This is a preprint of: “Period function of planar turning points”, Renato Huzak, David Rojas, Electron. J. Qual. Theo., vol. 2021(16), 1–21, 2021. DOI: [10.14232/ejqtde.2021.1.16] Period function of planar turning points Renato Huzak Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium David Rojas Departament d’Informàtica, Matemàtica Aplicada i Estadı́stica, Universitat de Girona, Girona 17003, Catalonia, Spain Abstract This paper is devoted to the study of the period function of planar generic and non-generic turning points. In the generic case (resp. non-generic) a non-degenerate (resp. degenerate) center disappears in the limit ε → 0, where ε ≥ 0 is the singular perturbation parameter. We show that, for each ε > 0 and ε ∼ 0, the period function is monotonously increasing (resp. has exactly one minimum). The result is valid in an ε-uniform neighborhood of the turning points. We also solve a part of the conjecture about a uniform upper bound for the number of critical periods inside classical Liénard systems of fixed degree, formulated by De Maesschalck and Dumortier in 2007. We use singular perturbation theory and the family blow-up. Keywords: critical periods, family blow-up, period function, slow-fast systems. 2020 Mathematics Subject Classification: 34E15, 34E17. 1 Introduction We consider slow-fast polynomial Liénard equations of center type  Pl  ẋ = y − x2n + k=1 P ak x2n+2k  Xε,η : m (1) ẏ = ε2n −x2n−1 + k=1 bk x2n+2k−1 , where l, m, n ≥ 1, η := (a1 , . . . , al , b1 , . . . , bm ) is kept in a compact set K of Rl+m and ε ≥ 0 is the singular perturbation parameter kept small. System Xε,η is invariant under the symmetry (x, t) → (−x, −t) and has a center at the origin for all ε > 0, ε ∼ 0, and for all η ∈ K. The center is non-degenerate when n = 1 or nilpotent when n > 1. In the limit ε = 0, we encounter drastic changes in the dynamics of (1): the system has a curve of singular points, given Pl by {y = x2n + k=1 ak x2n+2k }, passing through the origin, and horizontal regular orbits (see Figure 1). A portion of the curve of singularities near the origin consists of the normally attracting part {x > 0}, the normally repelling 1 Figure 1: Dynamics of X0,η with 5 contact points. part {x < 0} and the contact point (x, y) = (0, 0) between them. We call the contact point a turning point because closed orbits surrounding the center, for ε > 0 small, pass from the attracting part to the repelling part of the curve of singularities. When n = 1, the turning point at the origin is generic (sometimes called simple). When n > 1, we deal with a non-generic or degenerate turning point. The period function of a center assigns to each periodic orbit its minimal period. Isolated critical points of the period function are called critical periods (or critical periodic orbits) and are central in the qualitative study of the period function. One can note that critical periods do not depend on the parametriza- tion of the set of periodic orbits used. The main purpose of this paper is to give a complete local study of the period function of Xε,η , near the center at the origin, in both the generic and non-generic case. The study is valid in a small fixed neighborhood of the turning point that is independent of (ε, η). Thus, the neighborhood does not shrink to the origin as ε → 0. In the generic (resp. non- generic) case, the period function of the center in Xε,η is strictly monotonous increasing (resp. has exactly one critical period which is a minimum). More precisely, let us denote by T (y; ε) the period function of the center at the origin of system (1) with ε > 0, ε ∼ 0, parametrized by the positive y-axis. Then we have: Theorem 1. Let l, m ≥ 1 and n = 1 (resp. n > 1) be fixed. For any compact K ⊂ Rl+m there exist ε0 > 0 and y0 > 0 small enough such that the period function T (y; ε) of the center of system (1) is strictly monotonous increasing (resp. has a global minimum) in the interval ]0, y0 ], for all ε ∈]0, ε0 ] and η ∈ K. We prove Theorem 1 in Section 3.4. To prove Theorem 1, we use a blow- up at the origin in the (x, y, ε)-space to desingularize system (1). Roughly speaking, after the blow-up we distinguish between “very small”, “small” and “intermediate” closed orbits surrounding the center (x, y) = (0, 0). The period function of the center of system (1) cannot be studied uniformly in these three regions and we have to use different techniques for each type of closed orbits. To treat the period function near the very small closed orbits (the ones closest to the center), we use Chicone and Jacobs [2], in the generic case, and generalized polar coordinates, in the non-generic case. The small closed orbits can be treated using the monotonicity criterion due to Schaaf [11], in the generic case, and a result due to Sabatini [10], in the non-generic case. The size of the very small 2 and small closed orbits tends to zero as ε → 0. In order to obtain the result in an (ε, η)-uniform neighborhood of (x, y) = (0, 0), the period function near the passage from the small closed orbits to detectable closed orbits in the (x, y)- space has to be studied. In this passage, we encounter the so-called intermediate closed orbits. The period function near such intermediate closed orbits, in both the generic and non-generic case, will be studied using techniques from [6, 8], where small-amplitude limit cycles in an ε-uniform neighborhood of slow-fast Hopf points have been investigated (the slow-fast Hopf points correspond to the generic case). For more details we refer to Section 2. We point out that Theorem 1 can be proved in a more general framework of smooth Liénard systems. More precisely, the same local result is true  if we replace (1) with {ẋ = y − x2n + O(x2n+2 ), ẏ = ε2n −x2n−1 + O(x2n+1 ) } where O(x2n+2 ) (resp. O(x2n+1 )) is an even (resp. odd) C ∞ -perturbation term that may depend on parameters kept in a compact set. The proof in this