Nekhoroshev Theorem Research Articles

Bounds on the growth of energy for particles on the torus with unbounded time dependent perturbations

We prove a C∞ version of Nekhoroshev theorem for time dependent Hamiltonians in Rd×Td. Precisely, we prove a result showing that for all times the energy of the system is bounded by a constant times ⟨t⟩ɛ. We apply the result to the dynamics of a charged particle in Td subject to a time dependent electromagnetic field.

A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and the corresponding Hamiltonian vector field is analytic as a map from the energy space to itself. Let ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} be the size of the perturbation. We prove that for initial data close in energy norm to an N-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain O(ϵ12(N+1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(\\epsilon ^{\\frac{1}{2(N+1)}})$$\\end{document} close to their initial value for times exponentially long with ϵ-12(N+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon ^{-\\frac{1}{2(N+1)}}$$\\end{document}.

Ensemble reservoir computing for dynamical systems: prediction of phase-space stable region for hadron storage rings

We investigate the ability of an ensemble reservoir computing approach to predict the long-term behaviour of the phase-space region in which the motion of charged particles in hadron storage rings is bounded, the so-called dynamic aperture. Currently, the calculation of the phase-space stability region of hadron storage rings is performed through direct computer simulations, which are resource- and time-intensive processes. Echo State Networks (ESN) are a class of recurrent neural networks that are computationally effective, since they avoid backpropagation and require only cross-validation. Furthermore, they have been proven to be universal approximants of dynamical systems. In this paper, we present the performance reached by ESN based on an ensemble approach for the prediction of the phase-space stability region and compare it with analytical scaling laws based on the stability-time estimate of the Nekhoroshev theorem for Hamiltonian systems. We observe that the proposed ESN approach is capable of effectively predicting the time evolution of the extent of the dynamic aperture, improving the predictions by analytical scaling laws, thus providing an efficient surrogate model.

Nekhoroshev estimates for the orbital stability of Earth’s satellites

We provide stability estimates, obtained by implementing the Nekhoroshev theorem, in reference to the orbital motion of a small body (satellite or space debris) around the Earth. We consider a Hamiltonian model, averaged over fast angles, including the J_2 geopotential term as well as third-body perturbations due to Sun and Moon. We discuss how to bring the Hamiltonian into a form suitable for the implementation of the Nekhoroshev theorem in the version given by Pöschel, (Math Z 213(1):187–216, 1993) for the ‘non-resonant’ regime. The manipulation of the Hamiltonian includes (i) averaging over fast angles, (ii) a suitable expansion around reference values for the orbit’s eccentricity and inclination, and (iii) a preliminary normalization allowing to eliminate particular terms whose existence is due to the nonzero inclination of the invariant plane of secular motions known as the ‘Laplace plane’. After bringing the Hamiltonian to a suitable form, we examine the domain of applicability of the theorem in the action space, translating the result in the space of physical elements. We find that the necessary conditions for the theorem to hold are fulfilled in some nonzero measure domains in the eccentricity and inclination plane (e, i) for a body’s orbital altitude (semimajor axis) up to about 20 000 km. For altitudes around 11 000 km, we obtain stability times of the order of several thousands of years in domains covering nearly all eccentricities and inclinations of interest in applications of the satellite problem, except for narrow zones around some so-called inclination-dependent resonances. On the other hand, the domains of Nekhoroshev stability recovered by the present method shrink in size as the semimajor axis a increases (and the corresponding Nekhoroshev times reduce to hundreds of years), while the stability domains practically all vanish for a>20{,}000 km. We finally examine the effect on Nekhoroshev stability by adding more geopotential terms (J_3 and J_4) as well as the second-order terms in J_2 in the Hamiltonian. We find that these terms have only a minimal effect on the domains of applicability of Nekhoroshev theorem and a moderate effect on the stability times.

Chaos and subdiffusion in infinite-range coupled quantum kicked rotors

We map the infinite-range coupled quantum kicked rotors over an infinite-range coupled interacting bosonic model. In this way we can apply exact diagonalization up to quite large system sizes and confirm that the system tends to ergodicity in the large-size limit. In the thermodynamic limit the system is described by a set of coupled Gross-Pitaevskij equations equivalent to an effective nonlinear single-rotor Hamiltonian. These equations give rise to a power-law increase in time of the energy with exponent $\gamma\sim 2/3$ in a wide range of parameters. We explain this finding by means of a master-equation approach based on the noisy behaviour of the effective nonlinear single-rotor Hamiltonian and on the Anderson localization of the single-rotor Floquet states. Furthermore, we study chaos by means of the largest Lyapunov exponent and find that it decreases towards zero for portions of the phase space with increasing momentum. Finally, we show that some stroboscopic Floquet integrals of motion of the noninteracting dynamics deviate from their initial values over a time scale related to the interaction strength according to the Nekhoroshev theorem.

