The Period Function of Hamiltonian Systems with Separable Variables

In this paper we study the period function of those planar Hamiltonian differential systems for which the Hamiltonian function H(x, y) has separable variables, i.e., it can be written as H(x,y)=F1(x)+F2(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(x,y)=F_1(x)+F_2(y)$$\end{document}. More concretely we are concerned with the search of sufficient conditions implying the monotonicity of the period function, i.e., the absence of critical periodic orbits. We are also interested in the uniqueness problem and in this respect we seek conditions implying that there exists at most one critical periodic orbit. We obtain in a unified way several sufficient conditions that already appear in the literature, together with some other results that to the best of our knowledge are new. Finally we also investigate the limit of the period function as the periodic orbits tend to the boundary of the period annulus of the center.


Introduction and Definitions
The present paper deals with the class of planar Hamiltonian differential systems where the Hamiltonian function has separable variables, i.e., it has the special form H (x, y) = F 1 (x) + F 2 (y).
For i = 1, 2 we suppose that F i (z) is an analytic function on R with a local minimum at z = 0, so that ẋ = −F 2 (y), has a critical point at the origin of center type. Recall that a critical point p of a planar differential system is a center if it has a punctured neighbourhood that consists entirely of periodic orbits surrounding p. The period annulus is the largest punctured neighbourhood with this property and we shall denote it by P. The solution curves of (1) are inside the energy levels of H . We can assume without loss of generality that F i (0) = 0 and, accordingly, It is easy to see on the other hand that the energy level of the Hamiltonian parametrizes the set of periodic orbits in P. Hence for each h ∈ (0, h 0 ) we denote by γ h the periodic orbit of P inside the energy level H = h. We are concerned with the period function of the center, which assigns to each periodic orbit in P its period. In order to study it we consider h −→ T (h) := period of γ h , that can be written as for each h ∈ (0, h 0 ).
(Here, and in what follows, we take the oval γ h clockwise oriented.) This is an analytic map that provides the qualitative properties of the period function that we are interested in. Particularly the existence of critical periods, which are isolated critical points of this function, i.e., those valuesĥ ∈ (0, h 0 ) such that T (h) = α(h −ĥ) k + o((h −ĥ) k ) with α = 0 and k 1. In this case we shall say that γĥ 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 in P used. We say that the period function of the center is monotonous increasing (respectively, decreasing) if T (h) is strictly positive (respectively, negative) for all h ∈ (0, h 0 ). The problem of bounding the number of critical periodic orbits is analogous to the problem of bounding the number of limit cycles, which is related to the well known Hilbert's 16th Problem (see [1,8,26,34] and references therein) 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 [6,17], monotonicity [2,30] or bifurcation of critical periodic orbits [5,27].
The paper is organized as follows. In Sect. 2 we prove some auxiliary tools that will be used henceforth. More concretely, we first use the Gelfand-Leray derivation formula to obtain an expression for T (h) given in terms of an Abelian integral, see Lemma 2.3. Next we prove a general result, namely Proposition 2.7, that enables us to write this type of Abelian integral more conveniently in order to take advantage of the involutions associated to the Hamiltonian. Section 3 is devoted to obtain sufficient conditions for the monotonicity of the period function.
To this aim we begin by proving Proposition 3.2, which particularized yields to well-known monotonicity conditions, see Corollaries 3.4 and 3.5, that were previously obtained by other authors (see [2,9,25,32]). Proposition 3.7 is the main result in Sect. 3 and to the best of our knowledge it constitutes a new result. Section 4 is addressed to the problem of uniqueness of critical periodic orbits and in this regard we prove Theorems 4.1 and 4.11, which provide conditions implying the existence of at most one critical periodic orbit. Finally Sect. 5 deals with the limit of the period function T (h) as h tends to the endpoints of its domain (0, h 0 ), see Theorem 5.1. This has been studied previously by several authors under different settings (see [4,7,15,18,31] and references therein). Our contribution, Theorem 5.1, is motivated by a property used by Kaplan and Yorke [13] in their proof on the existence of periodic solutions of differential-delay equations. Once we prove our result we will make some comments concerning a delicate point in their proof that we think did not receive the required attention, see Remark 5.2.

