This is a preprint of: “Periodic oscillators, isochronous centers and resonance”, Rafael Ortega, David Rojas, Nonlinearity, vol. 32, 800–832, 2019. Periodic oscillators, isochronous centers and resonance Rafael Ortega and David Rojas Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain Abstract. An oscillator is called isochronous if all motions have a common period. When the sys- tem is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are un- bounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators. 1 Introduction Consider an oscillator with equation ẍ + V 0 (x) = 0, x ∈ R (1) and assume that it has an isochronous center at the origin. This means that x = 0 is the only equilibrium of the equation and the remaining solutions are periodic with a fixed period, say T = 2π. We are interested in the phenomenon of resonance for periodic perturbations. More precisely, we ask for the class of 2π-periodic functions p(t) such that all the solutions of the non-autonomous equation ẍ + V 0 (x) = εp(t) (2) are unbounded. Here ε 6= 0 is a small parameter. The simplest isochronous center is produced by the harmonic oscillator, V (x) = 21 n2 x2 , n = 1, 2, . . . In this case the previous question has a well-known answer: resonance occurs whenever the integral Z 2π In (p) := p(t)eint dt 0 does not vanish. After this example the study of resonance for general isochronous oscillators seems natural. As far as we know this question was first raised by Prof. Roussarie in the Open Problems Session of the II Symposium on Planar Vector Fields (Lleida, 2000). Concrete examples of functions p(t) producing resonance were presented in [17]. See also [4]. The goal of the present paper is to identify a general class of forcings leading to resonance. Our main result can be interpreted as a nonlinear version of the condition In (p) 6= 0. To explain this in more precise terms we consider a cylinder C = (R/2πZ) × [0, ∞) with coordinates (θ, r). The solution of (1) with initial conditions x(0) = r, ẋ(0) = 0 will be denoted by ϕ(t, r). The complex-valued solution of the linear problem ÿ + V 00 (ϕ(t, r))y = 0, y(0) = 1, ẏ(0) = i 2010 Mathematics Subject Classification. 34C15; 34C10, 34D05, 34D10, 34D23. Key words and phrases: Isochronous center, oscillator, resonance, perturbation. All the authors are partially supported by the MINECO/FEDER grant MTM2017-82348-C2-1-P. D. Rojas is also partially supported by the MINECO/FEDER grant MTM2017-86795-C3-1-P. Email addresses: rojas@ugr.es (D. Rojas, corresponding author), rortega@ugr.es (R. Ortega). 1 is denoted by ψ(t, r). Then we define a function on the cylinder Z 2π 1 Φp : C → C, Φp (θ, r) = p(t − θ)ψ(t, r)dt 2π 0 and prove, under certain assumptions on the potential V , that all solutions of (2) are unbounded when the condition inf |Φp (θ, r)| > 0 (3) C holds. This is a sufficient condition for resonance but it is not too far from being also necessary. A partial converse of the main theorem holds: a periodic solution exists when the function Φp has a non-degenerate zero. Note that periodic solutions are bounded and so resonance is excluded. 1 2 2 Let us now discuss the form of Φp in some particular cases. For the linear oscillator, V (x) = 2n x , n = 1, 2, . . . elementary computations lead to the estimates 1 1 |In (p)| 6 |Φp (θ, r)| 6 |In (p)|. 2πn 2π The condition In (p) 6= 0 is equivalent to (3). Another example that has been widely studied is the asymmetric oscillator associated to the potential 1  V (x) = α(x+ )2 + β(x− )2 2 where x+ = max{x, 0}, x− = max{−x, 0} and α, β are positive constants with √1α + √1β = 2π n , n = 1, 2, . . . In this case the homogeneity of the equation (1) implies that Φp (θ, r) = Φp (θ, 1). The function Φp (θ, 1) appeared first in the work of Dancer on the periodic problem (see [8]). Assuming that the zeros of the function Φp (θ, 1) were non-degenerate, it was proved in [2] that all solutions with large initial condition were unbounded. Soon after, it was proved in [11] that all solution are bounded when Φp (θ, 1) does not vanish. This boundedness condition improved a previous result in [16] of more local nature. The boundedness of all solutions has been also treated in [3, 10]. These papers deal with a class of isochronous oscillators with the same asymptotic behavior as the asymmetric oscillator. For a general isochronous oscillator checking the condition (3) can be difficult. We have analysed in detail the equation   1 1 ẍ + x+1− = εp(t), x ∈ (1, +∞). 4 (x + 1)3 In this case the functions ϕ(t, r) and ψ(t, r) can be obtained explicitly and the resonance condition (3) can be reformulated in terms of the Fourier coefficients of p(t). In particular we obtain an extension of the result in [4] for p(t) = sin t. This equation has a singularity at x = −1 and the motion is constrained to a proper and unbounded interval. It can be seen as a prototype of the class of isochronous oscillators with one asymptote. Incidentally we note that there are no isochronous oscillators having two asymptotes so that the motion is constrained to a bounded interval. See [19] for more details. Some boundedness results for isochronous oscillators with singularities can be found in [6, 12]. Up to now all non-autonomous perturbations have been of additive type. This is very natural if we have in mind mechanical oscillators but different perturbations can appear in other contexts. Motivated by a geometrical problem, Ai, Chou and Wei considered in [1] the equation R(t) ẍ + x = , x>0 (4) x3 where R(t) is T -periodic. They proved the existence of T -periodic solutions when R is a positive C 2 -function and T < π. When R is a positive constant this equation is equivalent to (6) and an isochronous center with minimal period π appears. This suggests that the condition T < π in the result in [1] should be essential due to the appearance of resonance for T = π. We have analysed an example that somehow shows that this 2 is the case, although our function R(t) is not smooth. We thank Prof. Pedro J. Torres for bringing to our attention the result in [1]. The rest of the paper is organized in five sections. The main results of the paper are stated in Section 2. Those readers who are only interested on the results for oscillators without singularities can go directly to Section 5 to find the proofs. Section 3 is concerned with some preliminary remarks on Sturm theory. In Section 4 we present a detailed analysis of the asymptotic behaviour of the action-angle coordinates associated to (1). These results will be employed in Section 6 to prove the main theorems concerning oscillators with singularities. The paper is finished with an Appendix inspired by [4]. 2 Statement of the main results Let us start with a potential V ∈ C 2 (R) defined on the whole real line and satisfying V (0) = 0, xV 0 (x) > 0 if x 6= 0. In addition we assume that all the solutions of equation (1) are 2π-periodic. In particular this implies that V 00 (0) = N 2 , N = 1, 2, . . . where N depends upon the minimal period. See [21] for more details and the construction of concrete examples. There will be no particular restrictions on the forcing term p(t) and we will just assume that it is 2π-periodic and locally integrable. This will be indicated by p ∈ L1 (T). Theorem A. Assume that V satisfies the previous conditions and V 00 is bounded over the whole real line. In addition the condition (3) holds for some p ∈ L1 (T). Then the equation (2) is resonant for small ε 6= 0. The proof of this result will be presented in Section 5.1. We observe that the potential associated to the asymmetric oscillator is not C 2 . The reader is invited to modify the proof of the Theorem so that it includes this example. In the introduction we defined resonant equation as an equation where all solutions are unbounded. This means that each solution x(t) satisfies |x(tn )| + |ẋ(tn )| → +∞ for some sequence {tn }. We will prove that a slightly stronger notion of resonance also holds, every solution x(t) of (2) satisfies lim (|x(t)| + |ẋ(t)|) = +∞. |t|→+∞ Remark 2.1. We point out that all the solutions of (2) are globally defined. Indeed, if x(t) is solution of (2) and E(t) = 21 ẋ(t)2 + V (x(t)) then E(t) satisfies the differential inequality p Ė(t) 6 ε |p(t)| |ẋ(t)| 6 ε |p(t)| 2E(t). Thus, Z p p ε t E(t) − E(0) 6 √ |p(s)| ds (5) 2 0 and so the energy E(t) cannot blow up in finite time.  