Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation \[\mathcal{H}_\varepsilon(p,q,I,\varphi,t)=h(I)+\sum_{i=1}^n\pm \left(\frac{1}{2}p_i^2+V_i(q_i)\right)+\varepsilon H_1(p,q,I,\varphi, t), \] where $(p,q)\in \mathbb{R}^n\times\mathbb{T}^n$, $(I,\varphi)\in\mathbb{R}^d\times\mathbb{T}^d$ with $n, d\geq 1$, $V_i$ are Morse potentials, and $\varepsilon$ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations $H_1$. Indeed, the set of admissible $H_1$ is $C^\omega$ dense and $C^3$ open (a fortiori, $C^\omega$ open). Our perturbative technique for the genericity is valid in the $C^k$ topology for all $k\in [3,\infty)\cup\{\infty, \omega\}$.


INTRODUCTION
The goal of this paper is to study the Arnold diffusion problem for analytic perturbations of a given a priori unstable Hamiltonian system. Arnold diffusion is a phenomenon of instability in Hamiltonian systems with more than two degrees of freedom. This problem arises in the study of the effect of small perturbations on integrable systems, and has attracted a lot of attention both in mathematics and in physics due to its importance for the applications.
For a nearly integrable Hamiltonian system, the celebrated Kolmogorov-Arnold-Moser (KAM) theory asserts that the most part (in the measure-theoretic sense) of the phase space is filled with KAM invariant tori carrying quasi-periodic dynamics. Arnold diffusion asks for the large scale motions in the complement of KAM tori. The first example is constructed by V. I. Arnold in [1]. He also conjectured that the diffusive phenomena occur for generic systems: the typical case in a multidimensional problem is topological instability: through an arbitrarily small neighborhood of any point there passes a phase trajectory along which the action variables go away from the initial values by a quantity of order one [2,Chapter 6]. One of the main problems in this conjecture is the genericity in some appropriate function space (e.g. C r -differentiable, analytic). The genericity in the C r -differentiable topology is now well understood. However, as pointed out in a recent survey [14], it remains a deep open problem to prove Arnold diffusion in the analytic category. This issue is of great interest since many Hamiltonians with physical significance are analytic. 2010 Mathematics Subject Classification. 37J40, 37J25, 70H08, 70H33.
Key words and phrases. Arnold diffusion, genericity, scattering map, Melnikov method.
In this paper we give an affirmative answer to the C ω -genericity issue of Arnold diffusion for a priori unstable Hamiltonian systems. The a priori unstable Hamiltonian system consists of a rotorpendulum system plus a time periodic perturbation, and it can be viewed as a scaled approximation on the dynamics near simple resonances of the a priori stable systems [17].
Note that the proof presented in this paper works even if the functions V i (q i ) : T → R, i = 1, · · · , n in the unperturbed part are small (weak hyperbolicity), so our result applies to some a priori stable systems, see Remark 3 for more explanation. Our approach for the proof follows a recent geometric mechanism established in [38]. This mechanism relies on the presence of normally hyperbolic invariant manifold (NHIM), with transverse intersection between the associated stable and unstable manifolds. Indeed, the Melnikov method will be used to show transverse homoclinic orbits in the perturbed system. In this setting, we can then use the theory of scattering maps to compute the effect of homoclinic excursions. Heuristically, the scattering map gives the future asymptotic of an orbit as a function of its past asymptotic [19,23]. By shadowing the pseudo-orbits of this map, it allows to show instability for the original dynamics.
Here, we give a brief overview of the previous works and approaches on the genericity problem of Arnold diffusion. The scattering map has become an effective tool to study the phenomena of instability in concrete examples or generic systems. By exploiting this geometric tool, Arnold diffusion has been proved to occur for generic perturbations in the C r -differentiable topology, see for instance [19,24,26,38].
In particular, the geometric mechanism developed in [38] requires almost no information of the inner dynamics on the NHIM. Only recurrence of the motion in the NHIM is needed, and it is automatically satisfied in the Hamiltonian case by Poincaré recurrence theorem if the motions in the NHIM are bounded (Of course, if the motions in the NHIM are not bounded, one has diffusion in the NHIM!).
The main hypothesis of the mechanism of [38] is some explicit transversality conditions, which are implied by checking that some Melnikov-type functions have non-degenerate critical points. Sometimes the dynamics of a single scattering map may have difficulties moving long distances. But if several scattering maps are available, one can iterate these scattering maps to find large scale motions.
For applications of the mechanism of [39] in celestial mechanics see e.g. [9,28].
Another mechanism assuming mainly recurrence -but assuming some separation of time scales appears in [36]. Geometric methods that use NHIM but assume that there are some other invariant objects in the NHIM (e.g secondary tori) appear in [19,20,21,26,22,25,24] and applications to celestial mechanics and other concrete models appear in [26,30,29]. Another important geometric method based on separatrix maps to study Arnold diffusion can be found in [51,52,53], etc. It is worth mentioning that the variational method is also an effective approach to study the diffusion problem. The techniques and ideas developed by J. Mather [48,49] have significant influence. Applying global variational methods to convex Hamiltonian systems, several authors have established the genericity in the C r -differentiable topology, see [15,4,16,43,11,12,5,13,44]. Note, however that in some of these papers, the notion of genericity is redefined and that, of course, a convexity condition is needed. This paper does not aim to review the rich history on this very active area. There are many other related works, we mention here [8,7,6,18,45,34,42,54,36,40,47,10,29,35] and references therein.
Thus, as mentioned above, the genericity of Arnold diffusion in the C r -differentiable category has been well studied. However, the most difficult case is the analytic genericity, which is still an open problem. The difficulty lies in that most of the previous works require the use of non-analytic techniques (e.g. bump functions) for perturbations.
