Periodic, Subharmonic, and Quasi-periodic Oscillations under the Action of a Central Force

We consider planar systems driven by a central force which depends periodically on time. If the force is sublinear and attractive, then there is a connected set of subharmonic and quasi-periodic solutions rotating around the origin at different speeds; moreover, this connected set stretches from zero to infinity. The result still holds allowing the force to be attractive only in average provided that an uniformity condition is satisfied and there are no periodic oscillations with zero angular momentum. We provide examples showing that these assumptions cannot be skipped.


Introduction
The motion of a particle subjected to the influence of an (autonomous) central force field in the plane may be mathematically modelled as a system of differential equations: Many phenomena of the nature obey to laws of this type. For instance, the newtonian equation for the motion of a particle subjected to the gravitational attraction of a sun which lies at the originẍ = − c x |x| 3 , corresponds to the choice f (r) := c/r 2 for some positive constant c > 0. If, on the contrary, c < 0, equation (2) still has a relevant physical meaning, as it may be used to model Rutherford's scattering of α particles by heavy atomic nuclei. If the force field (1) is attractive, i.e., f > 0, then there is a collection of solutions of (1) which rotate around the origin at constant angular speeds. Indeed, direct computations show that x(t) = r(cos(ωt), sin(ωt)) satisfies (1) if and only if |ω| = f (r)/r. These solutions, which we shall call copernican in what follows, are all of them periodic; however, excepting the case in which f (r) = c r is linear, the period 2π/|ω| = 2π r/f (r) will depend on the solution.
On the other hand, already Newton [12], in his study of Kepler's Second Law considered the problem of a central force divided into instantaneous impulses which take place periodically in time. Leaving aside the issue of the discreteness (which would require to work with measure-type forces), this motivates us to consider the following question: What is left from the copernican orbits when the central force field depends periodically on time?
More specifically, in this paper we shall study systems of the form where the L 1 −Carathéodory function is T −periodic in the time variable t for some T > 0. It may have a singularity at r = 0; consequently, the solutions of our equation (3) 'live' on the punctured plane R 2 \{0}. Throughout this paper, continuous functions x : R → R 2 \{0} will be routinely decomposed in polar coordinates, x(t) = r x (t)(cos θ x (t), sin θ x (t)). Such a function will be called T -radially periodic if the modulus r x is T -periodic and there exists some number ω ∈ R such that θ x (t) − ωt is T -periodic. In this case, ω = (θ x (T ) − θ x (0))/T , so that this number may be interpreted as the average angular speed of x. It will be called the rotation number of x, and denoted by ω = rot(x).
For instance, x ω (t) = (2+sin t)(cos ωt, sin ωt) is 2π-radially periodic, independently of the value of the parameter ω, which is the rotation number of x. Already in this first example we observe that radially periodic curves may not be periodic of any period; this is indeed the case of x ω if ω ∈ Q. Actually, an arbitrary T -radially periodic curve is T -periodic if and only if its rotation number is an integer multiple of 2π/T . If the rotation number belongs instead to (2π/T )Q, then the curve will not be T -periodic, but a subharmonic. On the other hand, if the modulus r x is not constant and rot(x) ∈ (2π/T )Q, then the T -radially-periodic curve x will not be periodic of any period; instead, it will be quasi-periodic on the two frequencies ω 1 = 2π/T and ω 2 = rot(x). This is easy to check as, in complex notation, x(t) = r(t)e iθ(t) may be decomposed as the product of the T −periodic function r(t)e iθ(t)−i rot(x)t and the 2π/ rot(x)−periodic function e i rot(x)t .
Without further assumptions, our equation (3) may not have any bounded solution at all; this is actually the case, if, for instance, the force field is repulsive. This intuitive statement may be checked by introducing polar coordinates x(t) = r(t)(cos θ(t), sin θ(t)) in (3), thus obtaining the systemr where µ = r 2θ is the angular momentum (which remains constant along solutions, cf. [3]). Now, if f < 0, the first equation above implies that r is strictly convex, and, consequently, it cannot be globally bounded. Thus, we shall assume, in a first approach, that our force is attractive: Even under this assumption, our equation may not have bounded orbits if f is allowed to grow linearly on r. To check this fact, it suffices to consider forcing terms of the form f (t, r) := h(t)r, giving rise to the following Hill's type equation on the punctured plane: Observe that x = (x 1 , x 2 ) : R → R 2 \{0} is a solution if and only if both components x i solve the corresponding one-dimensional Hill's equation Assumption (H 1 ) holds provided that h > 0. But the T -periodic function h > 0 may be chosen so that the equation is hyperbolic, preventing the existence of nontrivial bounded solutions, see [11]. We owe this observation to R. Ortega.
