Quasi-Periodic Attractors in Celestial Mechanics

Quasi–periodic attractors in celestial mechanics Alessandra Celletti Luigi Chierchia Dipartimento di Matematica Dipartimento di Matematica Università di Roma Tor Vergata Università “Roma Tre” I-00133 Roma (Italy) I-00146 Roma (Italy) ([email protected]) ([email protected]) 13 March, 2007 Abstract We prove that KAM tori smoothly bifurcate into quasi–periodic attractors in dissipative mechanical models, provided external parameters are tuned with the frequency of the motion. An application to the dissipative “spin–orbit model” of celestial mechanics (which actually motivated the analysis in this paper) is presented. Keywords: Quasi–periodic attractors, small divisors, celestial mechanics, spin–orbit problem, nearly Hamiltonian systems, dissipative systems, KAM theory, Nash–Moser theorem. MSC2000 numbers: 34C27, 34C30, 37C70, 37J40, 70F10, 70F15, 70F40, 70K43. Contents 1 Introduction and results 1.1 Dissipative nearly–integrable flows on R × T2 . . . . . . . . . . . . . . . . 1.2 The dissipative spin–orbit model . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A dissipative Nash–Moser Theorem . . . . . . . . . . . . . . . . . . . . . . 2 2 5 8 2 Proofs 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Newton scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 KAM Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 16 20 1 2.4 2.5 2.6 2.7 2.8 Convergence of the Nash–Moser algorithm Local uniqueness . . . . . . . . . . . . . . Proof of Theorem 3 . . . . . . . . . . . . . Proof of Theorem 1 . . . . . . . . . . . . . Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 29 32 34 35 37 1 Introduction and results In physical examples, Hamiltonian dynamics, typically, arises through dissipative systems with very small dissipation. It is therefore natural to ask which part of the Hamiltonian dynamics smoothly persists when the dissipation is turned on. In particular, if the reference Hamiltonian system is nearly–integrable, a basic question is to discuss the fate of quasi–periodic trajectories, which, as KAM theory shows (see, e.g., [1]), are common in the purely Hamiltonian nearly–integrable regime. In this paper we show that – under suitable assumptions relating external (physical) parameters with the motion frequencies – KAM tori smoothly bifurcate into quasi–periodic attractors when dissipative effects are taken into account. Dissipative nearly–integrable flows on R × T2 1.1 Motivated by the “spin–orbit model” of celestial mechanics (see § 1.2 below), we first consider dissipative, nearly–integrable flows on R × T2 , where T2 denotes the standard flat 2–torus R2 /(2πZ2 ). Namely, we consider the differential equation ẍ + η(ẋ − Ω) + εfx (x, t) = 0 , (1.1) where: • dot stands for time derivative and x = x(t); • η is the “dissipation parameter”: for η > 0 the system is dissipative, while for η = 0 the system is conservative (Hamiltonian); values η < 0 are also allowed; • Ω is an “external parameter” (in the spin–orbit problem it will be related to the eccentricity of the reference Keplerian orbit); • ε measures the size of the perturbation (for ε = 0 the system is integrable); 2 • “the potential” f is a given real–analytic function1 on T2 . As mentioned above, for η = 0, (1.1) is the Lagrange equation for the nearly–integrable Lagrangian ẋ2 − εf (x, t) , (ẋ, x, t) ∈ R × T2 , Lε (ẋ, x, t) := 2 or, equivalently, corresponds to the Hamiltonian equation ẏ = −∂x Hε , ẋ = ∂y Hε , 2 for the nearly–integrable Hamiltonian Hε (y, x, t) := y2 + εf (x, t) defined on R × T2 . In the conservative case η = 0, standard KAM theory (see, e.g., [1]) implies that, if ε is small enough, (1.1) admits many quasi–periodic solutions, i.e., solutions of the form x(t) = ωt + u(ωt, t) with u(θ) = u(θ1 , θ2 ) real–anlytic on T2 and ω Diophantine: κ , ∀ (n1 , n2 ) ∈ Z2 , n1 6= 0 , |ωn1 + n2 | ≥ τ |n1 | (1.2) (1.3) for some κ, τ > 0. Furthermore, such solutions are analytic in ε and are Whitney smooth in2 ω. In the following, Dκ,τ denotes the set of Diophantine numbers in R satisfying3 (1.3). On the other hand, when η 6= 0 and ε = 0, the general solution of (1.1) is given by x(t) = x0 + Ω t + 1 − exp(−ηt) (v0 − Ω) , η showing that the periodic (remember that x is an angle) solution x = cost + Ωt, ẋ ≡ Ω is a global attractor for the dynamics. The leitmotiv is that KAM quasi–periodic solutions, for η 6= 0, smoothly bifurcate into quasi–periodic attractors provided the “external frequency” Ω is tuned with the “internal frequency” ω in (1.2). This is the content of the first result: 1 “Finitely differentiable function” suffices, but we focus on the real–analytic case in view of our application to celestial mechanics (and for simplicity). 2 k A function f : A ⊂ Rn → Rm is Whitney C k or CW if it is the restriction on A of a C k (Rn ) function; for the original definition by H. Whitney and for relevance in dynamical system, see, e.g., [2]. Incidentally, we mention that Whitney smoothness was discussed for the first time in the framework of conservative dynamical systems in 1982 in [7] and, independently, in [11]. 3 Observe that if (1.3) holds, then 0 < κ < 1 and τ ≥ 1. In fact, taking n1 = 1 and n2 = −[ω] ([x] = integer part of x) in (1.3) shows that κ < 1, while the fact that τ ≥ 1 comes from Liouville’s theorem on rational approximations (“For any ω ∈ R\Q and for any N ≥ 1 thereSexist integers p and q with |q| ≤ N such that |ωq − p| < 1/N ”). Finally, we recall that, when τ > 1, κ>0 Dκ,τ is a set of full Lebesgue measure. 3 Theorem 1 Fix 0 < κ < 1 ≤ τ and η0 > 0. Then, there exists 0 < ε0 < 1 such that for any ε ∈ [0, ε0 ], any η ∈ I0 := [−η0 , η0 ] and any ω ∈ Dκ,τ , there exists a unique function4 Z dθ u =0, u = uε (θ; η, ω) = O(ε) , hui := 2 T2 (2π) such that x(t) in (1.2) solves (1.1) with   Ω = ω 1 + h(uθ1 ) i . 2 (1.4) Furthermore, the function uε is smooth in the sense of Whitney in all its variables, is real–analytic in θ ∈ T2 and ε, C ∞ in η and Whitney C ∞ in ω. Remark 1 (i) Uniqueness has to be understood in the following sense: if one is given a second solution x̃(t) = ωt + ũ(ωt, t) of (1.1) for some Ω ∈ R, with ũ = ũε (θ; η, ω) = O(ε) real–analytic in (θ, ε) and having vanishing average over T2 , then ũ ≡ u and Ω is as in (1.4). (ii) Time–derivative for x(t) corresponds to the directional derivative ∂ω := ω ∂ ∂ + , ∂θ1 ∂θ2 (1.5) for the function u(θ), so that, being the flow θ ∈ T2 → θ + (ωt, t) dense in T2 , one sees that x(t) as in (1.2) is a quasi–periodic solution of (1.1) if and only if u solves the following PDE on T2 : ∂ω2 u + η ∂ω u + ε fx (θ1 + u, θ2 ) + γ = 0 , γ = η(ω − Ω) . (1.6) This equation will actually be the main object of investigation of this paper. (iii) Theorem 1 implies that the 2–torus Tε,η (ω) := {(x, t) = (θ1 + uε (θ; η, ω), θ2 ) : θ ∈ T2 } , (1.7) is a quasi–periodic attractor for the dynamics on the phase space R × T2 associated to (1.1) with Ω as in (1.4) and that the dynamics on Tε,η (ω) is analytically conjugated to the linear flow θ → θ + (ωt, t); compare, also, point (i) of Remark 3 below. (iv) The result is perturbative in ε but it is uniform in η. It is particularly noticeable the smooth dependence of uε on η as η → 0, which shows that the invariant KAM torus Tε,0 (ω) smoothly bifurcates into the attractor (1.7) as η 6= 0. 