In this paper, a new perturbative technique is introduced to solve the analytic genericity of Arnold diffusion. Following [38], the scattering map is used as an essential tool. We will take advantage of the Poincaré-Melnikov method and the family of periodic potential functions to verify the genericity of some transversality hypotheses. As will see below, the novelty of our technique is as follows: (I) It is valid in both the C r -differentiable topology and the C ω topology; (II) We can also obtain the genericity in the sense of Mañé [46], that is, the genericity ( actually, C ω dense and C 3 open ) in the space of periodic potential functions.
The basic idea of our method is as follows. The work of [38] shows that it suffices to verify some transversality conditions for the zeros of a rather explicit Melnikov integral. Following the standard procedure in transversality, we show that, if there are some degenerate situations, more or less arbitrary perturbations break the degeneracy. See Section 4.
Note that the families of perturbations we choose are rather arbitrary. Hence, the result is stronger than density. We show that the transversality -and hence the diffusion -can only fail in an infinite codimension set. See Remark 10. For practical applications, we note that the only condition to check is a very explicit (and rapidly convergent) integral, so that given a concrete system (e.g. in celestial mechanics), one can verify the result with a finite precision calculation and obtain quantitative information on the location of the diffusing orbits.
where R 1 and R 2 are suitably large, and · is the standard Euclidean norm.
We consider the following a priori unstable system with a time-periodic perturbation: Here, p = (p 1 , · · · , p n ) and q = (q 1 , · · · , q n ) are symplectically conjugate variables, I = (I 1 , · · · , I d ) and ϕ = (ϕ 1 , · · · , ϕ d ) are symplectically conjugate variables, and T = R/2πZ. The unperturbed Hamiltonian H 0 is given by Here, the symbol ± in (2.2) means that one can take either the plus sign " + " or the minus sign " − " in front of each pendulum 1 2 p 2 i + V i (q i ). The whole perturbation term εH 1 is assumed to be real analytic and periodic on time t with a period 2π. The unperturbed part H 0 represents a d -degree-offreedom rotator plus n pendulums. H 0 is not necessarily convex, and the induced inner dynamics is not necessarily a twist map.
Throughout this paper, we assume the following conditions on H 0 : (H2) For each i , the function V i : T → R has a unique maximum point which is non-degenerate in the sense of Morse. Without loss of generality and to simplify the notation, we may always assume the maximum point q max = 0.
It is clear that the nondegeneracy condition (H2) is C 2 -open and C ω -dense. We also remark that the approach used in this paper is also applicable to those systems whose unperturbed part is h(I ) + n i =1 P i (p i , q i ), as long as each P i has a hyperbolic equilibrium and a homoclinic orbit.
Of course, the potentials satisfying Morse non-degeneracy are generic.
The phase space is M := D × T n × B × T d , endowed with the standard symplectic form. The corresponding Hamilton's equations arė It is clear that the dynamics of the unperturbed system H 0 is integrable. Hence, the diffusion phenomena may occur only if ε = 0. Denoting the extended phase space the perturbation function H 1 in (2.1) is assumed to be real analytic on M , which means the analyticity can extend to a complex neighborhood of M .
For each κ > 0 we denote by M κ the set of all points (p, q, I , ϕ, t ) ∈ C n ×C n /(2πZ) M κ is an open domain in the complex space. In order to discuss the genericity of Arnold diffusion in the real analytic category, we introduce the following space of bounded analytic functions on M κ , is a Banach space. Sometimes, for simplicity, we use C ω κ instead of C ω ( M κ ) when there is no confusion. dense set U ⊂ C ω κ , and for each H 1 ∈ U we can find ε 0 = ε 0 (H 1 ) > 0 and ρ = ρ(H 1 ) > 0 satisfying the following property: for each ε ∈ (−ε 0 , ε 0 )\{0}, the Hamiltonian flow of H ε = H 0 +εH 1 admits a trajectory whose action variables I (t ) satisfy sup t >0 where the initial condition I (0) ∈ V I 0 .
Remark 1. The above result states that we can find one diffusing orbit whose initial condition I (0) in the action space is just some point in the neighborhood V I 0 . This initial condition I (0) in general may not be I 0 . To construct diffusion orbits one needs to pose some hypotheses on H 1 , see (H3a)-(H3b) in Section 3. Moreover, as we will show in Section 4, the set of H 1 satisfying (H3a)-(H3b) is indeed C ω κ dense and C 3 open. These conditions of genericity in H 1 are rather explicit. They are given by conditions (which only require finite precision calculation) on a rapidly convergent integral. Hence, they can be verified in concrete models (e.g. in celestial mechanics). Also, we point out that ρ does not depend on ε, so Theorem 2.1 implies that for generic systems, through an arbitrarily small neighborhood of a given point there passes a trajectory whose action coordinates go away from the initial values by O (1) with respect to the size of the perturbation.
Remark 2. κ stands for the size of analytic extension. It is worth noting that our result of analytic genericity holds for any κ > 0.
Remark 3. As we will see from the proof in the following sections, our method also allows that V i are weakly hyperbolic, that is, V i can be replaced by δV i for a small δ > 0, and the perturbation parameter ε ≪ δ. This is similar to Arnold's example [1]. Of course, in that case the threshold value ε 0 (H 1 ) ≪ δ shrinks to zero as δ tends to zero. This is, very typical of the approches to diffusions near integrable systems. Of course one expects that making ε larger will generate more diffusion.
We can also interpret the genericity result stated above for system H 0 + H 1 without using the parameter ε. More precisely, let S be the unit sphere in the space (C ω κ , · κ ) with κ > 0. Then there exists a non-negative function ǫ 0 : S → [0, +∞) taking positive values on an open-dense subset of S, such that for each H 1 in the ǫ 0 -ball B = λP P ∈ S, λ ∈ 0, ǫ 0 (P ) , the Hamiltonian H 0 + H 1 admits Arnold diffusion.