To avoid this second pathology, we shall restrict our study to forcing terms f which are sublinear near infinity, i.e., there exists some function h ∈ L 1 loc (R) and some number r 0 > 0 such that Let us consider the set of all T -radially periodic curves in the punctured plane R 2 \{0}. It becomes a topological space after being endowed with the topology of the uniform convergence on compact intervals of time.
Theorem 1.1. Assume (H 1 ) and (H 2 ). Then, there exists a connected set C of T -radially periodic solutions of (3) which goes from zero to infinity, meaning that Assumption (H 1 ) may be criticized on the grounds that it is probably too restrictive. One might consider the possibility of extending this theorem to forcing terms f which are positive only in average, i.e., However, this is not enough, and we shall give a counterexample in Section 6. It will exhibit C ∞ regularity (not just Carathéodory's) and will be bounded (not just sublinear). Proposition 1.2. There exists a bounded and C ∞ function f : (R/2πZ)× ]0, +∞[→ R verifying (W 1 ), but such that, for any connected set C of 2π-radially periodic solutions of (3), the interval {|x(t)| : t ∈ R, x ∈ C} is bounded.
Condition (W 1 ) turned out to be too weak. But the idea of formulating a generalization of Theorem 1.1 in which f might change sign provided that it is, in some sense, positive in average, is not completely erroneous. We define, for any ρ ≥ 1, the function Observe that f 1 = f . For ρ > 1 one still has the inequality f ρ ≤ f .
Assume also (H 2 ). Then, there is a connected set C of T -radially periodic solutions to (3) coming from infinity in the sense that the interval {min |x| : x ∈ C} is unbounded from above.
Assumption (M 1 ) is certainly weaker than (H 1 ), and thus, Theorem 1.3 may be seen as a generalization of Theorem 1.1 (actually, only a partial generalization, since also the thesis (7) is weaker than than (5)). The assumptions of Theorem 1.3 apply, for instance, when f has the form provided that the L 1 -Carathéodory function f * : R× ]0, +∞[→ R verifies assumptions (H 1 ) and (H 2 ), and the L 1 (R/T Z)-function e has nonnegative mean: Notice however that, under suitable additional conditions, assumption (W 1 ) may imply (M 1 ). This is for instance the case if f is monotone in r, meaning that f (t, ·) is either increasing for a.e. t ∈ R of decreasing for a.e. t ∈ R (we remark that the definition does not include those functions which are sometimes increasing and sometimes decreasing depending on the value of t). Consequently, if f is monotone in r and verifies (W 1 ) and (H 2 ), Theorem 1.3 holds for equation (3).
A third class of equations where Theorem 1.3 applies corresponds to forces f of the type where −∞ < γ < 1 and c, e ∈ L 1 (R/T Z) are T -periodic and verify as it may be easily checked. Nonlinearities of these forms (8,10) were previously considered in [6], under an extra assumption at infinity, which was needed to parameterize a family of largeamplitude solutions with the angular momentum. In this paper we avoid that assumption by using as parameter the distance from our solution to the origin at time zero. As we already observed, Theorem 1.3 is not a true generalization of Theorem 1.1 because it does not state anymore that the connected set C of radially periodic solutions approaches the origin. It motivates the question of whether one may find actual examples where C cannot be continued up to the origin. Somehow, this is expectable, since the assumptions (M 1 ) and (H 2 ) only refer to the behavior of f in a neighborhood of infinity. We have studied in detail equations with the form (10) and e ≡ 0 , assuming that c ∈ C(R/T Z) and T 0 c(t)dt > 0. We get the following result: Proposition 1.4. If 0 ≤ γ < 1, then (11) has a connected set C of T -radially periodic solutions going from zero to infinity in the sense of (5). On the other hand, for every γ < 0 there exists some continuous and 1-periodic function c γ with positive mean and such that (11) does not have 1-radially periodic solutions x with min R |x| = 1.
