Periodic oscillators, isochronous centers and resonance

An oscillator is called isochronous if all motions have a common period. When the system 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 unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators.


Introduction
Consider an oscillator with equationẍ 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 are unbounded. Here ε = 0 is a small parameter.
The simplest isochronous center is produced by the harmonic oscillator, V (x) = 1 2 n 2 x 2 , n = 1, 2, . . . In this case the previous question has a well-known answer: resonance occurs whenever the integral I n (p) := 2π 0 p(t)e int dt 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 [13]. See also [3].
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 I n (p) = 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 y + V (ϕ(t, r))y = 0, y(0) = 1,ẏ(0) = i is denoted by ψ(t, r). Then we define a function on the cylinder Φ p : C → C, Φ p (θ, r) = 1 2π 2π 0 p(t − θ)ψ(t, r)dt and prove, under certain assumptions on the potential V , that all solutions of (2) are unbounded when the condition inf C |Φ p (θ, r)| > 0 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 Φ p has a non-degenerate zero. Note that periodic solutions are bounded and so resonance is excluded.
The condition I n (p) = 0 is equivalent to (3). Another example that has been widely studied is the asymmetric oscillator associated to the potential where x + = max{x, 0}, x − = max{−x, 0} and α, β are positive constants with 1 √ α + 1 √ β = 2π n , n = 1, 2, . . . In this case the homogeneity of the equation (1) implies that Φ p (θ, r) = Φ p (θ, 1). The function Φ p (θ, 1) appeared first in the work of Dancer on the periodic problem (see [6]) and later it has been shown that it also plays a relevant role in the problem of boundedness (see [2,8,12]).
For a general isochronous oscillator checking the condition (3) can be difficult. We have analysed in detail the equationẍ + 1 4 x + 1 − 1 (x + 1) 3 = εp(t), x ∈ (1, +∞). 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 [3] 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 [15] for more details.
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 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 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 [3].

Statement of the main results
Let us start with a potential V ∈ C 2 (R) defined on the whole real line and satisfying In addition we assume that all the solutions of equation (1) are 2π-periodic. In particular this implies that V (0) = N 2 , N = 1, 2, . . . where N depends upon the minimal period. See [17] 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 ∈ L 1 (T).
Theorem A. Assume that V satisfies the previous conditions and V is bounded over the whole real line. In addition the condition (3) holds for some p ∈ L 1 (T). Then the equation (2) is resonant for small ε = 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(t n )| + |ẋ(t n )| → +∞ for some sequence {t n }. We will prove that a slightly stronger notion of resonance also holds, every solution x(t) of (2) satisfies lim |t|→+∞ (|x(t)| + |ẋ(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 Thus, and so the energy E(t) cannot blow up in finite time.
Let us now assume that the function p ∈ L 1 (T) is such that the condition (3) does not hold. This means that Φ p has a zero at some point (θ * , r * ) of the cylinder C or that it vanishes at infinity. That is, either Φ p (θ * , r * ) = 0 or Φ p (θ n , r n ) → 0 for some sequence (θ n , r n ) with r n → +∞. The next result shows that resonance cannot occur if Φ p has a non-degenerate zero.
Proposition 2.2. Assume that V is in the conditions of Theorem A and that Φ p has a non-degenerate zero (θ * , r * ) with r * > 0. Then the equation (2) has a 2π-periodic solution for small ε.
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 Φ p (θ, r) = 0 for each point (θ, r) ∈Ū \ {(θ * , r * )} and the Brouwer degree deg(Φ p , 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 Φ p is crucial for the understanding of the resonance problem. In concrete examples it can be rather difficult to analyse the properties of Φ p . 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 L 1 (T) as a Banach space with norm Proposition 2.3. Assume that V is in the conditions of Theorem A and define R V as the set of forcing terms satisfying (3), Then R V is open and non-empty.
This result will be proved in Section 5.3. By now we observe that if {p n } is a sequence in L 1 (T) converging in a weak topology to the Dirac measure (p n − δ) then p n ∈ R V for large n. This is consistent with the example in [13]. The convergence p n − δ means (a) sup n p n L 1 (T) < ∞, and 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 defined for all x ∈ (−1, +∞). This equation can be solved explicitly (see [16]). In particular ϕ(t, r) = −1 + λ 2 cos 2 (t/2) + 1 λ 2 sin 2 (t/2) with λ := 1 + r 1. Let ψ(t, r) be the solution of the associated variational equation with initial conditions ψ(0, r) = 1,ψ(0, r) = i. A computation shows that The perturbed equationẍ was considered in [3]. The authors proved that all solutions are unbounded if ε = 0 is small enough. Next we present an analogous result valid for general periodic perturbations of (6).
Theorem B. Let p ∈ L 1 (T) be a function satisfying (3) with ψ(t, r) defined by (8). Then all the solutions of equationẍ are unbounded for sufficiently small ε = 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 [3] 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 [3] from Theorem B we present a corollary for the class of equations (10) with p(t) = a 0 + a 1 cos t + b 1 sin t.
Corollary 2.4. Assuming that p(t) is a trigonometric function as above, the equation (10) is resonant if Note that this result extends the resonance result in [3] for the case a 0 = 0.
We finish this Section with a result on the equation (4).
Theorem C. Consider the π-periodic function with c > 0. Then all solutions of (4) are unbounded if c = 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).

