This is a preprint of: “The period function of Hamiltonian systems with separable variables”, Jordi Villadelprat, Xiang Zhang, J. Dyn. Diff. Equat., 1–27, 2019. DOI: [10.1007/s10884-019-09759-w] The period function of Hamiltonian systems with separable variables Jordi Villadelprat and Xiang Zhang Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain Email: jordi.villadelprat@urv.cat School of Mathematical Sciences, MOE–LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China Email: xzhang@sjtu.edu.cn February 20, 2019 Abstract. 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). 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. 1 Introduction and definitions The present paper deals with the class of planar Hamiltonian differential systems  ẋ = −Hy (x, y), ẏ = Hx (x, y), where the Hamiltonian function has separable variables, i.e., it has the special form H(x, y) = F1 (x) + F2 (y). For i = 1, 2 we suppose that Fi (z) is an analytic function on R with a local minimum at z = 0, so that  ẋ = −F2′ (y), (1) ẏ = F1′ (x), 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. Keywords: Hamiltonian differential system, center, period function, critical periodic orbit. 2010 MSC: 34C25, 37J35, 70H05, 70K42, 70K70. 1 The solution curves of (1) are inside the energy levels of H. We can assume without loss of generality that Fi (0) = 0 and, accordingly, H(0, 0) = 0. Thus H(P) = (0, h0 ) for some h0 ∈ R>0 ∪ {+∞}. 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, h0 ) we denote by γh 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 7−→ T (h) := period of γh , that can be written as Z dx T (h) = ′ for each h ∈ (0, h0 ). γh F2 (y) (Here, and in what follows, we take the oval γh 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 values ĥ ∈ (0, h0 ) such that T ′ (h) = α(h − ĥ)k + o((h − ĥ)k ) with α 6= 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, h0 ). 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 Section 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 Section 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 Section 5 deals with the limit of the period function T (h) as h tends to the endpoints of its domain (0, h0 ), 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. 2 Auxiliary results Let us fix that Fi (z) = αi z ki + o(z ki ) for some even number ki and positive real number αi . In addition, let I1 and I2 denote, respectively, the projection of P on y = 0 and x = 0. Note that Ii = (e− + i , ei ) with − + ei < 0 < ei and zFi (z) > 0 for all z ∈ Ii \ {0}. Each Fi defines an analytic involution σi on Ii by means ′ the relation Fi (z) = Fi (σi (z)) for all z ∈ Ii . Recall that a function σ is said to be an involution if σ ◦ σ = Id and σ 6= Id. In this respect observe that σi (0) = 0 and σi′ (z) < 0 for all z ∈ Ii . In the statement of our first result, and in what follows, the 2 multiplicity of an analytic function f at x = x̂ is denoted by mult(f, x̂). Lemma 2.1. Let a be an analytic function on I1 with mult (a, 0) > k1 − 1 and let ℓ be a function such that F2′ ℓ is analytic on I2 . Then Z Z  ′ a a(x)ℓ(y)dx = (x)ℓ̂(y)dx, where ℓ̂′ = ℓF2′ . γh γh F1′ Proof. Let R(x) and S(y) be analytic functions on I1 and I2 , respectively. If (x, y) ∈ γh then F1′ (x) d (S(x)R(y)) = S ′ (x)R(y)dx + S(x)R′ (y)dy = S ′ (x)R(y)dx − S(x)R′ (y) dx F2′ (y) and, consequently, Z Z   R′ S ′ (x)R(y)dx = (SF1′ ) (x) (y)dx. γh γh F2′ R′ The result follows from this equality taking a = SF1′ and ℓ = F2′ , which leads to S = a F1′ and R′ = ℓF2′ . Lemma 2.2. If R and S are analytic functions on I1 and I2 , respectively, then Z Z  ′ d S R(x)S(y)dx = R(x) ′ (y)dx. dh γh γh F2 Proof. This follows by the Gelfand-Leray derivation formula (see for instance [12, Theorem 26.32]), which asserts that Z Z d ω= η dh γh γh provided that dω = dH ∧ η. In all over the paper for the sake of shortness we will use the notation Fi ℓi := for i = 1, 2. Fi′ Since Fi (z) = αi z ki + o(z ki ), note that ℓi is an analytic function on Ii = (e− + i , ei ) with ℓi (z) = 1 ki z + o(z). Lemma 2.3. The period function of the center at the origin of the differential system (1) verifies Z ′ 1  dx T (h) = ℓ′ (x) + ℓ′2 (y) − 1 ′ . h γh 1 F2 (y) Proof. Note that, since h = F1 (x) + F2 (y) for all (x, y) ∈ γh , Z Z Z  ′ Z   dx dx F1 F2 hT (h) = F1 (x) ′ + F2 (y) ′ = ′ (x)ydx + (y)dx, γh F2 (y) γh F2 (y) γh F1 γh F2′ where in the second equality we applied Lemma 2.1. Then by Lemma 2.2 we can assert that Z  ′ Z  ′ F1 dx F2 dx (hT (h))′ = ′ (x) ′ + ′ (y) ′ . γh F1 F2 (y) γh F2 F2 (y) Thus Z  ′ Z  ′ Z ′ F1 dx F2 dx dx hT (h) = (x) ′ + (y) − γh F1′ F2 (y) γh ′ F2 ′ F2 (y) ′ γh F2 (y) and so the result follows. 3 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. Lemma 2.4. For i = 1, 2, suppose that Fi (z) = αi,2 z 2 + αi,3 z 3 + αi,4 z 4 + o(z 4 ). Assume furthermore that ∆ := 5(α21,3 + α22,3 ) − 4(α1,2 α1,4 + α2,2 α2,4 ) 6= 0. Then the period function is locally monotonous increasing (respectively, decreasing) near the center in case that ∆ is positive (respectively, negative). Proof. For i = 1, 2, let us define ai := ℓ′i − 1 2 for the sake of shortness. Then by Lemma 2.3 we can write Z Z   Z  ′ Z   dx a2 a1 a2 hT ′ (h) = a1 (x) + (y)dx = (x)ydx + ′ (y)dx, γh F2′ (y) γh F2′ γh F1 ′ γh F2 wherein the  second equality we applied Lemma 2.1 taking a1 (0) = 0 and F1 (0) 6= 0 into account. Therefore, ′′ since Fai′ (z) is analytic at z = 0 for i = 1, 2, we can apply Green’s Theorem (see [28]) to obtain i ZZ  ′  ′  a1 a2 hT ′ (h) = (x) + (y) dydx, Int(γ) F1′ F2′ where Int(γ) stands for the bounded connected   component of R \{γ}. (We shall use this notation hereafter.) 2 2 α 3 5αi,3 −4αi,2 αi,4 Now, since one can readily verify that ai Fi′ (z) = − 14 α2i,3 + 16 α3i,2 z + o(z), the result follows. i,2 Definition 2.5. Let σ be an analytic involution and consider a given function f. The σ-even part of f is f (x) − f (σ(x))σ ′ (x) Pσ (f )(x) := . 