Then, what happens with the connected set C? Why, in some cases, it does not continue up to the origin? We may obtain a further insight on the situation by going back to the equivalent formulation (4). Observe that if x = r(cos θ, sin θ) is a solution with angular momentum µ, thenx = r(cos(−θ), sin(−θ)) is another one, this time with angular momentum −µ. Thus, together with C, there is a second connected set of solutions coming from infinity: It may happen that both connected sets C andC coincide. This will be the situation if C contains some solutions with zero angular momentum µ = 0. Looking back to the second equation of (4) we see that such a solution must have zero rotation number, as it lives in a ray emanating from the origin. Precisely, it will have the form where |v| = 1 and r is a T -periodic solution of the one-dimensional equation Thus, if we do not want to allow the connected set C go back to infinity throughC, we have to prevent it to contain such solutions: Theorem 1.5. Assume (M 1 ) and (H 2 ), the assumptions of Theorem 1.3. Then, there exists a connected set C of T -radially periodic solutions of (3) coming from infinity, i.e. (7). Furthermore, C may be chosen so that either it approaches the origin in the sense of (5), or it contains some solutions with zero angular momentum. Theorem 1.5 may be considered as a generalization of Theorem 1.1, since f being positive implies that (12) does not have T -periodic solutions. This statement can be easily checked by integrating both sides of the equation, to geṫ so thatṙ(T ) <ṙ(0) and r cannot be T -periodic. The argument may be repeated to show that (12) does not have T -periodic solutions if f has the form (8) for some L 1 -Carathéodory function f * with (H 1 ) and (H 2 ), and some T -periodic function e ∈ L 1 loc (R) verifying (9). And a similar situation occurs if f is monotone in r and verifies (W 1 ) and (H 2 ). Thus, in all these cases there is a connected set C of T -radially periodic solutions going from zero to infinity in the sense of (5). We shall also rely on Theorem 1.5 to prove the first part of Proposition 1.4.
Remark that Theorem 1.5 includes Theorem 1.3. This result implies the existence of Tradially periodic solutions of (3) with big amplitudes, motivating the study the appearance of these solutions. It turns out that they look like copernican, meaning that the ratio between their maximum and minimum distances to the origin is close to one. Moreover, their angular speeds get small. This result, which is related to Lemmas 1,2 and 3 of [6], holds under the common requirements of Theorems 1.3 and 1.5: assumptions (M 1 ) and (H 2 ). Proposition 1.6. Assume (M 1 ) and (H 2 ). Then, for each > 0 there exists some number The combination of some elements taken from (the proof of) Theorem 1.5 and Proposition 1.6 will lead us to: . Then, there exists some numberω > 0 with the property that for every real number 0 < ω <ω there is a T −radially periodic solution andθ Choose some integer k ∈ N big enough so that ω k := 2π/(kT ) ∈ ]0,ω[ and observe that the T -radially periodic solution having rotation number ω k , which we shall now call x k , is actually kT -periodic, winding once around the origin on each period. We arrive to the following result, which may be seen as a generalization of Theorem 4 of [6]: Corollary 1.8. Assume (M 1 ) and (H 2 ). Then, there exists some k 1 ≥ 1 such that, for any integer k ≥ k 1 , equation (3) has some subharmonic solution x k = r k (cos θ k , sin θ k ) with minimal period kT , which makes exactly one revolution around the origin in the period time kT . These solutions verify When ω ∈ (2π/T )Q, the radially-periodic solution x ω given by Theorem 1.7 is quasiperiodic of the frequencies ω 1 = 2π/T , ω 2 = ω, this was already observed. We deduce: Corollary 1.9. Assume (M 1 ) and (H 2 ). Then, there exists someω > 0 such that, for any number 0 < ω <ω not commensurable with 2π/T , equation (3) has some quasiperiodic solution x ω = r ω (cos θ ω , sin θ ω ) of the frequencies ω 1 = 2π/T, ω 2 = ω. These solutions verify (14) and (15).
For a systematic treatment of non-radially symmetric systems with a singularity, by the use of variational methods, the reader can consult [1] and the references therein. See also [7] for some results obtained by the use of degree theory.
