This is a preprint of: “On the upper bound of the criticality of potential systems at the outer boundary using the Roussarie-Ecalle compensator”, David Rojas, J. Differential Equations, vol. 267, 3922–3951, 2019. DOI: [10.1016/j.jde.2019.04.021] On the upper bound of the criticality of potential systems at the outer boundary using the Roussarie-Ecalle compensator David Rojas Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain. Abstract. This paper is concerned with the study of the criticality of families of planar centers. More precisely, we study sufficient conditions to bound the number of critical periodic orbits that bifurcate from the outer boundary of the period annulus of potential centers. In the recent years, the new approach of embedding the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary has shown to be fruitful in this issue. In this work, we tackle with a remaining case that was not taken into account in the previous studies in which the Roussarie- Ecalle compensator plays an essential role. The theoretical results we develop are applied to study the bifurcation diagram of the period function of two different families of centers: the power-like family ẍ = xp − xq , p, q ∈ R with p > q; and the family of dehomogenized Loud’s centers. 1 Introduction The present paper deals with the bifurcation of critical periodic orbits of families of planar potential centers. Let Λ be an open set of Rd , d > 1, µ ∈ Λ be a parameter and consider a continuous family of analytic functions Vµ = Vµ (x) defined in an open interval Iµ ⊂ R that contains x = 0. It is well known that the system ẋ = y, ẏ = −Vµ0 (x) (1) has a non-degenerate center at the origin for each µ ∈ Λ if Vµ (0) = Vµ0 (0) = 0 and Vµ00 (0) > 0. That is, the origin has a punctured neighborhood entirely foliated by closed orbits surrounding it. The largest neighborhood with this property is called the period annulus of the center and we will denote it by Pµ . After embedding Pµ into RP2 , the boundary of the period annulus has two connected components: the center itself (which is called the inner boundary) and the outer boundary defined by Πµ := ∂Pµ \ {(0, 0)}. 2 The periodic orbits are inside the energy levels of the Hamiltonian H(x, y; µ) = y2 + Vµ (x). We have then H(Pµ ) = (0, h0 (λ)) with h0 (µ) ∈ R+ ∪ {+∞}. The inner boundary is inside the energy level h = 0 for all µ ∈ Λ and we shall say that h0 (µ) is the energy level of the outer boundary Πµ . The minimal period Tµ (h) of the periodic orbit γh,µ inside the energy level {H(x, y; µ) = h} is given by the Abelian integral Z dx Tµ (h) = . γh,µ y This function is analytic on (0, h0 (µ)) for each λ ∈ Λ and it can be extended analytically to h = 0 since the center is non-degenerate. Its derivative Tµ0 (h) is also given by an Abelian integral and this paper is concerned with its zeros near the energy level h = h0 (µ), which correspond to critical periodic orbits near Πµ . More concretely, for a fixed µ̂ ∈ Λ, we study the number of critical periodic orbits of system (1) that 2010 Mathematics Subject Classification. 34C07; 34C23; 34C25. Key words and phrases: Center, period function, critical periodic orbit, bifurcation, criticality, Chebyshev system. E-mail address: rojas@ugr.es. 1 http://www.gsd.uab.cat can emerge or disappear from Πµ̂ as we move slightly the parameter µ ≈ µ̂. 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 µ̂ ∈ Λ. Suppose that the outer boundary of the period annulus varies continuously at µ̂ ∈ Λ, meaning that for any ε > 0 there exists δ > 0 such that dH (Πµ , Πµ̂ ) 6 ε for all µ ∈ Λ with kµ − µ̂k 6 δ. Then, setting N (δ, ε) := sup {# critical periodic orbits γ of Xµ in Pµ with dH (γ, Πµ̂ ) 6 ε and kµ − µ̂k 6 δ} ,  the criticality of (Πµ̂ , Xµ̂ ) with respect to the deformation Xµ is Crit (Πµ̂ , Xµ̂ ), Xµ := inf δ,ε N (δ, ε).  In the previous definition dH stands for the Hausdorff distance between compact sets of RP2 . The criticality of (Πµ̂ , Xµ̂ ) may be infinite but in the case it is finite it gives the maximal number of critical periodic orbits of system Xµ that tend to the outer boundary Πµ̂ in the Hausdorff sense as the parameter µ approach µ̂. We stress the requirement of the assumption that the period annulus varies continuously, which ensures that the changes on the geometry of Pµ do not occur abruptly as we vary the parameters of the system (see [14] for details). On account of the previous definition, we say that a parameter µ̂ ∈ Λ is a local regular value of the period function at the outer boundary of the period annulus if Crit (Πµ̂ , Xµ̂ ), Xµ = 0. Otherwise we say that it is a local bifurcation value at the outer boundary. The study of critical periodic orbits is analogous to the study of limit cycles, the objects of main concern of the Hilbert’s 16th problem (see for instance [2, 7, 25, 31] and references there in). Questions related to the behavior of the period function have been extensively studied by a large number of authors. Let us quote for instance the problems of isochronicity (see [6, 10, 21]), monotonicity (see [3, 4, 27]) or bifurcation of critical periodic orbits (see [5, 26, 28]). In the collection of works [11, 12] tools that enable to bound the criticality at the outer boundary for the family of potential systems (1) were developed. These tools, and the ones we present here, allow to tackle the bifurcation problem in the following two situations: either h0 (µ) = +∞ or h0 (µ) < +∞ for all µ ≈ µ̂. (The case in which in any neighborhood of µ̂ there are µ1 and µ2 with h0 (µ1 ) = +∞ and h0 (µ2 ) < +∞ is not  considered.) For each one of these two situations, sufficient conditions in order that Crit (Πµ̂ , Xµ̂ ), Xµ 6 n for n ∈ N ∪ {0} were given in [11, Theorem A and Theorem B]. The idea in both cases was 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 2.1) on the interval (h0 (µ) − ε, h0 (µ)) if kµ − µ̂k < δ. In particular this implies that Tµ0 (h) has at most n zeros in (h0 (µ) − ε, h0 (µ)), counted with multiplicities, uniformly for all parameters µ ≈ µ̂ and so Crit (Πµ̂ , Xµ̂ ), Xµ 6 n. The problem consisted then to guarantee that the Wronskian (see Definition 2.2) of (φ1µ , φ2µ , . . . , φnµ , Tµ0 ) is different from zero for all h ≈ h0 (µ) and µ ≈ µ̂. In the present paper we extend the results [11, Theorems A and B] by considering a remaining case which at that moment the techniques did not cover (see Theorems E and F). To do so, additional tools to the ones in these works are developed in Section 2. The treatment of the necessary conditions to bound the criticality in this limit cases are presented in Section 3. As in the previous works, our testing ground is the two-parametric family of potential differential systems 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}. Note