Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations
aa r X i v : . [ m a t h . D S ] S e p D R A F T Negligibility of small divisor effects in the normal formtheory for nearly-integrable Hamiltonians withdecaying non-autonomous perturbations ∗ Alessandro Fortunati ( a ) Stephen Wiggins ( b ) School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
Abstract
The paper deals with the problem of the existence of a normal form for a nearly-integrablereal-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly)in time. In particular, in the case of an isochronous integrable part, the system can be castin an exact normal form, regardless of the properties of the frequency vector. The generalcase is treated by a suitable adaptation of the finite order normalization techniques usuallyused for Nekhoroshev arguments. The key point is that the so called “geometric part” is notnecessary in this case. As a consequence, no hypotheses on the integrable part are required,apart from analyticity.The work, based on two different perturbative approaches developed by A.Giorgilli et al., is ageneralisation of the techniques used by the same authors to treat more specific aperiodicallytime-dependent problems.
Keywords:
Non-autonomous Hamiltonian systems, Aperiodic time dependence.
Primary: 70H08. Secondary: 37J40, 37J25.
The problem of casting an analytic nearly-integrable Hamiltonian system into normal form isdeeply related to Poincaré’s challenging problème général de la dynamique [Poi92]. Nowadays,normal forms are still one of the main technical tools used to deal with the issue raised byPoincaré in this context.The particular case in which the unperturbed part is supposed to be linear in the actions(isochronous case), already investigated by Birkhoff (and for this reason also known as the
Birkhoff problem ) [Bir27], has a peculiar interest. The first rigorous statement concerning itsstability can be found in [Gal86]. The possibility to cast the considered Hamiltonian in normalform, up to some finite order r and to obtain, as a consequence, a stability time estimate “à laNekhoroshev”, is directly related to a particularly simple small-divisors analysis: the non-resonant ∗ This research was supported by ONR Grant No. N00014-01-1-0769 and MINECO: ICMAT Severo Ochoaproject SEV-2011-0087. ( a ) E-mail: [email protected] ( b ) E-mail: [email protected] It is easy to see that any attempt to consider the limit r → ∞ would imply the degeneration into a trivialproblem, (i.e. in which the allowed perturbation size reduces to zero, see also [GG85, formula (46), Pag. 105]). R A F T (Diophantine) hypothesis on the frequency vector ω of the unperturbed system is sufficient inorder to ensure the resolvability of the (standard) homological equation arising in the normal-ization algorithm. An extensive bibliography on this problem goes beyond the purposes of thispaper, we only mention the recent generalisations for the Planetary problem of [Pin13] and of[Bam05] for infinite dimensional systems.It is well known that the extension to the non-isochronous case requires a careful analysis (ge-ometric part, see [Nek77], [Nek79] and [BG86]) on the regions of the phase space in which theactions I are such that ω = ω ( I ) is non-resonant (non-resonant domains).The problem of dealing with time-dependent perturbations without any hypothesis on the timedependence (e.g. periodic or quasi-periodic) has peculiar technical difficulties. After the pioneer-ing works of [Pus74] and [GZ92], the interest for this class of problems has been recently renewedin [Bou13], [FW14a] and subsequent papers. Basically, the novelty consists in the treatment ofthe time-dependent homological equation. A first approach consists in keeping the terms involv-ing the time derivative of the generating function (also called extra-terms) in the normal formand then providing a bound for them. This approach, originally suggested in [GZ92] then usedin [FW14a], yields a normal form result for the case a of slow time dependence. This hypothesisprovides a smallness condition for the mentioned extra-terms. Alternatively, those terms can beremoved by including them into the homological equation, which turns out to be, in this way,a linear ODE in time. This has been profitably used in [FW14b], [FW15a] and in [FW15b] butrequires (except for a particular case described in [FW15b]) an important assumption. Moreprecisely, it is necessary to suppose that the perturbation, as a function of t , belongs to the classof summable functions over the real semi-axis . As in (3), those functions exhibiting a (slow)exponential decay will be used as a paradigmatic case. It will be shown that the consequencesof this assumption in the isochronous case are remarkable: the normalization algorithm can beiterated an infinite number of times by means of a superconvergent method borrowed from KAMtype arguments, see e.g. [Chi09]. The procedure leads to the so-called strong normal form i.e.in which the normalized Hamiltonian has the same form of the integrable part of the initialproblem. Furthermore, no restrictions are imposed on ω , hence flows with arbitrary frequenciespersist in the transformed system.As it would be likely to expect, this phenomenon has an important consequence also in thenon-isochronous case. The possibility to disregard the problems related to the small divisorsimplies that the well known geography of the resonances analysis, a key step of the Nekhoroshevtheorem, is not necessary in this case and the results that can be stated are purely “analytic”.In such a way, the classical assumptions on the unperturbed part of the Hamiltonian (such assteepness, convexity etc.), are no longer required. As a common