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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]

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 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]

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 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]

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]