4 As usual, f = O(xk ) means that f is a smooth function of x having equal to zero the first k derivatives at x = 0. 4 (v) Theorem 1 will be obtained as a corollary of a “dissipative Nash–Moser” theorem (see § 1.3 below), after having rewritten Eq. (1.6) as a functional equation. Indeed, the method of proof is rather robust and general and could be easily adapted to cover dissipative maps such as the “fattened Arnold family” studied in [3] or it could be extended to systems with more degrees of freedom. In this second case, however, it would be important, in our opinion, to motivate physically the form of the dissipation, a problem in itself difficult and intriguing. 1.2 The dissipative spin–orbit model We turn, now, to the mechanical problem that motivated this paper, namely, the dissipative spin–orbit problem. Such problem consists in studying the rotations of a triaxial non–rigid body (satellite), having its center of mass revolving on a given Keplerian ellipse, and subject to the gravitational attraction of a major body sitting on a focus of the ellipse. To simplify the analysis, we assume that the satellite is symmetric with respect to an “equatorial plane” and study motions having the equatorial plane coinciding with the Keplerian orbit plane (because of the assumed symmetry of the satellite, such motions belong to an invariant submanifold of the phase space). Under such hypotheses, the motions of the satellite are described by the angle x formed by, say, the direction of the major physical axis of the satellite (assumed to lie in the equatorial plane) with a fixed axis of the Keplerian orbit plane (say the direction of the semimajor axis of the ellipse). satellite .................. ..... ... ........................................................................... .... ............ ................. ... .......... ..... ............ .... .. .......... .......... . . . . . . . . . . . . ........... ..... . .. . . . . . . . . . . . ............. .... .... . . . . . . . . . . . . . .... ........... ... ..... . . . . . . . . ... .. ... ... ........ .... .... ..... ... ... ....... ..... ..... ... ........ ......... ..... .... ..... ... ......... .... ... ... . . ... . ... . ... .. .. ... . ... . . . ... ... .. .. .. . . . . ... ... .. .. .. . . . ... ... .... .... . . . .... ... ........... .. ... ... . ... . . . . ............................................................................................................................................................................................................................................................................ ... .. ... .... ... . . ... ... ... ... ... .. ... ... ... . . ..... ..... ..... .... ..... ..... ...... ..... ...... ..... . . ...... . . . . ....... ....... ........ ....... ......... ........ .......... ......... ............. ........... . . . . . . .................. . . . . . . . . .............................................................. • b ρe x 5 • ae fe Assuming that the non–rigidity of the planet (meant to reflect the averaged effect of tides) is modeled by the so–called “MacDonald’s torque” [9] the differential equation governing the motion of the satellite, in suitable units, is given by (1.1) with  3 B−A   η = KLe , Ω = Ωe , ε = 2 C , (1.8)     f = f (x, t; e) := − 1 3 cos 2x − 2 fe (t) , 2 ρe (t) where: • e ∈ [0, 1) is the eccentricity of the Keplerian orbit on which the center of mass of the satellite is revolving; • K ≥ 0 is a physical constant depending on the internal (non–rigid) structure of the satellite; • Le > 0, Ne > 0 and Ωe > 1 are known functions of the eccentricity e ∈ [0, 1) and are given by: 3 4 1 Le := 1 + 3e + e , 8 (1 − e2 )9/2  45 5  1 15 ; Ne := 1 + e2 + e4 + e6 2 8 16 (1 − e2 )6 Ne 3 Ωe := = 1 + 6e2 + e4 + O(e6 ) . Le 8  2 (1.9) (1.10) • 0 < A < B < C are the principal moments of inertia of the satellite; • ρe (t) and fe (t) are, respectively, the (normalized) orbital radius and the true anomaly of the Keplerian motion, which (because of the assumed normalizations) are 2π– periodic function of time t. The explicit expression for ρe and fe may be described as follows. Let u = ue (t) be the 2π–periodic function obtained by inverting (“Kepler′ s equation”) ; t = u − e sin u , (1.11) then ρe (t) = 1 − e cos ue (t) r 1 + e ue (t)  . tan fe (t) = 2 arctan 1−e 2 6 (1.12) Remark 2 (i) A detailed derivation of the equations of motions of the dissipative spin– orbit model (i.e., Eq.’s (1.1)&(1.8)) can be found, e.g., in [4] and in [10]. In particular, the conservative equation (K =p 0) is derived and discussed in [4]: compare Eq. (2.2) of [4] with the normalization n := (Gm)/a3 = 1. The dissipative term (K 6= 0) is derived in [10]: compare Eq. (21) of [10], where (as above) n = 1 and in view of our assumption about the spin axis being vertical, one has to take vanishing eX and eY components, i = 0 and ψm = x; K is the constant in front of the curly brackets in Eq. (21) of [10]; the functions Le and Ne are denoted in [10], respectively, by f1 (e) and f2 (e). The physical validity of the equations of motions (1.1)&(1.8) is confirmed in [8]. (ii) For K = 0, equations (1.1)&(1.8) correspond to the Hamiltonian flow associated to the one–and–a–half degree–of–freedom Hamiltonian   ε 1 2 cos 2x − 2 fe (t) , H(y, x, t) := y − 2 2 ρe (t)3 (y, x) being standard symplectic variables. The associated “spin–orbit” Hamiltonian system is non–integrable if ε > 0 and5 e > 0. (iii) The external frequency Ωe is a real–analytic invertible function of e mapping (0, 1) onto (1, ∞); we denote by Ω−1 : (1, ∞) → (0, 1) the inverse map (which is also real– analytic). The invertibility of the frequency map Ωe will play the rôle of a nondegeneracy condition, allowing to fix the eccentricities for which quasi–periodic attractors exist in the full dynamics. (iv) In many examples taken from the Solar system, both ε and K are small. For example, for the Earth–Moon