Chaotic behavior of rapidly oscillating Lagrangian systems

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS FRANCESCA ALESSIO, VITTORIO COTI ZELATI, AND PIERO MONTECCHIARI

1. Introduction Let us consider the Lagrangian system (Lω )

q¨ = α(ωt)V 0 (q),

t ∈ R, q ∈ RN ,

where ω > 0, α is a non constant, strictly positive, almost-periodic function such that α = inf R α > 0 and V ∈ C 2 (RN , R) satisfies (V1) V (x + ξ) = V (x) for all x ∈ RN and ξ ∈ ZN ; (V2) V (x) > V (0) = 0 for all x ∈ RN \ ZN ; 2 (V3) there exists ν > 0 such that hV 00 (0)x | xi ≥ ν |x| for all x ∈ RN . Using global variational methods it has been proved that (Lω ) has, for all such α and ω > 0, infinitely many homoclinic solutions (see for example [1, 12]). 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. S`er`e in [13]). Such a condition has been shown to be satisfied for a given α whenever ω is small enough (see [5, 1, 2]), and generically in α (see [2]). The main purpose of this paper is to describe some examples of (class of) functions α such that (Lω ) has a multibump solutions 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). Let us point out that we cannot handle any periodic α with our approach. One of the motivations in studying such a problem lies in the fact that the system (Lω ) is equivalent to the following 1 q¨ = 2 α(t)V 0 (q) ω (it is enough to perform the change of variable q(t) 7→ q(ωt)). This system can be seen as a small perturbation of the completely integrable system q¨ = 0 on the torus TN . 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 [11, 9, 8, 7, 6]. Let us point out that all the results we are aware of work for equations of the form q¨ = εp α(ωt)V 0 (q) for small ε and requires analyticity of V in q. 2. 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. Supported by MIUR, project “Variational Methods and Nonlinear Differential Equations”. 1

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F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

Given any α > α > 0, we will denote by  A = α ∈ C(R; R) 0 < α ≤ α(t) ≤ α, α almost periodic and we let, for α ∈ A and V which satisfy (V1)–(V3), 1 2 Lα (t, q, q) ˙ = |q| ˙ + α(t)V (q) 2 be the Lagrangian of the system q¨ = α(t)V 0 (q).

(Lα )

We also introduce the function space Z  2 1 E = q ∈ Hloc (R; RN ) |q(t)| ˙ dt < +∞ R

which becomes a Hilbert space with inner product Z (x | y) = hx(0) | y(0)i + hx(t) ˙ | y(t)i ˙ dt. R

Remark 2.1. Let us remark that (V3) implies that there is are r0 ∈ (0, 16 ) and ν˜ > 0 such that, for all |y| ≤ r0 and for all x ∈ RN ν 2 (2.1) hV 00 (y)x | xi ≥ |x| 2 ν 2 (2.2) hV 0 (y) | yi ≥ |y| 2 (2.3) |hV 0 (y) | xi| ≤ ν˜ |x| ν 2 (2.4) |y| ≤ V (y) ≤ ν˜ |y| 4 We then define, for all q ∈ E, Z fα (q) = Lα (t, q(t), q(t)) ˙ dt. R

We also define, for r ∈ (0, r0 ),  µr = inf V (x) x ∈ / Br (ZN ) ,

(2.5)

where Br (ZN ) = ∪ξ∈ZN Br (ξ). By assumption (V2) we have that µr > 0.   Lemma 2.2. Let r ∈ (0, r0 ). If q ∈ E is such that q(t) ∈ / Br (ZN ) for all t ∈ t, t then Z t (2.6) fα (q) ≥ Lα (t, q(t), q(t)) ˙ dt ≥ αµr (t − t); t

(2.7)

q(t) − q(t) ≤ √ 1 2αµr

Z t

t

Lα (t, q(t), q(t)) ˙ dt ≤ √

1 fα (q), 2αµr

for all α ∈ A. Proof. (2.6) is an immediate consequence of the definition of µr . (2.7) follows since Z t 1 fα (q) ≥ Lα (t, q(t), q(t)) ˙ dt ≥ |q(t) − q(t)|2 + αµr (t − t) 2(t − t) t p ≥ 2αµr |q(t) − q(t)|.  By Lemma 2.2, for any r > 0, any q ∈ E with fα (q) < ∞ must definitely enter in Br (ZN ). More precisely, one easily proves

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

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Lemma 2.3. If q ∈ E is such that fα (q) < +∞, then q ∈ L∞ (R; RN ) and there exist ξ − , ξ + ∈ ZN such that q(t) → ξ ± as t → ±∞. We define, for all ξ ∈ ZN and α ∈ A  Γξ = q ∈ E q(−∞) = 0, q(+∞) = ξ ,

 cξ (α) = inf fα (q) q ∈ Γξ .

Note that from (2.7) for all α ∈ A and q ∈ Γξ , we have r αµ1/4 ≡ c. fα (q) ≥ 2 Hence, using Lemma 2.2 we obtain Lemma 2.4. For all ξ ∈ ZN \ {0}, α ∈ A we have that cξ (α) ≥ c and cξ (α) → +∞ as |ξ| → +∞. Following [4], among the ξ ∈ ZN it is possible to select a finite number of them which generate all of ZN and have additional properties, as stated in the following lemma. Lemma 2.5. Let α ∈ A. There exist η1 , . . . , ηm ∈ ZN \ {0} such that ZN =  Pm nj ∈ N and if ξ1 , ξ2 ∈ ZN \ {0} are such that ξ1 + ξ2 = η ∈ j=1 nj ηj {η1 , . . . , ηm }, then cξ1 (α) + cξ2 (α) ≥ cη (α) + c.

