Chaotic behavior of rapidly oscillating Lagrangian systems

In the paper we prove that the Lagrangian system 
 
$ \ddot{q} = \alpha(\omega t) V'(q), \quad t \in \mathbb R, q \in \mathbb R^N,$ $\qquad\qquad 
 (L_\omega)$ 
 has, for some classes of functions $\alpha$, a chaotic 
 behavior---more precisely the system has multi-bump 
 solutions---for all $\omega$ large. These classes of functions 
 include some quasi-periodic and some limit-periodic ones, but not 
 any periodic function. We prove the result using global variational methods.

Using variational methods it has been proved that (L ω ) has, for all such α and ω > 0, infinitely many homoclinic solutions (see for example [1,15,3]). It is also known such a system has multibump solutions, and hence chaotic behavior, whenever a suitable nondegeneracy condition is satisfied (following ideas introduced by E. Sèrè in [16]). Such a condition has been shown to be satisfied for a given α whenever ω is small enough (see [8,1,2]), and generically in α (see [2]). See also [7] for a case in which α is the sum of a slowly oscillating part and a fast oscillating one.
The main purpose of this paper is to describe some examples of (class of) functions α such that (L ω ) has a multibump solution for all ω > 0 large. We will give three examples: a quasi periodic one (subsection 4.1), an almost periodic one (subsection 4.2) and a limit periodic one (subsection 4.3).
To prove the result we will need (see section 4) that for all ω > 0 one can split the function α(ωt) = α ω,1 (t) + α ω,2 (t) + α ω,3 (t) in such a way that α ω,1 is a nontrivial slowly oscillating function, α ω,2 is a fast oscillating function and α ω, 3 is small compared to α ω,1 . Here the time scale is given by the time it takes to the 688 F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI homoclinic solution ofq = αV (q) to go from a neighborhood of 0 to a neighborhood of ξ. For this reason we cannot handle the case in which α is a periodic function (since in this case there α ω,1 ≡ 0) or is a quasi-periodic function α(t) = β( γt) where β : T k → R is an analytic function (since in this case α ω, 3 is not sufficiently small compared to α ω,1 ).
One of the motivations in studying such a problem lies in the fact that the system (L ω ) is equivalent to the following q = 1 ω 2 α(t)V (q) (it is enough to perform the change of variable q(t) → q(ωt)). This system can be seen as a small perturbation of the completely integrable systemq = 0 on the torus T N . The result then shows that there are perturbations, however small, that change completely the dynamics of the system.
A particularly interesting case is the one in which α is a quasi-periodic function. Indeed in this case the problem is related with that of Arnold's diffusion. Results on this kind of problem have been obtained by various Authors; let us recall here [4,14,12,11,10,9]. Let us point out that all the results we are aware of work for equations of the formq = ε p α(ωt)V (q) for small ε and requires analyticity of V in q.

