Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most n⩾0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 0$$\end{document} critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family x¨=xp-xq,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{x}=x^p-x^q,$$\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} with 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}.


Introduction and setting of the problem
This paper is concerned with the period function of centers of planar differential systems. A singular point p of an analytic differential system ẋ = f (x, y), is a center if it has a punctured neighbourhood that consists entirely of periodic orbits surrounding p. The largest punctured neighbourhood with this property is called the period B J. Villadelprat jordi.villadelprat@urv.cat 1 annulus of the center and it will be denoted by P. Henceforth ∂P will denote the boundary of P after embedding it into RP 2 . Clearly the center p belongs to ∂P, and in what follows we will call it the inner boundary of the period annulus. We also define the outer boundary of the period annulus to be := ∂P\{ p}. Note that is a non-empty compact subset of RP 2 . The period function of the center assigns to each periodic orbits in P its period. Since the period function is defined on the set of periodic orbits in P, in order to study its qualitative properties usually the first step is to parametrize this set. This can be done by taking an analytic transverse section to the vector field X = f (x, y)∂ x + g(x, y)∂ y on P, for instance an orbit of the orthogonal vector field X ⊥ . If {γ s } s∈(0,1) is such a parametrization, then s −→ T (s) := {period of γ s } is an analytic map that provides the qualitative properties of the period function that we are interested in. In particular the existence of critical periods, which are isolated critical points of this function, i.e.ŝ ∈ (0, 1) such that T (s) = α(s −ŝ) k + o (s −ŝ) k with α = 0 and k 1. In this case we shall say that γŝ is a critical periodic orbit of multiplicity k of the center. One can readily see that this definition does not depend on the particular parametrization of the set of periodic orbits used.
The study of critical periodic orbits is analogous to the study of limit cycles, which are the main concern of the celebrated Hilbert's 16th problem (see [3,8,25,31] and references there in) and its various weakened versions. Questions related to the behaviour of the period function have been extensively studied by a number of authors. Let us quote for instance the problems of isochronicity (see [7,13,21]), monotonicity (see [4,5,27]) or bifurcation of critical periodic orbits (see [6,26,28]).
Our goal in the present paper is to study the bifurcation of critical periodic orbits from the outer boundary of the period annulus. Of course, as any bifurcation phenomenon, this occurs in case that X depends on a parameter, say μ ∈ ⊂ R d . Thus, for each μ ∈ , suppose that X μ is an analytic vector field on some open set U μ ⊂ R 2 with a center at p μ . Following the notation introduced previously, we denote by μ the outer boundary of its period annulus P μ . Concerning the regularity with respect to the parameter, we shall assume that {X μ } μ∈ is a continuous family, meaning that the map (x, y, μ) −→ X μ (x, y) is continuous in {(x, y, μ) ∈ R d+2 : (x, y) ∈ U μ , μ ∈ }. The present paper is addressed to study of the number of critical periodic orbits of X μ that can emerge or disappear from μ as we move slightly the parameter μ ≈μ. We call this number the criticality of the outer boundary and its precise definition is the following, where d H stands for the Hausdorff distance between compact sets of RP 2 . Definition 1.1 Consider a continuous family {X μ } μ∈ of planar analytic vector fields with a center and fix someμ ∈ . Suppose that the outer boundary of the period annulus varies continuously atμ ∈ , meaning that for any ε > 0 there exists δ > 0 such that d H ( μ , μ ) ε for all μ ∈ with μ −μ δ. Then, setting N (δ, ε) = sup # critical periodic orbits γ of X μ in P μ with d H (γ , μ ) ε and μ −μ δ , move μ ≈μ. The assumption that the period annulus varies continuously ensures that these changes do not occur abruptly. In this regard note that X μ = −y∂ x + (x + μx 3 + x 5 )∂ y , with μ ∈ R, form a continuous family of planar analytic vector fields with a center at the origin for which the outer boundary does not vary continuously at μ = 2. Indeed, the period annulus P μ is the whole plane for μ < 2, whereas is bounded for μ = 2 (see [22] for details).