2 Near infinity, radially periodic solutions are close to copernican In this section we exploit the sublinearity of f to obtain some insight on the solutions of (3) with big amplitude. Our main goal will consist in showing Proposition 1.6, the result stating that, as the amplitude grows to infinity, radially periodic solutions become similar to copernican, while spinning slower and slower. Along this Section we shall assume (M 1 ) and (H 2 ). It will be convenient to use polar coordinates x = r(cos θ, sin θ), and consequently, we go back to the equivalent system (4). If x is T -radially periodic, then r must be T -periodic. In combination with the first equation of (4), it leads us to the boundary value problem Observe that the parameter µ (the angular momentum of the solution) has not a prefixed value, and thus, solutions are couples (r, µ). For such a solution one has i.e., r is a lower solution of (12). And for these lower solutions, the first part of Proposition 1.6 holds (we let ρ = 1 + ): Proof. Using a contradiction argument, assume that the result were not true. Then, it would be possible to find a sequence {r n } n of lower solutions of (17) with max r n → +∞ and max r n min r n ≥ ρ 0 for some ρ 0 > 1 . Then, min r n ≤ (max r n )/ρ 0 , and we may find instants of time 0 ≤ s n < t n ≤ T such that see Figure 1(a). We use now Lagrange's Mean value Theorem and find some time c n ∈ ]s n , t n [ such thaṫ

which is a positive constant and verifieṡ
On the other hand,ṙ(s n ) = 0, and again by the Mean Value Theorem (this time in its integral form),ṙ We define now, for each n ∈ N, which is again continuous and T −periodic. Moreover, on [s n , c n ], r n andr n coincide, and maxr n = max r n . Using (19) we deduce that and then However, assumption (H 2 ) implies that the sequence f (t,r n (t))/r n (t) converges to zero pointwise on [0, T ], and is dominated by the integrable function h. Lebesgue's Convergence Theorem then implies that contradicting (20).
To continue, we remember assumption (M 1 ) and choose some number ρ * > 1 as given there. We define the function Observe that M is continuous. Moreover, and the combination of assumption (H 2 ) and Lebesgue's Theorem implies that i.e., M is subquadratic. The following result collects some properties of our equation and the function M which will be needed later. As before, ρ * > 1 is given by assumption (M 1 ): There exists some λ 0 > 0 with the following properties: (i) For any solution (r, µ) of (16) with max r ≥ λ 0 , one has that r(t 0 )/ρ * < r(t) < ρ * r(t 0 ) for any t 0 , t ∈ R.
(i): If (r, µ) is a solution of (16) with max r ≥ λ 0 , then max r ≥ λ 2 and for any time or, what is the same, Then, λ ≥ λ 1 , and the first part of (ii) follows from assumption (M 1 ). Concerning the second part, we recall (21) and observe that as claimed.
The following result states the boundedness of the set of solutions (r, µ) of (16) for which the minimum of r lies between two given bounds. It will follow from Lemma 2.2(i): Proof. After possibly replacing K by a bigger quantity, it is not restrictive to assume that K ≥ λ 0 , the constant appearing in Lemma 2.2. Then, any solution (r, µ) of (16) with k ≤ min r ≤ K verifies max r < M 1 := ρ * K .
Next, we remember that f is L 1 -Carathéodory and find some function h ∈ L 1 (R/T Z) such that |f (t, x)| ≤ h(t) for a.e. t ∈ R and all x ∈ [k, M 1 ]. We let Choose now a solution (r, µ) of (16) with k ≤ min r ≤ K. In view of (23), max r ≤ M 1 , and integrating both sides of the equation of (16) we get and we deduce that |µ| ≤ M 2 . It completes the proof.
Our next step will consist in showing that the angular momentum of radially periodic solutions with big amplitude may be bounded by the subquadratic function M applied to the distance from our solution to the origin. Lemma 2.4. There exists some λ 0 > 0 such that every solution (r, µ) of (16) with max r ≥ λ 0 verifies 0 < |µ| < M(r(t 0 )) for any t 0 ∈ R .