2 We say that f is σ-even if Pσ (f ) = f.  Observe that this definition coincides with the usual notion of evenness in case that σ = −Id. 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 = Pσ (g).  Proof. The necessity is obvious. In order to see the sufficiency, assume that f (z) = 21 g(z) − g(σ(z))σ ′ (z) for some g. Then      2Pσ (f )(z) =g(z) − g σ(z) σ ′ (z) − g σ(z) − g σ 2 (z) σ ′ σ(z) σ ′ (z)     g(z)  =g(z) − g σ(z) σ ′ (z) − g σ(z) − ′ σ ′ (z) = g(z) − g σ(z) σ ′ (z) = 2f (z), σ (z) which gives Pσ (f ) = f, as desired. Proposition 2.7. Let γ be an oval, clockwise oriented, inside the energy level F1 (x) + F2 (y) = h and define Z dx C := a(x)b(y) ′ , γ F2 (y) where a and b are analytic functions on I1 = (e− + − + 1 , e1 ) and I2 = (e2 , e2 ), respectively. Then, if we denote the first quadrant by Q1 , the following hold: Z Z   dx   dx (a) C = Pσ1 a (x)Pσ2 b (y) ′ =4 Pσ1 a (x)Pσ2 b (y) ′ . γ F2 (y) Q1 ∩γ F 2 (y) 4 ZZ       (b) hC = 4 Pσ1 (aℓ1 )′ (x)Pσ2 b (y) + Pσ1 a (x)Pσ2 (bℓ2 )′ (y) dydx. Q1 ∩Int(γ) Proof. Let us split the given oval as the concatenation γ = γ1 + γ4 + γ3 + γ2 , where  γi is the intersection of γ with the i-th quadrant. Define ψ1 (x, y) := σ1 (x), y and ψ2 (x, y) := x, σ2 (y) . Due to σi′ (z) < 0 for all z ∈ Ii , note that ψ1 and ψ2 are analytic diffeomorphisms on P that reverse orientation. On account of this, and the fact that ψ1 (γ2 ∪ γ3 ) = γ1 ∪ γ4 , the coordinate change (x, y) = ψ1 (u, v) yields Z Z dx  du a(x)b(y) ′ =− a σ1 (u) b(v)σ1′ (u) ′ . γ3 +γ2 F2 (y) γ1 +γ4 F2 (v) Hence, on account of Definition 2.5, Z Z Z dx dx dx C= a(x)b(y) ′ = a(x)b(y) ′ + a(x)b(y) ′ (2) γ F2 (y) γ1 +γ4 F 2 (y) γ3 +γ2 F2 (y) Z   Z  dx  dx = a(x) − a σ1 (x) σ1′ (x) b(y) ′ =2 Pσ1 a (x)b(y) ′ . γ1 +γ4 F2 (y) γ1 +γ4 F2 (y) On the other hand, since ψ2 (γ4 ) = γ1 , the change of coordinates (x, y) = ψ2 (u, v) yields Z Z Z  dx   du   du Pσ1 a (x)b(y) ′ =− Pσ1 a (u)b σ2 (v) ′  =− Pσ1 a (u)b σ2 (v) σ2′ (v) ′ , γ4 F2 (y) γ1 F2 σ2 (v) γ1 F2 (v) where in the second equality we used that F2′ (y) = F2′ (σ2 (y))σ2′ (y). Therefore, from (2), Z Z Z dx     dx   dx C = a(x)b(y) ′ =2 Pσ1 a (x) b(y) − b σ2 (y) σ2′ (y) ′ = 4 Pσ1 a (x)Pσ2 b (y) ′ , γ F2 (y) γ1 F2 (y) γ1 F2 (y) and this proves (a). Note in this respect  that, thanks to the above equality, in the definition  of C we can replace a by Pσ1 a and b by Pσ2 b because, on account of Lemma 2.6, Pσ Pσ (f ) = Pσ (f ) for any f . Let us turn next to the proof of (b). Due to F1 (x) + F2 (y) = h for all (x, y) ∈ γ we have Z Z   Z    dx  b bF2 hC = F1 (x) + F2 (y) a(x)b(y) ′ = aF1 (x) (y)dx + a(x) (y)dx. γ F2 (y) γ F2′ γ F2′ Thus by applying Lemma 2.1 we get Z ′ Z   Z   aF1 F2 b ′ hC = ′ (x)B(y)dx + a(x) (y)dx = aℓ 1 (x)B(y) + a(x) bℓ 2 )(y) dx, γ F1 γ F2′ γ where B ′ = b, and then the application of Green’s Theorem (see for instance [28]) yields ZZ   ′ hC = aℓ1 (x)b(y) + a(x) bℓ2 )′ (y) dydx. Int(γ) Similarly as before, we split Int(γ) = ∪4i=1 Ri where Ri is the intersection of Int(γ) with the i-th quadrant. Since ψ1 (R2 ∪ R3 ) = R1 ∪ R4 , the coordinate change (x, y) = ψ1 (u, v) yields ZZ   ZZ   ′ ′ aℓ1 (x)b(y) + a(x) bℓ2 )′ (y) dydx = − aℓ1 (σ1 (u))b(v) + a(σ(u)) bℓ2 )′ (v) σ1′ (u)dvdu R2 ∪R3 R1 ∪R4 Therefore, ZZ   ′  ′ hC = 2 Pσ1 (aℓ1 ) (x)b(y) + Pσ1 a (x) (bℓ2 ) (y) dydx. R1 ∪R4 5 On the other hand, since ψ2 (R4 ) = R1 , the change of coordinates (x, y) = ψ2 (u, v) yields ZZ     Pσ1 (aℓ1 )′ (x)b(y) + Pσ1 a (x) (bℓ2 )′ (y) dydx R4 ZZ   ′  ′ =− Pσ1 (aℓ1 ) (u)b(σ2 (v)) + Pσ1 a (u) (bℓ2 ) (σ2 (v)) σ2′ (v)dvdu R1 Finally taking Definition 2.5 into account once again we get ZZ   ′   ′ hC = 4 Pσ1 (aℓ1 ) (x)Pσ2 b (y) + Pσ1 a (x)Pσ2 (bℓ2 ) (y) dydx R1 as desired, and this concludes the proof of the result. 3 Monotonicity results for the period function  Proposition 3.1. Suppose that there exist η1 , η2 ∈ R with η1 +η2 = 1 such that, for i = 1, 2, Pσi ℓ′i (z)−ηi is positive (respectively, negative) for all z ∈ (0, e+ i ). 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 Z     dx hT ′ (h) = 4 Pσ1 ℓ′1 − η1 (x)Pσ2 1)(y) + Pσ1 1)(x)Pσ2 ℓ′2 − η2 (y) , Q1 ∩γh F2′ (y) where recall that Q1 stands for the first quadrant. Since Q1 ∩ γh ⊂ (0, e+ + 1 )×(0, e2 ) and, on the other hand, Pσi 1)(z) = 1 − σi (z) > 0, the result follows. ′ 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 Pσi ℓ′i − 12 does not vanish. Proposition  ′′3.2.  Assume that the center at the origin is non-degenerate, i.e., k1 = k2 = 2. If the function Fi Pσi (F ′ )2 is positive (respectively, negative) on (0, e+ i ) for i = 1, 2, then the period function of the i 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 η1 = η2 = 12 . With this aim in view note first that, due to Fi′ (z) = Fi′ (σi (z))σi′ (z),      ′ 1   ′ 1  ℓi − 2 ℓ − Pσi ℓ′i − 21 (z) = ℓ′i − 21 (z) − ℓ′i − 21 σi (z) σi′ (z) = Fi′ (z) F′ (z) − iF ′ 2 σi (z) . (3) i i   ℓ′i − 21 Observe in addition that, since Fi (z) = αi z 2 + o(z 2 ), we have ℓi (z) = 12 z + o(z). Therefore Fi′ (z) is an analytic function at z = 0 and  ′ 1  ′ 1  ℓi − 2 ℓi − 2 F ′ (z) − F′ σi (z) = 0. i i z=0  Consequently, from (3) and on account of Fi′ (z) > 0 for all z ∈ (0, e+ i ), a sufficient condition for Pσi ℓi − ′ 1 2 to be positive (respectively, negative) on (0, e+ i ) is that       d ℓ′i − 21 ℓ′i − 12  ℓ′i − 12 ′ dz Fi′ (z) − Fi′ σi (z) = Pσi F′ (z) i  ′ ℓ′i − 12 is positive (respectively, negative) for z ∈ (0, e+ i ). This proves the result because Fi′ = 1 2 Fi (Fi′ )2 . 6 In general one cannot expect to have the explicit expression of the involutions σi 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 Pσi (ℓ′i − ηi ) without knowing explicitly σi . This can be done for instance in case that Fi 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 Fi implying that Pσi (ℓ′i − ηi ) 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 (4) 5Fi′′′ (z)2 − 3Fi′′ (z)Fi (z) > 0 for any z ∈ (e− + ′′ i , ei ) with Fi (z) > 0 and Fi′ (z)Fi′′′ (z) < 0 for any z ∈ (e− + ′′ i , ei ) with Fi (z) = 0  hold then Pσi ℓ′i (z) − 21 > 0 for all z ∈ (0, e+ i ). Likewise Schaaf also proved that if (4) i , ei ) with Fi (z) > 0 5Fi′′′ (z)2 − 3Fi′′ (z)Fi (z) < 0 for any z ∈ (e− + ′′  then Pσi ℓ′i (z)− 21 < 0 for all z ∈ (0, e+ i ). 