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The diffusion mechanism in Hamiltonian systems, close to completely integrable, is usually connected with the existence of the so-called "transition chains". In this case slow diffusion occurs in a neighborhood of intersecting separatrices of hyperbolic periodic solutions (or, more generally, lower-dimensional invariant tori) of the perturbed system. In this note we discuss another diffusion mechanism that uses destruction of invariant tori of the unperturbed system with an almost resonant set of frequencies. We demonstrate this mechanism on a particular isoenergetically nondegenerate Hamiltonian system with three degrees of freedom. The same phenomenon also occurs for general higher-dimensional Hamiltonian systems. Drift of slow variables is shown using analysis of integrals of quasi-periodic functions of the time variable (possibly unbounded) with zero mean value. In addition, the proof uses the conditions of topological transitivity for cylindrical cascades. [ABSTRACT FROM AUTHOR]

The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness. [ABSTRACT FROM AUTHOR]

Dynamical systems subject to perturbations that decay over time are relevant in describing many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, and celestial mechanics. For this purpose, we consider time-dependent Hamiltonian vector fields that are the sum of two components. The first has an invariant torus supporting quasiperiodic solutions, and the second decays as time tends to infinity. The time decay is modelled by functions satisfying suitable conditions verified by a proper polynomial decay in time. We prove the existence of orbits converging as time tends to infinity to the quasiperiodic solutions associated with the unperturbed system. The proof of this result relies on a new strategy based on a refined analysis of the Banach spaces and the functionals involved in the resolution of suitable nonlinear invariant equations. This result is proved for finite differentiable and real-analytic Hamiltonians. Analogous statements for time-dependent vector fields on the torus are also obtained as a corollary. These results extend a previous work of Canadell and de la Llave, where only exponential decay in time is considered. The relaxation of the decay in time makes the results in the present paper suited for applications in many physical problems, such as celestial dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hamiltonian systems H(y, x, ξ, ε) = 〈ω(ξ),y〉+εP(y, x, ξ, ε), where ω and P are continuous about ξ. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under a certain transversality condition and a weak convexity condition for the frequency mapping ω. As a direct application, we prove a Kolmogorov-Arnold-Moser (KAM) theorem when the perturbation P holds arbitrary Hölder continuity with respect to the parameter ξ. The infinite-dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in the parameter beyond Hölder's type. [ABSTRACT FROM AUTHOR]

In this article, we define a circular restricted (N + 1) -body problem on the surfaces M κ 3 , with κ = ± 1 . The motion of the primaries corresponds to an elliptic relative equilibria studied in Diacu (Relative equilibria of the curved N-body problem. Atlantis Studies in Dynamical Systems, Atlantis Press, Paris, 2012), where N identical mass particles are rotating uniformly at the vertices of a regular polygon placed at a fixed parallel of a maximal sphere. By introducing rotating coordinates, this problem gives rise to a 3 d.o.f. Hamiltonian system. This problem has an equilibrium point placed at the poles of S 3 and the vertex of H 3 , for any value of the parameters. We give information about the linear and nonlinear stability of these equilibria. Finally, we carry out a study about the existence of periodic solutions and KAM tori. [ABSTRACT FROM AUTHOR]

In this paper we consider the persistence of degenerate lower-dimensional tori in reversible systems with a degenerate normal equilibrium point, including hyperbolic and elliptic types. Based on the method of introducing external parameters, KAM iteration and implicit function theorem, we prove that if the perturbations are sufficiently small and frequency ω 0 satisfies the Diophantine condition, the reversible system still has a lower-dimensional torus with frequency ω 0 . [ABSTRACT FROM AUTHOR]

This article is concerned with the geometry of germs of real analytic surfaces in (C 2 , 0) having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in (C 2 , 0) having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions { τ 1 , τ 2 } at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to { z 1 z 2 = c o n s t } (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of almost periodically forced harmonic oscillators x ¨ + τ 2 x = ϵ f (t , x) where τ ∈ A with A being a closed interval not containing zero, the forcing term f is real analytic almost periodic functions in t with the infinite frequency ω = (⋯ , ω λ , ⋯) λ ∈ Z . Using the modified Kolmogorov–Arnold–Moser (or KAM Arnold (Uspehi Mat. Nauk 18(5 (113)):13–40, 1963), Moser (Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962:1–20 1962), Kolmogorov (Dokl. Akad. Nauk SSSR (N.S.) 98:527–530 1954)) theory about the lower dimensional tori, we show that there exists a positive Lebesgue measure set of τ contained in A such that the harmonic oscillators has almost periodic solutions with the same frequencies as f. The result extends the earlier research results with the forcing term f being quasi-periodic. [ABSTRACT FROM AUTHOR]