Proof. Let λ 0 > 0 be as in Lemma 2.2 and let (r, µ) be a solution of (16) with max r ≥ λ 0 . Then Lemma 2.2(i) implies that (22) holds for λ = r(t 0 ) and any t 0 ∈ R. In particular, r(t 0 ) ≥ λ 0 /ρ * , and Lemma 2.2(ii) gives On the other hand, integrating on the equation of (16) we observe that and the result follows.
To close this Section, we prove Proposition 1.6. We have only to combine Lemmas 2.1 and 2.4.
Finally, pick some solution x = r(cos θ, sin θ) of (3) with max r ≥ λ 2 . We consider the angular momentum µ = r(t) 2θ (t), which does not depend on time. Thus, (r, µ) is now a solution of (16), and Lemma 2.1 implies that max r/ min r < 1 + . On the other hand, since max r ≥ λ 1 ≥ λ 0 , we have so that |θ(t)| < for any t ∈ R. This concludes the proof.
the previous Section, r = r x must be a solution of the periodic boundary value problem (16), the parameter µ = µ x = r 2θ being the angular momentum of the solution. Conversely, let now (r, µ) be a solution of (16). Then solves the second equation of system (4). It further verifies that θ r,µ (t + T ) = θ r,µ (t) + θ r,µ (T ) for any t ∈ R, and, consequently, is a T −radially periodic solution of (3), its rotation number being θ r,µ (T )/T . Since θ r,µ (0) = 0, this solution verifies i.e., at the initial time our solution crosses the horizontal axis on its positive side. This discussion leads us to consider the mappings the sets R and X being defined by Observe that Φ and Ψ are mutually inverse bijections. Moreover, if R and X are endowed, respectively, with the topologies inherited from R times the space C(R/T Z) of continuous and T −periodic functions on the real line, and the topology of uniform convergence on compact sets, then Φ and Ψ become mutually inverse homeomorphisms.
What relevance has this discussion with respect to Theorem 1.5? To answer this question we recall that, roughly speaking, this result states the existence of a 'large' connected set C of T -radially periodic solutions of (3). In principle, C may be not contained into X , as the functions of C are not obliged to cross the positive part of the horizontal axis at time zero. Observe, however, that our equation (3) is rotation-invariant. By this we mean that, if x = r x (cos θ x , sin θ x ) is a solution, and R is the rotation of some angle θ 0 around the origin, then Rx = r x (cos(θ x + θ 0 ), sin(θ x + θ 0 )) is again a solution. In this way, we may rotate the elements of C to build a second connected set C of solutions of (3) which is now contained inside X : Notice that {min |x| : x ∈ C } = {min |x| : x ∈ C}, so that there is no restriction in looking for the connected set C in the smaller space X . The homeomorphism Ψ will then send it into another connected set K ⊂ R, which will verify (K 1 ) the interval min r : (r, µ) ∈ K is unbounded (from above), and one of the following: In this way, the following result may be seen as a corollary of Theorem 1.5: Lemma 3.1. Assume (M 1 ) and (H 2 ). Then, there exists a connected set K ⊂ R verifying (K 1 ), and either (K 2a ) or (K 2b ) above.
Actually, Lemma 3.1 is equivalent to Theorem 1.5, because Φ and Ψ are homeomorphisms. And this will be the spirit of our approach; we shall prove Theorem 1.5 in the form given by Lemma 3.1. Before, it will be convenient to establish a functional analysis framework for our problem. This property plays an important role, as it will allow us to use the Leray-Schauder degree arguments. We look for solutions of the fixed-point equation For this, we further assume the existence of some λ 0 ∈ R and some open and bounded set Υ λ 0 ⊂ Y such that {λ 0 } × Υ λ 0 ⊂ Ω and (i) Equation (25) has no solutions (λ, y) ∈ Ω such that λ = λ 0 and y ∈ Y \Υ λ 0 .
To state the main result of this section it will be convenient to introduce some notation. Given any set A ⊂ Ω, we divide it into its (not necessarily disjoint) left and right pieces A ± . The right one A + is defined by while A − is given analogously after reversing the inequality. We also consider the set Σ of solutions of (25), Σ := (λ, y) ∈ Ω : y = H(λ, y) .