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 [7].
In this section we shall consider the equation y + a(t)y = 0 (12) with a ∈ L 1 (J) defined in an open interval J and such that J * a(t)dt > 0 for each sub-interval J * ⊂ J.
We know from Lebesgue differentiation theorem that for almost every t ∈ J. In consequence a(t) 0 almost everywhere.
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 = re iθ we have that the argument satisfieṡ 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.  c) y(t) reaches a local maximum (respectively, minimum) at t * ∈ J if and only ifẏ(t * ) = 0, y(t * ) > 0 (respectively, y(t * ) < 0).
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 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.
We finish this section with a result on symmetric equations. Proof. First we observe that the symmetry of the system with respect to t = 0 induces the symmetries on the solutions φ 1 (−t) = φ 1 (t) and φ 2 (−t) = −φ 2 (t). The function φ 1 (t) has a critical point at t = 0, say that it is a maximum. Since φ 2 has no critical point on J then, by Lemma 3.2, φ 1 has at most another critical point t * = 0 (a minimum). The function φ 1 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 φ 1 .

Potential centers and action-angle variables
In this section we work with the equationẍ where J = (a, b) is some interval with −∞ a < 0 < b +∞ and the following hypothesis on the potential is considered 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 θ = 2πτ 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 γ r be the orbit passing through (r, 0) with 0 < r < b. The region enclosed by γ r will be denoted by A r and it increases with r. Moreover 0<r<b A r = J × R and so I = meas(A r ) → +∞ as r → b. From these facts and the well-known theory of integrable systems we deduce that the map defines a symplectic diffeomorphism transforming the equation (15) intȯ where ω(I) = 2π T (I) and T (I) is the minimal period interpreted as a smooth function of the action. Let x(t, I) be the solution of (15) having action I > 0 and satisfying the conditions x(0, I) > 0, x(0, I) = 0. This section is concerned with the study of this function and the derivativesẋ = ∂x ∂t and ∂x ∂I . Letting r = x(0, I) we recall the classical formula 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.  Proof. The smoothness of x(t, I) follows from the connection with ϕ(t, r) via the identity (16). Indeed the functionsẋ(t, I) and ∂x ∂I (t, I) are solutions of the variational equation of (15) given bÿ From (16) we deduce that ∂x ∂I (t, I) satisfies the initial conditions ∂x The Rofe-Beketov formula (see [4], 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 The uniqueness of solution of the initial value problem implies that the identity (17)  Proof. By energy conservation we have 1 2ẋ (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.