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 a1 , a2 , . . . , an and b1 , b2 , . . . , bn be analytic functions on (0, e+ + 1 ) and (0, e2 ), respec- tively, and consider the annulus [ A = γh with 0 6 h1 < h2 6 h0 . (4) h∈(h1 ,h2 ) P n   P n Then Pσ1 ai (x)Pσ2 bi (y) is positive (respectively, negative) on Q1 ∩A if ai (x)bi (y) is positive i=1 i=1 (respectively, negative) on A .  (b) Pσi f is positive (respectively, negative) on (0, e+ i ) if f is a positive (respectively, negative) function on (e− i , e + i ) \ {0}.  R(Fi )f (c) Pσi f is positive (respectively, negative) on (0, e+ i ) if F′ is a smooth monotonous increasing i (respectively, decreasing) function on (e− + i , ei ), where R is any function with R(h) > 0 for all h ∈ (0, h0 ). Pn Proof. To show (a) assume for instance that i=1 ai (x)bi (y) > 0 for all (x, y) ∈ A . Let Ai denote the intersection of A with the i-th quadrant. PRecall that σi′ < 0 since Fi′ (σi (z))σi′ (z) = Fi′ (z) for all z ∈ (e− + i , ei ). n Thus, if (x, y) ∈ A1 ∪ A2 then −σ2 (y) i=1 ai (x)bi (σ2 (y)) > 0 because (x, σ2 (y)) ∈ A3 ∪ A4 ⊂ A . Hence, ′ by definition, n X n  X   2 ai (x)Pσ2 bi (y) = ai (x)bi (y) − ai (x)bi (σ2 (y))σ2′ (y) > 0 for all (x, y) ∈ A1 ∪ A2 . i=1 i=1 Pn  On account of this, if (x, y) ∈ A1 then, since (σ1 (x), y) ∈ A2 , we get −σ1′ (x) i=1 ai (σ1 (x))Pσ2 bi (y) > 0, which by definition implies n X n  X      2 Pσ1 ai (x)Pσ2 bi (y) = ai (x)Pσ2 bi (y) − ai (σ1 (x))Pσ2 bi (y)σ1′ (x) > 0. i=1 i=1 7 This proves the validity of (a). The assertion (b) is obvious from the definition and the fact that σi′ < 0. Finally in order to show (c) note that, on account of Fi (z) = Fi (σi (z)) for all z,   ′       F ′ (z) F 2Pσi f (z) = f (z) − f (σi (z)) F ′ (σi i (z)) = R(Fi i ) (z) R(F F i )f ′ (z) − R(Fi )f F ′ σi (z) . i i i Since σi (z) < 0 < z for all z ∈ (0, e+  + i ) and Fi ((0, ei )) = (0, h0 ), the assertion in (c) follows thanks to ′ Fi + R(Fi )f R(Fi ) (z) > 0 for all z ∈ (0, ei ) and the monotonicity assumption of F ′ . This proves the result. i 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]. Corollary 3.4. Suppose that there exist η1 , η2 ∈ R with η1 + η2 = 1 such that, for i = 1, 2, ℓ′i (z) − ηi is positive (respectively, negative) for all z ∈ (e− + i , ei ) \ {0}. 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) = 21 y 2 + V (x) and it was later extended in [9, 25] for general systems H(x, y) = F1 (x) + F2 (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-degenerate, i.e., k1 = k2 = 2. If (FF′i)2 is i positive (respectively, negative) on (e− + i , ei ) \ {0} for i = 1, 2, then the period function of the center at the origin of the differential system (1) is monotonous increasing (respectively, decreasing). A computation shows that  ′  ′′ F (F ′ )2 − 2F F ′′ F 2F F ′ F ′′′ − 6F (F ′′ )2 − 3(F ′ )2 F ′′ = and = . (F ′ )2 (F ′ )3 (F ′ )2 (F ′ )4 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.  Corollary 3.6. Suppose that there exist η1 , η2 ∈ R with η1 + η2 = 1 such that ℓi ℓ′i − ηi is monotonous increasing (respectively, decreasing) on (e− + i , ei ) 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. By Proposition 3.1 it suffices to show that, for i = 1, 2, Pσi ℓ′i (z) − ηi is positive (respectively, negative) for all z ∈ (0, e+ i ). In turn, by applying (c) in Lemma 3.3 with R(z) = z, a sufficient condition for this to hold is that FFi′ (ℓ′i − ηi ) is monotonous increasing (respectively, decreasing) on (e− + i , ei ). i 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 ZZ h2 T ′ (h) = 4 K(x, y)dydx, Q1 ∩Int(γh ) 8 where    ′  K(x, y) := 2Pσ1 ℓ′1 (x)Pσ2 ℓ′2 (y) + Pσ1 ℓ1 (ℓ′1 − 1) (x)Pσ2 1)(y)  ′   + Pσ2 ℓ2 (ℓ′2 − 1) (y)Pσ1 1 (x). Moreover there are no critical periodic orbits in the period annulus if K does not change sign on Q1 ∩ P. Finally, a sufficient condition for the latter to hold is that ′ ′ K̂(x, y) := 2ℓ′1 (x)ℓ′2 (y) + ℓ1 (ℓ′1 − 1) (x) + ℓ2 (ℓ′2 − 1) (y) does not change sign on P. Proof. By Lemma 2.3 we can assert that Z  dx hT ′ (h) = ℓ′1 (x) + ℓ′2 (y) − 1 . γh F2′ (y) Then the expression for h2 T ′ (h) follows by applying (b) in Proposition 2.7 three times, with {a = ℓ′1 , b = 1}, {a = 1, b = ℓ′2 } and {a = b = 1}, and performing afterwards some easy simplifications. From this expression it is clear that a sufficient condition for T ′ (h) 6= 0 for all h ∈ (0, h0 ) is that K does not change sign on Q1 ∩ P. In its turn, by (a) in Lemma 3.3, a sufficient condition for this to be verified is that K̂ does not change sign on P. This proves the result. Remark 3.8. For reader’s convenience let us note that  ′  ′  2 ′′ ′  2 ′′ ′ F1 F2 F1 F1 F2 F2 K̂(x, y) = 2 ′ (x) ′ (y) − ′ (x) − (y). F1 F2 (F1 )3 (F2′ )3 On the other hand, with regard to the non-vanishing assumption of the function K in Proposition 3.7, we note that if Fi (z) = αi z ki + o(z ki ) then one can check that K(0, 0) = η(η − 1), with η := k11 + k12 . Hence K(0, 0) is negative if the center is degenerate (i.e., when either k1 > 2 or k2 > 2) and zero otherwise (i.e., when k1 = k2 = 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.  4 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. Theorem 4.1. If there exist α, β ∈ R with α + β > 1 (respectively, α + β < 1) such that the function    ′  Q(x, y) := 2Pσ1 ℓ′1 (x)Pσ2 ℓ′2 (y) + Pσ1 ℓ1 (ℓ′1 − α) (x)Pσ2 1)(y)  ′     + Pσ2 ℓ2 (ℓ′2 − α) (y)Pσ1 1 (x) − βPσ1 1 (x)Pσ2 1 (y) is positive (respectively, negative) on Q1 ∩ 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 ′ ′ Q̂(x, y) := 2ℓ′1 (x)ℓ′2 (y) + ℓ1 (ℓ′1 − α) (x) + ℓ2 (ℓ′2 − α) (y) − β is positive (respectively, negative) on A . 