Proposition 4.1. Assume (C), (i), and (ii). Then, there exists a connected subset C ⊂ Σ such that each piece C ± lies under the following disjunctive: either or C ± goes up to the boundary of Ω, i.e. its distance to the boundary of Ω is zero. This result is well known, and even though we could not find it in the literature in the form presented here, related results may be found, for instance, in [2,4,8,9,10]. We shall actually need only a corollary of this result, which we describe next. It will be convenient to introduce first the following notation: given any subset Γ ⊂ Ω and any λ ∈ ]0, +∞[ we denote by Γ λ the vertical section Γ λ := {y ∈ Y : (λ, y) ∈ Γ}.
In the case we are interested in, Ω will be a cylinder where V is an open (and possibly unbounded) subset of Y . With other words, we are assuming that Ω λ = V is always the same set for every λ ∈ ]0, +∞[. As before, we shall also need (C), we fix some value λ 0 > 0 and we denote by Ω ± the sides of Ω to the left and the right of λ 0 . Proof. Applying Proposition 4.1 we deduce the existence of a connected set C for which each piece C ± lies under the disjunctive (a ± )-(b ± ). For instance, C + must verify either (a + ) or (b + ). But this time, our assumptions (1.) and (3.) prevent (b + ) from happening, so that we must have (a + ), and in view of (2.), we deduce that C λ = ∅ for any λ ≥ λ 0 , as claimed.
We still have the disjunctive (a − )-(b − ). Possibility (a − ) is now called (a). Concerning to possibility (b − ), one observes that the boundary of our cylindrical domain Ω may be decomposed into two parts: {0} ×V and ]0, +∞[×∂V . Thus, if the distance from C − to ∂Ω is zero, it is because either the distance to {0} ×V is zero or the distance to ]0, +∞[×∂V is zero. The first possibility is what we called (b ) and the second one is (b ). The proof is complete.
Finally, we state a one-sided variant of Corollary 4.2 which we shall need later. It is also well-known and the proof is skipped: Under the framework of Corollary 4.2, for any λ * ≥ λ 0 , there exists a connected subset C * ⊂ Σ such that

A connected set of solutions coming from infinity
Throughout this Section we assume (M 1 ) and (H 2 ). In order to apply the abstract framework developed in the previous section to find connected sets of solutions (r, µ) of (16), we have to reformulate this problem as a fixed point equation in a Banach space. The equation will depend on a one-dimensional parameter; however, this parameter will not be µ, but the value λ of our T -periodic unknown r at the integer multiples of the period. For this reason, our first step will be to rewrite r as where λ = r(0) andr = r/λ − 1. It follows now thatr(0) = 0, motivating us to consider the space which is endowed with the uniform norm · ∞ . This space has the following property: given any integrable function h ∈ L 1 (R/T Z) there exists a unique function Kh ∈ C 0 (R/T Z) which has W 2,1 regularity on [0, T ] and whose second derivative there is h. Moreover, the mapping K :  = (r,μ)).
Observe that, instead of the angular momentum µ we have used here a new letterμ. Well, problem (16) has the property that, if (r, µ) is a solution, then so is (r, −µ). We do not want this situation to be translated to our abstract framework, and that is the reason why we have introduced the new variableμ. Whenμ ≥ 0, thenμ = µ will be just the angular momentum, but ifμ < 0 then we will be looking for solutions with zero angular momentum µ = 0. It means that we are going to loose the solutions with negative angular momentum; on the other hand, if there is some solution with µ = 0, then it will be repeated for each value ofμ ∈ ] − ∞, 0]. Having this in mind, the 'modified Nemitski functional' for our problem (16) is defined by We rewrite (16) as a Lyapunov-Schmidt-type system on Ω: This system may be seen as a fixed point equation on V depending on the parameter λ: Observe that H : ]0, +∞[×V → Y is continuous; moreover, it verifies the compactness assumption which we labeled (C) in the previous Section. We want to apply Corollary 4.2, but before, we have to place ourselves inside the framework of this result. Thus, we fix some number ρ * > 1 as given by assumption (M 1 ), we recall the function M : ]0, +∞[→ R defined in (21), and we pick some number λ 0 > 0 verifying simultaneously the statements of Lemmas 2.2 and 2.4. Finally, we define Observe that Υ is an open subset of Ω + = λ, r µ ∈ Ω : λ ≥ λ 0 . The following result states that the other assumptions of Corollary 4.2 are also satisfied.  H(λ, ·). We let r := λ(1 +r) and µ = max{μ, 0}, so that (r, µ) is a solution of (16). Moreover, max r ≥ r(0) = λ ≥ λ 0 , and, for t 0 = 0, Lemma 2.2 (i) implies that λ/ρ * < r(t) < ρ * λ, or, equivalently, On the other hand, Lemma 2.4 states that 0 < |µ| < M(λ). It follows that µ =μ > 0 and (λ,r,μ) ∈ Υ, proving (3.).