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 (H 0 ) with J = (a, +∞), −∞ < a < 0. In addition we shall assume that V is convex near the asymptote, meaning that 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 [10]. 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 Lemma 4.4. Assume that V satisfies (H 0 ) and (H 1 ). Then T − (I) → 0 as I → +∞.
In the proof of this result we will employ the following elementary property. Proof. Assume M > 0. By the convexity property we have that for all 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 where r − (I) denotes the intersection of γ − (I) with the x-axis. The change of variable V − (x) 1 2 = −Ω(I) 1 2 sin θ transforms the previous integral into On the other hand, by hypothesis (H 1 ), 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 Therefore, on account of (19) and (20), there exists M > 0 such that (V−) . 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.
Proof. The functionsẋ(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 (x(t, I)) > 0 for all t ∈ A(I) andẋ(t, I) has no critical points for t ∈ A(I). Therefore the variational equation satisfies the condition (13) on A(I) and, by . Since I > I ζ we have a < r − (I) < a + ζ. Consequently, there exists η = η(I) ∈ (r − (I), a + ζ) such that 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 → −x.

Potential center with Lipschitz behaviour at infinity
In this section we consider a potential function defined on J = (a, +∞) with −∞ a < 0 and satisfying (H 0 ). In addition we assume that  where · denotes the matrix norm induced by the sup norm of R 2 . Given a matrix A = (a ij ) ∈ M 2 (R), for all t lying in the component of B(I) containing t = 0.

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 (H 0 ), (H 1 ) and (H 2 ). Moreover, In the above conditions it is not restrictive to assume that a + ζ = α. From now on, V ∞ := sup{|V (x)| : x > a + ζ}.
Here we have used that V (0) = 0. We can invoke the identity (16) to deduce the inequality Ω The inequality Ω(I) 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). 1 for all t ∈ P(I) and I > 0, |a| Ω(I) for all t ∈ N (I) and I > 0.
Proof. In order to prove (a) let us note that by Lemma 4.11, Ω(I) c 2 r. Since |x(t, I)| r for all t ∈ P(I) and I > 0, we have 1 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 By hypothesis (H 2 ) in (a + ζ, +∞) we have that B(I) = [0, T (I)] \ A(I) correspond to the set of times when V (x(t, I)) is bounded. Therefore, on account of Lemma 4.3 and Lemma 4.10 together with the general inequalities a √ for all t ∈ B(I).
Here we also used that 1 + V ∞ 2M .
On the other hand, if t ∈ A(I), by (a) in Lemma 4.7 we have where τ = τ + ζ (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.

