) with F? ≈ 2.34. The union of the bold curves correspond to the set of bifurcation parameters at the outer boundary. The union of dotted straight lines correspond to the set of unspecified parameters. The complementary of those two sets correspond to regular parameters. now, this have been proved for parameters in that bifurcation curve with F ∈ ( 32 , 43 ). In order to state the result properly, let us consider the parameter space Λ := {(D, F ) ∈ R2 : 1 < F < 23 , D < − 12 , D + F > 0}. Moreover, let 2 F1 (a, b; c; z) be the Hypergeometric function (see [1, Section 15]), α := (p2 − 1)/(p2 − p1 ) with p1 and p2 defined in (15), and let c(µ) be the function in the statement of Lemma 4.3 using the expression of Vµ in (18). Theorem B. Let {Xµ }µ∈Λ be the family of analytic potential systems (3) and consider the period function  of the center at the origin. Let µ0 = (D0 , F0 ) ∈ Λ satisfying D0 = G(F0 ). Then Crit (Πµ0 , Xµ0 ), Xµ = 1 in the following situations: (a) If F0 ∈ ( 65 , 34 ). (b) If F0 ∈ ( 98 , 56 ) and c(µ0 ) 6= 0. (c) If F0 ∈ (1, 98 ], c(µ0 ) 6= 0 and  2 F1 − 32 , 52 ; 72 − 4F0 ; α(µ0 )  6= 0. (4) Γ 72 − 4F0 The proof of the result is given in Sections 4.2.2 and 4.2.3. In that last Section, also a numerical manifestation that condition (4) seems to be fulfilled for F ∈ (1, 98 ] is given. The condition c(µ0 ) 6= 0 is a technical requirement for the techniques involved in the proof and it is conjectured to be non necessary for the criticality to be also exactly one for those parameters satisfying c(µ0 ) = 0. Regarding the equality c(µ0 ) = 0 we also show numerically that the equation has a unique solution (D0 , F0 ) ≈ (−0.56996, 1.00781) in Section 4.2.3. The parameter D0 = G( 43 ) is conjectured to have criticality exactly two at the outer boundary of the period annulus. In the forthcoming paper [13] the authors also obtain (4) as a sufficient condition for bifurcation param- eters in D = G(F ) to have criticality exactly one using a completely different approach. In that work the authors study the asymptotic development of the Dulac time function near hyperbolic saddle singularities of meromorphic planar centers. They also use the dehomogenized Loud’s centers as testing ground and reach the same result in Theorem B without the technical restriction c(µ0 ) 6= 0 and allowing F0 = 65 . 4 The rest of the paper is organized as follows. In Section 2 we study the asymptotic behaviour of an integral operator that will be useful for the proof of the dynamical results in the paper. In Section 3 we use these techniques to obtain two results that gives sufficient conditions in order to bound the criticality at the outer boundary of planar potential centers. Finally Section 4 is dedicated to the application of such sufficient conditions and so we prove Theorems A and B. The work is complemented with an Appendix that contain the more technical proofs. 2 Asymptotic behaviour of a certain integral operator Let c ∈ R+ ∪ {+∞} and consider the integral operator F : C[0, c) → C[0, c) defined by Z π 2 F [f ](x) := f (x sin θ)dθ. 0 Here, and in what follows, C[0, c) stands for the set of continuous functions on [0, c). In the collection of works [5,6,15] the previous operator is studied because of its relation with the bifurcation of critical periodic orbits. Indeed, the derivative of the period function of system (1) satisfies the equality √ 2 0 2 2h Tµ (h ) = F [fµ ](h), h ∈ (0, h0 (µ)) (5) p with fµ (x) = x(gµ−1 )00 (x) − x(gµ−1 )00 (−x) and gµ (x) := sgn(x) Vµ (x). Roughly speaking, the main objective in these works is to give sufficient conditions to the first term of the asymptotic development at x = c of the function fµ in order that the first term of the asymptotic development at x = c of F [fµ ] is obtained, uniformly on the parameters. These conditions are formulated using the following notions.  Definition 2.1. Let {fµ }µ∈Λ be a continuous  family of continuous functions  on a(µ), b(µ) , meaning that the map (x, µ) 7−→ fµ (x) is continuous on (x, µ) ∈ R × Λ : x ∈ a(µ), b(µ) . Assume that either b : Λ → R is continuous or b ≡ +∞ in Λ. Given µ0 ∈ Λ we say that {fµ }µ∈Λ is continuously quantifiable in µ0 at b(µ) by α(µ) with limit `(µ) if there exists an open neighbourhood U of µ0 such that for all µ̂ ∈ U , (i) If b(µ0 ) < +∞, then lim(x,µ)→(b(µ̂),µ̂) fµ (x)(b(µ) − x)α(µ) = `(µ̂) and `(µ̂) 6= 0. (ii) If b(µ0 ) = +∞, then lim(x,µ)→(+∞,µ̂) x−α(µ) fµ (x) = `(µ̂) and `(µ̂) 6= 0. For the sake of shortness, in the first case we write fµ (x) ∼b(µ) `(µ)(b(µ) − x)−α(µ) at µ0 , and in the second case fµ (x) ∼+∞ `(µ)xα(µ) at µ0 . We use the analogous definition for the left endpoint a(µ).  We point out that the map α : U → R in the previous definition is continuous at µ = µ0 (see [5, Remark 2.6]). From now on let us assume that fµ ∈ C[0, +∞). The purpose of this section is to deal with an specific situation that was not contemplated in the previous works. With this aim in view we first recover the main results in [5, 15]. Definition 2.2. The function defined for all x > 0 and α ∈ R by means of ( α+1 x −1 α+1 if α 6= −1, ω(x, α) := log x if α = −1, is called the Roussarie-Ecalle compensator. For the sake of brevity, we also define √  π Γ 1+α 2  G (α) := and Ω(x, α) := (α + 1)G (α)ω(x, α), 2 Γ 1 + α2 5 where Γ is the Gamma function. Following the notation in Definition 2.1, we write fµ (x) ∼+∞ `(µ)Ω(x, α(µ)) at µ0 if fµ (x) lim = `(µ0 ) 6= 0. (x,µ)→(+∞,µ0 ) Ω(x, α(µ))  Definition 2.3. Let f ∈ C[0, +∞). We call Z +∞ Mn [f ] := x2n−2 f (x)dx 0 the n-th momentum of f , whenever it is well defined. If n = 1 we simply say that M [f ] := M1 [f ] is the momentum of f .  The next result gathers Theorems 2.13 and 2.17 in [5] together with Theorem C in [15]. Theorem 2.4. Let Λ be an open subset of Rd and consider a continuous family {fµ }µ∈Λ of continuous functions on [0, +∞). Suppose that fµ (x) ∼+∞ a(µ)xα(µ) at µ0 . The following assertions hold: (a) If α(µ0 ) > −1 then F [fµ ](x) ∼+∞ a(µ)G (α(µ))xα(µ) at µ0 . (b) If α(µ0 ) = −1 then F [fµ ](x) ∼+∞ a(µ)Ω(x, α(µ)) x1 at µ0 . (c) If α(µ0 ) < −1 let us take m ∈ N such that α(µ0 ) + 2m ∈ [−1, 1). In this case: (c1) If Mi [fµ ] ≡ 0 for i = 1, . . . , ` − 1 and M` [fµ0 ] 6= 0 for some 1 6 ` 6 m, then F [fµ ](x) ∼+∞ M` [fµ ]x1−2` at µ0 . (c2) If Mi [fµ ] ≡ 0 for i = 1, . . . , m and α(µ0 ) + 2m ∈ / {−1, 0} then m Y α(µ) + 2i F [fµ ](x) ∼+∞ a(µ) G (α(µ) + 2m)xα(µ) at µ0 . i=1 α(µ) + 2i − 1 (c3) If Mi [fµ ] ≡ 0 for i = 1, . . . , m and α(µ0 ) + 2m = −1 then m Y α(µ) + 2i F [fµ ](x) ∼+∞ a(µ) Ω(x, α(µ) + 2m)x−2m−1 at µ0 . i=1 α(µ) + 2i − 1 In the previous result the first term in the asymptotic development at infinity of the function F [fµ ](x) is provided as soon as fµ (x) ∼+∞ a(µ)xα(µ) at µ0 except in the case when α(µ0 ) = −2m for some m ∈ N, m > 1, and Mi [f ] = 0 for i = 1, . . . , m. In this special situation the hypothesis of fµ to be quantifiable by α(µ) = −2m is not enough to quantify F [fµ ] at infinity. The following three functions exemplify this phenomena for n = 1 even in the non-parametric scenario: ( 5 ( ( x−2 + x− 2 x > 1, x−2 + x−5 x > 1, x−2 + x−3 x > 1, f (x) = 22 16 g(x) = 13 9 h(x) = (6) 3 x− 3 x ∈ [0, 1), 2 x− 2 x ∈ [0, 1), 7x − 5 x ∈ [0, 1). All these functions are quantifiable by α = −2 at infinity and it is a computation to show that their momenta vanish. One can verify that F [f ] and F [g] are quantifiable at infinity by − 25 and −3 respectively, and that x3 1 lim F [h](x) = . x→+∞ log x 2 These three examples subscribe the idea that more information on the asymptotic development of the function fµ is needed to quantify F [fµ ] in this situation. To address this problem we generalize Definition 2.1 as follows. 6 Definition 2.5. Let {fµ }µ∈Λ be a continuous family of continuous functions on (a(µ), b(µ)). Assume that either b : Λ → R is continuous or b ≡ +∞ in Λ. Given µ0 ∈ Λ we say that fµ has uniformly development at x = b(µ) n X fµ (x) ∼b(µ) ai (µ)(b(µ) − x)−αi (µ) in µ0 i=1 if there exists an open neighbourhood U of µ0 such that for all µ̂ ∈ U , k−1 ! X αk (µ) −αi (µ) lim (b(µ) − x) fµ (x) − ai (µ)(b(µ) − x) = ak (µ̂) 6= 0 (x,µ)→(b(µ̂),µ̂) i=1 for each k = 1, . . . , n. If b ≡ +∞ then n X fµ (x) ∼+∞ ai (µ)xαi (µ) in µ0 i=1 if for all µ̂ ∈ U , k−1 ! X lim x−αk (µ) fµ (x) − ai (µ)xαi (µ) = ak (µ̂) 6= 0. (x,µ)→(b(µ̂),µ̂) i=1 We use the analogous definition at a.  Similarly as in Definition 2.1, the functions µ 7→ αk (µ) are continuous at µ = µ0 . Definition 2.6. Let f ∈ C[0, +∞). Setting [f ]0 := f , we define Z x 2 [f ]m (x) := x [f ]m−1 (x) + x [f ]m−1 (s)ds 0 for all m > 1.    The functions F [f ] and F [f ]m are related by the following result. Lemma 2.7 (see [5]). Let f ∈ C[0, +∞). For any m > 1, 1   F [f ](x) = F [f ]m (x) for all x > 0. x2m In the following statement the assumption M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]`−2 ] ≡ 0 in assertion (a) is void in case that ` = 1. Proposition 2.8. Let {fµ }µ∈Λ be a continuous family of continuous functions on [0, +∞) satisfying PN fµ (x) ∼∞ i=1 ai (µ)x −2ni + b(µ)xβ(µ) at µ0 with 1 6 n1 < n2 < · · · < nN positive integers and β(µ0 ) < −2nN . The following holds: (a) If M [[fµ ]0 ] ≡ · · · ≡ M [[fµ ]`−2 ] ≡ 0 and M [[fµ0 ]`−1 ] 6= 0 with 1 6 ` 6 nN then, for m = 1, 2, . . . , ` − 1, N X m Y Y β(µ) + 2j m 2j − 2ni [fµ ]m (x) ∼∞ ai (µ)x2m−2ni + b(µ)xβ(µ)+2m at µ0 i=1 j=1 2j − 2n i − 1 j=1 β(µ) + 2j − 1 ni >m and [fµ ]` (x) ∼∞ M [[fµ ]`−1 ]x at µ0 . QnN β(µ)+2j β(µ)+2nN (b) If M [[fµ ]0 ] ≡ · · · ≡ M [[fµ ]nN −1 ] ≡ 0 then [fµ ]nN (x) ∼∞ j=1 β(µ)+2j−1 b(µ)x at µ0 . 