feature with the isochronouscase, the obtained normal form does not exhibit resonant terms , as these have been annihilatedin the normalization by using the time-dependent homological equation. This implies that, inthis case, the plane of fast drift (see e.g. [Gio03]) degenerates to a point.The paper uses in a concise but self-contained form, the tools developed in the above mentionedpapers of the same authors, especially of [FW15b] in which the concept of “family” of canonicaltransformations parametrised by t is introduced. The proofs are entirely constructed by usingthe language and the tools of the Lie series and Lie transform methods developed by Giorgilli etal., see e.g. [Gio03]. We stress that this hypothesis is usually not satisfied in the case of periodic or quasi-periodic time dependence. R A F T Consider the following nearly integrable Hamiltonian H ( I, ϕ, η, t ) = h ( I ) + η + ˆ εf ( I, ϕ, t ) , (1)with ( I, ϕ, η, t ) ∈ G × T n × R × R + , where G ⊂ R n and ˆ ε > is a small parameter, which isthe “autonomous equivalent” in the extended phase space of Hamiltonian H ( I, ϕ, t ) = h ( I ) +ˆ εf ( I, ϕ, t ) .We define, for all t ∈ R + := [0 , + ∞ ) , the following complexified domain D ρ,σ := G ρ × T nσ × S ρ ,where G ρ := S I ∈ G ∆ ρ ( I ) and ∆ ρ ( I ) := { ˆ I ∈ C n : | ˆ I − I | ≤ ρ } , T nσ := { ϕ ∈ C n : |ℑ ϕ | ≤ σ } , S ρ := { η ∈ C : |ℑ η | ≤ ρ } ,with ρ, σ ∈ (0 , . For all g : G ρ × T nσ × R + → C , write g = P k ∈ Z n g k ( I, t ) e ik · ϕ , then define the Fourier norm (parametrized by t ) k g k ρ,σ := X k ∈ Z n | g k ( I, t ) | ρ e | k | σ , (2)with |·| ρ is the usual supremum norm over G ρ and | k | := P nl =1 | k l | . For all w : G ρ × T nσ × R + → C n we shall set k w k ρ,σ := P nl =1 k w l k ρ,σ The standard framework (see eg. [BGGS84]) is the space C ρ,σ , of continuous functions on G ρ × T nσ , holomorphic in its interior for some ρ, σ and real on G × T n for all t ∈ R + . We shall suppose h ( I ) ∈ C ρ, · and f ∈ C ρ,σ while it is sufficient to assumethat, for all I ∈ G ρ , f k ( I, · ) ∈ C ( R + ) .Similarly to [FW15b], we introduce the following Hypothesis 2.1 (Time decay) . There exists M f > and a ∈ (0 , k f ( I, ϕ, t ) k ρ,σ ≤ M f e − at . (3)Set ε := ˆ εM f . We firstly state the following Theorem 2.2 (Strong aperiodic Birkhoff) . Consider Hamiltonian (1) with h ( I ) := ω · I , underthe Hypothesis 2.1 and the described regularity assumptions. Then, for all a ∈ (0 , there exists ε a > such that the following statement holds true. For all ε ∈ (0 , ε a ] , it is possible to find < ρ ∗ < ρ < ρ and < σ ∗ < σ < σ and an analytic, canonical, ε − close and asymptotic to theidentity change of variables ( I, ϕ, η ) = B ( I ( ∞ ) , ϕ ( ∞ ) , η ( ∞ ) ) , B : D ρ ∗ ,σ ∗ → D ρ ,σ for all t ∈ R + ,casting Hamiltonian (1) into the strong Birkhoff normal form H ( ∞ ) ( I ( ∞ ) , ϕ ( ∞ ) , η ( ∞ ) ) = ω · I ( ∞ ) + η ( ∞ ) . (4)Hence, in the new variables, the flow with frequency ω persists for all ω , regardless of thenumerical features of this vector, i.e. more specifically, no matter if it is resonant or not. Theabsence of a non-resonance hypothesis on ω implies also that (4) holds also if ω has an arbitrarynumber of zero components, implying the persistence of any lower dimensional torus .With a straightforward adaptation of the notational setting, the result in the general case statesas follows: In particular, if g ∈ C ρ,σ then | g k | ρ ≤ k g k ρ,σ exp( −| k | σ ) for all t ∈ R + . R A F T Theorem 2.3.
There exist ε ∗ a > and r ∈ N \ { } such that, for all ε ∈ (0 , ε ∗ a ] it is possible tofind an analytic, canonical, ε − close and asymptotic to the identity change of variables ( I, ϕ, η ) = N r ( I ( r ) , ϕ ( r ) , η ( r ) ) , N r : D ˜ ρ ∗ , ˜ σ ∗ → D ˜ ρ , ˜ σ for all t ∈ R + , casting Hamiltonian (1) under theHypothesis 2.1, into the normal form of order rH ( r ) ( I ( r ) , ϕ ( r ) , η ( r ) , t ) = h ( I ( r ) ) + η ( r ) + R ( r +1) ( I ( r ) , ϕ ( r ) , t ) , (5) where R ( r +1) is “exponentially small” with respect to r and vanishes for t → + ∞ . Moreover,for all I (0) ∈ G one has in (1): | I ( t ) − I (0) | ≤ √ ε ˜ ρ / for all t ∈ R + . Similarly to [FW15b] (and the mentioned previous papers), no lower bounds are imposed on a so that the decay can be arbitrary slow. The (natural) consequence is that either ε a or ε ∗ a decrease with a , see (15) and (56). Part I
Proof of Theorem 2.2
Given a function G := G ( I, ϕ, t ) , define the Lie series operator exp( L G ) := Id + P s ≥ (1 /s !) L sG ,where L G F := { F, G } ≡ F ϕ · G I − G ϕ · F I − F η G t . The aim is to construct a generating sequence { χ ( j ) } j ∈ N , such that the formal limit B := lim j →∞ B ( j ) ◦ B ( j − ◦ . . . ◦ B (0) , (6)where B ( j ) := exp( L χ ( j ) ) is such that B ◦ H is of the form (4). The following statement showsthat this is possible, at least at a formal level Proposition 3.1.
Suppose that for some j ∈ N Hamiltonian (1) is of the form H ( j ) = ω · I + η + F ( j ) ( I, ϕ, t ) . (7) Then H ( j +1) := B ( j ) ◦ H ( j ) is still of the form (7) with F ( j +1) = X s ≥ s ( s + 1)! L sχ ( j ) F ( j ) , (8) provided that χ ( j ) solves the homological equation χ ( j ) t + ω · χ ( j ) ϕ = F ( j ) . (9)Since Hamiltonian (1) is of the form (7), one can set H (0) := H with F (0) := ˆ εf . Thus, byinduction, the form (7) holds for all j ∈ N . Clearly, this does not guarantee that the objectsinvolved in the algorithm are meaningful for all j , as it is well known their sizes can growunboundedly as j increases, as a consequence of small divisors phenomena. The aim of Section4 (and in particular of Lemma 4.5) is to show that this is not the case: the key ingredient is thetime decay of f . See bound (54). R A F T Proof.