We mention that our genericity result can extend to the Hamiltonians of the form H 0 (p, q, I ) + εH 1 (p, q, I , ϕ, t ; ε) where H 1 (p, q, I , ϕ, t ; ε) also depends analytically on the parameter ε. In fact, our geometric method uses mainly the first-order analysis. The conditions (H3a)-(H3b), see Section 3, imposed on H 1 only involve the properties of H 1 (p, q, I , ϕ, t ; 0). Similar discussions can also be found in works such as [38,39].
It is exactly the set of all bounded real analytic functions on M . Note that C ω is a Fréchet space, we then have the following immediate consequence: and dense set V ⊂ C ω , and for each H 1 ∈ V we can find ε 0 = ε 0 (H 1 ) > 0 and ρ = ρ(H 1 ) > 0 satisfying the following property: for each ε ∈ (−ε 0 , ε 0 )\{0}, the Hamiltonian flow of H ε = H 0 +εH 1 admits a trajectory whose action variables I (t ) satisfy sup t >0 where the initial condition I (0) ∈ V I 0 .
As we will show in Section 4, the genericity is verified by taking advantage of perturbation functions depending only on (q, ϕ, t ). Thanks to the work of [38], we will see that two hypotheses formulated below as (H3a), (H3b) for a specific integral, imply diffusion. Hence, for us, it suffices to show that (H3a), (H3b) are generic. Therefore, we can even establish the genericity in the sense of Mañé [46], namely, the diffusive phenomenon occurs under generic periodic potential perturbations. More precisely, we denote by C ω κ (T n+d+1 ) the set of all real analytic functions which can extend analytically to the complex neighborhood {(q, ϕ, t ) ∈ C n+d+1 /(2πZ) n+d+1 : |Im ι| < κ, ι = q i , ϕ i , t }. Then we have where the initial condition I (0) ∈ V I 0 .
We end this section by giving a concluding remark on our result and approach.
(1) Our perturbative technique for the genericity is valid in both the C r -differentiable (3 ≤ r ≤ ∞) and the C ω topologies. Also, it applies to the genericity in the sense of Mañé.
(2) The unperturbed part H 0 is only needed to be C r smooth with r ≥ 3. We do not require the inner dynamics to satisfy a twist condition, and the diffusion mechanism used in the present paper only relies on the outer dynamics since invariant objects (e.g. primary KAM tori, Aubry-Mather sets) of the inner map are not used at all.
(3) Both the phase space of the rotator and the phase space of the pendulums can be of arbitrary dimensions.
2.3. Organization of the paper. In Section 3, we first review the results we use on the normally hyperbolic invariant manifolds and the scattering maps for the a priori unstable system (2.1). Then, we review the geometric program established in [38]. It allows us to obtain Arnold diffusion for the original dynamics by shadowing the pseudo-orbits of the scattering map. We provide more details for this geometric mechanism in Appendix C for the reader's convenience. Section 4 is devoted to the proofs of our results on analytic genericity. Appendix A and Appendix B give general introductions to the theory of NHIMs and the theory of scattering maps. The perturbative argument to break the possible degeneracies of the conditions in [38] is described in Section 4.

SCATTERING MAPS AND GEOMETRIC MECHANISM OF ARNOLD DIFFUSION
The main characteristic of an a priori unstable Hamiltonian system is that there exists a normally hyperbolic invariant manifold (NHIM) with unstable and stable invariant manifolds. The presence of these invariant objects plays an important role in the Arnold diffusion problem. The scattering map of the NHIM is an effective tool to quantify the homoclinic excursions. This map associates the orbit asymptotic in the past to the orbit asymptotic in the future. Using the perturbation theory and the Melnikov method one can estimate the effect of the perturbation on all the variables of the scattering map. See Appendix A and Appendix B for general introductions.
In this section, we first give some important results on the NHIM and the scattering map for our a priori unstable system H ε = H 0 + εH 1 . Then, we review a recent geometric mechanism of Arnold diffusion established in [38].
Recall that H ε satisfies conditions (H1)-(H2). From now on, it is convenient to fix two closed balls (suitably large) D * ⊂ D and B * ⊂ B, and study the dynamics on the following domain 3.1. Normal hyperbolicity of the unperturbed system. As the unperturbed system H 0 is given by we use Φ t ,0 to denote the corresponding autonomous C r −1 Hamiltonian flow on M = D ×T n ×B ×T d .
Condition (H2) implies that each pendulum 1 2 p 2 i + V i (q i ) has two homoclinic orbits. For each i , we choose and fix one homoclinic orbit (p 0 i (t ), q 0 i (t )). It converges exponentially to the hyperbolic equilibrium (0, 0) with characteristic exponent This is equivalent to saying The autonomous flow Φ t ,0 has a 2d -dimensional invariant manifold Λ 0 with boundary, Λ 0 is foliated completely by invariant tori, and hence the dynamics restricted on Λ 0 is integrable.
Sometimes, we need to work in the extended space M = M × T, which yields a (2d Λ 0 is a normally hyperbolic invariant manifold (see Appendix A). To verify it, we use (3.1) and take the normal exponents For the central exponents, we can take −λ c = µ c with the positive exponent µ c as close as desired to the value 0 since the dynamics on Λ 0 is completely integrable. Consequently, for everyx ∈ Λ 0 we have the invariant splitting of the tangent bundle Tx where the constant C > 1. The stable (resp. unstable) space E s x (resp. E ũ x ) is just the direct sum of the stable (resp. unstable) spaces at the hyperbolic equilibrium of each pendulum. In particular, Λ 0 is also On the other hand, there is also a family of homoclinic orbits parameterized by , · · · , n}, represents the time shift for the i -th homoclinic orbit. p 0 (τ + t1), q 0 (τ + t1) is asymptotic to (0, 0) in the future with an exponential rate at least µ s , and in the past with an exponential rate at least λ µ . Moreover, these homoclinic orbits form the stable manifold W s Λ 0 and the unstable manifold W u Λ 0 of the NHIM Λ 0 . In particular, the unstable and stable manifolds coincide, that is W s

Persistence of normally hyperbolic invariant manifolds.