Clearly the notion of criticality as defined in Definition 1.1 is meaningless in this situation. Definition 1. 2 We say thatμ ∈ is a local regular value of the period function at the outer boundary of the period annulus if Crit ( μ , Xμ), X μ = 0. Otherwise we say that it is a local bifurcation value of the period function at the outer boundary.
In this paper we develop tools that enable to bound the criticality at the outer boundary in case that we deal with a family of potential differential systems. These tools improve the ones that we obtained in [14] although to prove them we shall strongly rely on the results in that paper. In order to set all these results in a context we first recall some well-known facts about the period function of potential differential systems. So consider an analytic function V on some open interval I containing x = 0 such that V (0) = V (0) = 0 and V (0) > 0. Then the potential differential system X = −y∂ x + V (x)∂ y has a non-degenerated center at the origin with the periodic orbits inside the energy levels of the Hamiltonian function Clearly the inner boundary of P, i.e. the center at the origin, is inside the energy level h = 0. We say, by abuse of language, that the outer boundary of P is "inside" the energy level h = h 0 . (We remark that is a subset of RP 2 which may have points outside the vertical strip I × R, the domain of the Hamiltonian function.) The period T (h) of the periodic orbit γ h inside the energy level {H (x, y) = h} is given by the Abelian integral It is well known that T is an analytic function on (0, h 0 ) that, if the center is non-degenerated, can be extended analytically to h = 0. Its derivative T (h) is also given by an Abelian integral and we are interested in its zeros near h = h 0 , which correspond to critical period orbits near . Suppose now that we deal with a family of potential differential systems {X μ } μ∈ and that the problem is to compute Crit ( μ , Xμ), X μ for a givenμ ∈ . Note firstly that the energy level h 0 at the outer boundary depends on μ. The tools that we developed in [14] allow to tackle the problem in the following two situations: • either h 0 (μ) = +∞ for all μ ≈μ, • or h 0 (μ) < +∞ for all μ ≈μ.
(We do not treat the case in which in any neighbourhood ofμ there are μ 1 and μ 2 with h(μ 1 ) = +∞ and h(μ 2 ) < +∞.) For each one of these two situations, we gave a theoretical result for Crit ( μ , Xμ), X μ = 0 and another one for Crit ( μ , Xμ), X μ 1. It is to be noted that the approach we followed to prove those results, as well as their assumptions, are very different for the cases h 0 = +∞ and h 0 < +∞. In this paper we go further and prove Theorems A and B, addressed to cases h 0 = +∞ and h 0 < +∞, respectively, in which we give sufficient conditions in order that Crit ( μ , Xμ), X μ n for n ∈ N ∪ {0}. The idea in both cases is to find functions φ i μ (h), i = 1, 2, . . . , n, verifying that there exist δ, ε > 0 such that if μ −μ < δ, then (φ 1 μ , φ 2 μ , . . . , φ n μ , T μ ) is an extended complete Chebyshev system (ECT-system for short, see Definition 2.1) on the interval (h 0 (μ) − ε, h 0 (μ)). This implies in particular that T μ (h) has at most n zeros for h ∈ (h 0 (μ) − ε, h 0 (μ)), counted with multiplicities, for all μ ≈μ and, accordingly, Crit ( μ , Xμ), X μ n. We choose different type of functions φ i μ for the cases h 0 = +∞ and h 0 < +∞, but in both situations we take them simple enough in order that (φ 1 μ , φ 2 μ , . . . , φ n μ ) is an ECT-system on (0, h 0 (μ)). Taking this into account, the problem is then to guarantee that the Wronskian (see Definition 2.2) of (φ 1 μ , φ 2 μ , . . . , φ n μ , T μ ) is non-vanishing near h = h 0 (μ) for all μ ≈μ. Theorems A and B can be compared to the results obtained in the series of papers [17][18][19][20] by Mardešić et al. because both studies deal with the bifurcation of critical periodic orbits from the outer boundary of the period annulus. However striking differences exist. The first one is that their results apply to differential systems which need not be potential, but on the other hand their approach requires that the differential system has a meromorphic extension to (for instance, starting with a polynomial system and making its Poincaré compactification). The second one is due to the fact that we bound the criticality by embedding the derivative of the period function in an ECT-system, whereas their approach is to obtain the asymptotic expansion of the period function near the outer boundary and then compute the coefficient of the principal term. The testing ground for the results by Mardešić et al. is the family of Loud's centers. In this regard it is to be pointed 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 techniques. We expect to exploit this in a forthcoming paper that we hope will prove some aspects of the conjectural bifurcation diagram of the period function of the Loud's centers proposed in [19].