Auxiliary Results
Let us fix that F i (z) = α i z k i + o(z k i ) for some even number k i and positive real number α i . In addition, let I 1 and I 2 denote, respectively, the projection of P on y = 0 and x = 0. Note Each F i defines an analytic involution σ i on I i by means the relation Recall that a function σ is said to be an involution if σ • σ = I d and σ = I d. In this respect observe that σ i (0) = 0 and σ i (z) < 0 for all z ∈ I i . In the statement of our first result, and in what follows, the multiplicity of an analytic function f at x =x is denoted by mult( f ,x).

Lemma 2.1 Let a be an analytic function on I
Proof Let R(x) and S(y) be analytic functions on I 1 and I 2 , respectively. If (x, y) ∈ γ h then and, consequently, Proof This follows by the Gelfand-Leray derivation formula (see for instance [12,Theorem 26.32]), which asserts that In all over the paper for the sake of shortness we will use the notation The period function of the center at the origin of the differential system (1) verifies where in the second equality we applied Lemma 2.1. Then by Lemma 2.2 we can assert that Thus and so the result follows.
Taking advantage of the previous lemmas we can already obtain an expression that enables to study the monotonicity of the period function near a non-degenerate center.
Proof For i = 1, 2, let us define a i := i − 1 2 for the sake of shortness. Then by Lemma 2.3 we can write where in the second equality we applied Lemma 2.1 taking a 1 (0) = 0 and F 1 (0) = 0 into account. Therefore, since a i F i (z) is analytic at z = 0 for i = 1, 2, we can apply Green's Theorem (see [28]) to obtain where Int(γ ) stands for the bounded connected component of R 2 \{γ }. (We shall use this notation hereafter.) Now, since one can readily verify that a i F i , the result follows.

Definition 2.5
Let σ be an analytic involution and consider a given function f . The σ -even part of f is We say that f is σ - Observe that this definition coincides with the usual notion of evenness in case that σ = −I d. Likewise we have the following characterization of σ -evenness.
Lemma 2.6 f is σ -even if, and only if, there exits g such that f = P σ (g).

Proof
The necessity is obvious. In order to see the sufficiency, assume that f (z) = 1 2 g(z) − g(σ (z))σ (z) for some g. Then

Proposition 2.7
Let γ be an oval, clockwise oriented, inside the energy level F 1 (x)+ F 2 (y) = h and define where a and b are analytic functions on I 1 = (e − 1 , e + 1 ) and I 2 = (e − 2 , e + 2 ), respectively. Then, if we denote the first quadrant by Q 1 , the following hold: Proof Let us split the given oval as the concatenation γ = γ 1 + γ 4 + γ 3 + γ 2 , where γ i is the intersection of γ with the i-th quadrant. Define ψ 1 (x, y) := σ 1 (x), y and ψ 2 (x, y) := x, σ 2 (y) . Due to σ i (z) < 0 for all z ∈ I i , note that ψ 1 and ψ 2 are analytic diffeomorphisms on P that reverse orientation. On account of this, and the fact that Hence, on account of Definition 2.5, On the other hand, since ψ 2 (γ 4 ) = γ 1 , the change of coordinates (x, y) = ψ 2 (u, v) yields where in the second equality we used that F 2 (y) = F 2 (σ 2 (y))σ 2 (y). Therefore, from (2), and this proves (a). Note in this respect that, thanks to the above equality, in the definition of C we can replace a by P σ 1 a and b by P σ 2 b because, on account of Lemma 2.6, P σ P σ ( f ) = P σ ( f ) for any f . Let us turn next to the proof of (b).
Thus by applying Lemma 2.1 we get where B = b, and then the application of Green's Theorem (see for instance [28]) yields Similarly as before, we split On the other hand, since ψ 2 (R 4 ) = R 1 , the change of coordinates (x, y) = ψ 2 (u, v) yields Finally taking Definition 2.5 into account once again we get as desired, and this concludes the proof of the result.