We get exp( L χ ( j ) ) H ( j ) = I · ω + η + F ( j ) ( I, ϕ, t ) + L χ ( j ) ( ω · I + η ) + P s ≥ (1 /s !) L sχ ( j ) F ( j ) + P s ≥ (1 /s !) L sχ ( j ) ( ω · I + η ) . The sum between the third and fourth terms of the r.h.s. of thelatter equation vanishes due to (9). As for the last two terms, by setting F ( j +1) as the sum ofthem, one gets F ( j +1) = P s ≥ (1 /s !) L χ ( j ) [ F ( j ) + ( s + 1) − L χ ( j ) ( ω · I + η )] , which immediatelyyields (8) by using (9).The (formal) expansions χ ( j ) = P k ∈ Z n c ( j ) k ( I, t ) e ik · ϕ and F ( j ) = P k ∈ Z n f ( j ) k ( I, t ) e ik · ϕ yield(9) in terms of Fourier components ∂ t c ( j ) k ( I, t ) + iλ ( k ) c ( j ) k ( I, t ) = f ( j ) k ( I, t ) , (10)with λ ( k ) := ω · k . The solution of (10) is c ( j ) k ( I, t ) = e − iλ ( k ) t (cid:20) c ( j ) k ( I,
0) + Z t e iλ ( k ) s f ( j ) k ( I, s ) ds (cid:21) , (11)where c ( j ) k ( I, will be chosen later. The classical argument requires the construction of a sequence of nested domains D ρ j +1 ,σ j +1 ⊂D ρ j ,σ j ∋ ( I ( j ) , ϕ ( j ) , η ( j ) ) , such that B j : D j +1 → D j . The resulting progressive restriction isessential in order to use standard Cauchy tools, see Prop. 4.1. The estimates found in Lemma4.2, concerning the solution of equation (9), will be used to prove Lemma 4.5, providing in thisway the bound on F ( j ) defined in Prop. 3.1. This is achieved for a suitable sequence of domainsprepared in Lemma 4.4 via { ρ j } and { σ j } . This allows us to conclude that the perturbationterm is actually removed in the limit (6).The final step consists of showing that B defines an analytic map B : D ρ ∗ ,σ ∗ ∋ ( I ( ∞ ) , ϕ ( ∞ ) , η ( ∞ ) ) →D ρ ,σ ∋ ( I (0) , ϕ (0) , η (0) ) ≡ ( I, ϕ, η ) , where ρ ∗ ≤ ρ j and σ ∗ ≤ σ j for all j ∈ N . This property isshown in Lemma 4.6. As D ρ ∗ ,σ ∗ will be the domain of analyticity of the transformed Hamiltonianvia B , it will be essential to require that ρ ∗ , σ ∗ > . Proposition 4.1.
Let
F, G : G ρ × T nσ × R + → C such that k F k (1 − d ′ )( ρ,σ ) and k G k (1 − d ′′ )( ρ,σ ) arebounded for some d ′ , d ′′ ∈ [0 , . Then, defining δ := | d ′ − d ′′ | and ˆ d := max { d ′ , d ′′ } , for all ˜ d ∈ (0 , − ˆ d ) one has for all s ∈ N \ { }kL sG F k (1 − ˜ d − ˆ d )( ρ,σ ) ≤ s ! e (cid:18) e ˜ d ( ˜ d + ˜ δ s ) ρσ k G k (1 − d ′′ )( ρ,σ ) (cid:19) s k F k (1 − d ′ )( ρ,σ ) , (12) where ˜ δ s = δ if s = 1 and is zero otherwise.Proof. Straightforward from [Gio03, Lemmas 4.1, 4.2].
Lemma 4.2.
Suppose that F ( j ) satisfies (cid:13)(cid:13) F ( j ) (cid:13)(cid:13) [ˆ σ, ˆ ρ ] ≤ M ( j ) exp( − at ) for some M ( j ) > , ˆ ρ ≤ ρ and ˆ σ ≤ σ . Define C ω := 1 + | ω | , then for all δ ∈ (0 , the solution of (9) satisfies (cid:13)(cid:13)(cid:13) χ ( j ) (cid:13)(cid:13)(cid:13) (1 − δ )(ˆ ρ, ˆ σ ) ≤ M ( j ) a (cid:16) eδ ˆ σ (cid:17) n e − at , (cid:13)(cid:13)(cid:13) χ ( j ) t (cid:13)(cid:13)(cid:13) (1 − δ )(ˆ ρ, ˆ σ ) ≤ C ω M ( j ) a (cid:16) eδ ˆ σ (cid:17) n e − at . (13)5 R A F T Proof.
First of all, by hypothesis | f ( j ) k ( I, t ) | ≤ M ( j ) exp( −| k | ˆ σ − at ) , in particular, by choosing c ( j ) k ( I,
0) := − R R + exp( iλ ( k ) s ) f ( j ) k ( I, s ) ds we have that | c ( j ) k ( I, | < + ∞ for all I ∈ G ρ . Substi-tuting c ( j ) k ( I, in (11) one gets | c ( j ) k ( I, t ) | ≤ R ∞ t | f ( j ) k ( I, s ) | ds ≤ ( M ( j ) /a ) exp( −| k | ˆ σ − at ) whichyields the first of (13). As for the second of (13), it is sufficient to use (10), which implies, | ∂ t c ( j ) k ( I, t ) | ≤ ( M ( j ) /a )(1 + | ω || k | ) exp( −| k | ˆ σ − at ) then proceed similarly. Remark 4.3.
It is immediate to notice that a hypothesis of non-resonance on ω does notsubstantially improve the bounds (13). A more careful computation yields | c ( j ) k ( I, t ) | ≤ M ( j ) ( a + ( ω · k ) ) − e −| k | σ j − at ,Hence the estimate cannot be refined due to the presence of | c ( j )0 ( I, t ) | , no matter what theminimum value of ( ω · k ) is. Lemma 4.4.
Let { d j } j ∈ N be a (real valued) sequence such that ≤ d j ≤ / . Consider, for all j ∈ N , the following sequences ǫ j +1 := Ka − d − τj ǫ j , ( ρ j +1 , σ j +1 ) := (1 − d j )( ρ j , σ j ) , (14) with K > and τ := 2 n + 3 . Then, for all < ρ ≤ ρ , < σ ≤ σ and ǫ ≤ ε a where ε a ≤ aK − (2 π ) − τ , (15) it is possible to construct { d j } j ∈ N such that ( ρ ∗ , σ ∗ ) = (1 / ρ , σ ) , in particular they are strictlypositive. Furthermore lim j →∞ ǫ j = 0 .Proof. Choose ǫ j := ǫ ( j + 1) − τ (so that lim j →∞ ǫ j = 0 by construction). By the first of (14)one gets d j = ( ǫ Ka − ) τ ( j + 2) / ( j + 1) , (16)hence, by (15), d j ≤ π − ( j + 1) − . This implies P j ≥ d j ≤ / and then, trivially, d j ≤ / forall j ∈ N . Now we have ln Π j ≥ (1 − d j ) = P j ≥ ln(1 − d j ) ≥ − P j ≥ d j = − ln 2 , hence lim j →∞ ρ j = ρ Π j ≥ (1 − d j ) ≥ ρ / ρ ∗ . Analogously σ ∗ := σ / . Lemma 4.5.