Our testing ground is the two-parametric family of potential differential systems given by ẋ = −y, which has a non-degenerated center at the origin for all μ := (q, p) varying inside := {(q, p) ∈ R 2 : p > q}. Note that, for each μ ∈ , X μ : We became interested in this family because of the previous results by Miyamoto and Yagasaki in [23] concerning the monotonicity of the period function for q = 1 and p ∈ N. Later Yagasaki improved the result showing in [30] the monotonicity of the period function for q = 1 and any real number p > 1. (Another proof of this result can be found in [2].) We studied afterwards the whole family {X μ } μ∈ in [14,15].
To be more precise, in [15] 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 the isochronous centers. In [14] we studied the criticality of the outer boundary and it is precisely the result we obtained there for the family (1) the one that we seek to improve here. In short, see Fig. 1, we proved that Crit ( μ , Xμ), X μ = 0 ifμ ∈ \{ B ∪ U } and Crit ( μ , Xμ), X μ 1 ifμ ∈ B . Without going into detail for the sake of shortness, we proved moreover that the criticality is exactly one for parameters inside two segments in B .
By applying the general tools developed in this paper we can go further and prove the Theorem C Let {X μ } μ∈ be the family of potential vector fields in (1) and consider the period function of the center at the origin. Then the following hold: It is worth mentioning that the centers corresponding to the parameters μ = (− 1 2 , 0) and μ = (0, 1) are isochronous. In this regard the result in (a) and (b) is optimal for these particular parameters.
The paper has three additional sections organized in the following way. In Sect. 2 we introduce the notions of ECT-system and Wronskian and recall well-known properties that relate them. Furthermore we obtain the analytical tools that we shall later use to prove the results about the criticality, which is done in Sect. 3. To be more precise, we treat the case h 0 = +∞ and prove Theorem A in Sect. 3.1, whereas we consider the case h 0 < +∞ and show Theorem B in Sect. 3.2. Finally, to illustrate its applicability we prove Theorem C in Sect. 4.

Previous Analytic Results
This section is devoted to obtain the technical tools that we shall later use to prove the results concerning the criticality. To this end let us take a ∈ R + ∪ {+∞} and consider the integral operator Here, and in what follows, C ω ([0, a)) stands for the set of analytic functions on (0, a) that can be analytically extended to x = 0. The reason why we are interested in this operator is because we can relate it with the derivative of the period function of X = −y∂ x + V (x)∂ y . Indeed, it is well known (see for instance [4,16]) that the period T (h) of the periodic orbit γ h inside the energy level { 1 2 y 2 + V (x) = h} is given by . Next we recall the notions of Chebyshev system and Wronskian, that will be very useful 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). Then . . .
These two notions are closely related by the following well-known result (see for instance [12]). Our goal is to complete F [ f ] with some analytic functions g 0 , . . . , g n−1 in order that (g 0 , . . . , g n−1 , F [ f ]) form an ECT-system on (a − ε, a) for some ε > 0. In particular this will imply that F [ f ](x), and so the derivative of the period function, has at most n isolated zeroes for x ≈ a counted with multiplicities. Then to obtain the desired upper bounds on the criticality, the delicate point will be as usual to guarantee the uniformity with respect to the parameters of the system. Thus we aim to find sufficient conditions in terms of f in order that F [ f ] can be embedded into an ECT-system. These conditions will be formulated using the notions that we introduce next:

Definition 2.4 Let f be an analytic function on I = (a, b).
