Asymptotic Development of an Integral Operator and Boundedness of the Criticality of Potential Centers

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 x¨=xq-xp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{x}=x^q-x^p$$\end{document}, p,q∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q\in \mathbb {R}$$\end{document}, p>q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>q$$\end{document}; and the family of dehomogenized Loud’s centers.


Introduction
Consider a continuous family of planar potential systemṡ where μ ∈ is a parameter, is an open subset of R d , d 1, and V μ is an analytic function defined in an open interval I μ ⊂ R containing x = 0. In the case when V μ (0) = V μ (0) = 0 and V μ (0) > 0 Eq. (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 P μ . If we consider the embedding of P μ into RP 2 , its boundary, namely ∂P μ , 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 μ := ∂P μ \{(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) = 1 2 y 2 + V μ (x). Since V μ (0) = 0 by convention, we B David Rojas david.rojas@udg.edu 1 have that H (P μ ) = (0, h 0 (μ)), where h 0 (μ) ∈ R + ∪ {+∞} denotes the energy level of the outer boundary μ .
The object under study in this paper is the period function of the center. The minimal period T μ (h) of the periodic orbit γ h,μ inside the energy level {H (x, y; μ) = h} can be written as the Abelian integral This function is analytic on (0, h 0 (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 T μ (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 μ . That is, for a fixed μ 0 ∈ , we aim to control the number of critical periodic orbits of system (1) that may emerge or disappear from μ 0 as we move slightly the parameter μ ≈ μ 0 . This number is called the criticality of the outer boundary. Definition 1.1 Consider a continuous family {X μ } μ∈ of planar analytic vector fields with a center and fix some μ 0 ∈ . Suppose that the outer boundary of the period annulus varies continuously at μ 0 ∈ , meaning that for any ε > 0 there exists δ > 0 such that d H ( μ , μ 0 ) ε for all μ ∈ with μ − μ 0 δ. Then, setting N (δ, ε) := sup{#critical periodic orbits γ of X μ in P μ with d H (γ , μ 0 ) ε and μ − μ 0 δ}, the criticality of ( μ 0 , X μ 0 ) with respect to the deformation X μ is Crit ( μ 0 , X μ 0 ), X μ := inf δ,ε N (δ, ε).
In the previous definition d H stands for the Hausdorff distance between compact sets of RP 2 . 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 X μ tending to the outer boundary μ 0 in the Hausdorff sense as the parameter μ approaches μ 0 . The requirement of the continuity of P μ with respect to the parameters of the system ensures that the possible changes of P μ do not occur abruptly. We refer to [8] for details illustrating the necessity of this extra assumption.

