Pseudo-synchronous solutions for dissipative non-autonomous systems
Michele Bartuccelli, Livia Corsi, Jonathan Deane, Guido Gentile
aa r X i v : . [ m a t h . D S ] F e b Pseudo-synchronous solutionsfor dissipative non-autonomous systems
Michele Bartuccelli , Livia Corsi , Jonathan Deane and Guido Gentile Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK Dipartimento di Matematica e Fisica, Universit`a Roma Tre, Roma, I-00146, Italy
E-mail: [email protected], [email protected], [email protected], [email protected]
Abstract
In the framework of KAM theory, the persistence of invariant tori in quasi-integrablesystems is proved by assuming a non-resonance condition on the frequencies, such as thestandard Diophantine condition or the milder Bryuno condition. In the presence of dis-sipation, most of the quasi-periodic solutions disappear and one expects, at most, only afew of them to survive together with the periodic attractors. However, to prove that aquasi-periodic solution really exists, usually one assumes that the frequencies still satisfya Diophantine condition and, furthermore, that some external parameters of the systemare suitably tuned with them. In this paper we consider a class of systems on the one-dimensional torus, subject to a periodic perturbation and in the presence of dissipation,and show that, however small the dissipation, if the perturbation is a trigonometric poly-nomial in the angles and the unperturbed frequencies satisfy a non-resonance condition offinite order, depending on the size of the dissipation, then a quasi-periodic solution existswith slightly perturbed frequencies provided the size of the perturbation is small enough.If on the one hand the maximal size of the perturbation is not uniform in the degree of thetrigonometric polynomial, on the other hand all but finitely many frequencies are allowedand there is no restriction arising from the tuning of the external parameters. A physicallyrelevant case, where the result applies, is the spin-orbit model, which describes the rotationof a satellite around its own axis, while revolving on a Keplerian orbit around a planet, inthe case in which the dissipation is taken into account through the MacDonald torque.
Almost all satellites are locked in an orbital resonance with their planets: the period of revolutionis commensurable with the period of rotation. In many cases the two periods are equal to eachother (resonance 1:1), with the remarkable exception of Mercury, considered as a satellite of theSun (Mercury is entrapped in a 3:2 resonance).A simple model widely used to study the problem is the spin-orbit model , in which thesatellite is described as an ellipsoidal rigid body orbiting around its primary in a Keplerian orbit[21, 29, 36]. Thus, the only variable is the sidereal angle θ that the longest axis of the satelliteforms with respect to the line of the apsides of the orbit. If C denotes the maximal moment ofinertia of the satellite, the time evolution for θ is described by a second order non-autonomous1rdinary differential equation of the form C ¨ θ = T ( θ, ˙ θ, t ) , (1.1)in which the dots represent derivatives with respect to time t and the total torque T ( θ, ˙ θ, t )takes into account both the gravitational force – through the triaxial torque , depending only on θ and t – and the dissipation – through the tidal torque .The appearance of attractors is ultimately related to the presence of dissipation. Whilethere is a universally accepted expression for the triaxial torque, the expression for the tidaltorque is much more tricky. In the literature, up to very recent times, the MacDonald model[31, 29, 19, 39, 15, 3] was the model mainly used for the tidal torque. However, at least in somecases, the applicability of the MacDonald model appears doubtable. From a physical ponit ofview, the main drawback of the MacDonald model is that it predicts the existence of a pseudo-synchronous orbit, that is a solution in which the spin rate is not exactly a rational multiple ofthe mean motion of the satellite. Another disappointing aspect of the model is that, in someinstances, such as that of the Mercury, the probability of capture of the 3:2 resonance is very lowwith respect to those of the coexisting pseudo-synchronous orbit [29, 15, 2]. This runs counterto physical expectations; only assuming a chaotic evolution of Mercury can the probability ofcapture become higher [19].Recently, in the case of Mercury, the validity of the MacDonald model – as well as of anymodel based on constant time lag – was strongly questioned and a more realistic model wasproposed [22, 23, 42, 24, 33, 37]. The probability of capture of the attractor close to theresonance 3:2, as predicted by such a model, is much higher with respect to the MacDonaldmodel [32]. There is strong numerical evidence that the orbit is still pseudo-synchronous, butwith spin rate much closer to the value 3 /
2, and that it corresponds not to a KAM torus, but to aLagrangian torus inside a resonant gap [5]. By contrast, astronomical observations suggest thatthe orbit is purely periodic and with smaller amplitude with respect to the realistic model. Apossible explanation of this apparent discrepancy is that the large-amplitude pseudo-synchronousorbit originally captured the satellite, which, much later in time, because of modifications in itsinternal structure, evolved into the small-amplitude libration observed nowadays: despite thefact that the probability of capture of the final resonant attractor was very low, the satellite wasalready entrapped in its (small) basin of attraction when the nearby pseudo-synchronous orbitdisappeared. We stress that, as widely pointed out in the literature [23, 42, 24, 33, 25, 37], themodel to use for the tidal torque strongly depends on the system one is interested in; moreover,in order to detect the ultimate fate of a satellite, it is important to keep track of how thedissipation evolves in time, considering the huge time scales involved [2, 3, 43, 44].Although the MacDonald model turns out not to be right, at least for Mercury, the MacDon-ald model and the realistic model have one feature in common: both predict the existence of apseudo-synchronous orbit. From a mathematical point of view, a rigorous proof of the existenceof a solution which is not periodic is delicate. In fact, one has to deal with a small divisorproblem and, typically, some Diophantine condition must be assumed on the two-dimensionalfrequency vector; in particular this means that one looks for a quasi-periodic solution.The MacDonald model, because of its simplicity, is well suited for analytical computations(the tidal torque of the realistic model, which is only C and is characterised by narrow, largeamplitudes ‘spikes’ at certain values of ˙ θ , additionally make exhaustive numerical computationsmuch harder [4, 5]). The pseudo-synchronous attractor is expected to reduce to a KAM torusin the absence of dissipation, so that one may use KAM techniques for maximal tori. However,2n order to prove that a quasi-periodic solution exists, one has to adjust an external parameter(depending on the eccentricity), which appears in the MacDonald torque, as a function of thefrequency vector [15, 20]; for similar results see also refs. [10, 16] for the dissipative standard mapand refs. [40, 32] for generalisations of the spin-orbit model to higher dimensions. In principle onecould make the parameter equal to the physical one by using some implicit function argument,but this is hindered by the parameter not being a smooth function of the frequency vector (infact, it is defined on a set full of holes). So, with KAM arguments, one can prove that there arequasi-periodic solutions provided the external parameter is suitably tuned; whether the physicalvalues of the parameters are compatible with the tuning in general remains undecidable.In this paper we aim to study what happens when one looks for a pseudo-synchronoussolution, requiring no condition at all on the external parameter and assuming a non-resonancecondition as weak as possible on the frequency vector. The spin orbit model (1.1), with theMacDonald torque, is a particular case of the class of dynamical systems described by theordinary differential equation ¨ θ + γ ( ˙ θ − α ) + ε f ( θ, t ) = 0 , (1.2)where θ ∈ T = R / π Z , ε is the perturbation parameter (measuring the size of the triaxialtorque), γ is the dissipation parameter and α is the aforementioned external parameter [29, 19,15]. The function f depends periodically on both θ and t , and is real analytic.Most physically relevant is the weakly dissipative case; so we allow γ to be arbitrarily small,by setting γ = ε m γ , with γ independent of ε and m ∈ N . It is quite natural to assume f tohave zero mean, since it is, up to the sign, the derivative with respect to θ of a potential energywhich depends periodically on θ . The main assumption we make, additionally, on f is that it isa trigonometric polynomial of finite order N . Usually the Fourier sum of f is truncated to somefinite order and one expects the remainder to give negligible corrections. In the KAM approach,to deal with full Fourier series or trigonometric polynomials makes no real difference [15]. Bycontrast, for us this a crucial hypothesis.Set, here and henceforth, | a | := | a | = | a | + | a | if a = ( a , a ) ∈ R , and let · denote thestandard scalar product in R , i.e. a · b = a b + a b for any a = ( a , b ) and b = ( b , b ) in R . Hypothesis 1.
The function f is a zero-mean trigonometric polynomial of degree N , i.e. itsFourier components f ν vanish for all ν ∈ Z such that | ν | > N . Set Φ := max {| f ν | : 0 < | ν | ≤ N } . Hypothesis 2.
The vector ω := ( α, is non-resonant up to order m N , i.e. one has ω · ν = 0 ∀ ν ∈ Z such that < | ν | ≤ m N , for some m ∈ N . Theorem 1.
Consider (1.2) , with γ = ε m γ for some m ∈ N , and assume Hypotheses 1 and2. Then there exists ε > , depending on m , N , γ , α and Φ , such that for all | ε | ≤ ε there exists an analytic multi-periodic solution to (1.2) with frequency vector ω = ( α , , with | α − α | ≤ C | ε | , for a suitable constant C . Remark 1.1.
The value ε goes to 0 when either m or N tends to infinity. That ε tendsto vanish with γ is expected: without requiring any stronger non-resonance condition on thefrequency vector, such as the standard Diophantine or the Bryuno condition, the invariant torusbreaks up in the absence of dissipation, as numerical evidence [30, 41, 38, 17] and analyticalresults [26, 6] suggest. On the contrary, that ε becomes zero when f is a generic analyticalfunction might be a technical issue of our approach and we leave as an open problem whetherTheorem 1 can be extended to analytical torques containing an infinite number of harmonics.3or fixed values of ε and γ , when writing γ = ε m γ for some m ∈ N , it may happen thatthe constant γ be rather large. However, Theorem 1 may be straightened by only requiring γ to be of the form γ = | ε | s ¯ γ , with ¯ γ independent of ε and s an arbitrary positive constant. Todo this, one writes γ = ε m γ , where m := ⌈ s ⌉ , so that s = m − a , for some a ∈ [0 , ε does not dependon a as long as 0 ≤ a <
1. This leads to the following extension of Theorem 1.
Theorem 2.
Consider (1.2) , with γ = | ε | s ¯ γ for some s > , and assume Hypotheses 1 and 2,with m = ⌈ s ⌉ . Then there exists ε > , depending on m , N , ¯ γ , α and Φ , such that for all | ε | ≤ ε there exists an analytic multi-periodic solution to (1.2) with frequency vector ω = ( α , , with | α − α | ≤ C | ε | , for a suitable constant C . Remark 1.2.
Theorem 2 shows that, assuming only that the dissipation parameter γ is nottoo large with respect to ε (essentially it must be sublinear in ε ), for ε small enough there exists amulti-periodic solution with frequency vector close to ( α, γ is (significantly) smaller than ε , so one may take ¯ γ = 1 and fix s consequently. Remark 1.3.
Theorems 1 and 2 leave out what happens when α is a rational number p/q , withthe two natural numbers p and q not too large (more precisely such that one has p + q ≤ m N ).Although this is a very unlikely situation – since it corresponds to a set of values for the externalparameter which not only has zero measure but consists in a finite number of points – however,the estimates provided by the proof of Theorem 1 are not uniform on α : in fact, the closer α is to such a rational, the smaller the value of ε . In any case, when α is equal – or even close– to a rational number p/q , with q small, periodic solutions with frequency p/q are known toexist [2, 7]. In such a case, a periodic and a quasi-periodic attractor may coexist: if α is veryclose to a rational p/q , when q is small, the periodic attractor is expected to be dominant, whilethe opposite occurs when q is large: compare the cases of the the Moon, with α very close to1, and Mercury, with α not quite close to 3 /
2, with the continued fraction expansions giving α = [1; 55 , , , , . . . ] and α = [1; 3 , , , , . . . ], respectively [14, 3]. Remark 1.4.
In refs. [15, 40, 32], for ω = ( α ,
1) fixed to be Diophantine, a suitable function˜ α ( α , ε ) is proved to exist such that (1.2), with α := α + ε ˜ α ( ε, α ), admits a quasi-periodicsolution with frequency vector ω . However, the map α ˜ α ( ε, α ) is not smooth, so, for given α , there is no way to establish whether the equation α = α + ε ˜ α ( ε, α ) admits a solution α suchthat ω is Diophantine. On the contrary, under the Hypothesis 2 on α , the function ˜ α ( ε, α ) isfound to be smooth, so that the implicit function theorem applies. Remark 1.5.
Theorems 1 and 2 state the existence of a multi-periodic solution (which describesan invariant torus if it is quasi-periodic), but says nothing about its possible attractiveness. Inrefs [40, 32], the quasi-periodic solution with Diophantine frequency vector is obtained by aKAM algorithm, using the fact that the special form of the dissipation – and its isotropy inhigher dimensions – allows to extend the construction of the normal form to the dissipativecase, and hence to deduce that the solution is normally attractive. Assuming only Hypothesis2 on ω does not guarantee ω to satisfy any Diophantine condition, so that the KAM schemedoes not apply. Certainly, the multi-periodic solution is not a global attractor, since the systemadmits many attracting periodic orbits: essentially as many as the harmonics of the function f [7, 12, 14, 2], even though to prove that the analytical results hold for values of the parameterswhich fit the astronomical values is not quite trivial [1]. However, not all coexisting attractorsare equally relevant [14, 2] and, in general, the basins of attraction in dissipative forced systems4re strongly entwined and are apparently fractal [2, 43, 44, 45], which suggest that any analyticalinvestigation might be very difficult. Remark 1.6.
It would be interesting to extend the results to the case of the Lagrangian toriwhich, in the absence of dissipation, arise inside the small oscillatory islands which appear wherethe KAM tori are destroyed by the perturbation [8, 9, 35]. Indeed, in the case of Mercury, asalready stressed, both numerical and analytical computations for the the spin-orbit model withrealistic tidal torque suggest that the main attractor is a torus of this kind.The paper is organised as follows. In Section 2 we introduce an approximate solution to(1.2), up to order m in ε , and write the equation for the correction to the approximate solution.In Section 3 we first bound the approximate solution, and then we write the correction as a seriesexpansion and provide a graphical representation for its Fourier coefficients. In particular, weshow that the series expansion is well defined to any order, provided suitable counterterms areadded to the external parameter. In Section 4 we show that the series expansion converges. InSection 5 we show that the counterterm can be absorbed in the original value of the externalparameter, by slightly changing the frequency vector without modifying the properties of con-vergence of the series expansion. This concludes the proof of Theorem 1. Finally, in Section 6we show that, by writing γ := ¯ γ | ε | − a , the radius of convergence of the series expansion can beproved to be independent of a for all a ∈ [0 , a = m − s . Consider the non-autonomous ordinary differential equation in T ¨ θ + ε m γ ( ˙ θ − α ) + εf ( θ, t ) = 0 , (2.1)where ε ∈ R is a small parameter, m ∈ N and (cf. Hypothesis 1) f ( ψ ) = X ν ∈ Z < | ν |≤ N e iν · ψ f ν , ψ = ( θ, t ) , f ∗− ν = f ν . (2.2)Since Theorems 1 and 2 hold trivially if f identically zero, in the following, we assume Φ > Remark 2.1.