that, for each µ ∈ Λ, system (2) is analytic on {(x, y) ∈ R2 : x > −1}. Our interest in this family began because of the results by Miyamoto and Yagasaki [23] about the monotonicity of the period function for q = 1 and p ∈ N. Later Yagasaki [30] improved this result proving the monotonicity property remains if one consider any p > 1 real. Following that, we performed a more exhaustive study of the period function of the family (2) for all µ ∈ Λ in [11–13]. To be more precise, in [13] we were concerned with the monotonicity of the period function, the criticality of the inner boundary and the criticality of the interior of the period annulus of its isochronous centers. In [11, 12] we studied the criticality of the outer boundary and it is precisely the 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 to [12, Theorem E] and [11, Theorem C]. On the right, improvement of the bifurcation diagram according with Theorem A. In both figures, ΓB and ΓU stand, respectively, for the union of the solid and dotted lines. result we obtained there for the family (2) the one that we improve here. In short, see Figure 1a, we proved  that Crit (Πµ̂ , Xµ̂ ), Xµ = 0 if µ̂ ∈ Λ\{ΓB ∪ΓU ∪{(− 21 , p0 )} with p0 ≈ 1.20175, and Crit (Πµ̂ , Xµ̂ ), Xµ > 1 if µ̂ ∈ ΓB . Moreover, we showed that the criticality is exactly one for parameters µ̂ ∈ ΓB such that µ̂ = (0, p̂) with p̂ ∈ (0, +∞) \ { 21 , 1}, µ̂ = (q̂, 1) with q̂ < −3 and µ̂ = (q̂, −2q̂ − 1) with q̂ ∈ (− 53 , − 31 ) \ {− 12 }. By applying the new tools in this paper we can go further, see Figure 1b, and prove the following result, where 3+3p f (p) := ( 3+3p 2 ) 1+3p − 3p − 2. Theorem A. Let {Xµ }µ∈Λ be the family of potential vector fields in (2) and consider the period function of the center at the origin. Then the following hold: (a) If µ̂ = (− 13 , p̂) with p̂ ∈ (− 13 , +∞) \ {p1 }, where p1 ≈ 1.15685 is the unique zero of f on (− 13 , +∞), then Crit (Πµ̂ , Xµ̂ ), Xµ = 0.  (b) If µ̂ = (0, p̂) with p̂ ∈ (0, +∞) \ {1} then Crit (Πµ̂ , Xµ̂ ), Xµ = 1. We finish this section stating a second application of the techniques developed in this work. The results obtained in the series of papers [15,17,19,20,22,29] by Mañosas, Mardešić, Marı́n, Saavedra and Villadelprat deal also with the bifurcation of critical periodic orbits from the outer boundary of the period annulus. Their testing ground is the family of demohogenized Loud’s centers  ẋ = −y + xy, (3) ẏ = x + Dx2 + F y 2 , where µ := (D, F ) ∈ R2 . In this collection of works, the bifurcation diagram of the period function at the polycycle has been studied (see Figure 2). We refer the reader to the previous works for complete information and to [24] for a summary of the latest results. We point out that the Loud’s family can be brought to potential form by means of an explicit coordinate transformation, see [29, Lemma 2.2], and hence it is susceptible to be studied with our methods. In this paper we prove the following result. Theorem B. Let {Xµ }µ∈Λ be the family of potential vector fields in (3) and consider the period function  of the center at the origin. If µ̂ = (D̂, 2) with D̂ ∈ (−2, 0) \ {−1/2} then Crit (Πµ̂ , Xµ̂ ), Xµ = 1. 3 Figure 2: Bifurcation diagram of the period function at the polycycle according to [15, 17, 19, 20, 22, 29], 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. 2 Technical results In this section we develop the technical tools that we will use in Section 3 to prove the results concerning the criticality. Let a ∈ R+ ∪ {+∞} and consider the integral operator F : C ω [0, a) → C ω [0, a) defined by Z π 2 F [f ](x) := f (x sin θ)dθ. (4) 0 Here, and it what follows, C ω [0, a) stands for the set of analytic functions on (0, a) that can be extended analytically to x = 0. In [11] it was shown that the derivative of the period function of system (1) is related with this operator by means of the equality √ 2 0 2 2h Tµ (h ) = F [fµ ](h), (5) p where fµ (x) = x(gµ−1 )00 (x) − x(gµ−1 )00 (−x) and gµ (x) := sgn(x) Vµ (x). Note that fµ is in the class C ω [0, h0 (µ)). Let us recall the notions of Chebyshev system and Wronskian, that will be useful for our purposes. Definition 2.1. 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. (In these abbreviations, “T” stands for Tchebycheff, which in some sources is the transcription of the Russian name Chebyshev).  4 Definition 2.2. Let f0 , f1 , . . . , fk−1 be analytic functions on an open interval I of R. Then f0 (x) ··· fk−1 (x)   f00 (x) ··· 0 fk−1 (x) (i) W [f0 , f1 , . . . , fk−1 ](x) := det fj (x) = .. 06i,j6k−1 . (k−1) (k−1) f0 (x) · · · fk−1 (x) is the Wronskian of (f0 , f1 , . . . , fk−1 ) at x ∈ I.  The previous two notions are closely related by the following result (see [9]). Lemma 2.3. (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. With equality (5) in mind, the aim is to complete F [fµ ] with a collection of analytic functions φ1µ , . . . , φnµ in order that (φ1µ , . . . , φnµ , F [fµ ]) form an ECT-system on (a − ε, a) for some ε > 0, for all µ ≈ µ̂. Notice that, to obtain the desired upper bounds on the criticality, the crucial point is to guarantee the uniformity with respect to the parameters of the system. In the works [11, 12] sufficient conditions in terms of fµ in order that F [fµ ] can be embedded into an ECT-system were given. These conditions were formulated using the notions that we introduce next. Definition 2.4. Let f ∈ C ω (a, b). We say that f is quantifiable at b by α with limit ` in case that: (i) If b ∈ R, then limx→b− f (x)(b − x)α = ` and ` 6= 0. f (x) (ii) If b = +∞, then limx→+∞ xα = ` and ` 6= 0. We call α the quantifier of f at b. We shall use the analogous definition at a.  Definition 2.5. Let {fµ }µ∈Λ be a continuous family of analytic 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 b is either a continuous function from Λ to R or b(µ) = +∞ for all µ ∈ Λ. Given µ̂ ∈ Λ we shall say that {fµ }µ∈Λ is continuously quantifiable in µ̂ at b(µ) by α(µ) with limit `(µ̂) if there exists an open neighborhood U of µ̂ such that fµ is quantifiable at b(µ) by α(µ) with limit `(µ) for all µ ∈ U and, moreover, (i) In case that b(µ̂) < +∞, then lim(x,µ)→(b(µ̂),µ̂) fµ (x)(b(µ) − x)α(µ) = `(µ̂) and `(µ̂) 6= 0. fµ (x) (ii) In case that b(µ̂) = +∞, then lim(x,µ)→(+∞,µ̂) xα(µ) = `(µ̂) and `(µ̂) 6= 0. For the sake of shortness, in the first case we shall write fµ (x) ∼b(µ) `(µ)(b(µ) − x)−α(µ) at µ̂, and in the second case fµ (x) ∼+∞ `(µ)xα(µ) at µ̂. We shall use the analogous definition for the left endpoint a(µ).  