7 Proof. We shall prove the result by induction on m. To do so, we shall first assume m ∈ {1, . . . , n1 − 1}. (In the case n1 = 1 this assumption is void and we move to the next step.) Let us start considering m = 1. By definition of [f ]1 and elementary manipulations, we have Pk−1 2−2ni 2−2ni Pk−1 2−2ni Rx [fµ ]1 (x) − i=1 1−2ni ai (µ)x x2 [fµ ]0 (x) − i=1 1−2ni ai (µ)x 2−2ni +x 0 [fµ ]0 (s)ds = x2−2nk x2−2nk Pk−1 Rx Pk−1ai (µ) 1−2ni [fµ ]0 (x) − i=1 ai (µ)x −2ni 0 [fµ ]0 (s)ds − i=1 1−2ni x = −2n + x k x1−2nk for any k = 1, . . . , N . The hypothesis M [[fµ ]0 ] ≡ 0 together with ni > 1 imply that the numerator of the second quotient tends to zero as x tends to infinity. Therefore the second quotient is a 0/0-indeterminacy as (x, µ) → (+∞, µ̂) for any µ̂ ≈ µ0 . We apply the Uniform Hôpital’s Rule in [5, Proposition A.1] to deduce that Pk−1 2−2ni [fµ ]1 (x) − i=1 1−2n i ai (µ)x2−2ni 2 − 2nk lim = ak (µ̂) (x,µ)→(+∞,µ̂) x2−2nk 1 − 2nk for any k = 1, . . . , N . Similarly, PN 2−2ni 2−2ni [fµ ]1 (x) − i=1 1−2ni ai (µ)x β(µ) + 2 lim = b(µ̂). (x,µ)→(+∞,µ̂) xβ(µ)+2 β(µ) + 1 Therefore, N X 2 − 2ni β(µ) + 2 [fµ ]1 (x) ∼∞ ai (µ)x2−2ni + b(µ̂)xβ(µ)+2 at µ0 . i=1 1 − 2ni β(µ) + 1 This proves the result for m = 1. If m ∈ {2, . . . , n1 − 1} the result follows identically changing [fµ ]0 by [fµ ]1 . So we have N Y X m Y β(µ) + 2jm 2j − 2ni [fµ ]m (x) ∼∞ ai (µ)x2m−2ni + b(µ)xβ(µ)+2m at µ0 i=1 j=1 2j − 2n i − 1 j=1 β(µ) + 2j − 1 for m = 1, . . . , n1 − 1. Let us consider now that m ∈ {n1 , . . . , n2 − 1}, and let us start by taking m = n1 . The same procedure as before can be applied taking into account that the fist term in the asymptotic development disappear. Indeed, we have X n1 N Y Y 1 n 2j − 2ni β(µ) + 2j [fµ ]n1 (x) ∼∞ ai (µ)x2n1 −2ni + b(µ)xβ(µ)+2n1 at µ0 , i=2 j=1 2j − 2n i − 1 j=1 β(µ) + 2j − 1 where now the sum starts at i = 2 instead of the previous i = 1. Now using the first procedure the result follows for m = n1 , . . . , n2 − 1. Inductively the result holds for m = 1, . . . , ` − 1 provided that M [[fµ ]0 ] ≡ · · · M [[fµ ]`−2 ] ≡ 0. To finish the proof of (a), let us assume that M [[fµ0 ]`−1 ] 6= 0. Then  Z x  [fµ ]` (x) lim = lim x[fµ ]`−1 (x) + [fµ ]`−1 (s)ds = M [[fµ̂ ]`−1 ] 6= 0 (x,µ)→(+∞,µ̂) x (x,µ)→(+∞,µ̂) 0 for all µ̂ ≈ µ0 . Here we used that N X `−1 Y 2j − 2ni [fµ ]`−1 (x) ∼∞ ai (µ)x2`−2−2ni i=1 j=1 2j − 2n i − 1 ni >`−1 and, since ` 6 ni , x[fµ ]`−1 (x) → 0 as (x, µ) → (+∞, µ̂). This proves (a). To show (b) let us assume that M [[fµ ]0 ] ≡ · · · M [[fµ ]nN −1 ] ≡ 0. The procedure holds for all m = 1, . . . , nN in this case. For m = nN all 8 the terms in the asymptotic development associated with even powers disappear and so the only remaining term is the one with betas. Then, nN Y β(µ) + 2j [fµ ]nN (x) ∼∞ b(µ)xβ(µ)+2nN at µ0 . j=1 β(µ) + 2j − 1 This ends the proof of the result. Remark 2.9. A very useful tool for the computation of momenta was introduced in [5]. If f is quantifiable by α < −2n + 1, n ∈ N with n > 2, and M [[f ]i ] = 0 for i = 0, . . . , n − 2 then Z ∞ M [[f ]n−1 ] = Cn x2n−2 f (x)dx = Cn Mn [f ] 0 for some constant Cn 6= 0. We point out that this result do not contemplate the case when α = −2n and M [[f ]i ] = 0 for i = 0, . . . , n − 2. Therefore this simplification can not be used in Proposition 2.8.  From now on, for the sake of simplicity, we shall assume that the functions are analytic on [0, +∞). The reason is that in Section 3 the differential system (1) is assumed to be analytic and so the functions involved also are. However, the reader may notice that weaker regularity is allowed in the forthcoming definitions and results. Let us recall at this point the notions of Chebyshev system and its relation with the Wronskian, which are both key ingredients for our purposes. Definition 2.10. Let f0 , f1 , . . . fn−1 be analytic functions on an open real interval I. The ordered set (f0 , f1 , . . . fn−1 ) is an extended complete Chebyshev system (for short, a ECT-system) on I if, for all k = 1, 2, . . . n, any nontrivial linear combination α0 f0 (x) + α1 f1 (x) + · · · + αk−1 fk−1 (x) has at most k−1 isolated zeros on I counted with multiplicities. (Let us mention that, in these abbreviations, “T” stands for Tchebycheff, which in some sources is the transcription of the Russian name Chebyshev).  Definition 2.11. Let f0 , f1 , . . . , fn−1 be analytic functions on an open interval I of R. Then f0 (x) ··· fn−1 (x)   f00 (x) ··· 0 fn−1 (x) (i) W [f0 , f1 , . . . , fn−1 ](x) = det fj (x) = .. 06i,j6n−1 . (n−1) (n−1) f0 (x) · · · fn−1 (x) is the Wronskian of (f0 , f1 , . . . , fn−1 ) at x ∈ I.  These two notions are closely related by the following result (see for instance [4]). Lemma 2.12. (f0 , f1 , . . . , fn−1 ) is an ECT-system on I if and only if, for each k = 1, 2, . . . , n, W [f0 , f1 , . . . , fk−1 ](x) 6= 0 for all x ∈ I. In the study of bifurcation of critical periodic orbits from the outer boundary the main objective is to bound the zeros of the derivative of the period function for energy levels h ≈ h0 (µ) uniformly on the parameters close to a fixed µ0 ∈ Λ. Invoking equality (5), this problem is tackled by completing F [fµ ] with a collection of analytic functions φ1µ , . . . , φnµ in order that (φ1µ , . . . , φnµ , F [fµ ]) form an ECT-system on (h0 (µ) − ε, h0 (µ)) for some ε > 0 and all µ ≈ µ0 . Note that guarantee the uniformity with respect to the parameters of the system is mandatory to obtain the desired upper bounds. With this aim in view, and on account of the characterization in Lemma 2.12, given ν1 , ν2 , . . . , νn ∈ R, we consider the linear ordinary differential operator Lν n (µ) : C ω (0, +∞) → C ω (0, +∞) 9 defined by W [xν1 , xν2 , . . . , xνn , f (x)] Lν n (µ) [f ](x) := Pn . x i=1 (νi −i) Here, and in what follows, for the sake of shortness we use the notation ν n = (ν1 , ν2 , . . . , νn ). Furthermore we define Lν 0 = id in order that forthcoming statements contemplate the case n = 0 as well. We also denote by C ω (0, +∞) the analytic functions on (0, +∞) and by C ω [0, +∞) the functions on C ω (0, +∞) that can be extended analytically to x = 0. Proposition 2.13 (See [6]). For any f ∈ C ω (0, +∞) and ν1 , . . . , νn ∈ R, the following recurrence holds:  Lν n [f ](x) = cn xLν n−1 [f ]0 (x) − νn Lν n−1 [f ](x) , Qn−1 where c1 := 1 and cn := i=1 (νn −νi ) for n > 2. In particular, if f ∈ C ω [0, +∞), then Lν n [f ] ∈ C ω [0, +∞). Moreover, F ◦ Lν n = Lν n ◦ F . In the statement of the following result the assumptions M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]`−2 ] ≡ 0 in assertion (a) and Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , ` − 1 in assertion (d1) are void in case that ` = 1. Theorem C. Let Λ be an open subset of Rd and {fµ }µ∈Λ be a continuous family of analytic functions on [0, +∞). Assume that, in a neighbourhood of some fixed µ0 ∈ Λ there exist n > 0 continuous functions ν1 , ν2 , . . . , νn pairwise distinct at µ = µ0 such that the function Lµ := Lν n (µ) [fµ ] satisfies N X Lµ (x) ∼∞ ai (µ)x−2ni + b(µ)xβ(µ) at µ0 i=1 with 1 6 n1 < n2 < · · · < nN positive integers and β(µ0 ) < −2nN . The following holds: (a) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]`−2 ] ≡ 0 and M [[Lµ0 ]`−1 ] 6= 0 for some 1 6 ` 6 nN then (Lν n (µ) ◦ F )[fµ ](x) ∼∞ M [[Lµ ]`−1 ]x1−2` at µ0 . (b) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]nN −1 ] ≡ 0 and β(µ0 ) + 2nN = −1 then nN Y β(µ) + 2j Ω(x, β(µ) + 2nN ) (Lν n (µ) ◦ F )[fµ ](x) ∼∞ b(µ) at µ0 . j=1 β(µ) + 2j − 1 x1+2nN (c) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]nN −1 ] ≡ 0 and β(µ0 ) + 2nN > −1 then nN Y β(µ) + 2j (Lν n (µ) ◦ F )[fµ ](x) ∼∞ G (β(µ) + 2nN ) b(µ)xβ(µ) at µ0 . j=1 β(µ) + 2j − 1 (d) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]nN −1 ] ≡ 0 and β(µ0 ) + 2nN < −1, let us take m ∈ N such that β(µ0 ) + 2nN + 2m ∈ [−1, 1) and let us assume additionally that β(µ0 ) + 2nN + 2m 6= 0. In this case, (d1) If Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , ` − 1 and M` [[Lµ0 ]nN ] 6= 0 for some 1 6 ` 6 m, then (Lν n (µ) ◦ F )[fµ ](x) ∼∞ M` [[Lµ ]nN ]x1−2nN −2` at µ0 . (d2) If Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , m and β(µ0 ) + 2nN + 2m 6= −1 then nN Y +m β(µ) + 2j (Lν n (µ) ◦ F )[fµ ](x) ∼∞ b(µ)G(β(µ) + 2nN + 2m)xβ(µ) at µ0 . j=1 β(µ) + 2j − 1 10 (d3) If Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , m and β(µ0 ) + 2nN + 2m = −1 then nN Y +m β(µ) + 2j Ω(x, β(µ) + 2nN + 2m) (Lν n (µ) ◦ F )[fµ ](x) ∼∞ b(µ) at µ0 . j=1 β(µ) + 2j − 1 x2nN +2m+1 Proof. Let us start by considering n = 0. In this case Lµ = fµ and the assumptions of Proposition 2.8 are satisfied. Moreover, if (a) is satisfied then by (a) in Proposition 2.8 we have [fµ ]` (x) ∼∞ M [[fµ ]`−1 ]x at µ0 . Therefore, using Lemma 2.7 and Theorem 2.4 with [fµ ]` , since G (1) = 1, 1 F [fµ ](x) = F [[fµ ]` ](x) ∼+∞ M [[fµ ]`−1 ]x1−2` at µ0 . x2` This proves (a). Let us show (b), (c) and (d). By hypothesis we have M [[fµ ]i ] ≡ 0 for i = 0, . . . , nN − 1. Therefore by (b) in Proposition 2.8, nN Y β(µ) + 2j [fµ ]nN ∼+∞ b(µ)xβ(µ)+2nN at µ0 . j=1 β(µ) + 2j − 1 The result follows using Theorem 2.4 with [fµ ]nN and Lemma 2.7 again. This ends the proof for the case n = 0. Let us consider now n > 1. By Proposition 2.13 Lµ = Lν n (µ) [fµ ] is an analytic function on [0, +∞) for each µ ∈ Λ and (Lν n (µ) ◦ F )[fµ ](x) = (F ◦ Lν n (µ) )[fµ ](x). Then the result follows by applying the case n = 0 to the family {Lν n (µ) [fµ ]}µ∈Λ . Remark 2.14. Let us recover at this point the examples in (6). The function f satisfies the assumptions of Theorem C with n = 0, n1 = 1 and β = −5/2. Since β + 2 = −1 2 > −1 then assertion (b) of the Theorem states that √  π Γ 41 − 5 F [f ](x) ∼+∞  x 2. 