We say that f is quantifiable at b by α with limit in case that: x α = and = 0. We call α the quantifier of f at b. We shall use the analogous definition at a.

Definition 2.5
Let be an open subset of R d and suppose that, for each μ ∈ , f μ is an analytic function on some real interval I μ . Suppose also that the map ( for all μ ∈ U and, moreover, x α(μ) = and = 0. We shall use the analogous definition for the left endpoint of I μ .
Remark 2.7 Notice that the map α : U −→ R that appears in the previous definition must be continuous atμ, otherwise there exists a sequence {μ n } n∈N such that lim n→∞ α( x α(μ) is finite and different from zero by definition.
Proof The result follows by using Hôpital's Rule and the uniqueness of the quantifier (see Remark 2.7).

Definition 2.9
Let f be an analytic function on [0, +∞). Then, for each n ∈ N, we call Following the previous definitions and notation, the next result gathers Theorems 2.13 and 2.17 in [14].

Theorem 2.10 Let be an open subset of R d and consider a continuous family
The following assertions hold: 1). In this case: We point out that the hypothesis Given ν 1 , ν 2 , . . . , ν n ∈ R, we define the linear ordinary differential operator given by Here, and in what follows, for the sake of shortness we use the notation ν n = (ν 1 , . . . , ν n ). Furthermore we define L ν 0 = id in order that the statements of the next results contemplate the case n = 0 as well. The rest of the present section is devoted to study under which conditions the quantifier To this end some previous technical results about Wronskians are needed. The first two lemmas are well known (see, respectively, [16] and [11,24]).

Lemma 2.11
Let f 0 , f 1 , . . . , f n−1 be analytic functions. Then the following statements hold: for any analytic function g.

Lemma 2.13
Given ν 1 , ν 2 , . . . , ν n ∈ R, the following identity holds: Proof We prove the result by induction on n. Since the base case n = 1 is obvious, let us show the induction step. By applying (b) in Lemma 2.11 we get Let us denote β i := ν i − ν n for shortness. Then, using well-known properties of the determinant and the induction hypothesis, Consequently, substituting the previous equality in (5), we have where we used β i = ν i − ν n in the first equality and the second one follows by means of some easy manipulations. This shows the induction step and so the result is proved.
The previous lemma enables to write the differential operator under consideration as a quotient of Wronskians. Indeed, if ν 1 , ν 2 , . . . , ν n are pairwise distinct, then we have that At this point it is worth noting that the linear ordinary differential operator has already appeared in the literature in relation with the so called "Chebyshev asymptotic scales" (see [9,10] and references therein). Of course, it is also related to the divisionderivation algorithm (see [25] for instance) due to the fact that its kernel is spanned by Our next result shows that the integral operator F and the differential operator L ν n commute. This fact is the key point in order to prove our main results. Proposition 2.14 For any given f ∈ C ω ((0, +∞)) and ν 1 , . . . , ν n ∈ R, the following recurrence holds: where c 1 := 1 and c n : In particular, if f can be extended Proof We can suppose that ν 1 , ν 2 , . . . , ν n are pairwise distinct, otherwise there is nothing to be proved. The case n = 1 of the recurrence is straightforward because, by definition, Let us show now the case n 2. To this end take any k ∈ {1, 2, . . . , n − 1} and note that, by Lemma 2.12, Hence, some easy computations taking Lemma 2.13 into account show that By definition, see (4), we have on the other hand that Then, using (6) and the above equality, after some computations we get Thus, taking k = n − 1 we obtain the recurrence in the statement for n 2.