(2.8)

Proof. It follows from Lemma 2.4 that there is η1 ∈ Z \ {0} such that  cη1 (α) = min cξ (α) ξ ∈ ZN \ {0} .  Pj−1 n` ∈ that we have already chosen {η1 , . . . , ηj−1 }, and `=1 n` η` Suppose N N 6= Z . Then we can find ηj such that j−1 n X o n` η` n` ∈ N . cηj (α) = min cξ (α) ξ ∈ ZN \ `=1

If ξ1 + ξ2 = ηj , then max{cξ1 (α), cξ2 (α)} ≥ cηj (α), and we have that cξ1 (α) + cξ2 (α) ≥ cηj (α)+cη1 (α), so the lemma follows since, by Lemma 2.4, c ≤ cη1 (α).  Even if the set {η1 , . . . , ηm } may depend on α, the minimum levels cηi (α), i = 1, . . . , m, are uniformly bounded from above by a constant c independent from α ∈ A. Precisely Lemma 2.6. There exist c > 0 such that for all α ∈ A we have cηi (α) ≤ c. Proof. Let e1 , . . . , eN be the canonical base RN and denote ξ 1 = e1 , . . . , ξN = eN ,  Pof 2N n` ∈ N = ZN and we set ξN +1 = −e1 , . . . , ξ2N = −eN . Then, n ξ ` ` `=1 c = max{cξ1 (α), . . . , cξ2N (α)}. Letting α ∈ A, by construction we have cηi (α) ≤ · · · ≤ cηm (α) and to prove the to show that cηm (α) ≤ c. To this lemma it is sufficient  Pm−1  Pm−1 N aim note that n η n ∈ N = 6 Z and so there exists ξ ∈ / ` ` ` i `=1 `=1 n` η` n` ∈ N . Then n X  m−1 o N cηm (α) = min cξ (α) ξ ∈ Z \ n` η` n` ∈ N ≤ cξi (α) ≤ cξi (α) ≤ c. `=1

 For all δ ∈ (0, r0 ), let 1 λδ = ( + α˜ ν )δ 2 . 2

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F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

The next Lemma, which follows Lemma 2.5, describes the concentration properties satisfied by the functions in Γηi ∩ {fα ≤ cηi (α) + λδ }, i ∈ {1, . . . , m}, when δ is small enough. ¯ there exists Lemma 2.7. There exists δ¯ ∈ (0, r0 ) such that for all δ ∈ (0, δ), Lδ > 0, for which, for all α ∈ A, if q ∈ Γηi satisfy fα (q) ≤ cηi (α) + λδ for some i ∈ {1, . . . , m} then (i) if q(t) ∈ / Bδ (ZN ) for all t ∈ [a, b] then b −√a ≤ Lδ ; (ii) if√|q(a)| = |q(b) − ηi | = δ, then |q(t)| < 2 δ for all t ≤ a and |q(t) − ηi | < 2 δ for all t ≥ b; (iii) |q(t) − η| > δ for all η ∈ ZN \ {0, ηi } and t ∈ R. Proof. Let us choose δ¯ such that: c λδ¯ < ; 4 √ 1 2αν }; δ¯ < min{r02 , 3(1 + 2˜ ν α) 4 inf

x∈B / δ¯(ZN )

V (x) = inf V (x) ≥ |x|=δ¯

ν ¯2 δ . 4

¯ one has that Then for any r ∈ (0, δ) µr =

inf

x∈B / r (ZN )

V (x) =

V (x) ≥

inf r≤|x|≤r0

νr2 . 4

√

¯ let ρδ = 2 δ and take q ∈ Γη such that fα (q) ≤ cη (α) + λδ . Given δ ∈ (0, δ), i i First of all we note that (i) follows from (2.6) and Lemma 2.6. To prove (ii) let t− ∈ R be such that |q(t− )| = δ and let   if t < t− − 1, 0 q˜(t) = linear if t− − 1 ≤ t ≤ t− ,   q(t) if t > t− . Then, q˜ ∈ Γηi and so fα (˜ q ) ≥ cηi (α). With a direct computation we obtain Z

t−

Z

t−

Lα (t, q(t), q(t)) ˙ dt = fα (q) − fα (˜ q) + −∞

Lα (t, q˜(t), q˜˙(t)) dt ≤ 2λδ .

t− −1

Assume now s < t− such that |q(s)| = ρδ .√Then, √ by contradiction that there −exists + since δ < δ < ρδ < r0 , there exists (s ,√ s ) ⊂ (s, t− ) such √ that q(t) ∈ ( δ, ρδ ) − + − + for all t ∈ (s , s ) and |q(s ) − q(s )| = δ. By (2.4), if δ < |x| < ρδ we have V (x) ≥ ν4 δ. Then, by (2.7) and the choice of δ¯ we get r Z t− Z s+ ν 1 Lα (t, q(t), q(t)) ˙ dt ≥ Lα (t, q(t), q(t)) ˙ dt ≥ α δ ≥ 3( + ν˜α)δ 2 = 3λδ . 2 2 −∞ s− a contradiction which proves that |q(t)| < ρδ for all t < t− . Analogously one shows that if |q(t+ ) − ηi | = δ then |q(t) − ηi | < ρδ for all t > t+ and so (ii) follows. To prove (iii) assume that there exists t0 ∈ R and η ∈ ZN \ {0, ηi } such that |q(t0 ) − η| ≤ δ. We define     if t < t0 , if t < t0 − 1, q(t) η q1 (t) = linear if t0 ≤ t ≤ t0 + 1, and q2 (t) = linear if t0 − 1 ≤ t ≤ t0 ,     η if t > t0 + 1, q(t) if t > t0 .

