Relevant bibliographies by topics / Nekhoroshev Theorem

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Academic literature on the topic 'Nekhoroshev Theorem'

Author: Grafiati
Published: 9 March 2023

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Journal articles on the topic "Nekhoroshev Theorem"

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Wiggins, S., and A. M. Mancho. "Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori." Nonlinear Processes in Geophysics 21, no. 1 (February 4, 2014): 165–85. http://dx.doi.org/10.5194/npg-21-165-2014.

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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.
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Bambusi, Dario, and Beatrice Langella. "A $C^\infty$ Nekhoroshev theorem." Mathematics in Engineering 3, no. 2 (2021): 1–17. http://dx.doi.org/10.3934/mine.2021019.

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Gaeta, Giuseppe. "The Poincaré–Lyapounov–Nekhoroshev Theorem." Annals of Physics 297, no. 1 (April 2002): 157–73. http://dx.doi.org/10.1006/aphy.2002.6238.

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Guzzo, M. "The nekhoroshev theorem and long-term stabilities in the solar system." Serbian Astronomical Journal, no. 190 (2015): 1–10. http://dx.doi.org/10.2298/saj1590001g.

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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.
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FIORANI, EMANUELE. "GEOMETRICAL ASPECTS OF INTEGRABLE SYSTEMS." International Journal of Geometric Methods in Modern Physics 05, no. 03 (May 2008): 457–71. http://dx.doi.org/10.1142/s0219887808002886.

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We review some basic theorems on integrability of Hamiltonian systems, namely the Liouville–Arnold theorem on complete integrability, the Nekhoroshev theorem on partial integrability and the Mishchenko–Fomenko theorem on noncommutative integrability, and for each of them we give a version suitable for the noncompact case. We give a possible global version of the previous local results, under certain topological hypotheses on the base space. It turns out that locally affine structures arise naturally in this setting.
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Froeschlé, Claude, and Elena Lega. "On the Diffusion Along Resonant Lines in Continuous and Discrete Dynamical Systems." International Journal of Modern Physics B 17, no. 22n24 (September 30, 2003): 3964–76. http://dx.doi.org/10.1142/s0217979203023033.

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We detect and measure diffusion along resonances in a discrete symplectic map for different values of the coupling parameter. Qualitatively and quantitatively the results are very similar to those obtained for a quasi-integrable Hamiltonian system, i.e. in agreement with Nekhoroshev predictions, although the discrete mapping does not fulfill completely, a priori, the conditions of the Nekhoroshev theorem.
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Todorovic, N. "The role of a steepness parameter in the exponential stability of a model problem: Numerical aspects." Serbian Astronomical Journal, no. 182 (2011): 25–33. http://dx.doi.org/10.2298/saj1182025t.

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The Nekhoroshev theorem considers quasi integrable Hamiltonians providing stability of actions in exponentially long times. One of the hypothesis required by the theorem is a mathematical condition called steepness. Nekhoroshev conjectured that different steepness properties should imply numerically observable differences in the stability times. After a recent study on this problem (Guzzo et al. 2011, Todorovic et al. 2011) we show some additional numerical results on the change of resonances and the diffusion laws produced by the increasing effect of steepness. The experiments are performed on a 4-dimensional steep symplectic map designed in a way that a parameter smoothly regulates the steepness properties in the model.
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Henrici, Andreas. "Nekhoroshev Stability for the Dirichlet Toda Lattice." Symmetry 10, no. 10 (October 16, 2018): 506. http://dx.doi.org/10.3390/sym10100506.

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In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically observed in these chains, it has been a long-standing research aim to apply the theory of perturbed integrable systems to these chains, in particular to the Toda lattice which has been shown to be a completely integrable system. The Dirichlet Toda lattice can be treated mathematically by using symmetries of the periodic Toda lattice. Precisely, by treating the phase space of the former system as an invariant subset of the latter one, namely as the fixed point set of an important symmetry of the periodic lattice, the results already obtained for the periodic lattice can be used to obtain analogous results for the Dirichlet lattice. In this way, we transfer our stability results for the periodic lattice to the Dirichlet lattice. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of related theorems, and the lattice with fixed ends is more important for applications than the periodic one.
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Xue, Jinxin. "Continuous averaging proof of the Nekhoroshev theorem." Discrete & Continuous Dynamical Systems - A 35, no. 8 (2015): 3827–55. http://dx.doi.org/10.3934/dcds.2015.35.3827.