more general setting is analogous to the proof for polynomial Liénard equations presented in this paper. Theorem 1, in the generic case n = 1, can be used to solve a part of the following conjecture formulated in [4]: there exists a uniform upper bound on the number of critical periods of classical Liénard equations {ẋ = y − G(x), ẏ = −x} where G is an even polynomial of degree 2N , N ≥ 1, and G(0) = 0. Moreover, this upper bound is conjectured to be 2N − 2. Following Theorem 5 in [4], this can be reduced to the following problem: there exist a small ε0 > 0 and an integer M > 0 such that the slow-fast system  PN −1  ẋ = y − x2N + k=1 c2k x2k (2) ẏ = −εx has at most M critical periods, for all ε ∈]0, ε0 ] and (c2 , c4 , . . . , c2(N −1) ) ∈ SN −2 . The following result covers the case where the curve of singularities of (2), at level ε = 0, has only one contact point, the one at the origin (x, y) = (0, 0). Theorem 2. Let c02 > 0 be small and fixed and let N ≥ 1 be a fixed integer. Denote by C the set of all values (c2 , c4 , . . . , c2(N −1) ) ∈ SN −2 such that c2 ≥ c02 0 PN −1 and G x(x) > 0 for all x ∈ R, where G(x) = x2N + k=1 c2k x2k . For any compact set C̃, with C̃ ⊂ C, there exists a small ε0 > 0 such that system (2) has no critical periods for all ε ∈]0, ε0 ] and (c2 , c4 , . . . , c2(N −1) ) ∈ C̃. We prove Theorem 2 in Section 3.5. Note that keeping the parameter in a compact set C̃ ensures that the critical curve has no contact points other than the origin. The compact set C̃ = {(c2 , c4 , . . . , c2(N −1) ) ∈ SN −2 | c2 ≥ c02 and ci ≥ 0 for i = 4, . . . , 2(N − 1)} is always contained in the set C defined in Theorem 2. When N = 1, Theorem 2 implies that {ẋ = y − x2 , ẏ = −εx} has no critical periods for all ε ∈]0, ε0 ], for some small ε0 > 0. When N = 2, we have to deal with the slow-fast systems {ẋ = y − x4 ± x2 , ẏ = −εx}. From Theorem 2 follows that the system with the negative sign in front of x2 has no critical periods. The system with the positive sign in front of x2 is conjectured to have at most 2 critical periods (see [4]). As explained in [4], it is more difficult to deal with the part of the conjecture when the curve of singularities of (2) has more contact points. 3 When c2 is uniformly nonzero, Theorem 1 in the generic case implies that system (2) has no critical periods in an ε-uniform neighborhood of the origin in the (x, y)-space. It suffices to notice that the change of coordinates (x, y) → (c2 x, c2 y) transforms (2) into (1). See Section 3.5. 2 Blow-up and statement of results 2.1 Family blow-up at the origin in the (x, y, ε)-space To be able to study the period function near the turning point, uniformly in (ε, η) with ε > 0 small, we have to desingularize the system Xε,η near (x, y, ε) = (0, 0, 0) using the so-called family blow-up. The family blow-up is the following “singular” coordinate change with n ≥ 1: Ψ : R+ × S2+ → R3 : (r, (x̄, ȳ, ε̄)) 7→ (x, y, ε) = (rx̄, r2n ȳ, rε̄), ε̄ ≥ 0. ∂ We define the blown-up vector field as the pullback of Xε,η + 0 ∂ε divided by 2n−1 1 ∗ ∂  r : X̄η := r2n−1 Ψ Xε,η + 0 ∂ε . To study the blown-up vector field X̄η (or r2n−1 X̄η ) near the blow-up locus {0} × S2+ , it is convenient to use different charts with “rectified” coordinates, instead of the spherical coordinates. For our purposes, only the family chart {ε̄ = 1} and the phase directional chart {ȳ = 1} are relevant for the study of the period function since all closed orbits near the center (x, y) = (0, 0) are visible therein (see Figure 2). ȳ = 1 p+ p− ε̄ = 1 Figure 2: The family blow-up at the origin (x, y, ε) = (0, 0, 0) and dynamics on the blow-up locus. To study the period function of (1) in an (ε, η)-uniform neighborhood of (x, y) = (0, 0), it suffices to use the charts {ε̄ = 1} and {ȳ = 1}. In the family chart {ε̄ = 1}, we have (x, y, ε) = (rx̄, r2n ȳ, r) 4 with (x̄, ȳ) kept in an arbitrary but fixed compact set. In this chart, r = ε and system (1) becomes XF := ε2n−1 X̄F , where  Pl  x̄˙ = ȳ − x̄2n +P k=1 ak ε2k x̄2n+2k X̄F : m (3) ȳ˙ = −x̄2n−1 + k=1 bk ε2k x̄2n+2k−1 . System (3) is invariant under the symmetry (x̄, t) → (−x̄, −t) and has a center at the origin, for all ε ≥ 0, ε ∼ 0 and η ∈ K. When ε = 0, we are located on the blow-up locus and the vector field (3) becomes  x̄˙ = ȳ − x̄2n (4) ȳ˙ = −x̄2n−1 . A first integral of (4) is given by   −2nȳ 2n 1 H(x̄, ȳ) = e ȳ − x̄ + . (5) 2n 1 Note that the invariant curve {ȳ = x̄2n − 2n } is the boundary of the period annulus (see Figure 2). The main advantage of the family blow-up is that the blown-up vector field (3) has no curves of singularities. The (x̄, ȳ)-compact sets in which we will study system (3) (see Section 2.2) shrink to the origin in the (x, y)-space as ε → 0. To obtain (ε, η)-uniform results, we also have to study Xε,η in the phase directional chart {ȳ = 1}. In the chart {ȳ = 1}, we deal with the coordinate change (x, y, ε) = (RX, R2n , RE), where R ≥ 0 and E ≥ 0 are small and X is kept in any compact set. System (1) becomes XD := R2n−1 X̄D , where  Pl  1  Ẋ = 1 − X 2n + k=1 ak R2k X 2n+2k + 2n 2n XE F (X, R, η) 1 X̄D : Ṙ = − 2n RE 2n F (X, R, η) (6)  1 2n+1 Ė = 2n E F (X, R, η) Pm with F (X, R, η) = X 2n−1 − k=1 bk R2k X 2n+2k−1 . For R = E = 0, the sys- tem has semi-hyperbolic singularities at X = −1 (denoted by p+ ) and X = 1 (denoted by p− ). The singularity p+ (resp. p− ) has the X-axis as unstable (resp. stable) manifold and a two-dimensional center manifold, transverse to the X-axis. Using (3) and (6) we easily detect the singular polycycle Γ on the blow- up locus consisting of singularities p+ and p− and the regular orbits that are heteroclinic to them (see Figure 2). Note that p± are the end points of the 1 regular curve {ȳ = x̄2n − 2n }. It is clear now that the study of the period function of the center in (1), in a small ε-uniform neighborhood of (x, y) = (0, 0), can be divided into three parts: the study near the center (x̄, ȳ) = (0, 0) of (3), the study of the interior of the period annulus inside the family (3), away from (x̄, ȳ) = (0, 0) and Γ, and the study near Γ, combining systems (3) and (6). The results related to the first two parts (resp. the third part) are stated in Section 2.2 (resp. Section 2.3). 