Let us now assume that the function p ∈ L1 (T) is such that the condition (3) does not hold. This means that Φp has a zero at some point (θ∗ , r∗ ) of the cylinder C or that it vanishes at infinity. That is, either Φp (θ∗ , r∗ ) = 0 or Φp (θn , rn ) → 0 for some sequence (θn , rn ) with rn → +∞. The next result shows that resonance cannot occur if Φp has a non-degenerate zero. Proposition 2.2. Assume that V is in the conditions of Theorem A and that Φp has a non-degenerate zero (θ∗ , r∗ ) with r∗ > 0. Then the equation (2) has a 2π-periodic solution for small ε. 3 In the previous statement non-degeneracy is understood in the topological sense. That is, there exists a small open neighbourhood U of (θ∗ , r∗ ) such that Φp (θ, r) 6= 0 for each point (θ, r) ∈ Ū \ {(θ∗ , r∗ )} and the Brouwer degree deg(Φp , U, 0) does not vanish. We refer to Section 5.2 for the proof and more comments on Brouwer degree. The two previous results show that the function Φp is crucial for the understanding of the resonance problem. In concrete examples it can be rather difficult to analyse the properties of Φp . In the next result we show that, at least at a theoretical level, the condition (3) is applicable to any isochronous potential and it is persistent under small perturbations of the forcing term p(t). To be precise it is convenient to employ the language of Functional Analysis. Now we interpret L1 (T) as a Banach space with norm Z 2π kpkL1 (T) = |p(t)| dt. 0 Proposition 2.3. Assume that V is in the conditions of Theorem A and define RV as the set of forcing terms satisfying (3), RV = {p ∈ L1 (T) : inf |Φp (θ, r)| > 0}. C Then RV is open and non-empty. This result will be proved in Section 5.3. By now we observe that if {pn } is a sequence in L1 (T) converging in a weak topology to the Dirac measure (pn − * δ) then pn ∈ RV for large n. This is consistent with the example in [17]. The convergence pn − * δ means (a) supn kpn kL1 (T) < ∞, and R 2π (b) 0 pn (t)φ(t)dt → φ(0) for each function φ(t) that is continuous and 2π-periodic. In the previous results we have worked with oscillators defined on the whole real line but there are also oscillators producing an isochronous center and having a singularity. A well-known example is   1 1 ẍ + x+1− = 0, (6) 4 (x + 1)3 defined for all x ∈ (−1, +∞). This equation can be solved explicitly (see [20]). In particular r 1 ϕ(t, r) = −1 + λ2 cos2 (t/2) + sin2 (t/2) λ2 with λ := 1 + r > 1. Let ψ(t, r) be the solution of the associated variational equation   1 3 ψ̈ + 1+ ψ=0 (7) 4 (ϕ(t, r) + 1)4 with initial conditions ψ(0, r) = 1, ψ̇(0, r) = i. A computation shows that     ∂ϕ 1 cos2 t 2 − λ14 sin2 2t + 2i sin 2t cos t 2 ψ(t, r) = (t, r) − 0 ϕ̇(t, r)i = q   . (8) ∂r V (r) cos2 2t + λ14 sin2 2t The perturbed equation   1 1 ẍ + x+1− = ε sin t (9) 4 (x + 1)3 was considered in [4]. The authors proved that all solutions are unbounded if ε 6= 0 is small enough. Next we present an analogous result valid for general periodic perturbations of (6). 4 Theorem B. Let p ∈ L1 (T) be a function satisfying (3) with ψ(t, r) defined by (8). Then all the solutions of equation   1 1 ẍ + x+1− = εp(t) (10) 4 (x + 1)3 are unbounded for sufficiently small ε 6= 0. This resonance result deals with the specific equation (6) but the method of proof can be extended to a larger class of potentials V . In many cases the existence of a singularity of V at x = −1 determines the behaviour of V (x) as x → +∞. This interesting observation was made in [4] and somehow shows that the equation (6) can be seen as a paradigm for centers with singularity. For more details we refer to the Appendix. To show how to deduce the result for (9) in [4] from Theorem B we present a corollary for the class of equations (10) with p(t) = a0 + a1 cos t + b1 sin t. Corollary 2.4. Assuming that p(t) is a trigonometric function as above, the equation (10) is resonant if a21 + b21 > 9a20 . (11) Note that this result extends the resonance result in [4] for the case a0 = 0. We finish this Section with a result on the equation (4). Theorem C. Consider the π-periodic function (   1 if t ∈ 0, π2 , R(t) =   c if t ∈ π2 , π , with c > 0. Then all solutions of (4) are unbounded if c 6= 1. Since R(t) is discontinuous the equation (4) is understood in the Carathéodory sense. It would be interesting to construct similar examples for smooth functions R(t). 3 Variations on Sturm Theory In the forthcoming sections we study the variations of the solution of the autonomous system (1) with respect to the action-angle variables. Some qualitative properties of these solutions will be understood thanks to a variant of Sturm theory. The classical results in this theory deal with the zeros of solutions but it is known that in some cases these zeros can be replaced by the zeros of the derivatives (critical points). See [9]. In this section we shall consider the equation ÿ + a(t)y = 0 (12) with a ∈ L1 (J) defined in an open interval J and such that Z a(t)dt > 0 for each sub-interval J∗ ⊂ J. (13) J∗ We know from Lebesgue differentiation theorem that Z t+h 1 lim a(s)ds = a(t) h→0 h t for almost every t ∈ J. In consequence a(t) > 0 almost everywhere. 5 Our basic tool will be the argument θ(t) of any non-trivial solution y(t) of (12). Using the polar change of variables y + iẏ = reiθ we have that the argument satisfies θ̇ = −(sin2 θ + a(t) cos2 θ). (14) Critical points of y(t) correspond to θ ∈ πZ; that is, ẏ(t) = 0 is equivalent to θ(t) = nπ for some integer n. In the next result we prove that the trajectory (y(t), ẏ(t)) rotates around the origin in a clock-wise sense. This fact has useful consequences. Lemma 3.1. Let y(t) be a non-trivial solution of (12). Then the argument θ(t) is strictly decreasing. In consequence, (a) all critical points of y(t) are isolated, (b) a local maximum or minimum is reached at each critical point of y(t), (c) y(t) reaches a local maximum (respectively, minimum) at t∗ ∈ J if and only if ẏ(t∗ ) = 0, y(t∗ ) > 0 (respectively, y(t∗ ) < 0). Proof. Given t1 , t2 ∈ J with t1 < t2 we integrate the equation (14) over the interval J∗ = (t1 , t2 ) and observe that Z t2 Z t2 θ(t2 ) − θ(t1 ) = − (sin2 θ(t) + a(t) cos2 θ(t))dt 6 − min{1, a(t)}dt < 0. t1 t1 At this point we have employed the condition (13). Once we know that (y(t), ẏ(t)) rotates in a clock-wise sense we deduce that if t∗ ∈ J is such that θ(t∗ ) = nπ with n even, then there exists δ > 0 such that y(t) > 0, (t − t∗ )ẏ(t) < 0 if 0 < |t − t∗ | < δ. These inequalities are reversed when n is odd. Now it is easy to prove all the properties (a) − (c). Next we state a result on separation of critical points. Lemma 3.2. Let φ1 and φ2 be a fundamental pair of solutions of (12). Assume that φ1 reaches two local maxima (respectively, minima) at t1 , t2 ∈ J with t1 < t2 . Then there exists t0 ∈ (t1 , t2 ) such that φ2 reaches a local maximum (respectively, minimum) at t0 . Proof. We assume that a local maximum is reached at t1 and t2 . Otherwise we could replace φ1 by −φ1 . Let θi (t) be the argument function corresponding to φi (t). Both arguments are solutions of the same first order equation (14) and so they cannot cross. Without loss of generality we can assume that θ1 (t1 ) = 0 and θ2 (t1 ) ∈ (−2π, 0). Then θ1 (t) > θ2 (t) whenever t > t1 . Since θ1 (t) is decreasing we know that θ1 (t2 ) = −2nπ for some integer n > 1. This implies that θ2 (t2 ) < −2nπ and so there exists some t0 ∈ (t1 , t2 ) such that θ2 (t0 ) = −2π. Consequently, a local maximum of φ2 is reached at t0 . We finish this section with a result on symmetric equations. Lemma 3.3. Let φ1 and φ2 be a fundamental pair of solutions of (12). Assume that a(t) is defined in an open symmetric interval J = (−τ, τ ) such that a(t) = a(−t) for all t ∈ J. Moreover, assume that φ̇1 (0) = φ2 (0) = 0 and φ2 has no critical points in J. Then φ1 has no critical points lying in J \ {0}. Proof. First we observe that the symmetry of the system with respect to t = 0 induces the symmetries on the solutions φ1 (−t) = φ1 (t) and φ2 (−t) = −φ2 (t). The function φ1 (t) has a critical point at t = 0, say that it is a maximum. Since φ2 has no critical point on J then, by Lemma 3.2, φ1 has at most another critical point t∗ 6= 0 (a minimum). The function φ1 is even and therefore critical points must be symmetric with respect to t = 0. This excludes the possible existence of t∗ and so t = 0 is the only critical point of φ1 . 6 4 Potential centers and action-angle variables In this section we work with the equation ẍ + V 0 (x) = 0, x ∈ J (15) where J = (a, b) is some interval with −∞ 6 a < 0 < b 6 +∞ and the following hypothesis on the potential is considered (H0 ) V ∈ C 2 (J ), V (0) = V 0 (0) = 0, V 0 (x)x > 0 if x 6= 0, limx→∂J V (x) = +∞ and V 00 (0) 6= 0. In these conditions the equation (15) produces a center at the origin with J as a projection of the period annulus. Given a point (x, ẋ) in the punctured strip (J × R) \ {(0, 0)}, there exists a unique closed orbit γ passing through it. The action I = I(x, ẋ) is the area of the region enclosed by γ. The angle θ = θ(x, ẋ) can be defined mechanically as the quotient 2πτ θ= T where τ is the time employed to travel through γ from the horizontal axis {ẋ = 0} to the point and T is the minimal period of the orbit. Let γr be the orbit passing through (r, 0) S with 0 < r < b. The region enclosed by γr will be denoted by Ar and it increases with r. Moreover 0 0 and satisfying the conditions x(0, I) > 0, ẋ(0, I) = 0. This section is concerned with the study of this function and the derivatives ẋ = ∂x ∂x ∂t and ∂I . Letting r = x(0, I) we recall the classical formula Ω(I) = V (r) (16) where Ω is the primitive of ω with Ω(0) = 0. This identity allows to connect the function x(t, I) with the function ϕ(t, r) appearing in the resonance condition (3). Note that (16) also shows that Ω(I) coincides with the energy along the corresponding orbit. Definition 4.1. For each I > 0 we define by P(I) := {t ∈ [0, T (I)] : x(t, I) > 0} the set of times when x(t, I) is non-negative. We also define by N (I) := [0, T (I)] \ P(I) the set of times when x(t, I) is negative.  T (I) Notice that, due to the symmetry x(−t, I) = x(t, I), N (I) is an interval centered at t = 2 and P(I) is the union of two intervals. Lemma 4.2. The function x(t, I) belongs to C 1 (R × (0, ∞)) with Z  1 ∂x ẍ(t, I) t 1 − V 00 x(s, I) (ẋ(s, I)2 − ẍ(s, I)2 ) (t, I) = − + ẋ(t, I) 2 ds. (17) ω(I) ∂I ẋ(t, I)2 + ẍ(t, I)2 0 ẋ(s, I)2 + ẍ(s, I)2 ∂x T (I)  Moreover, ∂I 2 ,I < 0. 