9 R Proof. Let us consider A(h) := γh ydx for h ∈ (0, h0 ). Clearly A(h) is the area of the region Int(γh ) and a general well known result on the period function of Hamiltonian differential systems asserts that A′ (h) = T (h), see for instance [19, Theorem A]. As a matter of fact, for the particular Hamiltonians that we deal with, this can be shown by applying Lemma 2.2 because in doing so we get Z Z d dx A′ (h) = ydx = ′ = T (h). dh γh γh F2 (y) Note in addition that, by applying Green’s Theorem and taking Fi ◦ σi = Fi for i = 1, 2 into account, ZZ ZZ   A(h) = dydx = 4 Pσ1 1 (x)Pσ2 1 (y)dydx. Int(γh ) Q1 ∩Int(γh ) On the other hand, by applying (b) in Proposition 2.7 with {a = b = 1} we get ZZ     hT (h) = 4 Pσ1 (ℓ′1 )(x)Pσ2 1 (y) + Pσ1 1 (x)Pσ2 (ℓ′2 )(x) dydx. Q1 ∩Int(γh ) The two previous identities, together with the one given in Proposition 3.7, yield ZZ G(h) := h2 T ′ (h) + (1 − α)hT (h) − βA(h) = 4 Q(x, y)dydx Q1 ∩Int(γh ) because one can verify that       Q(x, y) = K(x, y) + (1 − α) Pσ1 (ℓ′1 )(x)Pσ2 1 (y) + Pσ1 1 (x)Pσ2 (ℓ′2 )(x) − βPσ1 1 (x)Pσ2 1 (y). We are now in position to prove the result. To this end assume α + β > 1 and that Q is positive on Q1 ∩ A (the other case follows exactly the same way). Then for any h, ĥ ∈ (h1 , h2 ) with h < ĥ we have ZZ G(ĥ) − G(h) = Q(x, y)dydx > 0. Q1 ∩(Int(γĥ )\Int(γh )) Accordingly G is a monotonous increasing function on (h1 , h2 ). Therefore, on account of A′ = T, G′ (h) = h2 T ′′ (h) + (2 + 1 − α)hT ′ (h) + (1 − α − β)T (h) > 0 for all h ∈ (h1 , h2 ). If γh⋆ is a critical periodic orbit in A , i.e, T ′ (h⋆ ) = 0, then h2⋆ T ′′ (h⋆ ) + (1 − α − β)T (h⋆ ) > 0, which implies h2⋆ T ′′ (h⋆ ) > (α + β − 1)T (h⋆ ) > 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 4.2. The sign of Q near the origin is relevant in case that Theorem 4.1 is applied with A = P. In this respect, since Fi (x) = ai xki + o(z ki ) with ki an even number, note that Q(0, 0) = η 2 − αη − β, where η = k11 + k12 ∈ (0, 1].  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 µs2 = µs2 (x, y) and shows that the Hamiltonian system has at most one critical periodic orbit if µs2 has constant sign on P. His result is in some way complementary to Theorem 4.1 because one can verify that µs2 is the function Q̂ 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. 10 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 L2 and L3 in that result imply the monotonicity of the period function. (This last assertion can be shown, for instance, by applying Corollary 3.4.) It occurs the same with Rothe’s result [25, Theorem 3], which provides a criterion  ′ ′ for T ′ (h) > 0 and hT (h) T (h) > 0 for all h ∈ (0, h0 ). 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) = F1 (x) + F2 (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 Fi 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) = F1 (x) + F2 (y) with 1 4 1 8 1 1 F1 (z) = z − z and F2 (z) = z 6 − z 10 . 4 8 6 10 It is easy to verify that Fi′ (z) = 0 if and only if z = 0 or z = ±1. Consequently the period annulus is bounded and, due F1 (1) > F2 (1), its outer boundary is given by {F1 (x) + F2 (y) = F2 (1)}, which is a polycycle that consists in two hyperbolic saddles located at (0, ±1) together with two trajectories connecting them. Hence limh→h− T (h) = +∞. Since on the other hand we have limh→0+ T (h) = +∞ due to the fact that the 0 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 Fi is an even function for i = 1, 2, note that σi = −Id and ℓi = FFi′ is odd. Then it turns out that i ′ ′ Q(x, y) = 2ℓ′1 (x)ℓ′2 (y) + ℓ1 (ℓ′1 − α) (x) + ℓ2 (ℓ′2 − α) (y) − β. Taking α = 5/2 and β = −1, one can verify that Q(x, y) = 1900 − 13600x4 + 50925x8 − 20592y 12 + 5125x16 + 3564y 16 − 33550x12 − 12400y 4 + 30600y 8 − 171600x4y 8 + 449790x8y 8 − 296580x12y 8 + 52590x16 y 8 + 79600x4y 4 − 245700x8y 4 + 114768x4y 12 − 21456x4y 16 − 300132x8y 12 + 63549x8y 16 + 161800x12y 4 + 197928x12y 12 − 41886x12y 16 − 26500x16 y 4 − 35172x16 y 12 + 7029x16 y 16 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 p0 ∈ P is a critical point of Q, i.e., ∂x Q(p0 ) = ∂y Q(p0 ) = 0, then Q(p0 ) > 0 and that, on the other hand, Q(p) > 0 for all p ∈ ∂P. Let us explain this skiping 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 Q1 ∩ P because the smallest positive root of F1 (x) = F2 (1) is x ≈ 0.7502. 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 F1 (x) + F2 (y) − F2 (1) and we verify next by Sturm’s Theorem that the polynomial in y that thus obtain does not vanish on (0, 1).  Example 4.5. Let us now study H(x, y) = F1 (x) + F2 (y) with z4 z4 F1 (z) = and F2 (z) = . 1 + 2z 6 1 + z8 11 One can check that Fi′ (z) = 0 if and only if z = 0 or z = ±1. In this case F1 (1) < F2 (1) and so the outer boundary of P is {F1 (x) + F2 (y) = F1 (1)}. Moreover, since the smallest positive root of F2 (y) = F1 (1) is y ≈ 0.7861, we can assert that Q1 ∩ 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 S(x,y) apply Theorem 4.1 taking α = 9/4 and β = −1 to obtain Q(x, y) = 16(x6 −1) 4 (y 8 −1)4 with S(x, y) = 2 + 13x6 + 16y 32 − 78y 24 − 89x18 + 26x24 + 782y 16 + 1020x12 + 488x6 y 8 − 4598x6y 16 − 5376x12y 8 + 14160x12y 16 + 1112x18y 8 − 5666x18 y 16 − 158x24 y 8 + 1154x24 y 16 + 1152x6 y 24 − 127x6 y 32 − 6480x12 y 24 + 1284x12y 32 + 1824x18 y 24 − 253x18 y 32 − 306x24 y 24 + 52x24 y 32 + 46y 8 . 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) = p0 (y) + p1 (y)x6 + p2 (y)x12 + p3 (y)x18 + p4 (y)x24 and we show that the lower bounds p0 (y) > 2, p1 (y) > 12, p2 (y) > 486, p3 (y) > −90 and p4 (y) > 20 hold for all y ∈ [0, 0.79]. Thus Q(x, y) > 2 + 12 x6 + 486 x12 − 90 x18 + 20 x24 for all (x, y) ∈ P and, on the other hand, by applying Sturm’s Theorem one can prove that the polynomial in x is positive on (0, 1).   ′′  Proposition 3.2 shows that if the center is non-degenerate and, for i = 1, 2, the function Pσi Fi (Fi′ )2 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]. Definition 4.6. Let f0 , f1 , . . . , fn−1 be analytic functions on an open interval L of R. (a) The ordered set (f0 , f1 , . . . , fn−1 ) is a complete Chebyshev system (in short, CT-system) on L 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 L. (b) The ordered set (f0 , f1 , . . . , fn−1 ) is an extended complete Chebyshev system (in short, ECT-system) on L 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 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]). Lemma 4.7. (f0 , f1 , . . . , fn−1 ) is an ECT-system on I if and only if none of the leading principal minors of its Wronskian f0 (x) ··· fn−1 (x)   f0′ (x) ··· ′ fn−1 (x) W f0 , f1 , . . . , fn−1 (x) = .. . (n−1) (n−1) f0 (x) · · · fn−1 (x) vanishes on I. 12 The following result is Theorem A in [11] and in its statement f0 , f1 , . . . , fn − 1 and g are analytic functions. For reader’s convenience we adapt the statement to the definitions we use in the present paper. Theorem 4.8. Let us consider the Abelian integrals Z Ii (h) = fi (x)g(y)dx, i = 0, 1, . . . , n − 1, γh where, for each h ∈ (0, h0 ), γh is the oval surrounding the origin inside the level curve {F1 (x) + F2 (y) = h}. For i = 1, 2, let σi be the involution associated to Fi and suppose ki = 2. Finally, setting g0 = g, let us define g′ gi+1 = Fi′ . Then (I0 , I1 , . . . , In−1 ) is an ECT-system on (0, h0 ) if the following hypothesis are satisfied: 2     (a) Pσ1 f0 , Pσ1 f1 , . . . , Pσ1 fn−1 is a CT-system on (0, e+ 1 ),     (b) Pσ2 F2′ g0 , Pσ2 F2′ g1 , . . . , Pσ2 F2′ gn−1 is a CT-system on (0, e+ 2 ) and g(y) − g(−y) = o(y 2(n−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. Proposition 4.9. Let σ be an analytic involution on L = (a, b) with σ(0) = 0 and consider an analytic σ-even function f on L. Then f has at most n zeros on (0, b) taking multiplicities into account if and only if there exist g0 , g1 , . . . , gn−1 analytic σ-even functions on L such that (g0 , g1 , . . . , gn−1 , f ) is an ECT-system on (0, b). Lemma 4.10. Assume that the center at the origin is non-degenerate, i.e., k1 = k2 = 2. Suppose moreover that either F2 (y) = F1 (y) for all y ∈ I2 or F2 (y) = F1 (−y) for all y ∈ I2 . Then the period function T (h) of the center at the origin of the differential system (1) verifies Z  ′′ ′ F1 hT (h) = (x)ydx. γh (F1′ )2 Proof. Let us set ai := ℓ′i − 1 2 for i = 1, 2. We claim that Z dx hT ′ (h) = 2 a1 (x) ′ . γh F2 (y) To see this we use that, by Lemma 2.3, Z ′ dx hT (h) = (a1 (x) + a2 (y)) . γh F2′ (y) Let us consider first the case F2 (y) = F1 (y). Then a2 (y) = a1 (y) and Z Z Z Z Z dx dx dy du du a2 (y) ′ = a1 (y) ′ =− a1 (y) ′ = a1 (u) ′ = a1 (u) ′ , γh F2 (y) γh F2 (y) γh F1 (x) γh F1 (v) γh F2 (v) dy F ′ (x) where in the second equality we used that dx = − F1′ (y) for all (x, y) ∈ γh and in the third one we make the 2 (orientating reversing) coordinate change {u = y, v = x}. Hence the claim is true in this case. Suppose now that F2 (y) = F1 (−y). Then a computation shows that a2 (y) = a1 (−y). Thus Z Z Z Z Z dx dx dy du du a2 (y) ′ = a1 (−y) ′ =− a1 (−y) ′ =− a1 (u) ′ = a1 (u) ′ , γh F2 (y) γh F2 (y) γh F 1 (x) γh F1 (−v) γh F2 (v) 13 dy F ′ (x) where in the second equality we use that dx = − F1′ (y) for all (x, y) ∈ γh and in the third one we make the 2 (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 F1 (x) = 12 x2 + o(x2 ), we can apply Lemma 2.1 to conclude Z Z  ′ dx a1 hT ′ (h) = 2 a1 (x) =2 (x)ydx, γh F2′ (y) γh F1′  ′ which completes the proof of the result because an easy computation shows that a1 F1′ = 1 2 F1 (F1′ )2 . 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., k1 = k2 = 2, and that either F2 (y) = F1 (y) for all y ∈ I2 or F2 (y) = F1 (−y) for all y ∈ I2 . 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:  ′′  (a) Pσ1 (FF′1)2 has at most one zero on (0, e+ 1 ), counted with multiplicities, and 1  2  ′′  F (b) Pσ1 1 (x) + Pσ1 (F 1′ )2 (x)Pσ1 (xF1′ (x)) 6= 0 for all x ∈ (0, e+ 1 ). 1 Proof. By applying Lemma 4.10, it is clear that it suffices to study the zeros of Z  ′′ ′ L(h) := hT (h) = f1 (x)ydx for h ∈ (0, h0 ), where we set f1 := (FF′1)2 . 1 γh We claim that the assumptions guarantee the existence of an analytic function I0 on (0, h0 ) such that (I0 , L) form an ECT-system on (0, h0 ). 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, h0 ) counting multiplicities. So the  result will follow once we prove the claim. To this end we will use first the hypothesis that ϕ1 := Pσ1 f1 has at most one zero on (0, e+ 1 ), counted with multiplicities. Note that, by Lemma 2.6, ϕ1 is a σ1 -even function. Hence, by applying Proposition 4.9, there exists another analytic σ1 -even function ϕ0 such that (ϕ0 , ϕ1 ) is an ECT-system on (0, e+ 1 ). In addition, by Lemma 2.6 once again, we can write ϕ0 = Pσ1 (f0 ) for some analytic function f0 on (e− + 1 , e1 ). We choose I0 to be Z I0 (h) := f0 (x)ydx γh and accordingly we must verify that (I0 , L) is indeed an ECT-system on (0, h0 ). To this  end we shall apply Theorem 4.8 with n = 2 and g(y) = y. So far we have showed that Pσ1 (f0 ), Pσ1 (f1 ) 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 Pσ2 yF2′ (y) , Pσ2 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    2Pσ2 yF2′ (y) = yF2′ (y) − σ2 (y)F2′ σ2 (y) σ2′ (y) = F2′ (y) y − σ2 (y) > 0 for all y ∈ (0, e+ 2 ),    we only need to check that the Wronskian W Pσ2 yF2′ (y) , Pσ2 1 does not vanish on the interval (0, e+ 2 ). In this regard, setting  2  ′′  F Li (z) := Pσi 1 (z) + Pσi (F i′ )2 (z)Pσi (zFi′ (z)) for i = 1, 2, i a computation shows that the Wronskian writes as    W Pσ2 yF2′ (y) , Pσ2 1 = −F2′ (y)L2 (y). 14 Recall at this point that by hypothesis either F2 (y) = F1 (y) or F2 (y) = F1 (−y), which imply σ2 (y) = σ1 (y) and σ2 (y) = −σ1 (−y), respectively. Therefore L2 (y) = L1 (y) for all y ∈ I2 in the first case, whereas one can check that L2 (y) = L1 (−y) for all y ∈ I2 in the second case. Since L1 (y) 6= 0 for all y ∈ (0, e+ 1 ) by assumption and L1 (σ1 (y)) = L1 (y) for all y ∈ I1 , this shows that the Wronskian does not vanish on (0, e+2 ). Accordingly we can apply Theorem 4.8 and assert that (I0 , L) is indeed an ECT-system on (0, h0 ). 