Monotonicity Results for the Period Function
. Then the period function of the center at the origin of the differential system (1) is monotonous increasing (respectively, decreasing).

Proof By applying Lemma 2.3 and (a) in Proposition 2.7 we have that
where recall that Q 1 stands for the first quadrant. Since and, on the other hand, Next result is addressed to non-degenerate centers and it is in fact a consequence of Proposition 3.1 because it provides a sufficient condition in order that P σ i i − 1 2 does not vanish.

Proposition 3.2 Assume that the center at the origin is non
is positive (respectively, negative) on (0, e + i ) for i = 1, 2, then the period function of the center at the origin of the differential system (1) is monotonous increasing (respectively, decreasing).
Proof The result will follow by applying Proposition 3.1 with Observe in addition that, since is an analytic function at z = 0 and Consequently, from (3) and on account of F i (z) > 0 for all z ∈ (0, e + i ), a sufficient condition for P σ i i − 1 2 to be positive (respectively, negative) on (0, e + i ) is that is positive (respectively, negative) for z ∈ (0, e + i ). This proves the result because i − 1 In general one cannot expect to have the explicit expression of the involutions σ i and this certainly diminishes the applicability of Propositions 3.1 and 3.2. There are however situations where we can bypass this obstruction and study effectively P σ i ( i − η i ) without knowing explicitly σ i . This can be done for instance in case that F i are algebraic functions with the aid of the multipolynomial resultant (see [10,11,20,21] for examples of application of this approach). Alternatively one can seek (explicit) conditions on F i implying that P σ i ( i − η i ) is non-vanishing. This is in fact the underlying idea in the monotonicity criterion obtained by Schaaf [31]. More concretely he showed that if the conditions . Interestingly enough, as the author points out, the first sufficient condition is related to the Schwarzian derivative. Later on Rothe [25] extended and studied systematically this type of sufficient conditions. Next we shall obtain two of these sufficient conditions in Corollaries 3.4 and 3.5. Before that we prove the following technical result.

Lemma 3.3 (a)
Let a 1 , a 2 , . . . , a n and b 1 , b 2 , . . . , b n be analytic functions on (0, e + 1 ) and (0, e + 2 ), respectively, and consider the annulus This proves the validity of (a). The assertion (b) is obvious from the definition and the fact that and the monotonicity assumption of This proves the result.
The following result is a consequence of Proposition 3.1 and (b) in Lemma 3.3. For non-degenerate centers it was previously obtained by Rothe [25,Theorem 1], see also [9,Proposition 10].
Then the period function of the center at the origin of the differential system (1) is monotonous increasing (respectively, decreasing).
Next result was proved initially by Chicone [2, Theorem A] for potential systems H (x, y) = 1 2 y 2 + V (x) and it was later extended in [9,25] for general systems H (x, y) = F 1 (x)+ F 2 (y). It follows by applying Proposition 3.2 and taking account of (b) in Lemma 3.3.

Corollary 3.5 Assume that the center at the origin is non
then the period function of the center at the origin of the differential system (1) is monotonous increasing (respectively, decreasing).
A computation shows that The following result is valid for degenerate centers as well and it constitutes our last application of Proposition 3.2. In this case we will use (c) in Lemma 3.3 taking R(z) = z. Let us remark that other choices for R will lead to new monotonicity criteria, certainly with longer expressions but perhaps more convenient in order to study specific systems.
Then the period function of the center at the origin of the differential system (1) is monotonous increasing (respectively, decreasing).
The monotonicity criteria that we obtained so far require that two functions of a single variable do not vanish in an interval. This provides sufficient conditions for monotonicity that are easy to verify. The disadvantage is that these conditions are perhaps excessively far away from being necessary. In what follows we will try to amend this by giving sufficient conditions that concern two-variable functions. The following is the first one of the results in this direction.