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Moszynski, M. "A quantum version of the Nekhoroshev theorem." Journal of Physics A: Mathematical and General 25, no. 8 (April 21, 1992): L443—L448. http://dx.doi.org/10.1088/0305-4470/25/8/011.

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Dissertations / Theses on the topic "Nekhoroshev Theorem"

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Schirinzi, Gabriella. "Investigation of new conditions for steepness from a former result by Nekhoroshev." Doctoral thesis, Università degli studi di Padova, 2014. http://hdl.handle.net/11577/3423527.

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This Thesis presents the construction of new sufficient conditions for the verification of a property of functions called steepness. It is a peculiar property required for the application of the Nekhoroshev Theorem to a quasi-integrable Hamiltonian system, and its formulation is given by Nekhoroshev in an implicit way. Therefore sufficient conditions are necessary for the verification of the steepness. Nekhoroshev formulated his celebrated Theorem in the seventies, providing under suitable hypothesis a strong stability result for those dynamical systems which are not integrable, but can be considered as a small perturbation of an integrable system. The Nekhoroshev Theorem is a fundamental result in the framework of the Perturbation Theory, especially for its important applications in Celestial Mechanics. For the construction of new sufficient conditions for steepness, a result proved by Nekhoroshev is used. The new conditions are weaker than the ones known up to now, hence they allow to detect a larger class of steep functions. In particular, the new conditions concern functions of two, three and four variables respectively. In the last Chapter of this Thesis a general algorithm for the verification of the steepness of functions of three or four variables is constructed. Moreover, in order to provide some concrete examples of applicability of the new conditions, such algorithm is applied to two physical systems: the Hamiltonian of the circular restricted three-body problem, and the Hamiltonian of a chain of four harmonic oscillators, with the potential energy of the Fermi-Pasta-Ulam problem. In both cases the new sufficient conditions allow to prove numerical evidence of the steepness.
In questa tesi viene presentata la costruzione di nuove condizioni sufficienti per la verifica di una proprietà delle funzioni denominata steepness. Tale proprietà è un’ipotesi fondamentale per l’applicazione del teorema di Nekhoroshev ad un sistema Hamiltoniano quasi integrabile, e la sua formulazione viene fornita da Nekhoroshev in maniera implicita. Per questo motivo è necessario avere a disposizione delle condizioni sufficienti per la verifica della stepness. Nekhoroshev formulò negli anni settanta il suo celebre teorema, il quale garantisce sotto opportune ipotesi una forte stabilità per quei sistemi dinamici che non sono integrabili, ma possono scriversi come una piccola perturbazione di un sistema integrabile. Il teorema di Nekhoroshev costituisce un risultato fondamentale nell’ambito della Teoria delle Perturbazioni, in particolar modo per le sue importanti applicazioni nella meccanica celeste. Per la costruzione delle nuove condizioni sufficienti per la steepness viene utilizzato un risultato dimostrato da Nekhoroshev. Le nuove condizioni sono più deboli di quelle conosciute fino ad ora, e di conseguenza permettono di individuare una classe più ampia di funzioni steep. In particolare, le nuove condizioni riguardano funzioni di due, tre e quattro variabili rispettivamente. Nell’ultimo capitolo di questa tesi viene costruito un algoritmo generale per la verifica della steepness di funzioni di tre o quattro variabili. Inoltre, allo scopo di fornire qualche esempio concreto di applicazione delle nuove condizioni, tale algoritmo viene applicato a due sistemi fisici: l’Hamiltoniana del problema dei tre corpi ristretto circolare, e l’Hamiltoniana di una catena di quattro oscillatori armonici, con l’energia potenziale del problema di Fermi-Pasta-Ulam. In entrambi i casi le nuove condizioni sufficienti permettono di dimostrare numericamente la steepness.
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LANGELLA, BEATRICE. "NORMAL FORM AND KAM METHODS FOR HIGHER DIMENSIONAL LINEAR PDES." Doctoral thesis, Università degli Studi di Milano, 2020. http://hdl.handle.net/2434/798372.