6 Γ 43 The function g satisfies the hypothesis with n = 0, n1 = 1 and β = −5. In this case β + 2 = −3 < −1 and Z ∞ ( 3 ! 4x 3 if x > 1 3 M [[g]1 ] = 3 2 dx = − . 0 4 x (13x − 12) if x ∈ [0, 1) 16 That is, g satisfies the hypothesis in assertion (c1) of the Theorem with j = 1. Therefore, 3 −3 F [g](x) ∼+∞ − x . 16 Finally, the function h satisfies the hypothesis with n = 0, n1 = 1 and β = −3. In this example assumptions in assertion (b) are satisfied and log x F [h](x) ∼+∞ . 2x3  We point out that Theorem C together with Theorem 2.4 cover all possible situations of the uniform asymptotic development of fµ except the case when all powers are negative even numbers at µ = µ0 . The last result of this Section is a recursive formula for the computation of a certain momentum that will be useful in the applications. 11 Lemma 2.15. Let f ∈ C ω [0, +∞), n > 1 and ν1 , ν2 , . . . , νn ∈ R. Then, Z Z ! R   3 (νn + 1)R2 3νn + 5 R 2 M [Lν n [f ]]1 = cn lim R Lν n−1 [f ](R) + Lν n−1 [f ](x)dx − x Lν n−1 [f ](x)dx R→+∞ 2 0 2 0 Qn−1 where c1 := 1 and cn := i=1 (νn − νi ) for n > 2. Proof. By Definition 2.6 we have Z ∞ Z R  Z x    M [Lν n [f ]]1 = [Lν n [f ]]1 (x)dx = lim x2 Lν n [f ](x) + x Lν n [f ](s)ds dx. 0 R→+∞ 0 0 Using the recursive expression in Proposition 2.13, Z R    M [Lν n [f ]]1 = cn lim x3 Lν n−1 [f ]0 (x) − νn x2 Lν n−1 [f ](x) dx + R→+∞ 0 Z R  Z x  !  + x sLν n−1 [f ]0 (s) − νn Lν n−1 [f ](s) ds dx . 0 0 The result then follows integrating by parts. 3 Criticality of the period function at the outer boundary In this section we apply Theorem C in order to obtain sufficient conditions to bound the number of critical periodic orbits that may bifurcate from the outer boundary of the period annulus in families of planar potential centers. We consider analytic differential systems (1) depending on a parameter µ ∈ Λ ⊂ Rd and we assume that the origin is a non-degenerate center for all µ. We denote by Iµ = (x` (µ), xr (µ)) the projection of the period annulus on the x-axis, x` < 0 < xr and by h0 (µ) the energy level at the outer boundary of the period annulus. Definition 3.1. We say that the family (1) verifies the hypothesis (H) in case that: (k) (a) For all k > 0, the map (x, µ) 7−→ Vµ (x) is continuous on {(x, µ) ∈ R × Λ : x ∈ Iµ }, (b) µ 7−→ xr (µ) is continuous on Λ or xr (µ) = +∞ for all µ ∈ Λ, (c) µ 7−→ x` (µ) is continuous on Λ or x` (µ) = −∞ for all µ ∈ Λ, (d) µ 7−→ h0 (µ) is continuous on Λ or h0 (µ) = +∞ for all µ ∈ Λ.  Lemma 3.2 (See [5]). Let {Xµ }µ∈Λ be a family of potential analytic differential systems p verifying p (H).   Then the map (z, µ) 7−→ gµ−1 (z) is continuous on the open set (z, µ) ∈ R × Λ : z ∈ − h0 (µ), h0 (µ) . This section is divided in two parts according with the dichotomy produced by h0 . First, Section 3.1 deals with the case h0 ≡ +∞. Second, Section 3.2 is dedicated to the case h0 finite. 3.1 Potential systems with infinite energy In this section we present sufficient conditions to bound the criticality at the outer boundary for potential systems satisfying h0 (µ) = +∞ for all µ ∈ Λ. Following the strategy in [5,6,15], we find sufficient conditions such that f Tµ0 can be embedded into the ECT-system (hν1 (µ) , hν2 (µ) , . . . , hνn (µ) ), where f is an analytic non- vanishing function. Next result is a combination of [6, Lemma 3.5] and [15, Lemma 3.3], and it is the key piece that connects the analytic tools studied in Section 2 with the dynamical results we are looking for. 12 Lemma 3.3. Let {Xµ }µ∈Λ be a family of potential analytic differential systems verifying (H) and such that h0 ≡ +∞. Assume that there exist n > 1 continuous functions ν1 , ν2 , . . . , νn in a neighbourhood of some fixed µ0 ∈ Λ, a continuous function α : Λ → R with α(µ0 ) = −1 and an analytic non-vanishing function f on (0, +∞) such that hνn (µ) lim W [hν1 (µ) , . . . , hνn−1 (µ) , f (h)Tµ0 (h)] = ` 6= 0 (h,µ)→(+∞,µ0 ) Ω(h, α(µ))m  with m ∈ {0, 1}. Then Crit (Πµ0 , Xµ0 ), Xµ 6 n − 1. The assumption requiring the existence of functions ν1 , ν2 , . . . , νn in the following statement is void in case that n = 0. The same happens to the assumptions M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]`−2 ] ≡ 0 in assertion (a) and Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , ` − 1 in assertion (c1) in case that ` = 1. Theorem D. Let {Xµ }µ∈Λ be a family of potential analytic differential systems verifying (H) with h0 ≡ +∞ and that there exist n > 0 continuous functions ν1 , ν2 , . . . , νn in a neighbourhood of some fixed µ0 ∈ Λ such that N X Lµ (x) := Lν n (µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] ∼+∞ ai (µ)x−2ni + b(µ)xβ(µ) at µ0 i=1  with 1 6 n1 < n2 < · · · < nN positive integers and β(µ0 ) < −2nN . Then Crit (Πµ0 , Xµ0 ), Xµ 6 n if one of the following assertions hold: (a) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]`−2 ] ≡ 0 and M [[Lµ0 ]`−1 ] 6= 0 for some 1 6 ` 6 nN . (b) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]nN −1 ] ≡ 0 and β(µ0 ) + 2nN > −1. (c) If M [[Lµ ]0 ] ≡ · · · ≡ M [[Lµ ]nN −1 ] ≡ 0 and β(µ0 ) + 2nN < −1, let us take m ∈ N such that β(µ0 ) + 2nN + 2m ∈ [−1, 1) \ {0}. In this case: (c1) If Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , ` − 1 and M` [[Lµ0 ]nN ] 6= 0 for some 1 6 ` 6 m. (c2) If Mi [[Lµ ]nN ] ≡ 0 for i = 1, . . . , m. Proof. For the sake of shortness let us denote fµ (x) := x(gµ−1 )00 (x) − x(gµ−1 )00 (−x). Lemma 3.2 and the hypothesis (H) imply that {fµ }µ∈Λ is a continuous family of analytic functions on [0, +∞). According to equality (5) the result will follow once we show that there exist M, ε > 0 in such a way F [fµ ] has at most n isolated zeros for h > M and kµ − µ0 k < ε, multiplicities taken into account. Let us assume that Lµ := Lν n (µ) [fµ ] satisfies one of the hypothesis of the statement. Therefore Lµ satisfies one of the hypothesis in Theorem C, so we can assert that either (Lν n (µ) ◦ F )[fµ ](h) ∼+∞ C(µ)hξ(µ) at µ0 or (Lν n (µ) ◦ F )[fµ ](h) ∼+∞ C(µ)Ω(h, α(µ))hξ(µ) at µ0 for some functions C, ξ, α with C(µ0 ) 6= 0 and α(µ0 ) = −1. Taking into account the definition of the operator Lν n (µ) , we have that either   W hν1 (µ) , . . . , hνn (µ) , F [fµ ](h) lim Pn = C(µ0 ), (x,µ)→(+∞,µ0 ) hξ(µ)+ i=1 (νi (µ)−i) or   W hν1 (µ) , . . . , hνn (µ) , F [fµ ](h) lim P n = C(µ0 ). (x,µ)→(+∞,µ0 ) hξ(µ)+ i=1 (νi (µ)−i) Ω(h, α(µ))  Therefore, on account of equality (5), by Lemma 3.3 we have that Crit (Πµ0 , Xµ0 ), Xµ 6 n as desired. 13 3.2 Potential systems with finite energy We assume in this section that the energy at the outer boundary of system (1) is finite for all parameters µ ∈ Λ. As in the previous works [5, 6, 15], in order to embed the function f Tµ0 into some ECT-system for an appropriate non-vanishing function f , the spirit of this section is to “translate” the case h0 < +∞ to the case h0 = +∞ so we can take advantage of Theorem C. This translation is given by the operator B : C ω [0, 1) −→ C ω [0, +∞) defined by   1  B[f ](x) := 1 − φ2 (x) f ◦ φ (x) = 2 f ◦ φ (x), (7) 1+x where φ(x) := √1+x x 2 . Given ν1 , . . . , νn ∈ R, B conjugates the operator Lν n with the linear ordinary differential operator Dν n : C ω (0, 1) −→ C ω (0, 1) defined by n(n+1) W [ψν1 , . . . , ψνn , f ] (z) Dν n [f ](z) := (z(1 − z 2 )) 2 Qn . (8) i=1 ψνi (z) Definition 3.4. Let f ∈ C ω [0, 1). We call Z 1  2n−2 f (z) z Nn [f ] := √ √ dz 0 1 − z2 1 − z2 the n-th momentum of f , whenever it is well defined. If n = 1 we simply say that N [f ] := N1 [f ] is the momentum of f .  Lemma 3.5. Consider ν1 , ν2 , . . . , νn ∈ R. Then the following hold: (a) B[ψνi ](x) = xνi for i = 1, 2, . . . , n. (b) B ◦ Dν n = Lν n ◦ B.  √  (c) F ◦ B [f ](x) = 1 + x2 B ◦ F [f ](x) for any f ∈ C ω (0, 1). (d) Nn = Mn ◦ B. Definition 3.6. Given a continuous function f on [0, 1) we define [f ]m (z) := 1 1−z 2 [B[f ]]m (φ −1 (z)).  Lemma 3.7. Let {fµ }µ∈Λ be a continuous family of analytic functions on [0, 1). Then n X n X fµ (z) ∼z=1 ai (µ)(1 − z 2 )−αi (µ) at µ0 if and only if B[fµ ](x) ∼+∞ ai (µ)x2αi (µ)−2 at µ0 . i=1 i=1 1 √ x Proof. By definition B[f ](x) = 1+x2 f (φ(x)) with φ(x) = 1+x2 . Therefore, for a fixed µ0 ∈ Λ, Pk−1 2αi (µ)−2 Pk−1 B[fµ ](x) − i=1 ai (µ)x fµ (φ(x)) − ai (µ)(1 + x2 )αi (µ) i=1 lim = lim . (x,µ)→(+∞,µ0 ) x2αk (µ)−2 (x,µ)→(+∞,µ0 ) (1 + x2 )αk (µ) Denoting z = φ(x), since φ(x) → 1 as x → +∞, we have that Pk−1 Pk−1 B[fµ ](x) − i=1 ai (µ)x 2αi (µ)−2 fµ (z) − i=1ai (µ)(1 − z 2 )−αi (µ) lim = lim . (x,µ)→(+∞,µ0 ) x k (µ)−2 2α (z,µ)→(1,µ0 ) (1 − z 2 )−αk (µ) Consequently the result follows on account of Definition 2.5. 