In the case of the spin-orbit model the sum is restricted to integer vectors ν ∈ Z of the form ν = (2 , − k ), with k = 0 [29, 36]. Specifically for the purpose of numericalsimulations, the sum is usually truncated by requiring N ≤ k ≤ N , for some N , N ∈ Z ;usually one takes N ≥ − N ≤ − and10 − , respectively [20, 19, 39, 14]: this means that, if one aims to describe the physical problemthrough the differential equation (2.1), one has to select m = 2 in (2.1).The system described by (2.1) has a fixed frequency, equal to 1. We look for a multi-periodicsolution with frequency vector ω = ( α , ∈ R , with α close to α to be determined.Set, for notational convenience, Z ∗ := Z \{ } , Z ( N ) = { ν ∈ Z : | ν | ≤ N } , Z ∗ ( N ) = { ν ∈ Z ∗ : | ν | ≤ N } , and define β := min {| ω · ν | : ν ∈ Z ∗ (4 m N ) } . (2.3)5 emark 2.2. If ω is non-resonant, i.e. if its components are rationally independent, one has ω · ν = 0 for all ν ∈ Z ∗ and hence β >
0. A vector ω is said to be non-resonant up to order N if ω · ν = 0 for all ν ∈ Z ∗ ( N ). Of course it is sufficient that ω be non-resonant up to order 4 m N for β to be positive: thus, assuming Hypothesis 2 implies β >
0. Furthermore, if ω = ( α, | ω · ν | ≥ β for all ν ∈ Z ∗ (4 m N ), then one has β ≥ β/ | α − α | is small enough.For ε = 0 any solution to (2.1) is of the form θ ( t ) = ¯ θ + α t , where both ¯ θ ∈ T and α ∈ R can be arbitrarily chosen; without loss of generality we can fix ¯ θ = 0. Set α = α + ε ˜ α , where ˜ α will be referred to as the counterterm , and define the m -th order approximate solution to (2.1),with f given by (2.2), as θ ( t ) + h ( ω t ; ε ) , h ( ω t ; ε ) := εh (1) ( ω t ) + . . . + ε m h ( m ) ( ω t ) , such that ¨ h ( k ) + f ( k ) ( ω t ) = 0 , k = 1 , . . . , m , (2.4)with f (1) ( ω t ) = f ( ω t ) and f ( k ) ( ω t ) := k − X p =1 p ! ∂ p ∂θ p f ( ω t ) X k ,...,k p ≥ k + ... + k p = k − h ( k ) ( ω t ) . . . h ( k p ) ( ω t ) , k = 2 , . . . , m . (2.5)Writing f ( k ) ( ψ ) = X ν ∈ Z ( pN ) e iν · ψ f ( k ) ν , k = 1 , . . . , m , one can immediately check by induction that f ( k ) ν – and hence h ( k ) ν – vanishes whenever | ν | > kN .If β = 0, the system (2.4) can be solved recursively and gives, for all k = 1 , . . . , m , h ( k ) ( ψ ) = X ν ∈ Z ( kN ) e iν · ψ h ( k ) ν , h ( k ) ν := f ( k ) ν ( ω · ν ) ∀ ν ∈ Z ∗ ( kN ) , (2.6)while f ( k )0 = 0 (cf. Lemma 3.4 below for details) and h ( k )0 is arbitrary. The coefficients h ( k ) ν arewell defined if ω is non-resonant up to order m N ; hence h ( k ) is a trigonometric polynomial ofdegree kN , for k = 1 , . . . , m , and h is a trigonometric polynomial of degree m N .Once the function h ( ψ ; ε ) has been determined, write θ ( t ) = α t + h ( ω t ; ε ) + ε m +1 H ( ω t ; ε ) , H ( ψ ; ε ) = X ν ∈ Z e iν · ψ H ν , (2.7)which, inserted into (2.1), provides a differential equation for H :¨ H + ε m γ ˙ H − γ ˜ α + 1 ε m F ( ω t, H ; ε ) = 0 , (2.8)with F ( ω t, H ; ε ) := ∞ X k = m +1 ε k − k − X p =1 p ! ∂ p ∂θ p f ( ω t ) X ≤ k ,...,k p ≤ m k + ... + k p = k − h ( k ) ( ω t ) . . . h ( k p ) ( ω t )+ ∞ X k =1 ∂ k ∂θ k f ( ω t ) X p ≥ ,q ≥ p + q = k p ! q ! ( h ( ω t ; ε )) p ( ε m +1 H ) q − ε m − γ ˙ h ( ω t ; ε ) . (2.9)6ote that, due to the constraints on the exponents, F is at least of order ε m . Moreover, since h + ε m +1 H can be absorbed into ¯ θ , we can and shall assume that both h and H vanish.In Fourier space, if we set F ν ( q ) := ( iν , ) q f ν + ∞ X p =1 p ! X ≤ k ,...,k p ≤ m ε k + ... + k p X ˜ ν ∈ Z ∗ ( N )˜ ν ,..., ˜ ν p ∈ Z ∗ ( m N )˜ ν +˜ ν + ... +˜ ν p = ν ( i ˜ ν , ) p + q f ˜ ν h ( k )˜ ν . . . h ( k p )˜ ν p , (2.10a) F ν := ∞ X p =1 p ! X ≤ k ,...,k p ≤ m k + ... + k p ≥ m ε k + ... + k p − m X ˜ ν ∈ Z ∗ ( N )˜ ν ,..., ˜ ν p ∈ Z ∗ ( m N )˜ ν +˜ ν + ... +˜ ν p = ν ( i ˜ ν , ) p f ˜ ν h ( k )˜ ν . . . h ( k p )˜ ν p , (2.10b) G ν := − ( iω · ν ) γ m X k =1 ε k − h ( k ) ν , (2.10c)then (2.8) gives for ν = 0 (cid:16) ( ω · ν ) − iε m γ ω · ν (cid:17) H ν = F ν + G ν + ∞ X q =1 q ! ε ( m +1) q − m X ν ,ν ,...,ν q ∈ Z ∗ ν + ... + ν q = ν F ν ( q ) H ν . . . H ν q , (2.11)provided one has for ν = 0 γ ˜ α = F + ∞ X q =1 q ! ε ( m +1) q − m X ν ,ν ,...,ν q ∈ Z ∗ ν + ... + ν q =0 F ν ( q ) H ν . . . H ν q . (2.12)In the following sections we prove that (2.11) admits a solution in the form of a convergentseries, provided ˜ α is fixed according to (2.12). For given α , the counterterm ˜ α is uniquelydefined by (2.12): this fixes α to the value α = α + ε ˜ α . However, the physical parameter is α , so what we really need to do is the opposite, i.e. to find, for fixed α , a value α such that α + ε ˜ α = α . This leads to solve an implicit function problem. A rooted tree ϑ is an oriented graph with no cycle, such that all the lines are oriented toward aunique vertex (the root ) which has only one incident line (the root line ℓ ϑ ). We call nodes allthe vertices in ϑ except the root. The orientation of the lines in ϑ induces a partial orderingrelation ( (cid:22) ) between the nodes. Given two nodes v and w , we shall write w ≺ v every time v isalong the path (of lines) connecting w to the root; we shall write w ≺ ℓ if w (cid:22) v , where v is theunique node that the line ℓ exits. If a line ℓ exits a node v we may write ℓ = ℓ v . For any node v denote by q v the number of lines entering v .Given a rooted tree ϑ we denote by N ( ϑ ) the set of nodes, by E ( ϑ ) the set of end nodes (or levaes ), i.e. nodes v with q v = 0, by I ( ϑ ) the set of internal nodes , i.e. nodes v with q v ≥
1, andby L ( ϑ ) the set of lines; by definition N ( ϑ ) = E ( ϑ ) ⊔ I ( ϑ ).7e associate with each node v ∈ N ( ϑ ) a mode label ν v ∈ Z and with each line ℓ ∈ L ( θ ) \{ ℓ ϑ } a momentum label ν ℓ ∈ Z ∗ , while we associate with the root line ℓ ϑ a momentum ν ℓ ϑ ∈ Z . Wealso impose the conservation law ν ℓ = X v ≺ ℓ ν v ∀ ℓ ∈ L ( ϑ ) . Lemma 3.1.
For any tree θ with k nodes one has X v ∈ I ( ϑ ) q v = k − , | I ( θ ) | ≤ k − . Proof . For each node v there are q v entering lines and all lines enter a node, the only exceptionbeing the root line. Moreover there is at least one end node. Lemma 3.2.
The number of unlabelled trees with k nodes is bounded by k .Proof . The number of trees with k nodes and with no labels is bounded by the number ofrandom walks with 2 k steps, that is 2 k . h Here and in the following, we provide a graphical representation in terms of labelled trees, asdefined in Section 3.1, for both the coefficients h ( k ) ν and H ( k ) ν . We start with the the coefficientsof the approximate solution h .Assume ω to be non-resonant up to order m N , so that β = 0. In Fourier space (2.5) gives,for k = 2 , . . . , m , f ( k ) ν = k − X p =1 p ! X k ,...,k p ≥ k + ... + k p = k − X ν ∈ Z ∗ ( N ) ( iν , ) p f ν X ν ,...,ν p ∈ Z ∗ ( m N ) ν + ... + ν p = ν − ν h ( k ) ν . . . h ( k p ) ν k . (3.1)A tree ϑ is said to be of order k if one has | N ( ϑ ) | = k . Given a tree ϑ , one associates witheach node v ∈ N ( ϑ ) a node factor I ∗ v and with each line ℓ ∈ L ( ϑ ) a propagator G ∗ ℓ , where I ∗ v and G ∗ ℓ are defined as I ∗ v := ( iν v, ) q v q v ! f ν v , G ∗ ℓ := ω · ν ℓ ) , ν ℓ ∈ Z ∗ ( m N ) , , ν ℓ = 0 . Define the value of the tree ϑ as V ∗ ( ϑ ) := (cid:16) Y v ∈ N ( ϑ ) I ∗ v (cid:17)(cid:16) Y ℓ ∈ L ( ϑ ) G ∗ ℓ (cid:17) , and let T ∗ k,ν denote the set of the labelled trees of order k and with momentum ν associatedwith the root line.Finally, set, for ν ∈ Z and k ∈ Z + , p ( ν ) := min { p ∈ Z + : | ν | ≤ ( p + 1) N } , p k ( ν ) = max { p ( ν ) − k, } , (3.2)and define the small divisors D ( ω · ν ) := ( ω · ν ) − iε m γ ω · ν − εF (1) . (3.3)8 emma 3.3. Let p ( ν ) and p k ( ν ) be defined as in (3.2) . Then one has p ( ν ) + p ( ν ) ≥ p ( ν + ν ) − , (3.4a) p ( ν ) + p k ( ν ) ≥ p k ( ν + ν ) − . (3.4b) Proof . One has p ( ν + ν ) N < | ν + ν | ≤ | ν | + | ν | ≤ ( p ( ν ) + 1) N + ( p ( ν ) + 1) N, which yields p ( ν + ν ) + 1 ≤ p ( ν ) + p ( ν ) + 2, since p ( ν ) is an integer for any ν ∈ Z . Thisproves (3.4a).If p ( ν ) ≥ k and p ( ν + ν ) ≥ k , then (3.4b) follows from (3.4a). If p ( ν + ν ) < k , (3.4b) holdstrivially, since both p ( ν ) and p k ( ν ) are non-negative. Finally, if p ( ν ) < k and p ( ν + ν ) ≥ k ,one has | ν + ν | ≤ | ν | + | ν | ≤ ( p ( ν ) + 1) N + kN and hence p ( ν + ν ) ≤ p ( ν ) + k + 1, whichyields p ( ν ) ≥ p k ( ν + ν ) − Lemma 3.4.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . For k = 1 , . . . , m , one has f ( k )0 := X ϑ ∈ T ∗ k, V ∗ ( ϑ ) , h ( k ) ν := X ϑ ∈ T ∗ k,ν V ∗ ( ϑ ) , ν ∈ Z ∗ ( kN ) , , | ν | > kN. Moreover, one has f ( k )0 = 0 for k = 1 , . . . , m and hence the function 2.6 solves (2.4) .Proof . The graphical representation in tems of trees is proved by induction, using the recursivedefinition (2.6) for the coefficients h ( k ) ν and the definition of the tree values.The vanishing of f ( k )0 is a symmetry property, due to the fact that, up to order m the systemis conservative [27, 18, 28]. Given a tree ϑ , if v is the node the root line exits, call ϑ ( v ) the treeobtained by detaching the root line from v and reattaching to the node v ∈ N ( ϑ ), and P ( v , v )the unique path of lines connecting the node v to the node v . f F ( ϑ ) denotes the family of alltrees obtained from ϑ by varying the node v which the root line is reattached to, one has X ϑ ∈ T k, V ∗ ( ϑ ) = X ϑ ∈ T k, | F ( ϑ ) | X ϑ ′ ∈ F ( ϑ ) V ∗ ( ϑ ′ ) . Since the mean of f is zero, one can express f ( θ, t ) as the derivative with respect to θ of afunction ϕ ( θ, t ) and write f ν = iν ϕ ν . Let Q w denote the number of lines incident on w ∈ N ( ϑ ),without counting the root line for the node v ; for any ϑ ( v ) ∈ F ( ϑ ) one has q v = Q v + 1 and q w = Q w for any other node w . Call ϑ ∗ the non-ordered graph obtained from ϑ by detachingthe root line, and set V ∗ ( ϑ ∗ ) = (cid:16) Y v ∈ N ( ϑ ∗ ) ( iν v, ) Q v f ν v Q v ! (cid:17)(cid:16) Y ℓ ∈ L ( ϑ ∗ ) G ℓ (cid:17) , with N ( ϑ ∗ ) := N ( ϑ ) and L ( ϑ ∗ ) := L ( ϑ ) \ { ℓ θ } . The value of tree θ ( v ) obtained from θ byreattaching the root line to v differs from V ∗ ( θ ) because the momenta of the lines ℓ ∈ P ( v , v )are reversed (i.e. ν ℓ is replaced by − ν ℓ ) and the combinatorial factors 1 /q v ! and 1 /q v ! of the9wo nodes v and v are replaced by 1 / ( q v − / ( q v + 1)!, respectively. However, thereare q v trees obtained from θ ∗ which have the same value as θ and q v + 1 trees obtained from θ ∗ trees which have the same value as θ ( v ), so that X ϑ ′ ∈I ( ϑ ) V ∗ ( ϑ ′ ) = X v ∈ N ( ϑ ) iν v, V ∗ ( ϑ ∗ ) = i V ∗ ( ϑ ∗ ) X v ∈ N ( ϑ ) ν v, = 0 , which implies that f ( k )0 = 0 for k = 1 , . . . , m . Lemma 3.5.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . For k = 1 , . . . , m , one has | h ( k ) ν | ≤ ( β − Φ ) k ∀ ν ∈ Z ∗ ( kN ) , Φ := 16 N Φ , with β as in (2.3) . Moreover one has h ( k ) ν = 0 for all ν ∈ Z such that p ( ν ) ≥ k .Proof . The value of any tree ϑ ∈ T ∗ k,ν , with k ≤ m and ν = 0, is bounded as | V ∗ ( ϑ ) | ≤ N k − Φ k β − k , where Lemma 3.1 has been used. The sum of the mode labels is bounded by (4 N ) k and thenumber of trees of order k is bounded by 4 k by Lemma 3.2. This yields the bound on h ( k ) ν .Finally, by construction (cf. Lemma 3.4), h ( k ) ν = 0 requires | ν | ≤ kN , i.e. p ( ν ) ≤ k − Lemma 3.6.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Thereexist positive constants A , K and ε such that, for all | ε | ≤ ε , one has (cid:12)(cid:12) F (1) + A ε (cid:12)(cid:12) ≤ K | ε | . Proof . From (2.10a), with q = 1 and ν = 0, one obtains F (1) = ε X ˜ ν ∈ Z ∗ ( N ) ( i ˜ ν , ) f ˜ ν h (1) − ˜ ν + ˜ F (1)with ˜ F (1) := m X k =2 ε k X ˜ ν ∈ Z ∗ ( N ) ( i ˜ ν , ) f ˜ ν h ( k ) − ˜ ν + ∞ X p =2 p ! X ≤ k ,...,k p ≤ m ε k + ... + k p X ˜ ν , ˜ ν ,..., ˜ ν p ∈ Z ∗ ( m N )˜ ν +˜ ν + ... +˜ ν p = ν ( i ˜ ν , ) p +1 f ˜ ν h ( k )˜ ν . . . h ( k p )˜ ν p , where we have (a) used that the first term in (2.10a) vanishes and (b) separated the rest of thesum, i.e. ˜ F (1), from the contribution of order ε . The latter, using (2.6), becomes ε X ˜ ν ∈ Z ∗ ( N ) ( i ˜ ν , ) f ˜ ν f − ˜ ν ( ω · ˜ ν ) = − ε C , < C := 12 X ˜ ν ∈ Z ∗ ( N ) (˜ ν , ) | f ˜ ν | ( ω · ˜ ν ) ≤ N β − Φ . (3.5)Moreover one has, by using Lemma 3.5, | ˜ F (1) | ≤ m X k =2 N Φ (cid:0) | ε | β − Φ (cid:1) k + ∞ X p =2 p m N X k = p X ≤ k ,...,k p ≤ m k + ... + k p = k N p +1 Φ(4 m N ) p ( | ε | β − Φ ) k ≤ m N Φ (cid:0) | ε | β − Φ (cid:1) + 3 N Φ (cid:0) m N | ε | β − Φ (cid:1) ≤ K | ε | , with K := 4 N Φ (cid:16) m N Φ β (cid:17) , (3.6)provided one takes | ε | ≤ ε , with ε := β m N Φ . (3.7)Therefore, one obtains (cid:12)(cid:12) F (1) + A ε (cid:12)(cid:12) ≤ K | ε | for all | ε | ≤ ε . Lemma 3.7.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Thereexists positive constants C and ε such that, for all | ε | ≤ ε , one has | D ( ω · ν ) | ≥ | ε | m γ | ω · ν | , ∀ ν ∈ Z ∗ , | D ( ω · ν ) | ≥ ( ω · ν ) + C | ε | ≥ max (cid:26) C | ε | , β (cid:27) ∀ ν ∈ Z ∗ | D ( ω · ν ) | ≥ β ∀ ν ∈ Z ∗ (4 m N ) . Proof . Define ˜ ε := C K = C ε N Φ , ε := min { ε , ˜ ε } , (3.8)with C , K and ε defined in (3.5), in (3.6) and in (3.7), respectively.One easily check from (2.10a) that F ( q ) is real for any q ∈ Z + , so that one can bound | D ( ω · ν ) | ≥ | ε m γ ω · ν | .Moreover one has | D ( ω · ν ) | ≥ | ( ω · ν ) − εF (1) | ≥ ( ω · ν ) + C | ε | ≥ max (cid:8) ( ω · ν ) , C | ε | (cid:9) , for all | ε | ≤ ε . Thus, one can also bound D ( ω · ν ) by C | ε | , whenever one has | ω · ν | < β / | D ( ω · ν ) | ≥ ( ω · ν ) ≥ β for all ν ∈ Z ∗ (4 m N ) by (2.3). Remark 3.8.