system and for the Sun–Mercury system ε is of the order of 10−4 , while K is of the order of 10−8 . (v) Mercury is observed in a nearly 3:2 spin–orbit resonance (i.e., it rotates three times on its spin axis, while making two orbital revolutions around the Sun) and is moving on a nearly Keplerian orbit with eccentricity e ≃ 0.2. However, recent numerical investigations show that “the chaotic evolution of Mercury’s orbit can drive its eccentricity beyond 0.325 during the planet’s history” ([8]). Now, if Ω−1 is the function introduced in point (iii) above, Ω−1 (3/2) = 0.284.... Thus, Theorem 1 for the dissipative spin–orbit model (spelled out in Theorem 2 below) might give new mathematical insight into the question of the so–called “capture in the 3:2 resonance” of Mercury (see [8]) suggesting that such capture is related to the existence of an underlying attractor with Diophantine frequency close to 3/2. For more information about this point, see [6]. (vi) As mentioned in point (ii), Remark 1, quasi–periodic trajectories for (1.1)&(1.8) When e = 0, u0 (t) = t = f0 (t), ρ0 = 1 so that H = 21 y 2 − integrable. 5 7 ε 2 cos(2x − 2t), which is easily seen to be correspond to solutions of the PDE   ε ∂ω2 u + KLe ∂ω u + sin 2(θ + u(θ)) − 2 f (θ ) = KNe − KLe ω . 1 e 2 ρe (θ2 )3 (1.13) Next result translates Theorem 1 in the case of the dissipative spin–orbit model. Theorem 2 Fix κ, r ∈ (0, 1) and τ ≥ 1. There exists ε0 > 0 such that for any ε ∈ [0, ε0 ], any K ∈ [0, 1] and any ω ∈ Dκ,τ ∩ [1 + r, 1/r], there exist unique functions6 eε = eε (K, ω) = Ω−1 (ω) + O(ε2 ) , u = uε (θ; K, ω) = O(ε) , R with T2 u dθ = 0, satisfying (1.13) with e = eε . The functions eε and uε are smooth in the sense of Whitney in all their variables and are real–analytic in θ ∈ T2 and ε, C ∞ in K and Whitney C ∞ in ω. 1.3 A dissipative Nash–Moser Theorem We, now, state the main technical result, namely, an existence and uniqueness7 Nash– Moser (or KAM) theorem for dissipative/conservative flows on a two torus. In this theorem the “potential” is not assumed to be small but, rather, we assume to start with a good enough approximate solution. Special care is devoted to the dependence of solutions upon the dissipative parameter, which appears explicitly in the small divisor problems involved. More specifically, Theorem 3 below deals with finding real–analytic “local” solutions u : T2 → R and a number γ such that8  2  ∂ω u + η ∂ω u + gx (θ1 + u, θ2 ) + γ = 0 , (1.14)  hui = 0 , 1 + uθ1 6= 0 . Remark 3 (i) While the first condition in the second line of (1.14) is just a normalization condition (needed for uniqueness), the second one implies that the map θ ∈ T2 → (θ1 + u(θ), θ2 ) ∈ T2 is a diffeomorphism and, therefore, if u solves (1.14), the set o n   2 (y, (x, t)) = ω + ∂ω u, (θ1 + u, θ2 ) : θ ∈ T 6 The map Ω−1 is the inverse map of e → Ωe defined in (1.10). While, KAM procedures are abundant in literature, local uniqueness is seldom discussed. R dθ 8 ∂ω is defined in (1.5) and hui := T2 u(θ) (2π) 2. 7 8 is a 2–dimensional torus embedded in the 3–dimensional phase space R × T2 , which is invariant for the dynamics generated by the differential equation ẍ + η (ẋ − ω) + gx (x, t) + γ = 0 , (1.15) meaning that, for each θ ∈ T2 , t → x(t; θ) := θ1 + ωt + u(θ1 + ωt, θ2 + t) is a solution of (1.15). (ii) As mentioned above, the unknowns of (1.14) are u and γ ∈ R. Indeed, we shall see below that u and γ are not independent, as they satisfy the relation η ω h(uθ1 )2 i + γ = 0 . (1.16) On the other hand, ω and η are regarded as external parameters taken, respectively in Dκ,τ (for some prefixed 0 < κ < 1 ≤ τ ) and in a compact interval [−η0 , η0 ] (for some prefixed η0 > 0). It will be useful to rewrite the differential equation in (1.14) as a functional equation involving parameters. For η ∈ R we define Dη : v ∈ C 1 (T2 ) → Dη v = ∂ω v + ηv , ∆η := Dη ∂ω = ∂ω Dη , Fη : (v, γ) ∈ C 2 (T2 ) × R → Fη (v; γ) := ∆η v + gx (θ1 + v, θ2 ) + γ (notice that D0 = ∂ω ). Then, problem (1.14) may be rewritten as   Fη (u; γ) = 0 ,  hui = 0 , (1.17) (1.18) 1 + uθ1 6= 0 . In order to state existence and uniqueness theorems for (1.18), we need to introduce suitable function spaces, which, because of our motivating model, will be spaces of real– analytic functions. Let us denote by Hξ the Banach space of continuous functions u : T2 → R with finite norm9 X kukξ := |un | ❡|n|ξ , n∈Z2 The letter ❡ denotes the Neper number exp(1) = 2.71828...; do not get confused with the letter e used for eccentricity and the letter e, which will be used, below, for the “error” function e(θ). 9 9 where un denotes the n–th Fourier coefficient and |n| = |n1 | + |n2 |. Let us also denote by H0ξ the closed subspace of Hξ formed by functions with zero average: H0ξ := {u ∈ Hξ s.t. u0 = hui = 0} . Remark 4 (i) The spaces {Hξ : ξ ≥ 0} form a nested family of Banach spaces as 0 ≤ ξ′ < ξ ′ Hξ & Hξ =⇒ and kukξ′ ≤ kukξ . (ii) Clearly, functions in Hξ can be analytically extended to the complex multi–strip T2ξ := {θ ∈ C2 : | Im θi | < ξ , i = 1, 2} and sup |u| ≤ kukξ . T2ξ Viceversa, if u is a real–analytic function on T2 , then its Fourier coefficients decay exponentially fast and therefore there exists ξ > 0 such that u ∈ Hξ . (iii) From the definition of the norm k · kξ , it follows that if u ∈ H0ξ then10 k∂θh ukξ ≤ k∂θk ukξ , ∀ h, k ∈ N2 : hi ≤ ki ; the same inequalities hold also for u ∈ Hξ when h 6= 0. ′ (iv) Dη is a “diagonal” operator on Fourier spaces mapping Hξ into Hξ for any11 0 ≤ ξ ′ < ξ: X  X Dη u = Dη un ❡in·θ = λη,n un ❡in·θ , n∈Z2 n∈Z2 where λη,n := i(ωn1 + n2 ) + η . (1.19) If η 6= 0, Dη is bounded (since |λη,n | ≥ |η|), invertible and its inverse, Dη−1 , maps Hξ onto itself: Dη−1 : Hξ → Hξ , ∀ η 6= 0 , ξ ≥ 0 X in·θ Dη−1 u = λ−1 . η,n un ❡ n∈Z2 Notice, however, that the limit η → 0, which is of particular interest to us, is singular (compare, also, next point). As usual, if h ∈ N2 , ∂θh := ∂θh11 ∂θh22 . 11 As usual n · θ denotes the usual inner product n1 θ1 + n2 θ2 . 