Definition 1.2 A parameter μ 0 ∈
is called a local regular value of the period function at the outer boundary of the period annulus if Crit ( μ 0 , X μ 0 ), X μ = 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 φ i μ (h), i = 1, 2, . . . , n, verifying that there exist δ, ε > 0 such that (φ 1 μ , φ 2 μ , . . . , φ n μ , T μ ) form an Extended Complete Chebyshev system (ECTsystem for short, see Definition 2.10) on the interval (h 0 (μ) − ε, h 0 (μ)) and μ − μ 0 δ. This fact implies that T μ (h) has at most n zeros in (h 0 (μ)−ε, h 0 (μ)), counting multiplicities, uniformly on the parameters μ ≈ μ 0 . In particular, Crit ( μ 0 , X μ 0 ), X μ 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 [φ 1 μ , φ 2 μ , . . . , φ n μ , T μ ](h) does not vanish for all (h, μ) ≈ (h 0 (μ 0 ), μ 0 ). The tools of the previous works, and also the ones we present here, allow to  (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, B 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 tackle this problem in the following two situations: either h 0 (μ) = +∞ or h 0 (μ) < +∞ for all μ ≈ μ 0 . That is, the case in which there exist μ 1 and μ 2 in any neighbourhood of μ 0 with h 0 (μ 1 ) = +∞ and h 0 (μ 2 ) < +∞ is not considered.
Roughly speaking, the results in the previous papers relate the first term in the asymptotic development of the potential V μ at the endpoints of I μ with the first term of the asymptotic development of W [φ 1 μ , φ 2 μ , . . . , φ n μ , T μ ](h) at h = h 0 (μ). This allows to control the sign of the Wronskian under consideration for (h, μ) ≈ (h 0 (μ 0 ), μ 0 ); that is, uniformly on the parameters μ ≈ μ 0 . However, there are some situations where these first terms of V μ are not enough to compute the first term of the Wronskian at h = h 0 (μ) and so more terms in the asymptotic development must be employed. Theorems D and E in Sect. 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 Sect. 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, which has a non-degenerate center at the origin for all μ := (q, p) varying in := {(q, p) ∈ R 2 : 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 Fig. 1a, we proved that parameters μ 0 ∈ \ B ∪{q + 1 = 0} ∪ {(− 1 2 , p 1 )} ∪ {(− 1 3 , p 2 )} , with p 1 ≈ 1.20175 and p 2 ≈ 1.15685, are local regular values of the period function at the outer boundary. In addition, Crit ( μ 0 , X μ 0 ), X μ 1 if μ 0 ∈ B and it is exactly one for parameters μ 0 = (q 0 , p 0 ) satisfying either q 0 = 0 and p 0 ∈ (0, +∞)\{1}, p 0 = 1 and q 0 < −3, or p 0 + 2q 0 + 1 = 0 and q 0 ∈ (− 3 5 , − 1 3 )\{− 1 2 }. Using the tools in the present paper, the bifurcation diagram in Fig. 1a is improved by the following result. (See Fig. 1b).
Theorem A Let {X μ } μ∈ be the family of analytic potential systems (2) and consider the period function of the center at the origin. If μ 0 = (q 0 , p 0 ) with either p 0 = 1 and q 0 ∈ (−3, −1)\{−2}, or p 0 + 2q 0 + 1 = 0 and q 0 ∈ (−1, The proof of this result is presented in Sect. 4.1.1 (see Proposition 4.2) and Sect. 4.1.2 (see Proposition 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 Fig. 1b) We point out that the line corresponds to parameters such that the energy at the outer boundary h 0 (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 Fig. 1b agrees with the global bifurcation diagram conjectured in [7].
The second application is on the family of quadratic polynomial planar centers. The literature classify quadratic centers in four families: Hamiltonian, reversible Q R 3 , codimension four Q 4 , and generalized Lotka-Volterra Q LV 3 . Chicone [2] conjectured that reversible centers have at most two critical periodic orbits 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 In [3] the authors show that if B = 0 the period of the center at the origin is globally monotone. When B = 0 one can reduce the system, by means of a rescaling, to B = 1. That is, This family is known as dehomogenized Loud's centers and it has a center at the origin for all parameters μ := (D, F) ∈ R 2 . 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 Fig. 2) We refer to the series of papers [8,[10][11][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 now, this have been proved for parameters in that bifurcation curve with F ∈ ( 3 2 , 4 3 ). In order to state the result properly, let us consider the parameter space . Moreover, let 2 F 1 (a, b; c; z) be the Hypergeometric function (see [1,Section 15]), α := ( p 2 − 1)/( p 2 − p 1 ) with p 1 and p 2 defined in (15), and let c(μ) be the function in the statement of Lemma 4.3 using the expression of V μ in (18).
Theorem B Let {X μ } μ∈ be the family of analytic potential systems (3) and consider the period function of the center at the origin. Let μ 0 = (D 0 , F 0 ) ∈ satisfying D 0 = G(F 0 ). Then Crit ( μ 0 , X μ 0 ), X μ = 1 in the following situations:  9 8 ], c(μ 0 ) = 0 and The proof of the result is given in Sects. 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, 9 8 ] is given. The condition c(μ 0 ) = 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(μ 0 ) = 0. Regarding the equality c(μ 0 ) = 0 we also show numerically that the equation has a unique solution (D 0 , F 0 ) ≈ (− 0.56996, 1.00781) in Sect. 4.2.3. The parameter D 0 = G( 4 3 ) 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 parameters 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(μ 0 ) = 0 and allowing F 0 = 6 5 . The rest of the paper is organized as follows. In Sect. 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 Sect. 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 Sect. 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.