5 2.2 Statement of results inside the vector field X̄F Let l, m ≥ 1 be fixed. For the vector field X̄F given in (3) we define by TF (ȳ; ε) the period function of the center at the origin parametrized by the ȳ-axis. As we will see in Sections 3.1 and 3.2, the function TF (ȳ; ε) is well defined in any compact interval [ȳ1 , ȳ2 ] and when the turning point is generic it can be extended analytically to ȳ = 0. We prove the following two results concerning the period function of system X̄F . The first one states the behaviour of the period function of the center of system (3) close to the equilibrium at the origin, whereas the second one is a global statement in the interior of the period annulus. Theorem 3. For any compact K ⊂ Rl+m there exist ȳ0 > 0 small enough and d d ε0 > 0 small enough such that dȳ TF (ȳ; ε) > 0 (resp. dȳ TF (ȳ; ε) < 0) for n = 1 (resp. n > 1) for all ȳ ∈]0, ȳ0 ], ε ∈ [0, ε0 ] and η ∈ K. Moreover, TF (ȳ; ε) → 2π (resp. +∞) as ȳ → 0+ for n = 1 (resp. n > 1). Theorem 4. For any compact K ⊂ Rl+m and any 0 < ȳ1 < ȳ2 < +∞ there d d2 exists ε0 > 0 small enough such that dȳ TF (ȳ; ε) > 0 (resp. dȳ 2 TF (ȳ; ε) > 0) for n = 1 (resp. n > 1) for all ȳ ∈ [ȳ1 , ȳ2 ], ε ∈ [0, ε0 ] and η ∈ K. We prove Theorem 3 in Section 3.1 and Theorem 4 in Section 3.2. A key fact in the proof of the previous results is that, when ε = 0, the vector field (3) becomes (4) with a first integral given by (5). As we will see, the period function of (3) is an ε-perturbation of the period function of (4). 2.3 Statement of results near Γ In both the generic and non-generic case, we have the following result about the period function of the center of the vector field r2n−1 X̄η , with r > 0, in an η-uniform neighborhood of Γ. Theorem 5. Let l, m ≥ 1 be fixed. For any compact K ⊂ Rl+m there exists ε0 > 0 small enough such that the period function of the center of system r2n−1 X̄η , with r > 0, near the polycycle Γ is monotonous increasing for all ε ∈]0, ε0 ] and η ∈ K. We prove Theorem 5 in Section 3.3. For a precise definition of a neighbor- hood of Γ in the family blow-up coordinates and the period function near Γ see Section 3.3. 3 Proof of Theorem 1–Theorem 5 First we prove Theorem 3, Theorem 4 and Theorem 5. Then we glue them together and prove Theorem 1 (see Section 3.4). Theorem 2 is proved in Section 3.5. 3.1 Proof of Theorem 3 Let us start considering the case n = 1. We define by TF (x̄; ε) the period function of system (3) parametrized by the x̄-axis. Notice that, since n = 1, the center at the origin is non-degenerate and therefore the period function can 6 be extended analytically to x̄ = 0. For ε ≥ 0 small system (3) is an analytic perturbation of the quadratic system  x̄˙ = ȳ − x̄2 (7) ȳ˙ = −x̄. Therefore we can consider the Taylor’s series development at ε = 0 of TF (x̄; ε), TF (x̄; ε) = T0 (x̄) + O(ε), where T0 (x̄) is the period function of system (7) parametrized by the x̄-axis. In d d particular, if dx̄ T0 (x̄) > 0 then dx̄ T (x̄; ε) > 0 for every ε ≥ 0 small enough. In consequence, the assertion concerning n = 1 in Theorem 3 will follow once we show that the period function T0 (x̄) of the quadratic system (7) is monotonous increasing near the origin. To do so we use Chicone and Jacobs [2] result on quadratic centers to deduce that, in a neighbourhood of the origin, T0 (x̄) = 2π + p2 (λ)x̄2 + O(x̄3 ), π where p2 (λ) = 12 (16λ22 +8λ2 λ5 +λ25 +18λ23 −12λ3 λ6 +9λ3 λ4 +10λ26 −λ4 λ6 +λ24 ), and λi stand for the coefficients of the Bautin’s normal form for quadratic systems  ẋ = −y − λ3 x2 + (2λ2 + λ5 )xy + λ6 y 2 ẏ = x + λ2 x2 + (2λ3 + λ4 )xy − λ2 y 2 . In our case system (7) can be brought to the Bautin’s normal form with the change of variable {ȳ 7→ −ȳ} and corresponds to the parameters λ2 = λ5 = λ6 = 0, λ3 = 1 and λ4 = −2. Consequently, for system (7) the period function near the origin can be written as π 2 T0 (x̄) = 2π + x̄ + O(x̄3 ). 3 This fact, together with the discussion at the beginning of the section, shows d that there exist ε0 , x̄0 > 0 small such that dx̄ TF (x̄; ε) > 0 for x̄ ∈]0, x̄0 ] and ε ∈ [0, ε0 ]. Since monotonicity is unaltered by parametrization, this finishes the proof of Theorem 3 for the case n = 1. For n > 1 the center at the origin becomes degenerate and Chicone-Jacobs procedure do not apply. With the aim of studying the period function of system (3) near the origin (x̄, ȳ) = (0, 0) for n > 1 we consider the change to generalized polar coordinates (x̄, ȳ) = (r cos θ, rn sin θ) with r ≥ 0 and θ ∈ T. After this change system (3) is written as    rn  ṙ = 2 2 cos θ sin θ − cos2n−1 θ sin θ + O(r) cos θ + n sin θ  rn−1   θ̇ = − n sin 2 θ − cos 2n θ + O(r) . 2 cos2 θ + n sin θ We note that terms with ε small are inside O(r) so the forthcoming arguments are uniform with respect to ε ∈ [0, ε0 ]. 7 For r > 0 small enough we have θ̇ < 0. Therefore we can parametrize the orbits near the origin by ϕ := −θ. We denote by TF (s; ε) the period of the solution r(ϕ, s) and for the sake of simplicity we write f (ϕ) := cos2 ϕ + n sin2 ϕ, α(ϕ) := cos2n−1 ϕ sin ϕ − cos ϕ sin ϕ, β(ϕ) := n sin2 ϕ + cos2n ϕ. Note that β(ϕ) > 0. Due to the symmetry of system (3) the function TF (s; ε) writes Z π Z π 2 dϕ 2 f (ϕ)dϕ TF (s; ε) = 2 =2 . −π 2 ϕ̇ −π 2 r(ϕ, s)n−1 β(ϕ) + O(r(ϕ, s)) Moreover, d r(ϕ, s) dϕ α(ϕ) α(ϕ) = + O(r(ϕ, s)) = + O(s), r(ϕ, s) β(ϕ) β(ϕ) where in the second equality we use r(ϕ, s) = O(s). Therefore, R ϕ α(φ)  R ϕ α(φ)  +O(s) dφ r(ϕ, s) = r(0, s)e 0 β(φ) = s e 0 β(φ) dφ + O(s) . Rϕ α(φ) We denote ρ(ϕ) := e 0 β(φ) dφ > 0. Substituting the previous equality in the expression of TF (s; ε) and taking into account that O(r(ϕ, s)) = O(s) we get Z π 2 2 f (ϕ)dϕ TF (s; ε) = n−1  sn−1 −π 2 ρ(ϕ) + O(s) β(ϕ) + O(s) Z π2   2 f (ϕ) = + O(s) dϕ sn−1 −π 2 ρ(ϕ)n−1 β(ϕ) Z π2 ! 