Proposition 3.7
The period function of the center at the origin of the differential system (1) verifies Moreover there are no critical periodic orbits in the period annulus if K does not change sign on Q 1 ∩ P. Finally, a sufficient condition for the latter to hold is that does not change sign on P.
Proof By Lemma 2.3 we can assert that Then the expression for h 2 T (h) follows by applying (b) in Proposition 2.7 three times, with {a = 1 , b = 1}, {a = 1, b = 2 } and {a = b = 1}, and performing afterwards some easy simplifications. From this expression it is clear that a sufficient condition for T (h) = 0 for all h ∈ (0, h 0 ) is that K does not change sign on Q 1 ∩ P. In its turn, by (a) in Lemma 3.3, a sufficient condition for this to be verified is thatK does not change sign on P. This proves the result.

Remark 3.8
For reader's convenience let us note that On the other hand, with regard to the non-vanishing assumption of the function K in Proposition 3.7, we note that if F i (z) = α i z k i + o(z k i ) then one can check that K (0, 0) = η(η − 1), with η := 1 k 1 + 1 k 2 . Hence K (0, 0) is negative if the center is degenerate (i.e., when either k 1 > 2 or k 2 > 2) and zero otherwise (i.e., when k 1 = k 2 = 2). Thus, in the first case the period function is monotonous decreasing for h ≈ 0, which is consistent with the well known fact that, for degenerate centers, T (h) −→ +∞ as h tends to zero.

Criteria for at Most One Critical Periodic Orbit
In the statement of our next result A stands for an open annulus in P as introduced in (4). By taking A to be the whole P we obtain a criterion for the existence of at most one critical periodic orbit.
is positive (respectively, negative) on Q 1 ∩ A , then there exists at most one critical periodic orbit inside A , multiplicities taking into account, and it is a minimum (respectively, maximum). In addition, a sufficient condition for this to hold is that Note in addition that, by applying Green's Theorem and taking F i • σ i = F i for i = 1, 2 into account, On the other hand, by applying (b) in Proposition 2.7 with {a = b = 1} we get The two previous identities, together with the one given in Proposition 3.7, yield Q(x, y)dydx because one can verify that We are now in position to prove the result. To this end assume α +β > 1 and that Q is positive on Q 1 ∩ A (the other case follows exactly the same way). Then for any h,ĥ ∈ (h 1 , h 2 ) with h <ĥ we have Accordingly G is a monotonous increasing function on (h 1 , h 2 ). Therefore, on account of A = T , > 0 thanks to the hypothesis α + β > 1 . This shows, simultaneously, that the critical periodic orbit must be a minimum and that there exists at most one critical periodic orbit in A , multiplicities taking into account. This proves the first assertion in the statement whereas the second one follows by (a) in Lemma 3.3. So the result is proved.

Remark 4.3
At this point it is to be referred Sabatini's paper [30], where to the best of our knowledge it is given the only criterion that appears in the literature to ensure the existence of at most one critical periodic orbit for Hamiltonian systems with separable variables. In short, the author introduces a function μ s2 = μ s2 (x, y) and shows that the Hamiltonian system has at most one critical periodic orbit if μ s2 has constant sign on P. His result is in some way complementary to Theorem 4.1 because one can verify that μ s2 is the functionQ taking α = 2 and β = −1. The approach in that paper is completely different to the one we follow here and it relies in the use of the so-called normalizers.
In this setting there is another paper by Sabatini that is worth to mention. Indeed he proved a result, see [29,Theorem 1], that provides sufficient conditions for the existence of at most one critical periodic orbit in an annulus A inside P. However, when applied to the case A = P, conditions L 2 and L 3 in that result imply the monotonicity of the period function.  ∈ (0, h 0 ). The second inequality easily implies the existence of at most one critical periodic orbit in P but certainly it does not constitute a uniqueness criterion for (non-monotonous) period functions.
To the best of our knowledge, in the literature there are no examples apart from the potential systems of Hamiltonian centers with H (x, y) = F 1 (x)+ F 2 (y) for which it has been established the existence of exactly one critical periodic orbit. Next we give two examples of this by applying Theorem 4.1. Let us remark that we tried to avoid technicalities due to computational issues and so although it would be possible to tackle examples with F i not being even functions or having parameters, we prefer not to do it.