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

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Noting that q1 ∈ Γη and q2 − η ∈ Γηi −η , we obtain Z t0 Lα (t, q(t), q(t)) ˙ dt ≥ fα (q1 ) − λδ ≥ cη (α) − λδ −∞

and Z

+∞

Lα (t, q(t), q(t)) ˙ dt ≥ fα (q2 − η) − λδ ≥ cηi −η (α) − λδ . t0

Hence, since λδ < λδ¯ < 4c , by (2.8) we obtain the contradiction fα (q) ≥ cη (α) + cηi −η (α) − 2λδ ≥ cηi (α) + c − 2λδ > cηi (α) + 2λδ .  These concentration properties imply compactness of the minimizing sequences in Γηi . Lemma 2.8. Let α ∈ A. ¯ Suppose that qn ∈ Γηi , fα (qn ) → cηi (α) and that there exist τ ∈ R and r ∈ (0, δ) such that |qn (τ )| = r for all n ∈ N or |qn (τ ) − ηi | = r for all n. Then, there exists q ∈ Γηi such that, along a subsequence, qn → q weakly in 1 Hloc (R; RN ) and fα (q) = cηi (α). Proof. We consider only the case |qn (τ )| = r√for all n ∈ N, the other case being similar. We choose δ ∈ (0, r) such that ρδ ≡ 2 δ ∈ (0, r). Since fα (qn ) → cηi (α) it is not restrictive to assume that fα (q) ≤ cηi (α) + λδ for all n ∈ N. By Lemma 2.7, for any n ∈ N, we have (i) |qn (t) − η| > δ for all η ∈ ZN \ {0, ηi } and t ∈ R; (ii) there exist an ∈ (τ − Lδ , τ ) and bn ∈ (τ, τ + Lδ ) such that |qn (an )| = |qn (bn ) − ηi | = δ and |qn (t)| ≤ ρδ for all t ≤ an and |qn (t) − ηi | ≤ ρδ for all t ≥ bn . By Lemma 2.2 and since fα (qn ) → cηi (α), we have that (qn ) is bounded in 1 1 Hloc (R, RN ) and then, up to a subsequence, it converges weakly in Hloc (R, RN ) ∞ N and strongly in Lloc (R, R ) to some q ∈ E. Since fα is plainly weakly lower semi1 continuous in Hloc (R, RN ), we have fα (q) ≤ cηi (α). Moreover, q satisfies (i) |q(t) − η| ≥ δ for all η ∈ ZN \ {0, ηi } and t ∈ R; (ii) |q(t)| ≤ ρδ for all t ≤ τ − Lδ and |q(t) − ηi | ≤ ρδ for all t ≥ τ + Lδ . Then, by Lemma 2.3, q ∈ Γηi and fα (q) = cηi (q).  Let us now state some consequence of the fact that α ∈ A is a almost-periodic function. We recall that Definition 2.9. A continuous function α : R → R is an almost-periodic function if for every ε > 0, there is Tε > 0 such that every interval [a, a + Tε ] ⊂ R contains τ such that |α(t + τ ) − α(t)| < ε for all t ∈ R. If this holds one say that τ is an ε-periods of α and that the ε-periods are Tε -dense in R. See, for the more information on almost periodic functions, [3, 10]. The almost periodicity of the function α implies that the functional fα 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. Lemma 2.10. For all α ∈ A, ε > 0 there is a M0 > 0 such that for all ηi , i = 1, . . . , m, and for all [a, b] ⊂ R with b − a > M0 we can find a function q ∈ Γηi such that q(t) = 0 for all t ≤ a, q(t) = ηi for all t ≥ b, fα (q) ≤ cηi (α) + ε.

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F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

¯ be such that λδ < ε. Then, choose qi ∈ Γη , Proof. Given any ε > 0, let δ ∈ (0, δ) i i ∈ {1, . . . , m}, such that fα (qi ) ≤ cηi (α) + λδ . By Lemma 2.7 for all i there is [ai , bi ], with bi − ai < Lδ , such that |qi (ai )| = δ and |qi (bi ) − ηi | = δ. Then, letting   0 t ≤ ai − 1     linear ai − 1 ≤ t ≤ ai  q˜i (t) = qi (t) ai ≤ t ≤ bi    linear bi ≤ t ≤ bi + 1    η bi ≤ t i we obtain q˜i ∈ Γηi and fα (˜ qi ) ≤ cηi (α) + 2λδ . Let now ε1 > 0 such that |fα (qi ) − fα (qi (· − τ ))| < ε for all i provided τ is an ε1 -period of α. Let T be such that ε1 /2-periods are T -dense. Take M0 = Lδ +2T +2 and let (a, b) ⊂ R be such that b−a ≥ M0 . Then we can find τi ∈ [ai − 1 − T, ai − 1] and τ ∈ [a, a + T ], ε1 /2-periods for α. Letting q¯i (t) = q˜i (t + τi − τ ), we obtain q¯i (t) = 0 for all t ≤ a, q¯i (t) = ηi for all t ≥ b. Since τi − τ is an ε1 -period for α, we have that fα (¯ qi ) ≤ fα (˜ qi ) + ε ≤ cηi (α) + 3ε and the lemma follows.  In the next two Lemma we characterize the behavior of solutions of (Lα ) which in certain time intervals remain in the configuration space nearby the stationary points ξ ∈ ZN . These results are consequences of the (V3) assumption as discussed in Remark 2.1. ¯ such that for all δ ∈ (0, δ0 ], Lemma 2.11. Let b > a. Then there is δ0 ∈ (0, δ) α ∈ A, ζ1 , ζ2 ∈ Bδ (0) the minimization problem Z  b inf Lα (t, q(t), q(t)) ˙ dt q ∈ H 1 ((a, b); RN ), q(a) = ζ1 , q(b) = ζ2 a

is achieved at a unique function q, which is a solution of (Lα ) satisfying |q(t)| ≤ r0 for all t ∈ [a, b]. 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)| ≤ r0 for all t ∈ [a, b] we note that if δ0 is sufficiently Rb small, it is enough to remark that the “cost” (as measured by a Lα (t, q(t), q(t)) ˙ dt) of going from ζ1 to ∂Br0 (0) back to ζ2 exceed, uniformly for α ∈ A, the cost of going from ζ1 to 0 back to ζ2 . Uniqueness then follows from the convexity of Lα (t, q, q) ˙ for |q| ≤ r0 .  Lemma 2.12. Let q be a solution of (Lα ) such that |q(t)| ≤ r0 for all t ∈ [a, b]. Assume that b > a. Then, for all t ∈ [a, b], 2

(2.9)

2

√

|q(a)| − |q(b)| e− αν(b−a) −√αν(t−a) √ e |q(t)| ≤ 1 − e−2 αν(b−a) √ 2 2 |q(b)| − |q(a)| e− αν(b−a) √αν(t−b) √ e , + 1 − e−2 αν(b−a) 2

in particular 2

|q(t)| < r0

for all t ∈ (a, b) .