We point out that the map α : U −→ R that appears in the previous definition must be continuous at µ̂ (see [11, Remark 2.6]). Given ν1 , ν2 , . . . , νn ∈ R, consider the linear ordinary differential operator Lν n : C ω (0, +∞) −→ C ω (0, +∞) given by W [xν1 , xν2 , . . . , xνn , f (x)] Lν n [f ](x) := Pn . (6) x i=1 (νi −i) Here, and in what follows, for the sake of shortness we use the notation ν n = (ν1 , . . . , νn ). Furthermore we define Lν 0 = id in order that the statements of the next results contemplate the case n = 0 as well. From now one and till the end of this section let us assume that a = +∞ and so fµ ∈ C ω [0, +∞). In [11, 12] sufficient conditions on the family {Lν n [fµ ]}µ∈Λ in order that the family {(Lν n ◦ F )[fµ ]}µ∈Λ 5 is continuously quantifiable at infinity were given. One of the major requirements was that the continuous family {Lν n [fµ ]}µ∈Λ satisfies Lν n [fµ ](x) ∼∞ `(µ)xα(µ) at µ̂ with α(µ̂) 6= −1 + 2m, m ∈ N ∪ {0}. This fact, among other specific assumptions, allowed to bound the number of zeros of (Lν n ◦ F )[fµ ](x) near infinity uniformly on the parameters µ ≈ µ̂. Consequently, criticality results were obtained on account of Lemma 2.3 (see [11, Proposition 2.16 and Theorem A]). In the present paper we aim to extend these results to the case α(µ̂) = −1 + 2m, m ∈ N ∪ {0}. This situation presents a much more complicated behavior than the general one. For the sake of simplicity in the forthcoming illustration, let restrict ourselves to the case n = 0 and m = 0, and consider the continuous family of analytic functions fµ (x) = xα(µ) . In [12, Theorem 2.13] it was proved that if α(µ̂) > −1 then F [fµ ](x) ∼∞ xα(µ) at µ̂. In addition, using [12, Theorem 2.17] one can easily show that if α(µ̂) ∈ (−2, −1) then F [fµ ](x) ∼∞ x1 at µ̂. These two results are uniformly with respect to the parameters. However, if one fix µ = µ̂ such that α(µ̂) = −1 then, see [12, Proposition 2.5], x lim F [fµ̂ ](x) = 1. x→+∞ log(x) This situation can not be covered uniformly on the parameters using the notions introduced in Definition 2.5 because of the appearance of a logarithm term. In order to deal with this, we shall use the so-called Roussarie-Ecalle compensator (see [25]). Definition 2.6. 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.  Here we centered the singular value of the parameter at α = −1 instead of the classical version at α = 0 for the sake of simplicity through this paper. A useful property that was proved in [25] and we shall use here is that lim ω(x, α) = +∞. (7) (x,α)→(+∞,−1) With the purpose to state the main result of this section properly, we denote √  π Γ 1+α2  G (α) := (8) 2 Γ 1 + α2 where Γ is the Gamma function and ( 1 G (α) − 1+α if α 6= −1, K (α) := (9) log(2) if α = −1, which is a continuous positive function on (−2, 0) (see Lemma 2.8). For the sake of brevity, we also define Ω(x, α) := (α + 1)G (α)ω(x, α), (10) which is also a continuous positive function for all x > 1 and α > −2 (see Lemma 2.8). Following the notation in Definition 2.5, we shall write fµ (x) ∼+∞ `(µ)Ω(x, α(µ)) at µ̂ if 1 lim fµ (x) = `(µ̂) 6= 0. (x,µ)→(+∞,µ̂) Ω(x, α(µ)) Definition 2.7. Let f ∈ C ω [0, +∞). Then, for each n ∈ N, we call Z +∞ Mn [f ] := x2n−2 f (x)dx 0 the n-th momentum of f , whenever it is well defined.  6 Qm α+2i Let us finally denote b0 (α) ≡ 1 and bm (α) := i=1 α+2i−1 for m > 1. In the next statement, as well as in the following one, the condition M1 ≡ M2 ≡ · · · ≡ Mm ≡ 0 is void in case that m = 0. Theorem C. Let Λ be an open subset of Rd and consider a continuous family {fµ }µ∈Λ of analytic functions on [0, +∞) such that fµ (x) ∼∞ a(µ)xα(µ) at µ̂ ∈ Λ. Let us assume that there exists m ∈ N ∪ {0} such that α(µ̂) + 2m = −1. If M1 [fµ ] ≡ M2 [fµ ] ≡ · · · ≡ Mm [fµ ] ≡ 0 then x2m+1 F [fµ ](x) ∼∞ a(µ)bm (α(µ))Ω(x, α(µ) + 2m) at µ̂. We point out at this point that Theorem C covers the uniformity on the parameters that lacked in the above illustration example. Indeed, let us consider again fµ (x) = xα(µ) and let us fix µ̂ ∈ Λ such that α(µ̂) = −1. According with Theorem C, 1 F [fµ ](x) ∼∞ Ω(x, α(µ)) at µ̂. x We invoke equality (10) and Definition 2.6 to show that  α(µ)+1  x −1   ∼+∞ xα(µ) if α(µ) > −1,   x(α(µ) + 1) 1  log x Ω(x, α(µ)) = if α(µ) = −1, x   x   xα(µ)+1 − 1 1   ∼+∞ if − 2 < α(µ) < −1. x(α(µ) + 1) x We finish this section stating a general version of Theorem C. This generalization gives necessary con- ditions on the family {Lν n [fµ ]}µ∈Λ that enables to quantify (Lν n ◦ F )[fµ ] at x = +∞. In the following statement ν1 , ν2 , . . . , νn are not real numbers any more but continuous functions on Λ. For shortness, we keep using the notation ν n (µ) = (ν1 (µ), . . . , νn (µ)). Again, the condition of the existence of continuous function ν1 , ν2 , . . . , νn is void in case that n = 0. Theorem D. Let Λ be an open subset of Rd and consider a continuous family {fµ }µ∈Λ of analytic functions on [0, +∞). Assume that, in a neighborhood of some fixed µ̂ ∈ Λ, there exist n > 0 continuous functions ν1 , ν2 , . . . , νn , with ν1 (µ̂), . . . , νn (µ̂) pairwise distinct, and m ∈ N ∪ {0} such that Lν n [fµ ](x) ∼∞ a(µ)xα(µ) at µ̂ with α(µ̂) + 2m = −1. If M1 [Lν n [fµ ]] ≡ M2 [Lν n [fµ ]] ≡ · · · ≡ Mm [Lν n [fµ ]] ≡ 0 then x2m+1 (Lν n ◦ F )[fµ ](x) ∼∞ a(µ)bm (α(µ))Ω(x, α(µ) + 2m) at µ̂. 2.1 Proof of Theorem C This section is devoted to the proof of Theorem C. To this end, we first prove a collection of technical lemmas that will be useful for this purpose. Lemma 2.8. The following holds: (a) For every fixed x ∈ (1, +∞) the function α 7→ Ω(x, α) = (1 + α)G (α)ω(x, α) is positive and monotonous increasing for all α ∈ (−2, +∞). Moreover, limα→−1 Ω(x, α) = log(x) for all x > 1. (b) The function α 7→ K (α) is continuous and positive for all α ∈ (−2, +∞). Proof. Let us fix x ∈ (1, +∞). In order to prove (a) we compute the derivative of the compensator using the expression on Definition 2.6. We have d (α + 1) ω = ((α + 1)ω + 1) log x − ω. (11) dα (Here we omit the dependence of ω with respect to x and α for the sake of brevity.) Multiplying this expression by (α + 1) and deriving again,   d 2 d (α + 1) ω = (α + 1)((α + 1)ω + 1) log2 x. dα dα 7 Since (α + 1)ω + 1 > 0 for all α > −2, the previous expression vanishes only at α = −1. This, together d with the expression in (11) implies that (α + 1)2 dα ω > 0 for all α ∈ R and the equality holds if and only if α = −1. Deriving the expression in Definition 2.6 we have that d 1 lim ω = log2 x 6= 0. α→−1 dα 2 This shows the monotonicity of the map α 7→ ω(x, α) for all α ∈ (−2, +∞). Moreover, ω(x, −2) = 1− x1 > 0. This implies that ω(x, α) > 0 in (−2, +∞). Finally, on account of the properties of the Gamma function, the map  √ πΓ 3+α2  α 7→ (1 + α)G (α) = Γ 1 + α2 is a well defined positive monotonous increasing function in (−2, +∞). Therefore (1 + α)G (α)ω(x, α) is a positive monotonous increasing function on (−2, +∞). Moreover, the previous equality implies that lim (1 + α)G (α) = 1. (12) α→−1 Then limα→−1 Ω(x, α) = log(x) on account of Definition 2.6. This proves (a). (1+α)G (α)−1 Let us now prove (b). In order to see that K (α) = 1+α > 0 we show that √  √  πΓ 3+α 2  πΓ 3+α 2  > 1 if α ∈ (−1, +∞) and 0 < < 1 if α ∈ (−2, −1). Γ 1 + α2 Γ 1 + α2 A direct computation shows that √ ! √  d πΓ 3+α 2  π Γ 3+α 2  3+α   = ψ 2 − ψ(1 + α2 ) (13) dα Γ 1 + α2 2 Γ 1 + α2 where ψ(x) = Γ0 (x)/Γ(x) denotes the Digamma function (see [1, Section 6.3]). The function ψ(x) is √ πΓ( 3+α ) increasing for all x > 0. Then, on account of the expression of the derivative, the function Γ 1+ α2 is ( 2) increasing since it is positive for all α > −2. The function K (α) is positive then since √  √  πΓ 3+α 2  πΓ 3+α2  lim α = 0 and lim = +∞. α→−2+ Γ 1 + α→+∞ Γ 1 + α 2 2 1  To show that limα→−1 G (α) − 1+α = log(2) we invoke the limit (12) and, applying Hôpital’s rule,    ! 