Isochronous oscillators with an asymptote and a bouncing problem
In this section we assume that −∞ < a < 0 < b = +∞ and the potential V satisfies (H i ), 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. 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: (iii) ψ 0 and ψ(t) = 0 if and only if t = (2n + 1)π, (iv)ψ(π + 0) = −ψ(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 (i ) ψ : R → R is continuous and 2π-periodic, the restriction ψ| [−π,π] belongs to C 1 [−π, π] and the derivative ψ 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 will present a jump discontinuity at the points t = (2n + 1)π, n ∈ Z.
Theorem 4.14. Let (I n ) n 0 be a sequence of actions such that I n → +∞ as n → +∞. Then there exist a subsequence (I n k ) k 0 ⊂ (I n ) n 0 , γ ∈ L ∞ (T) and a solution ψ of the γ − BP such that (a) x(t, I n k ) I n k converges uniformly to ψ, and (b)ẋ (t, I n k ) I n k converges uniformly toψ in [−π + δ, π − δ] for all δ > 0.
Moreover the function γ satisfies almost everywhere.
An analogous result holds for ∂x ∂I . Theorem 4.15. In the same conditions of the previous theorem there exists a subsequence (I m k ) k 0 ⊂ (I n ) n 0 , γ 1 ∈ L ∞ (T) and a solution ψ 1 of the γ 1 − BP such that The function γ 1 satisfies the condition 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 γ 1 do not necessarily coincide even when a common subsequence of (I n ) 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 φ n ∈ L ∞ (J) and φ ∈ L ∞ (J), we say that φ n converges to φ in the weak * sense, Lemma 4.16. Let (φ n ) n 0 be a sequence of functions, φ n ∈ L ∞ (0, π), and C > 0 such that for each Proof. Let (δ k ) k 0 be a monotone sequence tending to zero and let us denote X k = L ∞ (0, π − δ k ). After extending by zero any function of X k to (0, π) we have X 0 ⊂ X 1 ⊂ · · · ⊂ X k ⊂ · · · ⊂ L ∞ (0, π).
Remark 4.17. The next example shows that the stronger conclusion φ n k * − φ in L ∞ (0, π) cannot be obtained. Let φ n be the continuous function which vanishes on [0, π− 1 n ] and grows linearly from π− 1 n to π in such a way that π 0 φ n = 1. By construction, for all δ > 0 there exists N = N (δ) such that φ n L ∞ (0,π−δ) 0 if n N . We prove that there is no subsequence φ n k converging in the weak * sense in L ∞ (0, π). Otherwise there should existφ ∈ L ∞ (0, π) with φ n k * − φ . Testing this convergence with the constant function χ = 1 we deduce that π 0φ = lim k→+∞ π 0 φ n k = 1. We can also use the characteristic function of the interval [0, π − 1 N ] as the test function χ. In this case we obtain Since N is arbitrary these two facts are not compatible and soφ cannot exist. such that φ n → +∞ almost uniformly as n → +∞. This means that there exists a sequence of numbers (M n ) with M n → +∞ and such that φ n (t) M n almost everywhere. In addition assume that, for some for almost every t ∈ J.
The choice of α, β and the convergence to infinity of φ n imply that f •φ n belongs to Ω α,β if n is large enough. The set Ω α,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 γ belongs to Ω α,β . The class of functions described by the inequalities (25) is precisely the set α,β Ω α,β .
Proof of Theorem 4.14.
Let us denote f I (t) := 1 √ I x(t, I) and let (I n ) n 0 be a sequence of actions such that I n → +∞ as n → +∞. The function f In satisfies the Cauchy problem where r n = x(0, I n ). Since Ω(I) = I, according to (a) and (b) in Proposition 4.12 we have that |f In (t)| and |ḟ In (t)| √ 2 are uniformly bounded in T for all n 0. Thus the family {f In } n 0 is equicontinuous, bounded and K-Lipschitz with K √ 2 so by Arzelà-Ascoli theorem there exists a subsequence (I n k ) k 0 ⊂ (I n ) n 0 such that f In k → ψ uniformly for some function ψ ∈ W 1,∞ (T) with Lipschitz constant [ψ] Lip = K √ 2. This proves by now that ψ ∈ W 1,∞ (T).
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 For the sake of simplicity we consider (I n k ) 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(I n k ) for sufficiently large k. We have, on account of equation in (26) It is at this point where property (28) plays an important role in the passage to the limit. Indeed, if u n → u uniformly in L ∞ (T) and v n * Therefore, since f In k converges uniformly to ψ and using the weak * convergence in (28), That is, ψ satisfies the ordinary differential equationψ + 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.
Passing to the limit we obtain the identity ψ(π + t) = ψ(π − t) which impliesψ(π + t) = −ψ(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 f In → ψ uniformly, there exists a constant µ J > 0 such that f In (t) µ J if n is large enough and t ∈ J. We apply Lemma 4.18 with f (x) = V (x) x and φ n (t) = f In (t) √ I n . Then φ n → +∞ almost uniformly on J with M n = µ J √ I n . 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).
By now we have proved that ψ 1 ∈ 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 (x) and φ n (t) = x(t, I n ). For each compact interval J ⊂ (−π, π) the sequence φ n → +∞ uniformly in J.

Resonance of isochronous oscillators
This section is dedicated to prove some of the results presented in Section 2.