Proof. Take ν given by assumption (V3) and let L(p) = −¨ p + ανp,

∀p ∈ C 2 (R, R),

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

7

and 2

√

2

2

√

2

|q(a)| − |q(b)| e− αν∆ −√αν(t−a) |q(b)| − |q(a)| e− αν∆ √αν(t−b) √ √ e w(t) = e + 1 − e−2 αν∆ 1 − e−2 αν∆ where ∆ = b − a. It is immediate to check that w solves the boundary value problem   in [a, b] L(w) = 0 2 (2.10) w(a) = |q(a)|   2 w(b) = |q(b)| . Let q be a solution of (Lα ) such that |q(t)| ≤ r0 for all t ∈ [a, b]. Then, using (Lα ) and (2.2), we deduce that 2

2

2

L(|q(t)| ) = −2 |q(t)| ˙ − 2hq(t) | q¨(t)i + αν |q(t)| 2

≤ −2hq(t) | α(t)V 0 (q(t)))i + αν |q(t)| 2

≤ −2αhV 0 (q(t)) | q(t)i + αν |q(t)| 2

2

≤ −αν |q(t)| + αν |q(t)| = 0 for all t ∈ [a, b]. Then ( 2 L(w − |q| ) ≥ 0 2 w(t) − |q(t)| = 0

in (a, b) for t = a, b.

By the maximum principle we obtain 2

|q(t)| ≤ w(t)

for all t ∈ [a, b] . 

Remark 2.13. The same arguments used in the proof of Lemma 2.12 show that (2.11)

2 √αν(t−a)

2

|q(t)| ≤ |q(a)| e

for all t ≤ a

whenever q is a solution of (Lα ) such that |q(t)| ≤ r0 for all t ≤ a, and that (2.12)

2 √αν(b−t)

2

|q(t)| ≤ |q(b)| e

for all t ≥ b

whenever q is a solution of (Lα ) such that |q(t)| ≤ r0 for all t ≥ b. By periodicity, analogous estimates hold for q(t) − ηi if |q(t) − ηi | ≤ r0 for all t ∈ [a, b]. The following final two Lemmas give further properties of solutions of (Lα ) which are minimal for fα in Γηi . 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. Lemma 2.14. There exists V > 0 such that for all α ∈ A, if q ∈ Γηi satisfies fα (q) = cηi (α) some i ∈ {1, . . . , m}, then Z (2.13) V (q(t)) dt ≥ V . R

Proof. To prove (2.13), one can find [a, b] ⊂ R such that q(t) ∈ / Br0 (ZN ) for all 1 t ∈ [a, b] and such that 2 ≤ |q(a) − q(b)|. Then one simply remarks that Z b 1 ≤ |q(a) − q(b)| ≤ |q(t)| ˙ dt 2 a q Z b  12 √ √ 2 ≤ b−a |q(t)| ˙ dt ≤ b − a 2cηi (α). a

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F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

Then b − a ≥ 8cη1(α) and hence, by Lemma 2.6, i Z Z b µr0 µr V (q(t)) dt ≥ V (q(t)) dt ≥ (b − a)µr0 ≥ ≥ 0. 8c 8¯ c η R a i  √

Let δ0 be fixed as in Lemma 2.11 and denote ρ0 = 2 δ0 and L0 = Lδ0 . Then we have Lemma 2.15. For any d > 0 there exists M (d) > 0 such that for all α ∈ A, if q ∈ Γηi satisfies fα (q) = cηi (α) for some i ∈ {1, . . . , m} and |q(τ )| = δ0 or |q(τ ) − ηi | = δ0 , then Z Z V (q(t)) dt < d V (q(t)) dt. |t−τ |≥M (d)

R

Proof. We will assume |q(τ )| = δ0 , the other case can be handled in the same way. First we prove that for every L > 0, Z √ ν˜δ 2 (2.14) V (q) dt ≤ 2 √ 0 e− ανL , αν R\[τ −L,τ +L0 +L] To this aim, note that since q satisfy fα (q) = cα (ηi ), it follows from Lemma 2.7 that |q(t)| ≤ ρ0 < r0 for all t ≤ τ . Then, noting that q is a solution of (Lα ), by Remark 2.13, we obtain that |q(t)| < δ0 for all t < τ . The same argument shows that |q(t)| > δ0 for all t > τ . Therefore, again by Lemma 2.7, |q(t) − ηi | < δ0 for all t > τ + L0 . Using Lemma 2.12 (see also Remark 2.13), we have √

2

|q(t)| ≤ δ02 e

αν(t−τ ) √

2

|q(t) − ηi | ≤ δ02 e

t≤τ

αν(τ +L0 −t)

t ≥ τ + L0 .

Then, by (2.4) √

2

V (q(t)) ≤ ν˜ |q(t)| ≤ ν˜δ02 e 2

V (q(t)) ≤ ν˜ |q(t) − ηi | ≤ We deduce that Z τ −L Z V (q(t)) dt ≤ −∞

τ −L

αν(t−τ )

2

ν˜ |q(t)| ≤ ν˜δ02

−∞

and

t≤τ

√ ν˜δ02 e αν(τ +L0 −t)

Z

τ −L √

e

t ≥ τ + L0 .

αν(t−τ )

−∞

√ ν˜δ 2 dt = √ 0 e− ανL αν

+∞

√ ν˜δ 2 V (q(t)) dt ≤ √ 0 e− ανL . αν τ +L0 +L

Z

Hence (2.14) follow. ν ˜δ02 −√ανL Now, fixed any d > 0 let L > 0 be such that 2 √αν < dV . Then, by e Lemma 2.14, the lemma follows setting M (d) = L + L0 .  3. Multibump solutions In this section we will show that under an additional assumption our system exhibits a chaotic behavior. Let us fix δ0 as in Lemma 2.11. We say that α ∈ A satisfies (∗) if (∗)

there is τ ∈ R such that q ∈ Γηi and |q(τ )| = δ0 or |q(τ ) − ηi | = δ0 implies that fα (q) > cηi (α),

for some i ∈ {1, ..., m}.