Example 4.4 We begin by studying the degenerate center given by H (x, y)
It is easy to verify that F i (z) = 0 if and only if z = 0 or z = ±1. Consequently the period annulus is bounded and, due F 1 (1) > F 2 (1), its outer boundary is given by {F 1 (x)+ F 2 (y) = F 2 (1)}, which is a polycycle that consists in two hyperbolic saddles located at (0, ±1) together with two trajectories connecting them. Hence lim h→h − 0 T (h) = +∞. Since on the other hand we have lim h→0 + T (h) = +∞ due to the fact that the center is degenerate, we can already assert the existence of at least one critical periodic orbit. By applying Theorem 4.1 we shall prove that there exists exactly one. Since F i is an even function for i = 1, 2, note that σ i = −I d and i = F i F i is odd. Then it turns out that and hence the problem reduces to check that this two variable polynomial is positive on P. This can be proved analytically with the help of an algebraic manipulator in several different ways. One possibility is to show that if p 0 ∈ P is a critical point of Q, i.e., ∂ x Q( p 0 ) = ∂ y Q( p 0 ) = 0, then Q( p 0 ) > 0 and that, on the other hand, Q( p) > 0 for all p ∈ ∂P. Let us explain this skipping the computational details for the sake of shortness. The possible critical points can be isolated in arbitrarily small boxes by computing the two resultants between ∂ x Q and ∂ y Q. In doing so we get twelve boxes inside the rectangle [0, 0.76] × [0, 1], which certainly contains Q 1 ∩ P because the smallest positive root of Then we prove, also analytically, that Q has a positive lower bound in each one of these twelve small boxes. Finally to show that Q( p) > 0 for all p ∈ ∂P we compute the resultant with respect to x between Q(x, y) and F 1 (x) + F 2 (y) − F 2 (1) and we verify next by Sturm's Theorem that the polynomial in y that thus obtain does not vanish on (0, 1). H (x, y)

Example 4.5 Let us now study
One can check that F i (z) = 0 if and only if z = 0 or z = ±1. In this case F 1 (1) < F 2 (1) and so the outer boundary of P is {F 1 (x) + F 2 (y) = F 1 (1)}. Moreover, since the smallest positive root of F 2 (y) = F 1 (1) is y ≈ 0.7861, we can assert that Q 1 ∩ P is inside the rectangle [0, 1] × [0.79]. Exactly as in the previous example, there exists at least one critical periodic orbit because the center is bounded and degenerate. We apply Theorem 4.1 taking α = 9/4 and β = −1 to obtain Q(x, y) =
As before the problem reduces to show that this polynomial is positive in the rectangle [0, 1] × [0.79] but in this case we argue differently. We collect it as Q(x, y) = p 0 (y) + p 1 (y)x 6 + p 2 (y)x 12 + p 3 (y)x 18 + p 4 (y)x 24  Proposition 3.2 shows that if the center is non-degenerate and, for i = 1, 2, the function does not vanish on (0, e + i ) then there are no critical periodic orbits. Next we will show that, under some additional hypothesis, if this function has exactly one zero on (0, e + i ) then the center has at most one critical periodic orbit. To see this we shall appeal to some tools developed in [11,20] and to this end some definitions are needed. The first one is the following, see for instance [14].  ( f 0 , f 1 , . . . , f n−1 ) is a complete Chebyshev system (in short, CT-system) on L if, for all k = 1, 2, . . . , n, any nontrivial linear combination has at most k − 1 isolated zeros on L.
The ordered set ( f 0 , f 1 , . . . , f n−1 ) is an extended complete Chebyshev system (in short, ECT-system) on L if, for all k = 1, 2, . . . , n, any nontrivial linear combination has at most k − 1 isolated zeros on L counted with multiplicities.
It is clear from the previous definitions that any ECT-system is in particular a CT-system. The first ones have an easy characterization in terms of Wronskians, as the next well known result shows (see again [14]). f 0 , f 1 , . .