In the theory of normally hyperbolic invariant manifolds, it is well known that the NHIM along with its stable and unstable manifolds persist under small perturbations [31,33,41]. In general, the NHIM will only be finitely differentiable. The optimal regularity depends on the ratio of the normal exponents and the central exponents. Following Appendix A.1.2, we set Remark 4. Note that ℓ s and ℓ u are only finite even when r = ∞ or ω. Taking the central exponents λ c and µ c sufficiently small if necessary, we can always let the indices ℓ s ≥ 2, ℓ u ≥ 2, and hence ℓ ≥ 2.
In particular, in the case of r ∈ [3, ∞), we can have ℓ = ℓ s = ℓ u = r − 1 for ε sufficiently small since λ c and µ c can be chosen as close as desired to 0.
The argument of [38], is a transversality argument that only requires a few derivatives of the invariant manifolds and the perturbations involved.
We give a sketch of the proof of Proposition 3.1 for the reader's convenience. We also refer to [21,24,37] for more details.
To prove Proposition 3.1, we first point out that the NHIM Λ 0 of the unperturbed equations has non-empty boundary on which the flow Φ t ,0 is invariant, but the invariance on the boundary will be destroyed under perturbations in general. Then, just as pointed out in [31,32], a standard treatment is to consider a slightly modified Hamiltonian. More precisely, we take two open domains U 1 and U 2 close enough to D and B respectively, and be a C ∞ smooth bump function such that ρ| U 1 ×U 2 ≡ 1, and ρ(p, I ) = 0 for those points (p, I ) outside D × B. Then we define the modified Hamiltonian G ε as follows normally hyperbolic and invariant under the flow Φ t ,G 0 . Note that Λ G 0 has no boundary. Thus we can apply Theorem A.2 to the perturbed system G ε to obtain a unique NHIM Λ G ε of the flow Φ t ,G ε .
In general, the manifold Λ G ε constructed in this way depends on the nature of the modification on the boundary. Anyway, orbits that never pass through the modified region behave identically to those of the unmodified Hamiltonian H ε . Since Φ t ,G ε agrees with the original flow Φ t ,ε on the domain U 1 × T n ×U 2 ×T d ×T, we therefore obtain a normally hyperbolic manifold Λ ε that is locally invariant under Φ t ,ε . Note that Λ ε is in general not unique, as its construction depends on the modified Hamiltonian G ε . Nevertheless, any one of them can be used to prove our following results because the conditions (H1)-(H2) are only on the unperturbed part H 0 . The choice only affects the smallness of ε 0 . See also [23] for more discussion.
The next step is to check the smoothness of Λ ε . Observe that Λ ε is normally hyperbolic with slight changes on the normal and central exponents given in (3.3). We denote the perturbed exponents by They are O(ε)-close to those in (3.3). This implies that the index ℓ defined in (3.5) would remain unchanged as long as ε is small enough. Hence, Λ ε is C ℓ smooth.
Finally, with the persistent manifold Λ ε , we obtain the local stable manifold W s,l oc In our model, for ε = 0 the stable and unstable manifolds of the flow Φ t ,0 coincide: do not coincide in general, and possibly do not intersect transversely along homoclinic manifolds. To measure the splitting of manifolds we introduce the Poincaré function (or Melnikov potential), where ω(I ) = (ω 1 (I ), · · · , ω n (I )) = Dh(I ) ∈ C r −1 ,1 = (1, . . . , 1) ∈ R n and the orbits (p 0 , q 0 ) is given in (3.4). It is a convergent improper integral of the perturbation evaluated along homoclinic orbits of the unperturbed system. We stress that this integral is absolutely convergent, because p 0 (τ + t1), q 0 (τ + t1) converges exponentially fast to (0, 0) as t → ±∞.
By definition it is easily seen that In particular, for the lower-dimensional case n = 1, which implies that the Poincaré function L(τ, I , ϕ, s) is periodic or quasi-periodic with respect to τ.
Remark 5 (Regularity of L). Note that the integral (3.7) is evaluated not on the perturbed homoclinic orbit but only on the unperturbed one. As H 1 is real analytic, it is not difficult to check that L(τ, I , ϕ, s) is C r −1 smooth. More precisely, the dependence on the variable τ is C r , the dependence on the variable I is C r −1 and the dependence on the variables (ϕ, s) is C ω . In particular, in the case when r = ∞ (resp. ω), the function L(τ, I , ϕ, s) is also C ∞ (resp. C ω ).
To verify the mechanism in [38], it suffices to verify two assumptions on L. Assumption (H3a) predicts that the perturbations generate homoclinic intersections and assumption (H3b) implies that the homoclinic intersections indeed generate changes in the action. Rather remarkably, both assumptions amount to properties of L.
The non-degenerate critical points of L would yield the existence of transverse homoclinic orbits for the perturbed system.
has a non-degenerate critical point τ * . By the implicit function theorem, τ * is locally given by We stress that in the present paper the size of the domain U − in (H3a) does not need to be too large, but it is independent of ε since the expression we need to study does not involve ε. In applications, it suffices to verify the existence of non-degenerate critical point for a fixed point (I 0 , ϕ 0 , s 0 ),then by the implicit function theorem there is a neighborhood U − of (I 0 , ϕ 0 , s 0 ) whose size is independent of ε, such that (H3a) holds.