Let us turn to the proof of F • L ν n = L ν n • F . We show it by induction on n 0 taking advantage of the recurrence we have just proved. The base case n = 0 is clear because L ν 0 = id. To show the induction step take any g ∈ C ω ((0, +∞)) and note that Thus, deriving the induction hypothesis, we get where in the second equality we use twice the recurrence, taking f = g and f = F [g], in the third one the linearity of F , and in the fourth one the induction hypothesis. Hence and so the induction step follows. This concludes the proof of the result.

Lemma 2.15 Let f be an analytic function on
where c 1 := 1 and c n := n−1 i=1 (ν n − ν i ) for n 2.
Proof By using the recurrence in Proposition 2.14 and the definition of the momentum, Moreover, by Proposition 2.14, L ν n−1 [ f ] is analytic at x = 0. So integrating by parts we get and this proves the result.
We point out that the assumption Recall in addition that, by definition, L ν 0 (μ) = id. Thus Proposition 2.16 with n = 0 gives Theorem 2.10 as a particular case.

Criticality of the Period Function at the Outer Boundary
This section is devoted to prove the two main theoretical results of the paper. We consider analytic potential differential systems ẋ = −y, depending on a parameter μ ∈ ⊂ R d . Here, for each fixed μ ∈ , V μ is an analytic function on a certain real interval I μ that contains x = 0. In what follows sometimes we shall also use the vector field notation X μ := −y∂ x + V μ (x)∂ y to refer to the above differential system. We suppose V μ (0) = 0 and V μ (0) > 0, so that the origin is a non-degenerated center and we shall denote the projection of its period annulus P μ on the x-axis by I μ = (x (μ), x r (μ)). Thus x (μ) < 0 < x r (μ). The corresponding Hamiltonian function is given by H μ (x, y) = 1 2 y 2 + V μ (x), where we fix that V μ (0) = 0, and we set the energy level of the outer boundary of P μ to be h 0 (μ), i.e. H μ (P μ ) = (0, h 0 (μ)). Note then that h 0 (μ) is a positive number or +∞. In addition we define (2) and (3), the period for all h ∈ (0, √ h 0 (μ)). Finally, it is also well known that T μ is an analytic function on (0, h 0 (μ)) which can be analytically extended to h = 0. Definition 3.1 Following the notation introduced just before, we say that the family of potential analytic differential systems {X μ } μ∈ verifies the hypothesis (H) in case that: be a family of potential analytic differential systems verifying (H). Then the outer boundary of its period annulus varies continuously in the sense of Definition 1.1. Indeed, to show this let γ h,μ be the periodic orbit of X μ inside the energy level which tends to zero as h → h 0 (μ) and μ →μ thanks to the hypothesis (a) and (d) in (H).
Next two results are proved in [14].
Next two sections are concerned with the criticality at the outer boundary of potential systems verifying the hypothesis (H). Section 3.1 is devoted to prove Theorem A, that deals with the case h 0 ≡ +∞, whereas in Sect. 3.2 we prove Theorem B, that tackle the case in which h 0 is finite.

Potential Systems with Infinite Energy
In this section we shall study the criticality at the outer boundary of the period annulus for families of potential systems such that h 0 (μ) = +∞ for all μ ∈ . The idea is to take a nonvanishing function f and find sufficient conditions in order that f T μ can be embedded into the simplest ECT-system we can consider, namely (h ν 1 (μ) , h ν 2 (μ) , . . . , h ν n (μ) ). We precise this in the following result. Proof Note first that ν 1 , ν 2 . . . , ν n−1 must be pairwise distinct at μ =μ because (μ) = 0. Thus, by continuity, for each k = 1, 2, . . . , n − 1 we have that W [h ν 1 (μ) , . . . , h ν k (μ) ] = 0 for all h > 0 and μ ≈μ. On the other hand, by the uniformity of the limit as h tends to +∞ and the assumption (μ) = 0, there exist M > 0 and a neighbourhood U ofμ such that Then, by Lemma 2.3 we can assert that (h ν 1 (μ) , . . . , h ν n−1 (μ) , f (h)T μ (h)) is an ECT-system on (M, +∞) for all μ ∈ U. In particular, since f is a unity, T μ has no more than n − 1 isolated zeros on (M, +∞) for μ ≈μ, counted with multiplicities. We claim that this implies Crit ( μ , Xμ), X μ n −1, see Definition 1.1. To show this notice first that, by Remark 3.2, the outer boundary of the period annulus varies continuously. Suppose, by contradiction, that there exist n sequences {γ k μ i } i∈N , k = 1, 2, . . . , n, where each γ k μ i is a critical periodic orbit of X μ i , such that μ i →μ and d H (γ k μ i , μ ) → 0 as i → +∞. Then, due to we have that d H (γ k μ i , μ i ) tends to zero as i → +∞. This contradicts that, for all μ ∈ U, T μ has no more than n − 1 isolated zeros on (M, +∞). So the claim is true and the result follows.