1 √ d Γ 3+α 2  lim G (α) − = lim π α→−1 1+α α→−1 dα Γ 1 + α2 if this last limit exists. On account of equality (13), the result follows using that ψ(1) − ψ(1/2) = 2 log(2) (see [1, 6.3.3]). Lemma 2.9. Let Λ be an open subset of Rd and consider a continuous family {fµ }µ∈Λ of analytic functions on [0, +∞). Let α(µ) be a continuous function such that α(µ̂) = −1 for some fixed µ̂ ∈ Λ. Then for every ε > 0 and M > 0 there exist x0 > M and δ > 0 such that Z arcsin(M/x) x fµ (x sin θ)dθ 6 ε Ω(x, α(µ)) 0 for all x > x0 and kµ − µ̂k < δ. 8 Proof. Let us fix ε, M > 0 and let δ > 0 such that α(µ) ∈ (−2, 0) for all kµ − µ̂k 6 δ. Since {fµ }µ∈Λ is a continuous family of analytic functions on [0, +∞), N = N (M, δ) := max{|fµ (x)| : x ∈ [0, M ], kµ − µ̂k 6 δ} is well-defined. Then, for all x > M , Z arcsin(M/x) x N x arcsin(M/x) |fµ (x sin θ)| dθ 6 . Ω(x, α(µ)) 0 Ω(x, α(µ)) On account of the identity (7) and the limit (12) we have that lim Ω(x, α) = lim (1 + α)G (α)ω(x, α) = +∞. (x,α)→(+∞,−1) (x,α)→(+∞,−1) The result follows then on account of the continuity of α and due to limx→+∞ x arcsin(M/x) = M . Lemma 2.10. For any M > 0, Ω(x, α) + K (α) − ω(M, α) lim = 1. (x,α)→(+∞,−1) Ω(x, α) Proof. The result is straightforward using identity (7) together with the limit (12) and the fact that both ω(M, α) and K (α) are bounded functions for α ≈ −1. The next result requires the introduction of the Gauss hypergeometric series X∞ (a)n (b)n z n 2 F1 (a, b; c; z) = n=0 (c)n n! for z, a, b, c ∈ C, where (a)n = a(a + 1) · · · (a + n − 1). The radius of convergence of this series is 1. In what follows we consider z, a, b, c ∈ R. The series is absolute convergent when c − a − b > 0, divergent when c − a − b 6 −1 and convergent when −1 < c − a − b 6 0 provided that z 6= 1. Notice that the series is not defined when c is equal to −m, (m = 0, 1, 2, . . . ), provided a or b is not a negative integer n with n < m (see [1]). d Lemma 2.11. dz 2 F1 (a, b; a + 1; z) = az ((1 − z)−b − 2 F1 (a, b; a + 1; z)). Proof. This is straightforward by using the formulae in [1]. Indeed, it shows that d a z 2 F1 (a, b; a + 1; z) = az a−1 2 F 1(a, b; a; z) = az a−1 (1 − z)−b , dz where the first equality is a particular case of 15.2.4 and the second one follows by applying 15.1.8. Then an easy manipulation yields to the desired equality after deriving the product on the left. The following result is the key tool on the proof of Theorem C. Proposition 2.12. Let Λ be an open subset of Rd , M > 0 and α : Λ → R be a continuous function such that α(µ̂) = −1 for some fixed µ̂ ∈ Λ. Then Z π 2 xα(µ)+1 (sin θ)α(µ) dθ ∼∞ Ω(x, α(µ)) at µ̂. arcsin(M/x) Proof. First we claim that, for all x > M , q   Z π2   1− M 2 1 1−α 3 M2 x2 2 F 1 2,  2 ; 2; 1 − x2 if α 6= −1, α (sin θ) dθ = q arcsin(M/x)  arctanh 1− M 2 if α = −1. x2 9 In order to prove the claim, let us start considering the case α = −1. We can write Z π2 Z 1 dθ dz = √ arcsin(M/x) sin θ M/x z 1 − z2 d √ 1 for any M > 0 and the result follows using that dz arctanh( 1 − z 2 ) = z√1−z 2 . In the case α 6= −1, on account of Lemma 2.11, we observe that     d 1 1−α 3 − 2 F1 , ; , cos2 θ cos θ = (sin θ)α . dθ 2 2 2 The claim follows, in this second  case, evaluating the primitive p at the endpoints of the interval of integration using that 2 F1 21 , 1−α 3 2 ; 2 , 0 = 1 and cos(arcsin(M/x)) = 1 − M 2 /x2 . Let us now prove the result. To do so, we take advantage of equality 15.3.6 on [1]. That is, Γ(c)Γ(c − a − b) 2 F1 (a, b; c; z) = 2 F1 (a, b; a + b − c + 1; 1 − z) Γ(c − a)Γ(c − b) Γ(c)Γ(a + b − c) + (1 − z)c−a−b 2 F1 (c − a, c − b; c − a − b + 1; 1 − z) Γ(a)Γ(b) for all z ∈ C with |arg(1 − z)| < π and c 6= a + b ± m, m ∈ N. In the particular case that we are concerned with, the previously equality yields to     1 1 − α(µ) 3 M2 x M 1+α(µ) α(µ) 3 + α(µ) M 2 2 F1 , ; ; 1 − 2 = G (α(µ)) √ − 2 F1 1, 1 + ; ; 2 2 2 2 x x2 − M 2 (1 + α(µ))x1+α(µ) 2 2 x provided that α(µ) 6= −1. Here we used that 2 F1 (a, b; b; z) = (1 − z)−a (see 15.1.8 in [1]). The result will follow once we prove that Z π2 α(µ)+1 1 x (sin θ)α(µ) dθ = Ω(x, α(µ)) + K (α) − ω(M, α(µ)) + 2 (c + r(x, µ)) arcsin(M/x) x for some constant c and some smooth function r(x, µ) satisfying lim(x,µ)→(+∞,µ̂) r(x, µ) = 0. Indeed, if it is so, we would have Z π2 xα(µ)+1 Ω(x, α(µ)) + K (α) − ω(M, α(µ)) c + r(x, µ) (sin θ)α(µ) dθ = + 2 . Ω(x, α(µ)) arcsin(M/x) Ω(x, α(µ)) x Ω(x, α(µ)) Since, by Lemma 2.10, Ω(x, α(µ)) + K (α(µ)) − ω(M, α(µ)) lim = 1, (x,µ)→(+∞,µ̂) Ω(x, α(µ)) the result would follow using the limits (7) and (12), which imply lim(x,µ)→(+∞,µ̂) x2 Ω(x, α(µ)) = +∞. We claim at this point that r     M2 α(µ) 3 + α(µ) M 2 1 + α(µ) M2 1 − 2 2 F1 1, 1 + ; ; 2 =1+ + r(x, µ) , x 2 2 x x2 2(3 + α(µ)) with lim(x,µ)→(+∞,µ̂) r(x, µ) = 0. Indeed, from equality 15.3.3 in [1] we have that r     M2 α(µ) 3 + α(µ) M 2 1 1 + α(µ) 3 + α(µ) M 2 1 − 2 2 F1 1, 1 + ; ; 2 = 2 F1 , ; ; 2 . x 2 2 x 2 2 2 x By definition,   ∞  1+α(µ)  2n (1 + α(µ)) M 2 X 1 1 1 + α(µ) 3 + α(µ) M 2 2 n 2 M 2 F1 , ; ; 2 =1+ + 3+α(µ)  n x−2n . 2 2 2 x 2(3 + α(µ)) x2 n=2 n! 2 n 10 Notice that, in the previous expression, there is always a factor (1 + α(µ)) on the numerator for n > 1. Moreover, it is easy to show that, after extracting (1 + α(µ)) as a common factor, the sequence has positive monotone decreasing coefficients for α(µ) ≈ −1. This follows from the identity (a)n+1 = (an )(a + n). Therefore,   ! 1 1+α(µ) 3+α(µ) M 2 1 M 2 2 F1 2 , 2 ; 2 ; x 2 −1 1 ∞ M 2 X 2n −2n 6 6 + M x . 2(3 + α(µ)) x2 1 + α(µ) 2(3 + α(µ)) x2 n=2 P∞ The claim follows then on account that the series n=2 z 2n is convergent for |z| < 1 and taking the limit as (x, µ) → (+∞, µ̂). Here we point out that the constant M is fixed. On account of the previous claim, after some elementary manipulations, we have, for α(µ) 6= −1, Z π   1+α(µ) 2 α(µ) α(µ)+1 M 1+α(µ) M 3+α(µ) 1 x (sin θ) dθ = x G (α(µ)) − − + r(x, µ) arcsin(M/x) 1 + α(µ) x2 2(3 + α(µ))   1 M 3+α(µ) = Ω(x, α(µ)) + K (α(µ)) − ω(M, α(µ)) − 2 + r(x, µ) . x 2(3 + α(µ)) In the case α(µ) = −1, by the Taylor’s series expansion, Z π r ! 