With Proposition 3.1, the following result is well known. See for instance [21,24,37] for more details.
and the stable and unstable manifolds W s,u Λ ε intersect transversally along Γ ε . Also, Γ ε is transverse to the foliations of the stable/unstable manifolds. Moreover, Γ ε can C ℓ -smoothly extend to a manifold Γ 0 := As we can see from Remark 13 in Appendix B, taking U − suitably small if necessary we can ensure which are projections along the stable and unstable leaves (see Appendix B). Then, the wave maps restricted on Γ ε , namely Ω ε ± | Γ ε , are diffeomorphisms. In addition, by recalling the normal exponents in (3.6), for eachx ∈ Γ ε we can find a unique pointx + = Ω ε + (x) and a unique pointx Consequently, each Γ ε with 0 < |ε| ≤ ε 0 is a homoclinic channel. This enables us to define the scattering map σ ε := σ Γ ε associated to the homoclinic channel Γ ε , that is The scattering map σ ε above is a C ℓ diffeomorphism. By Proposition 3.2 the homoclinic manifold Γ ε can extend smoothly to a limiting manifold Γ 0 . Even Γ 0 is not a transversal intersection, σ ε can still C ℓ -smoothly extend to the identity map σ 0 = Id, as ε → 0.
3.4. Perturbative formulas for the scattering maps. The Melnikov method can also be used to estimate the effect of the perturbations on the scattering map. As was shown in [23], the symplectic property allows to give perturbative formulas for the Hamiltonian which generates the deformation of a family of symplectic scattering maps. This requires the dimension of the NHIM to be even while our Λ ε mentioned above is of odd dimensions. To overcome this difficulty, it is standard to consider an autonomous Hamiltonian defined by where (A, s) ∈ R × T are symplectically conjugate variables. The extended phase space is endowed with Then, the motions of the conjugate variables (A, t ) are governed bẏ However, the variable A does not play any dynamical role, because A does not appear in any of the ODEs for any of the coordinates, including itself. Consequently, by abuse of notation, we continue to use Φ t ,0 to denote the unperturbed flow and use to denote the normally hyperbolic invariant manifold, which is (2d +2)-dimensional. Since A does not play any dynamical role, the results obtained in the previous sections remain true for H ε . Then we continue to use Λ ε and W s,u Λ ε , respectively, to denote the NHIM and the associated stable and unstable manifolds for the flow Φ t ,ε . Also, we have the scattering map σ ε := σ Γ ε associated to the homoclinic channel Γ ε . Now that Λ ε has even dimensions, the map σ ε is symplectic (see [23]).
The perturbed NHIM Λ ε can be described in terms of the coordinates (I , ϕ, A, s) ∈ Λ 0 . In fact, there is a unique C ℓ -smooth family of symplectic parametrization k ε : Λ 0 → Λ ε , with k 0 = Id, satisfying Then we can express the scattering map σ ε on the reference manifold Λ 0 by: It is clear that s ε are the expression of σ ε in the same coordinate system Λ 0 , and hence s ε ∈ C ℓ with ℓ ≥ 2. Moreover, using the deformation theory this family of symplectic maps s ε can be generated by a Hamiltonian vector field [23]: there exists a Hamiltonian function S ε such that where S 0 (I , ϕ, A, s) := −L τ * (I , ϕ, s), I , ϕ, s .
Here, L is the Melnikov potential and τ * is given in (H3a). The function L(τ * (I , ϕ, s), I , ϕ, s) : U − → R is defined in a domain U − whose size is independent of ε. Thus, we compute the perturbed scattering map s ε up to the first order with respect to the size of the perturbation: with J the canonical matrix of the symplectic form ω 0 . See [23].
We infer from ∂L ∂τ (τ * (I , ϕ, s) Similarly, Here and subsequently, we use ∂L ∂I to denote the partial derivative of the Poincaré function L(τ, I , ϕ, s) with respect to the second variable, ∂L ∂ϕ to denote the partial derivative with respect to the third variable, and ∂L ∂s to denote the partial derivative with respect to the fourth variable. Now, formula (3.10) becomes there are manifolds Λ ε and symplectic parameterizations k ε such that In fact, Λ ε is exactly, by omitting the variable A since it does not play any dynamical role, the intersection of Λ ε and the section {s = 2π}. k ε is exactly the restriction of k ε to the section {s = 2π}. Thus, Λ ε is the normally hyperbolic (locally) invariant manifold for the map f ε .
Analogously, f ε has the corresponding homoclinic channel Γ ε and the scattering map σ ε := σ Γ ε : . As mentioned before, it is more convenient to describe the scattering map in the same coordinate (I , θ) via: Clearly, s ε are still symplectic, and one can choose a common domain Dom(L * ) for all s ε . Therefore, combining formula (3.11) and Remark 6 we conclude the following result: where the O symbol means estimates in the C ℓ−1 sense of the reminder.
See [21,23] for more details. In view of Proposition 3.3 we can formulate the following assumption: That is equivalent to saying ∂L * ∂θ (I 0 , θ 0 ) = 0. (3.15) It is worth noting that assumption (H3b) ensures that the vector fieldẋ = J∇L * (x) always has a trajectory along which the action variables I move a quantity independent of ε.
It is remarkable that both assumptions (H3a) and (H3b) amount to properties of the Melnikov potential L in (3.7). The gist of the genericity argument is that, if they happen to fail for some H 1 , a small modification of the H 1 will make them true. Both can be considered as transversality properties on the functions. We also remark that the perturbations needed to restore (H3a), (H3b) are themselves rather arbitrary. Hence, the assumptions can only fail for H 1 inside a manifold of infinite codimension in the space of maps.
3.5. Geometric construction of the diffusing orbits. In this paper, the construction of diffusing orbits is based on the geometric mechanism in [38]. This mechanism differs from earlier works, because it relies only on the outer dynamics. There are almost no assumptions on the inner dynamics (only the Poincaré recurrence is needed), because its invariant objects (e.g., primary and secondary tori, Aubry-Mather sets) on the NHIM are not used at all. The basic idea of this new mechanism is as follows: Assume that the Poincaré recurrence holds. Given any pseudo-orbit, generated by the successive iter- Through the parameterization k ε , the restriction of f ε to Λ ε can be expressed on the same reference manifold Λ 0 . We use f ε | Λ 0 to denote this restriction map:
We provide a sketch of the proof in Appendix C for the reader's convenience. One can also refer to [38,Theorem 3.11] for a complete proof.