The proof of the following result is a straightforward application of [14,Lemma 3.5]. For the sake of brevity we omit it here but we refer the reader to the proof of Lemma 3.11, which follows similarly.
We can now state our result concerning the criticality at the outer boundary for the case h 0 ≡ +∞. In its statement, and from now on, for a given function f : (−a, a) . Let us also remark that the assumption requiring the existence of functions ν 1 , ν 2 , . . . , ν n is void in case that n = 0.
Since ξ(μ) is the quantifier of {L ν n (μ) [ f μ ]} μ∈ at +∞, by applying Proposition 2.16 we can assert that {(L ν n (μ) • F )[ f μ ]} μ∈ is continuously quantifiable inμ at +∞ by ν n+1 (μ) := ξ(μ), in cases (a) and (b2), and by ν n+1 (μ) := 1 − 2 j, in case (b1). Then, taking account of the definition of L ν n (μ) , see (4), in these cases we get that , by applying Lemma 3.5 we have that Crit ( μ , Xμ), X μ n, as desired. This proves the first part of the result. Let us turn now to the proof of second part of the result. With this aim in view we note that if φ is any analytic function on (−a, a), then for all x ∈ (0, a). Let us set (μ) 3 and V g −1 (z) = z 2 , by applying Lemma 2.11 some computations show that at +∞ and at −∞ will "generically" determine the quantifier of L ν n (μ) [ f μ ] at +∞. Henceforth, for the sake of shortness, we omit the unessential dependence with respect to μ. On account of (i) and (ii) it follows that {S μ } μ∈ is continuously quantifiable respectively. Finally, again from (8) . This completes the proof of the result because one can easily verify that (n+1)(n+2)

Potential Systems with Finite Energy
In this section we shall study the criticality at the outer boundary of the period annulus for families of potential systems with h 0 (μ) < +∞ for all μ ∈ . If we proceed the same way as for the case h 0 = +∞, we would take an appropriate non-vanishing function f and try to embed f T μ into some easy ECT-system. To this end the natural candidate is ). However we did not succeed with such an approach. Instead we shall take advantage of Proposition 2.16, which is in fact addressed to the case h 0 = +∞. This forces us to "translate" the case h 0 < +∞ to the case h 0 = +∞ and gives rise to some technicalities that make things more complicated than it should be. With this aim in view we define next a differential operator which is conjugated to L ν n . The conjugation is precisely the tool that enables us to translate the case h 0 < +∞ to the case h 0 = +∞ and apply Proposition 2.16. Given ν 1 , . . . , ν n ∈ R, in this section we consider the linear ordinary differential operator where as usual we use the notation ν n = (ν 1 , . . . , ν n ) and In addition we define D ν 0 := id for the sake of convenience. Setting we also consider the operator B : We will show next that B conjugates D ν n and L ν n . This fact eventually will enable us to take advantage of Proposition 2.16. Before proving it we introduce the following definition.
the n-th momentum of f , whenever it is well defined.