14 The following is an analogous version of Lemma 3.3 for the case h0 finite. It gathers [6, Lemma 3.10] and [15, Lemma 3.8]. Lemma 3.8. Let {Xµ }µ∈Λ be a family of potential analytic differential systems verifying (H) and such that µ 7→ h0 (µ) is continuous on Λ. Assume that there exist n > 1 continuous functions ν1 , ν2 , . . . , νn in a neighbourhood of some fixed µ0 ∈ Λ, a continuous function α : Λ → R with α(µ0 ) = −1 and an analytic non-vanishing function f on (0, 1) such that (1 − z)νn (µ) lim  m W [ψν1 (µ) (z), . . . , ψνn−1 (µ) (z), f (z)Tµ0 (z 2 h0 (µ))] = ` 6= 0 (z,µ)→(1,µ0 ) z Ω √ 1−z 2 , α(µ)  with m ∈ {0, 1}. Then Crit (Πµ0 , Xµ0 ), Xµ 6 n − 1. In the same way as in the main result of the previous section, we stress that the assumption requiring the existence of functions ν1 , ν2 , . . . , νn in the following statement is void in case that n = 0. Also are void the assumptions N [[Dµ ]0 ] ≡ · · · ≡ N [[Dµ ]`−2 ] ≡ 0 in assertion (a) and Ni [[Dµ ]nN ] ≡ 0 for i = 1, . . . , ` − 1 in assertion (c1) in case that ` = 1. Theorem E. Let {Xµ }µ∈Λ be a family of potential analytic differential systems verifying (H) with h0 (µ) < +∞ for all µ ∈ Λ and that there exist n > 0 continuous functions ν1 , ν2 , . . . , νn in a neighbourhood of some fixed µ0 ∈ Λ such that the function p p p p fµ (z) := z h0 (µ)(gµ−1 )00 (z h0 (µ)) − z h0 (µ)(gµ−1 )00 (−z h0 (µ)) satisfies N X Dµ (z) := Dν n (µ) [fµ ](z) ∼z=1 ai (µ)(1 − z 2 )ni + b(µ)(1 − z 2 )β(µ) at µ0 i=1  with 0 6 n1 < n2 < · · · < nN integers and β(µ0 ) > nN . Then Crit (Πµ0 , Xµ0 ), Xµ 6 n if one of the following assertions hold: (a) If N [[Dµ ]0 ] ≡ · · · ≡ N [[Dµ ]`−2 ] ≡ 0 and N [[Dµ0 ]`−1 ] 6= 0 for some 1 6 ` 6 nN . (b) If N [[Dµ ]0 ] ≡ · · · ≡ N [[Dµ ]nN −1 ] ≡ 0 and β(µ0 ) − nN 6 12 . (c) If N [[Dµ ]0 ] ≡ · · · ≡ N [[Dµ ]nN −1 ] ≡ 0 and β(µ0 )−nN > 12 , let us take m ∈ N such that β(µ0 )−nN −m ∈ (− 12 , 12 ] \ {0}. In this case: (c1) If Ni [[Dµ ]nN ] ≡ 0 for i = 1, . . . , ` − 1 and N` [[Dµ0 ]nN ] 6= 0 for some 1 6 ` 6 m. (c2) If Ni [[Dµ ]nN ] ≡ 0 for i = 1, . . . , m. Proof. According with Lemma 3.2 and the hypothesis (H), the family {fµ }µ∈Λ is a continuous family of analytic functions on [0, 1). On account of equality (5), after the appropriate rescaling, √ F [fµ ](z) = 2h0 (µ)z 2 Tµ0 (h0 (µ)z 2 ) for all z ∈ (0, 1). (9) Therefore the result will follow if there exist ε > 0 and a neighbourhood U of µ0 such that F [fµ ](z) has at most n zeros for all z ∈ (1 − ε, 1) and µ ∈ U , multiplicities taken into account. To this end, we shall use that the operator B commutes the linear differential operators Dν n (µ) , defined for functions in C ω [0, 1), with Lν n (µ) , defined for functions in C ω [0, +∞). Then, as we proceeded in Theorem D, we aim to apply Theorem C in this case to the family B[fµ ]. First, by hypothesis we have that N X Dν n (µ) [fµ ](z) ∼z=1 ai (µ)(1 − z 2 )ni + b(µ)(1 − z 2 )β(µ) at µ0 , i=1 15 so, on account of Lemmas 3.5 and 3.7, we have N X (Lν n (µ) ◦ B)[fµ ](x) = (B ◦ Dν n (µ) )[fµ ](x) ∼+∞ ai (µ)x−2ni −2 + b(µ)x−2β(µ)−2 at µ0 . i=1 Moreover, on account of Lemma 3.5 and Definition 3.6,    1 −1    1 −1    Nn [Dµ ]m = Nn 1−z 2 [B[Dµ ]]m (φ (z)) = Mn B 1−z 2 [B[Dµ ]]m (φ (z)) = Mn [B[Dµ ]]m . That is,     Nn [Dµ ]m = Mn [(Lν n (µ) ◦ B)[fµ ]]m . Therefore, if Dµ satisfies one of the hypothesis of the statement, the function Lµ := (Lν n (µ) ◦B)[fµ ] satisfies one of the hypothesis in Theorem C. It turns out then that either (Lν n (µ) ◦ F ◦ B)[fµ ](x) ∼+∞ C(µ)xξ(µ) at µ0 or (Lν n (µ) ◦ F ◦ B)[fµ ](x) ∼+∞ C(µ)Ω(x, α(µ))xξ(µ) at µ0 for some functions C, ξ, α with C(µ0 ) 6= 0 and α(µ0 ) = −1. Let us note that p  (Lν n (µ) ◦ F ◦ B)[fµ ](x) = Lν n (µ) 1 + x2 (B ◦ F )[fµ ](x)  1  = (Lν n (µ) ◦ B) (1 − z 2 )− 2 F [fµ ](z) (x)  1  = (B ◦ Dν n (µ) ) (1 − z 2 )− 2 F [fµ ](z) (x), √ with z = φ(x) = √ x , where we use (c) in Lemma 3.5 in the first quality, the identity 1 + x2 B[ϕ](x) = 1+x2 2 − 21 B[(1 − z ) ϕ(z)] with ϕ = F [fµ ] in the second equality, and (b) in Lemma 3.5 in the third equality. So we have   1 (B ◦ Dν n (µ) ) (1 − z 2 )− 2 F [fµ ](z) (x) ∼+∞ C(µ)Ω(x, α(µ))m xξ(µ) at µ0 with m ∈ {0, 1}. Using the definition of B, the previous equation yields to 1  1  2 Dν n (µ) (1 − φ(x)2 )− 2 F [fµ ](φ(x)) ∼+∞ C(µ)Ω(x, α(µ))m xξ(µ) at µ0 . 1+x Setting x = φ−1 (z) = √ z , the previous identity implies that 1−z 2 ξ(µ)  1  (1 − z 2 )1+ 2Dν n (µ) (1 − z 2 )− 2 F [fµ ](z) lim z  m = C(µ0 ). (z,µ)→(1,µ0 ) Ω √1−z 2 , α(µ) Thus, on account of the definition of Dν n (µ) in (8), ξ(µ) n(n+1)  1  (1 − z 2 )1+ 2 (z(1 − z 2 )) 2 W ψν1 (µ) (z), . . . , ψνn (µ) (z), (1 − z 2 )− 2 F [fµ ](z) lim z m Qn = C(µ0 ) (z,µ)→(1,µ0 ) Ω √1−z 2 , α(µ) i=1 ψνi (µ) (z) zν which, since ψν (z) = (1−z 2 )1+ν/2 , implies that (1 − z 2 )κ(µ)  1  lim z m W ψν1 (µ) (z), . . . , ψνn (µ) (z), (1 − z 2 )− 2 F [fµ ](z) = C(µ0 ) (z,µ)→(1,µ0 ) Ω √ , α(µ) 1−z 2 Pn with κ(µ) := 1 + ξ(µ) 2 + n(n+3) 2 + 12 i=1 νi (µ) and m ∈ {0, 1}. The result follows then by Lemma 3.8 and taking the identity (9) into account. 16 4 Applications In this Section we present the results obtained applying Theorems D and E to the two-parametric families of centers introduced in Section 1. 4.1 The family ẍ + (x + 1)p − (x + 1)q = 0 The family of vector fields (2) is an analytic potential system with potential function Z x+1 Vµ (x) = (up − uq )du. (10) 1 According with the dichotomy that produces h0 in hypothesis (H), in order to prove Theorem A we split the parameter space in two parts: Λ1 := {(q, p) ∈ Λ : q > −1} and Λ2 := {(q, p) ∈ Λ : q < −1}. As mentioned in the Introduction, the line {q + 1 = 0} correspond to parameters such that in any neighbourhood of them h0 = +∞ and h0 < +∞ coexist. This scenario is out of reach with the current techniques. If µ ∈ Λ1 system (2) has a non-degenerate center at the origin and the projection of the period annulus on  p−q 1 the x-axis is Iµ = (−1, ρ(µ) − 1) with ρ(µ) := p+1 q+1 . Moreover, the energy at the outer boundary is p−q h0 (µ) = (p+1)(q+1) < +∞ for all µ ∈ Λ1 . On the other hand, if µ ∈ Λ2 the origin is also a non-degenerate center but in this case Iµ = (−1, +∞) and h0 (µ) = +∞. (For more details in this direction we refer to [7].) In particular, in both cases hypothesis (H) in Definition 3.1 is fulfilled. The proof of Theorem A follows directly from Propositions 4.2 and 4.6. 4.1.1 The criticality is at most one in Λ2 The purpose of this Section is to apply Theorem D with n = 1 on family (2) to parameters µ0 = (q0 , 1) with q0 ∈ (−3, −1), q0 6= −2. According with the Theorem, we first need to compute the asymptotic expression of the function Lν(µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] at x = +∞. To this end, we follow the same strategy used on the proof of [6, Theorem A]. Since the function x(gµ−1 )00 (x) is analytic on R, we can write W [xν(µ) , x(gµ−1 )00 (x)] W [xν(µ) , −x(gµ−1 )00 (−x)] Lν(µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] = + (11) xν(µ)−1 xν(µ)−1 (V 0 )2 −2V V 00 for all x ∈ (0, +∞). On account of the equalities (g −1 )00 (x) = 2R(g −1 (x)) with R := (V 0 )3 and V (g −1 (x)) = x2 , equality (11) reads  Lν(µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] = 4 Sµ (gµ−1 (x)) + Sµ (gµ−1 (−x)) , (12) where ν(µ)  ν(µ)−1  Sµ (x) := Vµ0 (x)−1 Vµ (x)2− 2 W Vµ 2 , Rµ (x). Lemma 4.1. Let Sµ be defined as above with Vµ defined in (10). Taking µ0 = (q0 , 1), the following hold: 3+4p 1+3p q−4p−2 1−3p+2q p(1+3p)(p−q) (3p−2q−1)(1−p+q)(p−q) (a) Sµ (gµ−1 (x)) ∼+∞ q+1 (p + 1)− p+1 x− p+1 + q+1 (p + 1) p+1 x p+1 at µ0 . 2+3q 1−q (q−1)(p−q) (b) Sµ (gµ−1 (−x)) ∼+∞ p+1 (−q − 1)− q+1 x q+1 at µ0 . Proof. By means of some algebraic manipulations the function Sµ writes p Vµ (x) Sµ (x) = ψµ (x), 2Vµ0 (x)5 where ψ := −(V 02 − 2V V 00 )((ν − 1)V 02 + 6V V 00 ) − 4V 2 V 0 V 000 . Using the expression in (10) we can assert that ψµ is the sum of 12 monomials of the form c(µ)(x + 1)n1 p+n2 q+n3 with ni ∈ Z for i = 1, 2, 3 and c 17 a well defined rational function at µ = µ0 . The biggest exponent for µ ≈ µ0 is (x + 1)4p with coefficient (p−1)(p−1+ν(p+1)) (p+1)2 . Let us notice that, since the coefficient vanishes at p = p0 = 1, ψµ is not continuously quantifiable at infinity in µ = µ0 unless we fix ν = p−1 p+1 . With this choice of ν, the following two largest exponents for µ ≈ µ0 are (x + 1)3p−1 and (x + 1)3p+q . More precisely, we have p(1 + 3p)(p − q) 1 (3p − 2q − 1)(1 + q − p)(p − q) 1 Sµ (x) ∼+∞ 5 (x + 1)− 2 (1+3p) + 5 (x + 1)− 2 (3p−2q−1) . (1 + p) (1 + q) 2 (1 + p) (1 + q) 2 (x+1)p+1 The result in (a) follows using that gµ−1 (x) → +∞ as x → +∞ and gµ (x)2 = Vµ (x) ∼+∞ p+1 . The assertion (b) follows similarly now looking for the smallest exponent for µ ≈ µ0 . Proposition 4.2. Let {Xµ }µ∈Λ be the family of analytic potential systems (2) and consider the period  function of the center at the origin. If µ0 = (q0 , 1) with q0 ∈ (−3, −1)\{−2} then Crit (Πµ0 , Xµ0 ), Xµ = 1.  