7 Proof. The smoothness of x(t, I) follows from the connection with ϕ(t, r) via the identity (16). Indeed the functions ẋ(t, I) and ∂x ∂I (t, I) are solutions of the variational equation of (15) given by  ξ¨ + V 00 x(t, I) ξ = 0. (18) ∂x From (16) we deduce that ∂I (t, I) satisfies the initial conditions ∂x ω(I) ∂ ẋ (0, I) = 0 , (0, I) = 0. ∂I V (r) ∂I The Rofe-Beketov formula (see [5], page 24) implies that the function ξ(t) given by the right hand side of (17) is a solution of (18) and a direct computation shows that 1 1 ˙ ξ(0) = − = 0 , ξ(0) = 0. ẍ(0, I) V (r) The uniqueness of solution of the initial value problem implies that the identity (17) holds. The inequality ∂x T (I)  T (I)  ∂I 2 , I < 0 follows from the expression of ∂x∂I (t, I) with t = 2 using that ẋ T (I) 2 ,I = 0 and  ẍ T (I) 2 , I = −V 0 x T (I) 2 , I > 0. p Lemma 4.3. In the previous notations, |ẋ(t, I)| 6 2Ω(I) for all t ∈ R and I > 0.  Proof. By energy conservation we have 21 ẋ(t, I)2 = Ω(I) − V x(t, I) . Then, taking into account that V (x) > 0 the result holds. The following sections are concerned with the behaviour of the periodic solutions x(t, I) of system (15) and its partial derivatives with respect to the action-angle variables. Section 4.1 is devoted to potential centers that are not globally defined by an asymptote of the potential function. In Section 4.2 we consider the case when the second derivative of the potential is bounded. Section 4.3 is concerned with a combination of the previous situations, an asymptote on one side and a bounded second derivative of V on the other side. Finally, in Section 4.4 we continue with this situation but in addition we assume that the center is isochronous. We describe the asymptotic behaviour of x(t, I), ẋ(t, I) and ∂x∂I (t, I) as I → +∞. The limit functions are solutions of a Bouncing Problem that will be analyzed in detail. 4.1 Potential center with an asymptote In this section we shall consider that the potential function V in (15) presents an asymptote at x = a. That is, we shall assume that V satisfies (H0 ) with J = (a, +∞), −∞ < a < 0. In addition we shall assume that V is convex near the asymptote, meaning that (H1 ) V 00 (x) > 0 for all x ∈ (a, a + ζ). where ζ is some number satisfying 0 < ζ < |a|. The first result is concerned with the limit as I → +∞ of the negative semi-period of x(t, I). It is based on a similar result in [14]. In order to state it properly, denote by T− (I) the negative semi-period for each I ∈ (0, +∞). That is, the length of the interval N (I). Note that   T (I) T− (I) T (I) T− (I) N (I) = − , + . 2 2 2 2 Lemma 4.4. Assume that V satisfies (H0 ) and (H1 ). Then T− (I) → 0 as I → +∞. In the proof of this result we will employ the following elementary property. Lemma 4.5. Let f : [M, +∞) → R be a convex and decreasing function. In addition assume that f has derivative f 0 (x) everywhere and limx→+∞ f (x) exists and is finite. Then lim xf 0 (x) = 0. x→+∞ 8 Proof. Assume M > 0. By the convexity property we have that for all x, y ∈ [M, +∞), f (y) > f (x) + f 0 (x)(y − x). Taking x > 2M and y = x/2, the previous inequality yields to xf 0 (x) > 2(f (x) − f (x/2)) and, using that f is decreasing, |xf 0 (x)| 6 2 |f (x) − f (x/2)| for all x > 2M . Then, since f (x) has a finite limit as x → +∞ the result holds. Proof of Lemma 4.4 We denote by γ− (I) the portion of the orbit described by x(t, I) on the half negative plane and by V− the restriction of the potential V on the negative semi-axis. Then, since Ω(I) is the energy of the orbit, the negative semi-period is written as Z Z 0 dx √ dx T− (I) = = 2 p , γ− (I) y r− (I) Ω(I) − V− (x) 1 1 where r− (I) denotes the intersection of γ− (I) with the x-axis. The change of variable V− (x) 2 = −Ω(I) 2 sin θ transforms the previous integral into ! √ Z 0 1 (V− ) 2  T− (I) = −2 2 0 V−−1 (Ω(I) sin2 θ) dθ. −2π V− From hypothesis (H0 ) we have that 1 (V− ) 2 (x) 1 lim 0 = −p . (19) x→0− V− (x) 2V 00 (0) On the other hand, by hypothesis (H1 ), V−−1 is a convex and decreasing function defined in (V− (a + ζ), +∞) with limit a < 0 at infinity so, by Lemma 4.5 and setting y = V− (x), we have that y(V−−1 )0 (y) 1 (V− ) 2 (x) lim 0 = lim 1 = 0. (20) x→a V− (x) y→+∞ y2 1 (V− ) 2 Therefore, on account of (19) and (20), there exists M > 0 such that 0 V− ◦ V−−1 (y) 6 M for all y ∈ (0, +∞). Since Ω(I) → +∞ as I → +∞ then V−−1 (Ω(I) sin2 θ) → a for every θ ∈ [− π2 , 0). Thus, by Lebesgue’s dominated convergence theorem and on account of (20), T− (I) → 0 as I → +∞. The previous result states that the presence of an asymptote of the potential at the negative x-axis implies that the periodic solution x(t, I) tends to stay on the positive half-plane as the action tends to infinity. A significant role will be played by the derivative of the periodic solutions with respect to the action on the interval N (I). The next result will help to control its behaviour. Before stating it we introduce some more notation. Definition 4.6. Let Iζ be defined by the equation Ω(Iζ ) = V (a + ζ). This is the value of the action corresponding to the energy at the point (a + ζ, 0). For each I > 0 we define A(I) := (τ − (I), τ + (I)) = {t ∈ [0, T (I)] : x(t, I) < a + ζ} the interval of time when x(t, I) belongs to the interval (a, a + ζ).    Due to the symmetry x t + T (I) T (I) 2 , I = x −t + 2 , I , when I > Iζ then A(I) is an interval centered at t = T (I) 2 and A(I) = { T (I) 2 } if and only if I = Iζ . Otherwise A(I) is empty. Lemma 4.7. Let x(t, I) be a solution of system (15) satisfying (H0 ) and (H1 ). Then the following holds: ∂x ∂x ∂x + (a) (T (I)/2, I) 6 (t, I) 6 (τ (I), I) for all t ∈ A(I). ∂I ∂I ∂I 9 ∂x ω(I) (b) 0 < − (T (I)/2, I) 6 |a| for all I > Iζ . ∂I I − Iζ Proof. The functions ẋ(t, I) and ∂x ∂I (t, I) are a fundamental pair of solutions of the variational equation ¨ ξ + V (x(t, I))ξ = 0. By definition of A(I), we have that V 00 (x(t, I)) > 0 for all t ∈ A(I) and ẋ(t, I) has 00 no critical points for t ∈ A(I). Therefore the variational equation satisfies the condition (13) on A(I) and, T (I) by Lemma 3.3, the only critical point of ∂x ∂I (t, I) in A(I) is t = 2 . Moreover, by Lemma 4.2 we have ∂x T (I)  T (I) ∂I 2 , I < 0 so t = 2 is a minimum as a consequence of Lemma 3.1 and so (a) is proved.    To prove (b) let us denote r− (I) := x T (I) 0 2 , I . We have then r− (I) = ∂I ∂x T (I) 2 , I due to ẋ T (I) 2 , I = 0. T (I) ω(I) On the other hand, by conservation of energy V (r− (I)) = Ω(I) and so ∂x ∂I 2 , I = V 0 (r− (I)) . Since I > Iζ we have a < r− (I) < a + ζ. Consequently, there exists η = η(I) ∈ (r− (I), a + ζ) such that I − Iζ = V (r− (I)) − V (a + ζ) = V 0 (η)(r− (I) − a − ζ) for all I > Iζ . On account of hypothesis (H1 ), |V 0 (r− (I))| > |V 0 (η)| for all I > Iζ . Hence, ∂x ω(I) a + ζ − r− (I) |a| (T (I)/2, I) 6 0 6 ω(I) 6 ω(I) ∂I |V (η)| I − Iζ I − Iζ for all I > Iζ as we desired. Remark 4.8. In the above results the potential function V (x) has an asymptote at x = a, a < 0. Analogous results can be obtained when a = −∞ and the asymptote is located at b ∈ (0, +∞). Indeed it is enough to use the change of variables x 7→ −x.  4.2 Potential center with Lipschitz behaviour at infinity In this section we consider a potential function defined on J = (a, +∞) with −∞ 6 a < 0 and satisfying (H0 ). In addition we assume that (H2 ) V 00 is bounded in J where J is a fixed interval of type J = [α, +∞) with a < α 6 0. We denote kV 00 k∞ := sup{|V 00 (x)| : x ∈ J} < ∞. Definition 4.9. Let x(t, I) be a solution of system (15) satisfying hypothesis (H0 ) and (H2 ). For each I > 0, we define B(I) := {t ∈ [0, T (I)] : x(t, I) ∈ J}.  We observe that either B(I) is the whole interval [0, T (I)] or it is the union of two intervals containing respectively t = 0 and t = T (I). Lemma 4.10. Let x(t, I) be a solution of system (15) satisfying (H0 ) and (H2 ). Then 1 −2T (I)M ẋ(t, I)2 + ẍ(t, I)2 > e ẍ(0, I)2 > 0 4 for all t ∈ B(I), where M := max{1, kV 00 k∞ }. Note that |ẍ(0, I)| = V 0 (r). Proof. The identity x(T (I) − t, I) = x(t, I) implies that it is sufficient to prove the estimate  on the compo- nent of B(I) containing t = 0. Let us consider the variational equation ξ¨ + V 00 x(t, I) ξ = 0. Clearly ẋ(t, I) ˙ X := (ξ, η)∗ and is a solution of this equation. If we set η := ξ,   0 1 A(t, I) := , −V 00 (x(t, I)) 0 10 the variational equation is written as Ẋ = A(t, I)X . Let Φ(t, I) be the fundamental matrix of the variational equation with Φ(0, I) = id for all I > 0. The Gronwall’s inequality states that Z t  kΦ(t, I)k 6 exp kA(s, I)kds , 0 where k · k denotes the matrix norm induced by the sup norm of R2 . Given a matrix A = (aij ) ∈ M2 (R), kAk = max{|a11 | + |a12 | , |a21 | + |a22 |}. Since t ∈ B(I) we have kA(t, I)k = max{1, kV 00 k∞ } = M and so kΦ(t, I)k 6 eT (I)M for all t ∈ B(I).  On the other hand, since the matrix Φ(t, I) is symplectic, Φ(t, I)−1 = −JΦ(t, I) J, where J = −1 0 . Consequently, kΦ(t, I)−1 k 6 kΦ(t, I)∗ k 6 2kΦ(t, I)k and so ∗ 0 1 kX (0)k 6 kΦ(t, I)−1 kkX (t)k 6 2eT (I)M kX (t)k for all t lying in the component of B(I) containing t = 0. 4.3 Potential center with an asymptote and Lipschitz behaviour at infinity In this section we consider potential functions V ∈ C 2 (J ), J = (a, +∞) with −∞ < a < 0 satisfying the hypotheses (H0 ), (H1 ) and (H2 ). Moreover, V 0 (x) (H3 ) lim inf > 0 and V 00 (0) > 0. x→+∞ x In the above conditions it is not restrictive to assume that a + ζ = α. From now on, kV 00 k∞ := sup{|V 00 (x)| : x > a + ζ}. Lemma 4.11. Assume that (Hi ), i = 0, 1, 2, 3, hold. Then there exists c > 0 such that c 2 kV 00 k∞ 2 r 6 Ω(I) 6 r , I > 0. 