Since the recurrence assumption is satisfied automatically in our Hamiltonian model if there are no unbounded orbits in the manifold, under hypotheses (H3a), (H3b), that either there are unbounded orbits in the NHIM or that there are sequences of homoclinic excursions that follow the sequence. In either of the two sides of the alternative, there is diffusion. The proof is postponed to Appendix C, which comes mainly from [38]. Finally, we also refer to [39] for the result using accessibility and several scattering maps.
Before proving it, we need some lemmas.

Lemma 4.2.
For each κ > 0 and each a ∈ R \ {0}, there is a sequence of real analytic functions f l ∈ C ω κ (T), l = 1, · · · , n, satisfying where e i at = cos at +i sin at , and q 0 l (t ) is the q l -coordinate of the unperturbed homoclinic orbit (p 0 , q 0 ).
Proof. For simplicity we only verify it for l = 1 and the others are similar. It is sufficient to prove that there exists a function f 1 in the set X := {cos k x, sink x : k ∈ Z}, such that Suppose that for all f ∈ X , ] · e i at d t = 0, then using the theory of Fourier analysis it is not difficult to prove that On the other hand, recalling that q 0 1 (t ) converges exponentially to 0 as t tends to ±∞, we can find a small closed interval J ⊂ T \ {0}, such that the orbit q 0 1 (t ) passes through J when and only when t ∈ [t − σ, t + σ] for some t ∈ R, σ > 0. By narrowing the interval J if necessary, we can let σ < π 4a . Let us pick a non-negative function h ∈ C 1 (T) satisfying h(0) = 0 and the support supph = J , then  For the case where ∂L ∂τ (τ,Î ,φ,ŝ) is always non-zero, we assume by contradiction that there is δ > 0 such that ∂L ∂τ (τ,Î ,φ,ŝ) ≥ δ, for all τ ∈ R n .
which is finite and bounded. Using the mean value theorem to (4.1), for every τ ∈ R n we have where C is a constant, and the last inequality is a consequence of the fact that (p 0 i (t ), q 0 i (t )) converges exponentially to (0, 0) as t → ±∞. Similarly, as the derivative (ṗ 0 i (t ),q 0 i (t )) also converges exponentially to (0, 0) as t → ±∞, we deduce that where C ′ > 0 is a constant.
Recall that (Î ,φ,ŝ) is fixed, then we consider an auxiliary differential equation: From (4.6) we see that the vector field above is bounded, which implies the flow is complete, that is all solutions are well defined for t ∈ R. We pick one solution x(t ) : R → R n and consider the onedimensional function t −→ L(x(t ),Î ,φ,ŝ). Then, for any T > 0 we obtain Here, we have used (4.4) in the last inequality. As T can be arbitrarily large, the estimate (4.7) yields sup τ∈R n |L(τ,Î ,φ,ŝ)| = +∞. This contradicts the boundedness estimate (4.5). Therefore, we finish the proof of (4.3).
Proof of Theorem 4.1. The openness is evident. In fact, a non-degenerate critical point for the map τ → L(τ, I , ϕ, s) remains non-degenerate after a sufficiently small C 2 perturbation, which therefore gives C 3 -openness and also C ω κ -openness. Here, C 3 -smoothness is necessary because H 1 needs to satisfy the lowest regularity (i.e. C 3 ) of the unperturbed system H 0 so that all the results obtained in Section 3 are still valid.
Then it remains to show that the existence of non-degenerate critical points is a dense property in the C ω κ topology. The proof splits into two steps.
Step 1: Fix the point (I 0 , ϕ 0 , s 0 ). If the map τ −→ L(τ, I 0 , ϕ 0 , s 0 ) has no critical points, we will show that there is an arbitrarily small perturbation to H 1 to create critical points. For this purpose, we take a small number δ > 0 and add a small perturbation δ 2 H 2 to H 1 with δ 2 ∈ (0, δ/2), and H 2 is of the form the coefficients {b i } n i =1 and the analytic functions { f i ∈ C ω κ (T)} n i =1 will be determined later. By multiplying a constant if necessary, we can always let H 2 κ < 1. Hence the new Melnikov potential, denoted by L δ 2 , associated to the Hamiltonian where for each i = 1, · · · , n, the constants A i ,1 , A i ,2 are given by . We can ensure each A 2 i ,1 + A 2 i ,2 = 0 by suitably choosing f i (see Lemma 4.2). This gives rise to In particular, for the point (I 0 , ϕ 0 , s 0 ) we can invoke Lemma 4.3 to find a point τ * satisfying ∂L ∂τ (τ * , I 0 , ϕ 0 , s 0 ) < min 1≤i ≤n Then for each i we can find b i ∈ [0, 2π] such that x i (τ * , I 0 , ϕ 0 , s 0 ) = 0, which therefore yields ∂L δ 2 ∂τ (τ * , I 0 , ϕ 0 , s 0 ) = 0.
Step 2: We have already shown the existence of critical points is a dense property. In the above argument, we may pick H 2 = 0 whenever the map τ → L(τ, I 0 , ϕ 0 , s 0 ) already has critical points.
Now, let us turn to check the non-degeneracy of the critical points. If (τ * , I 0 , ϕ 0 , s 0 ) is a degenerate critical point of L δ 2 , i.e., then we have to continue to add a small perturbation to create non-degeneracy. Indeed, we may pick a perturbation δ 3 H 3 to H 1 + δ 2 H 2 where δ 3 ∈ (0, δ/2) and The value δ 3 , the coefficients {c i ∈ R} n i =1 and the analytic functions {g i ∈ C ω κ (T)} n i =1 will be determined later. Without loss of generality we let H 3 κ < 1. Using arguments analogous to (4.8), the new Melnikov potential, denoted by L δ 2 ,δ 3 , associated to the Hamiltonian where the constants B i ,1 , B i ,2 are . Here, we can ensure each B 2 i ,1 + B 2 i ,2 = 0 by suitably choosing g i , see Lemma 4.2.