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In this thesis an approach to linear PDEs on higher dimensional spatial domains is proposed. I prove two kinds of results: first I develop an algorithm which enables to obtain reducibility for linear PDEs which depend quasi-periodically on time, and I apply it to a quasilinear transport equation of the form ∂_t u= ν•∇u+ ε P (ωt)u on the d-dimensional torus T^d, where ε is a small parameter, ν and ω are Diophantine vectors, P (ωt)=V(x,ωt)•∇+W(ωt), V is a smooth function on T^(d+n) and W(ωt) is an unbounded pseudo-differential operator of order strictly less than 1. The strategy is an extension of the methods originally developed in the context of quasilinear one dimensional equations. It consists in first using quantum normal form techniques in order to conjugate the original system to a new one with a smoothing perturbation, and then exploiting the smoothing nature of the new perturbation in order to balance the effects of the small denominators, which in this problem accumulate very fast to 0. The quantum normal form procedure developed in order to obtain reducibility for the above transport equation is global in phase space. In order to overcome such a limitation, the second problem I tackle in this thesis is that of developing a local quantum normal form procedure, which could be applied to much more general systems. As the simplest relevant model containing all the difficulties of the general case, I consider the operator H=-∆+V(x) with Floquet boundary conditions on the flat torus T^d_Γ, where T^d_Γ is the manifold obtained as quotient between the d-dimensional space R^d and an arbitrary d-dimensional lattice Γ, with the purpose of adapting the quantum normal form procedure to deal with this operator. As a result, I prove for the operator H a Structure Theorem à la Nekhoroshev, and I characterize the asymptotic behavior of all its eigenvalues. The asymptotic expansion is in |λ|^{-δ}, with δ ∊ (0, 1) for most of the eigenvalues λ (stable eigenvalues), while it is a "directional expansion" for the remaining eigenvalues (unstable eigenvalues).
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Fontanari, Daniele. "Quantum manifestations of the adiabatic chaos of perturbed susperintegrable Hamiltonian systems." Thesis, Littoral, 2013. http://www.theses.fr/2013DUNK0356/document.

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Dans cette thèse nous étudions un système quantique, obtenu comme un analogue d'un système classique superintégrable perturbé au moyen de la quantification géométrique. Notre objectif est de mettre en évidence la présence des phénomènes analogues à ceux qui caractérisent la superintégrabilité classique, notamment la coexistence des mouvements réguliers et chaotiques liés aux effets des résonances ainsi que la régularité du régime non-résonant. L'analyse est effectuée par l'étude des distributions du Husimi des états quantiques sélectionnés, avec une attention particulière aux états stationnaires et à l'évolution des états cohérents. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Bien que cette thèse devrait être considérée comme une étude préliminaire, dont l'objectif est de créer le socle des études futures, nos résultats donnent des indications intéressantes sur la dynamique quantique. Par exemple, il est démontré comment les résonancees classiques exercent une influence considérable sur le spectre du système quantique et comment il est possible, dans le comportement quantique, de trouver une trace de l'invariant adiabatique dans le régime de résonance
The abundance, among physical models, of perturbations of superintegrable Hamiltonian systems makes the understanding of their long-term dynamics an important research topic. While from the classical standpoint the situation, at least in many important cases, is well understood through the use of Nekhoroshev stability theorem and of the adiabatic invariants theory, in the quantum framework there is, on the contrary, a lack of precise results. The purpose of this thesis is to study a perturbed superintegrable quantum system, obtained from a classical counterpart by means of geometric quantization, in order to highlight the presence of indicators of superintegrability analogues to the ones that characterize the classical system, such as the coexistence of regular motions with chaotic one, due to the effects of resonances, opposed to the regularity in the non resonant regime. The analysis is carried out by studying the Husimi distributions of chosen quantum states, with particular emphasis on stationary states and evolved coherent states. The computation are performed using both numerical methods and perturbative schemes. Although this should be considered a preliminary work, the purpose of which is to lay the fundations for future investigations, the results obtained here give interesting insights into quantum dynamics. For instance, it is shown how classical resonances exert a considerable influence on the spectrum of the quantum system and how it is possible, in the quantum behaviour, to find a trace of the classical adiabatic invariance in the resonance regime
L'abbondanza, fra i modelli fisici, di perturbazioni di sistemi Hamiltoniani superintegrabili rende la comprensione della loro dinamica per tempi lunghi un importante argomento diricerca. Mentre dal punto di vista classico la situazione, perlomeno in molti case importanti, è ben compresa grazie all'uso del teorema di stabilità di Nekhoroshev e della teoria degli invariantiadiabatici, nel caso quantistico vi è, al contrario, una mancanza di risultati precisi. L'obiettivo di questa tesi è di studiare un sistema superintegrabile quantistico, ottenuto partendo da un corrispettivo classico tramite quantizzazione geometrica, al fine di evidenziare la presenza di indicatori di supertintegrabilità analoghi a quelliche caratterizzano il sistema classico, come la coesistenza di moti regolari e caotici, dovuta all'effetto delle risonanze, in contrapposizione con la regolarità nel regime non risonante. L'analisi è condotta studiando le distribuzioni di Husimi di stati quantistici scelti, con particolare enfasi posta sugli stati stazionari e sugli stati coerenti evoluti. I calcoli sono effettuati sia utilizzando tecniche numeriche che schemi perturbativi. Pur essendo da considerardi questo un lavoro preliminare, il cui compito è di porre le fondamenta per analisi future, i risultati qui ottenuti offrono interessanti spunti sulla dinamica quantistica. Per esempio è mostrato come le risonanze classiche abbiano un chiaro effeto sullo spettro del sistema quantistico, ed inoltre comesia possibile trovare una traccia, nel comportamento quantistico, dell'invarianza adiabatica classica nel regime risonante
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Books on the topic "Nekhoroshev Theorem"