2 f (ϕ)dϕ = + O(s) . sn−1 −π2 ρ(ϕ)n−1 β(ϕ) Since f , ρ and β are positive, the last equality shows that TF (s; ε) → +∞ as s → 0+ for n > 1. Moreover, Z π2 ! d 2(n − 1) f (ϕ)dϕ 2 TF (s; ε) = − n n−1 + O(s) + n−1 O(1) ds s −π 2 ρ(ϕ) β(ϕ) s Z π2 ! 1 f (ϕ)dϕ = n −2(n − 1) + O(s) . s π ρ(ϕ)n−1 β(ϕ) −2 d The last equality shows that ds TF (s; ε) → −∞ as s → 0+ . This ends the proof of Theorem 3 for n > 1. 3.2 Proof of Theorem 4 In order to study the global behaviour of the period function of system (3) uniformly on ε ≥ 0 small in a compact set inside the period annulus it is enough to study the period function of the system (4), that is when ε = 0. We denote by T0 (ȳ) the period function of system (4) parametrized by the positive ȳ-axis, and we consider ȳ inside an arbitrary compact interval [ȳ1 , ȳ2 ] with 0 < ȳ1 < ȳ2 < +∞. By continuity with respect to the small parameter ε of system (3), taking ε small enough the ȳ-axis is also transversal to all orbits of 8 (3), which are also periodic for ȳ ∈ [ȳ1 , ȳ2 ]. We can then define TF (ȳ; ε) as the period function of system (3) parametrized by the same ȳ as T0 . The function TF (ȳ; ε) is analytic for ε ≥ 0, ε ∼ 0, and so we can consider its Taylor’s series development at ε = 0, TF (ȳ; ε) = T0 (ȳ) + O(ε). Then, since the center of system (4) is not isochronous, properties of the period function of system for ε = 0 are reflected for ε ≥ 0 small enough. In particular, d d2 d dȳ T0 (ȳ) > 0 and dȳ 2 T0 (ȳ) > 0 for all ȳ ∈ [ȳ1 , ȳ2 ] will imply dȳ TF (ȳ; ε) > 0 and d2 > 0 for all ȳ ∈ [ȳ1 , ȳ2 ] and ε ≥ 0 small, respectively. For this reason, dȳ 2 TF (ȳ; ε) Theorem 4 is a consequence of the following result. Proposition 1. The period function of the center (4) is strictly monotone in- creasing for n = 1 and it is strictly convex for n > 1. The first part of the proof of Proposition 1 relies on the application of the following monotonicity criterion due to Schaaf [11]. Theorem 6. Consider a Hamiltonian system of the form u̇ = −v, v̇ = g(u), where function g satisfy the following assumptions: 1. g : R → R is three times continuously differentiable with g(0) = 0 and g 0 (0) > 0.  2. For all u ∈ R where g 0 (u) > 0: 5(g 00 )2 − 3g 0 g 000 (u) > 0. 3. If g 0 (u) = 0 then g(u)g 00 (u) < 0. Then the origin is a center and the period function is strictly increasing in the whole period annulus. One of the key elements to prove the second part of Proposition 1 is to show that at most one critical period can exist in the interior of the period annulus. To do so we use the following result due to Sabatini [10]. For the sake of shortness in the statement, we define the following operator for smooth functions defined on an interval I: 3g 2 g 002 − 3gg 02 g 00 − g 2 g 0 g 000 K[g] := . g 04 Theorem 7. Consider a Hamiltonian of the form H(u, v) = G(u)+F (v), where G(u) = αu2k + o(u2k ) ∈ C ∞ (IG ), F (v) = βv 2` + o(v 2` ) ∈ C ∞ (IF ), 0 ∈ IG ∩ IF , 0 < k, ` ∈ N, α, β > 0. Here IG and IF denote the maximal interval of definition of G and F , respectively. Then the origin is a center and if   GG00 F F 00 µs2 := 4 1 + 2 02 + K[G] + K[F ] >0 G F 02 then the period function is strictly convex in the whole period annulus. Proof of Proposition 1. The change of variables {u = ln(1+2n(ȳ − x̄2n )), v = x̄} transforms (4) into the Hamiltonian system with separable variables  u̇ = −2nv 2n−1 (8) v̇ = Vn0 (u), 9 1 where Vn (u) = 2n (eu − u − 1). We notice that both periodic functions of system (8) and (4) are the same through the change of variable. We shall prove the results for (8). Let us prove the first assertion of the statement. To do so, we apply Schaaf’s criterion to system (8) with n = 1. After a positive constant rescaling of time and taking g = V10 we have that the assumptions in Theorem 6 are fulfilled since g 0 (u) = V100 (u) = 12 eu > 0 for all u ∈ R and 5(g 00 )2 − 3g 0 g 000 (u) = 12 e2u > 0 for all u ∈ R. Therefore the period function of system (8) is strictly increasing and d so dȳ T0 (ȳ) > 0 for all ȳ > 0. This proves the assertion concerning n = 1. Let us consider n > 1. With the aim of applying Theorem 7 we denote 1 G(u) = Vn (u) = 2n (eu − u − 1) and F (v) = v 2n . Clearly the first part of the 1 2 assumptions of the theorem are fulfilled since Vn (u) = 4n u + o(u2 ). We claim 1 that µs2 ≥ n2 > 0 for all n ≥ 2. After showing the inequality, the result follows by direct application of Theorem 7. Using the expressions of F and G we have that 1 4eu µ̂s2 (u, n) := µs2 (u, n) − = η(u, n), n2 n(eu − 1)4 where η(u, n) = nu2 + (1 + 3n)u + 2n + 1 + (2nu2 − 2u − 3n − 3)eu + ((1 − 3n)u + 3)e2u + (n − 1)e3u . A direct computation shows that d 4(eu − u − 1)eu µ̂s2 (u) = >0 dn n2 (eu − 1)2 for all u ∈ R. Therefore to prove the claim it is enough to show that η(u, 2) ≥ 0. We perform a derivation-division procedure with respect to eu achieving the following equality: 3  3  −u d −u d e e η(u, 2) = 216eu − 40u − 156. du3 du3 The previous expression has exactly two simple negative zeros. Indeed, its derivative is zero only at u = ln(5/27), the image at u = 0 is positive and the limits u → ±∞ are both +∞. A sequence of simple arguments of continuity, number of zeros of the derivative, the values at u = 0 and the values of the limits at ±∞ yields to show that η(u, 2) ≥ 0 for all u ∈ R. This finishes the proof of the claim. 