2 α(µ) M2 (sin θ) dθ = arctanh 1− 2 arcsin(M/x) x   1 M2 = log(2x) − log(M ) − + r(x) x2 4   1 M2 = Ω(x, α(µ)) + K (α(µ)) − ω(M, α(µ)) − + r(x) x2 4 with limx→+∞ r(x) = 0. Therefore, for any α(µ), we have Z π   1+α(µ) 2 α(µ) 1 M 3+α(µ) x (sin θ) dθ = Ω(x, α(µ)) + K (α(µ)) − ω(M, α(µ)) − 2 + r(x, µ) arcsin(M/x) x 2(3 + α(µ)) with lim(x,µ)→(+∞,µ̂) r(x, µ) = 0. The result follows then by Lemma 2.10. Definition 2.13. Let Λ be an open subset of Rd and {fµ }µ∈Λ be a continuous R x family of analytic functions on [0, +∞). Setting f0 (x, µ) := fµ (x), we define fm (x, µ) := fm−1 (x, µ)x2 + 0 fm−1 (s, µ)ds for all m > 1.  Lemma 2.14 (see [12]). Let Λ be an open subset of Rd and {fµ }µ∈Λ be a continuous family of analytic functions on [0, +∞). Then, for any m ∈ N, 1 F [fµ ](x) = F [fm (·, µ)](x) for all x > 0. x2m Next result was proved in [12, Proposition 2.16]. We point out that in its original statement the re- Qm α(µ)+2i quirement α(µ̂) 6= −2m − 1 is not needed. This requirement was assumed for bm (α(µ)) = i=1 α(µ)+2i−1 to be well defined in a neighborhood of µ̂. However, the reader may easily check that this is true for α(µ̂) = −2m − 1. Proposition 2.15. Let Λ be an open subset of Rd and consider a continuous family {fµ }µ∈Λ of analytic functions on [0, +∞) such that fµ (x) ∼∞ a(µ)xα(µ) at any µ ∈ Λ. Assume that for some µ̂ ∈ Λ, α(µ̂) 6 −1 and let m ∈ N ∪ {0} be such that α(µ̂) ∈ [−2m − 1, −2m + 1). If M1 [fµ ] ≡ M2 [fµ ] ≡ · · · ≡ Mm [fµ ] ≡ 0 for all µ ∈ Λ and α(µ̂) + 2m 6= 0 then fm (x, µ) ∼∞ a(µ)bm (α(µ))xα(µ)+2m . 11 Proof of Theorem C. Let us fix m ∈ N ∪ {0} such that α(µ̂) + 2m = −1 and let us assume that M1 [fµ ] ≡ M2 [fµ ] ≡ · · · Mm [fµ ] ≡ 0. Moreover, let us set â := a(µ̂)bm (α(µ̂)) for the sake of shortness. The result will follow once we show that for any ε > 0 there exist x0 > 0 and δ0 > 0 such that Z π2 x fm (x sin θ, µ)dθ − â < ε Ω(x, α(µ) + 2m) 0 for all x > x0 and kµ − µ̂k < δ0 . Indeed, if it is so we have that xF [fm (·, µ)](x) ∼∞ a(µ)bm (α(µ))Ω(x, α(µ) + 2m) at µ̂. Consequently, on account of Lemma 2.14, x2m+1 F [fµ ](x) ∼∞ a(µ)bm (α(µ))Ω(x, α(µ) + 2m) at µ̂, as we desired. By Proposition 2.15 we have that fm (x, µ) ∼∞ a(µ)bm (α(µ))xα(µ)+2m in a neighborhood of µ̂ and α(µ̂) + 2m = −1. Then, for any fixed ε > 0, there exist M > 0 and δ1 > 0 such that ε x−α(µ)−2m fm (x, µ) − â < (14) 8 for all x > M and kµ − µ̂k < δ1 . Moreover we can assume that α(µ) + 2m ∈ (−2, 0) for all kµ − µ̂k < δ1 . Thus, by Lemma 2.9, there exists x1 > M such that Z arcsin(M/x) x ε fm (x sin θ, µ)dθ 6 (15) Ω(x, α(µ) + 2m) 0 2 for all x > x1 . From Proposition 2.12 we have that there exist x0 > x1 and 0 < δ0 < δ1 such that Z π2 âx ε (x sin θ)α(µ)+2m dθ − â 6 (16) Ω(x, α(µ) + 2m) arcsin(M/x) 4 and Z π x 2 (x sin θ)α(µ)+2m dθ 6 2 (17) Ω(x, α(µ) + 2m) arcsin(M/x) for all x > x0 and kµ − µ̂k < δ0 . Lastly, for all x > x0 and kµ − µ̂k < δ0 , Z π2   x fm (x sin θ, µ) − â (x sin θ)α(µ)+2m dθ 6 Ω(x, α(µ) + 2m) arcsin(M/x) (x sin θ)α(µ)+2m Z π2 ε x ε 6 (x sin θ)α(µ)+2m dθ 6 , 8 Ω(x, α(µ) + 2m) arcsin(M/x) 2 where we used (14) in the first inequality and (17) in the second one. The previous inequality together with (15) and (16) proves the Theorem. 2.2 Proof of Theorem D In this section we prove Theorem D as a corollary of Theorem C. We first recall a result that was already shown in [11]. Proposition 2.16. Let f ∈ C ω (0, +∞). If f can be extended analytically to x = 0, then Lν n [f ] can be extended analytically to x = 0. Moreover, F ◦ Lν n = Lν n ◦ F . Proof of Theorem D. By Proposition 2.16 Lν n (µ) [fµ ] is an analytic function on [0, +∞) for each µ ∈ Λ and Z π2 (Lν n (µ) ◦ F )[fµ ](x) = (F ◦ Lν n (µ) )[fµ ](x) = Lν n (µ) [fµ ](x sin θ)dθ. 0 Then the result follows by applying Theorem C to the family {Lν n (µ) [fµ ]}µ∈Λ . 12 3 Criticality of the period function at the outer boundary This section is devoted to the dynamical results of the paper. Consider the family of analytic potential dif- ferential systems (1) depending on a parameter µ ∈ Λ ⊂ Rd and suppose that the origin is a non-degenerate center for all µ. We denote the projection of the period annulus on the x-axis by Iµ = (x` (µ), xr (µ)), x` (µ) < 0 < xr (µ), and by h0 (µ) the energy level at the outer boundary of the period annulus. Definition 3.1. We say that the family of systems (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.  p Next result is proved in [12]. We recall that gµ (x) = sgn(x) Vµ (x). Lemma 3.2. Assume that the family of systems (1) verifies hypothesis (H). Then the map (x, µ) 7−→ gµ−1 (x) p p is continuous on the open set {(x, µ) ∈ R × Λ : x ∈ (− h0 (µ), h0 (µ))}. The following sections are concerned with sufficient conditions to bound the criticality at the outer boundary for potential systems (1) verifying hypothesis (H). Section 3.1 deals with the case h0 ≡ +∞ (see Theorem E), whereas Section 3.2 tackles the case h0 finite (see Theorem F). 3.1 Potential systems with infinite energy In this section let us consider that the energy at the outer boundary is h0 (µ) = +∞ for all µ ∈ Λ. Following the strategy in [11], we plan to 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 analogous to [11, Lemma 3.5] where only the power hνn (µ) appears multiplying the Wronskian. The reader may check that the same proof there can be applied here, so we skip it for the sake of shortness. 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 neighborhood of some fixed µ̂ ∈ Λ, a continuous function α : Λ → R with α(µ̂) = −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)→(+∞,µ̂) Ω(h, α(µ))  Then Crit (Πµ̂ , Xµ̂ ), Xµ 6 n − 1. We are now in position to state the main result concerning the criticality at the outer boundary for the case h0 ≡ +∞. In its statement, and from now on, for a given function f : (−a, a) −→ R, a > 0, we denote P[f ](x) := f (x) + f (−x). Let us also remark that the assumption requiring the existence of functions ν1 , ν2 , . . . , νn is void in case that n = 0. The same happens for the assumption of M1 ≡ M2 ≡ · · · ≡ Mm ≡ 0 when m = 0. Theorem E. 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 neighborhood of some fixed µ̂ ∈ Λ such that (Lν n (µ) ◦ P)[z(gµ−1 )00 (z)] ∼∞ a(µ)xξ(µ) at µ̂. For each i ∈ N, let Mi (µ) be the i-th momentum of (Lν n (µ) ◦ P)[z(gµ−1 )00 (z)], whenever it is well defined. If  ξ(µ̂) = −1 − 2m for some m ∈ N ∪ {0} and M1 ≡ M2 ≡ · · · ≡ Mm ≡ 0 then Crit (Πµ̂ , Xµ̂ ), Xµ 6 n. 13 00 Proof. Let us denote fµ (z) := P[z gµ−1 (z)]. By Lemma 3.2 and the hypothesis (H) we have that {fµ }µ∈Λ is a continuous family of analytic functions on (0, +∞) that extends analytically to z = 0. Since ξ(µ) is the quantifier of {Lν n (µ) [fµ ]}µ∈Λ at infinity, M1 ≡ M2 ≡ · · · ≡ Mm ≡ 0 and ξ(µ̂) + 2m = −1, applying Theorem D we can assert that x2m+1 (Lν n (µ) ◦ F )[fµ ](x) ∼∞ a(µ)bm (µ)Ω(x, ξ(µ) + 2m) at µ̂. Then, on account of the definition of Lν n (µ) , see (6), we have that   h W hν1 (µ) , . . . , hνn (µ) , F [fµ ](h) lim Pn 6= 0. (h,µ)→(+∞,µ̂) Ω(h, ξ(µ) + 2m) h i=1 (νi (µ)−i) The result follows then by Lemma 3.3 and using equality (5). 