By comparing (3.8) with (3.6) and (3.7), one finds˜ ε ≤ C ε N Φ , C ε N Φ ≤ N β − Φ N Φ β m N (16 N Φ) ≤ m N ) , so that ˜ ε ≤ m N ) − ε . Thus, one has ε = ˜ ε .11 .4 Bounds on the the coefficients F ν ( q ), F ν and G ν . Lemma 3.9.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Let ε be as in Lemma 3.6. For all | ε | ≤ ε , all q ≥ and all ν ∈ Z , one has | F ν ( q ) | ≤ N q Φ | D ε | p ( ν ) , D := 8 m N β − Φ , where Φ is as in Lemma 3.5.Proof . According to (2.10a) and Lemma 3.5, we have, if p ( ν ) ≥ f ν = 0, | F ν ( q ) | ≤ ∞ X p =1 p ! N p + q Φ X ˜ ν ,..., ˜ ν p ∈ Z ∗ ( m N ) p Y i =1 X k i ≥ p (˜ ν i )+1 (cid:0) | ε | β − Φ (cid:1) k i ! ≤ N q Φ ∞ X p =1 N p (cid:0) m N (cid:1) p p (cid:0) | ε | β − Φ (cid:1) max { p ( ν ) ,p } , where we have used that p (˜ ν ) = 0, p (˜ ν ) , . . . , p (˜ ν p ) ≥ p (˜ ν )+ p (˜ ν )+ . . . + p (˜ ν p ) ≥ p ( ν ) − p by Lemma 3.3, so that we obtain | F ν ( q ) | ≤ N q Φ (cid:0) | ε | β − Φ (cid:1) p ( ν ) p ( ν ) X p =1 (cid:0) m N (cid:1) p + ∞ X p = p ( ν )+1 (cid:0) m N | ε | β − Φ (cid:1) p ! ≤ N q Φ (cid:16) (cid:0) m N | ε | β − Φ (cid:1) p ( ν ) + (cid:0) m N | ε | β − Φ (cid:1) p ( ν ) (cid:17) , provided we take | ε | ≤ ε , with ε given by (3.7).If p ( ν ) = 0, instead, we have | F ν ( q ) | ≤ N q Φ + N q Φ ∞ X p =1 N p (cid:0) m N (cid:1) p p (cid:0) | ε | β − Φ (cid:1) p ≤ N q Φ , because we have to take into account also the first summand in (2.10a). Remark 3.10.
For ν = 0, the bound in Lemma 3.9 might be improved into 2 N q Φ | D ε | , since p ( ν ) = 0 and f = 0 vanishes. However, in the following, we do not need the improved bound,except when q = 1, for which Lemma 3.6 applies and, in particular, gives | F (1) | ≤ A | ε | , ifone assumes the stronger condition | ε | ≤ ε . Lemma 3.11.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Define ε and D as in Lemmas 3.6 and 3.9, respectively, and set Γ := 2 γβ − Φ D − m . Then, for all | ε | ≤ ε one has | F ν | ≤ D m | D ε | p m ( ν ) , ν ∈ Z ∗ , | F | ≤ D m | D ε | , | G ν | ≤ Γ D m , ν ∈ Z ∗ ( m N ) , while one has G ν = 0 for ν ∈ Z \ Z ∗ ( m N ) . roof . For | ε | ≤ ε , with ε as in Lemma 3.6, according to (2.10b) one has | F ν | ≤ ∞ X p =1 p ! N p Φ X k ≥ max { m ,p } X k ,...,k p ≥ k + ... + k p = k (4 m N ) p | ε | k − m (cid:0) β − Φ (cid:1) k ≤ Φ | ε | − m m X p =1 ∞ X k = m (4 m N ) p (cid:0) | ε | β − Φ (cid:1) k + Φ | ε | − m ∞ X p = m +1 ∞ X k = p (4 m N ) p (cid:0) | ε | β − Φ (cid:1) k ≤ | ε | − m ∞ X k = m (4 m N ) m (cid:0) | ε | β − Φ (cid:1) k + 2Φ | ε | − m ∞ X p = m +1 (4 m N ) p (cid:0) | ε | β − Φ (cid:1) p ≤ | ε | − m (cid:0) m N | ε | β − Φ (cid:1) m , which yields the first bound for ν ∈ Z ∗ ( m N ) . Since the sum over the Fourier labels in (2.10b)has the constraint ν = ˜ ν + ˜ ν + . . . + ˜ ν p , by Lemma 3.4 one has | ν | ≤ k X p =0 | ˜ ν k | ≤ N + k N + . . . + k p N ≤ ( k + 1) N, so that k ≥ p ( ν ). If | ν | > m N , then p ( ν ) ≥ m and hence | F ν | ≤ ∞ X p =1 p ! N p Φ X k ≥ max { p ( ν ) ,p } X k ,...,k p ≥ k + ... + k p = k (4 m N ) p | ε | k − m (cid:0) β − Φ (cid:1) k , ≤ | ε | − m ∞ X k = p ( ν ) (4 m N ) p ( ν ) (cid:0) | ε | β − Φ (cid:1) k + 2Φ | ε | − m ∞ X p = p ( ν )+1 (4 m N ) p (cid:0) | ε | β − Φ (cid:1) p ≤ | ε | − m (cid:0) m N | ε | β − Φ (cid:1) p ( ν ) , which shows that the bound still holds for ν ∈ Z \ Z ( m N ).For ν = 0, the contribution with k + · · · + k p = m in (2.10b) is f ( m +1)0 := ∞ X p =1 p ! X ≤ k ,...,k p ≤ m k + ... + k p = m X ˜ ν ∈ Z ∗ ( N )˜ ν ,..., ˜ ν p ∈ Z ∗ ( m N )˜ ν +˜ ν + ... +˜ ν p =0 ( i ˜ ν , ) p f ˜ ν h ( k )˜ ν . . . h ( k p )˜ ν p , which vanishes (the proof is the same as the proof that f ( k )0 = 0 for k = 1 , . . . , m in Lemma 3.5),while the contributions with k + · · · + k p ≥ m + 1 can be bounded by the same reasoning as wasused to obtain the first bound, with the difference that now k ≥ max { p, m + 1 } . By collectingthe two results one obtains the second bound.The last bound follows from (2.10c), by noting that, for | ε | ≤ ε , one has, for ν ∈ Z ∗ ( m N ), | G ν | ≤ β − | ε | − γ m X k =1 β − k − | ε | k Φ k ≤ γβ − Φ , since one has | h ( k ) ν | ≤ | ω · ν | − β − k − Φ k and | ε | β − Φ ≤ /
2. Finally, by Lemma 3.5, one has G ν = 0 if either ν = 0 or | ν | > m N . 13 .5 Perturbation series for the function H In order to study (2.11) and (2.12), it is convenient to introduce an auxiliary parameter µ , bywriting, for ν = 0, (cid:16) ( ω · ν ) − iε m γ ω · ν − εF (1) (cid:17) H ν = µ F ν + G ν + ε X ν ∈ Z ∗ F ν (1) H ν − ν + ∞ X q =2 ε ( m +1) q − m X ν ,...,ν q ∈ Z ν + ... + ν q = ν F ν ( q ) H ν . . . H ν q ! , while requiring, for ν = 0, that γ ˜ α = µ F + ε X ν ∈ Z ∗ F ν (1) H − ν + ∞ X q =2 ε ( m +1) q − m X ν ,ν ,...,ν q ∈ Z ν + ... + ν q =0 F ν ( q ) H ν . . . H ν q ! . Note that, for µ = 1, the two equations reduce to (2.11) and (2.12), respectively.If one writes H ( ψ ; ε ) = ∞ X k =1 µ k H ( k ) ( ψ ) = ∞ X k =1 X ν ∈ Z µ k e iν · ψ H ( k ) ν , ˜ α = ∞ X k =1 µ k α ( k ) , (3.9)then one obtains the recursive equations D ( ω · ν ) H (1) ν = F ν + G ν (3.10a) γα (1) = F , (3.10b)for k = 1, D ( ω · ν ) H (2) ν = ε X ν ∈ Z ∗ F ν (1) H (1) ν − ν , (3.11a) γα (2) = ε X ν ∈ Z ∗ F ν (1) H (1) − ν , (3.11b)for k = 2, and, for k ≥ D ( ω · ν ) H ( k ) ν = ε X ν ∈ Z ∗ F ν (1) H ( k − ν − ν + ∞ X q =2 ε ( m +1) q − m X ν ,...,ν q ∈ Z ν + ... + ν q = ν F ν ( q ) X k ,...,k q ≥ k + ... + k q = k − H ( k ) ν . . . H ( k q ) ν q , (3.12a) γα ( k ) = ε X ν ∈ Z ∗ F ν (1) H ( k − − ν + ∞ X q =2 ε ( m +1) q − m X ν ,ν ,...,ν q ∈ Z ν + ... + ν q =0 F ν ( q ) X k ,...,k q ≥ k + ... + k q = k − H ( k ) ν . . . H ( k q ) ν q . (3.12b)The quantities D ( ω · ν ) are bounded from below by Lemma 3.7. By looking at the recursiveequations (3.10)-(3.12) and using that the coefficients F ν ( q ) and F ν ( q ) are well defined anddecay exponentially in ν (by Lemmas 3.9 and 3.11), one easily proves by induction that thecoefficients H ( k ) ν are well defined for all k ∈ N and all ν ∈ Z ∗ .14 .6 Tree representation for the function H Hereafter we consider trees which, in addition to the constraints considered in Section 1, aresuch that for any v ∈ V ( θ ), if q v = 1, then one has ν v = 0; this implies that if there is only oneline ℓ entering a node, then ℓ cannot have the same momentum as the line exiting that node.We assign a further label λ v = F, G to each node v ∈ E ( ϑ ). We call λ v the end node label and set E λ ( ϑ ) := { v ∈ E ( ϑ ) : λ v = λ } , so that E ( ϑ ) = E F ( ϑ ) ⊔ E G ( ϑ ). Let T k,ν denote the setof all labelled trees with | N ( ϑ ) | = k such that the root line has momentum ν .We aim to show also that also the coefficients H ( k ) ν of the function H admit a graphicalrepresentation in terms of trees, with the further constraint mentioned above: of course, whatchanges, with respect to the coefficients of h , are the node factors and the propagators to beassociated with the nodes and the lines, respectively.We associate with each node v ∈ N ( ϑ ) a node factor I v , if v ∈ I ( ϑ ), and E v , if v ∈ E ( ϑ ),and with each line ℓ ∈ L ( ϑ ) propagator G ℓ , with I v := 1 q v ! F ν v ( q v ) , E v := F ν v , λ v = F,G ν v , λ v = G, G ℓ := D ( ω · ν ℓ ) , ν ℓ = 0 ,γ − ν ℓ = 0 . Define the value of the tree ϑ as V ( ϑ ) := (cid:16) Y v ∈ I ( ϑ ) ε ( m +1) q v − m I v (cid:17)(cid:16) Y v ∈ E ( ϑ ) E v (cid:17)(cid:16) Y ℓ ∈ L ( ϑ ) G ℓ (cid:17) . By construction only the root line may have zero momentum. Thus, an end node v ∈ E ( ϑ ) mayhave ν v = 0 if and only if ϑ ∈ T , , and in such a case one has V ( ϑ ) = γ − F . Lemma 3.12.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . For | ε | ≤ ε , with ε defined in Lemma 3.7, and or any k ∈ N one has H ( k ) ν := X ϑ ∈ T k,ν V ( ϑ ) , ν ∈ Z ∗ , α ( k ) := X ϑ ∈ T k, V ( ϑ ) . Proof . By induction, as in the proof of Lemma 3.4.
Define, for any tree ϑ ∈ T k,ν , with ( k, ν ) ∈ N × Z \ { (1 , } , W ( ϑ ) := (cid:16) Y v ∈ I ( ϑ ) | D ε | ( m +1) q v − m + p ( ν v ) (cid:17)(cid:16) Y v ∈ E ( ϑ ) | D ε | p m ( ν v ) (cid:17)(cid:16) Y ℓ ∈ L ( ϑ ) |G ℓ | (cid:17) , so that, by taking into account Lemma 3.1, one can bound, for any tree T k,ν with ( k, ν ) = (1 , V ( ϑ ) ≤ W ( ϑ ) D m − ( k − N k − (cid:16) Y v ∈ I ( ϑ ) ( D | ε | ) p ( ν v ) (cid:17)(cid:16) Y v ∈ E F ( ϑ ) ( D | ε | ) p m ( ν v ) (cid:17)(cid:16) Y v ∈ E G ( ϑ ) Γ (cid:17) . (4.1)15 emark 4.1. If | ε | ≤ ε , with ε as in Lemma 3.7, then one has | D ε | ≤ /
2. Indeed, bydefinition of D and ε , one has D ε = 1 /
2, so that, for | ε | ≤ ε , by using Remark 3.8, oneobtains | D ε | ≤ − | D ε | ≤ − . Lemma 4.2.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Let D be defined as in Lemma 3.9. There exists a positive constant B such that, for all k ∈ N , all ν ∈ Z ∗ and all | ε | ≤ ε , with ε as in Lemma 3.7, one has W ( ϑ ) ≤ B k | D ε | k − for any tree ϑ ∈ T k,ν .Proof . The proof is by induction on k . If ϑ ∈ T ,ν , then I ( ϑ ) is empty, E ( ϑ ) contains only onenode v and L ( ϑ ) contains only the root line ℓ . One has W ( ϑ ) = | D ε | p m ( ν ) |G ℓ | , where ν = ν v . If p ( ν ) < m , one has |G ℓ | ≤ β − , while if p ( ν ) ≥ m (and hence p m ( ν ) ≥ m ), | D ε | p m ( ν ) |G ℓ | ≤ | D ε | m C − D | D ε | − < C − D , so that the bound holds if one takes B ≥ max { β − , C − D } = C − D .If ϑ ∈ T k,ν , with k >
1, let v be the node which the root line exits and let ϑ , . . . , ϑ q , with q ≥
1, denote the subtrees which have v as root. By construction one has W ( ϑ ) = | D ε | ( m +1) q − m + p ( ν v ) |G ℓ | q Y i =1 W ( ϑ i ) .