10 10 (v) If η = 0, then ′ D0 = ∂ω : Hξ → H0ξ , (0 ≤ ξ ′ < ξ) ; since ω ∈ Dκ,τ , D0 is invertible on H0ξ : X ′ in·θ D0−1 : u ∈ H0ξ −→ λ−1 ∈ H0ξ , η,n un ❡ (0 ≤ ξ ′ < ξ) . n∈Z2 \{0} (vi) We, finally, recall two elementary properties concerning the product and the composition of functions in Hξ . If u, v ∈ Hξ , then uv ∈ Hξ and kuvkξ ≤ kukξ kvkξ . As for composition, if 0 ≤ ξ < ξ¯ and f ∈ Hξ̄ , v ∈ Hξ with kvkξ ≤ ξ¯ − ξ, then one has θ → F (θ) := f (θ1 + v(θ), θ2 ) ∈ Hξ and kF kξ ≤ kf kξ̄ . For the proof, see, e.g., [5], p. 426, 427. We are now ready to formulate the main technical result. Theorem 3 Let 0 < ξ∗ < ξ < ξ¯ ≤ 1; let 0 < κ < 1 ≤ τ ; let I0 := [−η0 , η0 ] for some η0 > 0; let ω ∈ Dκ,τ and (x, t) ∈ T2 → g(x, t) ∈ Hξ̄ ; let M > 0 be such that k∂x3 gkξ̄ ≤ M ; (1.20) let, also, 0 < ν < ξ¯ − ξ, 0 < α < 1 and 0 < σ < 1. Then, there exist a constant k = k(ξ, ξ∗ , κ, τ, η0 , M, ν, α, σ) > 1 such that the following holds. Assume that there exist functions v = v(θ; η) ∈ C ∞ (T2 × I0 ) and β = β(η) ∈ C ∞ (I0 ) (“initial approximate solution”) satisfying the following hypotheses: (H1) for each η ∈ I0 , the function θ → v(θ; η) belongs to H0ξ ; (H2) kvθ1 kξ ≤ σν for any η ∈ I0 ; (H3) define12 V := 1 + vθ1 , 2 W := V , hW −1 Dη−1 vθ1 i ρ := η , hW −1 i (1.21) and assume that, for any η ∈ I0 , |ρ| ≤ σα ; 12 Since σν < 1, hypothesis (H2) implies that V > 0 (and hence W > 0) for all θ and η. 11 (1.22) (H4) assume that, for any η ∈ I0 , the “error function13 ” e = e(θ; η) := Fη (v; β) , (1.23) satisfies the smallness condition k kekξ ≤ 1 . (1.24) Under the hypotheses (H1)÷(H4), there exist functions u = u(θ; η) ∈ C ∞ (T2 × I0 ) and γ = γ(η) ∈ C ∞ (I0 ) such that (1.18) holds. Furthermore, for each η ∈ I0 , u(·; η) ∈ H0ξ∗ and, if V∗ , W∗ and ρ∗ are defined as in (1.21) with v replaced by u, then, for each η ∈ I0 , one has kuθ1 kξ∗ ≤ ν , |ρ∗ | ≤ α , |γ − β| , kuθ1 − vθ1 kξ∗ , |ρ − ρ∗ | , kW − W∗ kξ∗ ≤ k kekξ , (1.25) (1.26) The functions u and γ are unique in the following sense. If u′ = u′ (θ; η) ∈ C ∞ (T2 × I0 ) and γ ′ = γ ′ (η) ∈ C ∞ (I0 ) are also solutions of (1.18), i.e., Fη (u′ ; γ ′ ) = 0 and u′ is such that, for any η ∈ I0 , θ → u′ (θ; η) ∈ H0ξ∗ , ku′θ kξ∗ ≤ ν , 1 k ku′ − vkξ∗ ≤ 1 (1.27) then u′ = u and γ ′ = γ. Remark 5 (i) The proof of this theorem is fully constructive: the solution is gotten as a limit (u, γ) = lim(vj , βj ) where (v0 , β0 ) = (v, β) is the starting approximate solution and (vj , βj ) are quadratically better and better approximations to the solution (u, γ). Details on how to evaluate the constant k are given along the proof. (ii) The assumptions that all the ξ’s are smaller than one and that ν < 1 are not needed and are made only to simplify the exposition. It would suffice to assume that 1 + vθ1 never vanishes. (iii) The smooth dependence upon η around η = 0 (the conservative case) is one of the main points of the theorem: it shows that KAM tori in the conservative case bifurcate smoothly into dissipative attractors, keeping the dynamics conjugated to the linear flow θ → θ + (ωt, t). Indeed, also the dependence upon the frequencies ω ∈ Dγ,τ is smooth, as explained briefly in Remark 7 below. The rôle of the bifurcation parameter is played by γ and in the application to the spin–orbit problem by the eccentricity e of the reference Keplerian orbit. 13 Recall the definitions given in (1.17). 12 2 2.1 Proofs Preliminaries Here, we discuss some more properties of the smal divisor operators Dη and ∆η and prove the compatibility condition (1.16). Lemma 1 (i) If u ∈ C 2 (T2 ), then hDη ui = ηhui , huθ1 ∆η ui = ηωh(uθ1 )2 i . (2.28) (2.29) (ii) If v, V ∈ C 2 (T2 ) and V (θ) 6= 0 ∀ θ ∈ T2 then  v  V ∆η v − v∆η V = Dη V 2 D0 ( ) . (2.30) V (iii) Let ω ∈ Dκ,τ ; let ξ > ξ ′ ≥ 0; let η ∈ R and let p and s be non negative integers. Then, for any u ∈ H0ξ , X ′ in·θ Dη−1 : u ∈ H0ξ −→ λ−1 ∈ H0ξ , η,n un ❡ n∈Z2 \{0} kDη−s p ∂θ ukξ′ ≤ σp,s (ξ − ξ ′ ) kukξ , (2.31) 1 where λη,n = i(ωn1 + n2 ) + η (compare (1.19)) and   −s p −δ|n| . |i(ωn1 + n2 ) + η| |n1 | ❡ σp,s (δ) := σp,s (δ; ω, η) := sup n∈Z2 \{0} Furthermore,  sτ + p sτ +p 1 . (2.32) κs δ sτ +p ❡ Finally, if p > 0, (2.31) holds for any u ∈ Hξ . (iv) Let B := T2 × I0 . Let (θ, η) ∈ B → u(θ; η) belong to C ∞ (B) and θ → u(θ; η) belong to Hξ for some ξ > 0 and any η ∈ I0 . Assume that σp,s (δ) ≤ u0 (0) := hu(·; 0)i = 0 , ′ Dη−1 u and let ω ∈ Dκ,τ . Then, ∈ C ∞ (B); θ → Dη−1 u(θ; η) belongs to Hξ for any 0 ≤ ξ ′ < ξ and any η ∈ I0 . Furthermore, u0 Dη−1 u = + Dη−1 (u − u0 ) , η −1 Dη u(θ; 0) = −∂η u0 (0) + D0−1 u(θ; 0) , (2.33) u0 (η) kDη−1 u(·; η)kξ′ ≤ + σ0,1 (ξ − ξ ′ ) ku(·, ·; η) − u0 (η)kξ . η 13 Proof Equality (2.28) is obvious. The operator D02 ∂θ1 is skew–symmetric, hence, by integration by parts, D uθ1 ∆η u E = huθ1 D02 ui + ηhuθ1 D0 ui = hD02 uθ1 ui + ηhuθ1 D0 ui = ηhuθ1 D0 ui = ηωh(uθ1 )2 i , which is (2.29). (ii) Relation (2.30) follows from the definitions of Dη and ∆η : V ∆η v − v∆η V = V Dη D0 v − vDη D0 V = V D02 v − vD02 V + η(V D0 v − vD0 V ) ; on the other hand one has v Dη (V 2 D0 ( )) = Dη (V D0 v − vD0 V ) V = V D02 v − vD02 V + η(V D0 v − vD0 V ) , from which (2.30) follows. (iii) (2.31) follows immediately from the definitions of Dη−1 and of σp,s : k∂θp Dη−s ukξ′ = 1 X n∈Z2 \{0}   ′ |n1 |p |i(ωn1 + n2 ) + η|−s ❡−|n|(ξ−ξ ) |un | ❡|n|ξ ≤ σp,s (ξ − ξ ′ ) kukξ . Since |i(ωn1 + n2 ) + η| ≥ |ωn1 + n2 | , (2.32) follows at once from the Diophantine estimate (1.3) and from the evaluation   a a  sup xa ❡−x = , ❡ x>0 valid for any a ≥ 0. (iv): The Fourier coefficients 1 un (η) = (2π)2 Z T2 u(θ; η) ❡−in·θ dθ 14 are C ∞ (I0 ) and, by assumption, ku(·; η)kξ := X |un (η)| ❡|n|ξ < ∞ , ∀η ∈ I0 . n∈Z2 Therefore, by the assumption on ω, it follows immediately that Dη−1 u belongs to C ∞ (B) ′ and that Dη−1 u(·, ·) belongs to Hξ for any η 6= 0 and ξ ′ < ξ; the evaluations in the first two lines of (2.33) show that the same is true for any η ∈ I0 (and any ξ ′ < ξ). Last estimate in (2.33) follows at once from point (iii). Corollary 1 Let Fη be as in (1.17). If u ∈ C 2 (T2 ) and η ∈ R then D E (1 + uθ1 )Fη (u; γ) = ηω h(uθ1 )2 i + γ . (2.34) In particular, if Fη (u; γ) = 0, then (1.16) holds. Proof First observe that by (2.28) h∆η ui = hD0 (Dη u)i = 0 . Observe also that h(1 + uθ1 )gx (θ1 + u, θ2 )i = h∂θ1 · g(θ1 + u, θ2 )i = 0 . By these observations and (2.29), D E D E (1 + uθ1 )Fη (u; γ) = (1 + uθ1 )∆η u + (1 + uθ1 )gx (θ1 + u, θ2 ) + (1 + uθ1 )γ D E = uθ1 ∆η u + ∂θ1 · g(θ1 + u, θ2 ) + γ D E = uθ1 ∆η u + γ = ηω h(uθ1 )2 i + γ , proving the claims. 15 2.2 Newton scheme We describe, now, the Newton (KAM) scheme, on which the proof of Theorem 3 is based. We start with a simple lemma on the differential dF of the operator Fη (v; β) = ∆η v + gx (θ1 + v, θ2 ) + β , which is given by14 dFη,v = ∆η + gxx (θ1 + v, θ2 ) . (2.35) Lemma 2 Let v ∈ C 2 (T2 ) and β ∈ R. Assume that V := 1 + vθ1 6= 0 , ∀ θ ∈ T2 , (2.36) and define e := e(θ; η) := Fη (v; β) , W := V 2 , Aη,v : w ∈ C 2 (T2 ) → Aη,v w := V −1 Dη   W D0 (V −1 w) ∈ C(T2 ) . (2.37) (2.38) Then, for every w ∈ C 2 (T2 ) and any η ∈ R, dFη,v (w) = Aη,v w + V −1 eθ1 w . (2.39) Notice that from (2.30) it follows that Aη,v w = ∆η w − V −1 ∆η V w . (2.40) Proof (of Lemma 2) From the definition of e(θ) it follows that eθ1 = ∆η V + gxx (θ1 + v, θ2 )V . (2.41) Then, by (2.35), (2.40) and (2.41) one sees that dFη,v (w) = ∆η w + gxx (θ1 + v, θ2 )w   = Aη,v w + V −1 ∆η V + gxx (θ1 + v, θ2 )V w = Aη,v w + V −1 eθ1 w . 