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Giancarlo, Benettin, Henrard J, Kuksin Sergej B. 1955-, Giorgilli Antonio, Centro internazionale matematico estivo, and European Mathematical Society, eds. Hamiltonian dynamics theory and applications: Lectures given at the C.I.M.E.-E.M.S. Summer School, held in Cetraro, Italy, July 1-10, 1999. Berlin: Springer, 2005.

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Henrard, Jacques, Giancarlo Benettin, and Sergei Kuksin. Hamiltonian Dynamics - Theory and Applications: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 1-10, 1999 (Lecture Notes in Mathematics / Fondazione C.I.M.E., Firenze). Springer, 2005.

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Benettin, Giancarlo, Jacques Henrard, Sergej B. Kuksin, and Antonio Giorgilli. Hamiltonian Dynamics - Theory and Applications: Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, July 1-10 1999. Springer London, Limited, 2010.

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Book chapters on the topic "Nekhoroshev Theorem"

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Benettin, Giancarlo. "Physical Applications of Nekhoroshev Theorem and Exponential Estimates." In Lecture Notes in Mathematics, 1–76. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/978-3-540-31541-4_1.

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Morbidelli, Alessandro, and Massimiliano Guzzo. "The Nekhoroshev Theorem and the Asteroid Belt Dynamical System." In The Dynamical Behaviour of our Planetary System, 107–36. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-011-5510-6_8.

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Guzzo, Massimiliano. "Long-Term Stability Analysis of Quasi Integrable Degenerate Systems through the Spectral Formulation of the Nekhoroshev Theorem." In Modern Celestial Mechanics: From Theory to Applications, 303–23. Dordrecht: Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-017-2304-6_19.

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Niederman, Laurent. "Nekhoroshev Theory." In Perturbation Theory, 291–305. New York, NY: Springer US, 2009. http://dx.doi.org/10.1007/978-1-0716-2621-4_352.

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Guzzo, Massimiliano, Zoran Knežević, and Andrea Milani. "Probing the Nekhoroshev Stability of Asteroids." In Modern Celestial Mechanics: From Theory to Applications, 121–40. Dordrecht: Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-017-2304-6_8.

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Delshams, Amadeu, and Pere Gutiérrez. "Nekhoroshev and KAM Theorems Revisited via a Unified Approach." In Hamiltonian Mechanics, 299–306. Boston, MA: Springer US, 1994. http://dx.doi.org/10.1007/978-1-4899-0964-0_29.

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Fortunati, Alessandro, and Stephen Wiggins. "The Kolmogorov-Arnold-Moser (KAM) and Nekhoroshev Theorems with Arbitrary Time Dependence." In Essays in Mathematics and its Applications, 89–99. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-31338-2_5.

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Morbidelli, A. "Bounds on Diffusion in Phase Space: Connection Between Nekhoroshev and Kam Theorems and Superexponential Stability of Invariant Tori." In Hamiltonian Systems with Three or More Degrees of Freedom, 514–17. Dordrecht: Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-011-4673-9_68.

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