Variational setting and preliminary results.
In this first section we discuss some preliminary results which will be basic in the proof of our main Theorem. Even if most of these properties are well known (see for example [1]) we will give some of the proofs.
Given any α > α > 0, we will denote by and we let, for α ∈ A and V which satisfy (V1)-(V3), be the Lagrangian of the systemq We also introduce the function space which becomes a Hilbert space with inner product Remark 2.1. Let us remark that (V3) implies that there is are r 0 ∈ (0, 1 6 ) andν > 0 such that, for all |y| ≤ r 0 and for all We then define, for all q ∈ E, We also define, for r ∈ (0, r 0 ), By assumption (V2) we have that µ r > 0.
for all α ∈ A.
Proof. (2.6) is an immediate consequence of the definition of µ r .
(2.7) follows since By Lemma 2.2, for any r > 0, any q ∈ E with f α (q) < ∞ must definitely enter in B r (Z N ). More precisely, one easily proves We define, for all ξ ∈ Z N and α ∈ A Note that from (2.7) for all α ∈ A and q ∈ Γ ξ , we have Hence, using Lemma 2.2 we obtain Following [6], among the ξ ∈ Z N it is possible to select a finite number of them which generate all of Z N and have additional properties, as stated in the following lemma.
Then, there exists q ∈ Γ ηi such that, along a subsequence, q n → q weakly in Proof. We consider only the case |q n (τ )| = r for all n ∈ N, the other case being similar. We choose δ ∈ (0, r) such that loc (R, R N ) and then, up to a subsequence, it converges weakly in H 1 loc (R, R N ) and strongly in L ∞ loc (R, R N ) to some q ∈ E. Since f α is plainly weakly lower semi- Let us now state some consequence of the fact that α ∈ A is a almost-periodic function. We recall that If this holds one say that τ is an ε-periods of α and that the ε-periods are T ε -dense in R. See, for the more information on almost periodic functions, [5,13].
The almost periodicity of the function α implies that the functional f α has similar properties. This fact is exploited in next Lemma in which we construct infinitely many test functions which will be used in a cutting and pasting procedure in the proof of Theorem 3.4 below.
Then, lettingq In the next two Lemma we characterize the behavior of solutions of (L α ) which in certain time intervals remain in the configuration space nearby the stationary points ξ ∈ Z N . These results are consequences of the (V3) assumption as discussed in Remark 2.1.
Proof. It is a standard fact that such a minimum exist and that it is a solution of the problem. To show that |q(t)| ≤ r 0 for all t ∈ [a, b] we note that if δ 0 is sufficiently small, it is enough to remark that the "cost" (as measured by b a L α (t, q(t),q(t)) dt) of going from ζ 1 to ∂B r0 (0) back to ζ 2 exceed, uniformly for α ∈ A, the cost of going from ζ 1 to 0 back to ζ 2 . Uniqueness then follows from the convexity of L α (t, q,q) for |q| ≤ r 0 .
Then, for all t ∈ [a, b], in particular Proof. Take ν given by assumption (V3) and let and It is immediate to check that w solves the boundary value problem (2.10) Let q be a solution of (L α ) such that |q(t)| ≤ r 0 for all t ∈ [a, b]. Then, using (L α ) and (2.2), we deduce that Then By the maximum principle we obtain Remark 2.13. The same arguments used in the proof of Lemma 2.12 show that whenever q is a solution of (L α ) such that |q(t)| ≤ r 0 for all t ≤ a, and that By periodicity, analogous estimates hold for The following final two Lemmas give further properties of solutions of (L α ) which are minimal for f α in Γ ηi . Even if in general, when α is almost periodic, one cannot say that these minimal solutions exist, these properties will be frequently used below in contradiction arguments.
Then one simply remarks that Then b − a ≥ 1 8cη i (α) and hence, by Lemma 2.6,