14 By definition dFη,v (w) := lim differential is independent of β. τ →0 Fη (v + τ w; β) − Fη (v; β) ; notice that since Fη is linear in β, its τ 16 The idea of a Newton scheme is to start with an approximate solution of the equation Fη (u; γ) = 0, namely, a function v : T2 → R and a number β such that e := Fη (v; β) := ∆η v + gx (θ1 + v, θ2 ) + β is small and, then, to find a “quadratically better approximation” v′ = v + w and β ′ = β + β̂ satisfying w = O1 = β̂ and Fη (v ′ ; β ′ ) = O2 where Ok = O(kekk ) , To find w and β̂, we define Q1 := gx (θ1 + v + w, θ2 ) − gx (θ1 + v, θ2 ) − gxx (θ1 + v, θ2 )w , Q2 := V −1 eθ1 w ; (2.42) notice that the Qi ’s are quadratic in w and e. Then, by Lemma 2, one has Fη (v ′ ; β ′ ) := Fη (v + w; β + β̂) = Fη (v; β) + β̂ + dFη,v (w) + Q1 = e + β̂ + Aη,v w + Q1 + Q2 . (2.43) Next result shows how to solve the equation e + β̂ + Aη,v w = 0 under suitable conditions upon the function v; in such equation v and (hence) e are given, while w and β̂ are unknowns. Proposition 1 Let g, ω and I0 be as in Theorem 3; let B := T2 × I0 . Let v ∈ C ∞ (B) and let θ → v(θ; b) belong to H0ξ for some ξ > 0 and any η ∈ I0 ; let β ∈ C ∞ (I0 ). Finally, let V and W be defined, respectively, as in (2.36) and (2.37) and assume that V (θ) 6= 0 for any θ ∈ T2 , hW −1 i = 6 0 and that ξ + kvθ1 kξ < ξ¯ , (2.44) |η hW −1 Dη−1 vθ1 i| < |hW −1 i| . (2.45) 17 Define: E(θ) := E(θ; η) := V e := V Fη (v; β) , E := E(η) := hEi , e e η) := E − E , E(θ) := E(θ; e − EhW −1 Dη−1 v i hW −1 Dη−1 Ei θ1 a := a(η) := , −1 −1 −1 hW i + ηhW Dη vθ1 i e hW −1 iE + η hW −1 Dη−1 Ei , β̂ := β̂(η) := −(E + η) a = − hW −1 i + ηhW −1 Dη−1 vθ1 i   e − a 1 + ηD−1 v − E D−1 v . E1 (θ) := E1 (θ; b) := Dη−1 E η η θ1 θ1 (2.46) Then: (i) all functions in (2.46) are C ∞ (B) (or C ∞ (I0 ) if do not depend on θ explicitly) ′ and, for any ξ ′ < ξ, they belong to Hξ , for all η ∈ I0 ; (ii) the following equalities hold: Dη E1 = E + β̂V , hW −1 E1 i = 0 . (2.47) (2.48) (iii) If we define ŵ(θ; η) := −V D0−1 (W −1 E1 ) , w := ŵ − V hŵi (2.49) then these functions are C ∞ (B) (or C ∞ (I0 ) if do not depend on θ explicitly) and, for any ′ ξ ′ < ξ, they belong to Hξ , for all η ∈ I0 . Furthermore, the following identities hold: hwi = 0 , e + β̂ + Aη,v w = 0 . (2.50) (2.51) From this statement15 , the definitions in (2.42) and from (2.43), there follows immediately the following Corollary 2 Under the assumptions of Proposition 1, if β̂ and w are defined as in (2.46) and (2.49), then e′ := Fη (v + w; β + β̂) = Q1 + Q2 . 15 Compare, especially, (2.51). 18 Proof (of Proposition 1) (i): The regularity properties of the functions defined in (2.46) follow from the assumption (2.45) and point (iv) of Lemma 1. Notice that, in view of point (vi) of Remark 4, assumption (2.44) implies that θ → gx (θ1 + v(θ), θ2 ) belongs to Hξ (and, hence, so does e(θ1 )). (ii): From the definitions of E1 and β̂ (and of the operator Dη ), there follows e − Dη a − η a v − Ev Dη E1 = E θ1 θ1 = E − (E + ηa)V = E + β̂V , proving (2.47). Again, from the definitions of E1 and a, there follows   −1 −1 −1 e −1 −1 −1 hW E1 i = hW Dη Ei − a hW i + ηhW Dη vθ1 i − EhW −1 Dη−1 vθ1 i = 0 , proving (2.48). (iii): The regularity claim is handled as above. Equation (2.50) follows at once from the definition of w (noting that hV i = 1). To check (2.51), first, observe that the definition of w implies that   D0 (V −1 w) = D0 − D0−1 (W −1 E1 ) − hŵi = −W −1 E1 . Whence, recalling (2.38) and (2.47) we find h i e + β̂ + Aη,v w = V −1 V e + β̂V + V Aη,v w  i h = V −1 E + β̂V + Dη W D0 (V −1 w) i h = V −1 E + β̂V − Dη E1 = 0. Remark 6 (i) Notice that a, β̂ = O(kek) so that, also, E1 , w = O(kek). Thus, Qi , e′ = O(kek2 ). Explicit estimates will be provided in the next paragraph. (ii) From (2.34) it follows that E := hV ei = h(1 + vθ1 )Fη (v; β)i = β + η ωh(vθ1 )2 i. 19 (2.52) (iii) In the conservative case (η = 0) we have that E = β and E = V F0 (v; β) = E0 + βV with (compare (2.52)) E0 := V F0 (v; 0) , hE0 i = 0 . Thus, from the definitions given in (2.46) and (2.49), there follows that e = E − β = E0 + βv , E θ1 hW −1 D0−1 E0 i , a= hW −1 i β̂ = −β , E1 = D0−1 E0 − a , and w as in (2.49). Thus, w is independent of β and so is the new approximate solution e′ = F0 (v + w; 0) , (recall that β ′ = β + β̂ = 0 in this case). This shows that in the conservative case one can always take β = 0. 2.3 KAM Estimates Here we collect the main estimates for the KAM algorithm described in Proposition 1. We start with the following Definition 1 Let ξ¯ > ξ > 0. Then, • Vξ denotes the set of functions v ∈ C ∞ (T2 × I0 ) such that, for all η ∈ I0 , θ → v(θ; η) ∈ H0ξ . • Vξξ̄ denotes the subset of v ∈ Vξ verifying, for all η ∈ I0 , V (θ) := 1 + vθ1 (θ) 6= 0 for all θ ∈ T2 , hW −1 i = 6 0 where W := V 2 and kvθ1 kξ < ξ¯ − ξ , |η hW −1 Dη−1 vθ1 i| < |hW −1 i| . • Wξ denotes the set Vξ × C ∞ (I0 ). • Wξξ̄ denotes the set Vξξ̄ × C ∞ (I0 ). 20 If (v, β) ∈ Wξξ̄ , we define K(v, β) := (v ′ , β ′ ) := (v + w, β + β̂) , where w and β̂ are defined as in Proposition 1. Thus, by Proposition 1, ∀ ξ¯ > ξ > ξ ′ > 0 . K : Wξξ̄ → Wξ′ , Lemma 3 (i) Let (v, β) ∈ Wξξ̄ ; let (w, β̂) := K(v, β) − (v, β) (compare Definition 1) and define hW −1 Dη−1 vθ1 i 2 . (2.53) V := 1 + vθ1 , W := V , ρ := η hW −1 i Assume that there exist ν, α such that, for any η ∈ I0 , kvθ1 kξ < ν < ξ¯ − ξ , |ρ| < α < 1 . (2.54) (2.55) Then, there exists s := s(τ ) > 1 and16 c := c(κ, τ, M, ν, α, η0 ) > 1 such that, for any 0 < δ < ξ and any η ∈ I0 , the following estimates hold: |β̂| , kwθ1 kξ−δ , kDη−1 wθ1 kξ−δ ≤ c δ −s kekξ . (2.56) (ii) Let W ′ and ρ′ be defined as, respectively, W and ρ with v replaced by v ′ := (v + w). If kwθ1 kξ−δ ≤ ν − kvθ1 kξ , (2.57) then kW ′ − W kξ−δ ≤ c δ −s kekξ ; (2.58) e′ := Fη (v ′ ; β ′ ) := Fη (K(v, β)) belongs to Hξ−δ , for any η ∈ I0 , and satisfies ke′ kξ−δ ≤ c δ −s kek2ξ . Finally, if then |W ′ − W |T2 ≤ 1 (1 − ν)4 , 4 (1 + ν)2 (1 − ν)2 , 2 h(W ′ )−1 i > (2.59) (2.60) (1 + ν)2 2 (2.61) sup |(W ′ )−1 − W −1 | , |ρ′ − ρ| ≤ c δ −s kekξ . (2.62) inf2 W ′ > T and T2 16 Recall the definition of M in (1.20). 