3.2 Potential systems with finite energy In this section let us consider that the energy at the outer boundary is finite for all µ ∈ Λ. With the intention of embedding f Tµ0 into some ECT-system for an appropriate non-vanishing function f , we proceed as in [11] and “translate” the case h0 < +∞ to the case h0 = +∞ so we can take advantage of Theorem D. With this aim in view, we define next a differential operator which is conjugated to Lν n (µ) . The conjugation is precisely the tool that enables this translation. Given ν1 , . . . , νn ∈ R, consider the linear ordinary differential operator Dν n : C ω (0, 1) −→ C ω (0, 1) defined by W [ψν1 , . . . , ψνn , f ] (x) n(n+1) Dν n [f ](x) := (x(1 − x2 )) 2 Qn , (18) i=1 ψνi (x)  ν 1 √ x where we use the notation ν n = (ν1 , . . . , νn ) and ψν (x) := 1−x 2 1−x2 . Furthermore we define Dν 0 := id for the sake of completeness. Setting φ(x) := √ x , we also consider the operator 1+x2 B : C ω [0, 1) −→ C ω [0, +∞) defined by   1  B[f ](x) := 1 − φ2 (x) f ◦ φ (x) = 2 f ◦ φ (x). (19) 1+x This operator is the conjugation mentioned before. Definition 3.4. Let f ∈ C ω [0, 1). Then, for each n ∈ N, we call Z 1  2n−2 f (x) x Nn [f ] := √ √ dx 0 1 − x2 1 − x2 the n-th momentum of f , whenever it is well defined.  Next results show the way B conjugates Dν n and Lν n . We refer the reader to [11] for the proofs. 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. 14 Lemma 3.6. Let {fµ }µ∈Λ be a continuous family of analytic functions on [0, 1). Then fµ (z) ∼z=1 a(µ)(1 − z)−α(µ) at µ̂ if and only if B[fµ ](x) ∼∞ a(µ)2α(µ) x2α(µ)−2 at µ̂. The next result is well known (see [16]). Lemma 3.7. Let f0 , f1 , . . . , fn−1 be analytic functions. Then the following statements hold: (n−1)n (a) W [f0 ◦ ϕ, . . . , fn−1 ◦ ϕ](x) = (ϕ0 (x)) 2 W [f0 , . . . , fn−1 ](ϕ(x)) for any analytic diffeomorphism ϕ. (b) W [gf0 , . . . , gfn−1 ](x) = g(x)n W [f0 , . . . , fn−1 ](x) for any analytic function g. The following Lemma is analogous to Lemma 3.3 for the case h0 finite. As before, the reader may check that the proof in [11, Lemma 3.10] is also valid for this generalization. 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 neighborhood of some fixed µ̂ ∈ Λ, a continuous function α : Λ → R with α(µ̂) = −1 and an analytic non-vanishing function f on (0, 1) such that (1 − z)νn (µ)   lim   W ψν1 (µ) (z), . . . , ψνn−1 (µ) (z), f (z)Tµ0 (z 2 h0 (µ)) = ` 6= 0. (z,µ)→(1,µ̂) Ω √ z , α(µ) 1−z 2  Then Crit (Πµ̂ , Xµ̂ ), Xµ 6 n − 1. Next we state the main result to bound the criticality at the outer boundary of the family of systems (1) in case that its energy level is finite. Again we stress that the assumptions requiring the existence of functions ν1 , ν2 , . . . , νn for n = 0 and N1 ≡ N2 ≡ · · · ≡ Nm ≡ 0 for m = 0 are void in its statement. Theorem F. Let us assume that the family of potential analytic systems (1) verifies hypothesis (H) with h0 (µ) < +∞ for all µ ∈ Λ and that there exist n > 0 continuous functions ν1 , ν2 , . . . , νn in a neighborhood of some fixed µ̂ ∈ Λ such that  p p  (Dν n (µ) ◦ P) z h0 (µ)(gµ−1 )00 (z h0 (µ)) ∼z=1 a(µ)(1 − z)ξ(µ) at µ̂.  p p  For each i ∈ N, let Ni (µ) be the i-th momentum of (Dν n (µ) ◦ P) z h0 (µ)(gµ−1 )00 (z h0 (µ)) , whenever it is well defined. If ξ(µ̂) = 21 − m for some m ∈ N ∪ {0} and N1 ≡ N2 ≡ . . . ≡ Nm ≡ 0 then Crit (Πµ̂ , Xµ̂ ), Xµ 6 n. p p Proof. Let us denote fµ (z) := P[z h0 (µ)(gµ−1 )00 (z h0 (µ))]. By Lemma 3.2 and the hypothesis (H) we have that {fµ }µ∈Λ is a continuous family of analytic functions on (0, 1) that extends analytically to z = 1. The identity (5), after the appropriate rescaling, yields to the identity √ F [fµ ](z) = 2h0 (µ)z 2 Tµ0 (h0 (µ)z 2 ) for all z ∈ (0, 1). (20) On account of this equality, to prove the result we must show that there exist ε > 0 and a neighborhood U of µ̂ such that F [fµ ](z) has at most n zeros for z ∈ (1−ε, 1), multiplicities taking intop p∈ U. By account, for all µ  hypothesis ξ(µ̂) = 2 − m, for some m ∈ N ∪ {0}, is the quantifier of (Dν n (µ) ◦ P) z h0 (µ)(gµ ) (z h0 (µ)) 1 −1 00 at z = 1 in µ = µ̂. By (b) and (d) in Lemma 3.5,       Ni Dν n (µ) [fµ ] = Mi (B ◦ Dν n (µ) )[fµ ] = Mi (Lν n (µ) ◦ B)[fµ ] for all i = 1, 2, . . . , m. Therefore the condition N1 ≡ N2 ≡ · · · Nm ≡ 0 in the statement is equivalent to the condition M1 ≡ M2 ≡ · · · Mm ≡ 0, where Mi is the i-th momentum of (Lν n (µ) ◦ B)[fµ ]. Recall at this point that, by (b) in Lemma 3.5, B ◦ Dν n = Lν n ◦ B. By assumption, Dν n (µ) [fµ ](z) ∼z=1 a(µ)(1 − z)ξ(µ) at µ̂. Then applying Lemma 3.6 we have that (Lν n (µ) ◦ B)[fµ ](x) ∼∞ a(µ)2ξ(µ) xη(µ) at µ̂, 15 where η(µ) := 2ξ(µ) − 2. Notice that η satisfies η(µ̂) + 2m = −1. Therefore Theorem D applied to the family {(Lν n (µ) ◦ B)[fµ ]}µ∈Λ shows that x2m+1 (Lν n (µ) ◦ F ◦ B)[fµ ](x) ∼∞ a(µ)2ξ(µ) bm (η(µ))Ω(x, η(µ) + 2m) at µ̂ with a(µ̂)bm (η(µ̂)) 6= 0. 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 equality, the identity 1 + x2 B[ϕ](x) = 1+x2 1 B[(1 − z 2 )− 2 ϕ(z)] with ϕ = F [fµ ] in the second one, and (b) in Lemma 3.5 in the third one. Note also that (Lν n (µ) ◦ F ◦ B)[fµ ] µ∈Λ is a continuous family of analytic functions on [0, +∞). On account of the definition of B in (19) and the previous equality, we have that x2m+1  1  Dν (µ) (1 − φ(x)2 )− 2 F [fµ ](φ(x)) ∼∞ a(µ)2ξ(µ) bm (η(µ))Ω(x, η(µ) + 2m) at µ̂. 1 + x2 n Setting x = φ−1 (z) = √ z , the previous identity yields to 1−z 2 (1 − z)−2m  1  lim   Dν n (µ) (1 − z 2 )− 2 F [fµ ](z) 6= 0. (z,µ)→(1,µ̂) Ω √ z , η(µ) + 2m 1−z 2 Thus, on account of the definition of Dν n (µ) in (18),  1  (1 − z)−2m 2 n(n+1) W ψν1 (µ) (z), . . . , ψνn (µ) (z), (1 − z 2 )− 2 F [fµ ](z) lim   (z(1 − z )) 2 Qn 6= 0, (z,µ)→(1,µ̂) Ω √ z , η(µ) + 2m i=1 ψνi (µ) (z) 1−z 2  ν 1 √ z which, since ψν (z) = 1−z 2 1−z 2 , implies that  (1 − z)κ(µ) 1  lim   W ψν1 (µ) (z), . . . , ψνn (µ) (z), (1 − z 2 )− 2 F [fµ ](z) 6= 0 (z,µ)→(1,µ̂) Ω √ z , η(µ) + 2m 1−z 2 n(n+3) 1 Pn with κ(µ) := −2m + 2 + 2 i=1 νi (µ). The result follows then by Lemma 3.8 and taking the identity (20) into account. 