2 2 Here x(t, I) is the solution of (15) and r = x(0, I). 0 Proof. The function W (x) = V x(x) , x ∈ (0, +∞) is positive and satisfies W (x) → V 00 (0) > 0 as x → 0+ . Since lim inf x→+∞ W (x) > 0 we conclude that W (x) has a positive lower bound, say c > 0. Once we know that V 0 (x) > cx if x > 0 (21) we deduce that V (x) > 2c x2 if x > 0. Here we have used that V (0) = 0. We can invoke the identity (16) to deduce the inequality Ω(I) = V (r) > 2c r2 . On the other hand, by (H2 ) we have that V 00 (x) 6 kV 00 k∞ for all x > 0. Then, taking into account that V (0) = V 0 (0) = 0, Z xZ s kV 00 k∞ 2 V (x) = V 00 (u)duds 6 x . 0 0 2 kV 00 k∞ 2 The inequality Ω(I) 6 2 r follows. We are now in position to prove the main result of this section. This result gives upper-bounds of x(t, I) and its partial derivatives in terms of the action. Since we are assuming α = a + ζ, the interval [0, T (I)] splits as A(I) ∪ B(I). Proposition 4.12. Let x(t, I) be a solution of system (15) satisfying (Hi ), i = 0, 1, 2, 3. Then the following holds: r   2 1  for all t ∈ P(I) and I > 0, (a) p x(t, I) 6 c Ω(I)  |a|  p for all t ∈ N (I) and I > 0. Ω(I) 11 1 √ (b) p ẋ(t, I) 6 2 for all t ∈ [0, T (I)] and I > 0. Ω(I)   ∆(I)( ) if t ∈ B(I), p ∂x p (c) Ω(I) (t, I) 6 |a| ω(I) Ω(I) ∂I  max ∆(I), if t ∈ A(I), I > Iζ I −I ζ √ √ 2M T (I)M 4 2M 2 T (I) 2T (I)M where ∆(I) := c e + c2 e and M := max{1, kV 00 k∞ }. p p Proof. In order to prove (a) let us note that by Lemma 4.11, Ω(I) > 2c r. Since |x(t, I)| 6 r for all t ∈ P(I) and I > 0, we have r 1 2 p x(t, I) 6 . Ω(I) c On the other hand, if t ∈ N (I) then a < x(t, I) < 0 so the assertion in (a) holds. The proof of (b) follows straightforward from Lemma 4.3. For proving (c), let us first start by considering t ∈ B(I). Indeed it is not restrictive to assume that t lies on the component containing 0. By Lemma 4.2 we have Z t  ∂x |ẍ(t, I)| 1 − V 00 x(s, I) ẋ(s, I)2 − ẍ(s, I)2 (t, I) 6 + |ẋ(t, I)| 2 ds. ∂I ẋ(t, I)2 + ẍ(t, I)2 0 ẋ(s, I)2 + ẍ(s, I)2 By hypothesis (H2 ) in (a + ζ, +∞) we have that B(I) = [0, T (I)] \ A(I) correspond to the set of times when V 00 (x(t, I)) is bounded. √ Therefore, on account of Lemma 4.3 and Lemma 4.10 together with the general inequalities a 6 a2 + b2 and a2 − b2 6 a2 + b2 we have Z t ∂x 1 |1 − V 00 (x(s, I))| (t, I) 6 1 + |ẋ(t, I)| 2 2 ds ∂I (ẋ(t, I)2 + ẍ(t, I)2 ) 2 0 ẋ(s, I) + ẍ(s, I) p Z 2eT (I)M 4 2Ω(I)e2T (I)M t 6 + |1 − V 00 (x(s, I))| ds. V 0 (r) V 0 (r)2 0 q 1 From (21) we deduce that V 0 (r) > cr > c kV 002k∞ Ω(I) 2 leading to √ √ p ∂x 2M T (I)M 4 2M 2 T (I) 2T (I)M Ω(I) (t, I) 6 e + e (22) ∂I c c2 for all t ∈ B(I). Here we also used that 1 + kV 00 k∞ 6 2M . On the other hand, if t ∈ A(I), by (a) in Lemma 4.7 we have   p ∂x p ∂x p ∂x Ω(I) (t, I) 6 max Ω(I) (π, I) , Ω(I) (τ, I) , ∂I ∂I ∂I where τ = τζ+ (I) is the right-hand endpoint of A(I). Notice that τ ∈ B(I), so the second function in the previous maximum satisfies the inequality in (22) and so the result follows by (b) in Lemma 4.7. 4.4 Isochronous oscillators with an asymptote and a bouncing problem In this section we assume that −∞ < a < 0 < b = +∞ and the potential V satisfies (Hi ), i = 0, 1, 2, 3. In addition it will be assumed that all non-constant solutions of equation (15) have minimal period 2π, that is Ω(I) = I. We will be interested in the asymptotic behaviour of the functions x(t, I), ẋ(t, I) and ∂x ∂I (t, I) as I → +∞. The following notion will be useful to describe this behaviour. 12 Definition 4.13. Given γ ∈ L∞ (T), we say that ψ is a solution of the γ−Bouncing problem (γ − BP ) if it satisfies the following properties: (i) ψ ∈ W 1,∞ (T) ∩ W 2,∞ (−π, π), (ii) ψ̈ + γ(t)ψ = 0 in (−π, π), (iii) ψ > 0 and ψ(t) = 0 if and only if t = (2n + 1)π, (iv) ψ̇(π + 0) = −ψ̇(π − 0).  The space L∞ (T) is composed by the measurable functions which are 2π-periodic and essentially bounded. W 1,∞ (T) is the Sobolev space modelled on L∞ (T). Indeed condition (i) can be formulated in more classical terms, without reference to Sobolev spaces. More precisely, we can replace it by (i0 ) ψ : R → R is continuous and 2π-periodic, the restriction ψ|[−π,π] belongs to C 1 [−π, π] and the derivative ψ 0 is Lipschitz-continuous on [−π, π]. The equation in (ii) can be understood in the Carathéodory sense. Finally we observe that, by periodicity, ψ also belongs to C 1 [π, 3π] and so the condition (iv) has a precise meaning. Typically ψ 0 will present a jump discontinuity at the points t = (2n + 1)π, n ∈ Z. Theorem 4.14. Let (In )n>0 be a sequence of actions such that In → +∞ as n → +∞. Then there exist a subsequence (Ink )k>0 ⊂ (In )n>0 , γ ∈ L∞ (T) and a solution ψ of the γ − BP such that x(t, Ink ) (a) p converges uniformly to ψ, and Ink ẋ(t, Ink ) (b) p converges uniformly to ψ̇ in [−π + δ, π − δ] for all δ > 0. Ink Moreover the function γ satisfies V 0 (x) V 0 (x) lim inf 6 γ(t) 6 lim sup (23) x→+∞ x x→+∞ x almost everywhere. ∂x An analogous result holds for ∂I . Theorem 4.15. In the same conditions of the previous theorem there exists a subsequence (Imk )k>0 ⊂ (In )n>0 , γ1 ∈ L∞ (T) and a solution ψ1 of the γ1 − BP such that p ∂x Imk (t, Imk ) converges uniformly to ψ1 in [−π + δ, π − δ] for all δ > 0. ∂I The function γ1 satisfies the condition lim inf V 00 (x) 6 γ1 (t) 6 lim sup V 00 (x) (24) x→+∞ x→+∞ almost everywhere. In view of the generalized L’Hôpital rule this condition is weaker than (23). It must be noticed that the functions γ and γ1 do not necessarily coincide even when a common subsequence of (In ) can be extracted. Before proving these results we need some preliminary lemmas about the weak∗ convergence in L∞ (J), where J is a bounded interval. We first recall this notion of convergence. Given a sequence of functions ∗ φn ∈ L∞ (J) and φ ∈ L∞ (J), we say that φn converges to φ in the weak∗ sense, φn − * φ in L∞ (J), if Z Z χφn → χφ for each χ ∈ L1 (J). J J 13 Lemma 4.16. Let (φn )n>0 be a sequence of functions, φn ∈ L∞ (0, π), and C > 0 such that for each δ > 0 there exists N = N (δ) satisfying kφn kL∞ (0,π−δ) 6 C if n > N . Then there exist a subsequence ∗ (φnk )k>0 ⊂ (φn )n>0 and φ̂ ∈ L∞ (0, π) such that φnk |(0,π−δ) − * φ̂|(0,π−δ) in L∞ (0, π − δ) for all δ > 0. Proof. Let (δk )k>0 be a monotone sequence tending to zero and let us denote Xk = L∞ (0, π − δk ). After extending by zero any function of Xk to (0, π) we have X0 ⊂ X1 ⊂ · · · ⊂ Xk ⊂ · · · ⊂ L∞ (0, π). We will employ a diagonal procedure. The sequence kφn kX0 is eventually bounded by C > 0 by hypothesis. Thus, Banach-Alaoglu Theorem states that there exist φ̂0 ∈ X0 and a subsequence (φσ0 (n) )n>0 ⊂ (φn )n>0 ∗ such that φσ0 (n) −* φ̂0 in X0 . Moreover, kφ̂0 kX0 6 C. The sequence kφn kX1 is also eventually bounded by C > 0 by hypothesis. Thus, again by Banach-Alaoglu Theorem, there exist φ̂1 ∈ X1 and a subsequence ∗ (φσ1 (n) )n>0 ⊂ (φσ0 (n) )n>0 such that φσ1 (n) − * φ̂1 in X1 . Moreover, kφ̂1 kX1 6 C and φ̂1 = φ̂0 almost everywhere in (0, π − δ0 ). Here we are using that (φσ1 (n) ) is a subsequence of (φσ0 (n) ) and the uniqueness of the limit in the weak∗ sense. Inductively we have that for each k > 0 there exist φ̂k ∈ Xk and a subsequence ∗ * φ̂k in Xk , kφ̂k kXk 6 C and (φσk (n) )n>0 ⊂ (φσk−1 (n) )n>0 ⊂ · · · ⊂ (φσ0 (n) )n>0 ⊂ (φn )n>0 such that φσk (n) − φ̂k = φ̂j almost everywhere in (0, π − δj ) for all 0 6 j < k. Let us consider the sequence (φσn (n) )n>0 and ∗ define φ̂ = φ̂k on (0, π − δk ). Thus φσn (n) − * φ̂ in Xk for all k > 0. Moreover kφ̂kXk 6 C for all k > 0 and so φ̂ : (0, π) → R is measurable and kφ̂k∞ 6 C. ∗ Remark 4.17. The next example shows that the stronger conclusion φnk − * φ̂ in L∞ (0, π) cannot be obtained. Let φnR be the continuous function which vanishes on [0, π− n ] and grows linearly from π− n1 to π in 1 π such a way that 0 φn = 1. By construction, for all δ > 0 there exists N = N (δ) such that kφn kL∞ (0,π−δ) 6 0 if n > N . We prove that there is no subsequence φnk converging in the weak∗ sense in L∞ (0, π). Otherwise ∗ there should exist φ̂ ∈ L∞ (0, π) with φnk − * φ̂. Testing this convergence with the constant function χ = 1 Rπ Rπ we deduce that 0 φ̂ = limk→+∞ 0 φnk = 1. We can also use the characteristic function of the interval R π− 1 R π− 1 [0, π − N1 ] as the test function χ. In this case we obtain 0 N φ̂ = limk→+∞ 0 N φnk = 0. Since N is arbitrary these two facts are not compatible and so φ̂ cannot exist.  Lemma 4.18. Let f : [0, +∞) → R be a continuous function and let (φn )n>0 be a sequence in L∞ (J) such that φn → +∞ almost uniformly as n → +∞. This means that there exists a sequence of numbers (Mn ) with Mn → +∞ and such that φn (t) > Mn almost everywhere. In addition assume that, for some ∗ γ ∈ L∞ (J), f ◦ φn − * γ in L∞ (J). Then lim inf f (x) 6 γ(t) 6 lim sup f (x) (25) x→+∞ x→+∞ for almost every t ∈ J. Proof. If lim inf x→+∞ f (x) is finite we take α ∈ R with α < lim inf x→+∞ f (x). Otherwise we define α = −∞. In the same way we take an upper bound β for lim supx→+∞ f (x). We define the set Ωα,β := {g ∈ L∞ (J) : α 6 g(t) 6 β a.e.