21 Proof The proof of this Lemma is (KAM) routine and it is based on a systematic use of point (iii) of Lemma 1 and of point (vi) of Remark 4. We begin by observing that (2.54) implies that µ0 := (1 + ν)−2 < |W −1 (θ)| < µ1 := (1 − ν)−2 , µ0 < W −1 (θ) < µ1 , ∀ θ ∈ S2ξ , ∀ θ ∈ T2 . (2.63) Next, from the definitions in (2.46) and (2.54) it follows immediately that From (2.31) it follows e ξ , kEkξ ≤ (1 + ν) kekξ < 2kekξ . |E| , kEk sup |Dη−1 vθ1 | ≤ kDη−1 vθ1 k0 ≤ σ0,1 (ξ)kvθ1 kξ < σ0,1 (ξ) ν T2 e ≤ kD−1 Ek e 0 ≤ σ0,1 (ξ)kekξ . sup |Dη−1 E| η T2 From (2.63) one has inf2 W > T 1 , µ1 hW −1 i > µ0 , so that hW −1 i + ηhW −1 Dη−1 vθ1 i ≥ hW −1 i(1 − |ρ|) > hW −1 i(1 − α) > µ0 (1 − α) . We are, now, ready to estimate |β̂|: in view of (2.32) and the above estimates, we get |β̂| ≤ c1 ξ −τ kekξ < c1 δ −τ kekξ with   η0 1 (1 + ν)2  η0 1 µ1  τ 2 + (τ / ❡ ) . 2 + (τ / ❡)τ = 1 − α µ0 κ 1 − α (1 − ν)2 κ Analogously, we find c1 := |a| ≤ c2 ξ −τ kekξ < c2 δ −τ kekξ , −1 −1 e kE1 kξ− δ ≤ kDη−1 Ek ξ− δ + |a|(1 + |η|kDη vθ1 kξ− δ ) + |E| kDη vθ1 kξ− δ 3 3 3 e ξ + |a|(1 + |η|σ0,1 (δ/3)ν) + |E|σ0,1 (δ/3)ν ≤ σ0,1 (δ/3)kEk −2τ ≤ c3 δ kekξ , 22 3 (2.64) for suitable constants ci = ci (κ, τ, η0 , ν, α) (which, being clear the ideas, we do not compute δ any more); notice that kE1 k has been estimated on an intermediate space Hξ− 3 so as to be able to control further applications of the operator Dη−1 or ∂θ1 . Now, recalling the definition of w and ŵ in (2.49), by point (iii) of Lemma 1, by (2.64), one gets kŵkξ− 2 δ ≤ (1 + kvθ1 kξ ) kD0−1 (W −1 E1 )kξ− 2 δ 3 3 ≤ (1 + ν)σ0,1 (δ/3) (1 − ν)−2 kE1 kξ− δ 3 ≤ c4 δ −3τ kekξ . Thus, since |hŵi| ≤ kŵkξ− 2 δ , we find 3 kwkξ− 2 δ ≤ c5 δ −3τ kekξ , 3 (2.65) and kwθ1 kξ−δ ≤ σ1,0 (δ/3) kwkξ− 2 δ ≤ c6 δ −(3τ +1) kekξ , 3 kDη−1 wθ1 kξ−δ ≤ σ0,1 (δ/3) kwθ1 kξ− 2 δ ≤ c7 δ −(4τ +1) kekξ , 3 (2.66) which proves part (i) of the Lemma. Now, by (2.57) and (2.66), kW ′ − W kξ−δ = k2wθ1 V + wθ2 kξ−δ 1 ≤ 2νkwθ1 kξ−δ + kwθ1 k2ξ−δ ≤ 3νkwθ1 kξ−δ ≤ c8 δ −(3τ +1) kekξ . (2.67) Inequality (2.57) guarantees, also, that e′ ∈ Hξ−δ ; compare point (vi), Remark 4. Then, by Corollary 2 and (2.65), one gets (for a suitable c9 > 1) ke′ kξ−δ ≤ kQ1 kξ−δ + kQ2 kξ−δ ≤ M kwk2ξ−δ + (1 − ν)−1 σ1,0 (δ)kekξ kwkξ−δ ≤ c9 δ −6τ kek2ξ . f := W ′ − W . From (2.60) it follows that Let W 4 f | ≤ 1 (1 − ν) = 1 µ0 < 1 , sup |W 4 (1 + ν)2 4 µ21 2µ1 T2 so that, on T2 one has (recall (2.63)) 1 f≤3 , ≤ 1 + W −1 W 2 2 1 1 (1 − ν)2 f W =W +W > − = >0. µ1 2µ1 2 ′ 23 (2.68) Let now Z := (W ′ )−1 − W −1 . Then, by (2.67) and (2.68) one finds (on T2 ) |Z| = W 2 (1 f W + f) W −1 W f | ≤ c10 min{δ −(3τ +1) kekξ , 1} ≤ 2µ21 |W (2.69) (for a suitable c10 > 1), proving also the first inequality in (2.62). Furthermore, by the first inequality in (2.68), f| ≥ h(W ′ )−1 i = hW −1 i + hZi ≥ hW −1 i − |hZi| ≥ µ0 − 2µ21 |W µ0 1 . = 2 2(1 + ν)2 Finally, using (2.69), (2.66) and (2.57) (in order to estimate kwθ1 k in terms of ν < 1), one obtains (W −1 + Z)(Dη−1 vθ1 + Dη−1 wθ1 ) W −1 Dη−1 vθ1 |ρ − ρ| ≤ η −η ≤ c11 δ −(4τ +1) kekξ , hW −1 i + hZi hW −1 i ′ proving also the second inequality in (2.62). The theses of the Lemma follow, now, by taking c := maxi ci and s = 6τ . 2.4 Convergence of the Nash–Moser algorithm Here we complete the quantitative description of the KAM procedure giving a sufficient condition in order for the algorithm to converge. ¯ we let For any i ≥ 0 and for 0 < ξ∗ < ξ < ξ, ξi := ξ∗ + ξ − ξ∗ , 2i δi+1 := ξi − ξi+1 = ξ − ξ∗ , 2i+1 (∀ i ≥ 0) ; as above (compare (2.63)) we let µ0 := (1 + ν)−2 and µ1 := (1 − ν)−2 ; we let 0 < µ0 < µ̄0 := inf2 W −1 ≤ µ̄1 := sup W −1 < µ1 ; T (2.70) T2 finally, we let (recall the definition of ρ in (1.21))   µ0 , µ1 − µ̄1 , µ̄0 − µ0 , α − |ρ| > 0 . µ := min ν − kvθ1 kξ , 2µ21 24 (2.71) Proposition 2 Under the same assumptions and notations of part (i) of Lemma 3, let 0 < ξ∗ < ξ and let ξi , µi , µ̄i and µ be as above; let C and m be positive numbers such that C ≥ c 4s (ξ − ξ∗ )−s , m ≤ 2s−1 µ , (2.72) and assume that 1 C kekξ ≤ ❡− ❡ m . (2.73) Then, (vi , βi ) := Ki (v, β) = (vi−1 + wi , βi−1 + β̂i ) ∈ Wξξ̄i for all i ≥ 1; the sequences {vi } and {βi } converge uniformly on, respectively, S2ξ∗ × I0 and I0 , defining a limit  (u, γ) := lim (vj , βj ) = v + j→∞ ∞ X wi , β + i=1 ∞ X i=1  β̂i ∈ Wξξ̄∗ , which is a solution of (1.18), i.e., Fη (u; γ) = 0 for all η ∈ I0 . Furthermore, if Wi and ρi are defined as, respectively, W and ρ in (1.21) with v replaced by vi , then W∗ := lim Wi ∈ Wξξ̄∗ , ρ∗ := lim ρi ∈ C ∞ (I0 ), and (for all θ ∈ T2 and any η ∈ I0 ) kuθ1 kξ∗ ≤ ν , |ρ∗ | ≤ α , |γ − β| , kuθ1 − vθ1 kξ∗ , |ρ − ρ∗ | , kW − W∗ kξ∗ ≤ C∗ kekξ , where C∗ := (2.74) (2.75) −1 C  − ❡1m . 1 − ❡ 2s Proof We claim that (1.24) implies that, for any i ≥ 1, (vi , βi ) := Ki (v, β) ∈ Wξξ̄i (2.76) −1 max{|β̂i |, k∂θ1 wi kξi , |ρi − ρi−1 |, sup |Wi − Wi−1 |, sup |Wi−1 − Wi−1 |} T2 T2 2i−1 ≤ (Ckekξ ) 2s (2.77) θ → ei (θ) := Fη (vi ; βi ) ∈ H ξi (Ckekξ )2 and kei kξi ≤ C 2s i i (2.78) where W0 := W , ρ0 := ρ. We prove the claim by induction. First of all, observe that (1.24) implies immediately that17 k 2k (Ckekξ )2 ≤ m , 17 ∀k≥0. (2.79) If x and y are positive numbers such that x ≤ exp(−1/(ey)), then txt ≤ y for any t > 0. In fact, let λ := log x−1 and observe that the hypothesis is equivalent to e1λ ≤ y. Then, txt = λ1 (λt) exp(−λt) ≤ 1 eλ ≤ y. 25 Now, let us check (2.76)÷(2.78) for i = 1. By Lemma 3, part (i), with δ := δ1 , ξ − δ = ξ1 , (v ′ , β ′ ) = (v1 , β1 ) = K(v, β) = (v + w1 , β + β̂1 ), by definition of C, δ1 , m and µ and by (2.79) (with k = 0), we have |β̂1 | , k∂θ1 w1 kξ1 ≤ cδ1−s kekξ = Ckekξ µ ≤ . 2s 2 (2.80) In particular ν − kvθ1 kξ , 2 which allows to apply part (ii) of Lemma 3, with W ′ = W1 , e′ = e1 , ρ′ = ρ1 , and to obtain, as above, k∂θ1 w1 kξ1 ≤ kW1 − W kξ1 ≤ µ Ckekξ ≤ , s 2 2 ke1 kξ1 ≤ (Ckekξ )2 C 2 kek = , ξ 2s C 2s (2.81) and, since µ ≤ µ0 /(2µ21 ), by (2.62) (recall the identity in (2.68)), we have also µ0 T 2 Ckekξ , sup |W1−1 − W −1 | , |ρ1 − ρ| ≤ 2s T2 inf2 W1 > 1 , 2µ1 hW1−1 i > which, together with (2.80) and (2.81), prove (2.77) and (2.78) for i = 1. Finally, from the above estimates and definitions, there follows ν − kvθ1 kξ < ν < ξ¯ − ξ , 2 µ α − |ρ| |ρ1 | ≤ |ρ| + |ρ1 − ρ| ≤ |ρ| + ≤ |ρ| + <α<1, 2 2 k∂θ1 v1 kξ1 ≤ kvθ1 kξ + k∂θ1 w1 kξ1 ≤ kvθ1 kξ + showing that (v1 , β1 ) ∈ Wξξ̄1 , proving also (2.76) for i = 1. Let, now, j ≥ 1 and assume that (2.76)÷(2.78) hold true for 1 ≤ i ≤ j; we want to prove (2.76)÷(2.78) for i = j + 1. Let 1 ≤ i ≤ j; then, by definition of vi , by (2.77) and (2.79), there follows k∂θ1 vi kξi = kvθ1 + i X ∂θ1 wk kξi k=1 ≤ kvθ1 kξ + i X k=1 26 k∂θ1 wk kξk k−1 i X (Ckekξ )2 ≤ kvθ1 kξ + 2s k=1 i m X 1 2s−1 k=1 2k ≤ kvθ1 kξ + i X 1 = kvθ1 kξ + µ 2k k=1 < kvθ1 kξ + µ < ν . (2.82) Analogously, for any 1 ≤ i ≤ j, |ρi | = |ρ + i X ≤ |ρ| + ρk − ρk−1 | k=1 i X |ρk − ρk−1 | k=1 < |ρ| + µ < α . (2.83) Thus, we can apply Lemma 3, part (i) (with δ := δj+1 , ξ = ξj , v, β, ρ replaced by vj , βj , ρj , (v ′ , β ′ ) = (vj+1 , βj+1 ) = K(vj , βj ) = (vj + wj+1 , βj + β̂j+1 )), which, in view of (2.56), the identity C −s (2.84) = s 2s i , c δi+1 2 and (2.78), yields −s |β̂j+1 | , k∂θ1 wj+1 kξj+1 ≤ c δj+1 kej kξj C = s 2s j kej kξj 2 j (Ckekξ )2 , ≤ 2s (2.85) which proves the first two bounds in (2.77) with i = j + 1. To apply part (ii) of Lemma 3, we have to check (2.57) with w and v replaced, respectively, by wj+1 and vj . Since k∂θ1 vj kξj + k∂θ1 wj+1 kξj+1 ≤ kvθ1 kξ + i+1 X k∂θ1 wk kξk , k=1 (2.85) shows that the inequalities in (2.82) hold also for i = j +1 so that (2.57) is satisfied. Thus, by (2.58) (with W ′ and W corresponding, respectively, to Wj+1 and Wj ), (2.84) 27 and by (2.78) with i = j, we see that j kWj+1 − Wj kξj+1 ≤ −s c δi+1 kej kξj (Ckekξ )2 C = s 2s i kej kξj ≤ , 2 2s (2.86) showing that (2.77) for kWi − Wi−1 k holds also with i = j + 1. Now, by (2.59), (2.84) and (2.78) with i = j, we find j+1 kej+1 kξj+1 C (Ckekξ )2 ≤ s 2s j 2 C 2 22s j j+1 (Ckekξ )2 = C 2s(j+1) , (2.87) i.e., (2.78) with i = j + 1. Next, from (2.86), (2.79), the fact that j ≥ 1 and the definition of the µ’s, there follows j (Ckekξ )2 2s µ m ≤ s j = j+1 2 2 2 µ0 µ ≤ 2 ≤ 4 4µ1 1 (1 − ν)4 = , 4 (1 + ν)2 kWj+1 − Wj kξj+1 ≤ showing that (2.60) holds in the present case. Therefore, by (2.61), inf2 Wj+1 > T (1 − ν)2 , 2 −1 hWj+1 i> (1 + ν)2 2 and j −1 sup |Wj+1 T2 − Wj−1 | , |ρj+1 − ρj | ≤ −s c δj+1 kej kξj (Ckekξ )2 , ≤ 2s (2.88) showing that (2.77) holds for i = j + 1. Finally, by (2.88), one sees that (2.83) holds also for i = j + 1, implying that |ρj+1 | < α < 1 , which shows that (vj+1 , βj+1 ) ∈ Wξξ̄j+1 . The claim has been completely proven. The (fast) decay of (Ckek)2 implies that (vj , βj ) converge uniformly to (u, γ) ∈ Wξξ̄∗ and that, for any η ∈ I0 , j Fη (u; γ) = lim Fη (vj ; βj ) = lim ei = 0 , j→∞ j→∞ 28 showing that (u, γ) is a solution of (1.18) for any η ∈ I0 . Furthermore, since k∂θ1 vi kξi < ν , |ρi | < α , we see, by taking limits, that (1.25) holds. As for the bounds in (1.26), we have, for example, that X kuθ1 − vθ1 kξ∗ ≤ k∂θ1 wi kξi i≥1 1 X i (Ckekξ )2 s 2 i≥1 1 X ≤ s (Ckekξ )i 2 i≥1 ≤ ≤ C∗ kekξ . which implies the second inequality in (1.26); the other inequalities are obtained in exactly the same way. 2.5 Local uniqueness Next, we prove a local uniqueness result for the solutions of (1.18). Such result is based upon the following simple observation on analytic functions. Lemma 4 Let w ∈ Hξ∗ and assume that there exist c, s > 0 such that kwkξ−δ ≤ c kwk2ξ δ −s , ∀ 0 < δ < ξ ≤ ξ∗ , (c 4s ξ∗−s ) kwkξ∗ ≤ 1 . (2.90) Then w ≡ 0. Proof Let, for all i ≥ 0, ξi := ξ∗ /2i and δi := ξ∗ /2i+1 . Then, by (2.89), one has kwkξi+1 ≤ B0 B1i kwk2ξi , (2.89) B0 := c 2s ξ∗−s , 29 B1 := 2s . Iterating this relation18 one gets, for all i ≥ 0,  Ckwkξ∗  2i , C := c 4s ξ∗−s , C 2s i P which, by (2.90), implies that kwk0 := |wn | = 0 so that w ≡ 0. kwk0 ≤ kwkξi ≤ Proposition 3 Let 0 < ξ∗ < ξ¯ and let (u, γ) ∈ Wξξ̄∗ be a solution of (1.18) (i.e., Fη (u; γ) = 0), satisfying also kuθ1 kξ∗ ≤ ν ≤ ξ¯ − ξ∗ , |ρ| ≤ α < 1 , b = C(κ, b τ, η0 , M, ν, α) > 1 for some ν, α ∈ (0, 1). Then, there exist ŝ := ŝ(τ ) > 1 and C ξ̄ such that, if (u′ , γ ′ ) ∈ Wξ∗ is also a solution of (1.18) (i.e. Fη (u′ ; γ ′ ) = 0) satisfying ku′θ kξ∗ ≤ ν , 1 then u ≡ u′ and γ = γ ′ . b ξ −ŝ ku′ − ukξ ≤ 1 C ∗ (2.91) Proof Let w := u′ − u , γ̂ := γ ′ − γ , and observe that, since e := Fη (u; γ) = 0, then, by (2.35) and (2.39), one has the following identities:   −1 −1 (2.92) Aη,u w := V Dη W D0 (V w) = dFη,u (w) = ∆η w + gxx (θ1 + u, θ2 )w , where V := 1 + uθ1 and W := V 2 . Thus, (since also Fη (u′ ; γ ′ ) = 0) 0 = Fη (u′ ; γ ′ ) = Fη (u; γ) + ∆η w + gx (θ1 + u′ , θ2 ) − gx (θ1 + u, θ2 ) + γ̂ = ∆η w + gxx (θ1 + u, θ2 ) w + γ̂   + gx (θ1 + u + w, θ2 ) − gx (θ1 + u, θ2 ) − gxx (θ1 + u, θ2 )w =: Aη,u w + γ̂ + Q1 (w) . 18 (2.93) If yi > 0 and {xi }i≥0 is a sequence of positive numbers satisfying xi+1 ≤ y0 y1i x2i , i then one also has xi ≤ (y0 y1 x0 )2 /(y0 y1i+1 ), as it follows multiplying both sides of the above inequality by y0 y1i+2 so as to obtain zi+1 ≤ zi2 with zi := y0 y1i+1 xi . 30 We, now, claim that, if we define E := V Q1 , then E := hEi , e := E − E , E e hW −1 iE + η hW −1 Dη−1 Ei . γ̂ = − hW −1 i + ηhW −1 Dη−1 uθ1 i In fact, from (2.92) and (2.93), there follows   e+E =0 . Dη W D0 (V −1 w) + γ̂ + uθ1 γ̂ + E (2.94) (2.95) (2.96) If η = 0, then taking average in (2.96) yields γ̂ = −E, which is (2.95) when η = 0. If η 6= 0, then, observing that Dη−1 1 = η1 , from (2.96) one obtains D0 (V −1 w) + −1 W −1 γ̂ e+W E =0; + γ̂W −1 Dη−1 uθ1 + W −1 Dη−1 E η η multiplying by η and taking average in the latter relation, yields, upon solving for γ̂, (2.95). Thus, w satisfies the equation Aη,u w = Q , Q := −γ̂ − Q1 , (2.97) with Q “quadratic” in w. In fact, observing that kQ1 kξ ≤ M kwk2ξ , ∀ ξ ≤ ξ∗ , (M being as in (1.20)), by the relations in (2.97), (2.95) and (2.94), one finds that |γ̂| , kQkξ ≤ c11 ξ −τ kwk2ξ , (2.98) for a suitable c11 = c11 (κ, τ, η0 , M, ν, α) > 1 and for any ξ ≤ ξ∗ . Next, from the definition of Aη,u , one finds the identity D0 (V −1 w) = W −1 Dη−1 (V Q) , (2.99) and, since hwi = 0 (as it follows from u, u′ ∈ H0ξ∗ ), one sees that (2.99) is equivalent to   w = ŵ − V hŵi , ŵ := V D0−1 W −1 Dη−1 (V Q) . 31 By Lemma 1, one gets the estimate kwkξ−δ ≤ c12 δ −2τ kQkξ , (0 < δ < ξ ≤ ξ∗ ) , which, by (2.98), implies kwkξ−δ ≤ c13 δ −3τ kwk2ξ , , (0 < δ < ξ ≤ ξ∗ ) b := c13 43τ and ŝ := 3τ , the thesis showing that w satisfies the estimate (2.89). Letting C follows from Lemma 4. The above analysis shows the smooth (C ∞ ) dependence upon the “dissipation” parameter η. We close this section with a brief remark on how solutions depend upon other eventual “external” parameters. Remark 7 (i) If the function g in (1.18) depends also in a real–analytic way on one (or more) external parameters ℘ ∈ J ⊂ Cm , then so do KAM solutions u(θ; η, ℘) provided the smallness condition (1.24) holds uniformly in ℘, i.e., provided such conditions holds with the k · kξ norm redefined as the norm  X kekξ := sup |en (℘)| ❡|n|ξ . n∈Z ℘∈J This claim follows from the uniform convergence of the KAM scheme and Weierstrass theorem on analytic limits of holomorphic function; for more details compare, e.g, with [5]. (ii) The dependence upon the frequency ω, as well known, is more delicate since it involves the small divisors λη,n : it is, however, standard to check that this dependence is C ∞ in the sense of Whitney on a bounded set of Diophantine numbers, say, Dκ,τ ∩ [1 + r, 1/r] for any prefixed 0 < r < 0; for more details on Whitney smoothness and proofs we refer the reader to [7], [11] and [2]. 