Analysis of the non-linear beam dynamics at top energy for the CERN Large Hadron Collider by means of a diffusion model

In this paper, the experimental results of the recent dynamic aperture at top energy for the CERN Large Hadron Collider are analysed by means of a diffusion model whose novelty consists of deriving the functional form of the diffusion coefficient from Nekhoroshev theorem. This theorem provides an optimal estimate of the remainder of perturbative series for Hamiltonian systems. As a consequence, a three-parameter diffusion model is built that reproduces the experimental results with a high level of accuracy. A detailed discussion of the physical interpretation of the proposed model is also presented.

Advances on the modeling of the time evolution of dynamic aperture of hadron circular accelerators

Determining a model for the time scaling of the dynamic aperture of a circular accelerator is a topic of strong interest and intense research efforts in accelerator physics. The motivation arises in the possibility of finding a method to reliably extrapolate the results of numerical simulations well beyond what is currently possible in terms of CPU time. In earlier work, a proposal for a model based on Nekhoroshev theorem and Kolmogorov--Arnold--Moser theory was made. This model has been studied in detail and proved successful in describing the evolution of the dynamic aperture in numerical simulations, however a number of shortcomings had been identified and new models are proposed in this paper, which solve the observed issues. The new models have been benchmarked against numerical simulations for a simple system, the 4D H\'enon map, as well as a realistic, non-linear representation of the beam dynamics in the LHC at 6.5 TeV providing in both cases excellent results.

Innovative method to measure the extent of the stable phase-space region of proton synchrotrons

The advent of circular accelerators based on superconducting magnets has revolutionized the field of beam dynamics, with particle motion turning from linear to nonlinear due to unavoidable high-order field errors generated by the ring magnets. Nonlinear dynamics was already well mastered, e.g., in the close field of celestial mechanics as similar problems had been considered and successfully tackled. Hence, several results were available to aid comprehension of the behavior of charged particle beams under the influence of nonlinear forces. Here, we discuss how concepts derived from the theory of dynamical systems, linked with the fundamental Kolmogorov--Arnold--Moser theory and Nekhoroshev theorem, can be successfully applied to the analysis of nonlinear motion of charged particles in a circular accelerator. Based on these ideas, an innovative method to measure the extent of the phase-space region within which bounded motion occurs is presented, which has been successfully tested for the first time at the CERN LHC.

Exponential Stability in the Perturbed Central Force Problem

We consider the spatial central force problem with a real analytic potential. We prove that for all analytic potentials, but the Keplerian and the Harmonic ones, the Hamiltonian fulfills a nondegeneracy property needed for the applicability of Nekhoroshev's theorem. We deduce stability of the actions over exponentially long times when the system is subject to arbitrary analytic perturbation. The case where the central system is put in interaction with a slow system is also studied and stability over exponentially long time is proved.

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.

Nekhoroshev theorem for perturbations of the central motion

In this paper we prove a Nekhoroshev type theorem for perturbations of Hamiltonians describing a particle subject to the force due to a central potential. Precisely, we prove that under an explicit condition on the potential, the Hamiltonian of the central motion is quasi-convex. Thus, when it is perturbed, two actions (the modulus of the total angular momentum and the action of the reduced radial system) are approximately conserved for times which are exponentially long with the inverse of the perturbation parameter.

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem which does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is non-resonant is more subtle. Our second result shows that for a generic perturbation, the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long interval of time, and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation, one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).

The Nekhoroshev theorem and the observation of long-term diffusion in Hamiltonian systems

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.

Dynamical tunneling versus fast diffusion for a non-convex Hamiltonian.