}. The choice of α, β and the convergence to infinity of φn imply that f ◦φn belongs to Ωα,β if n is large enough. The set Ωα,β is convex and closed with respect to the strong topology in L∞ (J). Thus, it is also closed with respect to the weak∗ topology. In consequence T γ belongs to Ωα,β . The class of functions described by the inequalities (25) is precisely the set α,β Ωα,β . Proof of Theorem 4.14. Let us denote fI (t) := √1I x(t, I) and let (In )n>0 be a sequence of actions such that In → +∞ as n → +∞. The function fIn satisfies the Cauchy problem  √  0 f¨ + V (fI√ n In ) fIn = 0, In fIn In (26)  fI (0) = √rn , f˙I (0) = 0, n I n n 14 where rn = √x(0, In ). Since Ω(I) = I, according to (a) and (b) in Proposition 4.12 we have that |fIn (t)| and |f˙In (t)| 6 2 are uniformly bounded √ in T for all n > 0. Thus the family {fIn }n>0 is equicontinuous, bounded and K-Lipschitz with K 6 2 so by Arzelà-Ascoli theorem there exists a subsequence (Ink )k>0 ⊂ (In )√ n>0 such that fInk → ψ uniformly for some function ψ ∈ W 1,∞ (T) with Lipschitz constant [ψ]Lip = K 6 2. This proves by now that ψ ∈ W 1,∞ (T). By Lemma 4.4, we have that N (In ) → {π} as n → +∞. Consequently, for any δ > 0 small enough there exists n0 (δ) such that [−π + δ, π − δ] ⊂ [−π, π] \ N (In ) for all n > n0 (δ). Thus, for each δ > 0, the function fIn is positive on [−π + δ, π − δ] for all n > n0 (δ) and so by hypothesis (H2 ) we have √ V 0 (fIn (t) In ) √ 6 kV 00 k∞ for all t ∈ [−π + δ, π − δ] and n > n0 (δ). (27) fIn (t) In Thus, it follows from (26) that there exists a constant C > 0 such that |f¨In (t)| 6 C for all t ∈ [−π + δ, π − δ], δ > 0 and n > n0 (δ). Again by Arzelà-Ascoli theorem, for each δ > 0 there exists a subsequence (Ink (δ) )k>0 such that f˙Ink (δ) → ψ̇ uniformly on [−π + δ, π − δ]. Taking a sequence (δk )k>0 tending to zero and arguing by a diagonal process we can consider the convergent subsequence independent from δ and so without loss of generality f˙Ink → ψ̇ uniformly in every compact subset of (−π, π). Up to now we have proved the convergence results (a) and (b). Moreover, we know by now that ψ ∈ W 1,∞ (T). Let us now show that ψ is a solution of a bouncing problem. The first step will be the construction of γ ∈ L∞ (T). On account of (27) and Lemma 4.16, there exists γ ∈ L∞ (T) with kγkL∞ (T) 6 kV 00 k∞ such that p V 0 (fInk Ink ) ∗ p −−* γ in L∞ (−π + δ, π − δ) for any δ > 0. (28) fInk Ink For the sake of simplicity we consider (Ink ) to be the same as the previous subsequence. From the previous limit we have that γ is 2π-periodic. Let us take δ > 0 and φ ∈ D(−π + δ, π − δ) a smooth test function with supp(φ) ⊂ [−π + δ, π − δ]. We have that supp(φ) ⊂ P(Ink ) for sufficiently large k. We have, on account of equation in (26), that Z π ( p ) V 0 (fInk (t) Ink ) fInk (t)φ̈(t) + p fInk (t)φ(t) dt = 0. −π fInk (t) Ink It is at this point where property (28) plays an important role in the passage to the limit. Indeed, if un → u ∗ R R uniformly in L∞ (T) and vn − * v in L∞ (T) with kvn kL∞ (T) bounded then T un vn → T uv. Therefore, since fInk converges uniformly to ψ and using the weak∗ convergence in (28), Z ( p ) Z π π V 0 (fInk (t) Ink ) fInk (t)φ̈(t) + p fInk (t)φ(t) dt → {ψ(t)φ̈(t) + γ(t)ψ(t)φ(t)}dt = 0. −π fInk (t) Ink −π That is, ψ satisfies the ordinary differential equation ψ̈ + γ(t)ψ = 0 in the sense of distributions in D(−π, π). Since we are dealing with an o.d.e., distributional solutions coincide with solutions in the Carathéodory sense and so ψ ∈ W 2,∞ (−π, π) and the equation holds almost everywhere in (−π, π). This shows properties (i) and (ii) of Definition 4.13. Let us now show the remaining properties. To show (iii) we select t ∈ [0, 2π], t 6= π. Then, since N (Ink ) → {π}, we know that fInk (t) > 0 for k large enough. In consequence ψ(t) = limk→+∞ fInk (t) is non-negative. Since ψ is continuous and periodic we deduce that ψ > 0 in T. Moreover, by Proposition 4.12 1 we have |fI (π)| 6 |a| I − 2 . Thus |ψ(π)| 6 0 and so ψ(π) = 0. On account of the 2π-periodicity we have then that ψ(t) = 0 if t = (2n + 1)π. On the other hand, if there exists t∗ ∈ (−π, π) such that ψ(t∗ ) = 0, since ψ > 0 we also have ψ̇(t∗ ) = 0. Notice that we already know that ψ is in W 2,∞ (−π, π) and so it also belongs to C 1 [−π, π]. Thus, on account of uniqueness of solution of the differential q equation ψ̈ + γ(t)ψ = 0 on (−π, π) we have that ψ ≡ 0. However, by Lemma 4.11 we have fIn (0) > kV 00 k∞ contradicting that ψ ≡ 0. 2 15 This proves (iii). The symmetry properties of the equation (15) lead to the identity fI (π + t) = fI (π − t). Passing to the limit we obtain the identity ψ(π + t) = ψ(π − t) which implies ψ̇(π + t) = −ψ̇(π − t) for all t ∈ [−π, π]. Note that, by periodicity, ψ also belongs to C 1 [π, 3π]. To complete the proof we must check the condition (23). Let us fix a compact interval J ⊂ (−π, π). Since ψ(t) is positive on J and fIn → ψ uniformly, there exists a constant µJ > 0 such that fIn (t) > µJ if n is large enough and t ∈ J. We apply V 0 (x) √ Lemma 4.18√ with f (x) = x and φn (t) = fIn (t) In . Then φn → +∞ almost uniformly on J with Mn = µJ In . Therefore (23) holds for almost every t ∈ J. The interval J ⊂ (−π, π) is arbitrary and the inequality (23) will also hold almost everywhere on (−π, π). Proof of Theorem 4.15. It is very similar to the proof of √ Theorem 4.14 and we will stress the points where there appear some differences. Let us denote gI (t) := I ∂x ∂I (t, I) and let (In )n>0 be a sequence of actions such that In → +∞ as n → +∞. The function gIn satisfies the Cauchy problem ( g̈In + V 00 (x(t, √ In ))gIn = 0, (29) gIn (0) = V 0 (rInn ) , ġIn (0) = 0. Since A(In ) ⊂ N (In ), by Lemma 4.4 we have A(In ) → {π} as In → +∞. Thus, for any δ > 0 small enough, there exists n0 (δ) such that [−π + δ, π − δ] ⊂ [−π, π] \ A(In ) for all n > n0 (δ). Therefore, by definition of A(I), |V 00 (x(t, In ))| 6 kV 00 k∞ for all t ∈ [−π + δ, π − δ] and, by (c) in Proposition 4.12, |gIn (t)| is uniformly bounded in T for all n > 0. Consequently, taking into account equation (29), we have that there exists some constant C > 0 such that kg̈In kL∞ ([−π+δ,π−δ]) 6 C and kġIn kL∞ ([−π+δ,π−δ]) 6 C for all n > n0 (δ). We point out here that C is independent from δ since the previous bounds are independent from δ. By Arzelà-Ascoli theorem, there exist a subsequence (Ink )k>0 of (In )n>n0 and ψ1 ∈ C 1 (−π, π) such that gInk → ψ1 and ġInk → ψ̇1 uniformly on compact subsets of (−π, π). Moreover |ψ1 (t)| 6 C and |ψ̇1 (t)| 6 C for all t ∈ (−π, π). In particular ψ1 is bounded and Lipschitz-continuous with [ψ1 ]Lip 6 C. This implies that ψ1 has a continuous extension to [−π, π] and, by periodicity, to the whole real line. By now we have proved that ψ1 ∈ W 1,∞ (T) and the convergence assertion on the statement. The rest of the proof goes along the same lines of the proof of Theorem 4.14. The only substantial difference is in the last step, when Lemma 4.18 is applied. Now f (x) = V 00 (x) and φn (t) = x(t, In ). For each compact interval J ⊂ (−π, π) the sequence φn → +∞ uniformly in J. 5 Resonance of isochronous oscillators This section is dedicated to prove some of the results presented in Section 2. 5.1 Proof of Theorem A We first study some simple properties of the linear equation ÿ + a(t)y = b(t) (30) where a ∈ L∞ (0, 2π), b ∈ L1 (0, 2π) and kakL∞ (0,2π) 6 A. (i) Let ψ(t) be the solution of (30) with b ≡ 0 and initial conditions ψ(0) = 1, ψ̇(0) = i. Then there exists C > 0, depending only on A, such that |ψ(t)| 6 C for each t ∈ [0, 2π]. (ii) Let y(t) be the solution of (30) with initial conditions y(0) = ẏ(0) = 0. Then |y(t)| 6 C 2 kbkL1 (0,2π) if t ∈ [0, 2π]. 0 1  To prove (i) we observe that the matrix ( uu̇ v̇v ) is a solution of the linear system Ẏ = −a(t) 0 Y , where u = Reψ, v = Imψ. The estimate is obtained as in Lemma 4.10. To prove (ii) we apply the formula of variation of constants, leading to Z t y(t) = [v(t)u(s) − v(s)u(t)]b(s)ds. 0 16 Cauchy-Schwarz inequality can be applied to the term in brackets to obtain 1 1 |v(t)u(s) − v(s)u(t)| 6 (u(t)2 + v(t)2 ) 2 (u(s)2 + v(s)2 ) 2 6 C 2 . The Second Massera’s Theorem [15] states that if all solutions of (2) are globally defined in the future and at least one of them is bounded in the future, then a 2π-periodic solution must exist. Our strategy will be to prove non-existence of 2π-periodic solutions of (2) for ε 6= 0 small. We proceed by contradiction and assume that {εn } is a sequence with εn 6= 0, εn → 0 such that the equation (2) has a 2π-periodic solution xn (t) for ε = εn . Let Xn (t) be the solution of (1) with the same initial conditions of xn (t) at t = 0, Xn (0) = xn (0) and Ẋn (0) = ẋn (0). The difference yn (t) = xn (t) − Xn (t) can be seen as a solution of the equation (30) with Z 1  a(t) = V 00 (1 − λ)xn (t) + λXn (t) dλ, b(t) = εn p(t). 0 From property (ii) we deduce that kyn kL∞ (T) 6 C 2 |εn |kpkL1 (T) . (31) Here A = supx∈R |V 00 (x)|. The function yn (t) can also be interpreted as a 2π-periodic solution of the linear periodic equation ÿ + V 00 (Xn (t))y = εn pn (t) − qn (t) with Z 1   qn (t) = yn (t) V 00 (1 − λ)xn (t) + λXn (t) − V 00 Xn (t) dλ. 0 Note that the coefficients of this equation are 2π-periodic because the oscillator defined by (1) is globally isochronous. All solutions of (1) are of the type ϕ(t − θ, r). In particular, Xn (t) = ϕ(t − θn , rn ) for some θn ∈ [0, 2π] and rn > 0. The function ψ(t − θn , rn ) is a 2π-periodic solution of the homogeneous equation ÿ + V 00 Xn (t) y = 0. Fredholm alternative can be applied to deduce that Z 2π Z 2π εn p(t)ψ(t − θn , rn )dt − qn (t)ψ(t − θn , rn )dt = 0. 