Proof. From [5] we already know that Crit (Πµ0 , Xµ0 ), Xµ > 1. Let us prove the opposite inequality. According with equality (12) and Lemma 4.1, the function Lν(µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] satisfies p(1 + 3p)(p − q) 1+3p 1+3p Lν(µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] ∼+∞ 5 (p + 1)− 2(p+1) x− p+1 + b(µ)xβ(µ) (p + 1) (q + 1) 2 at µ0 = (q0 , 1), where   ( 1−q 1 − 3p + 2q 1 − q q+1 if q ∈ (−3, −2), β(µ) := max , = 1−3p+2q µ≈µ0 p+1 q+1 p+1 if q ∈ (−2, −1), and ( (q−1)(p−q) 2+3q p+1 (−q − 1)− q+1 if q ∈ (−3, −2), b(µ) := (3p−2q−1)(1−p+q)(p−q) q−4p−2 q+1 (p + 1) p+1 if q ∈ (−2, −1). We point out that β(µ) is continuous but b(µ) changes sign at q = −2. On account of the previous computations, the quantifier at infinity of Lν(µ) [x(gµ−1 )00 (x) − x(gµ−1 )00 (−x)] at µ = µ0 is −2. In addition, the first momentum M [[Lµ ]0 ] on Theorem D vanishes identically. Indeed, Z ∞ Z +∞   +∞ 1 V V 00 V (x) x M [[Lµ ]0 ] = x(gµ−1 )00 (x)dx = − 0 2 (x) dx = 0 − =0 −∞ −1 2 (V ) V (x) 2 −1 for all µ ∈ Λ2 . Moreover, β(µ) ∈ (−3, −2) and b(µ) is continuous for µ ≈ µ0 . Therefore, condition (b) in Theorem D holds. This proves that the criticality at µ0 is at most one. 4.1.2 The criticality is at most one in Λ1 The energy at the outer boundary of the period annulus for parameters µ ∈ Λ1 is finite. Accordingly we shall use Theorem E with n = 1 to parameters µ0 = (q0 , p0 ) with p0 + 2q0 + 1 = 0, q0 ∈ (−1, − 21 ) in order to prove the second assertion of (b) in Theorem A. A similar argument as in the previous section shows that p p p p 4  p  p  Dν(µ) [z h0 (gµ−1 )00 (z h0 ) − z h0 (gµ−1 )00 (−z h0 )] = Sµ gµ−1 (z h0 ) − Sµ gµ−1 (−z h0 ) , (13) h0 where h i Vµ  2 ν 1 W h0 −Vµ , (h0 − Vµ )Vµ2 Rµ (x) Sµ (x) := ν ν . (h0 − Vµ (x))− 2 Vµ (x) 2 −1 Vµ0 (x) Here, and in what follows, we omit the dependence in µ of h0 for the sake of simplicity. Next result is general for any potential function Vµ with finite energy at the outer boundary. 18 Lemma 4.3. Let Sµ be defined as above and fix µ0 ∈ Λ. Assume that, for all µ in a neighbourhood of µ0 , h0 (µ) is finite and the right endpoint of the projection of the period annulus xr = xr (µ) is also finite and satisfies Vµ0 (xr ) 6= 0. Then 3 p  h02 (2 + ν)(2h0 Vµ00 (xr ) − Vµ0 (xr )2 ) Sµ gµ−1 (z h0 ) ∼1 + c(µ)(1 − z 2 ) at µ0 2Vµ0 (xr )3 with 3 h02 (12h20 (4 + ν)Vµ00 (xr )2 − 4h20 (4 + ν)Vµ000 (xr )Vµ0 (xr ) − 8h0 (5 + ν)Vµ00 (xr )Vµ0 (xr )2 + (8 + ν)Vµ0 (xr )4 ) c(µ) := . 4Vµ0 (xr )5 The analogous result is true for x` . Proof. The function Sµ can be written, after some algebraic manipulations, as p Vµ (x) Sµ (x) = ψµ (x), (14) 2Vµ0 (x)5   where ψ := − (V 0 )2 − 2V V 00 (V 0 )2 (h0 (ν − 1) + 3V ) + 6(h0 − V )V V 00 − 4V 2 (h0 − V )V 0 V 000 . Since the function Vµ (x) is analytic at x = xr the result follows by considering √ the Taylor’s expansion at x = xr of the previous expression and using the change of variable z = gµ (x)/ h0 . Lemma 4.4. Let Sµ be defined as above with Vµ defined in (10). Taking ν(µ) ≡ −1 and µ0 = (q0 , p0 ) with p0 + 2q0 + 1 = 0 and q0 ∈ (−1, − 12 ) then 1+3q − 2(1+q) 4+5q p  h (p − q)2 q(1 + q)− 1+q (1 + 3q) 1+2q Sµ gµ−1 (−z h0 ) ∼1 − 0 2 (1 − z 2 )− 1+q at µ0 . (1 + p) Moreover, the asymptotic expression in Lemma 4.3 holds with c(µ0 ) 6= 0. Proof. The first assertion of the result can be proven similarly as Lemma 4.1 so we skip the details for the sake or brevity. We only prove that c(µ0 ) 6= 0. Indeed, using the expression in (10) and substituting p0 = −2q0 − 1 some tedious but elementary computations show that (see Lemma 4.3) q 3 − 1+3q 4q  16− 1+3q h02 (1 + 3q)2 −1 + 1 1+q 3+8q  1+3q 1+q c(µ0 ) = −7q(1 + 3q)2 + 4 1+3q q(1 + q)(1 + 3q) −1 + 1 1+q 4q(1 + q)4 Vµ0 0 (ρ)5  4+8q  2(1+q) +9 × 2 1+3q (1 + q)2 −1 + 1 1+q 1+3q . We stress that each term inside the parenthesis is positive for q0 ∈ (−1, − 12 ). Also the multiplicative term is non-vanishing so c(µ0 ) 6= 0. The following result is the version of [6, Lemma 3.12] for ` = n = 1. Lemma 4.5. Let f ∈ C ω [0, 1) and ν ∈ R. Let us assume that f is quantifiable at z = 1 by ξ. If ξ < 1 2 then   N Dν [f ] = −(1 + ν)N [f ]. Proposition 4.6. Let {Xµ }µ∈Λ be the family of analytic potential systems (2) and consider the period 1 2  at the origin. If µ0 = (q0 , p0 ) with p0 + 2q0 + 1 = 0 and q0 ∈ (−1, − 2 ) \ {− 3 } then function of the center Crit (Πµ0 , Xµ0 ), Xµ = 1. 19  Proof. From [5] we already know that Crit (Πµ0 , Xµ0 ), Xµ > 1. Let us prove the opposite inequality. Let √ √ √ √ us denote fµ (z) := z h0 (gµ−1 )00 (z h0 ) − z h0 (gµ−1 )00 (−z h0 ) for the sake of simplicity. In [6, Lemma 4.4] it was shown that fµ satisfies  the hypothesis  on Lemma 4.5 for all µ ≈ µ0 . Applying Lemma 4.5 with ν(µ) ≡ −1 we have then N Dν(µ) [fµ ] = 0 for all µ ≈ µ0 . Moreover, on account of the equality (13) and Lemma 4.4, 3 h02 (2h0 Vµ00 (ρ) − Vµ0 (ρ)2 ) Dν(µ) [fµ ](z) ∼1 + b(µ)(1 − z 2 )β(µ) at µ0 , 2Vµ0 (ρ)3 where   ( 1 + 2q 1 if q ∈ (−1, − 32 ), β(µ) := min 1, − = µ≈µ0 1+q − 1+2q 1+q if q ∈ (− 23 , − 12 ), and   c(µ) if q ∈ (−1, − 32 ), 1+3q − 2(1+q) 4+5q b(µ) := h (p − q)2 q(1 + q)− 1+q (1 + 3q)  − 0 if q ∈ (− 23 , − 12 ), (1 + p)2 due to c(µ0 ) 6= 0 by Lemma 4.4. Let us apply now Theorem E with n = 1. According with the previous 2 1  3 , − 2 ) then assertion discussion, if p0 + 2q0 + 1 = 0 and q0 ∈ (− in (b) of Theorem E is satisfied  with N = 1, n1 = 0 and ` = 1 due to N Dν(µ) [fµ (z)] ≡ 0 and β(µ) − 1 6 12 . Then Crit (Πµ0 , Xµ0 ), Xµ 6 1 in this case. On the other hand, if q0 ∈ (−1, − 23 ) we apply Theorem E with N = 2, n1 = 0 and n2 = 1. By Lemma 5.4 we also have h 1 i N Dν [fµ0 ] 6= 0.  Then assertion (a) in Theorem E states that Crit (Πµ0 , Xµ0 ), Xµ 6 1 also in this case. 4.2 The family of dehomogenized Loud’s centers Through all this section we consider parameters µ = (D, F ) inside the open set Λ := {(D, F ) ∈ R2 : 1 < F < 23 , D < − 12 , D + F > 0}. The Loud system (3) has a first integral given by   −2F 1 2 Hµ (x, y) = (1 − x) y − qµ (x) 2 / {0, 1, 21 } (see for instance [11]) where qµ (x) = a(µ)x2 + b(µ)x + c(µ) with for all F ∈ D D−F +1 F −D−1 a= , b= and c = , 2(1 − F ) (1 − F )(1 − 2F ) 2F (1 − F )(1 − 2F ) and integrating factor κ(x) = (1 − x)−2F −1 . The line at infinity L∞ , the line {x = 1} and the conic Cµ = { 21 y 2 − qµ (x) = 0} are invariant curves of the differential system and, for parameters µ ∈ Λ, Cµ is a hyperbola intersecting the x-axis at √ √ −b − b2 − 4ac −b + b2 − 4ac x = p1 (µ) := and x = p2 (µ) := , (15) 2a 2a with 0 < p1 (µ) < p2 (µ). The outer boundary of the period annulus of the center at the origin of system (3) consists of the branch of the hyperbola Cµ passing through (p1 , 0) and the line at infinity Lµ , joined by two hyperbolic saddles (see Figure 3.) Although it is not relevant at this moment (but it will be nearly soon) we point out that there is a bifurcation on the phase portrait at D = −1 due to the fact that the branch of hyperbola passing through (p2 , 0) crosses the invariant line {x = 1}. In particular, p2 (µ) > 1 if and only if D > −1. 20 Πµ Πµ Figure 3: Phase portrait of (3) in the Poincaré disc for µ = (D, F ) ∈ Λ with D < −1 (left) and D > −1 (right). The center at (0, 0) is placed on the left of the centered invariant line {x = 1} for convenience. The invariant hyperbola Cµ is in boldface type. (Figure extracted from [11].) In [11] the authors give an implicit expression of the bifurcation curve D = G(F ) in Λ. More concretely, for each µ ∈ Λ and s > 0 small let P (s; µ) be the period of the periodic orbit of system (3) passing through the point (p1 (µ) − s, 0). Then, for parameters in Λ, the derivative of the period function P 0 (s; µ) tends to ∆(µ) as s tends to zero uniformly on compact subsets of Λ, where ( Z 1  2F −1 ! ) −1 − 23 2−2F 1 − p2 ∆(µ) := √ 2− (1 − u) u (u − 1) + 1 − 1 du . 2a(p2 − p1 )(1 − p1 ) 0 1 − p1 The bifurcation curve D = G(F ) is given by the zero set of ∆(µ) and the following properties are deduced: (a) −F < G(F ) < − 12 for all F ∈ (1, 32 ). (b) lim 3 G(F ) = − 32 , and F→2 (c) limF →1 G(F ) = − 12 . For the proof of these results we refer to [11, Theorem 3.6 and Proposition 3.11]. In particular the curve D = G(F ) is analytic on Λ and joins the parameters µ = (− 32 , 32 ) and µ = (− 12 , 1). The following result gives a more compact expression of the zero set of ∆(µ) and an additional property. The proof is given in the Appendix.   1  1 3 5 p2 −1 Proposition 4.7. For parameters µ ∈ Λ the zero sets of ∆(µ) and 7 2 F 1 − , ; 2 2 2 − 2F ; p2 −p1 Γ 2 −4F coincide. Moreover, lim 5 G(F ) = −1. F→4 The Loud system (3) is not a potential system and, at first glance, is not suitable to use the techniques developed in this paper to study the criticality at the outer boundary of its center. However we recall that it has a first integral quadratic in y and its integrating factor depends only on x. These two properties are the requirements of [3, Lemma 14], which shows that the change of variables z −F − 1 (u, v) = (φ(1 − x), (1 − x)−F y), with φ(z) := , F transforms system (3) to the potential system ( u̇ = −v, 1  1  (16) v̇ = (F u + 1) (F u + 1)− F − 1 D(F u + 1)− F − D − 1 . 