. , f n−1 ) is an ECT-system on I if and only if none of the leading principal minors of its Wronskian
. . .
vanishes on I .
The following result is Theorem A in [11] and in its statement f 0 , f 1 , . . . , f n − 1 and g are analytic functions. For reader's convenience we adapt the statement to the definitions we use in the present paper. (y)dx, i = 0, 1, . . . , n − 1, where, for each h ∈ (0, h 0 ), γ h is the oval surrounding the origin inside the level curve {F 1 (x) + F 2 (y) = h}. For i = 1, 2, let σ i be the involution associated to F i and suppose k i = 2. Finally, setting g 0 = g, let us define g i+1 = g i F 2
Next result (see [20,Proposition 2.2]) is the last ingredient that we need to borrow from the literature and it refers to the notion of σ -evenness as introduced in Definition 2.5.  (g 0 , g 1 , . . . , g n−1 , f ) is an ECT-system on (0, b).

Lemma 4.10
Assume that the center at the origin is non-degenerate, i.e., k 1 = k 2 = 2. Suppose moreover that either F 2 (y) = F 1 (y) for all y ∈ I 2 or F 2 (y) = F 1 (−y) for all y ∈ I 2 . Then the period function T (h) of the center at the origin of the differential system (1) verifies Proof Let us set a i := i − 1 2 for i = 1, 2. We claim that To see this we use that, by Lemma 2.3, Let us consider first the case F 2 (y) = F 1 (y). Then a 2 (y) = a 1 (y) and where in the second equality we used that dy dx = − F 1 (x) F 2 (y) for all (x, y) ∈ γ h and in the third one we make the (orientating reversing) coordinate change {u = y, v = x}. Hence the claim is true in this case. Suppose now that F 2 (y) = F 1 (−y). Then a computation shows that a 2 (y) = a 1 (−y). Thus where in the second equality we use that dy dx = − all (x, y) ∈ γ h and in the third one we make the (orientating reversing) coordinate change {u = −y, v = −x}. So the claim is true also in this case.
Finally, thanks to the claim and on account of F 1 (x) = 1 2 x 2 + o(x 2 ), we can apply Lemma 2.1 to conclude which completes the proof of the result because an easy computation shows that a 1 We are now in position to prove the following uniqueness result for critical periodic orbits.