0 0 In other words, Z 2π 1 qn (t) Φp (θn , rn ) = ψ(t − θn , rn )dt. 2π 0 εn The definition of qn together with the estimate (31) and the Dominated Convergence Theorem imply that ε1n qn (t) → 0 as n → +∞ for every t ∈ [0, 2π]. From the property (i) we know that kψ(·, rn )kL∞ (T) 6 C and the estimate (31) implies that kqn kL∞ (T) 6 2C 2 kpkL1 (T) kV 00 kL∞ (R) . |εn | Using once again dominated convergence, Φp (θn , rn ) → 0 as n → +∞. This is incompatible with the assumption (3) and proves Theorem A. A refinement of Massera’s Theorem shows that the equation (2) has a 2π-periodic solution if there exists a solution x(t) which is defined in [0, +∞) and satisfies lim inf (|x(2πn)| + |ẋ(2πn)|) < +∞. n→+∞ See [18]. In consequence, if the equation (2) is resonant all solutions satisfy lim (|x(2πn)| + |ẋ(2πn)|) = +∞. n→+∞ 17 A standard argument using Gronwall’s lemma and kV 00 k∞ < +∞ implies that there exists a constant C > 0 such that for every solution x(t), |x(t)| + |ẋ(t)| > C (|x(0)| + |ẋ(0)|) if t ∈ [0, 2π]. Therefore every solution of (2) satisfies |x(t)| + |ẋ(t)| → +∞ as t → +∞. 5.2 Proof of Proposition 2.2 The aim is to show that if Φp has a non-degenerate zero (θ∗ , r∗ ) then a 2π-periodic solution persists on the perturbed system (2) for small ε. At this point the authors want to emphasize that Melnikov method to study subharmonic bifurcations cannot be applied in this framework due to the isochronicity of the center. Instead, degree theory will be the key point on the proof of the result. For the sake of completeness we sketch Nagumo’s definition of Brouwer degree. There are other equivalent ways to define this degree, we refer the reader to [13] for more details. Let Ω be an open and bounded subset of Rd and a function f : Ω̄ → Rd which is continuous and does not vanish on the boundary of Ω. First we assume that f ∈ C 1 (Ω̄) and has a finite number of zeros x1 , . . . , xn ∈ Ω with det f 0 (xi ) 6= 0 for each i. Then the Brouwer degree of f in Ω is n X deg(f, Ω) = sign det f 0 (xi ). i=1 In the case f is continuous, we approximate it by functions fk in the previous conditions and define deg(f, Ω) = lim deg(fk , Ω). k→+∞ An important property of the degree is its invariance by homotopy, which plays a crucial role on the proof of the result under consideration. Proof of Proposition 2.2. Let x(t; x0 , v0 , ε) be the solution of (2) with initial conditions x(0) = x0 , ẋ(0) = v0 . Define (  1  ε x(t; x0 , v0 , ε) − x(t; x0 , v0 , 0) if ε 6= 0, ∆(t; x0 , v0 , ε) := ∂x ∂ε (t; x0 , v0 , 0) if ε = 0. For each ε ∈ R consider the planar map  ˙ Fε : R2 → R2 , Fε (x0 , v0 ) = ∆(2π; x0 , v0 , ε), ∆(2π; x0 , v0 , ε) . We are going to prove that F0 has a non-degenerate zero. Then, for small ε 6= 0, the map Fε will also have a zero. Clearly this zero will produce a 2π-periodic solution of (2). Let N > 1 be the integer such that all non-trivial solution of (1) have minimal period 2π N . Then ψ(·, r) has period 2π N and the map  Φ̃p (θ, r) := Φp Nθ , r is well defined on the cylinder C. The map (θ, r) 7→ (N θ, r) is a local diffeomorphism of the open cylinder C˙ = (R/2πZ) × (0, ∞). Let (θ∗ , r∗ ) be the non-degenerate zero of Φp given by assumption, then (N θ∗ , r∗ ) is a non-degenerate zero of Φ̃p . This can be deduced from the definition of degree or from Leray multiplication theorem. Let us now consider the diffeomorphism   η : (θ, r) ∈ C˙ 7→ ϕ − Nθ , r , ϕ̇ − Nθ , r ∈ R2 \ {0}. We are going to compute F0 ◦η using the formula of variation of constants. The function y(t) = ∆(t; η(θ, r), 0) is the solution of  ÿ + V 00 ϕ t − Nθ , r y = p(t), y(0) = ẏ(0) = 0. 18 Then   Z t  ∆(t; η(θ, r), 0)  u s− θ =M t− θ N ˙ η(θ, r), 0) ∆(t; N,r v s− θ p(s)ds 0 N v −u  where M = v̇ −u̇ and u(t, r) = Reψ(t, r), v(t, r) = Imψ(t, r). In consequence  F0 ◦ η(θ, r) = 2πM − Nθ , r Φ̃p (θ, r). Here we are identifying C and R2 so that Φ̃p takes values in R2 . The matrix M has determinant one and so the zeros of F0 ◦ η and Φ̃p coincide. For each λ ∈ [0, 1] define Hλ : C˙ → R2 by Hλ (θ, r) = M (− λθ N , r)Φ̃p (θ, r). 1  We observe that H0 = J Φ̃p and H1 = 2π F0 ◦ η where J = 01 −1 0 . Moreover, the zeros of Hλ are independent of λ. We conclude that (N θ∗ , r∗ ) is a non-degenerate zero of F0 ◦ η. Then (x∗0 , v0∗ ) = η(N θ∗ , r∗ ) is a non-degenerate zero of F0 . This completes the proof. 5.3 Proof of Proposition 2.3 Lemma 5.1. There exist C, c > 0 depending on kV 00 k∞ such that c 6 |ψ(t, r)| 6 C, |ψ̇(t, r)| 6 C if t ∈ [0, 2π] and r > 0. Proof. Assume that ψ = u + iv with u = Reψ, v = Imψ. Then ( uu̇ v̇v ) is a matrix solution of the linearized system and the upper estimates are obtained as in Lemma 4.10. In particular u̇(t)2 + v̇(t)2 6 C 2 if t ∈ [0, 2π]. From Liouville’s identity, 1 1 1 = uv̇ − u̇v 6 (u2 + v 2 ) 2 (u̇2 + v̇ 2 ) 2 6 C |ψ| 1 and we can take c = C. Proof of Proposition 2.3 Let p, q ∈ L1 (T). The fact that RV is open follows from Lemma 5.1 since C |Φp (θ, r) − Φq (θ, r)| 6 kp − qkL1 (T) . 2π Let us now show that RV is non-empty. To do so, we shall see that if we take a sequence pn ∈ L1 (T) with pn −* δ then pn ∈ RV for n large enough. In order to arrive at contradiction we consider a sequence {pn }n>0 such that for all n > 0, pn ∈ L1 (T), pn ∈ / RV and pn − * δ. From pn ∈ / RV it follows that for each (n) (n) (n) (n) n there exists a sequence (θk , rk ) ∈ C such that Φpn (θk , rk ) → 0. Arguing by a diagonal process we can consider Φpn (θn , rn ) → 0. On account of the bounds in Lemma 5.1, by Arzelà-Ascoli theorem, there exists a subsequence ψ(t + θnk , rnk ) that converges uniformly on [0, 2π] to some ψ̂ ∈ C(T, C) as k → +∞. Moreover, c 6 |ψ̂(t)| 6 C. Thus, Z 2π Z 2π   Z 2π 2πΦpnk (θnk , rnk ) = pnk (t)ψ(t+θnk , rnk )dt = pnk (t) ψ(t + θnk , rnk ) − ψ̂(t) dt+ pnk (t)ψ̂(t)dt. 0 0 0 The first integral tends to zero as k tends to infinity. Indeed, we have Z 2π   pnk (t) ψ(t + θnk , rnk ) − ψ̂(t) dt 6 kpnk kL1 (T) kψ(· + θnk , rnk ) − ψ̂kL∞ (T) → 0 0 as k → +∞ due to the boundedness of kpnk kL1 (T) and the convergence of ψ(t + θnk , rnk ) to ψ̂(t) uniformly in [0, 2π]. In addition, since pn − * δ, Z 2π pnk (t)ψ̂(t)dt → ψ̂(0) 0 as k → +∞ with |ψ̂(0)| > c > 0. Consequently, 1 c |Φpnk (θnk , rnk )| → |ψ̂(0)| > >0 2π 2π as k → +∞, reaching contradiction with the choice of pn ∈ / RV . 19 6 Two resonance results for a periodic perturbed isochronous of Pinney type This section is dedicated to prove Theorem B and C. 6.1 Proof of Theorem B We are concerned with the potential   1 1 1 V (x) = (x + 1)2 + − , x ∈ J = (−1, +∞). (32) 8 (x + 1)2 4   1 1 All the assumptions (Hi ) are satisfied in this case. From the first derivative V 0 (x) = 4 x+1− (x+1)3 0 it is easy to deduce that (H0 ) and (H3 ) hold. Note that V x(x) → 41 as x → +∞. The second derivative V 00 (x) = 14 + 34 (x + 1)−4 is positive and so (H1 ) also holds. Finally we observe that V 00 (x) 6 1 if x > 0 and this implies (H2 ). The next result could be stated for a general class of isochronous systems with strictly convex potential. For our purposes we can restrict to the potential defined by (32). Lemma 6.1. In the previous notations assume that x(t, I) is a solution of (6). Then ∂ ẋ (t, I) < 0 on (0, π). ∂I Proof. The functions ẋ(t, I) and ∂x ∂I (t, I) are a fundamental pair of solutions of the variational equation ¨ 00 ξ + V (x(t, I))ξ = 0. From equation (1) we deduce that the critical points of ẋ(t, I) coincide with the zeros of x(t, I). In consequence ẋ(t, I) has exactly two critical points in the interval [−π, π], say −π < t∗− < 0 < t∗+ < π. Moreover, by symmetry, t∗− = −t∗+ . We are going to apply Lemma 3.3 twice. First we take a(t) = V 00 (x(t, I)), φ1 (t) = ∂x ∗ ∗ ∗ ∂I (t, I), φ2 (t) = ẋ(t, I), τ = t+ . Since ẋ(t, I) has no critical points on (t− , t+ ) we ∂x ∗ ∗ deduce that ∂I (t, I) has no critical points on (t− , t+ ) \ {0}. Next we observe that x(t, I) has the symmetry, x(t + π, I) = x(−t + π, I). This allows us to apply again the Lemma with ã(t) = a(t + π), φ̃1 (t) = φ1 (t + π), φ̃2 (t) = φ2 (t + π), τ = π − t∗+ . Now we conclude that ∂x ∗ ∗ ∂I (t, I) has no critical points on (t+ , 2π + t− ) \ {π}. ∗ ∂x We also observe that t+ cannot be a critical point of ∂I (t, I) because this function and ẋ(t, I) are linearly independent. Summing up the previous discussion we conclude that ∂x ∂I (t, I) has no critical points on the interval (0, π). In consequence ∂I (t, I) is monotone on this interval. Finally, by Lemma 4.2, ∂x ∂x ∂I (0, I) > 0 and ∂x ∂I (π, I) < 0 and so ∂ ẋ ∂I (t, I) is decreasing. The conclusion follows. The next result describes the asymptotic behaviour of x(t, I) and its derivatives. It was already obtained in [4] but there are significant differences in the type of convergence and also in the proof. Proposition 6.2. Let x(θ, I) be a solution of system (6). Then, as I → +∞, √  (a) √1I x(t, I) → 2 2 cos 2t uniformly in T. √   (b) √1I ẋ(t, I) → − 2sign(cos 2t ) sin 2t uniformly in every compact subset of (−π, π). √ √  (c) I ∂x ∂I (t, I) → 2 cos 2t uniformly in T. Proof. We apply the results of Section 4.4. A key observation is that the inequality in (23) now becomes 0 an identity because the limit exists. Namely, limx→+∞ V x(x) = 14 . In consequence the function γ has to be the constant 41 . The next step will be to determine all the solutions of the Bouncing problem when γ = 14 . A direct computation shows that these solutions are of the type   t ψ(t) = λ cos with λ > 0. 