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

9

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, 12], it is enough to assume that there is a τ ∈ R such that q ∈ Γηi and |q(τ )| = δ0 implies fα (q) > cηi (α). Remark 3.2. Let us remark that fα has in Γηi uncountably many minimizers (and hence (Lα ) 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. Lemma 3.3. Suppose α satisfies (∗). Then there is λ > 0, ελ > 0 such that for all T ελ -periods for α and for all q ∈ Γηi such that |q(τ + T )| = δ0 or |q(τ + T ) − ηi | = δ0 we have that fα (q) ≥ cηi (α) + λ. Proof. Suppose not. Then there is a sequence Tn of n1 -periods and functions qn ∈ Γηi such that |qn (τ + Tn )| = δ0 or |qn (τ + Tn ) − ηi | = δ0 and fα (qn ) ≤ cηi (α) + n1 . But then, setting q˜n (t) = qn (t + Tn ), we find a sequence q˜n such that |˜ qn (τ )| = δ0 or |qn (τ ) − ηi | = δ0 and fα (˜ qn ) ≤ cηi (α) + C for some constant C. n N An application of Lemma 2.8 shows that q˜n → q˜ ∈ Γηi in L∞ loc (R; R ), with fα (˜ q ) = cηi (α) and |˜ q (τ )| = δ0 or |˜ q (τ ) − ηi | = δ0 , a contradiction with assumption (∗).  We now fix an integer k. Let λ be given by Lemma 3.3, and take M0 be as in Lemma 2.10 corresponding to ε = λ4 . We choose numbers τ1 , . . . , τ2k ∈ R in such a way that τj+1 − τj > M0 1 ≤ j < 2k (3.1) τj is an ε1 -period of α for all j = 1, . . . , 2k, where ε1 is chose in such a way that fα (q(· − τj )) ≤ cηi (α) + λ8 for all q ∈ Γηi such that fα (q) = cηi (α). We will also set τ0 = −∞, τ2k+1 = +∞. Finally we set  Mk = q ∈ E q(−∞) = 0, |q(t) − `ηi | ≤ δ0 for all t ∈ [τ2` + τ, τ2`+1 + τ ] , ` = 0, . . . , k and q(+∞) = kηi . We can now state the main result of this section Theorem 3.4. Assume V satisfies assumptions (V1), (V2) and (V3). Let α ∈ A, and assume that α satisfies (∗). Then for all k ∈ N and all choices of τ1 , . . . , τ2k satisfying (3.1) there is a solution qk of (Lα ) in Mk . Proof. The proof is simply based on a minimization argument. Indeed, let us consider the minimization problem  mk = min fα (q) q ∈ Mk . It is a standard fact that a minimum q ∈ Mk exists and that is a solution of system (Lα ) for all t with the exception of the times t ∈ [τ2` + τ, τ2`+1 + τ ], for some ` ∈ {0, . . . , k}, for which |q(t) − `ηi | = δ0 . Using Lemma 2.11 and Lemma 2.12 we deduce that |q(t) − `ηi | = δ0 for some t ∈ [τ2` + τ, τ2`+1 − τ ] only if t = τ2` + τ or t = τ2`+1 + τ . We will now assume that |q(τ2`+1 + τ ) − `ηi | = δ0 or |q(τ2`+2 + τ ) − (` + 1)ηi | = δ0 , and show that we reach a contradiction with (∗).

10

F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

First of all, let us consider q¯ as in Lemma 2.10 such that q¯ ∈ Γηi , q¯(t) = 0 for all t ≤ τ2`+1 + τ , q¯(t) = ηi for all t ≥ τ2`+2 + τ and fα (¯ q ) ≤ cηi + λ4 . We also let 2`+3 ) m1 = τ + (τ2` +τ2 2`+1 ) and m2 = τ + (τ2`+2 +τ . Then we define 2  q(t) t ≤ m1      linear m1 ≤ t ≤ m1 + 1      m1 + 1 ≤ t ≤ τ2`+1 + τ `ηi q˜(t) = q¯(t) + `ηi τ2`+1 + τ ≤ t ≤ τ2`+2 + τ   (` + 1)ηi τ2`+2 + τ ≤ t ≤ m2 − 1      linear m2 − 1 ≤ t ≤ m2    q(t) t ≥ m2 Since q˜ ∈ Mk , we have that 0 ≤ fα (˜ q ) − fα (q) Z m1 +1 Z ˙ = fα (¯ q) + Lα (t, q˜, q˜) dt +

m2

Lα (t, q˜, q˜˙) dt −

m2 −1

m1

m1

m2

Lα (t, q, q) ˙ dt m1

and hence, eventually taking M0 larger, Z m2 Z m1 +1 Z Lα (t, q, q) ˙ dt ≤ f (¯ q) + Lα (t, q˜, q˜˙) dt + Define now

Z

m2

λ Lα (t, q˜, q˜˙) dt ≤ cηi (α) + . 2 m2 −1

m1

  0      linear qˆ(t) = q(t) − `ηi    linear    η i

t ≤ m1 m1 ≤ t ≤ m1 + 1 m1 + 1 ≤ t ≤ m2 − 1 m2 − 1 ≤ m2 t ≥ m2

Then qˆ ∈ Γηi and Z

m2

fα (ˆ q) ≤

Lα (t, q, q) ˙ dt + m1

λ 3 ≤ cηi (α) + λ 4 4

and |ˆ q (τ2`+1 + τ )| = |q(τ2`+1 + τ ) − `ηi | = δ0 or |ˆ q (τ2`+2 + τ )| = |ˆ q (τ2`+2 + τ ) − (` + 1)ηi | = δ0 , Then also qˇ1 (t) = qˆ(· − τ2`+1 ) and qˇ2 (t) = qˆ(· − τ2`+2 ) ∈ Γηi , |ˇ q1 (τ )| = δ0 or |ˇ q2 (τ ) − ηi | = δ0 , and, since τ2`+1 is an ε1 -period, by the choice of ε1 we obtain λ 7 ≤ cηi (α) + λ, 8 8 a contradiction with Lemma 3.3 which proves the Theorem. fα (ˇ qi ) ≤ fα (ˆ q) +



Remark 3.5. Let (τj )j∈N ⊂ R be such that any τj is an ε1 -period of α with τj+1 − τj > M0 , for all j ∈ N. For all k ∈ N, let qk ∈ C 2 (R, RN ) be the solution to (Lα ) such that qk ∈ Mk . Then, one can easily see that for any compact K ⊂ R there exists a constant C = C(K) > 0 such that kqk kL∞ (K) ≤ C,

for all k ∈ N.