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) = 21 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  ′′ Fi −1 + 6x2 + 57x4 + 328x6 + 928x8 + 1592x10 + 1476x12 + 960x14 + 448x16 ′ 2 (x) = 6 (Fi ) (1 + 4x2 + 6x4 + 8x6 )4 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 σ1 = −Id, the assumption in (a) is fulfilled, whereas the condition in (b) writes as (xF1′ (x))′ = 2x + 16x3 + 36x5 + 64x7 6= 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, limh→+∞ T (h) = 0.  Finally we particularize Theorem 4.11 assuming additionally that F is an even function.  ′′ Corollary 4.13. Assume that F1 is even and F2 = F1 . If (FF′1)2 has at most one zero on (0, e+ 1 ), counted ′ + 1 with multiplicities, and xF1 (x) does not vanish on (0, e1 ), 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 Pσ1 (FF′1)2 = (FF′1)2  ′′  1 1 because F1 is an even function and σ1 = −Id. Hence Pσ1 (F ′ )2 F1 has exactly one zero on (0, e+1 ), 1 counted with multiplicities, thanks to the first assumption in the statement.  ′′ On  the other hand, using  2 F again that F1 is even, it follows that the condition Pσ1 1 (x) + Pσ1 (F 1′ )2 (x)Pσ1 (xF1′ (x)) 6= 0 for 1 all x ∈ (0, e+ + 1 ) simply writes as (xF1 (x)) 6= 0 for all x ∈ (0, e1 ), which is the second assumption in the ′ ′ statement. Hence Theorem 4.11 shows the validity of the result. 5 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) such that F1 (x) = F2 (−x). 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  ẋ = −f (y), (5) ẏ = −f (−x). This type of differential system is related with differential-delay equations because of the following result by Kapplan and Yorke [13, Theorem 1.1]. 15 Theorem (Kaplan-Yorke). Suppose that f : R −→ R is a continuous and odd function verifying xf (x) > 0 for all x 6= 0. Assume that limx→0 f (x) f (x) x = α and limx→∞ x = β exist (allowing either α or β to be 0 or ∞). Finally, suppose that limx→∞ F (x) = ∞. Then the differential-delay equation ẋ(t) = −f (x(t − 1)) has a nontrivial periodic solution x(t) of period four if either α < π2 < β or β < π2 < 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 limx→∞ 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 limh→0+ T (h) = 2π α and limh→+∞ T (h) = β , and then 2π 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 limx→∞ 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 f (x) f (x) f (x) α := lim , β− := lim and β+ := lim (6) x→0 x x→−∞ x x→+∞ x exist, allowing them to be zero or +∞. Since F ′ = f , the latter imply limx→±∞ Fx(x) 2 = 12 β± by L’Hôpital’s Rule (see [33]), so that h± := limx→±∞ F (x) ∈ R>0 ∪ {+∞} exist. Let us set H(x, y) = F (−x) + F (y) and recall that then H(P) = (0, h0 ), 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 satistying that the limits in (6) exist and that xf (x) > 0 for all x 6= 0. Then the period T (h) of the periodic orbit γh of (5) inside the energy level H = h verifies the following properties: (a)   if α > 0, 2π α lim+ T (h) = +∞ if α = 0, h→0  0 if α = +∞. (b) Assume that β+ = +∞ and β− = +∞. Then the origin is a global center and limh→+∞ T (h) = 0. (c) Assume that β+ < +∞ and β− = +∞ (respectively, β+ = +∞ and β− < +∞). (c1 ) If h+ (respectively, h− ) is finite then the center is non-global with h0 = h+ (respectively, h0 = h− ) and limh→h− T (h) = +∞. 0 (c2 ) If h+ = +∞ (respectively, h− = +∞), then the center is global and limh→+∞ T (h) is equal to π 2β+ when β+ > 0 and +∞ when β+ = 0 (respectively, 2βπ− when β− > 0 and +∞ when β− = 0). (d) Assume that β+ < +∞ and β− < +∞. (d1 ) If h+ or h− is finite then the center is non-global with h0 = min{h+ , h− } and limh→h− T (h) = +∞. 0 (d2 ) If h+ = h− = +∞, then the center is global, and   2  π lim T (h) = 2 √ 1 + √ 1 if β± ∈ (0, +∞), β+ β− h→+∞  +∞ if β− = 0 or β+ = 0. 16 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 (d2 ) 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), (c2 ) and (d2 ). Finally the fact that limh→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, h0 ). 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 = h0 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 = h0 , 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 (c2 ) and (d2 ) the periodic orbit γh 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. Proof of Theorem 5.1. On account of the hypothesis xf (x) > 0 for all x ∈ R \ {0}, the Hamiltonian verifies H(x, y) > 0 for all (x, y) ∈ R2 \ {(0, 0)} and the origin, which is a center, is the unique singular point of the differential system. Moreover the continuity of f implies H(0, 0) = 0. As in the previous section we denote by P the period annulus of the center and by γh the periodic orbit inside the energy level H = h. We also suppose that H(P) = (0, h0 ) with h0 ∈ R>0 ∪ {+∞}. An easy computation shows that the differential system (5) writes in polar coordinates as ( ṙ = R(r, θ) := − cos θf (r sin θ) − sin θf (−r cos θ), θ̇ = Θ(r, θ) := (− cos θf (−r cos θ) + sin θf (r sin θ))/r. Observe that, due to xf (x) > 0 for all x ∈ R \ {0}, Θ(r, θ) > 0 for all θ and r > 0. It is well known that this permits to express the radius of any solution of (5) in terms of the angle. Indeed, to this end we consider dr R(r, θ) = . dθ Θ(r, θ) For convenience (in order to take advantage of the symmetry), let r(θ; h) be the solution of this differential  rh rh equation with initial condition r(−π/4; h) = rh where rh is defined implicitly by means of H √ 2 , −√ 2 = h. Then, taking advantage of the symmetry of system (5) with respect to y = −x, we can express the period of the periodic orbit γh as the integral Z 3π 4 dθ T (h) = 2 . −π 4 Θ(r(θ; h), θ) Furthermore it is clear that T (h) = 2(T1 (h) + T2 (h) + T3 (h)), where Z 0 Z π Z 3π dθ 2 dθ 4 dθ T1 (h) := , T2 (h) := and T3 (h) := . −π 4 Θ(r(θ; h), θ) 0 Θ(r(θ; h), θ) π 2 Θ(r(θ; h), θ) We now proceed to prove separately each assertion in the statement. 