3.3 Proof of Theorem 5 We define a section Σ1 ⊂ {X = 0} parametrized by (R1 , E1 ) ∈ [0, R10 ] × [0, E10 ] for some small R10 , E10 > 0. The section Σ1 is defined using the coordinates (X, R, E) of (6) (we write (R1 , E1 ) instead of (R, E) to avoid confusion later). Similarly, we define Σ4 ⊂ {x̄ = 0} parametrized by (ȳ, ε), with ε ∈ [0, R10 E10 ], where (x̄, ȳ, ε) are the coordinates of (3). The sections Σ1 , Σ4 are transverse to the blown-up vector field X̄η and located near the polycycle Γ (see Figure 3). Since system (3) (resp. (6)) is invariant under the symmetry (x̄, t) → (−x̄, −t) (resp. (X, t) → (−X, −t)), it suffices to study the time spent between Σ1 and Σ4 , i.e. the half time period function of r2n−1 X̄η , denoted by H. Our goal is to prove that LH > 0 on Σ1 (for R10 , E10 > 0 small enough but fixed), with ε > 0, 10 Σ4 R=0 RE = ε Σ3 Σ1 Σ2 p+ RE = ε p− E R X E=0 (a) (b) Figure 3: (a) Closed orbits near the polycycle Γ, inside RE = ε, for a fixed ε > 0. Γ is located on the blow-up locus {r = 0} (it corresponds to {R = 0} in the phase directional chart). The center is visible in the family chart. (b) The study of the time spent inside {x ≥ 0} is divided into three parts: Σ1 → Σ2 , Σ2 → Σ3 and Σ3 → Σ4 . In the first two parts, we use the vector field XD , and in the last part we use XF . ∂ ∂ where L is the Lie-derivative along the vector field R ∂R − E ∂E (see Section 2n−1 3.3.5). This implies that r X̄η (r > 0) has no critical periods near Γ and the period function is monotonous increasing there. When ε = 0, system (1) has no center. We split up H in three parts: the time H1,2 spent between Σ1 and Σ2 (Section 3.3.2), the time H2,3 spent between Σ2 and Σ3 , near the semi-hyperbolic singularity p− (Section 3.3.1), and the time H3,4 between Σ3 and Σ4 (Section 3.3.3). In Section 3.3.4 we glue the local results together. Section 3.3.5 is devoted to the study of the Lie-derivative LH. 3.3.1 The study of H2,3 In this section we study the time H2,3 inside the family XD , i.e. X̄D multiplied by R2n−1 . First, we bring X eD := F (X, R, η)−1 X̄D , locally near p− = (1, 0, 0), to a normal form which simplifies the study of H2,3 (transverse sections Σ2,3 will be defined in the normal form coordinates). Since p− is partially hyperbolic for all η ∈ K, there exists a C k η-family of center manifolds at p− , given as a graph of X = 1 + ψ(R, E, η) with ψ(0, 0, η) ≡ 0. Following [5] in the generic case or [3] in the non-generic case, an η-family of center manifolds can be chosen to be C ∞ (i.e. ψ can be C ∞ ). We fix such ψ. Using the coordinate change Z = X − (1 + ψ(R, E, η)), the fixed family of 11 eD changes to center manifolds becomes {Z = 0} and the vector field X   Ż = − (Φ(R, E, η) + O(Z)) Z 1 Ṙ = − 2n RE 2n (9)  1 Ė = 2n E 2n+1 , where Φ is a smooth function with Φ(0, 0, η) = 2n. We used the fact that the eD . Now, we can normally linearize family of center manifolds is invariant for X the vector field (9) using Theorem 1.1 of [7]: Theorem 8. There is a smooth family Πη : (Z, R, E) → (Z̄, R, E) of local diffeomorphisms, defined in an η-uniform neighborhood of the origin in the (Z, R, E)-space, which brings (9) into the normally linearized vector field   Ż = −Φ(R, E, η)Z 1 X̂D : Ṙ = − 2n RE 2n (10)  1 Ė = 2n E 2n+1 , where Φ is defined in (9) and where we denote Z̄ again by Z. The diffeomor- phisms Πη preserve {RE = const}: Πη (Z, R, E) = (Z(1 + Zπη (Z, R, E)), R, E) with a smooth family πη . Remark 1. The coordinate change in the normal linearization theorem from [7] is C ∞ -smooth and preserves the parameter η and the leaves of the foliation {RE = const} (the center variables R, E are preserved). In [6], this normal linearization theorem has been used in the generic case (see also Remark 1.2 in [7]). In the same way we apply it to the non-generic case. We point out that we could also use C k center manifolds and the normal linearization theorem of [1] with a C k -coordinate change that preserves η and {RE = const}. The size of the domain of the coordinate change may tend to zero as k → ∞. The finite smoothness is not a problem in our proof. We conclude that, in the normal form coordinates (Z, R, E) of (10), XD can be written as R2n−1 κ(Z, R, E, η)X̂D , (11) where κ(Z, R, E, η) = F (Z(1 + O(Z)) + 1 + ψ, R, η) and κ(0, 0, 0, η) = 1. In the normal form coordinates, we define Σ2 ⊂ {Z = −Z0 }, parametrized by (R2 , E2 ) ∈ [0, R20 ] × [0, E20 ] for some small constants Z0 , R20 , E20 > 0, and Σ3 ⊂ {E = E3 }, parametrized by (Z, R) with Z ∼ 0 and R ∈ [0, R3 ] for some small constants R3 , E3 > 0. All the constants are chosen such that the transverse sections Σ2,3 are located in the domain of Π−1 η and such that the passage w.r.t. X̂D between Σ2 and Σ3 is well-defined. We can now find the time H2,3 (R2 , E2 ) in (11), spent between Σ2 and Σ3 . Note that the orbit of X̂D (or (11) with R > 0) with the initial point (R2 , E2 ) > (0, 0) on Σ2 has the form R2 E2 (Z(E, R2 , E2 ), , E) E R E Φ( R2sE2 ,s,η)  with Z(E, R2 , E2 ) = −Z0 exp −2n E2 s2n+1 ds . Using this, the time H2,3 can be written as Z E3 2n dE H2,3 (R2 , E2 ) = . (12) (R2 E2 )2n−1 E2 E 2 κ(Z(E, R2 , E2 ), R2EE2 , E, η) 12 Since |Z(E, R2 , E2 )| ≤ Z0 for E ≥ E2 and κ is positive and bounded for (Z, R, E) ∼ (0, 0, 0) and η ∈ K, it is clear that (12) tends to +∞ as ε = R2 E2 → 0, uniformly in η. (Note that the integral in (12) is of order O( E12 ).) We will use the expression (12) in Section 3.3.4. We conclude this section with a result about the transition map of X̂D between Σ2 and Σ3 . Proposition 2. There is a C ∞ -function J in (R2 , E2 , E22 ln E2 , η) such that the transition map (R2 , E2 ) → (Z, R) along the trajectories of (10) between Σ2 and Σ3 is given by R = RE2 E3 2 and   1 2 Z = −Z0 exp − 2n J(R2 , E2 , E2 ln E2 , η) E2 with J(0, 0, 0, η) = 2n. Proof. When n = 1, the proof of the proposition can be found in [6] (Proposition 4.9). The proof of the case “n > 1” is analogous to the proof of the case “n = 1”. Proposition 2 implies that the transition map between Σ2 and Σ3 is C ∞ - smooth in (R2 , E2 , η). This will be used in the gluing process in Section 3.3.4. 