4 Applications 4.1 Proof of Theorem A In this section we resume the study that we began in [11,12] for the family of potential differential systems (2) with µ = (q, p) ∈ Λ := {(q, p) ∈ R2 : p > q}. Following the notation we introduced in Section 1, we define Z x+1 Vµ (x) := (up − uq )du. (21) 1 Theorem A illustrates the application of the criticality results we have obtained in the previous section. This Theorem collects some of the cases that the results in [11, 12] did not cover. To prove the result we first need to show a technical lemma concerning the function f (p) we have introduced in the introductory section. This Lemma ensures the uniqueness of the point p1 in Theorem A. 16 3+3p Lemma 4.1. The function f (p) = ( 3+3p 2 ) 1+3p − 2 − 3p has a unique zero on (− 1 , +∞). 3 x Proof. We point out that the result is equivalent to show that g(x) := ( x2 ) x−2 − x + 1 is monotonous decreasing on (2, +∞). Indeed, we have g(x) = f (x/3−1), limx→2 g(x) = e−1 > 0 and limx→+∞ g(x) = −∞. The result follows then by continuity of the function f . Let us show the monotonicity of g. Elementary computations yield to  x  x−2 x x − 2 + log(4) − 2 log(x) g 0 (x) = −1 + 2 (x − 2)2 and  x  x−2 2 4 + x2 − 2x(2 + log(2)2 ) + x log(x)(log(16) − 2 log(x)) g 00 (x) = − . 2 (x − 2)4 Since limx→2+ g 0 (x) < 0 it is enough to show that g 00 < 0 in (2, +∞). On account of the expression of g 00 , this fact is equivalent to say that the function κ(x) := 4 + x2 − 2x(2 + log(2)2 ) + x log(x)(log(16) − 2 log(x)) is positive. This last statement is clear since one can easily verify that κ(2) = κ0 (2) = κ00 (2) = 0 and κ000 (x) = 4 log(x/2) x2 > 0 for all x > 2. This shows the validity of the Lemma. Proof of Theorem A. Before start with the proof itself, notice that, since p and q are both different from −1, then the expression in (21) writes (x + 1)p+1 (x + 1)q+1 Vµ (x) = − + h0 (µ), (22) p+1 q+1 p−q where h0 (µ) := (p+1)(q+1) corresponds to the energy level at the outer boundary of the period annulus of the family (2) when q > −1. Observe that this is the situation in Theorem A. The projection of the period annulus on the x-axis is Iµ = (x` (µ), xr (µ)), with   p−q 1 p+1 x` (µ) = −1 and xr (µ) = − 1. q+1 Following the notation in Theorem F, from (22) and due to p > q, one can easily check that the family {h0 (µ) − Vµ (x)}µ∈Λ is continuously quantifiable in any µ̂ = (q̂, p̂) ∈ Λ at x = x` by β` (µ) with limit b` and at x = xr by βr (µ) with limit br , where 1 β` (µ) = −(q + 1), βr (µ) = −1, b` = and br = Vµ̂0 (xr ). (23) q̂ + 1 Let us prove assertion in (a) by applying Theorem F with n = 0. To do so, let µ̂ = (q̂, p̂) with q̂ = − 13 3+3p and p̂ ∈ (− 31 , +∞) \ {p1 }, where p1 is the unique zero of f (p) = ( 3+3p 1 2 ) p − 2 − 3p on (− 3 , +∞) (see 1+3p  p −1 00 Lemma 4.1). In order to obtain the quantifier ξ of P[z h0 (µ)(gµ ) (z h0 (µ))] µ∈Λ we shall use the   second part of [11, Theorem B]. This result, for n = 0, states that ξ(µ) = − min{ αβ`` (µ), αβrr (µ)} + 1,  1 where β` and βr are the functions in (23), and α` and αr are the quantifiers of (h0 − Vµ )Vµ 2 Rµ µ∈Λ , with (Vµ0 )2 −2Vµ Vµ00 Rµ := , at x` and xr , respectively. Using the expression in (22), it is a computation to show that (Vµ0 )3 1  23 the previous family is continuously quantifiable in µ̂ at x = x` by α` (µ) = q with limit a` = 2 2 − p̂+1 . On the other hand, since Vµ is analytic at x = xr and Vµ̂0 (xr ) 6= 0, the family is continuously quantifiable in µ̂ at x = xr by αr (µ) = −1 with limit p Vµ̂0 (xr )2 − 2h0 (µ̂)Vµ̂00 (xr ) ar = h0 (µ̂) , Vµ̂0 (xr )2 provided that Vµ̂0 (xr )2 − 2h0 (µ̂)Vµ̂00 (xr ) 6= 0. This inequality is equivalent to require that f (p̂) 6= 0, which is   satisfied since p̂ 6= p1 . Therefore, we have that αβ`` (µ̂) = 12 and αβrr (µ̂) = 1 and so the quantifier of the  p p q family P[z h0 (µ)(gµ−1 )00 (z h0 (µ))] µ∈Λ at z = 1 is ξ(µ) = max{ q+1 , −1} + 1. 17 1 We are in position to apply now Theorem F. To this end  note that ξ(µ̂) = 2 . Then, the application of (a) in Theorem F with n = 0 shows that Crit (Πµ̂ , Xµ̂ ), Xµ = 0. This proves the validity of (a).  this case we consider µ̂ = (q̂, p̂) with q̂ = 0 and p̂ ∈ (0, +∞)\{1}. Let us now prove the assertion in (b). In By [12, Theorem E], Crit (Πµ̂ , Xµ̂ ), Xµ > 1 and by [11, Theorem C], Crit (Πµ̂ , Xµ̂ ), Xµ = 1 if p̂ ∈ (0, +∞) \ { 12 , 1}. Then the result will follow by applying Theorem F with n = 1 and µ̂ = (0, 12 ). From the computations in the proof of [11, Theorem C] we have that the family  p p  (Dν 1 (µ) ◦ P) z h0 (µ)(gµ−1 )00 (z h0 (µ)) 2q with ν1 (µ) = q+1 is continuously quantifiable in µ̂ = (0, p̂) at z = 1 by ξ(µ̂) = − min{p̂ − 1, 0} = 1 − p̂. In the case p̂ = 21 we have ξ(µ̂) = 21 . Then, applying Theorem F, we can assert that Crit((Πµ̂ , Xµ̂ ), Xµ ) 6 1 as desired. 4.2 Proof of Theorem B In this section we apply the techniques developed in Section 3 to contribute in the study of the bifurcation of critical periodic orbits of the family of dehomogenized Loud’s centers. We consider equation (3) with µ = (D, F ) inside the set Λ := {(D, F ) ∈ R2 : F > 1, D < 0, D + F > 0}. / {0, 1, 21 } then the function It is known (see [19] for instance) that if F ∈  Hµ (x, y) = (1 − x)−2F 1 2 2y − qµ (x) is a first integral of system (3), where qµ (x) = a(µ)x2 + b(µ)x + c(µ) with D D−F +1 F −D−1 a(µ) = , b(µ) = and c(µ) = . 2(1 − F ) (1 − F )(1 − 2F ) 2F (1 − F )(1 − 2F ) The line at infinity, the conic Cµ = { 21 y 2 − qµ (x) = 0} and the line {x = 1} are invariant curves of the system. If µ ∈ Λ then Cµ is a hyperbola that intersects the x-axis at p p −b(µ) − b(µ)2 − 4a(µ)c(µ) −b(µ) + b(µ)2 − 4a(µ)c(µ) x = p1 (µ) := and x = p2 (µ) := , 2a(µ) 2a(µ) with 0 < p1 (µ) < p2 (µ). For these parameters, the outer boundary of the period annulus of the center at the origin is formed by the union of the branch of the hyperbola Cµ passing through the point (p1 (µ), 0) and the line at infinity joining two hyperbolic saddles (see Figure 3). We are concerned with the criticality at the outer boundary of the period function for parameters µ̂ = (D̂, 2) ∈ Λ. As we advanced in the Introduction, by applying [8, Lemma 14] it follows that the change of coordinates z −F − 1 (u, v) = (φ(1 − x), (1 − x)−F y), with φ(z) := , F transforms the differential system (3) into the potential system ( u̇ = −v, 1  1  (24) v̇ = (F u + 1) (F u + 1)− F − 1 D(F u + 1)− F − D − 1 . This potential system has a non-degenerated center at the origin and the projection of its period annulus is the interval Iµ := (− F1 , ur (µ)), where (1 − p1 (µ))−F − 1 ur (µ) := . F 18 Figure 3: Phase portrait of (3) in the Poincaré disc for µ = (D, F ) ∈ Λ with D < −1 (left) and D > −1 (right), where for convenience we place the center at (0, 0) on the left of the centred invariant line {x = 1}. The polycycle Πµ at the outer boundary of the period annulus is the same in both cases: two hyperbolic saddles at infinity and the heteroclinic orbits between them. The invariant hyperbola Cµ is in boldface type. We point out that Iµ is the image by x 7→ φ(1 − x) of (−∞, p1 (µ)). Setting z = φ−1 (u) = (F u + 1)−1/F one can check that D 1+2D D+1 Vµ (u) = h0 (µ) − z −2F V0 (z, µ) with V0 (z, µ) = 2−2F z2 + 2F −1 z − 2F , Vµ0 (u) = z −F V1 (z, µ) with V1 (z, µ) = (z − 1)(D(z − 1) − 1), Vµ00 (u) = V2 (z, µ) with V2 (z, µ) = D(F − 2)z 2 − (2D + 1)(F − 1)z + F (D + 1), (25) (3) F 2 Vµ (u) = z V3 (z, µ) with V3 (z, µ) = −2D(F − 2)z + (2D + 1)(F − 1)z, (4) 2F Vµ (u) =z V4 (z, µ) with V4 (z, µ) = 2D(F 2 − 4)z 2 − (2D + 1)(F 2 − 1)z, F −D−1 where h0 (µ) := 2F (F −1)(2F −1) is the energy level at the outer boundary of the period annulus for all µ ∈ Λ. Before the proof of Theorem B a required technical result is shown. Its proof follows a similar argument than [24, Lemma 3.4]. Lemma 4.2. Let µ = (D, 2) ∈ Λ. If D 6= − 12 then Vµ0 (ur (µ))2 + D−1 00 6 Vµ (ur (µ)) 6= 0. D−1 Proof. Let us denote L(z, µ) := z −2F V1 (z, µ)2 + 6 V2 (z, µ). From the expressions in (25) we have D − 1 00 Vµ0 (u)2 + Vµ (u) = L(φ−1 (u)). 