There exists K = K ( ρ , σ ) > such that, if ε ≤ ε a where ε a satisfies (15), then (cid:13)(cid:13)(cid:13) F ( j ) (cid:13)(cid:13)(cid:13) ( ρ j ,σ j ) ≤ ǫ j e − at , (17) for all j ∈ N . Hence, the transformed Hamiltonian B ◦ H is in the form (4). Recall (2), then use the inequality P k ∈ Z n exp( − δ | k | ˆ σ ) ≤ ( eδ − ˆ σ − ) n . Its variant P k ∈ Z n (1 + | ω || k | ) exp( − δ | k | ˆ σ ) ≤ C ω ( eδ − ˆ σ − ) n is used to obtain the second of (13). Use the inequality ln(1 − x ) ≥ − x ln 2 , valid for all x ∈ [0 , / . R A F T Proof.
By induction. Note that (17) is true for j = 0 setting ǫ := ε . The condition on ε ensuresthe validity of Lemma 4.4. Hence, supposing (17), by Lemma 4.2 and Lemma 4.4, we get (cid:13)(cid:13)(cid:13) χ ( j ) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ ǫ j ( e/σ ∗ ) n a − d − nj e − at . (18)By (8) and Prop. 4.1 with d ′ = d j , d ′′ = 0 and ˜ d = d j (the condition d j ≤ − d j holds as d j ≤ / ) (cid:13)(cid:13)(cid:13) F ( j +1) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ X s ≥ s ! (cid:13)(cid:13)(cid:13) L sχ ( j ) F ( j ) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ − Θ (cid:13)(cid:13)(cid:13) F ( j ) (cid:13)(cid:13)(cid:13) ( ρ j ,σ j ) , (19)where Θ := 2 ǫ j nC ω ( e/σ ∗ ) τ ρ − ∗ a − d − n − j e − at ≤ / (20)is a sufficient condition for the convergence of the operator exp( L χ ( j ) ) , from which P s ≥ Θ s ≤ .Hence, by (19), (20), then by (18) one gets (use also σ ∗ , ρ ∗ , d j < ) (cid:13)(cid:13)(cid:13) F ( j +1) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ ǫ j nC ω ( e/σ ∗ ) τ ρ − ∗ a − d − τj e − at . (21)The latter is valid a fortiori in D (1 − d j )( ρ j ,σ j ) .In conclusion, by choosing K := nC ω ( e/σ ∗ ) τ ρ − ∗ = 2 τ +1 nC ω ( e/σ ) τ ρ − , from the first of (14),we have that (17) is satisfied for j → j + 1 . Furthermore, by the first of (14), condition (20)yields ≥ ǫ j Kd j a − d − τj e − at = 4 d j ( ǫ j +1 /ǫ j ) e − at . The latter is trivially true for all t ∈ R + bythe monotonicity of ǫ j and as d j ≤ / . Furthermore this implies Θ ≤ d j e − at . (22)Hence exp( L χ ( j ) ) is well defined for all j ∈ N .In this way the value of ε a mentioned in the statement of Theorem 2.2 is determined onceand for all. Lemma 4.6.
The limit (6) exists, it is ε − close to the identity and satisfies | I ( ∞ ) − I | , | η ( ∞ ) − η | ≤ ( ρ / e − at , | ϕ ( ∞ ) − ϕ | ≤ ( σ / e − at , (23) in particular it defines an analytic map B : D ρ ∗ ,σ ∗ → D ρ ,σ and H ( ∞ ) is an analytic function on D ρ ∗ ,σ ∗ for all t ∈ R + .Proof. Let us start with I . Note that (cid:13)(cid:13)(cid:13) L χ ( j ) I ( j +1) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ n ( ed j ρ j ) − (cid:13)(cid:13) χ ( j ) (cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) by a Cauchy estimate (see [Gio03, Lemma 4.1]), so that the presence of n in (20) is justified.Hence use Prop. 4.1 with F ← L χ ( j ) I ( j +1) , s ← s − , obtaining (cid:13)(cid:13)(cid:13) L sχ ( j ) ϕ ( j +1) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ e − s !Θ s ρ . This implies | I ( j +1) − I ( j ) | ≤ e − X s ≥ (1 /s !) (cid:13)(cid:13)(cid:13) L sχ ( j ) I ( j +1) (cid:13)(cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) ≤ − Θ ρ ≤ d j ρ e − at , The reason for using nC ω in the definition of Θ will be clear in the proof of Lemma 4.6. R A F T by (22). In particular | I ( j +1) − I ( j ) | is ε − close to the identity by (16) for all j ∈ N , hence | I ( ∞ ) − I | ≤ P j ≥ | I ( j +1) − I ( j ) | is. It is now sufficient to recall P j ≥ d j ≤ / in order toconclude.The argument for ϕ is analogous while the variable η requires a slight modification. In particular,as one needs to set F ← L χ ( j ) η = − χ ( j ) t , the use of the second of (13) requires the contributionof C ω in (20).In conclusion, the obtained composition of analytic maps is uniformly convergent in any compactsubset of D ρ ∗ ,σ ∗ . This implies that B is analytic on D ρ ∗ ,σ ∗ by the Weierstraß Theorem and hencethe image of H via B is an analytic function in the same domain. In this section we consider two alternative examples of perturbation. The main purpose is toshow that the hypothesis of summability in time over the semi-axis is the only key requirementfor the argument beyond the proof of Theorem 2.2.In particular, we shall firstly consider a decay which is assumed to be quadratic in time, while inthe second example a perturbation exhibiting a finite number of (differentiable) bumps is exam-ined. The procedure is fully similar, with the exception of some bounds that will be explicitlygiven below.