1. If q ≥
2, one uses the inductive hypothesis to bound q Y i =1 W ( ϑ i ) ≤ q Y i =1 B k i | D ε | ki − = B k − | D ε | k − − q , since one has k + . . . + k q = k −
1. Thus, for q ≥
2, one obtains W ( ϑ ) ≤ | D ε | ( m +1) q − m max { β − , C − D | D ε | − } B k − | D ε | k − − q . By using that( m + 1) q − m − k − − q (cid:16) m + 34 (cid:17) q − ( m + 2) + k − > k − , the bound follows by requiring B ≥ C − D .2. If q = 1 one has W ( ϑ ) = | D ε | p ( ν v ) |G ℓ |W ( ϑ ) , so that, if | ω · ν | ≥ β /
4, the bound follows once more by the inductive hypothesis, since W ( ϑ ) ≤ | D ε | β − B k − | D ε | k − < β − B k − | D ε | k − , which implies the bound provided one has B ≥ β − . If | ω · ν | < β / |G ℓ | ≤ C − | ε | − by Lemma 3.7), then one has p ( ν ) ≥ m and hence p m ( ν ) ≥ m . Let ℓ denotethe root line of ϑ , v the node which ℓ exits, and ϑ ′ , . . . , ϑ ′ q ′ the subtrees which have v asroot. 16.1. If q ′ = 0 (i.e. k = 2) we distinguish between the following cases.2.1.1. If | ω · ν ℓ | ≥ β /
4, then, by Lemma 3.3, one bounds W ( ϑ ) ≤ | D ε | − ( p ( ν v )+ p m ( ν ℓ )) C − D β − ≤ B | D ε | m − < B | D ε | , if B ≥ C − D , since p ( ν v ) + p m ( ν ℓ ) ≥ p ( ν v + ν ℓ ) − ≥ m − | ω · ν ℓ | < β /
4, then p ( ν ℓ ) ≥ m and hence p m ( ν ℓ ) ≥ m . Moreover | ω · ν v | = | ω · ( ν − ν ℓ ) | ≤ | ω · ν | + | ω · ν ℓ | < β p ( ν v ) ≥ m , so that one bounds W ( ϑ ) ≤ | D ε | ( p ( ν v )+ p m ( ν ℓ )) C − | ε | − ≤ | D ε | m − C − D < B | D ε | , if B ≥ C − D .2.2. If q ′ ≥ | ω · ν ℓ | < β /
4, one bounds |G ℓ | ≤ C − | ε | − and one has p ( ν v ) ≥ m , otherwise onewould find β ≤ | ω · ν v | ≤ | ω · ν | + | ω · ν ℓ | < β , which would lead to a contradiction. Thus one finds W ( ϑ ) ≤ C − | ε | − | D ε | p ( ν v ) C − | ε | − | D ε | ( m +1) q ′ − m + p ( ν v ) q ′ Y i =1 W ( ϑ ′ i ) ≤ C − D B k − | D ε | ( m +1) q ′ +2 m − k − − q ′ , where one has m , q ′ ≥ m + 1) q ′ + 2 m − k − − q ′ ≥ k − , which yields the bound if one requires B ≥ C − D B − .2.2.2. If | ω · ν ℓ | ≥ β / q ′ ≥
2, one has W ( ϑ ) ≤ C − D | D ε | − | D ε | ( p ( ν v )+ p ( ν v ))+( m +1) q ′ − m β − q ′ Y i =1 |W ( ϑ ′ i ) |≤ C − D β − B k − | D ε | ( m +1) q ′ − m − k − − q ′ , where one has( m + 1) q ′ − ( m + 1) + k − − q ′ (cid:16) m + 34 (cid:17) q ′ − ( m + 1) + k − ≥ k − , so that the bound follows provided one has B ≥ C − D β − B − .2.2.3. If | ω · ν ℓ | ≥ β / q ′ = 1, let ℓ ′ denote the root line of ϑ ′ , v ′ the node which ℓ ′ exits, and ϑ ′′ , . . . , ϑ ′′ q ′′ the subtrees which have v ′ as root.17.2.3.1. If q ′′ = 0 (i.e. k = 3), we distinguish between two cases.2.2.3.1.1. If | ω · ν ℓ ′ | ≥ β / p ( ν ) ≥ m and hence p m ( ν ) ≥ m , one obtains, if B ≥ C − D , W ( ϑ ) ≤ C − D (16 β − ) | D ε | ( p ( ν v )+ p ( ν v )+ p m ( ν v ′ )) ≤ B | D ε | < B | D ε | , because, by Lemma 3.3, one has p ( ν v ) + p ( ν v ) + p m ( ν v ′ )) ≥ p m ( ν ) − ≥ m − | ω · ν ℓ ′ | ≤ β /
4, then one has p m ( ν v ′ ) ≥ m . Moreover one has | ω · ( ν v + ν v ) | = | ω · ( ν − ν ℓ ′ ) | ≤ | ω · ν | + | ω · ν ℓ ′ | < β , so that, since, by Lemma 3.3, one has p ( ν v ) + p ( ν v ) ≥ p ( ν v + ν v ) − ≥ m − W ( ϑ ) ≤ C − D β − | D ε | − ( p ( ν v )+ p ( ν v )+ p m ( ν v ′ )) ≤ B | D ε | m − < B | D ε | , provided that B ≥ C − D .2.2.3.2. If one has q ′′ ≥
1, we still have two cases to discuss.2.2.3.2.1. If one has | ω · ν ℓ ′ | ≥ β /
4, one obtains W ( ϑ ) ≤ C − D | D ε | − | D ε | (16 β − ) B k − | D ε | ( m +1) q ′′ − m + k − − q ′′ , where one has − m + 1) q ′′ − m + k − − q ′′ (cid:16) m + 34 (cid:17) q ′′ − m + k − > k − , so that the bound holds as soon as one requires B ≥ C − D (16 β − ) B − .2.2.3.2.2. If, instead, one has | ω · ν ℓ ′ | < β /
4, one obtains W ( ϑ ) ≤ ( C | ε | ) − β − | D ε | ( p ( ν v )+ p ( ν v )) B k − | D ε | ( m +1) q ′′ − m + k − − q ′′ , and, since ν = ν v + ν v + ν ℓ ′ , one has | ω · ( ν v + ν v ) | = | ω · ( ν − ν ℓ ′ ) | ≤ | ω · ν | + | ω · ν ℓ ′ | < β , which in turns implies p ( ν v + ν v ) ≥ m and hence p ( ν v ) + p ( ν v ) ≥ m − W ( ϑ ) ≤ C − D β − B k − | D ε | (4 m − − m +1) q ′′ − m + k − − q ′′ , where, using that m , q ′′ ≥
1, one has34 (4 m − − m + 1) q ′′ − m + k − − q ′′ > k − , so that the bound follows once more provided that one requires B ≥ C − D β − B − .By collecting together the conditions required on the constant B and taking into account Remark4.1, one may fix B as B = C − D . (4.2)This concludes the proof. 18 emma 4.3. Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Let D be defined as in Lemma 3.9. For all k ∈ N , with k ≥ , and all | ε | ≤ ε , with ε as in Lemma3.7, one has W ( ϑ ) ≤ γ − B k − | D ε | k +24 for any tree ϑ ∈ T k, , with B the same constant as in Lemma 4.2.Proof . Given a tree ϑ ∈ T k, , with k ≥
2, let v be the node which the root line of ϑ exits andlet ϑ , . . . , ϑ q , with q ≥
1, be the subtrees which have v as root. One has W ( ϑ ) = γ − | D ε | ( m +1) q − m + p ( ν v ) q Y i =1 W ( ϑ i ) , where each W ( ϑ i ) can be bounded by using Lemma 4.2, since the momentum of the correspond-ing root line is different from zero. Thus, one finds W ( ϑ ) ≤ γ − B k − | D ε | ( m +1) q − m + k − − q ≤ γ − B k − | D ε | k +24 , where we have used that( m + 1) q − m + k − − q k − q m ( q − ≥ k + 24 , which implies the bound. Remark 4.4.
The bound in Lemma 4.3 may be improved for k ≥
3. First of all, note that, for q ≥
2, the last inequality in the proof implies that the exponent of | D ε | is bounded by ( k + 9) / k . If q = 1 and k ≥
3, let ℓ be the line entering v , v be the node which ℓ exits, and θ ′ , . . . , θ ′ q ′ , with q ′ ≥
1, the subtrees which have v as root. We distinguish between two cases.1. If | ω · ν ℓ | < β /
4, one has W ( ϑ ) ≤ γ − | D ε | p ( ν v ) − m +1) q ′ − m C − D B k − | D ε | k − − q ′ ≤ γ − | D ε | m C − D B k − | D ε | k − ≤ γ − B k − | D ε | m + k − ≤ γ − B k − | D ε | k +94 , because one has p ( ν ℓ ) ≥ m and ν v + ν ℓ = 0.2. If | ω · ν ℓ | ≥ β /
4, one has W ( ϑ ) ≤ γ − | D ε | m +1) q ′ − m β − B k − | D ε | k − − q ′ ≤ γ − B k − | D ε | k +54 . Therefore, for k ≥ ϑ ∈ T k, , one can bound W ( ϑ ) ≤ γ − B k − | D ε | k +54 . Lemma 4.5.
There exists a positive constant M such that, for all k ∈ N , one has (cid:12)(cid:12)(cid:12)(cid:16) Y v ∈ I ( ϑ ) (cid:17)(cid:16) Y v ∈ E F ( ϑ ) (cid:17)(cid:16) Y v ∈ E G ( ϑ ) Γ (cid:17)(cid:12)(cid:12)(cid:12) ≤ M k . Proof . Define M := max { , , Γ } = max { , γβ − Φ D − m } , (4.3)and use that | I ( ϑ ) | + | E ( ϑ ) | = | N ( ϑ ) | = k . Then the assertion follows with M given by(4.3). 19 emma 4.6. Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Let p ( ν ) and p m ( ν ) be defined as in (3.2) . There exists a positive constant R such that for all | ε | ≤ ε , with ε as in Lemma 3.7, one has X ν ∈ Z ∗ | D ε | p ( ν ) ≤ X ν ∈ Z ∗ | D ε | p m ( ν ) ≤ R . Proof . For | ε | ≤ ε one has | D ε | ≤ / (cf. Remark 4.1), so that the first sum is less than thesecond one. For any x ∈ [0 ,
1) one has X ν ∈ Z ∗ x p m ( ν ) ≤ X ν ∈ Z ∗ p ( ν ) ≤ m ∞ X p = m +1 X ν ∈ Z ∗ p ( ν )= p x p − m ≤ m + 1) N + ∞ X p =0 x p +1 p + 1 + ( m + 1)) N ≤ N (cid:18) ( m + 1) + x (1 − x ) + ( m + 1) x (1 − x ) (cid:19) , so that, if one sets R := 4 N (cid:0) ( m + 1) + m + 3 (cid:1) , (4.4)the second sum is bounded by R . Note that 4 N ( m + 1) ≤ R ≤ N ( m + 1) . Lemma 4.7.
For all k ∈ N , all ν ∈ Z ∗ and all | ε | ≤ ε , with ε as in Lemma 3.7, one has X v ∈ V ( θ ) p ( ν v ) + X v ∈ E ( θ ) p m ( ν v ) ≥ max { p ( ν ) − m k − ( k − , } for all ϑ ∈ T k,ν .Proof . Since p ( ν ) ≥ p m ( ν ) ≥ ν ∈ Z ∗ , the bound is proved if we show that onehas X v ∈ I ( ϑ ) p ( ν v ) + X v ∈ E ( ϑ ) p m ( ν v ) ≥ p ( ν ) − m k − ( k − | ν | ≥ ( m k + ( k − N . By iteratively applying Lemma 3.3, one finds X v ∈ I ( ϑ ) p ( ν v ) + X v ∈ E ( ϑ ) p ( ν v ) ≥ p ( ν ) − ( k − , since | I ( ϑ ) | + | E ( ϑ ) | = k , so that X v ∈ I ( ϑ ) p ( ν v ) + X v ∈ E ( ϑ ) p m ( ν v ) ≥ X v ∈ I ( ϑ ) p ( ν v ) + X v ∈ E ( ϑ ) p ( ν v ) − m | E ( ϑ ) | ≥ p ( ν ) − ( k − m | E ( ϑ ) | ) , with | E ( ϑ ) | ≤ k . (In fact, one has E ( ϑ ) ≤ k − k ≥ Lemma 4.8.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Set C := 2 D − N M R B , with the constants D , B , M and R as in Lemmas 3.9, 4.2, 4.5 and4.6, respectively. For all k ∈ N and all | ε | ≤ ε , with ε as in Lemma 3.7, one has | H ( k ) ν | ≤ D m +10 N − C k | D ε | k − +max { ( p ( ν ) − m k − ( k − , } for all ν ∈ Z ∗ and | α (1) | ≤ γ − D m | D ε | , | α ( k ) | ≤ γ − D m +10 N − B − C k | D ε | k +24 , k ≥ . roof . According to Lemmas 3.12 and 4.5, we find H ( k ) ν ≤ X ϑ ∈ T k,ν |V ( ϑ ) | , |V ( ϑ ) | ≤ W ( ϑ ) D m − ( k − N k − M k (cid:16) Y v ∈ I ( ϑ ) | D ε | p ( ν v ) (cid:17)(cid:16) Y v ∈ E F ( ϑ ) | D ε | p m ( ν v ) (cid:17) , where we use have (4.1), and bound W ( ϑ ) as in Lemma 4.2. The sum over the end node labelsproduces at most a factor 2 k − and, writing14 p ( ν v ) = 18 p ( ν v ) + 18 p ( ν v ) , p m ( ν v ) = 18 p m ( ν v ) + 18 p m ( ν v ) , the sum over the mode labels of the product (cid:16) Y v ∈ I ( ϑ ) | D ε | p ( ν v ) (cid:17)(cid:16) Y v ∈ E F ( ϑ ) | D ε | p m ( ν v ) (cid:17) , neglecting the constraint that the sum of the modes equals ν , is dealt with by using Lemma 4.6.The remaining product is bounded as (cid:16) Y v ∈ I ( ϑ ) | D ε | p ( ν v ) (cid:17)(cid:16) Y v ∈ E F ( ϑ ) | D ε | p m ( ν v ) (cid:17) ≤ | D ε | ( p ( ν ) − m k − ( k − , by Lemma 4.7, if p ( ν ) > m k − ( k − H ( k ) ν then follows.The bounds on the coefficients α ( k ) , for k ≥
2, are discussed in a similar way, relying onLemma 4.3, while the bound on α (1) follows immediately from Lemma 3.11, recalling that V ( ϑ ) = γ − F if ϑ ∈ T , . Lemma 4.9.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Let D and C be defined as in Lemma 3.9 and 4.8, respectively. For all k ∈ N and all | ε | ≤ ε , with ε as in Lemma 3.7, the functions H ( k ) ( ψ ) in (3.9) , for ψ ∈ T , are bounded as | H ( k ) ( ψ ) | ≤ k C D m +10 N − C k | D ε | k − , for a suitable constant C .Proof . Write H ( k ) ( ψ ) as H ( k ) ( ψ ) = X ν ∈ Z ∗ p ( ν ) < m k +( k − e iν · ψ H ( k ) ν + X ν ∈ Z ∗ p ( ν ) ≥ m k +( k − e iν · ψ H ( k ) ν . For real ψ , the first sum is bounded by4 ( m k + k ) N D m +10 N − C k | D ε | k − , by Lemma 4.8, while the second sum is bounded by D m +10 N − C k | D ε | k − X ν ∈ Z ∗ p ( ν ) ≥ m k +( k − | D ε | ( p ( ν ) − m k − ( k − , X ν ∈ Z ∗ p ( ν ) ≥ p x p ( ν ) − p ≤ ∞ X p =0 X ν ∈ Z ∗ p ( ν )= p + p x p ≤ ∞ X p =0 x p p + p + 1) N ≤ N (cid:18) − x ) + p − x (cid:19) , with x = | D ε | and p = m k + ( k − N (cid:18) ( m k + k ) + 1(1 − − ) + m k + ( k − − − (cid:19) ≤ C k , with C := 4 N (cid:0) ( m + 1) + 2 + 2( m + 1) (cid:1) , (4.5)the assertion follows. Lemma 4.10.