3.3.2 The study of H1,2 In this section we deal with the time H1,2 , spent between Σ1 and Σ2 , inside the vector field XD . The smooth sections Σ1,2 are defined above. Note that the system (6) has no singularities between Σ1 and Σ2 (since the section Σ2 is located uniformly away from the singularity p− , the X-component of (6) is strictly positive between Σ1 and Σ2 , for all (R, E) ∼ (0, 0) and for all η kept in the compact set K). Since XD is (6), multiplied by R2n−1 , it can be seen that 1 H1,2 (R1 , E1 ) = I1 (R1 , E1 , η) , (13) R12n−1 where I1 is a strictly positive C ∞ -function. We conclude this section with Proposition 3. There exists a C ∞ -function J(R1 , E1 , η) such that the transi- tion map (R1 , E1 ) → (R2 , E2 ) along the trajectories of (6) between Σ1 and Σ2 is given by  (R2 , E2 ) = R1 (1 + E12n J(R1 , E1 , η)), E1 (1 + E12n J(R1 , E1 , η))−1 . Proof. In the generic case (n = 1), the proof of the proposition is given in [6] (Proposition 5.1). The proof of the non-generic case (n > 1) is analogous to the proof of the generic case. We use (13) and Proposition 3 in Section 3.3.4. 13 3.3.3 The study of H3,4 In this section we deal with the time H3,4 , spent between Σ3 and Σ4 , inside the vector field XF (XF is equal to (3), multiplied by a constant ε2n−1 = (RE)2n−1 ). The smooth sections Σ3,4 are defined above. If we parametrize Σ3 with (x̄, ε) ((x̄, ȳ, ε) are the coordinates of (3)), then we can write H3,4 as 1 H3,4 (x̄, ε) = I3 (x̄, ε, η) , (14) ε2n−1 where I3 is a strictly positive C ∞ -function. This follows from the fact that the vector field (3) is regular along Γ on the blow-up locus, between Σ3 and Σ4 (see Figure 3). 3.3.4 The study of H In this section we glue together the local results obtained in Sections 3.3.1–3.3.3 and find an expression for the half time period function H. We know that H(R1 , E1 ) = H1,2 (R1 , E1 ) + H2,3 (R2 , E2 ) + H3,4 (x̄, ε), where the orbit of r2n−1 X̄η (X̄η is the blown-up vector field defined in Section 2) with the initial point (R1 , E1 ) ∈ Σ1 intersects section Σ2 at the point (R2 , E2 ) and section Σ3 at the point (x̄, ε). From (12) follows that Z E3 2n dE H2,3 (R2 , E2 ) = , (15) (R1 E1 )2n−1 E2 E 2 κ(Z(E, R2 , E2 ), R1EE1 , E, η) where R2 and E2 are the C ∞ -functions of (R1 , E1 , η) given in Proposition 3. Here we used that ε = R1 E1 = R2 E2 . Now, we want express H3,4 (x̄, ε) in terms of (R1 , E1 ). Let us recall that the constant E3 > 0 comes from the definition of Σ3 . Using x̄ = EX3 and X = Z(1 + O(Z)) + 1 + ψ( RE1 E3 1 , E3 , η) on Σ3 (O(Z) is a C ∞ -function, see Section 3.3.1), and the fact that Z is a C ∞ -function in (R1 , E1 , η) (we combine Proposition 2 and Proposition 3), we see that x̄ is a C ∞ -function of (R1 , E1 , η) and ε = R1 E1 . This and (14) imply that 1 H3,4 (x̄, ε) = I˜3 (R1 , E1 , η) (16) (R1 E1 )2n−1 where I˜3 is a strictly positive C ∞ -function. Combining (13), (15) and (16), we finally get Z E3 ! 2n dE H(R1 , E1 ) = R 1 E1 + I(R1 , E1 , η) , (R1 E1 )2n−1 2 E2 E κ(Z(E, R2 , E2 ), E , E, η) (17) where I is a C ∞ -function (thus, bounded). Note that the H2,3 -contribution is dominant in (17) and that H(R1 , E1 ) tends to +∞ as ε = R1 E1 → 0, uniformly in η. We know that R2 = R1 (1 + o(1)) and E2 = E1 (1 + o(1)) where the o(1)- terms are C ∞ -functions of (R1 , E1 , η), equal to 0 when E1 = 0. In Section 3.3.5 we show that the Lie-derivative of the integral in (17) is of order O( E11 ). 14 3.3.5 Lie-derivative of H When we fix any value of (ε, η), with ε > 0 small, H is 1-variable function defined on interval {(R1 , E1 ) ∈ Σ1 | R1 E1 = ε} (see Figure 3(b)). To study critical periods of H on such intervals, we define the Lie-derivative of H along ∂ ∂ the vector field R1 ∂R 1 − E1 ∂E 1 (it is tangent to the intervals and without singularities there): ∂H ∂H LH := R1 − E1 . ∂R1 ∂E1 It can be easily seen that the Lie-derivative of a C ∞ -function in (R1 , E1 , η) (e.g. Ie in (17)) is a C ∞ -function in (R1 , E1 , η),  equal to zero when (R1 , E1 ) = (0, 0). We also have L (R1 E1 ) = 0 and L R1 E1 = (l1 − l2 )R1l1 E1l2 for l1 , l2 ∈ Z. For l1 l2 more details about the Lie-derivative we refer the reader to [6, 9]. The Lie-derivative of the time (17) can be written as 2n 1 + o(1) (LH)(R1 , E1 ) = (R1 E1 )2n−1 E1 κ(−Z0 , R1 (1 + o(1)), E1 (1 + o(1)), η) Z ! E3 ∂κ − ∂Z (Z(E, R2 , E2 ), R1EE1 , E, η) + 2 (LZ)(E, R1 , E1 )dE + o(1) , E2 E 2 κ(Z(E, R2 , E2 ), R1EE1 , E, η) (18) where o(1)-terms are C ∞ -functions of (R1 , E1 , η), equal to zero when (R1 , E1 ) = (0, 0), and LZ is given by (LZ)(E, R1 , E1 ) = Z ! 2nZ0 (Φ(R1 , E1 , η) + o(1)) E Φ( R1sE1 , s, η) exp −2n ds , (19) E12n E2 s2n+1 where o(1)-terms have the same property as above. We show that the first term in (18) is dominant. ∂κ Since κ is uniformly positive near the origin (κ(0, 0, 0, η) = 1) and ∂Z and Φ are bounded, we find an upper bound for the integral in (18): Z Z Z ! E3 αZ0 E3 1 Φ( R1sE1 , s, η) E ≤ 2n exp −2n ds dE E1 (1+o(1)) E1 E1 (1+o(1)) E2 E1 (1+o(1)) s2n+1 (20) for some constant α > 0 independent of Z0 . (We write E2 = E1 (1 + o(1)).) For E1 > 0 and E1 ∼ 0, we split up the integral on the right-hand side of (20) in two parts: Z E3 Z 2E1 Z E3 = + . E1 (1+o(1)) E1 (1+o(1)) 2E1 We denote the first integral by J1 and the second by J2 . We make in J1 the 15 change of variable E = E1 τ , getting Z Z ! 1 2 1 E1 τ Φ( R1sE1 , s, η) J1 = exp −2n ds dτ E1 1+o(1) τ2 E1 (1+o(1)) s2n+1 Z Z τ ! 1 2 1 2n Φ( Ru1 , E1 u, η) = exp − 2n du dτ E1 1+o(1) τ2 E1 1+o(1) u2n+1 Z 2   1 1 β ≤ exp − 2n (τ − 1 − o(1)) dτ E1 1+o(1) τ2 E1 ≤γE12n−1 (21) for some constants β, γ > 0 independent of Z0 . In the second step we used the change of variableRs = E1 u and in the third step we used the fact that the τ integrand function in 1+o(1) is uniformly positive (Φ(0, 0, η) = 2n). In the last step the term τ12 is bounded on the segment [1 + o(1), 2] and the integral of the exponential function is bounded by E12n , multiplied by a positive constant. Note also that the o(1)-terms in the last step are equal. Concerning the integral J2 we get Z E3 Z E ! 1 Φ( R1sE1 , s, η) J2 = 2 exp −2n ds dE 2E1 E E1 (1+o(1)) s2n+1 Z E3 Z 2E1 ! 1 Φ( R1sE1 , s, η) ≤ 2 exp −2n ds dE 2E1 E E1 (1+o(1)) s2n+1 Z E3 Z 2 ! 