For each i = 1, · · · , n we take then the Hessian matrix of the map τ −→ L δ 2 ,δ 3 (τ, I 0 , ϕ 0 , s 0 ) is where the second term on the right-hand side is a diagonal matrix, and as a result of (4.11), Denoting v(δ 3 ) := det ∂ 2 L δ 2 ,δ 3 ∂τ i τ j (τ * , I 0 , ϕ 0 , s 0 ) 1≤i , j ≤n In particular, v(0) = 0 as a consequence of (4.9). It is not difficult to check that the one-dimensional function v : δ 3 → R is a polynomial function of degree n, and the leading term of v is n i =1 λ i δ n 3 . Thanks to (4.12), the leading coefficient is non-zero, which implies that the polynomial v has at most n zeros. Consequently, we can choose arbitrarily small δ 3 > 0 such that v(δ 3 ) = 0.
In conclusion, we have constructed a perturbation δ 2 H 2 + δ 3 H 3 to H 1 where As δ > 0 can be arbitrarily small, the existence of non-degenerate critical points for τ → L(τ, I 0 , ϕ 0 , s 0 ) is a dense property in the C ω κ topology. Finally, using the implicit function theorem we can obtain an open neighborhood U − = I × J ⊂ B * × T d+1 of the point (I 0 , ϕ 0 , s 0 ), such that for each (I , ϕ, s) ∈ U − the map τ ∈ R n −→ L(τ, I , ϕ, s) has a non-degenerate critical point τ * = τ * (I , ϕ, s). This finishes our proof.
We have provided a constructive proof for Theorem 4.1. Next, we proceed to show the genericity of assumption (H3b). Proof. Since assumption (H3a) has already been proved to be open and dense, in the following proof we restrict our discussions to the case where assumption (H3a) always holds. As mentioned previously, τ * (I , θ, 0) ∈ R n denotes the non-degenerate critical point for the map τ −→ L(τ, I , θ, 0). Just like assumption (H3a), it is easy to see that the non-degeneracy assumption (H3b) is open in the C 3 topology, which directly gives C ω κ -openness. Thus, the only thing left is to verify the density of (H3b). In what follows, we fix a point (Î ,θ) ∈ Dom(L * ). If ∂L * /∂θ(Î ,θ) = 0, then we have finished.

Suppose now
∂L * ∂θ (Î ,θ) = 0, (4.13) we will add a small perturbation to create non-degeneracy. More precisely, we add a perturbation δH 2 to H 1 where the number δ > 0 is small enough and the analytic function H 2 will be determined later.
As we will see below, the values of C i (Î , τ * ) for i = 2, 3, · · · , d play no role in the following proof.
Proof of Theorem 2.3. As we can see from the proof of Theorem 4.1, the perturbation functions constructed by us depend only on (q, t ) ∈ T n × T. Meanwhile, in the proof of Theorem 4.4, the perturbation functions constructed by us depend only on (q, ϕ) ∈ T n × T d . This implies that the genericity of assumptions (H3a) and (H3b) are established by constructing potential perturbations that are independent of p and I . Therefore, assumptions (H3a)-(H3b) are also open-dense in the C ω κ (T n+d+1 ) space. The remaining proof is just the same as that of Theorem 2.1.
Remark 10. We have verified that, in case that there is some degeneracy, it can be removed by adding some cos functions.
Verifying that perturbations of this kind remove the degeneracy amounted to a determinant being non-zero. Clearly, if we modify the cos slightly, this condition will remain true. This justifies the observation that the degeneracy can only fail of H 1 in a submanifold of infinite codimension.
We will not pursue this line of reasoning, but it seems that indeed, the set of directions transversal to the manifold containing all the H 1 where diffusion fails is not only infinite dimensional, but also dense. This is indeed a very strong form of genericity.

APPENDIX A. NORMALLY HYPERBOLIC INVARIANT MANIFOLDS
Normally hyperbolic invariant manifold (NHIM) can be viewed as a natural generalization of hyperbolic set. The NHIM has not only stable and unstable directions, but also central directions (tangent to the manifold itself). The theory of normal hyperbolicity and the theory of partial hyperbolicity are closely related in their results and methods. We refer the reader to the standard references [31,32,41,50]. In this appendix, we only review some classical results, including the existence of NHIMs, the existence of the stable and unstable manifolds and their invariant foliations, and the smoothness and the persistence of these manifolds. The definition of normal hyperbolicity that we adopt below is based on [41].
A.1. The continuous case. Let M be a smooth Riemannian manifold and Φ t be an autonomous C r 0 where the constant C > 1, and the rates The superscripts c, u and s stand for "center", "unstable" and "stable", respectively.
Remark 11. For the Hamiltonian model considered in this paper, the rates λ c , µ c are close to zero.
For a NHIM, the expansion and contraction in the central directions are weaker than those in the normal directions. Then the dynamics on N is approximately neutral and the dynamics in the normal directions is hyperbolic. That is why we call it normally hyperbolic.
We point out that the manifold M is not necessarily compact. As remarked in [41,3], it suffices to assume that Φ t is C r 0 in a neighborhood of N with all the derivatives of order up to r 0 uniformly continuous and uniformly bounded.
The global stable and unstable manifolds can have a topological characterization: where "dist(·, ·)" is the distance induced by the Riemannian metric on M . In fact, the distance convergence in (A.2) is exponential.
For each x ∈ N , we can also construct the stable and unstable manifolds with basepoint x: where C y is a constant, and ε > 0 is any small number satisfying This tells us that the trajectories starting on W s x or W u x satisfy certain asymptotic growth rate conditions, and the growth rate shall be greater than that of the trajectories on N . have limited regularity even if the flow is C ∞ . Their regularity is dictated by the ratio of the normal hyperbolicity and the central hyperbolicity. More precisely, we introduce the following integers (see [50,Chapter 5] or [41]): Clearly, ℓ u , ℓ s and ℓ are all finite values even when r 0 = ∞. The index ℓ means that DΦ t expands E u (resp. contracts E s ) at rates at least ℓ times of its expansion (resp. contraction) rate in T N . Thus, such a manifold N is also called ℓ-normally hyperbolic.