Lemma 3.8 Consider ν 1 , ν 2 , . . . , ν n ∈ R. Then the following hold: Proof Let us show (b) because (a) follows straightforward. So take f ∈ C ω (0, 1) and note that ) by definition. On the other hand, by applying Lemma 2.11 we get where in the second equality we use (a) and that xφ ( . Consequently, since on account of (a) we have ψ ν φ(x) = x ν 1−φ(x) 2 , the combination of the two previous indented equalities gives as desired. Let us turn now to the proof of (c). Take any s ∈ (0, 1) and h ∈ C ω ([0, 1)) and note that the change of variable u = s sin θ gives If f is any analytic function on [0, 1), then performing the change of variable z = φ(x) it follows that Accordingly, by applying above the equality in (11) with h = f and h = B[ f ], we get Finally the composition with B on both sides of this equality and an easy computation yields to which shows (c). Finally let us prove (d). If f is an analytic function on [0, 1), then by means of the change of variable z = φ(x) once again we get as desired. This completes the proof of the result. [0, 1).

Lemma 3.9 Let { f μ } μ∈ be a continuous family of analytic functions on
Therefore, for a given μ ∈ , where we used that φ −1 (z) = z √ 1−z 2 . Since the first limit is different from zero if and only if the last one is different from zero, the result follows.
We shall bound the criticality at the outer boundary by means of the following result. Lemma 3.10 Let {X μ } μ∈ be a family of potential analytic differential systems verifying (H) and such that μ −→ h 0 (μ) is continuous on . Assume that there exist n 1 continuous functions ν 1 , ν 2 . . . , ν n in a neighbourhood of some fixedμ ∈ and an analytic nonvanishing function f on (0, 1) such that uniformly in μ ≈μ, and (μ) = 0. Then Crit ( μ , Xμ), X μ n − 1.
Proof We show the result for z = 1 (the case z = −1 follows exactly the same way). By Lemma 3.4, we know that g −1 μ (z √ h 0 (μ)) tends to x r (μ) uniformly on μ as z −→ 1. Therefore, since g 2 uniformly on μ. Taking anyμ ∈ , this shows that and so the result follows.
The following is our main result in order to study the criticality of the outer boundary in case that its energy level is finite. As usual we point out that, in its statement, the assumptions requiring the existence of functions ν 1 , ν 2 , . . . , ν n for n = 0 and that N 1 ≡ N 2 ≡ . . . ≡ N j−1 ≡ 0 for j = 1 are void.
Theorem B Let {X μ } μ∈ be a family of potential analytic systems verifying (H) such that h 0 (μ) < +∞ for all μ ∈ . Assume that there exist n 0 continuous functions ν 1 , ν 2 , . . . , ν n in a neighbourhood of some fixedμ ∈ such that the family , whenever it is well defined. The following assertions hold: 2 + 1 if the following conditions are verified: with limits b (μ) and b r (μ), respectively, , the function is continuously quantifiable at x (μ) by α (μ) and at x r (μ) by α r (μ) with limits a (μ) and a r (μ), respectively, (iii) and either α β (μ) = α r β r (μ) or, otherwise, a r (b r ) h 0 (μ)) for shortness. Then, the hypothesis (H) and Lemma 3.3 guarantee that { f μ } μ∈ is a continuous family of analytic functions on [0, 1). Furthermore, see (7), recall that √ . Hence the obvious rescaling yields to the identity So we must show that there exist ε > 0 and a neighbourhood U ofμ such that F [ f μ ](z) has at most n zeros for z ∈ (1 − ε, 1), multiplicities taking into account, for all μ ∈ U. Recall that, by (b) in Lemma 3.8, B • D ν n = L ν n • B. This will allow us to transfer the assumptions on the family {D ν n (μ) [ f μ ]} μ∈ , which is defined on [0, 1), to another family defined on [0, +∞) and then apply Proposition 2.16 as we did in the proof of Theorem A. With this aim in view we first note that where we use (c) in Lemma 3.8 in the first equality, the identity We claim that if (L ν n (μ) • F • B)[ f μ ] μ∈ is continuously quantifiable at +∞ inμ, then the criticality of X μ at the outer boundary of the period annulus is at most n. Indeed, to show this suppose that the quantifier is η(μ). Then, on account of the previous equality and Lemma 3.9, D ν n (μ) ( Thus, according to the definition of D ν n in (9), where κ(μ) := 1 2 η(μ) + (n + 1)(n + 2) + n i=1 ν i (μ) . Now the claim follows by applying Lemma 3.10 and taking (12) into account.