2.6 Proof of Theorem 3 We start by observing that the hypotheses (H1)÷(H3) of Theorem 3 imply that (v, β) ∈ Wξξ̄ (recall Definition 1) and that (2.54) and (2.55) hold. Next, because of (H2) and (H3), we have that, on T2 , one has 1 1 ≥ W −1 ≥ , 2 (1 − σν) (1 + σν)2 32 so that (recall the definitions of µ0 , µ1 , µ̄0 and µ̄1 in (2.63) and (2.70)) we find (1 + ν)2 − (1 + σν)2 =: µ̂0 > 0 , µ̄0 − µ0 ≥ (1 + ν)2 (1 + σν)2 (1 − σν)2 − (1 − ν)2 =: µ̂1 > 0 . µ1 − µ̄1 ≥ (1 − ν)2 (1 − σν)2 Thus, the number µ defined in (2.71) is bounded below by n o (1 − ν)2 µ ≥ µ∗ = µ∗ (ν, α, σ) := min (1 − σ)ν , , µ̂ , µ̂ , (1 − σ)α >0. 0 1 2(1 + ν)2 Therefore, taking m = m∗ := 2s−1 µ∗ we see that condition (2.73) may be rewritten as k1 kekξ ≤ 1 , with 1 k1 = k1 (ξ, ξ∗ , κ, τ, η0 , M, ν, α, σ) := C ❡ ❡ m∗ (ν,α,σ) , while (2.75) holds with C∗ equal to k2 with −1 1 C  − ❡ m (ν,α,σ) ∗ . k2 = k2 (ξ, ξ∗ , κ, τ, η0 , M, ν, α, σ) := s 1 − ❡ 2 (2.100) Thus, if k is taken to be not smaller than max{k1 , k2 }, we see that (1.24) implies (2.73) so that, by Proposition 2, the claims about existence of the solution (u, γ) and the estimates (1.25) and (1.26) hold. b as in Proposition 3 and define Let us turn to uniqueness. Let ŝ and C b ∗−ŝ , k3 = k3 (ξ∗ , κ, τ, η0 , M, ν, α) := 2Cξ k4 := k2 k3 , and assume that k4 kekξ∗ ≤ 1 . (2.101) Then, if u′ and γ ′ solve (1.18), i.e., Fη (u′ ; γ ′ ) = 0 for each η ∈ I0 , and if k3 ku′ − vkξ∗ ≤ 1 , (2.102) then, by (2.102), (2.75), the definition of k2 in (2.100), (2.101) , we see that ku′ − vkξ∗ ≤ ku′ − ukξ∗ + ku − vkξ∗ ≤ 1 k2 2 1 1 , + k2 kekξ∗ ≤ + = = b ∗−ŝ k3 k3 k4 k3 Cξ 33 showing that (2.91) is satisfied so that, by Proposition 3, u′ = u and γ ′ = γ. Thus (since k4 is greater than k2 and k3 ), we see that all claims in Theorem 3 follow by taking k := max{k1 , k4 }. 2.7 Proof of Theorem 1 We now show how Theorem 1 can be obtained as a corollary of Theorem 3. Since f in (1.6) is assumed to be real–analytic, there exists a ξ¯ > 0 such that f ∈ Hξ̄ (point (ii) of Remark 4). Assuming, as we shall henceforth do, that |ε| ≤ ε0 < 1 we can take the constant M in Theorem 3 to be M := k∂x3 f kξ̄ . In this section, k · kξ denotes the norm (compare (i), Remark 7)  X khkξ := sup |hn (ε)| ❡|n|ξ , J := {ε ∈ C , |ε| ≤ ε0 } . n∈Z ε∈J The numbers ξ∗ , ξ, ν, α and σ can be chosen arbitrarily as long as they satisfy 0 < ξ∗ < ξ < ξ¯ , 0 < ν < ξ¯ − ξ , 0<α<1, 0<σ<1. (2.103) Finally, we choose, as initial approximate solution, the trivial couple (v, β) := (0, 0) . Then, the error function e defined in (1.23) is simply given by e = e(θ; ε) := Fη (0; 0) = ε ∂x f (θ) , kekξ ≤ ε0 M (2.104) and the functions defined in (1.21) are given by V =1, W =1, ρ=0. Thus, (H1)÷(H3) of Theorem 3 are trivially satisfied and in order to meet (H4), i.e., the smallness condition (1.24), it suffices to require n 1 o . (2.105) ε0 ≤ ε∗ := min 1 , kM 34 Thus, if (2.105) holds, by Theorem 3 and Remark 7, there exist unique functions u = u(θ; η) = uε (θ; η, ω) and γ = γ(η) = γε (η, ω) such that Fη (u; γ) = 0, for all η ∈ I0 , and θ → u(θ; η) ∈ H0ξ∗ . Furthermore, u and γ are Whitney C ∞ in all their variables (θ, η, ε, ω) in the domain T2ξ∗ × I0 × J × Dκ,τ , they are C ∞ in (θ, η, ε) and real–analytic in (θ; ε) ∈ T2ξ∗ × J. The solution (u, γ) satisfies the bounds (1.25) and (1.26). In particular (by (1.26) with v = 0), it holds kuθ1 kξ∗ ≤ ε0 kM , which, together with analyticity in ε, implies that u = O(ε), i.e., u|ε=0 = 0. Finally, the relation between γ and ω in (1.6) and Eq. (1.16) imply (1.4), completing the proof of Theorem 1. 2.8 Proof of Theorem 2 Also the proof of Theorem 2 is based upon Theorem 3 along the lines of § 2.7, but, first, we have to investigate the analytical properties of the spin–orbit potential defined in (1.8). For this purpose, we denote19 1 e1 := Ω−1 (1 + r) , e2 := Ω−1 , (2.106) r where 0 < r < 1 is a prefixed number as in Theorem 2 and Ω−1 is the real–analytic function (inverse of e → Ωe ) defined in point (iii) of Remark 2. Clearly, 0 < e1 < e2 < 1 . It is also clear that ρe (t) and fe (t) (defined in (1.11) and (1.12)) are real–analytic function of (e, t) ∈ (0, 1) × S1 , where S1 := R/(2πZ). Thus, there exist positive numbers 0 < ξ¯ < 1 , 0 < d < min{e1 , 1 − e2 } , (2.107) such that the functions ρe (t) and fe (t) may be analytically continued into the complex domain Er,d × S1ξ̄ , where Er,d := [ {e ∈ C : |e − e′ | ≤ d} , ¯ . S1ξ̄ := {t ∈ C : | Im t| < ξ} (2.108) e′ ∈[e1 ,e2 ] 19 Again: do not confuse the letter e, which stands for eccentricity, with the letter e, which denotes the error function. 35 Therefore, for any ε0 > 0, which will be henceforth assumed to be smaller or equal than 1, the function g(x, t; ε, e) := εf (x, t; e) (2.109) is real–analytic for ((x, t), (ε, e)) ∈ T2ξ̄ × J (2.110) J := {ε ∈ C : |ε| ≤ ε0 } × Er,d ⊂ C2 . (2.111) where, now, Clearly, in the present situation the k · kξ denotes the norm  X khkξ := sup |hn (ε, e)| ❡|n|ξ . n∈Z (ε,e)∈J Finally, we choose η0 as20 η0 := Le2 , and let h 1i , ω ∈ Dκ,τ ∩ 1 + r, r which guarantees that, as e ∈ [e1 , e2 ] then Ωe ∈ [1 + r, 1/r]. At this point, we can proceed as in the previous section (with the same choices of M , ξ∗ , ξ, ν, α and σ, (v, β)) and deduce from Theorem 3 the existence and uniqueness of functions u = u(θ; η) = u(θ; η, ε, e, ω) and γ = γ(η) = γ(η, ε, e, ω) such that Fη (u; γ) = 0, for all η ∈ I0 , and θ → u(θ; η) ∈ H0ξ∗ . As above, u and γ are Whitney C ∞ in all their variables (θ, η, (ε, e), ω) in the domain  h 1 i T2ξ∗ × I0 × J × Dκ,τ ∩ 1 + r, , r they are C ∞ in (θ, η, (ε, e)) and real–analytic in (θ; (ε, e)) ∈ T2ξ∗ × J; (u, γ) satisfies the bounds (1.25), (1.26) and kuθ1 kξ∗ ≤ ε0 kM , which, together with analyticity in ε, implies that u = O(ε), or u|ε=0 = 0. Therefore, relation (1.16) implies that γ = O(ε2 ) and we can write γ =: −ηω ε2 γ̃(η, ε, e, ω) , with γ̃ Whitney C ∞ in all its variables, C ∞ in η and real–analytic in (ε, e); the minus sign accounts for the fact that γ̃ ≥ 0 for real values of its arguments. To finish the proof of Theorem 2 we have to discuss the parameter relations (compare (1.13)) η = KLe , γ = KLe ω − KNe . (2.112) 20 Recall the definition of Le in (1.9) and note that K will be taken in the interval [−1, 1]. 36 By definition of Ωe and γ̃, we can rewrite (2.112) as η = KLe , ωε2 γ̃(η, ε, e, ω) = Ωe − ω . (2.113) Letting γ̂(K, ε, e, ω) := γ̃(KLe , ε, e, ω) , the second relation in (2.113) can be rewritten as   h(e, ε, K, ω) := Ωe − ω 1 + ε2 γ̂(K, ε, e, ω) = 0 . This last equation may be solved by the standard Implicit Function Theorem: Let e0 (ω) := Ω−1 (ω), then h(e0 (ω), 0, K, ω) = 0 , he (e0 (ω), 0, K, ω) = ∂e Ωe |e=e0 (ω) > 0 . 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