2 20 To determine the admissible values of λ we first compute a couple of limits. Recalling that x(0, I) = r with V (r) = I, r > 0, and using the concrete form of the potential given by (32) we deduce that 1 √ lim √ x(0, I) = 2 2. I→+∞ I ∂x 1 Also, from the identity ∂I (0, I) = V 0 (r) , √ ∂x √ lim I (0, I) = 2. (33) I→+∞ ∂I The existence √ of these  limits implies that the functions ψ and ψ1 are uniquely determined. Actually ψ(t) = 2 2 cos 2t and ψ1 (t) = 21 ψ(t). In consequence, as I → +∞, there is convergence of the func- √ tions √1I x(·, I), √1I ẋ(·, I) and I ∂x ∂I (·, I). The convergence must be understood in the senses specified by Theorems 4.14 and 4.15. Assertions (a) and (b) are√already proved but some additional work is required to prove (c). In view of Theorem 4.15 we know that I ∂x ∂I (t, I) converges to ψ1 (t) for each t ∈ (−π, π). Moreover, by item (b) of Lemma 4.7, √ ∂x lim I (π, I) = ψ1 (π) = 0. I→+∞ ∂I √ By symmetry we know that there is pointwise convergence of I ∂x ∂I (t, I) to ψ1 (t) for each t ∈ R. We must prove that this convergence√ is uniform and it is enough to consider the interval [0, π]. We know by Lemma 6.1 that the function I ∂x (·, I) is monotone in [0, π]. A classical result says that if a continuous ∂I √ function ψ1 (t) is the pointwise limit of a sequence of monotone functions I ∂x ∂I (·, I), then the convergence is also uniform. This result can be proved in a more direct way using the explicit formulas for ϕ(t, r) and ψ(t, r) given in Section 2. The proof above is more flexible and can be adapted to general families of potentials having (32) as a prototype. We finish these preliminary results with some estimates for x(t, I) and its derivatives. Proposition 6.3. Assume that V is given by (32) and let x(t, I) be the solution corresponding to (1). Then there exists a constant C > 0 independent of I such that 1 1 √ ∂x √ |x(t, I)| + √ |ẋ(t, I)| + I (t, I) 6 C I I ∂I it t ∈ [0, 2π] and I > 0. Proof. The bounds on |x(t, I)| and |ẋ(t, I)| are direct consequences of items (a) and (b) in Proposition 4.12. To get the bound on ∂x ∂I (t, I) we can combine item (c) with the continuity of the function √ ∂x (t, I) ∈ [0, 2π] × [0, +∞) 7→ I (t, I) ∈ R. ∂I Note that this continuity is a consequence of the theorem of continuous dependence with respect to param- eters and initial conditions when it is applied to the equation ÿ + V 00 (x(t, √ I))y = 0.√The continuity at I = 0 is also valid because limI→0+ x(t, I) = 0 uniformly in t and limI→0+ I ∂x ∂I (0, I) = 2. Proof of Theorem B We write the equation (2) as the first order system ẋ = y, ẏ = −V 0 (x) + εp(t). This system is defined on D = {(x, y) ∈ R2 : x > −1} and all solutions are global. In fact we can repeat the energy arguments of Remark 2.1 for the potential V (x) given by (32). The estimate (5) now implies that, in finite time, the norm of the solutions |x(t)| + |y(t)| cannot blow up and also that x(t) cannot approach the vertical line x = −1. Therefore the maximal solutions (x(t), y(t)) are defined for all t ∈ R. 21 Second Massera’s Theorem is not directly applicable because the system is not defined in the whole plane. To overcome this difficulty we transport the system from D to the plane (z, y) with x + 1 = ez . In the new variables the system is ż = e−z y, ẏ = −V 0 (ez − 1) + εp(t) and Massera’s Theorem is applicable. We can conclude that if the equation (2) has no 2π-periodic solutions then every solution x(t) of (2) satisfies   1 lim sup |x(t)| + |ẋ(t)| + = +∞. (34) t→+∞ x(t) + 1 We claim that every solution of (2) satisfying (34) has to be unbounded, for otherwise there should exist a number C > 0, a sequence tn → +∞ and a solution x(t) with |x(t)| + |ẋ(t)| 6 C if t > 0 and x(tn ) → −1. Then, for each h ∈ [0, 1], 1 + x(tn + h) 6 1 + x(tn ) + Ch and so Z tn +1 Z 1 dt dh > → +∞. tn (x(t) + 1)3 0 (x(tn ) + Ch + 1)3 After integrating the equation (2) over the interval [tn , tn + 1] we obtain Z tn +1 Z tn +1 Z tn +1 1 1 dt ẋ(tn + 1) − ẋ(tn ) + (x(t) + 1)dt = ε p(t)dt + . 4 tn tn 4 tn (x(t) + 1)3 This identity cannot hold for large n because the last term goes to infinity while the rest remain bounded. The previous discussions allow us to conclude that the equation (2) will be resonant as soon as it has no 2π-periodic solutions. Our strategy will be to prove non-existence of 2π-periodic solutions of (2) for ε 6= 0 small. We will proceed in several steps. First we prove that the initial condition of a 2π-periodic solution cannot be close to the origin. Claim 1: There exist σ > 0 and ε∗ > 0 such that if 0 < |ε| < ε∗ then every 2π-periodic solution of (2) satisfies |x(0)| + |ẋ(0)| > σ. To prove this claim we denote by x(t; x0 , v0 , ε) the solution of (2) with initial conditions x(0) = x0 , ẋ(0) = v0 and define   x(2π; x0 , v0 , ε) − x0 Fε (x0 , v0 ) := . (35) ẋ(2π; x0 , v0 , ε) − v0 The initial condition of a 2π-periodic solution corresponds to a zero of Fε . After differentiating with respect to the parameter ε we obtain the expansion Fε (x0 , v0 ) = εY (2π; x0 , v0 ) + o(ε), ε → 0 (36) where Y (t; x0 , v0 ) is the solution of the linear equation     0 1 0 Ẏ = Y + , Y (0) = 0. −V 00 (x(t; x0 , v0 )) 0 p(t) Moreover this expansion is uniform on (x0 , v0 ) ∈ K if K is a compact subset of D. For the origin (x0 , v0 ) = (0, 0) this system can be solved explicitly and a simple computation shows that Z 2π   − sin(t) Y (2π; 0, 0) = p(t) dt. 0 cos(t) The condition (3) at r = 0 implies that this vector does not vanish and, by continuous dependence, we can find σ > 0 such that 1 kY (2π; x0 , v0 )k > kY (2π; 0, 0)k > 0 2 if |x0 | + |v0 | 6 σ. The expansion (36) allows to find ε∗ > 0 such that Fε (x0 , v0 ) 6= 0 if 0 < |ε| < ε∗ and |x0 | + |v0 | 6 σ, proving the veracity of the claim. 22 Once we have proved the above claim we will employ action-angle variables (θ, I). In the punctured plane R2 \ {0} the Hamiltonian system (2) is transformed into  θ̇ = 1 + εp(t) ∂x (θ, I),  ∂I (37)  ∂x I˙ = −εp(t) (θ, I), ∂θ with Hamiltonian function H (θ, I, t) = I + εp(t)x(θ, I), where x(t, I) is the solution of (15) introduced in Section 4. In view of the Claim 1 every 2π-periodic solution of (2) corresponds to a solution (θ(t), I(t)) of (37) but the periodicity of this new solution has to be proved. Claim 2: There exists ε∗∗ > 0 such that if 0 < |ε| < ε∗∗ and x(t) is a 2π-periodic solution of (2) then (θ(t), I(t)) is well defined for each t ∈ R and satisfies θ(t + 2π) = θ(t) + 2π, I(t + 2π) = I(t). Let [0, ω) be the maximal interval of (θ(t), I(t)) when it is understood as a solution of (37). In view of the second equation (37) and Proposition 6.3 we deduce that, for each t ∈ [0, ω), d p C ( I(t)) 6 |ε| |p(t)| . (38) dt 2 Integrating this inequality and assuming that t ∈ [0, ω) ∩ [0, 2π], p p C I(t) − I(0) 6 |ε| kpkL1 (T) . (39) 2 This estimate shows that I(t) cannot blow up on the interval [0, 2π]. Let us show also that I(t) cannot touch the singularity I = 0. For this purpose we select a closed orbit γ of (15) contained in |x| + |ẋ| < σ and denote by σ∗ > 0 the area enclosed by γ. Assuming that C2 |ε|kpkL1 (T) < 21 σ∗ we deduce from Claim 1 and the condition (39) that 1 I(t) > σ∗ . (40) 2 In principle this estimate is valid for t ∈ [0, ω) ∩ [0, 2π] but now it is clear that ω > 2π. Since (θ(t), I(t)) comes from a 2π-periodic solution of (2) we know that θ(2π) = θ(0) + 2πM, I(2π) = I(0) for some integer M . The system (37) is 2π-periodic in t and θ and we can conclude that (θ(t), I(t)) can be extended to a global solution satisfying θ(t + 2π) = θ(t) + 2πM, I(t + 2π) = I(t), t ∈ R. To complete the proof of the claim we must show that M = 1. From the first equation in (37), Proposition 6.3 and (40) we deduce r d C|ε| |p(t)| 2 (θ(t) − t) 6 p 6 C|ε| |p(t)| , t ∈ [0, 2π]. (41) dt I(t) σ∗ For small ε the function θ(t) − t − θ(0) must be small on [0, 2π] and this implies that M = 1. Claim 3: For small ε 6= 0 the equation (2) has no 2π-periodic solutions. By an indirect argument we assume that {εn }n>0 , εn 6= 0, is a sequence tending to zero such that xn (t) is a 2π-periodic solution of (2) with ε = εn . Let (θn (t), In (t)) be the corresponding solution of (37). After extracting a subsequence we can assume In (0) → I∗ ∈ [0, +∞] and θn (0) → θ∗ ∈ [0, 2π]. In view of Claim 2 we know that θn (t + 2π) = θn (t) + 2π, In (t + 2π) = In (t). Let us assume first that I∗ < +∞. From Claim 1 we know that I∗ > σ∗ and the estimates (39) and (41) imply that In (t) → I∗ and θn (t) − t → θ∗ uniformly in t ∈ [0, 2π]. In addition, on account of equations in system (37) and the 2π-periodicity of In (t) and θn (t) − t in T, for all n > 0 we have Z 2π Z 2π ∂x ∂x p(t) (θn (t), In (t))dt = p(t) (θn (t), In (t))dt = 0. (42) 0 ∂I 0 ∂θ 23 Passing to the limit in the previous equalities we conclude that Z 2π Z 2π ∂x ∂x p(t) (θ∗ + t, I∗ )dt = p(t) (θ∗ + t, I∗ )dt = 0. 