Theorem 4.11
Assume that the center at the origin is non-degenerate, i.e., k 1 = k 2 = 2, and that either F 2 (y) = F 1 (y) for all y ∈ I 2 or F 2 (y) = F 1 (−y) for all y ∈ I 2 . Then the period function of the center at the origin of system (1) has at most one critical period, multiplicities taking into account, if the following conditions are satisfied: has at most one zero on (0, e + 1 ), counted with multiplicities, and Proof By applying Lemma 4.10, it is clear that it suffices to study the zeros of We claim that the assumptions guarantee the existence of an analytic function I 0 on (0, h 0 ) such that (I 0 , L) form an ECT-system on (0, h 0 ). This implies in particular, recall Definition 4.6, that L (and so the derivative of the period function) has at most one zero on (0, h 0 ) counting multiplicities. So the result will follow once we prove the claim. To this end we will use first the hypothesis that ϕ 1 := P σ 1 f 1 has at most one zero on (0, e + 1 ), counted with multiplicities. Note that, by Lemma 2.6, ϕ 1 is a σ 1 -even function. Hence, by applying Proposition 4.9, there exists another analytic σ 1 -even function ϕ 0 such that (ϕ 0 , ϕ 1 ) is an ECT-system on (0, e + 1 ). In addition, by Lemma 2.6 once again, we can write ϕ 0 = P σ 1 ( f 0 ) for some analytic function f 0 on (e − 1 , e + 1 ). We choose I 0 to be and accordingly we must verify that (I 0 , L) is indeed an ECT-system on (0, h 0 ). To this end we shall apply Theorem 4.8 with n = 2 and g(y) = y. So far we have showed that is an ECT-system on (0, e + 1 ). Since any ECT-system is in particular a CT-system, the hypothesis (a) in Theorem 4.8 is fulfilled. Due to g(y) = y, the hypothesis (b) requires P σ 2 y F 2 (y) , P σ 2 1 to be a CT-system on (0, e + 2 ). We will show, by applying Lemma 4.7, that it is an ECT-system. To this end, since we only need to check that the Wronskian W P σ 2 y F 2 (y) , P σ 2 1 does not vanish on the interval (0, e + 2 ). In this regard, setting a computation shows that the Wronskian writes as Recall at this point that by hypothesis either F 2 (y) = F 1 (y) or F 2 (y) = F 1 (−y), which imply σ 2 (y) = σ 1 (y) and σ 2 (y) = −σ 1 (−y), respectively. Therefore L 2 (y) = L 1 (y) for all y ∈ I 2 in the first case, whereas one can check that L 2 (y) = L 1 (−y) for all y ∈ I 2 in the second case. Since L 1 (y) = 0 for all y ∈ (0, e + 1 ) by assumption and L 1 (σ 1 (y)) = L 1 (y) for all y ∈ I 1 , this shows that the Wronskian does not vanish on (0, e + 2 ). Accordingly we can apply Theorem 4.8 and assert that (I 0 , L) is indeed an ECT-system on (0, h 0 ). This shows the claim and concludes the proof of the result. We give next an example of a center with a monotonous period function for which the sufficient condition given in Proposition 3.2 is not fulfilled.

Example 4.12
Consider now H (x, y) = F(x) + F(y) with F(z) = 1 2 z 2 + z 4 + z 6 + z 8 . In this case one can readily show that the center at the origin is global. Furthermore, by applying Lemma 2.4, the period function T (h) is decreasing near h = 0 since = −4. We will show that the period function is globally monotonous decreasing. To this end one can first try to apply Proposition 3.2, which requires to be non-vanishing on (0, +∞). By applying Sturm's Theorem it turns out however that this function has one positive root counted with multiplicities. Thus Proposition 3.2 does not apply but the desired result will follow by Theorem 4.11. Indeed, on account of σ 1 = −I d, the assumption in (a) is fulfilled, whereas the condition in (b) writes as (x F 1 (x)) = 2x + 16x 3 + 36x 5 + 64x 7 = 0 for all x > 0, which is obvious. Hence the period function has at most one critical period counting multiplicities. Finally we can discard the existence of one critical period by noting that, thanks to (b) in Theorem 5.1, lim h→+∞ T (h) = 0.
Finally we particularize Theorem 4.11 assuming additionally that F is an even function.

Corollary 4.13 Assume that F 1 is even and F
has at most one zero on (0, e + 1 ), counted with multiplicities, and x F 1 (x) does not vanish on (0, e + 1 ), then the period function of the center at the origin of system (1) has at most one critical period, multiplicities taking into account.
Proof The result will follow by applying Theorem 4.11. To this aim note that P σ 1 because F 1 is an even function and σ 1 = −I d.
has exactly one zero on (0, e + 1 ), counted with multiplicities, thanks to the first assumption in the statement. On the other hand, using again that F 1 is even, it follows that the condition , which is the second assumption in the statement. Hence Theorem 4.11 shows the validity of the result.