21 The potential system above has a non-degenerated center at the origin and all the properties that we derive of its period function are directly transmitted to the period function of the Loud’s center. In these new variables, the projection of the period annulus is Iµ := (− F1 , ur (µ)), where (1 − p1 (µ))−F − 1 ur (µ) := φ(1 − p1 (µ)) = . (17) F Let Hµ (u, v) = 21 v 2 + Vµ (u) by the Hamiltonian function associated to system (16) with Vµ (0) = 0. If we set z = φ−1 (u) = (F u + 1)−1/F the following expressions will be useful to simplify the forthcoming computations: D Vµ (u) = h0 (µ) − z −2F V0 (z; µ) with V0 (z; µ) = 2−2F z 2 + 1+2D D+1 2F −1 z − 2F , Vµ0 (u) = z −F V1 (z; µ) with V1 (z; µ) = (z − 1) D(z − 1) − 1), (18) Vµ00 (u) = V2 (z; µ) with V2 (z; µ) = D(F − 2)z 2 − (2D + 1)(F − 1)z + F (D + 1), Vµ000 (u) = z F V3 (z; µ) with V3 (z; µ) = −2D(F − 2)2 + (2D + 1)(F − 1)z. −D−1 Here h0 (µ) = 2F (FF−1)(2F −1) denotes the energy of the outer boundary of the center for all µ ∈ Λ. In particular, h0 (µ) is finite in Λ. 4.2.1 Quantification of the functions involved With the aim of applying Theorem E with n = 1 to parameters µ = (D, F ) ∈ Λ satisfying D = G(F ), we need to compute the first terms of the asymptotic development of Dµ (z) at z = 1. We recall that the previous is reduced to study the asymptotic development of h i Vµ  2 ν 1 W h0 −Vµ , (h0 − Vµ )Vµ2 Rµ (u) Sµ (u) := ν ν (h0 − Vµ (u))− 2 Vµ (u) 2 −1 Vµ0 (u) at u = − F1 and u = ur (µ) due to equality (13). Here, and in what follows, we omit the dependence in µ of h0 and ν for the sake of brevity. Moreover we use u instead of x to be consistent with the notation in (16). Lemma 4.8. Let Sµ and Vµ be defined as above. Taking ν ≡ −1 we have that p  3 1 2−F Sµ gµ−1 (−z h0 ) ∼1 a1 (µ)(1 − z 2 )− 2 + 2(F −1) + a2 (µ)(1 − z 2 ) 2F −2 at µ0 where F −1+ 2(F1−1) F 1 D 2−2F (1 + D − F )2 (2 − F )(2F − 3)h0 D 2−2F (1 + D − F )(3 − F )h02(F −1) a1 := 3 1 and a2 := 2−F . 4F 2 (2F − 1)2 (F − 1)3 (2 − 2F ) 2 − 2(F −1) 4F (2F − 1)(F − 1)3 (2 − 2F ) 2(F −1) √  Proof. In order to get the desired asymptotic expression of Sµ gµ−1 (−z h0 ) at z = 1 we study the asymptotic expression√of Sµ (u) at u = u` = −1/F . The result will follow then on account of the change of variable z = −gµ (u)/ h0 . To do so we invoke the expression in (14) and perform the change of variable z = φ−1 (u) = (F u + 1)−1/F . Taking advantage of the relations in (18) we can write Sµ (φ(z)) in terms of the polynomials Vi (z; µ) and some powers of z depending on the parameter F . Due to the fact that φ(z) → −1/F as z → +∞, we are concerned about the asymptotic expression of Sµ (φ(z)) at z = +∞. An analogous study than the one done in Lemma 4.1 shows in this case that √ √ (2 − F )(D + 1 − F )2 (2F − 3) h0 3F −4 (3 − F )(D + 1 − F ) h0 −2+F Sµ (φ(z)) ∼+∞ z + z at µ0 . 4F 2 D2 (F − 1)3 (2F − 1)2 4F D(F − 1)3 (2F − 1) The result follows undoing the change of variable. 22 4.2.2 Proof of Theorem B  From the results in [11] we already know that Crit (Πµ0 , Xµ0 ), Xµ > 1 for parameters µ0 satisfying D0 = G(F0 ). The strategy for showing the opposite inequality is to apply Theorem E with n = 1 to the family (16). To do so, let fµ be defined as in the statement of the Theorem. The first thing to notice is that fµ satisfies hypothesis (H) with h0 (µ) < +∞ for all µ ∈ Λ. Let us fix ν(µ) ≡ −1. On account of equality (13) and Lemmas 4.3 and 4.8 we have that 3 h02 (2h0 Vµ00 (ρ) − Vµ0 (ρ)2 ) Dν(µ) [fµ ](z) ∼1 + b(µ)(1 − z 2 )β(µ) at µ0 , 2Vµ0 (ρ)3 where   ( 3 1 1 if F ∈ (1, 56 ), β(µ) := min 1, − + = µ≈µ0 2 2(F − 1) − 23 + 1 2(F −1) if F ∈ ( 65 , 43 ), and ( c(µ) if F ∈ (1, 56 ), b(µ) := a1 (µ) if F ∈ ( 65 , 43 ). If F0 ∈ [ 45 , 43 ) then β(µ0 ) 6 1/2 and assertion (b) in Theorem E is fulfilled with N = 1, n1 = 0 and ` = 1. Therefore, assertion (a) in Theorem B holds. If F0 ∈ (1, 45 ) then β(µ0 ) > 1/2. Let us split this interval in two parts, namely (1, 65 ) and ( 65 , 54 ). The parameter F0 = 56 is excluded due to the noncontinuity of the function β(µ). If F0 ∈ ( 65 , 54 ) then the coefficient of (1 − z 2 )β(µ) in the previous asymptotic expansion is the nonvanishing function a1 (µ) on Lemma 4.8. Moreover, by Lemma 4.5, N [Dν(µ) [fµ ]] = 0 for all µ ≈ µ0 . Consequently, we apply Theorem E with N = 2, n1 = 0 and n2 = 1. Using Lemma 5.5 we have that h  1 i 3 5 7 2 F1 − 2 , 2 ; 2 − 4F0 ; α(µ0 ) N Dν [fµ0 ] = 0 if and only if  = 0. Γ 72 − 4F0 Proposition 4.9 ensures that the previous Hypergeometric function do not vanishes for F0 ∈ ( 98 , 54 ). Con- sequently, hypothesis of Theorem E are fulfilled and so the criticality at the outer boundary for those parameters is exactly one. Let us finally consider F0 ∈ (1, 56 ). In this case the coefficient of (1 − z 2 )β(µ) on the asymptotic expansion above is c(µ) given in Lemma 4.3. Here then we invoke the assumption in the statement of Theorem B regarding c(µ0 ) 6= 0. As before we can apply Theorem E with N = 2, n1 = 0 and n2 = 1 and the hypothesis are satisfied whenever condition (4) is fulfilled. That is, whenever the hypergeometric function above do not vanish. Again Proposition 4.9 ensures that this is true for parameters F0 ∈ ( 89 , 54 ). Consequently, the result in assertion (b) of Theorem (B) holds. For parameters F0 ∈ (1, 98 ] condition (4) is required and so the result follows if it is satisfied. This ends the proof of the first assertion in Theorem B. 4.2.3 Some additional comments This section is devoted to discuss some technicalities involved in the proof of Theorem B. More concretely, we show that condition (4) is fulfilled when F0 ∈ ( 98 , 54 ) and we also give some numerical intuition to the fact that the condition should be fulfilled for every F0 ∈ (1, 54 ). 9 Proposition 4.9. If µ = (D, F ) satisfy D = G(F ) with < F < 45 then 8 3 5 7  2 F1 − 2 , 2 ; 2 − 4F ; α(µ)  6= 0. Γ 72 − 4F Proof. From the fact that the parameters µ = (D, F ) satisfy D = G(F ) with 89 < F < 5 4 it follows that −1 D > −1 and so p2 (µ) > 1. Moreover p1 < 1. Consequently, α(µ) = pp22−p 1 ∈ (0, 1). 23 Consider the function  2 F1 − 23 , 52 ; 72 − 4F ; z ϕ := (z, F ) −→  . Γ 27 − 4F From [1] the derivative of ϕ(z, F ) with respect to z writes dϕ 15 1 7 9  (z, F ) = − 9  2 F1 − , ; − 4F ; z . dz 4Γ 2 − 4F 2 2 2 Using the series expression of the Hypergeometric function it turns out that all its coefficients are positive for F ∈ ( 89 , 45 ). Consequently, for each F ∈ ( 98 , 54 ) we have that dϕ (z, F ) > 0 if z > 0. dz Hence for each F ∈ ( 98 , 54 ) the function ϕ(·, F ) is increasing for z > 0. In addition, 1 ϕ(0, F ) = 7  >0 Γ 2 − 4F if F ∈ ( 89 , 45 ). Therefore, for each F ∈ ( 98 , 54 ), the function ϕ(z, F ) is positive for all z > 0. Taking z = α(µ) and on account of 0 < α(µ) < 1 for all parameters under consideration, the result follows. Figure 4a exhibits the curve D = G(F ) together with the two connected components of the zero set level of the function  3 5 7 2 F1 − 2 , 2 ; 2 − 4F ; α(µ)  . (19) Γ 72 − 4F As the previous result states, there is no crossing of the zero set with the curve D = G(F ) for parameters with F ∈ ( 89 , 54 ). Indeed Proposition 4.9 shows that condition (4) is satisfied in this situation. Numerics seems to show that the non-crossing property is still verified till F = 1. However, we have not been able to prove it analytically. We end this section with also a numerical intuition about the equation c(µ0 ) = 0 in the statement of Theorem B. Although again no analytical prove is provided, it seems that this equation has a unique solution with is approximately (D0 , F0 ) = (−0.56996, 1.00781). 5 Appendix: Computation of momenta The purpose of this section is to compute the momenta used to apply Theorem E on the two-parametric potential family and the Loud’s family (see Proposition 4.6 and Section 4.2.2.) In both cases the momentum we are interested in is h 1 i N Dν [fµ ] with ν = −1. Lemma 3.5 together with Definition 3.6 imply that h 1 i    N Dν [fµ ] = M (Lν ◦ B)[fµ ] 1 . Using the formula in Lemma 2.15 on the right-hand side of the equality above, we get the equality Z ! h 1 i R 3 2 N Dν [fµ ] = lim R B[fµ ](R) − x B[fµ ](x)dx . R→+∞ 0 24 (a) (b) Figure 4: (a) The curve joining the points (− 32 , 32 ) and (− 21 , 1) correspond to the bifurcation parameters D = G(F ). The other two bold lines are the two connected components of the zero set of the function (19). (b) Zoom of the previous figure in the parameter region (D, F ) ∈ (−1, − 12 ) × (1, 89 ) and the curve c(µ) = 0. r Finally, the change of variable R = φ−1 (r) = √1−r 2 yields to h  3 Z r 2  1 i r fµ (r) z fµ (z) N Dν [fµ ] = lim √ − dz 1 − r2 2 3 r→1 0 (1 − z ) 2  3 Z r Z r  r fµ (r) fµ (z) fµ (z) = lim √ − dz + dz 1 − r2 2 3 2 1 r→1 0 (1 − z ) 2 0 (1 − z ) 2  3 Z r  r fµ (r) fµ (z) = lim √ − dz + N [fµ ]. 