We are now in position to prove (a) and (b) in the first part of the statement. To this end recall that, by assumption, the family D ν n (μ) [ f μ ] μ∈ is continuously quantifiable in at z = 1 by ξ(μ). On the other hand, by (b) in Lemma 3.8, Hence by applying Lemma 3.9 we can assert that the family (L ν n (μ) • B)[ f μ ] μ∈ is continuously quantifiable in at +∞ by 2ξ(μ) − 2. (This is precisely the family defined on [0, +∞) that in the beginning of the proof we refer to.) • If ξ(μ) > 1 2 , then 2ξ(μ) − 2 > −1 and so Proposition 2.16 applied to {B[ f μ ]} μ∈ guarantees that (L ν n (μ) • F • B)[ f μ ] μ∈ is continuously quantifiable in a neighbourhood ofμ at +∞ by 2ξ(μ) − 2. This, thanks to the previous claim, proves that Crit ( μ , Xμ), X μ n and hence (a) follows. • To show (b) we use that, by (b) and (d) in Lemma 3.8, Then the result follows straightforward by applying Proposition 2.16 and taking the previous claim into account again.
Let us turn next to the proof of the second part of the result. Set (μ) := n i=1 (ν i (μ) − i) for the sake shortness. For the same reason, from now on we omit the dependence on μ when it is not essential. That being said note that, due to V μ (g −1 μ (z)) = z 2 , . Consequently, taking Lemma 2.11 also into account, some computations show that Similarly, due to g(z) = − √ V (z) for z < 0, for all z ∈ (0, 1) we have that for all z ∈ (0, 1). On account of the assumptions in (i) and (ii), we can assert that {S μ } μ∈ is continuously quantifiable at x by γ := α + β ( 2 + 3n(n+1)

Application
In this section we resume the study that we began in [14] for the family of potential differential systems {X μ } μ∈ , where and = {(q, p) ∈ R 2 : p > q}. Following the previous notation, we define We will prove Theorem C to illustrate the application of the theoretical results we have obtained so far. Before we need to show several lemmas. The first one will in particular ensure the uniqueness of p 0 as introduced in the statement of Theorem C. For the parameter values under consideration in Theorem C, it is easy to show that the projection of the period annulus on the x-axis is I μ = (−1, ρ(μ)), with ρ(μ) := p + 1 q + 1 1 p−q − 1, and that the energy level at the outer boundary is h 0 (μ) := p−q ( p+1)(q+1) . Then, with notation introduced in Section 3.2, we have the following result: Proof For the sake of shortness we shall omit the nonessential dependence on μ. That being said, the change of variable x = g −1 (z √ h 0 ) gives formally (The fact that this expression is different from zero follows by (b) in Lemma 4.1.) Accordingly, by the second part of Theorem B, the family P[z √ h 0 (μ)(g −1 μ ) (z √ h 0 (μ))] μ∈ is continuously quantifiable inμ at z = 1 by ξ(μ) = max{ q q+1 , −1} + 1. We apply next the first part of Theorem B. To this end note that ξ(μ) = 0 and that, by (a) in Lemma 4.4, the first momentum of the even part of z −→ z h 0 (μ)(g −1 μ ) (z h 0 (μ)) does not vanish. Then, the application of (b1) in Theorem B with n = 0 and j = 1 shows that Crit ( μ , Xμ), X μ = 0. This proves the validity of (a).