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

11

Hence, since any qk is a solution to (Lα ), we obtain that for any compact K ⊂ R ˜ there exists a constant C˜ = C(K) > 0 such that ˜ k¨ qk kL∞ (K) ≤ C,

for all k ∈ N.

Using the Ascoli-Arzela Theorem, we obtain that along a subsequence, (qk )k∈N 1 converges in Cloc (R, RN ) to a solution q to (Lα ), which verify |q(t) − `ηi | ≤ δ0 for all t ∈ [τ2` + τ, τ2`+1 + τ ] , ` ∈ N. 4. On the condition (∗) In this section we discuss some situation in which condition (∗) with τ = 0 is verified by system (Lω ) 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. In the three cases the argument of the proof has the same structure which we can roughly describe as follows. Fixed any ω > 0 we denote αω (t) = α(ωt) and we argue by contradiction, assuming that, for some i, there exists q ∈ Γηi such that fαω (q) = cηi (αω ) and |q(0)| = δ0 or |q(0) − ηi | = δ0 . Then we show that we can split αω in the sum of three non negative functions αω = α1 + α2 + α3 in such a way that there exist δ > 0 and τ0 ∈ R for which the following conditions are verified Z Z (α1) [α1 (t + τ0 ) − α1 (t)]V (q(t)) dt < −δ V (q(t)) dt; Z R ZR δ V (q(t)) dt; (α2) [α2 (t + τ0 ) − α2 (t)]V (q(t)) dt < 4 R R δ (α3) kα3 k∞ < . 4 We get a contradiction simply noting that (τ0 ∗ q)(t) ≡ q(t − τ0 ) ∈ Γηi and then fαω (τ0 ∗ q) ≥ cηi (αω ) = fαω (q), while by (α1), (α2) and (α3) Z fαω (τ0 ∗ q) − fαω (q) = [α1 (t + τ0 ) − α1 (t)]V (q(t)) dt R Z + [α2 (t + τ0 ) − α2 (t)]V (q(t)) dt ZR + [α3 (t + τ0 ) − α3 (t)]V (q(t)) dt R Z δ δ ≤ (−δ + + ) V (q(t)) dt < 0. 4 4 R 4.1. A quasi-periodic case. We recall that the function α is a quasi-periodic function if α(t) = F (γ1 t, γ2 t) for some function F (x, y) periodic both in x and y and for some constants γ1 , γ2 ∈ R. Then X F (x, y) = ck1 k2 ei(k1 x+k2 y) k1 ,k2 ∈Z

and α(t) = F (γ1 t, γ2 t) =

X

ck1 k2 ei(k1 γ1 +k2 γ2 )t

k1 ,k2 ∈Z

As in the paper [6], we take γ1 = 1, γ2 = γ = α(t) = 1 +

∞ X k=1

√ 1+ 5 2

ak cos(ωk t).

and consider

12

F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

 where (ωk ) is a sequence in k1 + γk2 k1 , k2 ∈ 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 ωk we will assume that ωk = F`k − γF`k−1 ,

k ∈ N,

where ` ∈ N and Fn is the n Fibonacci number (defined by F0 = 0, F1 = 1, Fn+1 = Fn + Fn−1 for all n ≥ 1). Let us also recall that Fn =

γ n − (−1)n γ −n √ 5

so that as n → ∞ Fn − γFn−1 = (−1)n

CF −3 + O(Fn−1 ), Fn−1

CF =

1 , γ + γ −1

see [7]. As a consequence, we have that there exists `1 ∈ N (independent of k) such that |ωk | =

(4.1)

C2 CF + O(γ −3`k+3 ) ≤ `k−1 γ `k−1 γ

for all ` ≥ `1 and k ≥ 1. Theorem 4.1. Suppose α(t) = 1 +

∞ X

ak cos(ωk t)

1 A B where there exist 0 < A ≤ 1, B > 0 and β > 1 + 4B A such that β k ≤ ak ≤ β k . Then there is `0 ∈ N such that for all ` ≥ `0 and for all ω > 0 the function αω (t) = α(ωt) satisfies condition (∗) with τ = 0.

Proof. To prove the Theorem we need some preliminary estimates. Let us first note that from the assumption on ak it follows that for any j ∈ N ∞ X k=j

ak ≤ B

∞ X

β −k =

k=j

B 1 A aj−1 ≤ β −j+1 ≤ β j−1 β − 1 4 4

In particular ∞ X k=1

ak ≤

A 1 ≤ 4 4

Hence for any ` ∈ N and ω > 0, we have that 34 ≤ αω (t) ≤ 54 . We will drop the dependence on `, and denote fω (q) = fαω (q). Arguing by contradiction, we will assume that for some i there is a function qω ∈ Γηi with |qω (0)| = δ0 or |qω (0) − ηi | = δ0 such that fω (qω ) = cηi (αω ). 1 Let M1 be given by Lemma 2.15 with d = 10 and τ = 0, i.e. be such that Z Z 1 V (q(t)) dt < V (q(t)) dt. 10 |t|≥M1 R We then define k¯ ∈ N (which depends on ω and `) by  1 k¯ = min k ∈ N ωωk M1 < , 8

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

13

and split αω setting α1 (t) = ak¯ cos(ωωk¯ t) α2 (t) = 1 +

¯ k−1 X

ak cos(ωωk t)

k=1

α3 (t) =

∞ X

ak cos(ωωk t)