1 − r2 2 3 r→1 0 (1 − z ) 2 p p p p p Let fµ (z) := z h0 (µ)(gµ−1 )00 (z h0 (µ)) − z h0 (µ)(gµ−1 )00 (−z h0 (µ)) with gµ := sgn(x) Vµ (x). In both families, for parameters µ = µ0 under consideration the momentum N [fµ0 ] vanishes. We point out that this fact is not a coincidence. Indeed, in both cases we are interested to bound the criticality of bifurcation parameters such that limh→h0 (µ0 ) Tµ0 0 (h) = 0. In other words, the bifurcation occurs in such a way that limh→h0 (µ) Tµ0 (h) is convergent for all µ ≈ µ0 and changes sign at µ = µ0 . From the definition of the period function, it turns out that Tµ0 (h) N [fµ ] = lim √ (20) h→h0 (µ) 2h0 (µ) and so we have N [fµ0 ] = 0 (we refer to [5, Corollary 3.12] for more details in this direction.) Hence, for parameters in such situation, h  3 Z r  1 i r fµ (r) fµ (z) N Dν [fµ ] = lim √ − dz . (21) 1 − r2 2 3 r→1 0 (1 − z ) 2 Next two lemmas are useful for the forthcoming computations. See for instance (6.2.1) and (6.2.2) of [1] for the first one. The first assertion of the second lemma was proved in [5, Lemma A.3]. The second assertion follows similarly. Lemma 5.1. Let α and β be any complex numbers with strictly positive real part. Then Z 1 Z ∞ Γ(α)Γ(β) uα−1 (1 − u)β−1 du = uα−1 (1 + u)−(α+β) du = , 0 0 Γ(α + β) 25 where Γ denotes the Gamma function. Lemma 5.2. Let α and β be real numbers such that α + β + 1 6= 0. Then Z Z β 1 uα (1 + u)β du = uα (1 + u)β−1 du + uα+1 (1 + u)β , α+β+1 α+β+1 and Z Z α 1 uα (1 − u)β du = uα−1 (1 − u)β − uα (1 − u)β+1 . α+β+1 α+β+1 Lemma 5.3. Let Φµ (x) := h0 − Vµ (x). The following hold: (a) If Vµ with µ = (q, p) is defined by (10) and q ∈ (−1, − 53 ) then Z s 3 1 3 Z k(s) − 32 2(p + 1) 2 (q + 1)s− 2 (3q+1) 1 2(p + 1) 2 (λ − 1) 1 Φµ (x) dx = u(s)− 2 + λ y 2 −λ (1 + y)λ−2 dy, 0 (p − q)(p + 1) (p − q) p+1 0 q+1 and Z s 3 1 1 1 5 2(p + 1) 2 (q + 1)s− 2 (5q+3) 3 4(p + 1) 2 (q + 1)2 (ω − 1)s− 2 (5q+3) 1 Φµ (x)− 2 dx = u(s)− 2 + u(s)− 2 0 3(p − q) 3(p − q) 5 Z 4(p + 1) 2 (ω − 1)(ω − 2) k(s) 3 −ω + ω y 2 (1 + y)µ−3 dy, 3(p − q) p+1 q+1 0 p+1 1 1 3p+1 1 5p+3 where u(s) := q+1 − sp−q , k(s) := u(s) s p−q , λ := 2 p−q and ω := 2 p−q . (b) If Vµ with µ = (D, F ) is defined by (18) then Z 3 3 Z ! s 1 − 32 (2 − 2F ) 2 (1 − p1 )2F − 2 − 21 − 32 2−2F − 23 Φµ (x) dx = 3 3 2(v(s) − 1) + y ((1 − y) (1 − αy) − 1)dy − F1 D 2 (p2 − p1 ) 2 v(s) and Z s 5 5  5 (2 − 2F ) 2 (1 − p1 )4F − 2 2 3 1 Φµ (x)− 2 dx = 5 (v(s)− 2 − 1) + (8F − 8 + 5α)(v(s)− 2 − 1) 5 − F1 D (p2 − p1 ) 2 3 2 Z 1 !   − 52 4−4F − 52 5 + y (1 − y) (1 − αy) − 1 − (4F − 4 + 2 α)y dy v(s) 1 (F s+1)− F +p1 −1 p2 −1 where v(s) := 1 , α := p2 −p1 and p1 , p2 are defined in (15). (F s+1)− F Proof. Let us compute the integrals in (a). Using the expression of Vµ in (10) we have that Z s Z s  − 23 − 23 3 − 32 (q+1) p+1 Φ(x) dx = (p + 1) 2 x − xp−q dx. 0 0 q+1 xp−q We perform the change of variable y = k(x) = p+1 p−q . Then the integral above writes q+1 −x Z s 3 Z k(s) − 32 (p + 1) 2 1 Φ(x) dx =  y 2 −λ (1 + y)λ−1 dy, p+1 λ 0 (p − q) q+1 0  1 where λ := 21 3p+1 p+1 p−q p−q . This improper integral is divergent as s tends to ρ = q+1 , which will be the case under consideration. We shall make explicit the order of the singularity of this function using Lemma 5.1. 26 In order to apply it, we note that the power of y in the integrand should be greater than −1 and the power of (1 + y) negative. The first condition is satisfied for q ∈ (−1, − 35 ), p > q, but λ − 1 = 1+p+2q 2(p−q) is positive for some parameters. (In fact vanishes for the bifurcation parameters.) To overpass this situation, we apply Lemma 5.2, getting Z s 3 3 Z k(s) 3 2(p + 1) 2 3 2(p + 1) 2 (λ − 1) 1 Φ(x)− 2 dx = −λ λ k(s) 2 (1 + k(s)) λ−1 + λ y 2 −λ (1 + y)λ−2 dy. p+1 0 (p − q) q+1 (p − q) p+1 q+1 0 p+1 1 p+1 1 Using u(s) := q+1 − sp−q we can write k(s) = sp−q u(s) and 1 + k(s) = q+1 u(s) so, Z s 3 1 3 Z k(s) − 32 2(p + 1) 2 (q + 1)s− 2 (3q+1) 1 2(p + 1) 2 (λ − 1) 1 Φ(x) dx = u(s)− 2 + λ y 2 −λ (1 + y)λ−2 dy 0 (p − q)(p + 1) (p − q) p+1 0 q+1 as desired. Similarly we can obtain the expression for the second integral in (a). In this case, we need to apply Lemma 5.2 twice in order that the power of (1 + y) of the integrand is negative. We omit the details for the sake of brevity. Let us now prove (b). Using the expression of Vµ in (18) together with the change of variable z = 1 φ−1 (x) = (F x + 1)− F the first integral on item (b) is written as Z +∞ − 23 z 2F −1 h0 (µ) − Vµ (φ(z)) dz. φ−1 (s) By definition of p1 and p2 in (15) the difference on the integrand above decomposes as D h0 (µ) − Vµ (φ(z)) = (z − 1 + p1 )(z − 1 + p2 ). 2 − 2F −1 1 −1 Let us now consider the change of variable y = v(x) = z+pz1 −1 = φ φ(x)+p −1 (x) . In this new variable, we have that Z s 3 3 Z 3 (2 − 2F ) 2 (1 − p1 )2F − 2 1 − 3 3 Φµ (x)− 2 dx = 3 3 y 2 (1 − y)2−2F (1 − αy)− 2 dy −F1 D 2 (p 2 − p 1 ) 2 v(s) −1 with α := pp22−p 1 . As in (a), we point out here that v(s) tends to zero as s tends to xr (see (17)) which will be the case under study. In particular, the above improper integral is divergent as v(s) tends to zero. 3 To deal with this situation, we add and subtract the value at y = 0 of (1 − y)2−2F (1 − αy)− 2 inside the integral, which turns out to be exactly 1. That is, Z s 3 3 Z 1 ! − 32 (2 − 2F ) 2 (1 − p1 )2F − 2 − 21 − 23 2−2F − 23 Φµ (x) dx = 3 3 2(v(s) − 1) + y ((1 − y) (1 − αy) − 1)dy − F1 D 2 (p2 − p1 ) 2 v(s) as desired. Similarly we obtain the expression for the second integral in (b). In this case, we need to add 3 and subtract 1 + (4F − 4 + 52 α)y to the function (1 − y)2−2F (1 − αy)− 2 , which turns out to be its Taylor’s development up to degree one. p−q Lemma 5.4. Let Vµ be defined as in (10) and h0 (µ) = (p+1)(q+1) . If µ = (q, 1) with q ∈ (−1, − 53 ) and ν = −1 then   √  − 1+5q √ 5q+3 h 1 i 2 −2q −2q 1+3q π Γ − 2(p−q) N Dν [fµ ] =   . q(q + 1)(3q + 1) q + 1 Γ 1 − 5q+3 2 2(p−q) Proof. For the parameters under consideration the equality deduced in (21) holds. For the sake of simplicity we shall omit the dependence on the parameter µ from now on. We stress that although the dependence on 27 the parameters is not shown all the limits in the proof are uniform with respect to the parameters. Using the definition of f , that is p p p p f (z) = z h0 (g −1 )00 (z h0 ) − z h0 (g −1 )00 (−z h0 ) p √ with g(x) = sgn(x) V (x), and the change of variable z = g(x)/ h0 , the following equality holds Z Z √ r g −1 (r h0 ) f (z)dz η1 (x) 3 = 2h0 √ 3 dx, 0 (1 − z2) 2 g −1 (−r h0 ) (h0 − V (x)) 2 1 V V 00 √ 3 with η1 = 2 − (V 0 )2 . The function g −1 (−r h0 ) tends to zero as r → 1. Moreover, η1 (x)(h0 − V (x))− 2 has 5 a singularity at x = 0 of order x− 2 (q+1) , which is integrable since q < − 53 . Consequently, Z √ Z √ g −1 (r h0 ) g −1 (r h0 ) η1 (x) η1 (x) lim √ 3 dx = lim 3 dx. r→1 g −1 (−r h0 ) (h0 − V (x)) 2 r→1 0 (h0 − V (x)) 2 Let us define 1 3 1 Φ(x) := h0 − V (x), l(x) := −, h(x) := h0 Φ(x)− 2 − Φ(x)− 2 . (22) V 0 (x) 3 3 √ We have η1 (x)(h0 − V (x))− 2 = 12 Φ(x)− 2 − l0 (x)h(x). Denoting s = g −1 (r h0 ) and integrating by parts, the previous discussion leads to the equality Z r Z s Z s f (z)dz − 23 = h0 Φ(x) dx + 2h 0 l(x)h0 (x)dx − 2h0 l(s)h(s). 2 3 0 (1 − z ) 2 0 0 Here we used that limx→0 l(x)h(x) = 0, which is a direct computation. By definition, 3 5 1 3 l(x)h0 (x) = − h0 Φ(x)− 2 + Φ(x)− 2 . 2 2 Then, Z Z Z r s s f (z)dz 3 5 3 = 2h0 Φ(x)− 2 dx − 3h20 Φ(x)− 2 dx − 2h0 l(s)h(s). (23) 0 (1 − z2) 2 0 0 p+1 From now on let us denote u(s) := q+1 − sp−q . Using the expressions of l and h we have 1 1 1 1 (p + 1) 2 s− 2 (3q+1) 1 (p − q)(p + 1) 2 s− 2 (5q+3) 3 l(s)h(s) = p−q u(s)− 2 − u(s)− 2 . (24) s −1 (q + 1)(sp−q − 1) Substituting (24) and the integrals in assertion (a) of Lemma 5.3 into (23), and taking into account that 1 1 3 1 ! 1 1 2h0 (p − q)(p + 1) 2 s− 2 (5q+3) 2h20 (p + 1) 2 (q + 1)s− 2 (5q+3) − 23 2h0 (p + 1) 2 s− 2 (5q+3) 1 p−q − u(s) = p−q u(s)− 2 (q + 1)(s − 1) p−q s −1 we obtain Z r 3 Z f (z)dz 4h0 (p + 1) 2 (λ − 1) k(s) 1 −λ 3 =  λ y 2 (1 + y)λ−2 dy 0 (1 − z 2 ) 2 (p − q) p+1 q+1 0 5 Z 4h20 (p + 1) 2 (ω − 1)(ω − 2) k(s) 3 −ω − ω y 2 (1 + y)ω−3 dy (p − q) p+1 0 q+1   1 − 12 (5q+3) 2(q + 1)sq+1 1 − sq+1 2h0 (q + 1)2 (ω − 1) 1 + 2h0 (p + 1) s2 + p−q − u(s)− 2 . p−q s −1 p−q (25) 28 3 Let us now compute √r f (r). Using again the expression of f the previous function writes 1−r 2 √ √ √ √ r3 f (r) r4 h0 (g −1 )00 (r h0 ) r4 h0 (g −1 )00 (−r h0 ) √ = √ − √ . (26) 1 − r2 1 − r2 1 − r2 √ First, the change of variable s = g −1 (−r h0 ) on the second quotient of the previous equality yields to √ √  r4 h0 (g −1 )00 (−r h0 ) 2V (s)2 V 0 (s)2 − 2V (s)V 00 (s) √ = p . 1 − r2 h0 V 0 (s)3 h0 − V (s) √ The function s = g −1 (−r h0 ) → 0 as r → 1 and the function in the right hand side of the previous equality 1 has a singularity at s = 0 of order s− 2 (5q+3) . Since q ∈ (−1, − 53 ) we have that √ √  r4 h0 (g −1 )00 (−r h0 ) 2V (s)2 V 0 (s)2 − 2V (s)V 00 (s) lim √ = lim p = 0. r→1 1 − r2 s→0 h0 V 0 (s)3 h0 − V (s) √ Second, the change of variable s = g −1 (r h0 ) yields to the same expression as before, √ √  r4 h0 (g −1 )00 (r h0 ) 2V (s)2 V 0 (s)2 − 2V (s)V 00 (s) √ = p . 1 − r2 h0 V 0 (s)3 h0 − V (s) p+1 On account of the expression of V , expanding in series on u = q+1 − sp−q we have that  1 1 2V (s)2 V 0 (s)2 − 2V (s)V 00 (s) 2h0 (p + 1) 2 s− 2 (5q+3) q+1 p−q  1 1 p = p−q − 1)3 s (s − 1)2 − 2h0 (psp−q − q) u− 2 + o(u 2 ). 