Asymptotic Behaviour of a Certain Integral Operator
Let c ∈ R + ∪ {+∞} and consider the integral operator 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 √ . 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 f μ in order that the first term of the asymptotic development at x = c of F [ f μ ] is obtained, uniformly on the parameters. These conditions are formulated using the following notions.
We point out that the map α : U → R in the previous definition is continuous at μ = μ 0 (see [5,Remark 2.6]).
From now on let us assume that f μ ∈ 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 is called the Roussarie-Ecalle compensator. For the sake of brevity, we also define and (x, α) : where is the Gamma function. Following the notation in Definition 2.1, we write  1). In this case: In the previous result the first term in the asymptotic development at infinity of the function In this special situation the hypothesis of f μ to be quantifiable by α(μ) = −2m is not enough to quantify F [ f μ ] at infinity. The following three functions exemplify this phenomena for n = 1 even in the non-parametric scenario: 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 − 5 2 and −3 respectively, and that lim x→+∞ These three examples subscribe the idea that more information on the asymptotic development of the function f μ is needed to quantify F [ f μ ] in this situation. To address this problem we generalize Definition 2.1 as follows.
We use the analogous definition at a.
for all m 1.
The functions F [ f ] and F [ f ] m are related by the following result.

Proposition 2.8 Let { f μ } μ∈ be a continuous family of continuous functions on
at μ 0 with 1 n 1 < n 2 < · · · < n N positive integers and β(μ 0 ) < −2n N . The following holds: Proof We shall prove the result by induction on m. To do so, we shall first assume m ∈ {1, . . . , n 1 − 1}. (In the case n 1 = 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 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) → (+∞,μ) for anyμ ≈ μ 0 . We apply the Uniform Hôpital's Rule in [5, Proposition A.1] to deduce that Therefore, This proves the result for m = 1. If m ∈ {2, . . . , n 1 − 1} the result follows identically Let us consider now that m ∈ {n 1 , . . . , n 2 − 1}, and let us start by taking m = n 1 . The same procedure as before can be applied taking into account that the fist term in the asymptotic development disappear. Indeed, we have where now the sum starts at i = 2 instead of the previous i = 1. Now using the first procedure the result follows for m = n 1 , . . . , Here we used that The procedure holds for all m = 1, . . . , n N in this case. For m = n N all the terms in the asymptotic development associated with even powers disappear and so the only remaining term is the one with betas. Then, 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 for some constant C n = 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 Sect. 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.
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).
These two notions are closely related by the following result (see for instance [4]).
is an ECT-system on I if and only if, for each k = 1, 2, . . . , n, 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 ≈ h 0 (μ) uniformly on the parameters close to a fixed μ 0 ∈ . Invoking equality (5), this problem is tackled by completing for some ε > 0 and all μ ≈ μ 0 . 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 ν 1 , ν 2 , . . . , ν n ∈ R, we consider the linear ordinary differential operator .
Proposition 2.13 (See [6]) For any f ∈ C ω (0, +∞) and ν 1 , . . . , ν n ∈ R, the following recurrence holds: where c 1 := 1 and c n : In the statement of the following result the assumptions with 1 n 1 < n 2 < · · · < n N positive integers and β(μ 0 ) < −2n N . The following holds: 1) and let us assume additionally that β(μ 0 ) + 2n N + 2m = 0. In this case, Proof Let us start by considering n = 0. In this case L μ = f μ and the assumptions of Proposition 2.8 are satisfied. Moreover, if (a) is satisfied then by (a) in Proposition 2.8 we have Therefore, using Lemma 2.7 and Theorem 2.