¯ k=k+1

We immediately have that kα3 k∞ ≤

∞ X

ak
0

Proof. Using the fact that qω is a solution, a simple integration by part shows that: Z

Z 1 cos(νt)V (qω (t)) dt = − sin(νt)hV 0 (qω (t)) | q˙ω (t)i dt ν R R Z 1 2 =− 2 cos(νt)[hV 00 (qω (t))q˙ω (t) | q˙ω (t)i − |V 0 (qω (t))| αω (t)] dt ν R 2

Since V 00 (x) is bounded and |V 0 (x)| ≤ CV (x) for some positive C, one deduces that Z ˜ Z 1 2 cos(νt)V (qω (t)) dt ≤ C ν 2 [ 2 |q˙ω (t)| + αω (t)V (qω (t))] dt R R ˜ η (αω ) C˜ Cc i = 2 fω (qω ) ≤ ν ν2 and the claim follows by Lemmas 2.14 and 2.6.

Now we remark that, thanks to (4.2), Z ak−1 ¯

Z [cos(ωωk−1 (t + τ0 )) − cos(ωωk−1 t)]V (qω (t)) dt ≤ 2ak−1 ε V (qω (t)) dt ¯ ¯ ¯ R R Z δ ≤ V (qω (t)) dt 8 R

provided ε has been chosen small enough.



CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

15

Then it follows from Lemma 4.3 that (4.3) Z [α2 (t + τ0 )−α2 (t)]V (qω (t)) dt ≤ R

≤

k−2 X k=1

Z ak [cos(ωωk (t + τ0 )) − cos(ωωk t)]V (qω (t)) dt R

Z + ak−1 [cos(ωωk−1 (t + τ0 )) − cos(ωωk−1 t)]V (qω (t)) dt R

≤

k−2 X k=1

=

C3 3ak 2 2 ω ωk

k−2 X k=1

Z

δ V (qω (t)) dt + 8 R

δ C3 3ak 2 2 + ω ωk 8

Z V (qω (t)) dt R

Z V (qω (t)) dt R

Pk−2 We want to show that the term k=1 3ak C3 ω −2 ωk−2 is small for all ω provided ` is large enough. Some care is necessary since k depends on ω and `. From the definition of k we know that ωωk M1 < 1/8 and that ωωk−1 M1 ≥ 1/8, so that ω −1 ≤ 8ωk−1 M1 and, using (4.1), k−2 X k=1

3ak

k−2 k−2 X ak C4 X C3 2 2 ≤ ak γ 2`k ≤ 3 · 64C M ω 3 1 k−1 2`(k−1) ωk2 ω 2 ωk2 γ k=1 k=1

≤

BC4 ¯ γ 2`(k−1)

¯ k−2 ¯ BC4 1−k¯ X k C5 1 − θ2−k γ 2`k = θ θ = ¯ βk β k−1 β k¯ θ − 1 k=1 k=1

k−2 X

where θ = γ 2` /β → +∞ as ` → ∞. Then, there exists `0 ≥ `1 (independent of ω ¯ such that for all ` ≥ `0 we have and k) ¯

C5 1 − θ2−k A a¯ δ < ≤ k = 8 8 β k¯ θ − 1 8β k¯ and we deduce from (4.3) that Z δZ V (qω (t)) dt. [α2 (t + τ0 ) − α2 (t)]V (qω (t)) dt ≤ 4 R R Then, also assumption (α2) is verified. As explained above, this gives the contradiction which proves Theorem 4.1.  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. Theorem 4.4. Suppose α(t) =

∞ X

ak b(ωk t)

k=1

where (B1) b ∈ C(R; R) is T -periodic,

1 2

≤ b(t) ≤ 2;

16

F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

(B2) (B3) (B4) (B5)

There isP t0 > 0 such that b is strictly decreasing in [−t0 , t0 ]; ∞ ak > 0, k=1 ak < +∞; ωk > ωk+1 > 0, ωk →0; every finite subset of ωk k ∈ N is rationally independent.

Then, for all ω > 0, αω (t) = α(ωt) satisfies (∗) with τ = 0. Proof. Let b0 =

min b(t) − max b(t) > 0. t0 t0 t t∈[ 20 ,t0 ] 4 , 4 ]

t∈[−

As in the proof of Theorem 4.1, we denote fω (q) = fαω (q). Arguing by contradiction, assume there is, for a given ω > 0, a function q ∈ Γη such that fω (q) = cη and |q(0)| = δ0 or |q(0) − η| = δ0 . Let M1 be fixed in Lemma 2.15 with d = b80 and τ = 0 and note that since b0 ≤ 23 we have Z Z b0 V (q(t)) dt < V (q(t)) dt, 8 R |t|≥M1 Z Z 1 V (q(t)) dt ≥ V (q(t)) dt. 2 R |t|≤M1 Take k¯ ≥ 1 such that ωωk¯ ≤

t0 8M1

and k˜ ≥ k¯ such that ∞ X

ak¯ b0 . 32

ak
0, αω (t) = α(ωt) satisfies (∗) with τ = 0.

18

F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

Proof. By assumption (C2), b0 =

b(t) −

min t∈[− T8 , T8 ]

max

b(t) > 0.