Let us suppose that (3) is modified as k f ( I, ϕ, t ) k ρ,σ ≤ M f ( t + 1) − .In the same framework, it is immediate to show that the analogous of Lemma 4.2 yields thefollowing estimates (cid:13)(cid:13)(cid:13) χ ( j ) (cid:13)(cid:13)(cid:13) (1 − δ )(ˆ ρ, ˆ σ ) ≤ M ( j ) ( eδ − ˆ σ − ) n ( t + 1) − , (cid:13)(cid:13)(cid:13) χ ( j ) t (cid:13)(cid:13)(cid:13) (1 − δ )(ˆ ρ, ˆ σ ) ≤ M ( j ) C ω ( eδ − ˆ σ − ) n ( t + 1) − .Clearly, in this case, the integration has led to a “loss of a power” in the decay. This is harmlessas, by (19), (cid:13)(cid:13) F ( j +1) (cid:13)(cid:13) (1 − d j )( ρ j ,σ j ) = O ( F j ) O ( χ ( j ) )+ h.o.t. and then F ( j +1) ∼ ( t +1) − ≤ ( t +1) − so that the scheme can be iterated .The rest of the proof is analogous provided that the term e − at is replaced with in the remainingestimates. Let L ∈ N \ { } and h > . Consider an increasing sequence { t l } l =1 ,...,L ∈ R + such that t l +1 − t l > h , then the following function ξ l ( t ) := (cid:26) ( a l /h )[( t − t l + h )( t − t l − h )] t ∈ [ t l − h, t l + h ]0 otherwise A similar (and even stronger) phenomenon could have been noticed in the original setting. Namely, supposeby induction that (cid:13)(cid:13)(cid:13) F ( j ) (cid:13)(cid:13)(cid:13) ( ρ j ,σ j ) ≤ ǫ j exp( − a j t ) . By Lemma 4.2 and (19), one finds that (cid:13)(cid:13)(cid:13) F ( j +1) (cid:13)(cid:13)(cid:13) ( ρ j +1 ,σ j +1 ) ≤ ǫ j +1 exp( − a j t ) and so on. This leads to a remarkable rate of decay ( a j = 2 j a ) but not to a substantial improve-ment of the estimates and of the threshold (15) of ε a , as these are uniform in j . R A F T where a l ∈ R . Considering a function ˜ f ( I, ϕ ) ∈ C ρ,σ , we set as f ( I, ϕ, t ) := ˜ f ( I, ϕ ) L X l =1 ξ l ( t ) .In such case we find (cid:13)(cid:13)(cid:13) χ ( j ) (cid:13)(cid:13)(cid:13) (1 − δ )(ˆ ρ, ˆ σ ) ≤ AM ( j ) h ( eδ − ˆ σ − ) n , (cid:13)(cid:13)(cid:13) χ ( j ) t (cid:13)(cid:13)(cid:13) (1 − δ )(ˆ ρ, ˆ σ ) ≤ M ( j ) C ω ( eδ − ˆ σ − ) n ,with A := P Ll =1 | a l | . The remaining part of the proof is straightforward with the obviousmodifications. In particular, as for the proof of Lemma 4.5, one finds K = 2 nC ω ( e/σ ∗ ) τ hAρ − ∗ . Part II
Proof of Theorem 2.3
In order to simplify the notation, we shall use ( ρ H , σ H ) in place of ( ρ, σ ) and ( ρ, σ ) in place of (˜ ρ , ˜ σ ) from now on. As in [Gio03], we write Hamiltonian (1) in the form H ( I, ϕ, η, t ) = H ( I, η ) + H ( I, ϕ, t ) + H ( I, ϕ, t ) + . . . where H ( I, η ) := h ( I ) + η, H s ( I, ϕ, t ) := X k ∈ Λ s f k ( I, t ) e ik · ϕ ,where Λ s := { k ∈ Z n : ( s − N ≤ | k | < sN } and N ∈ N \ { } is meant to be determined.Given a sequence of functions { χ ( s ) } s ≥ : C ρ,σ → C , the Lie transform operator is defined as T χ := X s ≥ E s , E s := Id s = 01 s s X j =1 j L χ ( j ) E s − j s ≥ . (24)Let r ∈ N \ { } to be determined. A finite generating sequence of order r , denoted with χ [ r ] , issuch that χ ( s ) ≡ for all s > r . Our aim is to determine it in such a way the effect of H , . . . , H r is removed, i.e. H ( r ) := T χ [ r ] H = H + R ( r +1) ( I, ϕ, t ) , (25)where the remainder R ( r +1) contains H >r and a moltitude of terms produced during the nor-malization, which Fourier harmonics lie on Λ >r . The smallness of the remainder is an immediateconsequence of the decay property of the coefficients of an analytic function. The procedure isstandard: condition (25), with the use of (24), yields a well known diagram which s − th level isof the form E s := E s H + s − X l =1 E s − l H l + H s = 0 , (26) Namely, those terms of the diagram which Fourier harmonics belong to Λ s . R A F T if s = 2 , . . . , r and E H + H = 0 if s = 1 . As sum of all the “non-normalised” levels, theremainder easily reads as R ( r +1) = X s>r E s . (27)By writing the first term of (26) in the form E s = L χ ( s ) + P s − j =1 ( j/s ) L χ ( j ) E s − j and using themanipulation described in [Gio03, Chapter 5], one obtains a remarkable cancellation of thecontribution of H . In this way, the generating sequence is determined as a solution of L H χ ( s ) = Ψ s , Ψ s := H s = 1 H s + s − X j =1 js E s − j H j s ≥ . (28)A formal expansion of χ ( j ) and of Ψ s := P k ∈ Z n ψ ( s ) k ( I, t ) e ik · ϕ yields for all s = 1 , . . . , r∂ t c ( s ) k ( I, t ) + i ( ω ( I ) · k ) c ( s ) k ( I, t ) = ψ ( s ) k ( I, t ) , k ∈ Λ s , (29)where, as usual, ω ( I ) := ∂ I h ( I ) . Remark 6.1.
As a substantial difference with the isochronous case, the function ω ( I ) is acomplex valued vector as I ∈ G ρ . In this way the exponent λ ( k ) t appearing in formula (11) isno longer purely complex. More precisely, one finds a term of the form exp(( ω C ( I ) · k ) t ) , havingdenoted ω ( I ) = ω R ( I )+ iω C ( I ) , ω R,C ( I ) ∈ R n . The size of this term cannot be controlled withouta cut-off on k . By restricting the analysis on the levels Λ s and using the fact that | ω C ( I ) | → as ρ → , a loss “of part of time decay” at each step (see Lemma 7.1) will be the key ingredientto overcome this difficulty. The mentioned elements are clear obstructions to the limit r → ∞ . The use of the analytic tools requires the usual construction of a sequence of nested domains.We shall choose, for all s = 1 , . . . , r , the rule d s := d ( s − /r , (30)with d ∈ (0 , / . Clearly d s < d for all s = 1 , . . . , r . Consider also the monotonically decreasingsequence of non-negative real numbers { a s } defined as follows a s +1 := a s (2 r − s ) / (2 r ) , a := a . (31)Given the analyticity domain of H expressed by ( ρ H , σ H ) , set σ := σ H / . Now consider thefunction Ω( ρ ) := sup I ∈G ρ | ω C ( I ) | , clearly Ω(0) = 0 . From now on we shall suppose that ρ satisfies the following condition rN Ω( ρ ) ≤ a . (32)The analyticity of h ( I ) implies the existence of C h ∈ [1 , + ∞ ) such that the value of ρ can bedetermined as ρ := min { ρ H , a (4 rN C h ) − } , (33) Obviously, Ω( ρ ) ≡ for all ρ in the case of an isochronous system, so that (32) would impose no restrictionson ρ . R A F T once r and N will be chosen.The scheme is constructed in such a way one can set (˜ ρ ∗ , ˜ σ ∗ ) := (1 − d )( ρ, σ ) .As a consequence of Hypothesis 2.1 and of the standard properties of analytic functions, one has k H m k ρ,σ ≤ F h m − e − at , m ≥ , (34)with F := ε ˜ F , where (see [Gio03, Lemma 5.2]) ˜ F := [(1 + exp( − σ/ / (1 − exp( − σ/ n and h := exp( − N σ/ . (35) Lemma 7.1.