We consider small perturbations of analytic non-twist area preserving mappings, and prove the existence of invariant curves with prescribed frequency by KAM iteration. Generally speaking, the frequency of invariant curve may undergo some drift, if the twist condition is not satisfied. But in this paper, we deal with a degenerate situation where the unperturbed rotation angle function r → w + r2n+1 is odd order degenerate at r = 0, and prove the existence of invariant curve without any drift in its frequency. Furthermore, we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders. [ABSTRACT FROM AUTHOR]

For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G , we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n -colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n -coloring if and only if $n=3$. We finally suggest generalizations of the latter results to n -colorings of ascending HNN-extensions of finitely generated abelian groups. [ABSTRACT FROM AUTHOR]

This paper focuses on the vector fields on the quasi-periodically forced circle flow φ ˙ = ρ + f (φ , θ) , θ ˙ = α where the forcing term f is real analytic in its arguments and small enough, the frequency vector α ∈ R d (d ≥ 2) is beyond Brjuno frequency. We prove that the flow above is reducible analytically provided fibered rotation number ρ f : = ρ (ρ + f (φ , θ)) is Diophantine with respect to the base frequency α. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite-dimensional systems with multi-dimensional frequency weaker than Brjuno frequency. [ABSTRACT FROM AUTHOR]

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mechanical counterexample which satisfies all conditions of Moser's twist theorem except the smoothness condition. The model (also called ping-pong model) is the vertical motion of a bouncing ball on a plate which moves in the vertical direction as a C 1 + α -smooth periodic function with 0 ≤ α < 1 / 3 . We construct an unbounded orbit of this simple mechanical model to present a mechanical phenomenon violating Moser's twist theorem. [ABSTRACT FROM AUTHOR]

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at times t where these models have validity, namely t = O(h−2), where h is proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously. [ABSTRACT FROM AUTHOR]

We study the integrability of the Hamiltonian normal form of 1:2:2 resonance. It is known that this normal form truncated to order three is integrable. The truncated to order four normal form contains many parameters. For a generic choice of parameters in the normal form up to order four, we carry on non-integrability analysis, based on the Morales–Ramis theory using only first variational equations along certain particular solutions. The non-integrability obtained by algebraic proofs produces dynamics illustrated by some numerical experiments.We also isolate a non-trivial case of integrability. [ABSTRACT FROM AUTHOR]

Cornerstone models of physics, from the semi-classical mechanics in atomic and molecular physics to planetary systems, are represented by quasi-integrable Hamiltonian systems. Since Arnold's example, the long-term diffusion in Hamiltonian systems with more than two degrees of freedom has been represented as a slow diffusion within the 'Arnold web,' an intricate web formed by chaotic trajectories. With modern computers it became possible to perform numerical integrations which reveal this phenomenon for moderately small perturbations. Here we provide a semi-analytic model which predicts the extremely slow-time evolution of the action variables along the resonances of multiplicity one. We base our model on two concepts: (i) by considering a (quasi-)stationary-phase approach to the analysis of the Nekhoroshev normal form, we demonstrate that only a small fraction of the terms of the associated optimal remainder provide meaningful contributions to the evolution of the action variables. (ii) We provide rigorous analytical approximations to the Melnikov integrals of terms with stationary or quasi-stationary phase. Applying our model to an example of three degrees of freedom steep Hamiltonian provides the speed of Arnold diffusion, as well as a precise representation of the evolution of the action variables, in very good agreement (over several orders of magnitude) with the numerically computed one. [ABSTRACT FROM AUTHOR]

A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange–d'Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given toward the end. [ABSTRACT FROM AUTHOR]

We consider geometric properties of 3-jet non-degenerate functions in connection with Nekhoroshev theory. In particular, after showing that 3-jet non-degenerate functions are "almost quasi-convex", we prove that they are steep and compute explicitly the steepness indices (which do not exceed 2) and the steepness coefficients. [ABSTRACT FROM AUTHOR]

We review V. I. Arnold's 1963 celebrated paper [1] Proof of A. N. Kolmogorov's Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold's scheme, one can get "sharp" asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed. [ABSTRACT FROM AUTHOR]