This paper attempts to resolve the issue of the nature of the 0.01-0.1 cm(-1) peak splittings observed in high-resolution IR spectra of polyatomic molecules. One hypothesis is that these splittings are caused by dynamical tunneling, a quantum-mechanical phenomenon whereby energy flows between two disconnected regions of phase-space across dynamical barriers. However, a competing classical mechanism for energy flow is Arnol'd diffusion, which connects different regions of phase-space by a resonance network known as the Arnol'd web. The speed of diffusion is bounded by the Nekhoroshev theorem, which guarantees stability on exponentially long time scales if the Hamiltonian is steep. Here we consider a non-convex Hamiltonian that contains the characteristics of a molecular Hamiltonian, but does not satisfy the Nekhoroshev theorem. The diffusion along the Arnol'd web is expected to be fast for a non-convex Hamiltonian. While fast diffusion is an unlikely competitor for longtime energy flow in molecules, we show how dynamical tunneling dominates compared to fast diffusion in the nearly integrable regime for a non-convex Hamiltonian, as well as present a new kind of dynamical tunneling.

A Reversible Nekhoroshev Theorem for Persistence of Invariant Tori in Systems with Symmetry

We prove a variant of a theorem by Nekhoroshev on persistence of invariant tori in systems with symmetry. The new proof applies to reversible non Hamiltonian systems equivariant under the action of an Abelian group and is much simpler then the original one.

The nekhoroshev theorem and long-term stabilities in the solar system

The Nekhoroshev theorem has been often indicated in the last decades as the reference theorem for explaining the dynamics of several systems which are stable in the long-term. The Solar System dynamics provides a wide range of possible and useful applications. In fact, despite the complicated models which are used to numerically integrate realistic Solar System dynamics as accurately as possible, when the integrated solutions are chaotic the reliability of the numerical integrations is limited, and a theoretical long-term stability analysis is required. After the first formulation of Nekhoroshev?s theorem in 1977, many theoretical improvements have been achieved. On the one hand, alternative proofs of the theorem itself led to consistent improvements of the stability estimates; on the other hand, the extensions which were necessary to apply the theorem to the systems of interest for Solar System Dynamics, in particular concerning the removal of degeneracies and the implementation of computer assisted proofs, have been developed. In this review paper we discuss some of the motivations and the results which have made Nekhoroshev?s theorem a reference stability result for many applications in the Solar System dynamics.

Continuous averaging proof of the Nekhoroshev theorem

In this paper we develop the continuous averaging method of Treschev to work on the simultaneous Diophantine approximation and apply the result to give a new proof of the Nekhoroshev theorem. We obtain a sharp normal form theorem and an explicit estimate of the stability constants appearing in the Nekhoroshev theorem.

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev

The application of the Nekhoroshev theorem to many problems arising in different fields of Physics and Astronomy depends on a non-degeneracy property, called steepness, that a suitable Hamiltonian approximation must satisfy. Since steepness is implicitly defined, we have the problem of recognizing whether a given function is steep or not. For this purpose, we here consider some sufficient conditions for steepness provided by Nekhoroshev in 1979, based on the solvability of a collection of systems depending on the number n of degrees of freedom, the derivatives of the function up to a certain order r, and some auxiliary parameters. These conditions are really explicit only for r = 2, corresponding to quasi-convexity , and for r = 3. Instead, for r ⩾ 4, the conditions are implicit, since they require an elaborate computation of the closure of a certain set. In this paper, we first revisit Nekhoroshev's result and we show that the number of parameters in the collections of systems can be suitably reduced. Then, we show that for r = 4 Nekhoroshev's result is interesting only for n = 2, 3, and 4, and in these cases we find explicit conditions for steepness which are formulated in a purely algebraic form.

The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems

The celebrated KAM and Nekhoroshev theorems provide essential informations about the long term dynamics of quasi-integrable Hamiltonian systems. In particular, long-term instability of the action variables can be observed only in the so-called Arnold web, which is the complement in the phase-space of all KAM invariant tori, and only on the very long times which depend exponentially on an inverse power of the perturbation parameter. Though the structure of the Arnold's web was clearly explained already on Arnold's 1963 article, its numerical detection with a precision sufficient to reveal exponentially slow diffusion of the actions through the web itself has become possible only in the last decade with the extensive computation of dynamical indicators. In this paper, we first review the detection method that allowed us to compute the Arnold web, and then we discuss its use to study the long-term diffusion through the web itself. We also show that the Arnold web of a quasi-integrable Hamiltonian system is useful to track the diffusion of orbits of weakly dissipative perturbations of the same Hamiltonian system.