Suppose that k Ψ s k (1 − d s )( ρ,σ ) ≤ M ( s ) exp( − a s t ) , for some M ( s ) > . Then thesolution of (28) satisfies a (cid:13)(cid:13)(cid:13) χ ( s ) (cid:13)(cid:13)(cid:13) (1 − d s +1 / )( ρ,σ ) , (cid:13)(cid:13)(cid:13) ∂ t χ ( s ) (cid:13)(cid:13)(cid:13) (1 − d s +1 / )( ρ,σ ) ≤ C r M ( s ) e − a s +1 t , (36) where C r := 2 n +4 ( r/d ) n .Proof. Use (29). Similarly to Lemma 4.2, we choose c ( s ) k ( I,
0) := − R R + exp( − ( ω ( I ) · k ) τ ) ψ ( s ) k ( I, τ ) dτ .Note that | c ( s ) k ( I, | ≤ M ( s ) exp( − (1 − d s ) | k | σ ) R R + exp( | ω C ( I ) || k |− a ) τ ) dτ < + ∞ on Λ s by (32).By using again (32) one gets | c ( s ) k ( I, t ) | ≤ M ( s ) e − (1 − d s ) | k | σ e ass r t Z ∞ t e a s ( s − r r ) τ dτ ≤ a M ( s ) e − (1 − d s ) | k | σ e − a s ( − s r ) t . (37)The first of (36) is easily recognised by (31). The second of (36) follow from (37) and from(29). Lemma 7.2.
Let A, Γ , τ > and consider the real-valued sequences { κ s } s ≥ and { γ l } l ≥ definedas κ l := Aτ l − + Γ s − X j =1 τ j − κ l − j , γ l := Γ l X j =1 τ j − γ l − j , (38) where κ and γ are given. Define ∆ := τ + Γ , then for all s ≥ and l ≥ κ s = (Γ κ + τ A )∆ s − , γ l = γ Γ∆ l − . (39) Proof.
We shall denote with (38a) and (38b) the first and the second of (38), respectively. Thesame for (39). Let us suppose for a moment that (39a) is proven, then choose A = Γ γ and κ = Γ γ = γ . By substituting in (39a) one immediately gets (39b). Hence we need only toprove (39a).For this purpose we use the well-known generating function method (see e.g. [Wil06]). Namely,define g ( z ) := P ∞ n =1 w n z n , multiply each equation obtained from (38a) by z s as s varies, then“sum” all the equations. This leads to g ( z ) = [1 − ∆ z ] − ( κ ( z − τ z ) + Aτ z ) = (1 + ∆ z + ∆ z + . . . )( κ ( z − τ z ) + Aτ z ) = κ z + (Γ κ + τ A ) P n ≥ ∆ n − z n , which is the (39a). Use the inequality P | k |≥ ( s − N exp( − δ | k | σ ) ≤ exp( − Nδn ( s − σ )( P + ∞ m =0 exp( − δmσ )) n ≤ (2 /δ ) n , where inthis case δ := d s + − d s = d/ (2 r ) . R A F T Proposition 7.3.
For all s ≤ r , the following estimate holds k χ s k (1 − d s +1 / )( ρ,σ ) ≤ (4 a ) − C r β s F e − a s +1 t , (40) where the sequence { β s } s =1 ,...,r ∈ R + is determined by the following system β s = h s − + Γ s s − X j =1 jθ s − j θ l = Γ l l X j =1 jβ j θ l − j (41) with { θ l } l =0 ,...,r − ∈ R + and Γ := 16 nr C r F ( ad ρσ ) − , (42) under the conditions β = θ = 1 . First of all note that by (24) and (34), one has k Ψ k (1 − d )( ρ,σ ) ≤ F exp( − a t ) and k E H m k (1 − d )( ρ,σ ) ≤F h m − exp( − a t ) (recall (31)). Hence, given by s ≤ r , we can suppose by induction to know β , . . . , β s − and ˜ θ ,m , . . . , ˜ θ s − ,m , for all m ≥ , with β = 1 and ˜ θ ,m = h m − , such that the thefollowing bounds hold for all j = 1 , . . . , s − and l = 0 , . . . , s − k Ψ j k (1 − d j )( ρ,σ ) ≤ β j F e − a j t , (43a) k E l H m k (1 − d l +1 )( ρ,σ ) ≤ ˜ θ l,m F e − a l +1 t , (43b)By (43a) and Lemma 7.1, the bound (40) holds with j in place of s . Hence by Prop. 4.1with G = χ ( j ) , F = E s − j − H m then ˆ d = max j =1 ,...,s − { d j +1 / , d s − j } = d s − / and finally ˜ d := d s − d s − / = d/ (2 r ) , one has (by setting δ = 0 ) (cid:13)(cid:13)(cid:13) L χ ( j ) E s − j − H m (cid:13)(cid:13)(cid:13) (1 − d s )( ρ,σ ) ≤ r ( ed ρσ ) − (cid:13)(cid:13) χ ( j ) (cid:13)(cid:13) (1 − d j +1 / )( ρ,σ ) k E l − j H k (1 − d l − j +1 / )( ρ,σ ) ≤ Γ F β j γ l − j e − a l +1 t (44)where the property a j +1 + a l − j +1 ≥ a l +1 has been used. Recalling (24), we have that (43b) holdsalso for l = s − , where ˜ θ l,m = Γ l l X j =1 jβ j ˜ θ l − j,m . (45)Furthermore, it is easy to show from the latter that ˜ θ l,m = h m − ˜ θ l, in such a way, defined θ l := ˜ θ l, one gets ˜ θ l,m = h m − θ l , and then the second of (41), provided θ = 1 . In conclusion, byusing (34), and the second of (41) in the definition of Ψ s as in (28), we get that (43a) is satisfiedif β s is defined as in the first of (41). Bound (40) follows from Lemma 7.1. Proposition 7.4.
The sequence β s defined by (41) satisfies β s ≤ τ s − /s , (46) for s = 1 , . . . , r , if τ := eh, Γ ≤ h/ (2 r ) . (47) From a “computational” point of view, first compute θ then proceed with β s , θ s for all s = 2 , . . . , r . R A F T Proof.