Asymptotic Results for the Period Function
In this section, motivated by the tools employed by Kaplan and Yorke [13], we study the class of Hamiltonian differential systems with separable variables (1) In other words, getting rid of the subscripts that are unnecessary in this case and setting F = f , we consider planar Hamiltonian differential systems of the form This type of differential system is related with differential-delay equations because of the following result by Kaplan  x = β exist (allowing either α or β to be 0 or ∞). Finally, suppose that lim x→∞ F(x) = ∞. Then the differential-delay equationẋ(t) = − f (x(t − 1)) has a nontrivial periodic solution x(t) of period four if either α < π 2 < β or β < π 2 < α. Furthermore, if y(t) is defined to be x(t − 1), this periodic solution satisfies (5).
This seminal result, that goes back to 1973, was generalized and extended by Nussbaum (see [22][23][24]) without the hypothesis that f is odd and lim x→∞ F(x) = ∞. (The approach in this series of papers is completely different to the one in [13] because it relies in very sophisticated fixed point theorems in Banach spaces). As a matter of fact our interest on the issue arises from the proof of the result by Kaplan and Yorke rather than the result itself. This is so because their hypothesis imply that system (5) has a global center at the origin and the proof consists in proving the existence of a periodic orbit of period 4. To this end they show that the period function of the center verifies lim h→0 + T (h) = 2π α and lim h→+∞ T (h) = 2π β , and then the result follows by the intermediate value theorem. Our aim in this section is to study the limit of T (h) at the endpoints of its domain without assuming lim x→∞ F(x) = ∞ (which forces the center to be global) and that f is odd. That being said, we point out that the regularity assumptions in this section, contrary to the rest of the paper, is that f is merely continuous. In addition to this, the standing hypothesis will be that the limits exist, allowing them to be zero or +∞. Since F = f , the latter imply lim x→±∞ F(x) x 2 = 1 2 β ± by L'Hôpital's Rule (see [33]), so that h ± := lim x→±∞ F(x) ∈ R >0 ∪ {+∞} exist. Let us set H (x, y) = F(−x) + F(y) and recall that then H (P) = (0, h 0 ), where P stands for the period annulus of the center at the origin of system (5). We are now in position to state our main result in this section.
Theorem 5.1 Suppose that f is a continuous function on R satisfying that the limits in (6) exist and that x f (x) > 0 for all x = 0. Then the period T (h) of the periodic orbit γ h of (5) inside the energy level H = h verifies the following properties: (b) Assume that β + = +∞ and β − = +∞. Then the origin is a global center and lim h→+∞ T (h) = 0. (c) Assume that β + < +∞ and β − = +∞ (respectively, β + = +∞ and β − < +∞).
There are some previous results in the literature related with Theorem 5.1 that should be referred. As we already mentioned, Kaplan and Yorke show the assertions (b) and (d 2 ) in their proof of [13, Theorem 1.1] under the additional assumption that f is odd. In this respect we refer the reader to Remark 5.2 for some further comments. On the other hand, [7,Theorem C] gives the first term in the asymptotic expansion of T (h) at h = +∞ for general Hamiltonian differential systems with separable variables, and its application yields to the assertions in Theorem 5.1 with regard to the global center case, i.e., (b), (c 2 ) and (d 2 ). Finally the fact that lim h→0 + T (h) can be given in terms of the linear part of the center is a classical result even for general differential systems and we include the assertion in (a) for completeness.
Theorem 5.1 gives the limit of T (h) as h tends to the endpoints of its domain of definition (0, h 0 ). Thus, in combination with the intermediate value theorem, it can be used to prove the existence of periodic orbits with prescribed periods. Likewise, taking the monotonicity of T (h) near h = 0 or h = h 0 into account, it can also be used to prove the existence of critical periodic orbits (see for example the proof of [3, Theorem A] or [16,Theorem 5.2]). Of course to this end it is necessary to compute the period constants (cf. Lemma 2.4), that give the monotonicity of T (h) at h = 0, or to study the asymptotic development of T (h) at h = h 0 , which constitutes a much more difficult problem (cf. [15,Theorem A]). It is also worth to remark that the proof of Theorem 5.1 shows that in cases (c 2 ) and (d 2 ) the periodic orbit γ h undergoes a kind of slow-fast phenomenon as h tends to +∞. Example 5.4 shows an explicit Hamiltonian differential system that exhibits this type of motion. Slow-fast oscillations occur typically in singular perturbation problems and so we think that it is an interesting issue for further research.