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS 695
Let δ 0 be fixed as in Lemma 2.11 and denote ρ 0 = 2 √ δ 0 and L 0 = L δ0 . Then we have Proof. We will assume |q(τ )| = δ 0 , the other case can be handled in the same way. First we prove that for every L > 0, To this aim, note that since q satisfy f α (q) = c α (η i ), it follows from Lemma 2.7 that |q(t)| ≤ ρ 0 < r 0 for all t ≤ τ . Then, noting that q is a solution of (L α ), by Remark 2.13, we obtain that |q(t)| < δ 0 for all t < τ. The same argument shows that |q(t)| > δ 0 for all t > τ. Therefore, again by Lemma 2.7, |q(t) − η i | < δ 0 for all t > τ + L 0 . Using Lemma 2.12 (see also Remark 2.13), we have Then, by (2.4) We deduce that Hence (2.14) follow. Now, fixed any d > 0 let L > 0 be such that 2ν √ αν e − √ ανL < dV . Then, by Lemma 2.14, the lemma follows setting M (d) = L + L 0 .
3. Multibump solutions. In this section we will show that under an additional assumption our system exhibits a chaotic behavior.
Remark 3.1. Let us remark that, in order to prove results like the ones contained in Theorem 3.4, one could actually assume a little bit less then ( * ). Indeed, as one can see for example in [1,15], it is enough to assume that there is a τ ∈ R such that q ∈ Γ ηi and |q(τ )| = δ 0 implies f α (q) > c ηi (α).
Remark 3.2. Let us remark that f α has in Γ ηi uncountably many minimizers (and hence (L α ) has uncountably many solutions) whenever ( * ) is not satisfied.
In order to state the result of this section, let us present some consequence of assumption ( * ) and introduce some notation.
We will now assume that and show that we reach a contradiction with ( * ).
Remark 3.5. Let (τ j ) j∈N ⊂ R be such that any τ j is an ε 1 -period of α with τ j+1 − τ j > M 0 , for all j ∈ N. For all k ∈ N, let q k ∈ C 2 (R, R N ) be the solution to (L α ) such that q k ∈ M k . Then, one can easily see that for any compact K ⊂ R there exists a constant C = C(K) > 0 such that Hence, since any q k is a solution to (L α ), we obtain that for any compact K ⊂ R there exists a constantC =C(K) > 0 such that Using the Ascoli-Arzela Theorem, we obtain that along a subsequence, (q k ) k∈N converges in C 1 loc (R, R N ) to a solution q to (L α ), which verify 4. On the condition ( * ). In this section we discuss some situation in which condition ( * ) with τ = 0 is verified by system (L ω ) for any ω > 0. We study three different examples of almost periodic functions, a case in which α is quasi periodic, a case in which it is limit periodic and a case in which it is only almost periodic.

A quasi-periodic case.
We recall that the function α is a quasi-periodic function if α(t) = F (γ 1 t, γ 2 t) for some function F (x, y) periodic both in x and y and for some constants γ 1 , γ 2 ∈ R. Then and c k1k2 e i(k1γ1+k2γ2)t As in the paper [9], we take γ 1 = 1, and consider where (ω k ) is a sequence in k 1 + γk 2 k 1 , k 2 ∈ Z . Then our function is a quasiperiodic function. Let us remark immediately that our assumptions will imply that F is not an analytic function.
On the frequencies ω k we will assume that where ∈ N and F n is the n Fibonacci number (defined by F 0 = 0, F 1 = 1, F n+1 = F n + F n−1 for all n ≥ 1). Let us also recall that see [10]. As a consequence, we have that there exists 1 ∈ N (independent of k) such that for all ≥ 1 and k ≥ 1.
Proof. To prove the Theorem we need some preliminary estimates.
Let us first note that from the assumption on a k it follows that for any Hence for any ∈ N and ω > 0, we have that 3 4 ≤ α ω (t) ≤ 5 4 . We will drop the dependence on , and denote f ω (q) = f αω (q). Arguing by contradiction, we will assume that for some i there is a function q ω ∈ Γ ηi with |q ω (0)| = δ 0 or |q ω (0) − η i | = δ 0 such that f ω (q ω ) = c ηi (α ω ).
Let M 1 be given by Lemma 2.15 with d = 1 10 and τ = 0, i.e. be such that We then definek ∈ N (which depends on ω and ) bȳ and split α ω setting α 1 (t) = ak cos(ωωkt) We immediately have that and (α3) is verified setting δ = ak. Note that the splitting and δ depend on ω and .
To deal with α 1 and α 2 we first observe that Lemma 4.2. For all k and ∈ N we have that ω k and ω k+1 are rationally independent.
Proof. Suppose there are i, j ∈ Z \ {0} such that jω k = iω k+1 . Then Since γ is irrational, necessarily and then, by definition of the Fibonacci's sequence In particular, for j = k, F = 0, a contradiction if ≥ 1.
As a consequence of this lemma, for all ε ∈ (0, 1), using the Kronecker's Theorem (see [13]), it is possible to find τ 0 ∈ R (which depends on k, and ω) such that Then, for all |t| < M 1 we have |ωωkt| < ωωkM 1 < 1 8 and then cos(ωωk(t + τ 0 )) − cos(ωωkt) = cos(ωωkt)(cos(ωωkτ 0 ) − 1) − sin(ωωkt) sin(ωωkτ 0 ) provided ε is sufficiently small. Moreover, for all t ∈ R, As a consequence and using Lemma 2.15 we have that and the condition (α1) holds true. To finally show that also (α2) holds, let us prove the following Lemma 4.3. There is C 3 = C 3 (V, α, α) such that, for all ∈ N, ω > 0 and for all q ω ∈ Γ ηi such that f ω (q ω ) = c ηi (α ω ) the following holds: Proof. Using the fact that q ω is a solution, a simple integration by part shows that: Since V (x) is bounded and |V (x)| 2 ≤ CV (x) for some positive C, one deduces that and the claim follows by Lemmas 2.14 and 2.6. Now we remark that, thanks to (4.2), provided ε has been chosen small enough.
Then it follows from Lemma 4.3 that We want to show that the term is small for all ω provided is large enough. Some care is necessary since k depends on ω and .