This proves (a). Let us show (b), (c) and (d). By hypothesis we have
The result follows using Theorem 2.4 with [ f μ ] n N and Lemma 2.7 again. This ends the proof for the case n = 0. Let us consider now n 1. By Proposition 2.13 Then the result follows by applying the case n = 0 to the family (6). The function f satisfies the assumptions of Theorem C with n = 0, n 1 = 1 and β = −5/2. Since β + 2 = −1 2 > −1 then assertion (b) of the Theorem states that

Remark 2.14 Let us recover at this point the examples in
The function g satisfies the hypothesis with n = 0, n 1 = 1 and β = −5. In this case β + 2 = −3 < −1 and That is, g satisfies the hypothesis in assertion (c1) of the Theorem with j = 1. Therefore, Finally, the function h satisfies the hypothesis with n = 0, n 1 = 1 and β = −3. In this example assumptions in assertion (b) are satisfied and We point out that Theorem C together with Theorem 2.4 cover all possible situations of the uniform asymptotic development of f μ except the case when all powers are negative even numbers at μ = μ 0 . The last result of this Section is a recursive formula for the computation of a certain momentum that will be useful in the applications.

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 ∈ ⊂ R d and we assume that the origin is a non-degenerate center for all μ. We denote by I μ = (x (μ), x r (μ)) the projection of the period annulus on the x-axis, x < 0 < x r and by h 0 (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: Lemma 3.2 (See [5]) Let {X μ } μ∈ be a family of potential analytic differential systems verifying (H) . Then the map (z, μ) This section is divided in two parts according with the dichotomy produced by h 0 . First, Sect. 3.1 deals with the case h 0 ≡ +∞. Second, Sect. 3.2 is dedicated to the case h 0 finite.

Potential Systems with Infinite Energy
In this section we present sufficient conditions to bound the criticality at the outer boundary for potential systems satisfying h 0 (μ) = +∞ for all μ ∈ . Following the strategy in [5,6,15], we find sufficient conditions such that f T μ can be embedded into the ECT-system (h ν 1 (μ) , h ν 2 (μ) , . . . , h ν n (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 Sect. 2 with the dynamical results we are looking for.
The assumption requiring the existence of functions ν 1 , ν 2 , . . . , ν n in the following statement is void in case that n = 0.
. Lemma 3.2 and the hypothesis (H) imply that { f μ } μ∈ 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 [ f μ ] has at most n isolated zeros for h > M and μ − μ 0 < ε, multiplicities taken into account.
Let us assume that L μ := L ν n (μ) [ f μ ] satisfies one of the hypothesis of the statement. Therefore L μ satisfies one of the hypothesis in Theorem C, so we can assert that either for some functions C, ξ, α with C(μ 0 ) = 0 and α(μ 0 ) = −1. Taking into account the definition of the operator L ν n (μ) , we have that either Therefore, on account of equality (5), by Lemma 3.3 we have that Crit ( μ 0 , X μ 0 ), X μ n as desired.

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 ∈ . As in the previous works [5,6,15], in order to embed the function f T μ into some ECT-system for an appropriate non-vanishing function f , the spirit of this section is to "translate" the case h 0 < +∞ to the case h 0 = +∞ so we can take advantage of Theorem C. This translation is given by the operator where φ(x) := x √ 1+x 2 . Given ν 1 , . . . , ν n ∈ R, B conjugates the operator L ν n with the linear ordinary differential operator the n-th momentum of f , whenever it is well defined. If n = 1 we simply say that N ν 1 , ν 2 , . . . , ν n ∈ R. Then the following hold:

Lemma 3.7 Let { f μ } μ∈ be a continuous family of analytic functions on
.
Consequently the result follows on account of Definition 2.5.
The following is an analogous version of Lemma 3.
In the same way as in the main result of the previous section, we stress that the assumption requiring the existence of functions ν 1 , ν 2 , . . . , ν n in the following statement is void in case that n = 0. Also are void the assumptions N Theorem E Let {X μ } μ∈ be a family of potential analytic differential systems verifying (H) with h 0 (μ) < +∞ for all μ ∈ and that there exist n 0 continuous functions ν 1 , ν 2 , . . . , ν n in a neighbourhood of some fixed μ 0 ∈ such that the function with 0 n 1 < n 2 < · · · < n N integers and β(μ 0 ) > n N . Then Crit ( μ 0 , X μ 0 ), X μ n if one of the following assertions hold:  (5), after the appropriate rescaling, Therefore the result will follow if there exist ε > 0 and a neighbourhood U of μ 0 such that F [ f μ ](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 D ν n (μ) , defined for functions in C ω [0, 1), with L ν 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[ f μ ].
First, by hypothesis we have that so, on account of Lemmas 3.5 and 3.7, we have Moreover, on account of Lemma 3.5 and Definition 3.6, Therefore, if D μ satisfies one of the hypothesis of the statement, the function L μ := (L ν n (μ) • B)[ f μ ] satisfies one of the hypothesis in Theorem C. It turns out then that either for some functions C, ξ, α with C(μ 0 ) = 0 and α(μ 0 ) = −1. Let us note that Lemma 3.5 in the first quality, the identity in the second equality, and (b) in Lemma 3.5 in the third equality. So we have Using the definition of B, the previous equation yields to 1 Thus, on account of the definition of D ν n (μ) in (8), and m ∈ {0, 1}. The result follows then by Lemma 3.8 and taking the identity (9) into account.

Applications
In this Section we present the results obtained applying Theorems D and E to the twoparametric families of centers introduced in Sect. 1. Familyẍ + (x + 1) p − (x + 1)

The
The family of vector fields (2) is an analytic potential system with potential function According with the dichotomy that produces h 0 in hypothesis (H), in order to prove Theorem A we split the parameter space in two parts: 1 := {(q, p) ∈ : q > −1} and 2 := {(q, p) ∈ : q < −1}. As mentioned in the Introduction, the line {q + 1 = 0} correspond to parameters such that in any neighbourhood of them h 0 = +∞ and h 0 < +∞ coexist. This scenario is out of reach with the current techniques. If μ ∈ 1 system (2) has a non-degenerate center at the origin and the projection of the period annulus on the x-axis is Moreover, the energy at the outer boundary On the other hand, if μ ∈ 2 the origin is also a non-degenerate center but in this case I μ = (−1, +∞) and h 0 (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.

The Criticality is at Most One in 3 2
The purpose of this Section is to apply Theorem D with n = 1 on family (2) to parameters μ 0 = (q 0 , 1) with q 0 ∈ (−3, −1), q 0 = −2. According with the Theorem, we first need to compute the asymptotic expression of the function To this end, we follow the same strategy used on the proof of [6, Theorem A]. Since the function x(g −1 μ ) (x) is analytic on R, we can write (11) for all x ∈ (0, +∞). On account of the equalities where Lemma 4.1 Let S μ be defined as above with V μ defined in (10). Taking μ 0 = (q 0 , 1), the following hold: Proof By means of some algebraic manipulations the function S μ writes Using the expression in (10) we can assert that ψ μ is the sum of 12 monomials of the form c(μ)(x + 1) n 1 p+n 2 q+n 3 with n i ∈ Z for i = 1, 2, 3 and c a well defined rational function at μ = μ 0 . The biggest . Let us notice that, since the coefficient vanishes at p = p 0 = 1, ψ μ is not continuously quantifiable at infinity in μ = μ 0 unless we fix ν = p−1 p+1 . With this choice of ν, the following two largest exponents for μ ≈ μ 0 are (x + 1) 3 p−1 and (x + 1) 3 p+q . More precisely, we have The result in (a) follows using that g −1 p+1 . The assertion (b) follows similarly now looking for the smallest exponent for μ ≈ μ 0 .
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 for all μ ∈ 2 . Moreover, β(μ) ∈ (−3, −2) and b(μ) is continuous for μ ≈ μ 0 . Therefore, condition (b) in Theorem D holds. This proves that the criticality at μ 0 is at most one.