1 2n Φ( Ru1 , E1 u, η) = 2 exp − 2n du dE 2E1 E E1 (1+o(1)) u2n+1   Z E3   β 1 γ β ≤ exp − 2n 2 dE ≤ exp − 2n (22) E1 2E1 E E1 E1 for some new constants β, γ > 0. Finally, combining inequalities (21) and (22) we obtain Z E3   αZ0 α 1 Z0 α2 β ≤ 2n (J1 + J2 ) ≤ + 2n+1 exp − 2n E1 (1+o(1)) E1 E1 E1 E1 for some constants α1 , α2 , β > 0. It is clear now that the first term in (18) is the leading term since κ1 > α1 Z0 (Z0 > 0 is as small as we want but fixed). We conclude that there are no critical periods for any fixed level ε > 0 on Σ1 with R10 , E10 > 0 small enough and fixed, uniformly in η. The Lie- derivative LH tends to +∞ as ε → 0, uniformly in η. Since LH > 0 and ∂ 2nȳ ∂∂ȳ + 0 ∂ε ∂ = R1 ∂R 1 ∂ − E1 ∂E 1 the period function is monotonous increasing (as large ȳ increases, i.e. as we go away from the center (x̄, ȳ) = (0, 0), the period function increases). This completes the proof of Theorem 5. 3.4 Proof of Theorem 1 Let n ≥ 1 and T (y; ε) be the period function of the center at the origin of system (1) with ε > 0, ε ∼ 0, parametrized by the positive y-axis, with y ∼ 0. We have 16 the following relation between the (x, y, ε)-coordinates, the family directional coordinates and the phase directional coordinates defined in Section 2.1: x = εx̄ = RX, y = ε2n ȳ = R2n , ε = RE. Note that the positive y-axis is given by {x = 0}. In the family chart (resp. the phase directional chart), it corresponds to {x̄ = 0} (resp. {X = 0}). For each ε > 0 and ε ∼ 0, we consider T in the following intervals: ]0, ε2n ȳ0 ], [ε ȳ1 , ε2n ȳ2 ] and [ε2n ȳ3 , y0 ] where ȳ0 , ȳ1 , y0 > 0 are small and independent of ε 2n and ȳ2 , ȳ3 > 0 are large and independent of ε. For ȳ0 , y0 small and ȳ3 large, it suffices to decrease ȳ1 and increase ȳ2 to cover the interval ]0, y0 ]. In the interval ]0, ε2n ȳ0 ] (resp. [ε2n ȳ1 , ε2n ȳ2 ] and [ε2n ȳ3 , y0 ]) we use Theorem 3 (resp. Theorem 4 and Theorem 5). Let us recall that the results of Theorem 5 are valid in a section Σ1 ⊂ {X = 0} parametrized by (R, E) ∈]0, R10 ]×]0, E10 ] where R10 , E10 > 0 are small enough and fixed (see Section 3.3). The interval ]0, R10 ] × {E10 } corresponds to y = ε2n (E 01)2n and ε ∈]0, R10 E10 ] (we denote (E 01)2n by ȳ3 ). The 1 1 interval {R10 }×]0, E10 ] is given by y = (R10 )2n (we denote (R10 )2n by y0 ) and ε ∈]0, R10 E10 ]. Theorem 3 is valid for ȳ ∈]0, ȳ0 ] and ε ∈]0, ε0 ] where ȳ0 , ε0 > 0 are small enough. In the (y, ε)-coordinates, it corresponds to y ∈]0, ε2n ȳ0 ] and ε ∈]0, ε0 ]. Finally, for any small ȳ1 > 0 and any large ȳ2 > 0, Theorem 4 is valid for ȳ ∈ [ȳ1 , ȳ2 ] and ε ∈]0, ε1 ] where ε1 > 0 is small enough. It corresponds to y ∈ [ε2n ȳ1 , ε2n ȳ2 ] and ε ∈]0, ε1 ] in the original coordinates. Note that the notion of critical period is independent of the chosen coordi- nates and the chosen transverse section (for example, if we work with the polar coordinates (r, θ) instead of (x̄, ȳ), we have the same number of critical periods, counting multiplicity). We consider two cases: n = 1 and n > 1. Suppose first that n = 1. Following Theorem 3, we have that ∂T 2 ∂y (y; ε) > 0 for all y ∈]0, ε ȳ0 ] and ε ∈]0, ε0 ]. Indeed, 1 y we know that T (y; ε) = ε TF ( ε2 ; ε) where TF (ȳ; ε) is the period function of the center of (3), parametrized by the positive ȳ-axis. Now, it suffices to see that ∂T 1 ∂TF y  (y; ε) = 3 ;ε (23) ∂y ε ∂ ȳ ε2 and that ∂T∂ ȳ (ȳ; ε) > 0 for all ȳ ∈]0, ȳ0 ] and ε ∈]0, ε0 ] (Theorem 3). On the F √ other hand, we know that T (y; ε) = TD ( y, √εy ) where TD (R, E) is the period function of rX̄η near the polycycle Γ (X̄η is the blown-up vector field). Note that ∂T 1 √ ε (y; ε) = (LTD )( y, √ ) ∂y 2y y and that LTD > 0 for all (R, E) ∈]0, R10 ]×]0, E10 ] (see Theorem 5). Thus, ∂T 2 ∂y (y; ε) > 0 for all y ∈ [ε ȳ3 , y0 ] and ε > 0 small. Finally, by taking ȳ1 < ȳ0 and ȳ2 > ȳ3 , we have that ∂T∂ ȳ (ȳ; ε) > 0 for all ȳ ∈ [ȳ1 , ȳ2 ] and ε > 0 small (see F ∂T Theorem 4) and thus ∂y (y; ε) > 0 for all y ∈ [ε2 ȳ1 , ε2 ȳ2 ] and ε > 0 small (see (23)). This ends the proof of Theorem 1 in the generic case. Suppose now that n > 1. The study of the non-generic case is similar to the study of the generic case. We have ∂T 2 ∂y (y; ε) < 0 for all y ∈]0, ε ȳ0 ] and for all ε > 0 small (see Theorem 3), and ∂T 2 ∂y (y; ε) > 0 for all y ∈ [ε ȳ3 , y0 ] and ε > 0 17 2 small (see Theorem 5). Using Theorem 4 we find that ∂∂yT2 (y; ε) > 0 for all y ∈ [ε2 ȳ1 , ε2 ȳ2 ] and ε > 0 small. This implies that at most one critical period can exist in ]0, y0 ]. Since ∂T ∂y goes from − to +, we conclude that precisely one critical period exists in ]0, y0 ]. This completes the proof of Theorem 1 in the non-generic case. 3.5 Proof of Theorem 2 We consider system (2) with N ≥ 1 and denote c := (c2 , c4 , . . . , c2(N −1) ) ∈ SN −2 (when N = 1, we don’t have the parameter c). When N ≥ 2, we assume that c2 ≥ c02 for some arbitrarily small and fixed c02 > 0. Let C and G be as defined in Theorem 2 and let C̃ be an arbitrary and fixed compact subset of C. Let c ∈ C̃. We replace ε in (2) by ε2 . It is clear that, if we can prove the result in a small interval in the new ε-space, then we have proved it in a small interval in the old ε-space. If we apply the scaling (x, y) = ( cx̃2 , cỹ2 ) to (2), we get  PN −1  ẋ = y − x2 + k=2 c̄2k x2k + c̄2N x2N (24) ẏ = −ε2 x, c2k 1 where c̄2k = c22k−1 , for k = 2, . . . , N − 1, and c̄2N = c2N −1 . (We use the old 2 notation (x, y) instead of (x̃, ỹ) for the sake of simplicity). Since c is kept in the compact set C̃, it is clear that c̄ = (c̄4 , . . . , c̄2N ) is also contained in a compact set, denoted by C̄, and that Ḡ0 (x) > 0, (25) x for all x ∈ R and c̄ ∈ C̄, where Ḡ denotes the polynomial in x in the first equation of (24). Note that Ḡ(x) = c2 G( cx2 ) and that system (24) is of type (1) with n = 1. Figure 4: Dynamics of (24). The curve of singularities, at level ε = 0, with indication of small-amplitude, detectable and large closed orbits, for ε > 0 and ε ∼ 