0 3 h0 V (s) h0 − V (s) (s   p−q 1 p+1 Let us recall ρ := q+1 . At this point we claim that 1 1 2h0 (p + 1) 2 s− 2 (5q+3) q+1 p−q  1 lim p−q 3 s (s − 1)2 − 2h0 (psp−q − q) u(s)− 2 s→ρ (s − 1)    1 − 1 (5q+3) 2(q + 1)sq+1 1 − sq+1 2h0 (q + 1)2 (ω − 1) − 1 −2h0 (p + 1) 2 s 2 + p−q − u(s) 2 = 0. p−q s −1 p−q Indeed, the previous expression writes   1 1 1 2(q + 1)sq+1 2h0 (q + 1)2 (ω − 1) 2h0 (psp−q − q) 1 2h0 (p + 1) 2 s− 2 (5q+3) u− 2 p−q u+ − p−q 3 − p−q . (p − q)(s − 1) p−q (s − 1) s −1 Using the expression of h0 and ω appearing in Lemma 5.3, 2h0 (q + 1)2 (ω − 1) 2h0 (psp−q − q) 1 u   − p−q 3 − p−q = c1 + c2 sp−q + c3 s2(p−q) , p−q (s − 1) s −1 (p − q)(p + 1) for some c1 , c2 , c3 ∈ R[p, q]. Substituting the previous equality on the limit above and taking into account 29 p+1 that u(s) = q+1 − sp−q → 0 as s → ρ the claim follows. Therefore, h  3 Z r  1 i r f (r) f (z)dz N Dν [fµ ] = lim √ − 1 − r2 2 3 r→1 0 (1 − z ) 2   3 Z 2V (s)2 V 0 (s)2 − 2V (s)V 00 (s) 4h0 (p + 1) 2 (λ − 1) k(s) 1 −λ = lim  p − λ y 2 (1 + y)λ−2 dy s→ρ h0 V 0 (s)3 h0 − V (s) (p − q) p+1 q+1 0 2 5 Z k(s) 4h (p + 1) (ω − 1)(ω − 2) 2 3 + 0 ω y 2 −ω (1 + y)ω−3 dy (p − q) p+1 q+1 0    1 1 2(q + 1)sq+1 1 − sq+1 2h0 (q + 1)2 (ω − 1) 1 − 2h0 (p + 1) 2 s− 2 (5q+3) + p−q − u− 2 p−q s −1 p−q 2 5 Z k(s) 4h0 (p + 1) 2 (ω − 1)(ω − 2) 3 = lim p+1 ω y 2 −ω (1 + y)ω−3 dy s→ρ (p − q) q+1 0  3 Z k(s) 4h0 (p + 1) (λ − 1) 2 1 −  y 2 −λ (1 + y)λ−2 dy  . p+1 λ (p − q) q+1 0 Let us now fix p = −2q − 1. On account of the expression of λ we have that λ − 1 = 0. Finally, using the expressions of ω and h0 , the fact that k(s) → +∞ as s → +∞ and Lemma 5.1 we obtain √   h √  − 1+5q 5q+3 1 i 2 −2q −2q 1+3q π Γ − 2(p−q) N Dν [fµ ] =   q(q + 1)(3q + 1) q + 1 Γ 1 − 5q+3 2 2(p−q) as we desired. Lemma 5.5. Let Vµ be defined as in (18). The following holds: (a) If µ = (D, F ) satisfies F ∈ (1, 32 ), D ∈ (− 23 , − 12 ) and D + F > 0 then 3 3 √ h0 (2 − 2F ) 2 (1 − p1 )2F − 2 πΓ(3 − 2F ) 3 1 5 N [fµ ] =  2 F1 2, −2; 2 − 2F ; α). Γ 52 − 2F 3 3 D (p2 − p1 ) 2 2 (b) If µ = (D, F ) satisfies D = G(F ) and F ∈ (1, 45 ) then h 5 √ 1 i 5 4h2 (2 − 2F ) 2 (1 − p1 )4F − 2 π Γ(5 − 4F )  N Dν [fµ ] =− 0 5 5 7  2 F1 − 32 , 52 ; 27 − 4F ; α . D 2 (p2 − p1 ) 2 Γ 2 − 4F p2 −1 F −D−1 Here α := p2 −p1 , h0 = 2F (F −1)(2F −1) and p1 , p2 are defined in (15). Proof. The proof of the assertion in (a) follows the same lines as the proof of (b) using that Z r fµ (z)dz N [fµ ] = lim 1 . r→1 0 (1 − z 2 ) 2 Since the proof of (b) is richer in subtle technicalities, and for the sake of brevity, we decided to prove (b) and omit the proof of assertion in (a). Let us show (b). For the parameters µ = (D, F ) with D = G(F ) the equality in (21) holds. Again we omit the dependence on parameters for the sake of simplicity although all the limits are uniform with respect to the parameters. Following the same discussion as in the proof of Lemma 5.4 we arrive to the equality Z r Z s Z s f (z)dz − 23 5 = 2h0 Φ(x) dx − 3h2 0 Φ(x)− 2 dx − 2h0 l(s)h(s), (27) (1 − z 2 ) 32 1 1 0 −F −F 30 where the functions √ Φ, h and l are defined in (22). The only difference lies on the interval of integration due to g −1 (−r h0 ) → − F1 as r → 1 in this case. 1 Let us denote u = u(s) := φ−1 (s) − 1 + p1 = (F s + 1)− F − 1 + p1 . From the expressions of h and l in (22) and the definition of V and V1 in (18) we have 1 3 (u + 1 − p1 )2F (2 − 2F ) 2 −2 1 h0 (u + 1 − p1 )4F (2 − 2F ) 2 3 l(s)h(s) = 1 1 u − 3 3 u− 2 . (28) D V1 (u + 1 − p1 )(u + p2 − p1 ) 2 2 D V1 (u + 1 − p1 )(u + p2 − p1 ) 2 2 Moreover, equality (26) also holds and the limit of the second quotient of the right-hand side of the equality tends to zero as r tends to one. Therefore √ √ r3 f (r) r4 h0 (g −1 )00 (r h0 ) lim √ = lim √ . r→1 1 − r2 r→1 1 − r2 √ The change of variable s = g −1 (r h0 ) yields to √ √ r4 h0 (g −1 )00 (r h0 ) 2V (s)2 (V 0 (s)2 − 2V (s)V 00 (s)) √ = p . 1 − r2 h0 V 0 (s)3 h0 − V (s) 1 On account of the expressions in (18), if z = φ−1 (s) = (F s + 1)− F we have √ √ r4 h0 (g −1 )00 (r h0 ) 2z 4F (h0 − z −2F V0 (z))2 (z −2F V1 (z)2 − 2(h0 − z −2F V0 (z))V2 (z)) √ = 1 . 1 − r2 h0 V1 (z)3 V0 (z) 2 D Finally, using the factorization V0 (z) = 2−2F (z − 1 + p1 )(z − 1 + p2 ) and u = u(s) = z − 1 + p1 defined above, after some algebraic manipulations we arrive to √ √ 1  r4 h0 (g −1 )00 (r h0 ) 2h0 (2 − 2F ) 2 (u + 1 − p1 )−2F V1 (u + 1 − p1 )2 − 2h0 V2 (u + 1 − p1 ) − 1 1 √ = 1 1 u 2 + o(u 2 ). 1−r 2 3 D 2 V1 (u + 1 − p1 ) (u + p2 − p1 ) 2 (u + 1 − p1 ) −4F At this point we substitute the previous equality and (27) into (21), and we use the equality (28) and the integrals in item (b) of Lemma 5.3 to have an explicit expression for the momentum under consideration. On −1 account that v(s) = φ φ(s)−1+p −1 (s) 1 u = u+1−p 1 (see Lemma 5.3), we can collect the expression of the momentum as follows h  3 Z r  1 i r f (r) f (z)dz N Dν [fµ ] = lim √ − 1 − r2 2 3 r→1 0 (1 − z ) 2 3 3 3 1 2h0 (2 − 2F ) 2 (1 − p1 )2F − 2 = lim a(u)u− 2 + b(u)u− 2 + 3 3 I1 (u) u→0 D 2 (p2 − p1 ) 2 5 5 ! 3h20 (2 − 2F ) 2 (1 − p1 )4F − 2 + 5 5 I2 (u) , D (p2 − p1 ) 2 2 where Z 1 3 3 I1 (u) := 2 − y − 2 ((1 − y)2−2F (1 − αy)− 2 − 1)dy, u u+1−p1 Z 1   22 5 5 I2 (u) := − + 8F + 5α − y − 2 (1 − y)4−4F (1 − αy)− 2 − 1 − (4F − 4 + 25 α)y dy, 3 u u+1−p1 −1 and both a and b are analytic functions at u = 0. Here α = pp22−p . First, from the first part of the proof of h 1 i 1 Lemma 4.6 we know that N Dν [fµ ] exists and so the previous limit is finite. (This can also be deduced from the fact that at the moment when certain momentum in Theorem E needs to be computed, such 31 momentum exists and it is finite.) The previous fact, together with the analyticity of a and b, implies that 3 1 a(u)u− 2 + b(u)u− 2 → 0 as u → 0. Second, we point out that I1 (0) is the expression of the curve D = G(F ) found in [11] which vanishes for bifurcation parameters we are considering. Consequently, taking the limit on the equality above, h 1 i 3h20 (2 − 2F ) 25 (1 − p1 )4F − 52 N Dν [fµ ] = 5 5 I2 (0). D 2 (p2 − p1 ) 2 The result will follow once we prove that √ 4 π Γ(5 − 4F )  I2 (0) = − 7  2 F1 − 23 , 52 ; 72 − 4F ; α . 3 Γ 2 − 4F To do so we shall show that the power series of the functions at both sides of the equality coincides. We notice that for parameters under consideration, |α| < 1 and so the Hypergeometric function is analytic and well defined as a power series (see [1, Section 15]). First notice that using the Newton’s binomial the integrand of I2 (0) writes ∞ 3 X  −1 + (1 − y)4−4F + (4 − 4F )y 5(1 − y)4−4F − 5 4−4F − 25 2 +k 5 + 3 α + (1 − y) y 3 αk y k . (29) y2 2y 2 k=2 2 By the assertion in Lemma 5.1 and on account that F < 5/4, for each k > 2 we have Z 1  4−4F k− 25 Γ(5 − 4F )Γ − 32 + k (1 − y) y dy =  . 0 Γ 72 − 4F + k Since the function inside the integral is positive for each k > 2, Tonelli’s theorem states that summation and integral signs interchange. Thus we obtain Z 1X∞ 3  ∞ 3   X Γ(5 − 4F )Γ − 23 + k k 2 + k αk (1 − y)4−4F y k− 2 dy = 2 +k 5 3 3  α 0 k=2 2 2 Γ 72 − 4F + k k=2 ∞   (30) X Γ(5 − 4F )Γ − 23 + k Γ 25 + k αk =   . Γ 27 − 4F + k Γ 52 k=2 k! Let us now integrate the second term in (29). The integration of the first term follows similarly and we omit the computation for the sake of shortness. Taking ε > 0 small we have Z 1 Z 1 5(1 − y)4−4F − 5 5 3 1 3 αdy = α y − 2 (1 − y)4−4F dy + 5α(1 − ε− 2 ). ε 2y 2 2 ε With the aim of applying Lemma 5.1 in view we need the powers of y and (1 − y) to be greater than −1. Indeed, since F ∈ (1, 45 ) we already have 4 − 4F > −1. On the contrary, the power of y do not satisfy the assumptions of the lemma. To overcome this we use Lemma 5.2 and we get Z 1 Z 3  1 −1 1 y − 2 (1 − y)4−4F dy = −2 92 − 4F y 2 (1 − y)4−4F dy + 2ε− 2 (1 − ε)5−4F . ε ε Substituting this equality on the previous expression, tending ε to zero and using Lemma 5.1 we get Z 1 Z √ 5(1 − y)4−4F − 5 9  1 −1 4−4F 5α πΓ(5 − 4F ) αdy = 5α − 5α 2 − 4F y (1 − y) 2 dy = 5α −  . Γ 92 − 4F 3 0 2y 2 0 As we noticed before, the same procedure shows that Z 1 √ −1 + (1 − y)4−4F + (4 − 4F )y 22 4 πΓ(5 − 4F ) dy = − + 8F +  . 3Γ 27 − 4F 5 0 y2 3 32 In this last case Lemma 5.2 must be applied twice. Summing these two last expressions together with (30) we have that √ √ ∞   4 πΓ(5 − 4F ) 5α πΓ(5 − 4F ) X Γ(5 − 4F )Γ − 23 + k Γ 52 + k αk I2 (0) = −  +  −   3Γ 72 − 4F Γ 92 − 4F k=2 Γ 72 − 4F + k Γ 25 k! √ ∞    ! 4 πΓ(5 − 4F ) 15α X Γ − 32 + k Γ 25 + k Γ 72 − 4F αk =−  1+ +    3Γ 72 − 4F 2(8F − 7) k=2 Γ − 32 Γ 52 Γ 72 − 4F + k k! √ 4 πΓ(5 − 4F )  =− 7  2 F1 − 23 , 52 ; 72 − 4F ; α , 3Γ 2 − 4F where on the first equality we use common properties of the Gamma function and on the second equality we use the definition of the Hypergeometric function 2 F1 as a power series on account that |α| < 1. Remark 5.6. The Hypergeometric function 2 F1 a, b; c; z) can be continued analytically for any complex number with |z| > 1 along any path in the complex plane that avoids the branch points 1 and infinity. For parameters µ = (D, F ) satisfying F ∈ (1, 23 ), D ∈ (− 32 , − 21 ) and D + F > 0 the property |α(µ)| < 1 do not hold. However, it holds that α(µ) < 1. Thus the function 1 3 1 5 5  2 F1 2, −2; 2 − 2F ; α(µ)) Γ 2 − 2F in the statement of Lemma 5.5 is well defined as a power series.  Proof of Proposition 4.7 As it is shown in (20) the bifurcation curve D = G(F ) must coincide with the zero set of N [fµ ]. For the parameters under consideration, the first assertion in Lemma 5.5 shows that this −1 zero set coincide with the zero set of the function 2 F1 23 , − 12 ; 52 − 2F ; α) where α := pp22−p 1 . This proves the first assertion of the result. To show the second assertion notice that 3 1 5 lim 2 F1 2 , − 2 ; 2 − 2F ; α(µ)) = 1 (D,F )→(−1,5/4) Γ(3−2F ) since α(µ) → 0 as D → −1. Therefore the result follows by the limit lim(D,F )→(−1,5/4) 5 = 0. Γ 2 −2F Acknowledgements The author thanks D. Marin and J. Villadelprat for fruitful discussions and valuable comments during the development of this work. The author is partially supported by the MINECO/FEDER grant MTM2017- 82348-C2-1-P and the MINECO/FEDER grant MTM2017-86795-C3-1-P. 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