The Criticality is at Most One in 3 1
The energy at the outer boundary of the period annulus for parameters μ ∈ 1 is finite. Accordingly we shall use Theorem E with n = 1 to parameters μ 0 = (q 0 , p 0 ) with p 0 + 2q 0 + 1 = 0, q 0 ∈ (−1, − 1 2 ) in order to prove the second assertion of (b) in Theorem A. A similar argument as in the previous section shows that where .
Here, and in what follows, we omit the dependence in μ of h 0 for the sake of simplicity. Next result is general for any potential function V μ with finite energy at the outer boundary.

Lemma 4.3
Let S μ be defined as above and fix μ 0 ∈ . Assume that, for all μ in a neighbourhood of μ 0 , h 0 (μ) is finite and the right endpoint of the projection of the period annulus x r = x r (μ) is also finite and satisfies V μ (x r ) = 0. Then The analogous result is true for x .

Proof
The function S μ can be written, after some algebraic manipulations, as where ψ : Since the function V μ (x) is analytic at x = x r the result follows by considering the Taylor's expansion at x = x r of the previous expression and using the change of variable
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(μ 0 ) = 0. Indeed, using the expression in (10) and substituting p 0 = −2q 0 − 1 some tedious but elementary computations show that (see Lemma 4.3) .
The following result is the version of [6, Lemma 3.12] for = n = 1.

The Family of Dehomogenized Loud's Centers
Through all this section we consider parameters μ = (D, F) inside the open set The Loud system (3) has a first integral given by for all F / ∈ {0, 1, 1 2 } (see for instance [11]) where q μ (x) = a(μ) and integrating factor κ(x) = (1 − x) −2F−1 . The line at infinity L ∞ , the line {x = 1} and the conic C μ = { 1 2 y 2 − q μ (x) = 0} are invariant curves of the differential system and, for parameters μ ∈ , C μ is a hyperbola intersecting the x-axis at with 0 < p 1 (μ) < p 2 (μ). The outer boundary of the period annulus of the center at the origin of system (3) consists of the branch of the hyperbola C μ passing through ( p 1 , 0) and the line at infinity L μ , joined by two hyperbolic saddles (see Fig. 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 ( p 2 , 0) crosses the invariant line {x = 1}. In particular, p 2 (μ) > 1 if and only if D > −1.
In [11] the authors give an implicit expression of the bifurcation curve D = G(F) in . More concretely, for each μ ∈ and s > 0 small let P(s; μ) be the period of the periodic orbit of system (3) passing through the point ( p 1 (μ) − s, 0). Then, for parameters in , the derivative of the period function P (s; μ) tends to (μ) as s tends to zero uniformly on compact subsets of , where The bifurcation curve D = G(F) is given by the zero set of (μ) and the following properties are deduced: 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 and joins the parameters μ = (− 3 2 , 3 2 ) and μ = (− 1 2 , 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".

Proposition 4.7
For parameters μ ∈ the zero sets of (μ) and 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 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 Let H μ (u, v) = 1 2 v 2 + V μ (u) by the Hamiltonian function associated to system (16) with V μ (0) = 0. If we set z = φ −1 (u) = (Fu + 1) −1/F the following expressions will be useful to simplify the forthcoming computations: Here h 0 (μ) = F−D−1 2F(F−1)(2F−1) denotes the energy of the outer boundary of the center for all μ ∈ . In particular, h 0 (μ) is finite in .