The property (46) is trivially true for s = 1 , hence let us suppose it for j = 1 , . . . , s − and proceed by induction with τ to be determined. Define ˜ θ l := θ l ( β j ) | β j = τ j − /j , then ˆ θ l := ˜ θ l /l ,obtaining ˆ θ l = Γ P lj =1 τ j − ˆ θ l − j . Clearly θ l ≤ ˜ θ l ≤ ˆ θ l /l , furthermore θ = ˜ θ = ˆ θ = 1 . Hence,by Lemma 7.2 we have θ l ≤ Γ∆ l − /l . (48)Now choose τ, Γ as in (47). By using (34) and (48) in the first of (41) one gets that (46) issatisfied simply by checking that the inequality y ( s ) := s + ( s − r (cid:18) e + 12 r (cid:19) s − ≤ e s − (49)holds true for all s = 1 , . . . , r . From now on we shall suppose that h and ε are chosen in such a way eh ≤ (50a) r Γ ≤ √ εh (50b)In particular, by definition and by (47), this immediately implies that ≤ (51)As in [Gio03] it is used that, despite the generating sequence is finite, one can use the boundobtained from 7.3 (cid:13)(cid:13)(cid:13) χ ( s ) (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ (4 a ) − C r F β s e − a r +1 t , (52)with β s satisfying (46) for all s , as it would be, trivially, β >r = 0 . Proposition 7.5.
Define ( I ( r ) , ϕ ( r ) , η ( r ) ) := T χ [ r ] ( I, ϕ, η ) . Then the following estimates hold (cid:13)(cid:13)(cid:13) I − I ( r ) (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) , (cid:13)(cid:13)(cid:13) η − η ( r ) (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ dρ e − a r +1 t , (cid:13)(cid:13)(cid:13) ϕ − ϕ ( r ) (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ dσ e − a r +1 t . (53) Proof.
Let us start from the variable I . Firstly, note that (cid:13)(cid:13)(cid:13) I − T χ [ r ] I (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ P s ≥ k E s I k (1 − d )( ρ,σ ) .In addition k E I k (1 − d )( ρ,σ ) = (cid:13)(cid:13)(cid:13) ∂ ϕ χ (1) (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ nr ( edσ ) − (cid:13)(cid:13)(cid:13) χ (1) (cid:13)(cid:13)(cid:13) (1 − d / )( ρ,σ ) ≤ D σ F exp( − a r +1 t ) ,with D σ := nrC r / (2 dσa ) by Prop. 7.3. Hence suppose k E l I k (1 − d l +1 )( ρ,σ ) ≤ F u l exp( − a r +1 t ) forall l = 1 , . . . , s − with u = D σ and proceed by induction.The bound of E l I can be treated in the same way of (43b) with the difference that in this casethe term L χ ( l ) I appearing in E l I needs to be bounded separately by using (40) and a Cauchyestimate. This leads to u l = β l D σ + Γ /l P l − j =1 jβ j u l − j . By using the same procedure used inthe proof of Prop. 7.4 for θ l one gets u l ≤ ( D σ /l )∆ l − . The required bound easily followsas F P s ≥ u s ≤ F D σ ≤ Γ dρ ≤ √ εdρ/ , where the second inequality follows from (51) andthe last one from (50b) then from (50a). The procedure for the variables ϕ and η is similar.The analyticity of the transformation N r := T − χ [ r ] easily follows from the bounds (53) and theinvertibility of the Lie transform operator, see [Gio03]. Clearly (49) holds for s ≤ r if y ( r ) ≤ exp( r − for all r ≥ (let it be directly checked for r = 1 , ). Henceset r = n + 1 and prove that y ( r ) r = n +1 ≤ exp( n ) for all n ≥ , conclusion that is immediate as one can find that y ( n ) ≤ n + 1 + 3 e n / (4 n ) . R A F T Proposition 7.6.
Define A := 10 ˜ F then for all r ≥ (cid:13)(cid:13)(cid:13) R ( r +1) (cid:13)(cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ εAe − ( r + a r +1 t ) . (54) Proof.
Define ( ρ ′ , σ ′ ) := (1 − d )( ρ, σ ) . Now recall (27) and suppose by induction, for all l =1 , . . . , s − , m = 0 , . . . , s − with s ∈ N k E l H k (1 − ( l/s ) d )( ρ ′ ,σ ′ ) ≤ F ǫ l exp( − a r +1 t ) , k E m H n k (1 − ( m/s ) d )( ρ ′ ,σ ′ ) ≤ F ζ m,n exp( − a r +1 t ) .(55)Indeed one can set ζ ,n = h n − and ǫ = β = 1 as L χ (1) H = − Ψ by (28). We stress that,despite based on the same computations, the argument is conceptually different from the previousestimates as s ∈ ( r, + ∞ ) and the use of δ in (12) plays here a key role. More precisely, use Prop.4.1 with G = χ ( j ) and F = E s − j H hence d ′′ = 0 then ˆ d = d ′ = δ = d ( s − j ) /s from which ˜ d = ( j/s ) d . This leads to (cid:13)(cid:13)(cid:13) L χ ( j ) E s − j H (cid:13)(cid:13)(cid:13) (1 − d )( ρ ′ ,σ ′ ) ≤ Γ( s/j ) β j ǫ s − j exp( − a r +1 t ) , implying thatthe first of (55) holds for l = s provided ǫ s = β s + Γ P s − j =1 β j ǫ s − j = ∆ s − , the latter by Lemma7.2. This implies k P sl =1 E s − l H l k (1 − d )( ρ ′ ,σ ′ ) ≤ F ( s + 1)∆ s − exp( − a r +1 t ) by using (34) and thetrivial bound h ≤ ∆ . Similarly one finds ζ s,n = h n − ∆ s − , hence ( F e − a r +1 t ) − R ( r +1) ≤ X s>r (2 + s )∆ s − = ∆ r (cid:18) r + 31 − ∆ + 11 − ∆ (cid:19) ≤ r + 4)∆ r ,by (51). Noticing that D (1 − d )( ρ,σ ) ⊂ D (1 − d ) ( ρ,σ ) , the bound (54) easily follows from (51) andfrom the simple inequality ( r + 4) e r ≤ r ) . Let us discuss a possible choice of the parameters in such a way the convergence conditions aresatisfied. More precisely by (35), condition (50a) holds if N = ⌈ σ − (1 + 3 log 2) ⌉ , where ⌈·⌉ denotes the rounding to the greater integer. This implies that h ≥ / (16 e ) , hence (50b) holdsif er Γ ≤ √ ε . Hence, recalling (32) and (42), this condition is achieved by choosing (see also[GG85]) r := $(cid:18) ε ∗ a ε (cid:19) γ % , p ε ∗ a := a d n +2 ρ H σ n +19 enC h ˜ F , (56)where γ = 5 + n and ⌊·⌋ denotes the rounding to the lower integer. The condition ε ≤ ε ∗ a , asin the statement of Theorem 2.3, clearly ensures that r ≥ . The final value of ρ is determinedwith (33).Let us write the usual bound | I ( t ) − I (0) | ≤ | I ( t ) − I ( r ) ( t ) | + | I ( r ) ( t ) − I ( r ) (0) | + | I ( r ) (0) − I (0) | .The first and third term of the r.h.s. are bounded by √ εdρ/ by (53). As for the second one,from the equations of motion ˙ I ( r ) = − ∂ ϕ H ( r ) = − ∂ ϕ R ( r +1) , furthermore (cid:13)(cid:13) ∂ ϕ R ( r +1) (cid:13)(cid:13) (1 − d )( ρ,σ ) ≤ εA ( edσ ) − exp( − ( r + a r +1 t )) by a Cauchy estimate and by (54). Hence | I ( r ) ( t ) − I ( r ) (0) | ≤ εA ( edσ ) − e − r Z t e − a r +1 s ds ≤ εA ( adeσ ) − (2 /e ) r , (57) The use of (12) with δ = 0 would have given ( s/j ) instead of ( s/j ) , producing in this way a troublesomefactorial in the estimates. Note that the threshold ε ∗ a takes into account of the condition (33) as we have used the obvious lower bound ρ ≥ aρ H (4 rNC h ) − , immediate from (33). R A F T as a r +1 = a (2 r − r − . . . ( r ) / (2 r ) r > a − r . Remark 7.7.