4.2.
An almost-periodic case. We now present a different situation, in which the function α is an almost-periodic function (see definition 2.9) which is not quasiperiodic. Intuitively an almost-periodic function can be seen as a combination of harmonics having countably many rationally independent frequencies, in contrast to quasi-periodic ones (which have harmonics of only finitely many rationally independent frequencies) and periodic (which have harmonics of just one frequency). As for the quasi-periodic example, we just present here a rather simple situation just to illustrate our method.
As in the proof of Theorem 4.1, we denote f ω (q) = f αω (q). Arguing by contradiction, assume there is, for a given ω > 0, a function q ∈ Γ η such that f ω (q) = c η and |q(0)| = δ 0 or |q(0) − η| = δ 0 . Let M 1 be fixed in Lemma 2.15 with d = b0 8 and τ = 0 and note that since Takek ≥ 1 such that Then, we consider the following splitting of α ω as First we have and so (α3) holds true letting δ = ak b0 4 . By uniform continuity, let ε > 0 be such that (4.4) By Kronecker's Theorem we can find τ 0 ∈ R and n 1 , . . . , nk ∈ N such that So we can estimate Since for all |t| ≤ M 1 we have that |ωωkt| ≤ |ωωkM 1 | ≤ t0 8 , by the choice of M 1 we obtain so that (α1) holds. Moreover, by the choice of τ 0 and (4.4), setting ε k = ωω k τ 0 − n k T , we have and also (α2) holds. This completes the proof of the Theorem.

A limit-periodic case.
We give now a third example. We consider here a limit-periodic function α. By definition, a limit-periodic function is a continuous function α : R → R which is the uniform limit of periodic functions.
Proof. By assumption (C2), As in the proof of Theorem 4.1, we denote f ω (q) = f αω (q) and, by contradiction, assume that there is, for a given ω > 0, a function q ∈ Γ ηi such that f ω (q) = c ηi (α ω ) and |q (0) Noting that ω k → 0, takek ≥ 1 such that Let τ 0 = T 2ωωk . Then Since for all |t| ≤ M 1 we have that |ωωkt| ≤ |ωωkM 1 | ≤ T 8 , by the choice of M 1 we obtain so that (α1) holds with δ = ak b0 4 . Moreover, we have that From assumption (C4) it follows that for all k = 1, . . . ,k − 1 there is an integer n k such that 2n k ωk = ω k , so that ωω k τ 0 = T ω k 2ωk = T n k and Moreover for all k >k and for all |t| ≤ M 1 we have ωω k |t| ≤ ωωkM 1 ≤ T 8 and ωω k τ 0 = ω k 2ωk T = T 2n for a certain n ∈ N. Then from assumption (C2) we also deduce that for all k >k and for all |t| ≤ M 1 b(ωω k t + ωω k τ 0 ) − b(ωω k t) ≤ max and the theorem follows.