Quantification of the Functions Involved
With the aim of applying Theorem E with n = 1 to parameters μ = (D, F) ∈ satisfying D = G(F), we need to compute the first terms of the asymptotic development of D μ (z) at z = 1. We recall that the previous is reduced to study the asymptotic development of at u = − 1 F and u = u r (μ) due to equality (13). Here, and in what follows, we omit the dependence in μ of h 0 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 S μ and V μ be defined as above. Taking ν ≡ −1 we have that and . Proof In order to get the desired asymptotic expression of S μ g −1 μ (−z √ h 0 ) at z = 1 we study the asymptotic expression of S μ (u) at u = u = −1/F. The result will follow then on account of the change of variable z = −g μ (u)/ √ h 0 . To do so we invoke the expression in (14) and perform the change of variable z = φ −1 (u) = (Fu + 1) −1/F . Taking advantage of the relations in (18) we can write S μ (φ(z)) in terms of the polynomials V i (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 S μ (φ(z)) at z = +∞. An analogous study than the one done in Lemma 4.1 shows in this case that The result follows undoing the change of variable.

Proof of Theorem B
From the results in [11] we already know that Crit ( μ 0 , X μ 0 ), X μ 1 for parameters μ 0 satisfying D 0 = G(F 0 ). The strategy for showing the opposite inequality is to apply Theorem E with n = 1 to the family (16). To do so, let f μ be defined as in the statement of the Theorem. The first thing to notice is that f μ satisfies hypothesis (H) with h 0 (μ) < +∞ for all μ ∈ . Let us fix ν(μ) ≡ −1. On account of equality (13) and Lemmas 4.3 and 4.8 we have that if F ∈ 6 5 , 4 3 , If F 0 ∈ [ 5 4 , 4 3 ) then β(μ 0 ) 1/2 and assertion (b) in Theorem E is fulfilled with N = 1, n 1 = 0 and = 1. Therefore, assertion (a) in Theorem B holds.
Let us finally consider F 0 ∈ (1, 6 5 ). 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(μ 0 ) = 0. As before we can apply Theorem E with N = 2, n 1 = 0 and n 2 = 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 F 0 ∈ ( 9 8 , 5 4 ). Consequently, the result in assertion (b) of Theorem (B) holds. For parameters F 0 ∈ (1, 9 8 ] condition (4) is required and so the result follows if it is satisfied. This ends the proof of the first assertion in Theorem B.

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 F 0 ∈ ( 9 8 , 5 4 ) and we also give some numerical intuition to the fact that the condition should be fulfilled for every F 0 ∈ (1, 5 4 ). 2 ) × (1, 9 8 ) and the curve c(μ) = 0 As the previous result states, there is no crossing of the zero set with the curve D = G(F) for parameters with F ∈ ( 9 8 , 5 4 ). 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(μ 0 ) = 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 (D 0 , F 0 ) = (−0.56996, 1.00781).
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.

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 Sect. 4.2.2.) In both cases the momentum we are interested in is Lemma 3.5 together with Definition 3.6 imply that Using the formula in Lemma 2.15 on the right-hand side of the equality above, we get the equality Finally, the change of variable R = φ −1 (r ) = r √ 1−r 2 yields to In both families, for parameters μ = μ 0 under consideration the momentum N [ f μ 0 ] 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 lim h→h 0 (μ 0 ) T μ 0 (h) = 0. In other words, the bifurcation occurs in such a way that lim h→h 0 (μ) T μ (h) is convergent for all μ ≈ μ 0 and changes sign at μ = μ 0 . From the definition of the period function, it turns out that and so we have N [ f μ 0 ] = 0 (we refer to [5,Corollary 3.12] for more details in this direction.) Hence, for parameters in such situation, 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. 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 h 0 , the fact that k(s) → +∞ as s → +∞ and Lemma 5.1 we obtain as we desired.

Proof
The proof of the assertion in (a) follows the same lines as the proof of (b) using that 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 finite. (This can also be deduced from the fact that at the moment when certain momentum in Theorem E needs to be computed, such momentum exists and it is finite.) The previous fact, together with the analyticity of a and b, implies that a(u)u − 3 2 + b(u)u − 1 2 → 0 as u → 0. Second, we point out that I 1 (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, Substituting this equality on the previous expression, tending ε to zero and using Lemma 5. As we noticed before, the same procedure shows that