Proof of Theorem A
We first study some simple properties of the linear equation where a ∈ L ∞ (0, 2π), b ∈ L 1 (0, 2π) and a L ∞ (0,2π) A.
(ii) Let y(t) be the solution of (30) with initial conditions y(0) =ẏ(0) = 0. Then |y(t)| To prove (i) we observe that the matrix ( u v uv ) is a solution of the linear systemẎ = Cauchy-Schwarz inequality can be applied to the term in brackets to obtain The Second Massera's Theorem [11] 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 = 0 small. We proceed by contradiction and assume that {ε n } is a sequence with ε n = 0, ε n → 0 such that the equation (2) has a 2π-periodic solution x n (t) for ε = ε n . Let X n (t) be the solution of (1) with the same initial conditions of x n (t) at t = 0, X n (0) = x n (0) andẊ n (0) =ẋ n (0). The difference y n (t) = x n (t) − X n (t) can be seen as a solution of the equation (30) with From property (ii) we deduce that Here A = sup x∈R |V (x)|.
The function y n (t) can also be interpreted as a 2π-periodic solution of the linear periodic equation 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, X n (t) = ϕ(t − θ n , r n ) for some θ n ∈ [0, 2π] and r n 0. The function ψ(t − θ n , r n ) is a 2π-periodic solution of the homogeneous equation y + V X n (t) y = 0. Fredholm alternative can be applied to deduce that In other words, The definition of q n together with the estimate (31) and the Dominated Convergence Theorem imply that 1 εn q n (t) → 0 as n → +∞ for every t ∈ [0, 2π]. From the property (i) we know that ψ(·, r n ) L ∞ (T) C and the estimate (31) implies that Using once again dominated convergence, Φ p (θ n , r n ) → 0 as n → +∞.

Proof of Proposition 2.2
The aim is to show that if Φ p 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 [9] for more details.
Let Ω be an open and bounded subset of R d and a function f :Ω → R d which is continuous and does not vanish on the boundary of Ω. First we assume that f ∈ C 1 (Ω) and has a finite number of zeros x 1 , . . . , x n ∈ Ω with det f (x i ) = 0 for each i. Then the Brouwer degree of f in Ω is In the case f is continuous, we approximate it by functions f k in the previous conditions and define 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; x 0 , v 0 , ε) be the solution of (2) with initial conditions For each ε ∈ R consider the planar map We are going to prove that F 0 has a non-degenerate zero. Then, for small ε = 0, the map F ε 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 mapΦ p (θ, r) := Φ p θ N , r is well defined on the cylinder C. The map (θ, r) → (N θ, r) is a local diffeomorphism of the open cylindeṙ C = (R/2πZ) × (0, ∞). Let (θ * , r * ) be the non-degenerate zero of Φ p given by assumption, then (N θ * , r * ) is a non-degenerate zero ofΦ p . This can be deduced from the definition of degree or from Leray multiplication theorem.
Proof. Assume that ψ = u + iv with u = Reψ, v = Imψ. Then ( u v uv ) is a matrix solution of the linearized system and the upper estimates are obtained as in Lemma 4.10. In particularu(t) 2 +v(t) 2 C 2 if t ∈ [0, 2π]. From Liouville's identity, and we can take c = 1 C .
Proof of Proposition 2.3 Let p, q ∈ L 1 (T). The fact that R V is open follows from Lemma 5.1 since Let us now show that R V is non-empty. To do so, we shall see that if we take a sequence p n ∈ L 1 (T) with p n − δ then p n ∈ R V for n large enough. In order to arrive at contradiction we consider a sequence {p n } n 0 such that for all n 0, p n ∈ L 1 (T), p n / ∈ R V and p n − δ. From p n / ∈ R V it follows that for each n there exists a sequence (θ Arguing by a diagonal process we can consider Φ pn (θ n , r n ) → 0. On account of the bounds in Lemma 5.1, by Arzelà-Ascoli theorem, there exists a subsequence ψ(t + θ n k , r n k ) that converges uniformly on [0, 2π] to someψ ∈ C(T, C) as k → +∞. Moreover, c |ψ(t)| C. Thus, 2πΦ pn k (θ n k , r n k ) = 2π 0 p n k (t)ψ(t+θ n k , r n k )dt = 2π 0 p n k (t) ψ(t + θ n k , r n k ) −ψ(t) dt+ 2π 0 p n k (t)ψ(t)dt.
The first integral tends to zero as k tends to infinity. Indeed, we have 2π 0 p n k (t) ψ(t + θ n k , r n k ) −ψ(t) dt p n k L 1 (T) ψ(· + θ n k , r n k ) −ψ L ∞ (T) → 0 as k → +∞ due to the boundedness of p n k L 1 (T) and the convergence of ψ(t + θ n k , r n k ) toψ(t) uniformly in [0, 2π]. In addition, since p n − δ, as k → +∞, reaching contradiction with the choice of p n / ∈ R V .