Assume Hypothesis 1, and assume ω to be non-resonant up to order m N . Thereexists a positive constant ε such that, for all | ε | ≤ ε , the series expansions (3.9) converge whensetting µ = 1 .Proof . The power series (3.9) converge for | µ | ≤ µ , with µ such that C | D ε | µ = 2 D − N BM R | D ε | µ < . Therefore, if one chooses | ε | ≤ ε , with˜ ε := 1(2 C ) D = D (4 N BM R ) , ε := min { ε , ˜ ε } = min { ˜ ε , ˜ ε } , (4.6)one can take µ = µ = 1. Remark 4.11.
By using that one has B ≥ C − D = 2 A − D (cf. (4.2)), M ≥
6Φ (cf. (4.3))and R ≥ N ( m + 1) (cf. the proof of Lemma 4.6), one finds4 N BM R ≥ N Φ D A N ( m + 1) ≥ N ( m + 1) ˜ ε and hence, by Remark 4.1,˜ ε ≤ D ˜ ε (24 N ( m + 1) ) ≤ N ( m + 1) ) (cid:18) ˜ ε ε (cid:19) ˜ ε ≤ N ( m + 1) ) (cid:18) m N ) (cid:19) ˜ ε . Therefore, in (4.6), one has ε = ˜ ε < ˜ ε < ε . Remark 4.12.
So far, constructing a solution to (2.1) of the form (2.7), we have fixed theparameter α . In the following we need to consider α varying in a small interval around a fixedvalue α . Write β = β ( α ) and C = C ( α ) in (2.3), (3.5) and (3.8), respectively, to stress thedependence on α ; then, for given β >
0, if one defines A ( β ) := { α ∈ R : β ( α ) ≥ β/ } , C ∗ ( β ) := min { C ( α ) : α ∈ A ( β ) } , all the bounds of this and the previous sections still hold for all α ∈ A ( β ), with C replacedwith C ∗ ( β ) and β replaced with β/
2. Indeed, for all α ∈ A ( β ), when ν ∈ Z ∗ is such such that | ω · ν | < β /
4, one can first bound | D ( ω · ν ) | ≥ C ( α ) | ε | – a property used at length in theproof of Lemma 4.2 – and then C ( α ) | ε | ≥ C ∗ ( β ) | ε | , while all the remaining factors appearingin V ( ϑ ) are proportional to inverse powers of β (cf. Section 3.6), so that, once C ( α ) has beenreplaced with C ∗ ( β ), the bounds which hold for α such that β ( α ) = β/ α such that β ( α ) ≥ β/
2. This allows us to have uniform bounds for all α ∈ A ( β ).22 The implicit function problem
By construction the counterterm ˜ α , as well as H ( ψ ; ε ), depends on ε and α (through thefrequency vector ω ); thus, we write ˜ α = ˜ α ( ε, α ).Given α in (2.1), set ω := ( α,
1) and define β := min {| ω · ν | : ν ∈ Z ∗ (4 m N ) } . (5.1) Lemma 5.1.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . There exist positiveconstants ε , η and a , such that for all | ε | ≤ ε and all | α − α | ≤ η the counterterm ˜ α ( ε, α ) depends continuously on both ε and α , and one has | ˜ α ( ε, α ) | ≤ a | ε | .Proof . For fixed α , let β = β ( α ) be defined as in (2.3). The value ˜ ε defined in (4.6) dependson β through the quantities D , B and M . To make explicit such a dependence we write D = D ( β ), B = B ( β ) and M = M ( β ), and set˜ ε ( β ) := 1(2 C ) D ( β ) = D ( β )(4 N BM ( β ) R ) . Define also B ∗ ( β ) := max { β − , ( C ∗ ( β )) − D ( β ) } . with C ∗ ( β ) defined in Remark 4.12. For fixed β , as long as β ≥ β/
2, the series in (3.9) whichdefines the counterterm converges provided one has | ε | ≤ ε , with ε = ε ( β ) := D ( β/ N B ∗ ( β/ M ( β/ R ) , (5.2)as discussed in Remark 4.12. Note that ε differs from ˜ ε ( β ) as, first, B = B ( β ) is replacedwith B ∗ ( β/
2) and, then, the remaining factors β are replaced with β/
2. By construction onehas ˜ ε ( β ) ≤ min { ˜ ε ( β ) : α ∈ A ( β ) } .Set D := D ( β/ M := M ( β/ B := B ∗ ( β/ α ∈ A ( β ), with A ( β ) definedin Remark 4.12, by Lemma 4.8 (and Remark 4.12) the coefficients α ( k ) are bounded as | α (1) | ≤ γ − D m | D ε | , | α ( k ) | ≤ γ − B − D m +11 N − C k | D ε | k +24 , k ≥ , with C := 2 D − N M R B . By Lemma 4.10 and Remark 4.12, the series expansion (3.9) for˜ α is well defined and converges for µ = 1, provided that | ε | ≤ ε . Indeed, one has | ˜ α | ≤ γ − D m | D ε | +( γ B N ) − D m +11 ∞ X k =2 C k | D ε | k +24 ≤ γ − D m (6Φ+8 N B D − M R ) | D ε | , since C | D ε | / < / | ε | ≤ ε . Then, defining a := γ − D m +11 (cid:18) N B M R D (cid:19) , (5.3)the bound | ˜ α | ≤ a | ε | follows as long as α ∈ A ( β ).On the other hand, setting ω = ( α,
1) and ω = ( α , | α − α | ≤ η , one has | ω · ν | ≥ | ω · ν | − | α − α | | ν | ≥ β − m N η ν ∈ Z ∗ (4 m N ). Thus, defining η := β m N , (5.4)for all α ∈ I ( η ) := [ α − η , α + η ] one has β ( α ) ≥ β/ α ∈ A ( β ). Therefore,if one fixes α so that β >
0, for all α ∈ I ( η ) and ε ∈ [ − ε , ε ], the function ˜ α ( ε, α ) dependscontinuously on α and satisfies the bound | α ( ε, α ) | ≤ a | ε | , with a as in (5.3). Continuity on ε trivially holds for ε = 0 and follows from the bounds of Lemma 4.8 for ε = 0. Remark 5.2.
The value of a given in Lemma 5.1 might be improved by computing explicitlythe contribution α (2) , and using the bounds in Lemma 4.7 for the contributions with k ≥ | D ε | / . Lemma 5.3.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . Let ε and η beas in Lemma 5.1. There exist positive constants a and a such that, for all | ε | ≤ ε and all | α − α | ≤ η , one has (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ α∂α ( ε, α ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ a | ε | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ α∂ε ( ε, α ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ a . Proof . The counterterm ˜ α ( ε, α ) is given by (3.9), with the coefficients α ( k ) as in Lemma 3.12.1. In order to bound the derivative of α ( k ) with respect to α , one has to bound the derivatives ∂F ν ∂α , ∂F ν ( q ) ∂α , ∂G ν ∂α , ∂ G ℓ ∂α . h ( k ) ν , which in turn depend on α throughthe propagators G ∗ ℓ (cf. Lemma 3.4). For any tree ϑ ∈ T ∗ k,ν and any ℓ ∈ L ( ϑ ), (cid:12)(cid:12)(cid:12)(cid:12) ∂ G ∗ ℓ ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ |G ∗ ℓ | | ν || ω · ν | ≤ β − kN |G ∗ ℓ | ≤ β − m N |G ∗ ℓ | , so that, by reasoning as in the proof of Lemma 3.5, one finds h ( k ) ν = 0 if k ≤ p ( ν ) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂h ( k ) ν ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ β − k N ( β − Φ ) k ≤ β − m N ( β − Φ ) k , k ≥ p ( ν ) + 1 . F (1) discussed in the proof of Lemma 3.6, when boundingits derivative with respect to α , one has to take into account that, in (2.10a) one has ∂∂α (cid:16) h ( k )˜ ν . . . h ( k p )˜ ν p (cid:17) = p X i =1 ∂h ( k i )˜ ν i ∂α ! p Y j =1 j = i h ( k j )˜ ν j so that one obtains, for | ε | ≤ ε , (cid:12)(cid:12)(cid:12)(cid:12) ∂F (1) ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ε ∂C ∂α (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ F (1) ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ β − N C | ε | + 2 β − m N m X k =2 | ε | k N Φ (cid:0) β − Φ (cid:1) k + 2 β − m N ∞ X p =2 pN p +1 Φ(4 m N ) p (2 | ε | β − Φ ) p ≤ β − N C | ε | + 8 β − m N K | ε | ≤ β − N C | ε | , K as in (3.6), since one has (cf. Remark 4.11) ε < C m K = 14 m ˜ ε , (5.5)with ˜ ε as in (3.8). This, together with the bounds in Lemma 3.7, implies the bound (cid:12)(cid:12)(cid:12)(cid:12) ∂ G ℓ ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ |G ℓ | (cid:18) | ν ℓ, || ω · ν ℓ − iε m γ | + (cid:12)(cid:12)(cid:12)(cid:12) ε ∂F (1) ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ |G ℓ | | ω · ν ℓ | + | ε | m γ + 8 β − N C | ε | ( ω · ν ℓ ) + | ε | C | ν ℓ |≤ |G ℓ | (cid:18) | ε |√ C + | ε | m γ | ε | C + 8 β − N (cid:19) | ν ℓ | ≤ Q | D ε | |G ℓ | | ν ℓ | , with Q := D (cid:18) √ C + γC | ε | m − + 8 β − N | ε | (cid:19) . The derivatives of F ν ( q ) can be bounded in the same way as was done for F (1). Again oneuses the fact that, when acting on some h ( k i )˜ ν i , the derivative with respect to α produces afurther factor 2 β − m N , so that one proceeds as in the proof of Lemma 3.9, by taking intoaccount that, in the sum over p , there is an extra factor 2 β − m N p . Eventually, insteadof the bound for F ν ( q ) of Lemma 3.9, one finds, for all | ε | ≤ ε , (cid:12)(cid:12)(cid:12)(cid:12) ∂F ν ( q ) ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ β − N m N q Φ | D ε | p ( ν ) . The same argument applies to F ν (0) as well. With respect to the bounds in Lemma 3.11,the derivatives are bounded as (cid:12)(cid:12)(cid:12)(cid:12) ∂F ν ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ β − N m D m | D ε | p m ( ν ) , ν ∈ Z ∗ , (cid:12)(cid:12)(cid:12)(cid:12) ∂F ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ β − N m D m D | ε | . Finally, the derivative of G ν (0) is bounded as (cid:12)(cid:12)(cid:12)(cid:12) ∂G ν ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ β − m N γβ − Φ . α of V ( ϑ ), for any tree ϑ ∈ T k, , oneobtains 2 k − k − G ℓ of a line different from the root line, and for each of them there is anextra factor Q | ν ℓ | / | D ε | with respect to the bound for V ( ϑ ); (b) the other k are producedwhen the derivative acts on a node factor, and each of them admits the same bound asbefore times a factor which is at most 48 β − m N . Moreover, for any line ℓ ∈ L ( ϑ ), onehas | ν ℓ | ≤ N ( ϑ ) , N ( ϑ ) := X v ∈ N ( ϑ ) | ν v | .
25f one writes in the bound of V ( ϑ )14 p ( ν v ) = 18 p ( ν v ) + 18 p ( ν v ) , p m ( ν v ) = 18 p m ( ν v ) + 18 p m ( ν v ) , and uses the fact that, by Lemmas 3.1 and 3.3, N ( ϑ ) ≤ ( P ( ϑ ) + m | E ( θ ) | + k ) N ≤ ( P ( ϑ ) + ( m + 1) k ) N, P ( ϑ ) := X v ∈ I ( ϑ ) p ( ν v ) + X v ∈ E ( ϑ ) p m ( ν v ) , one can bound N ( ϑ ) (cid:16) Y v ∈ I ( ϑ ) | D ε | p ( ν v ) (cid:17)(cid:16) Y v ∈ E ( ϑ ) | D ε | p m ( ν v ) (cid:17) ≤ ( m + 2) N k. α (1) = γ − F (0) can be computed explicitly, and gives (cid:12)(cid:12)(cid:12)(cid:12) ∂α (1) ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ β − N m γ − Φ D m | D ε | . α ( k ) , for k ≥
2, first of all one notes that, for | ε | ≤ ε ,by using also Remark 4.11 and (3.5), one has48 β − m N | D ε | ≤ β − m N D ε = 3 β D N Φ ≤ √ N D √ C ≤ Q and hence 48 β − m N ≤ Q / | D ε | .1.5.1. For k ≥
3, it is convenient to use the improved bound in Remark 4.4, which gives (cid:12)(cid:12)(cid:12)(cid:12) ∂α ( k ) ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( m + 2) N k (2 k − γ − Q B − D m +10 N − C k | D ε | k +14 , k ≥ . k = 2 needs a more careful analysis with respect to the discussionin Lemma 4.3 (cf. also Remark 5.2). The two nodes of any tree ϑ whose value contributesto α (2) have opposite mode labels − ν ℓ and ν ℓ (since the sum vanishes), where ν ℓ is themomentum of the line exiting the end node, so that | ω · ν ℓ | < β / p ( ν ℓ ) ≥ m and in that case W ( ϑ ) contains an extra factor | D ε | p ( ν ℓ ) ≤ | D ε | m ≤| D ε | . If | ω · ν ℓ | ≥ β /
4, then the derivative of the propagator admits a better bound,i.e. (cid:12)(cid:12)(cid:12)(cid:12) ∂ G ℓ ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ |G ℓ | | ω · ν ℓ | + | ε | m γ + 8 β − N C | ε | ( ω · ν ℓ ) + | ε | C | ν ℓ |≤ |G ℓ | (cid:0) β − + 2 β − | ε | m γ + β − N (cid:1) | ν ℓ | , where one easily checks that 8( β − + 2 β − | ε | m γ + β − N ) ≤ Q . This means that, withrespect to W ( ϑ ), the derivative with respect to α produces an extra factor which is lessthan 8 β − N m , when acting on a node factor, and less than max { Q | D ε | , Q } = Q ,when acting on the propagator of the line exiting the end node. In conclusion, oneobtains (cid:12)(cid:12)(cid:12)(cid:12) ∂α (2) ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( m + 2) N γ − Q B − D m +10 N − C | D ε | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ α∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ − D m (cid:18) m N Φ β + 48( m + 1) Q N BM R D (cid:18) N BM R D (cid:19)(cid:19) | D ε | . | α − α | < η , one has to reason as in theproof of Lemma 5.1: one requires | ε | ≤ ε and one has to replace β − , B , D and M with,respectively, 2 β − , B , D and M , as defined in the proof of Lemma 5.1, and Q with Q := 2 D p C ∗ ( β ) + γ C ∗ ( β ) | ε | m − + 16 β − N | ε | ! . Then, provided one has | α − α | < η and | ε | ≤ ε , the first bound follows with a := γ − D m +11 (cid:18) m N Φ β + 48( m + 1) Q N B M R D (cid:18) N B M R D (cid:19)(cid:19) . (5.6)2. The derivative with respect to ε of α ( k ) can be bounded in a similar way.2.1. First of all we recall that h ( k ) ν = 0 if k ≤ p ( ν ) and observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ε k h ( k ) ν ∂ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k | ε | ( | ε | β − Φ ) k ≤ m | ε | ( | ε | β − Φ ) k , k ≥ p ( ν ) + 1 , as Lemma 3.5 and the definition of the function h immediately imply. Therefore, in orderto bound the derivatives with respect to ε of the coefficients F ν ( q ), F (0), F ν (0), oneobserves that, with respect to the bounds in the proofs of Lemmas 3.9 and 3.11, a furtherfactor p m / | ε | appears for each summand in the sum over p , so that eventually, one obtains (cid:12)(cid:12)(cid:12)(cid:12) ∂F ν ( q ) ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ m | ε | N q Φ | D ε | p ( ν ) , (cid:12)(cid:12)(cid:12)(cid:12) ∂F ν ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ m | ε | D m | D ε | p m ( ν ) , ν ∈ Z ∗ , (cid:12)(cid:12)(cid:12)(cid:12) ∂F ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ m | ε | | D | m | D ε | . In a similar way one obtains, for | ε | ≤ ε , using also (5.5), (cid:12)(cid:12)(cid:12)(cid:12) ∂G ν ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ m | ε | γβ − Φ , (cid:12)(cid:12)(cid:12)(cid:12) ∂F (1) ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + 4 m K | ε | ≤ C , with K as in (3.6). The latter bound also implies (cid:12)(cid:12)(cid:12)(cid:12) ∂ G ℓ ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ |G ℓ | (cid:12)(cid:12)(cid:12)(cid:12) − i m ε m − γ ω · ν ℓ − ε ∂F (1) ∂ε − F (1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ |G ℓ | m | ε | m − γ | ω · ν ℓ | + 4 C | ε | + 4 C | ε | max { ( ω · ν ℓ ) + | ε | C , | ε | m γ | ω · ν ℓ |} ≤ m + 8 | ε | |G ℓ | , ε of V ( ϑ ), for any tree ϑ ∈ T k, , one obtains 2 k − k − Q / | ε | with respect to the bound for V ( ϑ ); the27ther k are produced when the derivative acts on a node factor, and each of them admitsthe same bound as before times a factor which is bounded by D (cid:18) m | D ε | + ( m + 1) q − m | D ε | (cid:19) = D (( m + 1) q + 3 m ) | D ε | , where the factor q cancels out thanks to the factorial q ! appearing in the node factor I v . If | ε | < ε , then, for any ϑ ∈ T k, , the derivative with respect to ε of V ( ϑ ) admits the samebound as found for V ( ϑ ) in Lemma 4.8 times an extra factor (2 k − Q / | ε | , with Q = max { m + 8 , m + 1 } . Therefore one obtains (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ α∂ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ m γ − Φ D m +10 + 24 Q N γ − D m BM R . | α − α | ≤ η are given by replacing B , D and M with B , D and M , respectively. In conclusion, the second bound follows as well, with a := 24 γ − m Φ D m +11 (cid:18) Q N B M R m Φ D (cid:19) , (5.7)provided one has | α − α | ≤ η and | ε | ≤ ε .Therefore the bounds follow for all | α − α | ≤ η and all | ε | ≤ ε . Remark 5.4.