Before going into the proof of (ii) we first notice that, as a consequence of our sublinearity assumption (H 2 ) and the function M being subquadratic, on the set Υ we have: lim λ→+∞ N (λ,r,μ) L 1 = 0 as λ → +∞ , uniformly with respect to (r,μ) .
We are going to prove now Theorem 1.7. In order to do so, we first establish an elementary result on a priori bounds for the first derivative of T -radially periodic functions whose second derivative is controlled: Proof. Since both |x| and |ẋ| are T -periodic, we may find points t 0 , s 0 ∈ R such that |ẋ(t 0 )| = max |ẋ|, |x(s 0 )| = min |x|, On the other hand, remembering that |ẍ| ≤ c, We integrate in both sides of this inequality, to obtain and thus, max |ẋ| = |ẋ(t 0 )| ≤ |x(s 0 + T )| + |x(s 0 )| + T c L 1 = 2 min |x| + T c L 1 the first half of the statement. We choose now s 1 ∈ [s 0 , s 0 + T ] such that |x(s 1 )| = max |x| and observe that concluding the proof.
We consider now the set which is an interval, because C is connected and rot is continuous. From (38) we see that I ⊂ ]0, +∞[. On the other hand, (37) and Proposition 1.6 imply that for someω > 0. With other words, for any 0 < ω <ω there exists some element x ω ∈ C * such that rot(x ω ) = ω. The first part of (15) follows now from (38).
To conclude the proof, it will suffice to show that lim ω→0 max |x ω | = +∞, since (14) and the second part of (15) will then follow from Proposition 1.6. Thus, we use a contradiction argument and assume that there is some sequence ω n → 0 such that max |x ω n | is bounded, say, by some constant K > 0. Taking into account the second part of (36) we observe that But f was assumed to be L 1 -Carathéodory, and then, there is some function c ∈ L 1 (R/Z) such that |f (t, r)| ≤ c(t) for a.e. t ∈ R/Z and every r ∈ [1, K]. From the equation (3) we conclude that |ẍ ω n (t)| ≤ c(t) for a.e. t ∈ R and all n ∈ N .
Inequality (39) states that the sequence {x ωn } is uniformly bounded, and Lemma 5.2 states that also {ẋ ωn } is uniformly bounded. In particular, {x ωn } is equicontinuous, and, in view of (40), also {ẋ ωn } is equicontinuous. Thus, Ascoli-Arzelà's Theorem applies and states that, after possibly passing to a subsequence, one may assume that there exists some C 1 function x * : R → R 2 such that {x ω n } → x * and {ẋ ω n } →ẋ * uniformly on compact sets. It immediately follows that x * is T -radially periodic. On the other hand, max |x ω n | ≥ λ 2 for any n, and we deduce that max |x * | ≥ λ 2 . Finally rewriting our equation (3) in its integral form (for x = x ω n ) x ω n (t) =ẋ ω n (0) + t 0 f (s, |x ω n (s)|) x ω n (s) |x ω n (s)| ds , t ∈ R , and taking limits as n → ∞, we see that also x * is a solution of (3). But then, (36) implies that rot x * = 0, contradicting the continuity of rot. It concludes the proof.
In this last Section we construct the examples announced in Propositions 1.2 and 1.4. We start with Proposition 1.2, and, with this aim, we consider the following increasing sequence of positive, 2π-periodic functions: Our proof for this result will be based on the following elementary result of real analysis: We shall not give a detailed proof of Lemma 6.1 here, as it may be considered an exercise; we just point out that f may be chosen with the form f (t, r) = u(r − (1/4) sin t) for some suitable function u : R → R.