Two resonance results for a periodic perturbed isochronous of Pinney type
This section is dedicated to prove Theorem B and C.

Proof of Theorem B
We are concerned with the potential All the assumptions (H i ) are satisfied in this case. From the first derivative V (x) = 1 it is easy to deduce that (H 0 ) and (H 3 ) hold. Note that V (x) x → 1 4 as x → +∞. The second derivative V (x) = 1 4 + 3 4 (x + 1) −4 is positive and so (H 1 ) also holds. Finally we observe that V (x) 1 if x 0 and this implies (H 2 ).
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
The next result describes the asymptotic behaviour of x(t, I) and its derivatives. It was already obtained in [3] but there are significant differences in the type of convergence and also in the proof.  Proof. We apply the results of Section 4.4. A key observation is that the inequality in (23) now becomes an identity because the limit exists. Namely, lim x→+∞ In consequence the function γ has to be the constant 1 4 . The next step will be to determine all the solutions of the Bouncing problem when γ = 1 4 . A direct computation shows that these solutions are of the type 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 Also, from the identity ∂x ∂I (0, The existence of these limits implies that the functions ψ and ψ 1 are uniquely determined. Actually ψ(t) = 2 √ 2 cos t 2 and ψ 1 (t) = 1 2 ψ(t). In consequence, as I → +∞, there is convergence of the functions 1 √ I x(·, I), 1 √ Iẋ (·, I) and √ I ∂x ∂I (·, I). The convergence must be understood in the senses specified by Theorems 4.14 and 4.15.
By symmetry we know that there is pointwise convergence of √ I ∂x ∂I (t, I) to ψ 1 (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 (·, I) is monotone in [0, π]. A classical result says that if a continuous function ψ 1 (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 Note that this continuity is a consequence of the theorem of continuous dependence with respect to parameters and initial conditions when it is applied to the equationÿ + V (x(t, I))y = 0. The continuity at I = 0 is also valid because lim I→0 + x(t, I) = 0 uniformly in t and lim I→0 Proof of Theorem B We write the equation (2) as the first order systemẋ = y,ẏ = −V (x) + εp(t). This system is defined on D = {(x, y) ∈ R 2 : 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.
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 = e z . In the new variables the system isż = e −z y,ẏ = −V (e z − 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 lim sup t→+∞ |x(t)| + |ẋ(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 t n → +∞ and a solution x(t) with |x(t)| + |ẋ(t)| C if t 0 and x(t n ) → −1.
Then, for each h ∈ [0, 1], 1 + x(t n + h) 1 + x(t n ) + Ch and so After integrating the equation (2) over the interval [t n , t n + 1] we obtaiṅ 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 = 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; x 0 , v 0 , ε) the solution of (2) with initial conditions x(0) = x 0 , The initial condition of a 2π-periodic solution corresponds to a zero of F ε . After differentiating with respect to the parameter ε we obtain the expansion where Y (t; x 0 , v 0 ) is the solution of the linear equatioṅ , Y (0) = 0.
Moreover this expansion is uniform on (x 0 , v 0 ) ∈ K if K is a compact subset of D. For the origin (x 0 , v 0 ) = (0, 0) this system can be solved explicitly and a simple computation shows that The condition (3) at r = 0 implies that this vector does not vanish and, by continuous dependence, we can find σ > 0 such that The expansion (36) allows to find ε * > 0 such that F ε (x 0 , v 0 ) = 0 if 0 < |ε| < ε * and |x 0 | + |v 0 | σ, proving the veracity of the claim.