A more careful analysis of the coefficient α (3) would permit an improvement tothe bound of the derivative with respect to α . Indeed the derivative of α (3) turns out to be morethan linear, and the derivatives of the coefficients α ( k ) , with k ≥
4, are bounded proportionallyto | D ε | / , as it follows from Remark 4.4 and from the fact that the derivative with respect to α produces at most the loss of a power ε . Lemma 5.5.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . Then there existconstants α and ε such that, setting ω := ( α , , β := min {| ω · ν | : ν ∈ Z ∗ (4 m N ) } , for all | ε | ≤ ε there exists a counterterm ˜ α ( ε, α ) , differentiable in both ε and α , and afunction H ( ψ ; ε ) , continuous in ε and analytical in ψ , such that α + ε ˜ α ( ε, α ) = α , β ≥ β/ and H ( ω t ; ε ) solves (2.8) .Proof . If α is such that β is positive, the counterterm ˜ α ( ε, α ) is well defined and differentiableas long as | ε | ≤ ε and | α − α | ≤ η , by Lemma 5.3. Define ε = min { ε , ˜ ε } , ˜ ε := min (cid:26)r a , r η a + a ) (cid:27) . (5.8)For fixed α , one needs α to be such α + ε ˜ α ( ε, α ) = α . Consider the implicit function problem F ( ε, α ) = 0 , F ( ε, α ) := α + ε ˜ α ( ε, α ) − α. One has F (0 , α ) = 0 and, if one takes | ε | ≤ ε , with ε as in (5.8), ∂F∂α ( ε, α ) = 1 + ε ∂ ˜ α∂α ( ε, α ) ≥ − a | ε | ≥ , (cid:12)(cid:12)(cid:12)(cid:12) ∂F∂ε ( ε, α ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˜ α ( ε, α ) + ε ∂ ˜ α∂ε ( ε, α ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( a + a ) | ε | , a , a and a given by (5.3), (5.6) and (5.7), respectively.One has, for | ε | ≤ ε and for suitable constants α ∗ ∈ ( α, α + η ) and ε ∗ ∈ (0 , ε ), F ( ε, α + η ) = F ( ε, α + η ) − F (0 , α ) = F ( ε, α + η ) − F ( ε, α ) + F ( ε, α ) − F (0 , α )= η ∂F∂α ( ε, α ∗ ) + ε ∂F∂ε ( ε ∗ , α ) ≥ η − ( a + a ) ε ≥ η , by (5.8). In a similar way one proves that F ( α − η , ε ) ≤ − η / | ε | ≤ ε . Therefore, oneconcludes that for any | ε | ≤ ε , there exists a unique value α = α ( ε ) such that F ( ε, α ( ε )) = 0.The function H ( ψ ; ε ), given by (3.9) with µ = 1, for all | ε | ≤ ε and | α − α | ≤ η , is welldefined and continuous in α as well as in ε , by construction, and, since α + ε ˜ α = α , it solves(2.8). Lemma 4.9, together with Lemma 4.10, only proves that H ( ψ ; ε ), as a function of ψ , isbounded. To prove analyticity, one relies once more on the bound | H ( k ) ν | ≤ D m N − C k | D ε | k − +max { ( p ( ν ) − m k − ( k − , } , which follows, for all k ∈ N and all ν ∈ Z ∗ , from Lemma 4.8 and Remark 4.12. Write theFourier coefficients of H ( ψ ; ε ) as H ν = ∞ X k =1 H ( k ) ν , and, for any ν ∈ Z ∗ , define k ( ν ) := (cid:22) p ( ν ) + 1 m + 1 (cid:23) . Then one has, if k ( ν ) ≥ | H ν | ≤ k ( ν ) X k =1 H ( k ) ν + ∞ X k = k ( ν )+1 H ( k ) ν ≤ D m N − k ( ν ) X k =1 C k | D ε | k − + m +18 ( k ( ν ) − k ) + D m N − ∞ X k = k ( ν )+1 C k | D ε | k − ≤ D m N − | D ε | m +18 k ( ν ) − (cid:16) C | D ε | − ( m − (cid:17) k ( ν ) + 2 D m N − C (cid:16) C | D ε | (cid:17) k ( ν ) ≤ D m N − C k ( ν )1 | D ε | k ( ν ) − + 2 D m N − C k ( ν )+11 | D ε | k ( ν )4 ≤ D m N − C k ( ν )1 | D ε | k ( ν ) − (cid:16) C | D ε | (cid:17) ≤ D m N − C k ( ν )1 | D ε | k ( ν ) − , which implies, for | ν | ≥ N ( m + 1), that | H ν | ≤ D m N − C ( C | D ε | ) | ν | N ( m +1) . For fixed ε , the Fourier coefficients decay, for | ν | large enough, as | H ν | ≤ D m N − e − ξ ε | ν | , ξ ε := 12 N ( m + 1) log 1 C | D ε | ,
29o that, one finds that, uniformly for all | ε | ≤ ε , the function H ( ψ ; ε ) is analytic in ψ in a stripof width ξ := 2 log 2 N ( m + 1) , since C | D ε | ≤ /
16. Then the assertion follows.
Remark 5.6.
According to (5.8), ε = min { ε , ˜ ε } . For fixed ε and m , if γ tends to 0, ε tendsto a finite value, while ˜ ε tends to 0. Lemma 5.7.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . Let ε be definedas in Remark 5.6. Then for all | ε | ≤ ε there is a multi-periodic solution to (2.1) , with f givenby (2.2) , of the form θ ( t ) = θ + α t + Θ( ω t ; ε ) , with θ ∈ T and ω = ( α , , such that lim ε → α = α, lim ε → Θ( ψ ; ε ) = 0 . Proof . The result follows from Lemma 5.5, fixing ˜ α accordingly and using that h ( ω t ; ε ) and H ( ω t ; ε ) solve (2.4) and (2.8), respectively. Remark 5.8.
Assuming Hypothesis 2, one has β > α and 1 are either commensurate or incommensurate according to the valueof ε . Therefore, for a given value of ε we cannot exclude the possibility that the solution isperiodic. Of course, for a full measure set of values of ε in [ − ε , ε ] the solution is expected tobe quasi-periodic. The value ε in Lemma 5.7 depends on γ through the constants M = M ( β/ ε in (5.2), and the constants a , a and a – which in turn depend on M too – appearing in ˜ ε in (5.8). In particular, for γ large enough, M is proportional to γ and hence ˜ ε is proportionalto γ − . However the bounds in Lemma 4.8 may be improved with a more careful analysis,showing that, with the exception of the tree of order 1, each time an end node v carries a label λ v = G a further power of ε arises. This will allow us to write γ = | ε | − a ¯ γ , with a ∈ [0 ,
1) and¯ γ independent of ε , so that Theorem 2 follows as soon as one fixes a so that m := s + a is aninteger.First of all the bound in Lemmas 4.2 and 4.3 can be refined as follows. Lemma 6.1.
Assume Hypotheses 1, and assume ω to be non-resonant up to order m N . Let D and ε be defined as in Lemma 3.9 and in Lemma 3.7, respectively, and assume that | ε | ≤ ε .For all ϑ ∈ T k,ν , with k ≥ and ν ∈ Z ∗ and | E G ( ϑ ) | ≥ , one has W ( ϑ ) ≤ B k | D ε | k − | D ε | | E G ( θ ) | , k ≥ , | E G ( ϑ ) | ≥ , B | D ε | , k = 2 , | E G ( ϑ ) | = 1 , B k = 1 , | E G ( ϑ ) | = 1 . Proof . The bounds are checked by direct computation for k = 1 , ,
3, and are proved by inductionfor k ≥
4. If v is the node which the root line ℓ of the tree ϑ exits, let ℓ , . . . , ℓ q denote the q lines entering v , with q ≥
0, and let ϑ , . . . , ϑ q be the subtrees whose root lines are ℓ , . . . , ℓ q ,30espectively. Up to a reordering, one can assume that the first q of such subtrees contain eitherat least two end nodes with end node labels G or none; the following q subtrees contain onlyone node with end node label G and are of order 2; the last q subtrees contain only one nodewith end node label G and are of order 1. By construction, one has q = q + q + q and | E G ( ϑ ) | = q + q if q = 0, otherwise, if q ≥ | E G ( ϑ ) | = q X i =1 | E G ( ϑ i ) | + q + q . (6.1)1. For k = 1 one has W ( ϑ ) ≤ β − ≤ B , since | ν ℓ | ≤ m N .2. For k = 2, one has q = 1 and the set N ( ϑ ) contains the nodes v and v , where v is the endnode which ℓ exits, so that λ v = G . Then | ν ℓ | ≤ m N and hence | ω · ν ℓ | ≥ β . Therefore,if | ω · ν ℓ | ≥ β /
4, one finds W ( ϑ ) ≤ β − | D ε | p ( ν v ) β − ≤ B | D ε | , while, if | ω · ν ℓ | < β /
4, one obtains W ( ϑ ) ≤ C − D | D ε | − p ( ν v ) β − ≤ B | D ε | , since p ( ν v ) ≥ m in such a case. Therefore one can bound W ( ϑ ) ≤ B | D ε | .3. For k = 3, one has either q = 1 or q = 2.3.1. If q = 2, the set N ( ϑ ) contains v and the two end nodes v and v which the two lines ℓ and ℓ exit, respectively, such that λ v = G .3.1.1. If λ v = F and λ v = G , one has W ( ϑ ) ≤ |G ℓ || D ε | m +1) − m + ( p ( ν v )+ p m ( ν v )) |G ℓ | B . We distinguish among four possible subcases.3.1.1.1. If | ω · ν ℓ | , | ω · ν ℓ | ≥ β / W ( ϑ ) ≤ (16 β − ) B | D ε | m +1) − m ≤ B | D ε | m +2 ≤ B | D ε | . | ω · ν ℓ | ≥ β / | ω · ν ℓ | < β /
4, one has W ( ϑ ) ≤ C − D β − B | D ε | − m +1) − m + m ≤ B | D ε | , since | ν ℓ | > m N implies p m ( ν v ) ≥ m .3.1.1.3. If | ω · ν ℓ | < β / | ω · ν ℓ | ≥ β /
4, one has W ( ϑ ) ≤ C − D β − B | D ε | − m +1) − m + (2 m − ≤ B | D ε | , since | ν ℓ | > m N and | ν ℓ | ≤ m N imply | ν v + ν v | > m N and hence p ( ν v + ν v ) ≥ m ,so that, by Lemma 3.3, one deduces p ( ν v ) + p m ( ν v ) ≥ m − | ω · ν ℓ | and | ω · ν ℓ | are less than β / | ω · ( ν v + ν v ) | ≤ β / | ν v + ν v | > m N , which, since | ν v | ≤ m N , implies | ν v | > m N ,i.e. p ( ν v ) ≥ m , and (b) | ω · ν v | > m N , i.e. p ( ν v ) ≥ m and hence p m ( ν v ) ≥ m .Thus one has W ( ϑ ) ≤ ( C − D ) B | D ε | − m +1) − m + m + m ≤ B | D ε | , λ v = λ v = G , one has W ( ϑ ) ≤ |G ℓ || D ε | m +1) − m + p ( ν v ) B . We distinguish between two possible subcases.3.1.2.1. If | ω · ν ℓ | ≥ β / W ( ϑ ) ≤ β − B | D ε | m +1) − m ≤ B | D ε | . | ω · ν ℓ | < β / W ( ϑ ) ≤ C − D B | D ε | − m +1) − m + m ≤ B | D ε | . since both | ν ℓ | and | ν ℓ | are ≤ m N , while | ν ℓ | > m N , so that one has p ( ν v ) ≥ m .3.2. If q = 1, the set N ( ϑ ) contains v , a second internal node v and an end node v ′ , such that λ v ′ = G . One has W ( ϑ ) ≤ |G ℓ || D ε | m +1 − m + p ( ν v ) |G ℓ || D ε | m +1 − m + p ( ν v ) B . We distinguish between four subcases.3.2.1. If | ω · ν ℓ | , | ω · ν ℓ | ≥ β /
4, one obtains W ( ϑ ) ≤ B | D ε | . | ω · ν ℓ | < β / | ω · ν ℓ | ≥ β /
4, then one has p ( ν v ) + p ( ν v ) ≥ m −
1, so that W ( ϑ ) ≤ B | D ε | − (3 m − ≤ B | D ε | . | ω · ν ℓ | > β / | ω · ν ℓ | < β /
4, then one has p ( ν v ) ≥ m , which gives W ( ϑ ) ≤ B | D ε | − m ≤ B | D ε | . | ω · ν ℓ | , | ω · ν ℓ | < β /
4, then one has p ( ν v ) ≥ m and p ( ν v ) ≥ m , so that W ( ϑ ) ≤ B | D ε | − m + m ≤ B | D ε | . Summarising, for k = 3, one has W ( ϑ ) ≤ B | D ε | + | E G ( ϑ ) | for all possible values of | E G ( ϑ ) | ,i.e. | E G ( ϑ ) | = 1 and | E G ( ϑ ) | = 2.4. If k ≥
4, one has, by the inductive hypothesis and (6.1), W ( ϑ ) ≤ |G ℓ || D ε | ( m +1) q − m + p ( ν v ) B k − | D ε | k − − q | D ε | | E G ( ϑ ) |− q − q | D ε | q , ≤ B k | D ε | k − | D ε | | E G ( ϑ ) | (cid:16) B − |G ℓ || D ε | p ( ν v )+ ρ ( q ,q ,q ) (cid:17) , with ρ ( q , q , q ) := ( m + 1) q − m − q − q − q + 34 q = (cid:16) m + 34 (cid:17) q + (cid:16) m + 12 (cid:17) q + (cid:16) m − (cid:17) q − m . | ω · ν ℓ | ≥ β /
4, then one has B − |G ℓ | ≤
1. Moreover, if q + q = 0, one needs q ≥ k ≥
4, so that ρ (0 , , q ) ≥ m − / >
0, while, if q + q ≥
1, one has ρ ( q , q , q ) ≥ / | ω · ν ℓ | < β /
4, one has B − |G ℓ | ≤ | D ε | − , we distinguish among the following cases.4.2.1. If q + q = 0, once more k ≥ q ≥
3. If q ≥
4, then ρ (0 , , q ) − ≥ m − ≥ q = 3, then the fact that one has | ν ℓ | , | ν ℓ | , | ν ℓ | ≤ m N and | ν ℓ | > m N implies that | ν v | > m N and hence p ( ν v ) ≥ m , so that34 p ( ν v ) + ρ (0 , , − ≥ m − ≥ . q + q ≥
2, one has ρ ( q , q , q ) − ≥ m − ≥ q + q = 1, we distinguish between two cases.4.2.3.1. If q ≥
2, still one has ρ (0 , , q ) − ≥ m − ≥ q = 0 ,
1, then call ℓ ′ , . . . , ℓ ′ q ′ the q ′ lines entering v , with q ′ ≥ k ≥ ϑ ′ , . . . , ϑ ′ q ′ the subtrees whose root lines are ℓ ′ , . . . , ℓ ′ q ′ , respectively. Once more,assume, without loss of generality, that the first q ′ of such subtrees contain eitherat least two end nodes with end node labels G or none; the following q ′ subtreescontain only one node with end node label G and are of order 2; the last q ′ subtreescontain only one node with end node label G and are of order 1. By construction,one has q ′ = q ′ + q ′ + q ′ , while | E G ( ϑ ) | = q + | E G ( ϑ ′ ) | + . . . + | E G ( ϑ ′ q ′ ) | + q ′ + q ′ , if q ′ ≥