We remark that the condition above on the positivity of the integrals over horizontal lines may be improved. Indeed, one easily checks that given some constant M > 0, the function f can be chosen so that 2π 0 f (t, r)dt ≥ M for any r > 0. In this way, also Proposition 1.2 could be sharpened; given M > 0 the function f may be chosen so that T 0 f (t, r)dt ≥ M for any r > 0. Lemma 6.1 will lead us to Proposition 1.2. We see it below: Proof of Proposition 1.2. We choose f as given by Lemma 6.1 and observe that the sequence {r n } n is made of strict upper solutions for the equation −r = f (t, r). In other words, Let now C be a connected set of 2π-radially periodic solutions of (3). Using the results of Section 3 we see that K := Ψ(C) is a connected set of solutions (r, µ) of (16). We recall that our aim is to show that the interval |x(t)| : t ∈ R, x ∈ C = r(t) : t ∈ R, (r, µ) ∈ K is bounded. With this aim we choose some element (r , µ ) ∈ K and fix some n 0 ∈ N such that r n 0 (t) > r (t) for any t ∈ [0, T ]. We consider the sets A := (r, µ) ∈ K : r(t) < r n 0 (t) for all t ∈ R , B := (r, µ) ∈ K : r(t * ) > r n 0 (t * ) for some t * ∈ R .
Observe that A and B are open in K. Moreover, we claim that A ∪ B = K. Indeed, the contrary would mean the existence of some element (r, µ) ∈ K such that r ≤ r n 0 on R , r(t 0 ) = r n 0 (t 0 ) for some t 0 ∈ R , which is not possible because, in view of (17), r is a lower solution of −r = f (t, r) while, as observed in (41), r n 0 is an strict upper solution.
Since K is connected, one of these sets must be empty. But (r , µ ) ∈ A, and we deduce that B = ∅. It concludes the proof.
We undertake now the proof of Proposition 1.4. This result is divided in two parts, the first one concerning the case 0 ≤ γ < 1, and the second about the case γ < 0, and they will be treated separately.
The first part will follow from Theorem 1.5. We only have to check that the onedimensional equation −r = c(t)r γ , r > 0 , does not have T -periodic solutions if 0 < γ < 1 and c ∈ C(R/T Z) has positive mean. Our strategy will consist in building a continuous family of lower solutions for this equation. Well, not all of them exactly lower solutions, but something close to that:   Proof of Proposition 1.4 for 0 ≤ γ < 1. Using a contradiction argument, assume that r = r(t) were a T -periodic solution of (42) and define p * := min p > 0 such that the graphs of ψ(·, p) and r intersect .
Then, ψ(t, p * ) ≤ r(t) for every t ∈ R and, for some t 0 , one has the equality ψ(t 0 , p * ) = r(t 0 ), which is contradictory with the fact that r and ψ(·, p * ) are, respectively, a solution and a strict lower solution of (42). It completes the proof.
We show now the second part of Proposition 1.4. With this aim, we start by choosing some γ < 0 which will be henceforth fixed and observe that any solution x of the equation with min |x| = 1 verifies |ẍ(t)| ≤ |c(t)| for a.e. t ∈ R .
But then, Lemma 5.2 gives a priori bounds for x and |ẋ|, which, in our case, read: At this point we introduce a contradiction argument, and, from now on, we shall assume that  0 c n (t)dt > 0 such that {c n } → c in the L 1 sense. Correspondingly, for each n ∈ N we find an associated 1-radially periodic solution x n of (43) with c = c n such that min |x n | = 1. But {c n } is bounded in L 1 (R/Z), and (44) states that {x n } and {ẋ n } are uniformly bounded on R. In particular, {x n } is equicontinuous, and using Ascoli-Arzelà Theorem, we may assume, after possibly passing to a subsequence, that {x n } converges uniformly on compact sets to some continuous function x * : R → R 2 with min |x * | = 1. It is not restrictive to assume that, moreover, {ẋ n (0)} converges to some number β ∈ R. Since x n is 1-radially periodic for each n, we conclude that x * is also 1-radially periodic. Finally, from the integral form of our equationẋ n (t) =ẋ n (0) − t 0 c n (t)|x n (t)| γ x n (t) |x n (t)| dt , n ∈ N , it follows thatẋ n converges uniformly on compact sets to the function