A.3. Persistence and dependence on parameters. For applications, it is important to study the persistence of the NHIM under perturbations, and if the persistent manifold depends smoothly on the perturbation parameter.
Theorem A.2. [31,41,3] Let N ⊂ M be a submanifold without boundary and N is a ℓ-normally hyperbolic invariant manifold (ℓ is defined in (A.5)) for the C r 0 flow Φ t generated by the vector field X . Then for the vector field Y which is C 1 -close to X , there exists a unique normally hyperbolic and Φ Y t -invarnat manifold N Y , which, is C ℓ diffeomorphic and close to N . In particular, N Y is also ℓ-normally hyperbolic.
The local stable manifold W s,l oc N Y and local unstable manifold W u,l oc N Y are C ℓ close to those of N .
Remark 12. The C 1 -closeness between the vectors is enough because the change in the rates λ ι , µ ι , ι = s, c, u, can be controlled by the C 1 distance. The persistent manifold N Y is still C ℓ for the reason that the exponents in (A.1) for N Y is very close to those of N , and hence the index ℓ in (A.5) remains unchanged.
For the case where the submanifold N has non-empty boundary, a locally invariant and normally hyperbolic manifold persists. To prove it, one can construct a slightly modified system for which the normally hyperbolic manifolds are globally invariant under the modified flow. Then, by Theorem A.2 there is a unique persistent NHIM. This invariant manifold for the slightly modified system would be a locally invariant manifold for the original system. It also implies the locally invariant manifold is not unique in general. Nevertheless, there are some cases for which the uniqueness holds. For example, the locally invariant manifold has KAM tori bounding them. For α = 0, N 0 is a C ℓ NHIM of the flow Ψ 0 t = Φ t ,0 × Id. When α is fixed and sufficiently small, the flow Ψ α t is close enough to Φ t ,0 , then the persistence result implies that the flow Ψ α t has a NHIM N α ⊂ M, and N α is C ℓ close and diffeomorphic to N 0 . Note that N α can be decomposed into where each N αε is invariant under Φ t ,αε and depends C ℓ -smoothly on the parameter ε ∈ [0, 1]. Also, it is not difficult to check that each N αε is a NHIM. Therefore, we conclude that for ε ∈ [0, α], N ε is a normally hyperbolic and Φ t ,ε -invariant manifold depending C ℓ -smoothly on the parameter ε.

APPENDIX B. THE SCATTERING MAP
The scattering map is used to describe homoclinic excursions. This map is introduced explicitly in [19] and enjoys remarkable geometric properties [23].
Recall that W s N and W u N are, respectively, foliated by the stable leaves W s x and the unstable leaves W u x , x ∈ N . For any point x ∈ W s N (resp. x ∈ W u N ), there is a unique point x + ∈ N (resp. x − ∈ N ) such that x ∈ W s x + (resp. x ∈ W u x − ). Then we can define the wave maps which are projections along the the leaves: The wave maps are C ℓ smooth as a result of the C ℓ -foliation property (see Proposition A.1).
To define the scattering map, we need the following transversality conditions: (1) W s N and W u N have a transversal intersection along a homoclinic manifold Γ, i.e., for each z ∈ Γ, (2) Γ is transverse to the foliations of the stable/unstable manifolds at each point z ∈ Γ, that is where x ± are the uniquely defined points in N satisfying z ∈ W s 3) hold at some point z * ∈ W u N ∩ W s N , then by the implicit function theorem the transversality conditions are also satisfied for all z ∈ W u N ∩ W s N close to z * . Hence we can find a locally unique manifold Γ ∋ z * satisfying (B.2)-(B.3). In addition, Γ is C ℓ .
They are C ℓ local diffeomorphisms in general. Even if Ω ± | Γ are locally invertible, they could fail to be invertible in a domain with non-contractible loops, see [19,22] for more examples.
Following [23], we say Γ ⊂ W u N ∩ W s N a homoclinic channel if it satisfies (B.2)-(B.3) and Ω ± | Γ are C ℓ diffeomorphisms. Then, the scattering map σ Γ associated to the homoclinic channel Γ is Note that σ Γ is C ℓ smooth. Clearly, the definition of scattering map depends on the homoclinic channel. Sometimes we will omit Γ from the notation when there is no confusion. One can also expect to have infinitely many Γ, and each of which has a different scattering map.
Remark 14. We shall note that there is no actual orbit of the flow Φ t starting from x − to x + . If x + = σ Γ (x − ), then we infer from (A.3) that there is a point z ∈ Γ satisfying APPENDIX C. PROOFS OF THEOREM 3.4 AND THEOREM 3.5 For the reader's convenience we repeat the relevant material from [38] to give a sketch of the proof of Theorems 3.4-3.5.
We claim that all points of the sequence {x i } N i =0 lies in V as long as ε is small enough. Let us take a sequence y i := γ(i ε) with i = 0, · · · , N . Denoting by φ t the Hamiltonian flow associated to the equation (C.2), we invoke the Gronwall inequality to (C.2) to find a constant C 1 > 0 such that Recalling the scattering map s ε is, up to O(ε 2 ), the ε-flow of the Hamiltonian L * (I , θ), we obtain where C 2 > 0 is a constant. We also remark that C 1 and C 2 depend on H 1 .
Recalling that in Theorem 3.4 we use f ε to denote the time-2π map for the flow of the Hamiltonian H ε , and use f ε | Λ 0 = (k ε ) −1 • f ε | Λ ε • k ε to denote the parameterized map defined on Λ 0 . Denoting it is a f ε -invariant set in Λ 0 , i.e. f ε (V ) ⊂ V . The measure of V is either finite or +∞.
This completes the proof.