1, and | E G ( ϑ ) | = q + q ′ + q ′ , if q ′ = 0; moreover the sum of the orders of the q ′ subtrees equals k − − q . Then, using ( m + 1) q − m = ( m + 1) q + 1 for q + q = 1and q = 0 ,
1, one obtains W ( ϑ ) ≤ B k | D ε | k − | D ε | | E G ( ϑ ) | (cid:16) B − |G ℓ || D ε | ( p ( ν v )+ p ( ν v ))+ ρ ( q ′ ,q ′ ,q ′ ,q ) (cid:17) , with ρ ( q ′ , q ′ , q ′ , q ) := − m + 1) q + 1 + ρ ( q ′ , q ′ , q ′ ) − − q = (cid:16) m + 34 (cid:17) q ′ + (cid:16) m + 12 (cid:17) q ′ + (cid:16) m − (cid:17) q ′ − m −
54 + (cid:16) m − (cid:17) q . | ω · ν ℓ | ≥ β /
4, then one has B − |G ℓ | ≤
1. If q ′ + q ′ = 0 and either q = 1 (sothat q ′ ≥
1) or q = 0 (so that q ′ ≥ q + q ′ = 2, ρ ( q ′ , q ′ , q ′ , q ) + 34 ( p ( ν v ) + p ( ν v )) ≥ (cid:16) m − (cid:17) − m −
54 + 34 (2 m − ≥
52 ( m − , since in such a case one has | ν v + ν v | > m and hence p ( ν v ) + p ( ν v ) ≥ m − q + q ′ ≥ ρ ( q ′ , q ′ , q ′ , q ) + 34 ( p ( ν v ) + p ( ν v ) ≥ (cid:16) m − (cid:17) − m − ≥ m − . If q ′ + q ′ ≥
2, one has ρ ( q ′ , q ′ , q ′ , q ) ≥ (cid:16) m + 12 (cid:17) − m − ≥ m − ≥ . q ′ + q ′ = 1, then either one has q ′ ≥
1, so that ρ ( q ′ , q ′ , q ′ , q ) + 34 ( p ( ν v ) + p ( ν v ) ≥ (cid:16) m + 12 (cid:17) + (cid:16) m − (cid:17) − m − ≥ m − , or there is only one line ℓ ′ entering v . In the latter case, one iterates the constructiononce more: call ℓ ′′ , . . . , ℓ ′′ q ′′ the q ′′ lines entering the node v ′ which ℓ ′ exits, with q ′′ ≥
1, and ϑ ′′ , . . . , ϑ ′′ q ′′ the subtrees whose root lines are ℓ ′′ , . . . , ℓ ′′ q ′′ , respectively.Again, one may assume that the first q ′′ subtrees contain either at least two endnodes with end node labels G or none; the following q ′ subtrees contain only onenode with end node label G and are of order 2; the last q ′ subtrees containi onlyone node with end node label G and are of order 1. By construction, one has q ′′ = q ′′ + q ′′ + q ′′ , while | E G ( ϑ ) | = q + | E G ( ϑ ′′ ) | + . . . + | E G ( ϑ ′′ q ′′ ) | + q ′′ + q ′′ , if q ′′ ≥
1, and | E G ( ϑ ) | = q + q ′′ + q ′′ , if q ′′ = 0; finally the sum of the orders of the q ′′ subtrees equals k − − q . Then one finds W ( ϑ ) ≤ B k | D ε | k − | D ε | | E G ( ϑ ) | (cid:16) B − |G ℓ ′ || D ε | ( p ( ν v )+ p ( ν v ))+ ρ ( q ′′ ,q ′′ ,q ′′ ,q )+1 − (cid:17) . If | ω · ν ℓ ′ | ≥ β /
4, one has B − |G ℓ ′ | ≤ q ′′ + q ′′ ≥
1, one finds ρ ( q ′′ , q ′′ , q ′′ , q ) + 1 − ≥ (cid:16) m + 12 (cid:17) − m −
54 + 1 − ≥ , while, if q ′′ + q ′′ = 0, either one has q ′′ + q ≥
2, so that one finds ρ ( q ′′ , q ′′ , q ′′ , q ) + 1 − ≥ (cid:16) m − (cid:17) − m −
54 + 1 − ≥ m − , or q ′′ + q = 1, so that one has | ν v + ν v + ν v ′ | > m N and hence p ( ν v ) + p ( ν v ) + p ( ν v ′ ) ≥ m −
2, which gives ρ ( q ′′ , q ′′ , q ′′ , q ) + 1 − ≥ (cid:16) m − (cid:17) − m −
54 + 1 −
14 + 34 (3 m − ≥
94 ( m − . If instead | ω · ν ℓ ′ | < β /
4, then one has B − |G ℓ ′ | ≤ | D ε | − , but, in such a case,in order to have | ν ℓ ′ | < m N , either q ′′ + q ′′ ≥ p ( ν v ) ≥ max { (4 − q ′′ ) m , } .Moreover, if q = 1, then | ν v + ν v + ν ℓ | > m N , which, since | ν ℓ | ≤ m N , implies | ν v + ν v | > m N and hence p ( ν v ) + p ( ν v ) ≥ m −
1, by Lemma 3.3, while, if q = 0, then | ν v + ν v | > m N and hence p ( ν v ) + p ( ν v ) ≥ m −
1, so that, for bothvalues of q , one finds − ρ ( q ′′ , q ′′ , q ′′ , q ) + 1 −
14 + 34 ( p ( ν v ) + p ( ν v )) ≥ − (cid:26) m + 12 ,
34 (4 − q ′′ ) m + (cid:16) m − (cid:17) q ′′ (cid:27) − m −
54 + (cid:16) m − (cid:17) q + 1 −
14 + 34 ((4 − q ) m − ≥ m −
114 + m − q ≥ . | ω · ν ℓ | < β /
4, then one has B − |G ℓ | ≤ | D ε | − . On the other hand one has ν ℓ = ν v + ν ℓ + ν ℓ if q = 1, and ν ℓ = ν v + ν ℓ if q = 0; in the first case, since one34as | ν ℓ | ≤ m N and | ν v + ν ℓ | > m N , one finds | ν v | > m N and hence p ( ν v ) ≥ m ,while in the second case one | ν v | > m N and hence p ( ν v ) ≥ m . Moreover, one haseither q ′ + q ′ ≥ q ′ + q ≥ k ≥
4) and p ( ν v ) ≥ max { (4 − q ′ ) m , } . Insummary, one finds − ρ ( q ′ , q ′ , q ′ , q ) + 34 ( p ( ν v ) + p ( ν v )) ≥ min (cid:26)(cid:16) m + 34 (cid:17) q ′ + (cid:16) m + 12 (cid:17) q ′ ,
34 (4 − q ′ ) m + (cid:16) m − (cid:17) q ′ (cid:27) − m −
134 + (cid:16) m − (cid:17) q + 34 (4 − q ) m , so that one obtains, for both values of q , − ρ ( q ′ , q ′ , q ′ , q ) + 34 ( p ( ν v ) + p ( ν v )) ≥ (cid:16) m + 12 (cid:17) + 2 m − ≥ m − . This concludes the proof.
Lemma 6.2.
Assume Hypotheses 1. Let D and ε be defined as in Lemmas 3.9 and 3.11,respectively, and assume that | ε | ≤ ε . For all ϑ ∈ T k, , with k ≥ , ν ∈ Z ∗ and | E G ( ϑ ) | ≥ ,one has W ( ϑ ) ≤ ( γ − B k − | D ε | k +14 | D ε | | E G ( θ ) | , k ≥ , | E G ( ϑ ) | ≥ ,γ − B | D ε | , k = 2 , | E G ( ϑ ) | = 1 . Proof . Let ℓ , . . . , ℓ q denote the q lines entering v which the root line ℓ exits, and ϑ , . . . , ϑ q the subtrees whose root lines are ℓ , . . . , ℓ q , respectively. As in the proof of Lemma 6.1, assumefor definiteness that the first q subtrees contain either at least two end nodes with end nodelabels G or none; the following q subtrees contain only one node with end node label G andare at least of order 2; the last q subtrees contain only one node with end node label G and areof order 1. By construction, one has q = q + q + q ≥ E G ( ϑ ) ≥
1) and | E G ( ϑ ) | equals q + q if q = 0, while it is given by (6.1) if q ≥ k = 2 one has q = 1 and the ℓ exits an end node v with λ v = G . Then one has W ( ϑ ) ≤ γ − | D ε | B .2. For k ≥ | E G ( ϑ ) | ≥
1, one has W ( ϑ ) = γ − | D ε | ( m +1) q − m q Y i =1 W ( ϑ i ) ≤ γ − B k − | D ε | k + ρ ( q ,q ,q ) | D ε | E G ( ϑ ) | with ρ ( q , q , q ) := ( m + 1) q − m − q − q − q = (cid:16) m + 34 (cid:17) q + (cid:16) m + 12 (cid:17) q + (cid:16) m − (cid:17) q − m − . If q + q ≥ ρ ( q , q , q ) ≥ (cid:16) m + 12 (cid:17) ( q + q ) − m − ≥ m + 12 − m − ≥ , q + q = 0 then, in order to have k ≥
3, one needs q ≥ ρ ( q , q , q ) ≥ (cid:16) m − (cid:17) q − m − ≥ (cid:16) m − (cid:17) − m − ≥ m − ≥ , so that the last bound follows in both cases.Define T k ,k ,ν as the set of labelled trees θ ∈ T k + k ,ν such that | E F ( ϑ ) | + | I ( ϑ ) | = k and | E G ( ϑ ) | = k . Write H ( k ,k ) ν := X ϑ ∈ T k ,k ,ν V ( ϑ ) , ν ∈ Z ∗ , α ( k ,k ) := X ϑ ∈ T k ,k , V ( ϑ ) , so that (3.9) is replaced with H ( ψ ) = X k ,k ≥ k + k ≥ X ν ∈ Z µ k µ k e iν · ψ H ( k ,k ) ν , ˜ α = X k ,k ≥ k + k ≥ µ k µ k α ( k ,k ) , (6.2)where µ and µ are two parameters that eventually have to be set equal to 1. As previouslyfor µ , this is proved to be allowed by showing that the radius of convergence in both variables µ and µ is greater than 1, provided that ε is small enough.Finally, set C ∗ := 2 D − N R B , with the constants D , B and R as in the proof of Lemma5.1, and define ε := 1(12Φ C ∗ ) D = D (24 N Φ B R ) . (6.3)Note that ε does not depend on γ . Also set Γ := 4 γβ − Φ D − m , and write Γ = ¯Γ | D ε | − a ,with ¯Γ := 4¯ γβ − Φ D − m independent of ε and a = m − s , where ¯ γ and s are as in Theorem 2.Lemmas 4.2, 4.3, 6.1 and 6.2 immediately imply the following result. Lemma 6.3.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . For all k ∈ N , all | ε | ≤ ε and all | α − α | ≤ η , with ε and η defined in (6.3) and in (5.4) , respectively, onehas, | H ( k , ν | ≤ D m +11 N − ( C ∗ ) k (6Φ) k | D ε | k − , | H (0 , ν | ≤ D m +11 N − C ∗ Γ , | H (1 , ν | ≤ D m +11 N − ( C ∗ ) | D ε | , | H ( k ,k ) ν | ≤ D m +11 N − ( C ∗ ) k + k (6Φ) k Γ k | D ε | k k − | D ε | k , k ≥ , for all ν ∈ Z ∗ with p ( ν ) < m ( k + k ) + k + k − . The same bounds hold, but multiplied by | D ε | ( p ( ν ) − ( m +1)( k + k ) − ( k + k − , if p ( ν ) ≥ ( m + 1)( k + k ) + ( k + k − , and | α (1 , | ≤ γ − D m | D ε | , | α ( k , | ≤ γ − B − D m +11 N − ( C ∗ ) k (6Φ) k | D ε | k , k ≥ , | α (1 , | ≤ γ − B − D m +11 N − ( C ∗ ) | D ε | , | α ( k ,k ) | ≤ γ − B − D m +11 N − ( C ∗ ) k + k (6Φ) k Γ k | D ε | k k | D ε | k , k ≥ . Moreover there exists a positive constant ε ≤ ε such that, for all | ε | ≤ ε , the series (6.3) converge with µ = µ = 1 . roof . For fixed α , one reasons as in the proof of Lemma 4.8, using the bounds in Lemmas 6.1and 6.2. The result is then extended to any α ∈ A ( β ) as outlined in Remark 4.12. The proofthat the radius of convergence of the series (3.9) in both variables µ and µ is greater than1, proceeds as in the proof of Lemma 4.10 – by taking into account once more Remark 4.12 toextend the argument to any α ∈ A ( β ) – provided that | ε | ≤ ε , with ε := min { ˜ ε , ε } , ˜ ε := 1(2 C ∗ ¯Γ ) D , (6.4)with the condition | ε | ≤ ˜ ε guaranteeing summability over k . Lemma 6.4.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . Let η and ε be asin Lemmas 5.1 and 6.3, respectively. There exist positive constants a ∗ , a ∗ and a ∗ such that, forall | ε | < ε and all | α − α | < η , one has | ˜ α ( ε, α ) | ≤ a ∗ | ε | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ α∂α ( ε, α ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ a ∗ | ε | , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ˜ α∂ε ( ε, α ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ a . Proof . The first bound is proved as in Lemma 5.1, the only difference being that now there is adouble sum, over k and k , with k ≥ k ≥
0. The two sums converge, for all a ∈ [0 , | ε | ≤ ε , as discussed in the proof of Lemma 6.3. Thus, the first bound followswith a ∗ := γ − D m +11 (cid:18) K N B R D (cid:19) , (6.5)where K := 2(6Φ) + 6ΦΓ + 4(6Φ)Γ | D ε | . Note that γ − and K are proportional to | ε | a and to | ε | − a , respectively, so that a is boundedby an ε -independent constant.The second and third bounds are established as in the proof of Lemma 5.3. The only caveat is that, when differentiating with respect to α the propagator G ℓ of a line ℓ ∈ L ( ϑ ), even thoughone still bounds in general (i.e. for any ν ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ G ℓ ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ |G ℓ | (cid:18) | ε |√ A + | ε | m γ | ε | A + 4 β − N (cid:19) ≤ Q | D ε | |G ℓ | | ν ℓ | , with Q as in the proof of Lemma 5.3, hence proportional to | ε | m − − a , however, when | ω · ν ℓ | ≥ β /
4, the bound can be replaced by (cid:12)(cid:12)(cid:12)(cid:12) ∂ G ℓ ∂α (cid:12)(cid:12)(cid:12)(cid:12) ≤ Q ∗ |G ℓ || ν ℓ | , Q ∗ := 8 β + 16 | ε | m γβ + 4 Nβ , with Q ∗ uniformly bounded (in ε ) for a ∈ [0 , ϑ of order k is suchthat E G ( ϑ ) ≥
1, then either k = 2 and the line exiting the end node has momentum ≤ m N or k ≥ V ( ϑ ) contains at least an extra factor | D ε | .Eventually this leads to the bound a ∗ | ε | , with a ∗ := γ − D m +11 (cid:18) m N Φ β + 24( m + 1) N B R D (cid:18) K Q ∗ + 8 K Q N B (6Φ) R D (cid:19)(cid:19) , (6.6)where Q is defined as in the proof of Lemma 5.3, with γ = ¯ γ | ε | − a , and K := (6Φ) + 6Φ Γ , K := 2(6Φ) + 4(6Φ)Γ | D ε | .