The bound (57) is the key element beyond the perpetual stability, despite a normalform of finite order. The remainder, which is bounded by a constant in the classical Nekhoroshevestimate and then produces a linearly growing bound for the quantity | I ( r ) ( t ) − I ( r ) (0) | , is nowsummable over R + . Hence, a restriction to exponentially large times is no longer necessary.It is immediate from (57) that for all ε ≤ ε ∗ a one has | I ( r ) ( t ) − I ( r ) (0) | ≤ ε ∗ a A ( ade σ ) − which is clearly smaller than √ εdρ/ by (56). Hence | I ( t ) − I (0) | ≤ √ εdρ/ . Acknowledgements
The first author is grateful to Proff. D. Bambusi, L. Biasco, A. Giorgilli and T. Penati for veryuseful discussions on a preliminary version of this paper.
References [Bam05] D. Bambusi. Birkhoff normal form for some quasilinear Hamiltonian PDEs. In
XIVthInternational Congress on Mathematical Physics , pages 273–280. World Sci. Publ.,Hackensack, NJ, 2005.[BG86] G. Benettin and G. Gallavotti. Stability of motions near resonances in quasi-integrableHamiltonian systems.
J. Statist. Phys. , 44(3-4):293–338, 1986.[BGGS84] G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn. A proof of Kolmogorov’stheorem on invariant tori using canonical transformations defined by the Lie method.
Nuovo Cimento B (11) , 79(2):201–223, 1984.[Bir27] G.D. Birkhoff.
Dynamical Systems . American Mathematical Society colloquium pub-lications. American Mathematical Society, 1927.[Bou13] A. Bounemoura. Effective stability for slow time-dependent near-integrable hamilto-nians and application.
C. R. Math. Acad. Sci. Paris , 351(17-18):673–676, 2013.[Chi09] L. Chierchia. Kolmogorov-Arnold-Moser (KAM) theory. In Robert A. Meyers, editor,
Encyclopedia of Complexity and Systems Science , pages 5064–5091. Springer, 2009.[FW14a] A. Fortunati and S. Wiggins. Normal form and Nekhoroshev stability for nearlyintegrable Hamiltonian systems with unconditionally slow aperiodic time dependence.
Regul. Chaotic Dyn. , 19(3):363–373, 2014.[FW14b] A. Fortunati and S. Wiggins. Persistence of Diophantine flows for quadratic nearlyintegrable Hamiltonians under slowly decaying aperiodic time dependence.
Regul.Chaotic Dyn. , 19(5):586–600, 2014.[FW15a] A. Fortunati and S. Wiggins. A Kolmogorov theorem for nearly-integrable Poissonsystems with asymptotically decaying time-dependent perturbation.
Regul. ChaoticDyn. , 20(4):476–485, 2015.[FW15b] A. Fortunati and S. Wiggins. Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium. Accepted forthe publication on Discr. Cont. Dyn. Sys., 2015.15 R A F T [Gal86] G. Gallavotti. Quasi-integrable mechanical systems. In Phénomènes critiques, sys-tèmes aléatoires, théories de jauge, Part I, II (Les Houches, 1984) , pages 539–624.North-Holland, Amsterdam, 1986.[GG85] A. Giorgilli and L. Galgani. Rigorous estimates for the series expansions of Hamilto-nian perturbation theory.
Celestial Mech. , 37(2):95–112, 1985.[Gio03] A. Giorgilli. Exponential stability of Hamiltonian systems. In
Dynamical systems.Part I , Pubbl. Cent. Ric. Mat. Ennio Giorgi, pages 87–198. Scuola Norm. Sup., Pisa,2003.[GZ92] A. Giorgilli and E. Zehnder. Exponential stability for time dependent potentials.
Z.Angew. Math. Phys. , 43(5):827–855, 1992.[Nek77] N. N. Nekhoroshev. An exponential estimate on the time of stabilty of nearly-integrable Hamiltonian systems.
Russ. Math. Surveys , 32:1–65, 1977.[Nek79] N. N. Nekhoroshev. An exponential estimate on the time of stabilty of nearly-integrable Hamiltonian systems II.
Trudy Sem. Petrovs. , 5:5–50, 1979.[Pin13] G. Pinzari. Aspects of the planetary Birkhoff normal form.
Regul. Chaotic Dyn. ,18(6):860–906, 2013.[Poi92] H. Poincaré.
Les méthodes nouvelles de la mécanique céleste.
Gauthier-Villars, Paris,1892.[Pus74] L. D. Pustyl’nikov. Stable and oscillating motions in nonautonomous dynamicalsystems. A generalization of C. L. Siegel’s theorem to the nonautonomous case.
Mat.Sb. (N.S.) , 94(136):407–429, 495, 1974.[Wil06] H.S. Wilf.