5T t∈[ 3T 8 , 8 ]

As in the proof of Theorem 4.1, we denote fω (q) = fαω (q) and, by contradiction, assume that there is, for a given ω > 0, a function q ∈ Γηi such that fω (q) = cηi (αω ) and |q(0)| = δ0 or |q(0) − ηi | = δ0 . ˙ ∞ and let M1 be fixed in Lemma 2.15 with d = min{ b0 ; b0 ; 3 } Let M = kbk 16 8T M 2 and τ = 0. Then, Z Z b0 b0 } V (q(t)) dt V (q(t)) dt < min{ ; 16 8T M R |t|≥M1 Z Z 1 V (q(t)) dt V (q(t)) dt ≥ 2 R |t|≤M1 Noting that ωk → 0, take k¯ ≥ 1 such that ωωk¯ ≤

T 8M1

and split αω setting α1 (t) = ak¯ b(ωωk¯ t) α2 (t) =

∞ X

ak b(ωωk t)

k=1 ¯ k6=k

α3 (t) = 0. Let τ0 =

T . 2ωωk ¯

Then Z

[α1 (t + τ0 ) − α1 (t)]V (q(t)) dt Z = ak¯ [b(ωωk¯ t + ωωk¯ τ0 ) − b(ωωk¯ t)]V (q(t)) dt ZR T = ak¯ [b( + ωωk¯ t) − b(ωωk¯ t)]V (q(t)) dt 2 ZR T = ak¯ [b( + ωωk¯ t) − b(ωωk¯ t)]V (q(t)) dt+ 2 |t|≤M1 Z T + ak¯ [b( + ωωk¯ t) − b(ωωk¯ t)]V (q(t)) dt 2 |t|≥M1

I1 =

R

Since for all |t| ≤ M1 we have that |ωωk¯ t| ≤ |ωωk¯ M1 | ≤ T8 , by the choice of M1 we obtain Z Z I1 ≤ −ak¯ b0 V (q(t)) dt + 4ak¯ V (q(t)) dt |t|≤M1 |t|≥M1 Z Z Z a ¯ b0 a ¯ b0 a ¯ b0 ≤− k V (q(t)) dt + k V (q(t)) dt = − k V (q(t)) dt 2 4 4 R R R so that (α1) holds with δ = ak¯ b40 .

CHAOTIC BEHAVIOR OF RAPIDLY OSCILLATING LAGRANGIAN SYSTEMS

19

Moreover, we have that Z I2 = [α2 (t + τ0 ) − α2 (t)]V (q(t)) dt =

R ¯ k−1 X

Z [b(ωωk τ0 + ωωk t) − b(ωωk t)]V (q(t)) dt+

ak R

k=1

∞ X

+

Z [b(ωωk τ0 + ωωk t) − b(ωωk t)]V (q(t)) dt.

ak R

¯ k=k+1

From assumption (C4) it follows that for all k = 1, . . . , k¯ − 1 there is an integer nk ωk such that 2nk ωk¯ = ωk , so that ωωk τ0 = T 2ω = T nk and ¯ k

¯ k−1 X

Z [b(ωωk τ0 + ωωk t) − b(ωωk t)]V (q(t)) dt = 0

ak

k=1

R

Moreover for all k > k¯ and for all |t| ≤ M1 we have ωωk |t| ≤ ωωk¯ M1 ≤ T8 and ωk T T = 2n for a certain n ∈ N. Then from assumption (C2) we also ωωk τ0 = 2ω ¯ k deduce that for all k > k¯ and for all |t| ≤ M1 b(ωωk t + ωωk τ0 ) − b(ωωk t) ≤ max b(ωωk t + ωωk τ0 ) − min b(ωωk t) |t|≤M1

|t|≤M1

= b(−ωωk M1 + ωωk τ0 ) − b(ωωk M1 ) ≤ 0 from which I2 ≤

≤

=

∞ X ¯ k=k+1 ∞ X

Z [b(ωωk τ0 + ωωk t) − b(ωωk t)]V (q(t)) dt

ak |t|≥M1

Z M ak ωωk τ0

¯ k=k+1 ∞  X

V (q(t)) dt |t|≥M1

ak

¯ k=k+1

ωk  T M ωk¯ 2

Z V (q(t)) dt. |t|≥M1

Finally remark that, by assumption (C3) ∞ X ¯ k=k+1

ak

∞ X ωk 1 ≤ ak¯ ≤ ak¯ k− ωk¯ 2 k¯ ¯ k=k+1

so that, by the choice of M1 , Z Z Z TM ak¯ b0 δ I2 ≤ ak¯ V (q(t)) dt ≤ V (q(t)) dt = V (q(t)) dt 2 |t|≥M1 16 R 4 R and the theorem follows.

 References

[1] F. Alessio, M. L. Bertotti, and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys. 50 (1999), no. 6, 860–891. [2] F. Alessio, P. Caldiroli, and P. Montecchiari, Genericity of the multibump dynamics for almost periodic Duffing-like systems, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 5, 885–901. [3] A. S. Besicovitch, Almost periodic functions, Dover Publications Inc., New York, 1955. [4] S. V. Bolotin and V. Kozlov, Libration in systems with many degrees of freedom, J. Appl. Math. Mech. 42 (1978), 256–261.

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F. ALESSIO, V. COTI ZELATI, AND P. MONTECCHIARI

[5] V. Coti Zelati and P. H. Rabinowitz, Multichain-type solutions for Hamiltonian systems, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999) (San Marcos, TX), Southwest Texas State Univ., 2000, pp. 223–235 (electronic). [6] A. Delshams, V. Gelfreich, A. Jorba, and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys. 189 (1997), no. 1, 35–71. [7] , Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 1–10 (electronic). [8] A. Delshams, A. Jorba, T. M. Seara, and V. Gelfreich, Splitting of separatrices for (fast) quasiperiodic forcing, Hamiltonian systems with three or more degrees of freedom (S’Agar´ o, 1995), Kluwer Acad. Publ., Dordrecht, 1999, pp. 367–371. [9] A. Delshams and T. M. Seara, An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys. 150 (1992), no. 3, 433–463. [10] A. M. Fink, Almost periodic differential equations, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 377. [11] P. Holmes, J. Marsden, and J. Scheurle, Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations, Hamiltonian dynamical systems (Boulder, CO, 1987), Amer. Math. Soc., Providence, RI, 1988, pp. 213–244. [12] P. H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations 5 (1997), no. 2, 159–182. [13] E. S´ er´ e, Looking for the Bernoulli shift, Ann. Inst. H. Poincar´ e. Anal. Non Lin´ eaire 10 (1993), no. 5, 561–590. ` di Torino (Alessio) Dipartimento di Matematica, Universita E-mail address: [email protected] ` (Coti Zelati) Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universita di Napoli E-mail address: [email protected] ` di Ancona (Montecchiari) Dipartimento di Matematica, Universita E-mail address: [email protected]