37n particular, the constants γ , K and Q are proportional to | ε | − a (the last one for m = 1),while Q ∗ and K are bounded by an ε -independent constant for all a ∈ [0 ,
1) so that a ∗ is alsobounded independently of ε .The last bound is also discussed as in the proof of Lemma 5.3, and is proved to hold with a ∗ := 24 γ − m Φ D m +11 (cid:18) K Q N B (6Φ) R m Φ D (cid:19) , (6.7)with K as in (6.5) and Q as in the proof of Lemma 5.3.A result analogous to Lemma 5.5 follows, with only difference being that the further conditionto be required on ε involves the parameters a ∗ , a ∗ and a ∗ . Lemma 6.5.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . Then there existconstants α and ¯ ε such that, defining ω and β as in Lemma 5.5, for all | ε | ≤ ¯ ε there existsa counterterm ˜ α ( ε, α ) , differentiable in both ε and α , and a function H ( ψ ; ε ) , continuous in ε and analytic in ψ , such that α + ε ˜ α ( ε, α ) = α , β ≥ β/ and H ( ω t ; ε ) solves (2.8) .Proof . One reasons as in the proof of Lemma 5.5, by defining¯ ε = min { ε , ˜ ε } , ˜ ε := min (s a ∗ , r η a ∗ + a ∗ ) ) , (6.8)with a ∗ , a ∗ and a ∗ defined as in Lemma 6.3.The following result, analogous to Lemma 5.7, completes the proof of Theorem 2. Lemma 6.6.
Assume Hypotheses 1 and 2, and let β be defined as in (5.1) . Let ¯ ε be definedas in Lemma 6.5. Then for all | ε | ≤ ¯ ε there is a multi-periodic solution θ ( t ) to (2.1) , with f given by (2.2) , of the form θ ( t ) = θ + α t + Θ( ω t ; ε ) , with θ ∈ T and ω = ( α , , such that lim ε → α = α, lim ε → Θ( ψ ; ε ) = 0 . Proof . As for Lemma 5.7.
We have considered a class of systems on the one-dimensional torus T , described by (1.2), andhave shown that, as long as γ is sublinear in ε and f is a trigonometric polynomial, if α satisfies a non-resonance condition up to some finite order, depending on both the degree of thepolynomial and the size of γ , then for | ε | small enough there exists a multi-periodic solution(cf. Theorem 2). The frequency vector of the multi-periodic solution is ( α , f in the variable t and α is such that α + ε ˜ α ( ε, α ) = α, (7.1)for a suitable function ˜ α ( ε, α ). The function ˜ α ( ε, α ) is smooth in both ε and α , as long as ε is small and α is close to α , so that (7.1) can be solved by the implicit function theorem.38he multi-periodic solution has two frequencies α and 1; it is quasi-periodic if α is in-commensurate with 1 and periodic otherwise. In the latter case, the period is expected to belarge, so that, up to very large time scales, the solution, for any practical purpose, appears tobe quasi-periodic even when it is periodic; for this reason, it is said to be pseudo-synchronous.The maximum size ε allowed for ε depends on the degree N of the trigonometric polynomial,on the size of the dissipation and on the parameter α . For fixed values of γ , one may write γ = | ε | m − a ¯ γ , for some a ∈ [0 ,
1) and ¯ γ independent of ε , and require a non-resonance conditionof order 4 m N on the frequency vector ω = ( α, ε goes to 0 when either N or m goes toinfinity. The fact that ε goes to 0 with m going to infinity is unavoidable: when the dissipationgoes to zero, the system becomes a quasi-integrable Hamiltonian system and the resonant toriare destroyed.Thus, the result leaves several open problems, which would require further investigation:1. in principle, the assumption that f is a trigonometric polynomial might be removable,possibly by requiring additional hypotheses on α ;2. the quasi-periodic solutions proved to exist are expected to be local attractors, on bothphysical and mathematical grounds, but this should be studied explicitly;3. in order to apply the results to real situations, one should use for the parameters ε , γ and α in (1.2) the values arising from astronomical data, and check whether they are allowedfor the theorems to hold.4. while the quasi-periodic solutions investigated here bifurcate from an unperturbed maximalKAM torus, it would be interesting to extend the results to the Langrangean tori appearinginside the resonant gaps.With respect to the existing literature, the main advantage of Theorem 2 is that the externalparameter does not need to be tuned, but can be set equal to all except finitely many values,and that the frequencies are required to satisfy very mild non-resonance conditions. On theother hand, the main drawback is that only trigonometric polynomials can be considered. So,a natural question, according to item 1, is whether one can extend the result so as to includeanalytic functions with arbitrarily many harmonics, and, if the answer is positive, whether onemust require stronger non-resonance conditions on α than those allowed by Hypothesis 2.Already in the case discussed in this paper, attractiveness of the solution is not obvious (seeitem 2), since one cannot rely on the normal form construction used in refs. [40, 32], where thefrequency vector ω satisfies a Diophantine condition. Linearisation around the multi-periodicsolution leads to Hill’s equation with multi-periodic potential and no Diophantine assumptionon the frequency vector.As far as item 3 is concerned, we note that even in the case of the periodic attractors compar-ison with the astronomical data is quite non-trivial [1, 7]. Moreover, as said in the introduction,the attractors found in problems of celestial mechanics essentially are either periodic orbits orLagrangian tori, so that a direct application to real situations is far from obvious. The case ofKAM tori could be seen as a first, preliminary step before attacking the (harder) case of La-grangian tori (see item 4). Note that the Lagrangian tori which appear inside the gaps amongthe surviving tori are rather irrelevant in the conservative case, since they cover in aggregatea negligible fraction of the phase space (see refs. [35, 8] for recent results); in the dissipativecase, instead, they can play a fundamental role, since the results in the literature suggest that,when a more realistic model for the tidal torque is considered, a Lagrangian torus survives asthe dominant attractor [37, 5]. 39 ckowledgments. We thank Luigi Chierchia and Jessica Massetti for useful discussions.
References [1] F. Antognini, L. Biasco, L. Chierchia,
The spin-orbit resonances of the solar system: a mathemat-ical treatment matching physical data , J. Nonlinear Sci. (2014), no. 3, 473–492.[2] M. Bartuccelli, J. Deane, G. Gentile, Attractiveness of periodic orbits in parametrically forcedsystems with time-increasing friction , J. Math. Phys. (2012), no. 10, 102703, 27 pp.[3] M. Bartuccelli, J. Deane, G. Gentile, The high-order Euler method and the spin-orbit model , Ce-lestial Mech. Dynam. Astonom. (2015), no. 3, 233–260.[4] M. Bartuccelli, J. Deane, G. Gentile,
Fast numerics for the spin-orbit equation with realistic tidaldissipation and constant eccentricity , Celestial Mech. Dynam. Astronom. (2017), no. 4, 453–473.[5] M.V. Bartuccelli, J.H.B. Deane, G. Gentile
Periodic and quasi-periodic attractors for the spin-orbitevolution of Mercury with a realistic tidal torque , Monthly Notices Roy. Astronom. Soc. (2017),no. 1, 127-150.[6] U. Bessi,
An analytic counterexample to the KAM theorem , Ergodic Theory Dynam. Systems (2000), no. 2, 317–333.[7] L. Biasco, L. Chierchia, Low-order resonances in weakly dissipative spin-orbit models , J. DifferentialEquations (2009), no. 11, 4345–4370.[8] L. Biasco, L. Chierchia,
On the measure of Lagrangian invariant tori in nearly-integrable mechanicalsystems , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (2015), no. 4, 423–432.[9] L. Biasco, L. Chierchia, On the topology of nearly-integrable Hamiltonians at simple resonances ,Nonlinearity (2020), no. 7, 3526–3567.[10] H. Broer, C. Sim´o, J.C. Tatjer, Towards global models near homoclinic tangencies of dissipativediffeomorphisms , Nonlinearity (1998), no. 3, 667–770.[11] A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems ,Regul. Chaotic Dyn. (2009), no. 1, 49–63.[12] A. Celletti, L. Chierchia, Construction of stable periodic orbits for the spin-orbit problem of celestialmechanics , Regul. Chaotic Dyn. (1998), no. 3, 107–121.[13] A. Celletti, L. Chierchia, Hamiltonian stability of spin-orbit resonances in celestial mechanics ,Celestial Mech. Dynam. Astronom. (2000), no. 4, 229–240.[14] A. Celletti, L. Chierchia, Measures of basins of attraction in spin-orbit dynamics , Celestial Mech.Dynam. Astronom. (2008), no. 1-2, 159–170.[15] A. Celletti, L. Chierchia,
Quasi-periodic attractors in celestial mechanics , Arch. Ration. Mech.Anal. (2009), no. 2, 311–345.[16] A. Celletti, S. Di Ruzza,
Periodic and quasi-periodic orbits of the dissipative standard map , DiscreteContin. Dyn. Syst. Ser. B (2011), no. 1, 151–171.
17] M. Cencini, F. Cecconi, Vulpiani,
Chaos. From simple models to complex systems , World Scientific,Singapore, 2010.[18] L. Chierchia, C. Falcolini,
A direct proof of a theorem by Kolmogorov in Hamiltonian systems , Ann.Scuola Norm. Sup. Pisa Cl. Sci. (4) (1994), no. 4, 541–593.[19] A.C.M. Correia, J. Laskar, Mercury’s capture into the 3/2 spin-orbit resonance as a result of itschaotic dynamics , Nature (2004), 848–850.[20] A.N. Cox,
Allen’s Astrophysical Quantities , Springer, New York, 2002.[21] J.M.A. Danby,
Fundamentals of celestial mechanics , Macmillan, New York, 1962.[22] M. Efroimsky, J.G. Williams,
Tidal torques: a critical review of some techniques , Celestial Mech.Dynam. Astronom. (2009), no. 3, 257–289.[23] M. Efroimsky,
Bodily tides near spin-orbit resonances , Celestial Mech. Dynam. Astronom. (2012), no. 3, 283–330.[24] M. Efroimsky, V.V. Makarov,
Tidal friction and tidal lagging. Applicability limitations for a popularfomula for the tidal torque,
Astrophys. J. (2013), no. 1, 73, 10pp.[25] S. Ferraz-Mello,
Tidal synchronization of close-in satellites and exoplanets. A rheophysical ap-proach , Celestial Mech. Dynam. Astronom. (2013), no. 2, 109–140.[26] G. Forni,
Analytic destruction of invariant circles , Ergodic Theory Dynam. Systems (1994),no. 2, 267–298.[27] G. Gallavotti, Twistless KAM tori , Comm. Math. Phys. (1994), no. 1, 145–156.[28] G. Gentile, V. Mastropietro,
Methods for the analysis of the Lindstedt series for KAM tori andrenormalizability in classical mechanics. A review with some applications , Rev. Math. Phys. (1996), no. 3, 393–444.[29] P. Goldreich, S. Peale, Spin-orbit coupling in the solar system , Astronom. J. (1966), no. 6,425–438.[30] A.J. Lichtenberg, M.A. Lieberman, Regular and chaotic dynamics , Springer, New York, 1992.[31] G.J.F. MacDonald,
Tidal friction , Rev. Geophys. (1964), 467–541.[32] V.V. Makarov, Conditions of passage and entrapment of terrestrial planets in spin-orbit resonances ,Astrophys. J. (2012), no. 1, 73, 8pp.[33] V.V. Makarov, M. Efroimsky,
No pseudo-synchronous rotation for terrestrial planets and moons ,Astrophys. J. (2013), no. 1, 27, 12pp.[34] J.E. Massetti,
Normal forms for perturbations of systems possessing a Diophantine invariant torus ,Ergodic Theory Dynam. Systems (2019), no. 8, 2176–2222.[35] A.G. Medvedev, A.I. Neishtadt, D.V. Treschev, Lagrangian tori near resonances of near-integrableHamiltonian systems , Nonlinearity (2015), no. 7, 2105–2130.[36] C.D. Murray and S.F. Dermott, Solar System Dynamics , Cambridge University Press, Cambridge,1999[37] B. Noyelles, J. Frouard, V.V. Makarov and M. Efroimsky,
Spin-orbit evolution of Mercury revisited ,Icarus (21014), 26–44.
38] E. Ott,
Chaos in dynamical systems , Cambridge University Press, Cambridge, 2002.[39] S.J. Peale,
The free precession and libration of Mercury , Icarus (2005), 4–18.[40] L. Stefanelli, U. Locatelli,
Kolmogorov’s normal form for equations of motion with dissipativeeffects , Discrete Contin. Dyn. Syst. Ser. B (2012), no. 7, 2561–2593.[41] M. Tabor, Chaos and integrability in nonlinear dynamics , John Wiley, New York, 1989.[42] J.G. Williams and M. Efroimsky,
Bodily tides near the 1:1 spin-orbit resonance: correction toGoldreich’s dynamical model , Celestial Mech. Dynam. Astronom. (2012), no. 4, 387–414.[43] J.A. Wright, J.H.B. Deane, M. Bartuccelli, G. Gentile,
The effects of time-dependent dissipationon the basins of attraction for the pendulum with oscillating support , Nonlinear Dynam. (2014),no. 4, 1377–1409.[44] J.A. Wright, J.H.B. Deane, M. Bartuccelli, G. Gentile, Basins of attraction in forced systems withtime-varying dissipation , Commun. Nonlinear Sci. Numeri. Simul. (2015), 72–87.[45] J.A. Wright, M. Bartuccelli, G. Gentile, Comparisons between the pendulum with varying lengthand the pendulum with oscillating support , J. Math. Anal. Appl. (2017), no. 2, 1685–1707.(2017), no. 2, 1685–1707.