0 ∂I 0 ∂θ The functions ∂x ∂x ∂I and ∂θ form a fundamental pair of solutions of (7). In consequence ψ(t, I∗ ) can be expressed as a complex linear combination of them and therefore Φp (−θ∗ , I∗ ) = 0. This is a contradiction with the condition (3). Now let us assume that I∗ = +∞. In this case, on account of the estimates (39) and (41) we can ensure that θn (t) − t → θ∗ and In (t) → +∞ uniformly in t ∈ T. More precisely, IInn(0) (t) → 1 uniformly in √ ∂x  t ∈ T. On account of Proposition 6.2 we have that, as I → +∞, I ∂I (θ, I) → cos 2 uniformly in T θ   and √1I ẋ(θ, I) → − sin θ2 sign cos θ2 uniformly in every compact subset of T \ {π}. Thus, on account In (t) of In (0) → 1 uniformly in t ∈ T, as n → +∞,   p ∂x θ∗ + t p(t) In (0) (θn (t), In (t)) → p(t) cos ∂I 2 uniformly in T and      1 θ∗ + t θ∗ + t p(t) p ẋ(θn (t), In (t)) → −p(t) sin sign cos In (0) 2 2 uniformly in every compact subset of T \ {π − θ∗ }. Consequently, using (42) for this particular case, Z π Z π   p ∂x θ∗ + t 0= p(t) In (0) (θn (t), In (t))dt → p(t) cos dt −π ∂I −π 2 and Z π Z π      1 θ∗ + t θ∗ + t 0= p(t) p ẋ(θn (t), In (t))dt → − p(t) sin sign cos dt. −π In (0) −π 2 2 1 In view of Proposition 6.3 we observe that √ ẋ(θn (t), In (t)) is uniformly bounded. The convergence of In (0) the second integral follows from Lebesgue dominated convergence Theorem. These two integrals vanish and this is not compatible with condition (3). Indeed we can apply again dominated convergence to prove that Z 2π Z 2π         t t t lim p(t − θ)ψ(t, r)dt = p(t − θ) cos + 2i sin sign cos dt. r→+∞ 0 0 2 2 2 Here we are using (8). Proof of Corollary 2.4 The Fourier expansion of ψ(·, r) is of the type +∞ X ψ(t, r) = cm (r)eimt m=−∞ and we will derive some properties of the first coefficients. The symmetry ψ(−t, r) = ψ(t, r) implies that the numbers c0 (r), c1 (r) and c−1 (r) are real. Define Z π 1 1 d+ (r) = (c1 (r) + c−1 (r)) = [Reψ(t, r)] cos(t)dt, 2 2π −π Z π 1 1 d− (r) = (c1 (r) − c−1 (r)) = [Imψ(t, r)] sin(t)dt. 2 2π −π 24 From the explicit formula (8) we observe that Z π 1 sin2 (t) 1 d− (r) = q dt > d− (0) = . (43) 2π −π cos2 ( t ) + 1 sin2 ( t ) 2 2 λ4 2 Here λ = 1 + r. To obtain a lower estimate of d+ (r) we cannot apply the same strategy because the corresponding integrand changes sign. We use the symmetry (Reψ(·, r) is even) to reduce the interval of integration, Z Z π2 1 π 1 d+ (r) = [Reψ(t, r)] cos(t)dt = [Reψ(t, r) − Reψ(π − t, r)] cos(t)dt. (44) π 0 π 0 A direct computation shows that Reψ̇(t, r) < 0 on (0, π). In consequence Reψ(·, r) is decreasing and d+ (r) is positive. We can go further after observing that d0+ (r) > 0 for each r > 0. This is a consequence of long and direct computations coming from (8). Differentiating with respect to r, √  ∂ 2 4 sin4 ( 2t ) + 3λ4 sin2 ( 2t ) [Reψ(t, r)] = 3 . (45) ∂r λ3 2 + (λ4 − 1)(1 + cos(t)) 2 Next we differentiate this formula with respect toh t by quotient i rule and observe that all summands on ∂ ∂ the numerator remain positive on (0, π). Then ∂r Reψ̇(t, r) > 0 and ∂r [Reψ(·, r)] is increasing on (0, π). Differentiating with respect to r in (44) and looking at the integral over (0, π2 ) we conclude that d0+ (r) < 0. Hence, Z 1 π 2 d+ (r) > d+ (+∞) = cos( 2t ) cos(t)dt = . (46) π 0 3π The computation of d+ (+∞) involves a passage to the limit (r → +∞) in the integral. This is justified by the estimate |Reψ(t, r)| 6 2 implying that there is dominated convergence. With respect to the coefficient Z π 1 c0 (r) = Reψ(t, r)dt 2π −π we differentiate in r and apply (45) to deduce that c00 (r) > 0. Then, using again dominated convergence, Z 1 π 2 0 6 c0 (r) < c0 (+∞) = cos( 2t ) dt = . π 0 π After these preliminary estimates we are ready to prove that the condition (11) implies (3). From the definition of Φp (θ, r), 2πΦp (θ, r) = a0 c0 (r) + d+ (r)(a1 cos(θ) − b1 sin(θ)) + d− (r)(b1 cos(θ) + a1 sin(θ))i. This integral can be identified to a vector in R2 defined by w + DR[θ]v where       a1 a0 c0 (r) d+ (r) 0 v= , w= ,D = , b1 0 0 d− (r) and R[θ] is the matrix of a rotation in the counter-clockwise sense. In the next computations we employ the Euclidean norm in R2 , denoted by | · |. From (43) and (46) we deduce 2 |2πΦp (θ, r)| = |w + DR[θ]v| = D(D−1 w + R[θ]v) > D−1 w + R[θ]v . 3π In addition, in view of (43) and (46), q q c0 (r) |a0 | D−1 w + R[θ]v > |v| − D−1 w = a21 + b21 − > a21 + b21 − 3 |a0 | > 0. d+ (r) The last inequality comes from condition (11). Finally, q  1 |Φp (θ, r)| > 2 a21 + b21 − 3 |a0 | > 0, 3π as we desired. 25 6.2 Proof of Theorem C According to [20] the equation λ , x>0 ẍ + x = (47) x3 can be solved explicitly. The solution satisfying x(0) = x0 > 0, ẋ(0) = y0 is s λ x(t) = (x0 cos(t) + y0 sin(t))2 + 2 sin2 (t). x0 For λ positive this solution is defined everywhere and it is possible to consider the map  π π  Φλ : D → D, (x(0), ẋ(0)) 7→ x , ẋ 2 2 where D = (0, +∞) × R. After some computations we obtain   s −y0 y02 λ Φλ (x0 , y0 ) = x0 ϕλ (x0 , y0 ), with ϕλ (x0 , y0 ) = 2 + 4. ϕλ (x0 , y0 ) x0 x0 The solutions of the equation (4) are also defined everywhere and the Poincaré map P : D → D is given by P = Φc ◦ Φ1 . Here we are using that the equation (47) is autonomous. Some additional computations show that   y0 P (x0 , y0 ) = x0 Π(x0 , y0 ), Π(x0 , y0 ) q 2 2 x y +c where Π(x0 , y0 ) = x20y02 +1 . Due to the structure of P it is easy to check that the function I(x0 , y0 ) = x0 y0 0 0 is a first integral, meaning that I ◦ P = I. In consequence Π is also a first integral and we can compute explicitly the orbits of P , namely (xn , yn ) = P n (x0 , y0 ) = x0 Π(x0 , y0 )n , Π(x0y,y 0 0) n . When c > 1 the function Π takes values above 1 and so the sequence x(nπ) = xn is a geometric progression going to infinity. When c < 1 this is also the case of ẋ(nπ) = yn . In both cases all solutions are unbounded. 7 Appendix: a result on isochronous centers Next result is concerned with the behaviour at infinity of isochronous center when it presents a single asymptote and it is based in [4, Lemma 8]. Here we present a more general version by considering the convexity located at the endpoints of the interval of definition of V instead of the global convexity. Lemma 7.1. Let V : (a, +∞) → [0, +∞) be an analytic 2π-isochronous potential such that V (0) = V 0 (0) = 0 and xV 0 (x) > 0 if x 6= 0. Assume that limx→a V (x) = +∞ and limx→+∞ V (x) = +∞. In addition we assume that there exist  ζ, M > 0 such that V is convex in (a, a + ζ) with a + ζ < 0 and in (M, +∞). Then limx→+∞ V 0 (x) − x4 = −a/4. Proof. Let us denote by V− and V+ the restriction of V on the negative and positive axis, respectively. From [7, Theorem B] the fact that V is an analytic 2π-isochronous implies that V+ (x) = (x − σ(x))2 /8 (48) for all x > 0, where σ(x) := V−−1 (V+ (x)). Hence, for x positive, V+0 (x) = 41 (x − σ(x))(1 − σ 0 (x)). Due to V+ (x) → +∞ as x → +∞ and V−−1 (y) → a as y → +∞ we have that σ(x) → a as x → +∞. On the other hand, V−−1 is a convex decreasing function in (V−−1 (a + ζ), +∞) with limit a < 0 at infinity so by Lemma 4.5 we have that lim y(V−−1 )0 (y) = 0. (49) y→+∞ 26 xV 0 (x) We claim at this point that V++(x) is bounded as x → +∞. Indeed, V+ is a positive increasing convex function in (M, +∞) so for all x, y ∈ (M, +∞) we have V+ (y) > V+ (x) + V+0 (x)(y − x). Taking x > M and y = 2x we have then xV+0 (x) V+ (2x) 06 6 −1 V+ (x) V+ (x) for all x > M . According with expression in (48) and using σ(x) → a as x → +∞, we have V+ (2x) (2x − σ(2x))2 lim − 1 = lim −1=3 x→+∞ V+ (x) x→+∞ (x − σ(x))2 which proves the claim. By the expression of σ(x) we have that xV+0 (x) −1 0 xσ 0 (x) = x(V−−1 )0 (V+ (x))V+0 (x) = (V ) (V+ (x))V+ (x). V+ (x) − On account of equality (49) and using V+ (x) → +∞ as x → +∞ we have limx→+∞ (V−−1 )0 (V+ (x))V+ (x) = 0. xV 0 (x) On the other hand, V++(x) is bounded as x → +∞. Thus, xσ 0 (x) → 0 as x → +∞ (and, particularly, σ 0 (x) → 0 as x → +∞). Consequently,    0 x 1 1 0 1 0 a lim V+ (x) − = lim − σ(x) + σ(x)σ (x) − xσ (x) = − x→+∞ 4 x→+∞ 4 4 4 4 and so the result is proved. References [1] J. Ai, K.S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 13 (2001) 311–337. [2] J.M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations 143 (1998) 201–220. [3] D. Bonheure and C. Fabry, Littlewood’s problem for isochronous oscillators, Arch. Math. (Basel) 93 (2009) 379–388. [4] D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillators at resonance, Discrete Contin. Dyn. Syst. 8 (2002) 907–930. [5] B.M. Brown, M.S.P. Eastham and K.M. Schmidt, Periodic differential operators, Operator Theory: Advances and Applications 230, Birkhäuser/Springer Basel AG, Basel (2013). [6] A. Capietto, W. Dambrosio and B. Liu, On the boundedness of solutions to a nonlinear singular oscillator, Z. Angew. Math. Phys. 60 (2009) 1007–1034. [7] A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations 157 (1999) 373–413. [8] E.N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Aust. Math. Soc. 15 (1976) 321–328. [9] E. Kamke, A new proof of Sturm’s comparison theorems, Amer. Math. Monthly 46 (1939) 417–421. [10] X. Li and S. Jin, Boundedness in forced isochronous oscillators, J. Math. Anal. Appl. 460 (2018) 714–736. 27 [11] B. Liu, Boundedness in asymmetric oscillators, J. Math. An. App. 231 (1999) 355–373. [12] B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance, J. Differential Equations 246 (2009) 3471–3495. [13] N.G. Lloyd, Degree theory, Cambridge Tracts in Mathematics 73. Cambridge University Press, Cambridge-New York-Melbourne (1978). [14] F. Mañosas, D. Rojas and J. Villadelprat, The criticality of centers of potential systems at the outer boundary, J. Differential Equations 260 (2016) 4918–4972. [15] J. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950), 457–475. [16] R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. 53 (1996) 325–342. [17] R. Ortega, Periodic perturbations of an isochronous center, Qualitative Theory of Dynamical Systems 3 (2002) 83–91. [18] R. Ortega, Periodic differential equations in the plane: a topological perspective, to appear. [19] R. Ortega and D. Rojas, A proof of Bertrand’s theorem using the theory of isochronous potentials, to appear in J. Dyn. Diff. Equat. (2018). [20] E. Pinney, The nonlinear differential equation y 00 + p(x)y + cy −3 = 0, Proc. Amer. Math. Soc. 1 (1950) 681. [21] M. Urabe, The potential force yielding a periodic motion whose period is an arbitrary continuous func- tion of the amplitude of the velocity, M. Arch. Rational Mech. Anal. 11 (1962) 27–33. 28