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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 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]

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]

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]

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]

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]

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 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]

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]

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 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]

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]

In this article we prove an action-angle theorem for b -integrable systems on b -Poisson manifolds improving the action-angle theorem contained in [14] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b -Poisson manifolds. [ABSTRACT FROM AUTHOR]