17 (a) By the continuity of f at zero we have that limh→0+ r(θ; h) = 0 uniformly in θ. Thus, since f (−r cos θ) f (r sin θ) Θ(r, h) = cos2 θ + sin2 θ , (7) −r cos θ r sin θ the hypothesis limx→0 f (x) x = α implies that limh→0+ Θ(r(θ; h), θ) = α uniformly in θ. If α = 0 then by applying Fatou’s Lemma (see [28, Theorem 11.31] for instance) we get Z 3π   4 1 lim T (h) > 2 lim dθ = +∞. h→0+ −π 4 h→0+ Θ(r(θ; h), θ) If α ∈ R>0 ∪ {+∞} then there exist ε > 0 and K > 0 such that Θ(r(θ; h), θ) > K for all θ and r ∈ (0, ε). Consequently the integrand in T (h) is uniformly bounded by a constant and the application of the Lebesgue’s Dominated Convergence Theorem (see [28, Theorem 11.32] for instance) gives Z 3π    4 1 2π α if α ∈ R>0 , lim T (h) = 2 lim dθ = h→0+ −π h→0+ Θ(r(θ; h), θ) 0 if α = +∞, 4 as desired. (b) Note first that the hypothesis β± = +∞ implies limx→±∞ F (x) = +∞, which in turn implies that the center is global with h0 = +∞. Consequently, due to F (−r cos θ) + F (r sin θ)|r=r(θ;h) = h for all θ and h > 0, we have that r(θ; h) → +∞ as h → +∞ and θ → θ0 for any θ0 . Hence limh→+∞ r(θ; h) = +∞ uniformly in θ. This shows, on account of the lower bounds Θ(r, θ) > cos2 θ f (−r cos θ) −r cos θ > 1 f (−r cos θ) 2 −r cos θ for all θ ∈ [− π4 , π4 ] and Θ(r, θ) > sin2 θ f (r sin θ) r sin θ > 1 f (r sin θ) 2 r sin θ for all θ ∈ [ π4 , 3π 4 ], together with the assumptions limx→±∞ f (x) x = +∞, that there exist two positive real numbers k and M 4 ] and h > M . Exactly as before, since it is clear that such that Θ(r(θ; h), θ) > k for all θ ∈ [− π4 , 3π limh→+∞ Θ(r(θ; h), θ) = +∞ for all θ ∈ (− π4 , 3π 4 ), by applying the Lebesgue’s Dominated Convergence Theorem we can assert that Z 3π   4 1 lim T (h) = 2 lim dθ = 0. h→+∞ −π 4 h→+∞ Θ(r(θ; h), θ) (c) Let us consider for instance the case β+ < +∞ and β− = +∞ (the other one follows verbatim). Note in this case that β− = +∞ implies h− = limx→−∞ F (x) = +∞. With regard to the behaviour of F at +∞ there are two possibilities to consider: (c1 ) h+ = limx→+∞ F (x) < +∞ Consider the set of periodic orbits {γh }h∈(0,h0 ) inside the period annulus P and recall that each γh is inside the level curve H(x, y) = F (−x) + F (y) = h. We claim that these level curves are ovals surrounding the origin provided that h ∈ (0, h+ ), i.e., that H(P) = (0, h0 ) with h0 = h+ . In order to show the claim note first that F : (0, +∞) −→ (0, h+ ) and F : (−∞, 0) −→ (0, +∞) 18 are diffeomorphisms because xf (x) > 0 for all x 6= 0. Denote their inverses by F+−1 and F−−1 , respectively. Then F (−x) + F (y) = h gives two branches y = y± (x; h) := F±−1 (h − F (−x)) with y− < 0 < y+ . For each fixed h ∈ (0, h+ ) the map x 7−→ y+ (x; h) is well defined and continuous provided that F (−x) < h, i.e., for x such that x− (h) < x < x+ (h) where x± (h) := −F∓−1 (h). Furthermore F (−x) + F (y+ (x; h)) = h implies that y+ (x; h) tends to +∞ as (x, h) → (0, h+ ) because F (0) = 0. Hence the level set H = h+ is unbounded. It is also clear that, for each h ∈ (0, h+ ), the level set H = h is an oval surrounding the origin that can be parametrized by means of the graphs of x 7−→ y± (x; h) with x ∈ (x− (h), x+ (h)). This proves the claim. What is more, the notation introduced so far enables to write the period of the periodic orbit γh as Z x+ (h) Z x− (h) dx dx T (h) = + , x− (h) f (y+ (x; h)) x+ (h) f (y− (x; h)) where each summand is positive because xf (x) > 0 for all x 6= 0. Taking F (−x) + F (y± (x; h)) = h and limx→+∞ F (x) = h+ into account, it follows that lim y± (x; h) = 0 as (x, h) → (−∞, h+ ). Consequently, since f is continuous and f (0) = 0, there exist xc < 0 and hc ∈ (0, h+ ) such that 0 < f (y+ (x; h)) < 1 for all x ∈ (x− (h), xc ) with h ∈ (hc , h+ ). Accordingly, if h ∈ (hc , h+ ) then Z x+ (h) Z xc dx dx T (h) > > > xc − x− (h), x− (h) f (y+ (x; h)) x− (h) f (y+ (x; h)) and the result follows noting that x− (h) = −F+−1 (h) tends to −∞ as h → h+ . (c2 ) h+ = limx→+∞ F (x) = +∞ In this subscase we have h+ = h− = +∞ and so the center is global. Acordingly, exactly as we argue in (b), limh→+∞ r(θ; h) = +∞ uniformly in θ. f (x) Let us discuss the case β+ > 0 first. Then, on account of the hypothesis limx→+∞ x = β+ and the lower bound Θ(r, θ) > sin2 θ f (r sin θ) 1 f (r sin θ) r sin θ > 2 r sin θ for all θ ∈ [ 4 , 4 ], π 3π there exists M+ > 0 such that Θ(r, θ) > 14 β+ for all θ ∈ [ π4 , 3π 4 ] and r > M+ . Similarly, due to Θ(r, θ) > cos2 θ f (−r cos θ) −r cos θ > 1 f (−r cos θ) 2 −r cos θ for all θ ∈ [− π4 , π4 ], and the assumption limx→−∞ f (x) x = +∞, there exist two positive numbers k and M− such that Θ(r, θ) > k for all θ ∈ [− π4 , π4 ] and r > M− .Then, using these two lower bounds together with the fact that limh→+∞ r(θ; h) = +∞ uniformly in θ, we can assert that there exist ℓ, M > 0 such that Θ(r(θ; h), θ) > ℓ for all θ ∈ [− π4 , 3π4 ] and r > M. This gives a constant uniform upper bound for the integrands in Ti (h), i = 1, 2, 3, that enables to apply the Lebesgue’s Dominated Convergence Theorem once again. In doing so, since limh→+∞ Θ(r(θ; h), θ) = 0 for all θ ∈ (− π4 , π2 ) due to β− = +∞, we get that limh→+∞ T1 (h) = limh→+∞ T2 (h) = 0. Furthermore limh→+∞ T3 (h) = 4βπ+ because limh→+∞ Θ(r(θ; h), θ) = β+ for all θ ∈ ( π2 , 3π4 ). Therefore T (h) = 2(T1 (h) + T2 (h) + T3 (h)) tends to 2βπ+ as h → +∞. (In this case the periodic orbits γh exhibit a kind of slow-fast motion as h tends to +∞ because T3 (h) tends to 4βπ+ , whereas T1 (h) and T2 (h) tend to zero.) The case β+ = 0 is shorter because, by Fatou’s Lemma, limh→+∞ T3 (h) = +∞, which clearly suffices to show the validity of the assertion since T1 and T2 are positive. (d) Let us treat finally the case in which β+ and β− are both finite. To this end it suffices to consider the following two cases. 