0. It suffices to show that there exists ε0 > 0 small such that system (24) has no critical periods for all ε ∈]0, ε0 ] and c̄ ∈ C̄. Let T (y; ε) be the period function of the center at the origin of system (24) with ε > 0 and ε ∼ 0, parametrized d by the positive y-axis. In the rest of this section we prove that dy T (y; ε) > 0 on 18 {y > 0}, for all ε ∈]0, ε0 ] and c̄ ∈ C̄, for some ε0 > 0. This will imply that there are no critical periods uniformly in ε ∼ 0. We study the period function T in the following intervals: ]0, y0 ], [ρ, ρ1 ] and [y1 , ∞[, where y0 > 0 is small enough, y1 > 0 is large enough and ρ > 0 is arbitrarily small (see Figure 4). When we find y0 and y1 , we decrease ρ (i.e., increase the segment [ρ, ρ1 ]) to cover the entire {y > 0}. d Following Theorem 1, there exist ε0 > 0 and y0 > 0 such that dy T (y; ε) > 0 for all y ∈]0, y0 ] and (ε, c̄) ∈]0, ε0 ] × C̄. Consider now the period function T in the segment [ρ, ρ1 ], for any small and fixed ρ > 0. The reduced flow (sometimes called the slow system) of (24) along the critical curve {y = Ḡ(x)}, away from the contact point (x, y) = (0, 0), is given by x x0 = − 0 (or y 0 = −x). Ḡ (x) Note that the reduced flow is well-defined and uniformly negative for all x kept in large compact sets and c̄ ∈ C̄. Here we use (25). The orbit through the point y ∈ [ρ, ρ1 ] is attracted to the curve of singularities {x > 0}, follows the reduced flow directed towards the turning point at the origin and then goes back to the point y due to the symmetry. This implies that T is well-defined for y ∈ [ρ, ρ1 ]. Following [3] (Theorem 2.1.) or [4], the period function T and its derivative, restricted to the segment [ρ, ρ1 ], are given by   1 d 1 d T (y; ε) = 2 2 (T0 (y) + o(1)) and T (y; ε) = 2 2 T0 (y) + o(1) , ε dy ε dy with ε > 0 small enough, where T0 (y) is the transition time (at level ε = 0) of the reduced flow along the attracting part of the curve of singularities between the ω-limit of the point y ∈ [ρ, ρ1 ] and the turning point. Using the expressions for the reduced flow we have for y ∈ [ρ, ρ1 ] Z 0 Z 0 dỹ dỹ T0 (y) = − =− , y x̃ y g(ỹ) where x̃ = g(ỹ) represents the attracting part of the critical curve, i.e. ỹ = Ḡ(g(ỹ)). Finally, we get d 1 T0 (y) = >0 dy g(y) for all y ∈ [ρ, ρ1 ] and c̄ ∈ C̄. We conclude that dy d T (y; ε) > 0 for all y ∈ [ρ, ρ1 ], c̄ ∈ C̄ and ε ∈]0, ε0 ] for some small ε0 > 0. We point out that we are allowed to use the results of [3] because the reduced flow has no singularities. d It remains to show that dy T (y; ε) > 0 for y ∈ [y1 , ∞[, c̄ ∈ C̄ and ε ∈]0, ε0 ] for y1 > 0 large enough and ε0 > 0 small enough. To investigate the period function when y → ∞, we apply the coordinate change (x, y) = ( x̃q , q2N 1 ) to (24), where q > 0 is small and x̃ is kept in a compact set. In the new coordinates (24) becomes q2N1−1 X∞ where the vector field X∞ is given by  PN −1  x̃˙ = 1 − q 2N −2 x̃2 + k=2 c̄2k q 2N −2k x̃2k + c̄2N x̃2N + 2N 1 2 4N −2 2 ε q x̃ 1 2 4N −1 q̇ = 2N ε q x̃. (26) 19 On the line {q = 0} (it represents infinity in the (x, y)-phase space), system 1  2N 1 (26) has two semi-hyperbolic singularities x̃ = ± c̄2N (resp. x̃ = ±1) when N ≥ 2 (resp. N = 1). Note that c̄2N is uniformly positive and bounded. It suffices to look at the positive sign. When ε = 0, we have the curve of (semi- 1  2N 1 hyperbolic) singularities x̃ = c̄2N + O(q) (resp. x̃ = 1). The reduced flow is given by   2N 1 !   0 1 4N −1 1 0 1 3 q = q + O(q) resp. q = q . 2N c̄2N 2 Using a Takens normal form for C k -equivalence (see e.g. [5]), system X∞ near the semi-hyperbolic singularity on the line {q = 0} is C k -equivalent to  x̂˙ = −x̂ (27) q̂˙ = ε2 q̂ 4N −1 h(q̂, ε, c̄) where h is a positive C k -function. We denote system (27) by X̂∞ . We conclude that in the normal form coordinates (x̂, q̂) the vector field q2N1−1 X∞ can be written as 1 X̂∞ , (28) q̂ 2N −1 ĥ(x̂, q̂, ε, c̄) where ĥ is a positive C k -function. We choose two transverse sections Σ− ⊂ {x̂ = x̂0 }, parametrized by q̂, and Σ+ ⊂ {q̂ = q̂0 }, parametrized by x̂, for some small and fixed x̂0 , q̂0 > 0. We compute the time of (28) spent between Σ− and Σ+ , near (x̂, q̂) = (0, 0). The orbit of (27) or (28) starting at q̂1 ∈ Σ− , with q̂1 > 0, is given by Z ! 1 q̂ dz x̂(q̂, q̂1 ) = x̂0 exp − 2 . ε q̂1 z 4N −1 h(z, ε, c̄) Now is the time spent by the orbit given by Z q̂0 1 h̄(x̂(z, q̂1 ), z, ε, c̄)dz T (q̂1 ; ε) = ε2 q̂1 z 2N with a positive C k -function h̄. The derivative is given by Z q̂0 ∂ h̄ (x̂(z, q̂ ), z, ε, c̄) ∂ x̂ (z, q̂ ) d h̄(x̂0 , q̂1 , ε, c̄) 1 ∂ x̂ 1 ∂ q̂1 1 T (q̂1 ; ε) = − + 2 dz. (29) dq̂1 ε2 q̂12N ε q̂1 z 2N Now, we proceed exactly as in Section 3.3.5. The first term in (29) tends to −∞ as ε2 q̂12N → 0 and we show that it is a dominant term. We have Z Z q̂0 exp − 1 R z ds  1 q̂0 αx̂0 ε2 q̂1 s4N −1 h(s,ε,c̄) ≤ dz (30) ε2 q̂1 ε4 q̂14N −1 q̂1 z 2N with a positive constant α. We used the fact that ∂∂ h̄x̂ is bounded and h is uniformly positive. For the [q̂1 , 2q̂1 ]-part of the integral on the right hand side 20 of (30), we get Z Z R q̂1 ze  2q̂1 1 2 exp − ε12 q̂1 ds s4N −1 h(s,ε,c̄) = de z q̂1 q̂12N −1 1 ze2N R ze  Z 2 1 exp − ε2 q̂4N de s 1 1 −2 e4N −1 h(q̂1 s 1 s e,ε,c̄) = de z q̂12N −1 1 ze2N  Z 2 exp − εβ(ez −1) 1 2 q̂ 4N −2 ≤ 1 de z q̂12N −1 1 ze2N ≤ γε2 q̂12N −1 , (31) where β, γ > 0 are constants. (See Section 3.3.5 for each step.) On the other hand, we have Z q̂0 Z q̂0 exp − 1 R 2q̂1 ds  ε2 q̂1 s4N −1 h(s,ε,c̄) ≤ dz 2q̂1 2q̂1 z 2N R2  Z q̂0 exp − 4N 1 de s ε2 q̂1 −2 1 s e4N −1 h(q̂1 s e,ε,c̄) = dz 2q̂1 z 2N   Z q̂0 β dz ≤ exp − ε2 q̂14N −2 2q̂1 z 2N   γ β ≤ 2N −1 exp − (32) q̂1 ε2 q̂14N −2 for some new constants β, γ > 0. Combining (30), (31) and (32) we finally have Z   1 q̂0 α1 x̂0 α2 β ≤ + exp − ε2 q̂1 ε2 q̂12N ε4 q̂16N −2 ε2 q̂14N −2 for positive constants α1 , α2 , β. Now, it suffices to notice that x̂0 > 0 can be arbitrarily small but fixed. 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