We give an alternate proof of a theorem in Wang and You (Z Angew Math Phys 47: 943–952, 1996) which shows that all solutions are bounded for a periodically forced nonlinear oscillator. Our proof relies on constructing an analytic change of variables by a convergent Lie series transformation to simplify the system so that the period map has large invariant curves by Moser's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the reducibility problem for a class of analytic almost periodic linear Hamiltonian systems dx dt = J [ A + ε Q (t) ] x where A is a symmetric matrix, J is an anti-symmetric symplectic matrix, Q(t) is an analytic almost periodic symmetric matrix with respect to t, and ε is a sufficiently small parameter. It is also assumed that JA has possible multiple eigenvalues and the basic frequencies of Q satisfy the non-resonance conditions. It is shown that, under some non-resonant conditions, some non-degeneracy conditions and for most sufficiently small ε , the Hamiltonian system can be reduced to a constant coefficients Hamiltonian system by means of an almost periodic symplectic change of variables with the same basic frequencies as Q(t). [ABSTRACT FROM AUTHOR]

We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order r = 1 ) and prove that when M is a smooth compact manifold or R d , the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When M = R d , d ≥ 2 , we also prove that for r ≥ 2 small radiation solutions of the order-r normalized equation approximate solutions of the nonlinear NLKG up to times of order O (c 2 (r - 1)) . We also prove a global existence result uniform with respect to c for the NLKG equation on R 3 with cubic nonlinearity for small initial data and Strichartz estimates for the Klein–Gordon equation with potential on R 3 . [ABSTRACT FROM AUTHOR]

We consider the motion of a 2π-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter e and other parameters and is autonomous at e = 0. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced two-degree-of-freedom system in the vicinity of the resonance point in the three-dimensional parameter space is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of 2π-periodic motions of the satellite is solved, and conclusions on the existence of two- and three-frequency conditionally periodic motions are obtained. [ABSTRACT FROM AUTHOR]

In 1994, Jürgen Moser generalized Hénon's area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser's six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none.The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser's map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled Hénon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2d planes through the phase space and by 3d slices through the tori. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems H=h(y)+f(x,y,t),where y∈D⊆Rn with D being an open bounded domain, x∈Tn, f(x, y, t) is a real analytic almost periodic function in t with the frequency ω=(…,ωλ,…)λ∈Z∈RZ. As an application, we will prove the existence of almost periodic solutions and the boundedness of all solutions for the second-order differential equations with superquadratic potentials depending almost periodically on time. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark-Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. We specifically obtain the Gevrey index of families of invariant curves which is related to the smoothness of area preserving mappings and the exponent of small divisors condition. Moreover, we obtain a Gevrey normal form of area preserving mappings in a neighborhood of the union of the invariant curves. [ABSTRACT FROM AUTHOR]

This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53-83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419-1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré-Treshchev mechanism. [ABSTRACT FROM AUTHOR]

From KAM theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, 'primary' tori in a nearly integrable, real-analytic Hamiltonian system is $$O(\sqrt{\varepsilon })$$ , if $$\varepsilon $$ is the size of the perturbation. In this paper we discuss how the constant in front of $$\sqrt{\varepsilon }$$ depends on the unperturbed system and in particular on the phase-space domain. [ABSTRACT FROM AUTHOR]

The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts' attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2. [ABSTRACT FROM AUTHOR]

A nearly-integrable dynamical system has a natural formulation in terms of actions, y (nearly constant), and angles, x (nearly rigidly rotating with frequency Ω( y)).We study angleaction maps that are close to symplectic and have a twist, the derivative of the frequency map, DΩ( y), that is positive-definite. When the map is symplectic, Nekhoroshev's theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volume-preserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank-r resonances. A comparison with computations for a generalized Froeschl´e map in four-dimensions shows that this theory gives accurate results for the rank-one case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the high dimensional Schrödinger equation $ -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, $ where u( t) is a real analytic quasi-periodic symmetric matrix, $E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})$ is a diagonal matrix with λ >0, j=1,..., n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann's non-resonant condition, then for most of sufficiently large λ , j=1,..., n, there exist n pairs of conjugate quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

We consider a system of rotators subject to a small quasi-periodic forcing. We require the forcing to be analytic and satisfy a time-reversibility property and we assume its frequency vector to be Bryuno. Then we prove that, without imposing any non-degeneracy condition on the forcing, there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. [ABSTRACT FROM AUTHOR]