6 √ b+ b2 −4ac Then the result will follows once we show that L(z, µ) at z = φ−1 (ur (µ)) = 1 − p1 (µ) = 1 + 2a does not vanish for D 6= − 12 . We claim that if µ̂ = (D̂, 2) ∈ Λ is a zero of L(1 − p1 (µ), µ) then the derivative of the map D 7→ L(1 − p1 (µ), µ) is strictly negative at µ = µ̂. Indeed, first we notice that if µ̂ is a zero of L(1 − p1 (µ), µ) then (1 − D)V2 (1 − p1 (µ̂), µ̂) (1 − p1 (µ̂))−2F = . (26) 6V1 (1 − p1 (µ̂), µ̂) In addition,   d 2F V1 (z, µ)2 L(1 − p1 (µ), µ) = z −2F 2V1 (z, µ)∂z V1 (z, µ) − ∂D (1 − p1 (µ)) dD z z=1−p1 (µ) 1 + z −2F ∂D (V1 (z, µ)2 ) + ∂D ((D − 1)V2 (z, µ))|z=1−p1 (µ) . 6 d Substituting the equality (26) in the previous identity and evaluating at µ̂ = (D̂, 2), we obtain that dD L(1− p1 (µ̂), µ̂) can be written as an algebraic expression in D̂, p d r1 (µ̂) ∆(µ̂) + r2 (µ̂) L(1 − p1 (µ̂), µ̂) = p with ∆(µ) := (b2 − 4ac)(µ) dD r3 (µ̂) ∆(µ̂) + r4 (µ̂) 19 and ri ∈ R[µ] satisfying 2187 4 r1 (µ̂)2 ∆(µ̂) − r2 (µ̂)2 = − D̂ (D̂ + 1)3 (D̂ + 2)(108 − 486D̂ + 810D̂2 − 594D̂3 + 162D̂4 ) 2 and r3 (µ̂)2 ∆(µ̂) − r4 (µ̂)2 = −5668704D̂6 (D̂ − 1)2 (D̂ + 1)3 (D̂ + 2)3 . We point out that if D̂ ∈ (−2, 0) \ {−1} the previous two expressions does not vanish. Indeed, the first one d follows by Descartes’ rule of signs while the second is straightforward. This proves that dD L(1 − p1 (µ̂), µ̂) is well defined and non-vanishing for all µ̂ = (D̂, 2) ∈ Λ with D̂ 6= −1. In addition, one may explicitly check that d 1 L(1 − p1 (µ̂), µ̂) = − if µ̂ = (−1, 2). dD 12 d This shows that dD L(1 − p1 (µ̂), µ̂) is negative for all D̂ ∈ (−2, 0), proving the validity of the claim. The claim implies that the map D 7→ L(1 − p1 (µ), µ) as at most one zero for D ∈ (−2, 0) and F = 2. It is a simple computation to show that D = − 21 is a zero and this proves the result. Proof of Theorem B. The strategy of the proof will be to use Theorem F with n = 1 and µ̂ = (D̂, 2) with D̂ ∈ (−2, 0) \ {−1/2}. As in the previous section, in order to obtain the quantifier ξ of {(Dν 1 (µ) ◦  p p  P) z h0 (µ)(gµ−1 )00 (z h0 (µ)) }µ∈Λ at z = 1 in µ̂ we shall use the second part of [11, Theorem B]. In this case, for n = 1, it states that      α` αr 1 ξ(µ) = − min (µ), (µ) − ν1 (µ), (27) β` βr 2 where α` and αr are the quantifiers of "  21 ν(µ) # 1 Vµ 1 Ψµ (u) := 0 W , (h0 (µ) − Vµ )Vµ Rµ (u) 2 Vµ (u) h0 (µ) − Vµ (V 0 )2 −2Vµ V 00 with Rµ := µ (V 0 )3 µ , and β` and βr are the quantifiers of (h0 − Vµ ), at u = −1/F and u = ur (µ), µ respectively. The map ν : Λ → R is a continuous function to be determined in such a way that Ψµ is continuously quantifiable at u = −1/F for F ≈ 2. Firstly, in [24, Lemma 3.3] it was shown that D 2F −2 h0 (µ) − Vµ (u) ∼− F1 (F u + 1) F at µ̂ (28) 2 − 2F and h0 (µ) − Vµ (u) ∼ur (µ) Vµ0 (ur (µ))(ur (µ) − u) at µ̂ with Vµ0 (ur (µ)) 6= 0. With the notation introduced before, we have then 2 − 2F β` = and βr = −1. F Secondly, a computation shows that   ν(µ) 1 Vµ (u) 2 Ψµ (u) = p fµ (u), (29) 2 Vµ (u)Vµ0 (u)5 h0 (µ) − Vµ (u) where fµ := 4Vµ2 (Vµ − h0 (µ))Vµ0 Vµ000 − ((Vµ0 )2 − 2Vµ Vµ00 )((Vµ0 )2 (3Vµ + h0 (µ)(ν(µ) − 1)) − 6(Vµ − h0 (µ))Vµ Vµ00 ). To compute α` , taking (25) into account, one can verify with the help of an algebraic manipulator that z −6F fµ (u) = ψµ (z) , 2(F − 1)3 F 3 (2F − 1)3 z=φ−1 (u) 20 where ψµ is the sum of 25 monomials of the form κ(µ)z n1 +n2 F with ni ∈ Z for i = 1, 2, and κ : Λ → R a well-defined rational function at µ̂. Moreover, the monomial with highest exponent for µ ≈ µ̂ is D3 F (1 + D − F )2 (F − 2)(2F − 1)(2 + F (ν(µ) − 1) − ν(µ))z 6+4F . We point out that the coefficient of the previous monomial vanishes at F = 2 so ψµ is not continuously quantifiable at z = ∞ in µ̂ for arbitrary function ν(µ). In order to succeed with the continuous quantification, we must take F −2 ν(µ) := . F −1 In this case, the previous coefficient vanishes for all µ ≈ µ̂ and the monomial with highest exponent is D2 F (1 + 2D)(1 + D − F )2 (F 2 − 2F − 3)z 5+4F . Consequently, D2 (1 + 2D)(1 + D − F )2 (F 2 − 2F − 3) 2F −5 fµ (u) ∼− F1 (F u + 1) F at µ̂. 2F 2 (F − 1)3 (2F − 1) F −2 In addition, using the expressions in (25), we have that Vµ0 (u) ∼− F1 D(F u + 1) F . Using the previous quantifications together with (28) in the expression (29) we have 7−4F Ψµ (u) ∼− F1 C1 (µ)(F u + 1) F at µ̂, where   2(F F −2 (1 + 2D)(1 + D − F )2 (F 2 − 2F − 3) (2 − 2F )h0 (µ) −1) C1 (µ) := p . 4D3 F 2 (F − 1)3 (2F − 1) h0 (µ) D Finally, by simple computations from (29) using that u = ur (µ) is a regular value of Vµ and a simple zero of h0 (µ) − Vµ (u) with Vµ0 (ur ) 6= 0, we have that F −2 Ψµ (u) ∼ur (µ) C2 (µ)(ur (µ) − u) 2(F −1) at µ̂, where (D − 1)(6Vµ0 (ur (µ))2 + (D − 1)Vµ00 (ur (µ))) C2 (µ) := p . 12 3(1 − D)Vµ0 (ur (µ)) F −2 We point out that ν(µ) = F −1 and that 6Vµ̂0 (ur (µ̂))2 + (D − 1)Vµ̂00 (ur (µ̂)) does not vanish if D̂ 6= −1/2 by Lemma 4.2. In particular, 4F − 7 2−F α` = and αr = . F 2(F − 1)  p p  Consequently, using the equality in (27), we have that {(Dν(µ) ◦P) z h0 (µ)(gµ−1 )00 (z h0 (µ)) }µ∈Λ with −2 ν(µ) = F F −1 is continuously quantifiable at z = 1 in µ̂ by   4F − 7 F − 2 F −2 1 ξ(µ) = − min , − = . 2 − 2F 2(F − 1) 2(F − 1) F =2 2 The result follows then applying Theorem F with n = 1, proving that Crit((Πµ̂ , Xµ̂ ), Xµ ) 6 1 as desired. Acknowledgements The author want to thank Prof. Pavao Mardešić for the short stay of four months at the Institut de Mathématiques de Bourgogne, Dijon, that gave rise to the start of this work; and to Prof. Francesc Mañosas and Dr. Jordi Villadelprat for the fruitful discussions that led to some corrections of this paper. The author is partially supported by the MINECO/FEDER grants MTM2017-82348-C2-1-P and MTM2017-86795-C3- 1-P. 21 References [1] M. Abramowith, I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55. 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