Once we have proved the above claim we will employ action-angle variables (θ, I). In the punctured plane R 2 \ {0} the Hamiltonian system (2) is transformed into 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.
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), Integrating this inequality and assuming that t ∈ [0, ω) ∩ [0, 2π], 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 C 2 |ε| p L 1 (T) < 1 2 σ * we deduce from Claim 1 and the condition (39) that 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 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 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 d dt (θ(t) − t) C|ε| |p(t)| For small ε the function θ(t) − t − θ(0) must be small on [0, 2π] and this implies that M = 1.
Let us assume first that I * < +∞. From Claim 1 we know that I * σ * and the estimates (39) and (41) imply that I n (t) → I * and θ n (t) − t → θ * uniformly in t ∈ [0, 2π]. In addition, on account of equations in system (37) and the 2π-periodicity of I n (t) and θ n (t) − t in T, for all n 0 we have Passing to the limit in the previous equalities we conclude that 2π 0 p(t) ∂x ∂I (θ * + t, I * )dt = 2π 0 p(t) ∂x ∂θ (θ * + t, I * )dt = 0.
The functions ∂x ∂I and ∂x ∂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 Φ p (−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 θ n (t) − t → θ * and I n (t) → +∞ uniformly in t ∈ T. More precisely, In(t) In(0) → 1 uniformly in t ∈ T. On account of Proposition 6.2 we have that, as I → +∞, √ I ∂x ∂I (θ, I) → cos θ 2 uniformly in T and 1 √ Iẋ (θ, I) → − sin θ 2 sign cos θ 2 uniformly in every compact subset of T \ {π}. Thus, on account of In(t) In(0) → 1 uniformly in t ∈ T, as n → +∞, uniformly in T and I n (0)ẋ (θ n (t), I n (t)) → −p(t) sin θ * + t 2 sign cos θ * + t 2 uniformly in every compact subset of T \ {π − θ * }. Consequently, using (42) for this particular case, In view of Proposition 6.3 we observe that 1 √ In(0)ẋ (θ n (t), I n (t)) is uniformly bounded. The convergence of 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 Here we are using (8).
Proof of Corollary 2.4 The Fourier expansion of ψ(·, r) is of the type ψ(t, r) = +∞ m=−∞ c m (r)e imt and we will derive some properties of the first coefficients. The symmetry ψ(−t, r) = ψ(t, r) implies that the numbers c 0 (r), c 1 (r) and c −1 (r) are real. Define [Imψ(t, r)] sin(t)dt.
From the explicit formula (8) we observe that 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, d + (r) = 1 π [Reψ(t, r) − Reψ(π − t, r)] cos(t)dt.
For λ positive this solution is defined everywhere and it is possible to consider the map Φ λ : D → D, (x(0),ẋ(0)) → x π 2 ,ẋ π 2 where D = (0, +∞) × R. After some computations we obtain Φ λ (x 0 , y 0 ) = x 0 ϕ λ (x 0 , y 0 ), −y 0 ϕ λ (x 0 , y 0 ) with ϕ λ (x 0 , y 0 ) = y 2 0 The solutions of the equation (4) are also defined everywhere and the Poincaré map P : D → D is given by P = Φ c • Φ 1 . Here we are using that the equation (47) is autonomous. Some additional computations show that P (x 0 , y 0 ) = x 0 Π(x 0 , y 0 ), y 0 Π(x 0 , y 0 ) where Π(x 0 , y 0 ) = x 2 0 y 2 0 +c x 2 0 y 2 0 +1 . Due to the structure of P it is easy to check that the function I(x 0 , y 0 ) = x 0 y 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 (x n , y n ) = P n (x 0 , y 0 ) = x 0 Π(x 0 , y 0 ) n , y0 Π(x0,y0) n . When c > 1 the function Π takes values above 1 and so the sequence x(nπ) = x n is a geometric progression going to infinity. When c < 1 this is also the case ofẋ(nπ) = y n . In both cases all solutions are unbounded.

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 [3,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.
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 + (x)(y − x). Taking x > M and y = 2x we have then for all x > M . According with expression in (48) and using σ(x) → a as x → +∞, we have which proves the claim.