We consider a multi-scale, nearly integrable Hamiltonian system. With proper degeneracy involved, such a Hamiltonian system arises naturally in problems of celestial mechanics such as Kepler problems. Under suitable non-degenerate conditions of Bruno-Rüssmann type, the persistence of the majority of non-resonant, quasi-periodic invariant tori has been shown in Han et al. (Ann. Henri Poincaré 10(8):1419-1436, 2010). This paper is devoted to the study of splitting of resonant invariant tori and the persistence of certain class of lower-dimensional tori in the resonance zone. Similar to the case of standard nearly integrable Hamiltonian systems (Li and Yi in Math. Ann. 326:649-690, 2003, Proceedings of Equadiff 2003, World Scientific, 2005, pp 136-151, 2005), we show the persistence of the majority of Poincaré-Treschev non-degenerate, lower-dimensional invariant tori on a the given resonant surface corresponding to the highest order of scale. The proof uses normal form reductions and KAM method in a non-standard way. More precisely, due to the involvement of multi-scales, finite steps of KAM iterations need to be firstly performed to the normal form to raise the non-integrable perturbation to a sufficiently high order for the standard KAM scheme to carry over. [ABSTRACT FROM AUTHOR]

The long-term diffusion properties of the action variables in real analytic quasiintegrable Hamiltonian systems is a largely open problem. The Nekhoroshev theorem provides bounds to such a diffusion as well as a set of techniques, constituting its proof, which have been used to inspect also the instability of the action variables on times longer than the Nekhoroshev stability time. In particular, the separation of the motions in a superposition of a fast drift oscillation and an extremely slow diffusion along the resonances has been observed in several numerical experiments. Global diffusion, which occurs when the range of the slow diffusion largely exceeds the range of fast drift oscillations, needs times larger than the Nekhoroshev stability times to be observed, and despite the power of modern computers, it has been detected only in a small interval of the perturbation parameter, just below the critical threshold of application of the theorem. In this paper we show through an example how sharp this phenomenon is. [ABSTRACT FROM AUTHOR]

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $$(S^2)^n$$ . In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $$T^*\mathbf {R}^{2n}$$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity. [ABSTRACT FROM AUTHOR]

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamiltonian, we estimate the size of the transformation that maps this Hamiltonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale. [ABSTRACT FROM AUTHOR]

In this paper we develop a new KAM technique to prove two general KAM theorems for nearly integrable Hamiltonian systems without assuming any nondegeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some nondegeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies. [ABSTRACT FROM AUTHOR]

We return to the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields. At the classical level, this is a Hamiltonian system with the phase space S × S. Its principal order Hamiltonian H was known already to Pauli in 1926. H defines an isochronous system with a linear flow on S × S and with frequencies depending on the external fields. Small perturbations of H due to higher order terms can be studied by further normalization, either resonant or nonresonant. We study the question, raised previously, of how to decide for given parameters of the fields what normalization should be used and with regard to which resonances. We base this analysis on the Nekhoroshev theory-a branch of the Hamiltonian perturbation theory that complements the Kolmogorov-Arnold-Moser theorem. Our answer depends on the a priori choice of the maximal order N of resonances that are going to be taken into account (a cutoff). For any given N, there is a decomposition of the parameter space into resonant and nonresonant zones, and a normal form with a remainder of order $\exp (-N)$ may be consistently constructed in each of such zones. [ABSTRACT FROM AUTHOR]

In this paper we prove the persistence of invariant tori for analytic perturbation of constant vector field under weaker non-degeneracy condition. In the proof we introduce a parameter q and make the steps of KAM iteration infinitely small in the speed of function $${q^{n} \epsilon}$$ , $${0 < q < 1}$$ , rather than super exponential function. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the Kolmogorov theorem of persistence of Diophantine flows for nearly integrable Poisson systems associated to a real analytic Hamiltonian with aperiodic time dependence, provided that the perturbation is asymptotically vanishing. The paper is an extension of an analogous result by the same authors for canonical Hamiltonian systems; the flexibility of the Lie series method developed by A. Giorgilli et al. is profitably used in the present generalization. [ABSTRACT FROM AUTHOR]

We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev's theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4-L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degrees-of-freedom Hamiltonian system obtained from the Fermi-Pasta-Ulam problem by setting the number of particles equal to four. [ABSTRACT FROM AUTHOR]