19 (d1 ) h+ or h− is finite: If h+ 6 h− (which in particular implies h+ < +∞) then the result follows verbatim the proof of (c1 ) because in that case, although we assume h− = +∞, it is only used that h− > h+ . That being said, if h− 6 h+ then one can reduce to the previous case by means of the change (x, y) 7−→ (−x, −y). and considering F ◦ (-Id) instead of F. (d2 ) h+ = h− = +∞: In this case the center is global, H(P) = (0, +∞) and, exactly as we argue in (b), it follows that limh→+∞ r(θ; h) = +∞ uniformly in θ. Taking this into account we proceed with the proof by distinguishing the three cases that can occur: 1. {β+ = 0 and β− = 0} Consequently, recall (7), limx→±∞ f (x) x = β± = 0 implies that limh→+∞ Θ(r(θ; h), θ) = +∞ for each fixed θ. Then, by Fatou’s Lemma, Z 3π   4 1 lim T (h) > 2 lim dθ = +∞. h→+∞ −π 4 h→+∞ Θ(r(θ; h), θ) 2. {β+ > 0 and β− > 0} f (x) Thus, thanks to the hypothesis limx→+∞ x = β+ and the lower bound Θ(r, θ) > sin2 θ f (r sin θ) r sin θ > 1 f (r sin θ) 2 r sin θ for all θ ∈ [ π4 , 3π 4 ], there exists M+ > 0 such that Θ(r, θ) > 41 β+ for all θ ∈ [ π4 , 3π 4 ] and r > M+ . Likewise, Θ(r, θ) > cos2 θ f (−r cos θ) −r cos θ > 1 f (−r cos θ) 2 −r cos θ for all θ ∈ [− π4 , π4 ] and limx→−∞ f (x) x = β− , imply the existence of M− > 0 satisfying that Θ(r, θ) > 4 β− for all 1 θ ∈ [− 4 , 4 ] and r > M− . Hence there exists M > 0 so that Θ(r(θ; h), θ) > 4 βm for all h > M π π 1 4 ]. For each i = 1, 2, 3, this provides a constant uniform upper bound for the and θ ∈ [− π4 , 3π integrand in Ti (h), which enables to apply the Lebesgue’s Dominated Convergence Theorem to compute limh→+∞ Ti (h). In doing so, since limh→+∞ Θ(r(θ; h), θ) = β− for each θ ∈ (− π4 , 0), it turns out limh→+∞ T1 (h) = 4βπ− . Similarly limh→+∞ Θ(r(θ; h), θ) = β+ for each θ ∈ ( π2 , 3π 4 ) implies limh→+∞ T3 (h) = 4βπ+ . Finally, since limh→+∞ Θ(r(θ; h), θ) = β− cos2 θ + β+ sin2 θ for each θ ∈ (0, π2 ) and, on the other hand, Z π 2 dθ π 2 = p , 0 β− cos2 θ + β+ sin θ 2 β+ β− we get that limh→+∞ T2 (h) = √ π . Therefore, taking T = 2(T1 + T2 + T3 ) into account, 2 β+ β− ! !2 π 1 2 1 π 1 1 lim T (h) = +p + = p +p . h→+∞ 2 β− β+ β− β+ 2 β+ β− 3. {β− = 0 and β+ > 0} or {β− > 0 and β+ = 0} Let us consider the first case since the second one follows similarly. Then, exactly as we did in the study of the preceding cases, limh→+∞ T (h) > limh→+∞ T1 (h) = +∞ by Fatou’s Lemma. (In contrast to limh→+∞ T1 (h) = +∞, the application of the Lebesgue’s Dominated Convergence Theorem shows that limh→+∞ T3 (h) = 4βπ+ . Hence this is another case in which the periodic orbits γh undergo a slow-fast phenomenon as h → +∞.) 20 Figure 1: Level curves for the Hamiltonian function considered in Example 5.3. This completes the proof of the result. Remark 5.2. To prove their result about differential-delay equations stated at the beginning of the section, Kaplan and Yorke show that if f is an odd function (which implies β+ = β− ) and h± = +∞ (which implies that the center is global) then limh→+∞ T (h) = β2π± . Following our notation, their proof relies in the identity Z T (h)  Θ r(t; h), θ(t; h) dt = 2π for all h ∈ (0, +∞), (8) 0 where r(t; h) and θ(t; h) are respectively the radial and angular coordinates of the solution of the differential system (5) inside the energy level H = h. (This identity follows straighforward by taking dθ dt = Θ(r, θ) into account.) Then, using that limh→+∞ r(t; h) = +∞ for each fixed t, together with the fact that f (r cos θ) f (r sin θ) Θ(r, θ) = cos2 θ + sin2 θ −→ β± as r tends to +∞ r cos θ r sin θ for each fixed θ ∈ 2 }, they claim that by making h → +∞ in (8) one gets β± limh→+∞ T (h) = 2π. / {0, π2 , π, 3π However this reasoning assumes the interchange of limit and integration, which is a delicate point.  Example 5.3. If we consider H(x, y) = F (−x) + F (y) with F (x) = 1 − e−x (1 + x) then we obtain  ẋ = −ye−y , ẏ = xex , which corresponds to the case (c1 ) in Theorem 5.1 because β+ = 0, β− = +∞, h+ = 1 and h− = +∞. Therefore the center is non-global with h0 = h+ = 1 and limh→1− T (h) = +∞. Figure 1 displays a numerical plot of the level curves H(−3, 0) ≈ 0.801 and H(−9, 0) ≈ 0.999.  Example 5.4. Likewise, the choice F (x) = 21 (1 + x2 − (1 + x)e−x ) yields to the differential system  ẋ = −y (1 + e−y ), ẏ = x (1 + ex ). 21 Figure 2: Level curves for the Hamiltonian function considered in Example 5.4. It satisfies β+ = 1, β− = +∞ and h± = +∞, so that it corresponds to the case (c2 ) in Theorem 5.1. Thus the center is global and limh→+∞ T (h) = π2 , see Figure 2. Recall that to prove this case we split the period function as T = 2(T1 + T2 + T3 ) and it occurs that limh→+∞ T1 (h) = limh→+∞ T2 (h) = 0, whereas limh→+∞ T3 (h) = 4βπ+ . Therefore the intermediate time that spends γh to cross the second quadrant tends to 2βπ+ as h → +∞, while the intermediate time to cross the other quadrants tends to zero. This kind of slow-fast phenomenon is certainly interesting to be studied.  Example 5.5. We give now two last examples for which the period function can be computed explicitly. The first one follows taking F (x) = 14 x4 , so that α = 0 and β± = +∞. Some easy computations show that T (h) = kh−1/2 with k = √8π Γ(5/4)2 , where Γ is the Gamma function. For the second one we choose F (x) = 43 x4/3 , so that α = +∞ and β± = 0. In this case T (h) = kh1/2 with k = √6 Γ(3/4)2 . 3π  Acknowledgments The first author is partially supported by MTM2017-86795-C3-2-P. The second author is partially supported by NNSF of China grants 11671254 and 11871334. The first author would like to thank the School of Mathematical Sciences of the Shanghai Jiao Tong University in People’s Republic of China for the kind hospitality during his research stay. References [1] T.R. Blows and N.G. Lloyd, The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 215–239. [2] C. 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