Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 2: 1-17. doi: 10.3934/mine.2021019
Previous Article Next Article
Research article Special Issues

A $C^\infty$ Nekhoroshev theorem

  • Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, I-20133 Milano, Italy
  • Received: 10 February 2020 Accepted: 24 May 2020 Published: 14 July 2020

  • We prove a $C^\infty$ version of the Nekhoroshev's estimate on the stability times of the actions in close to integrable Hamiltonian systems. The proof we give is a variant of the original Nekhoroshev's proof and it consists in first conjugating, globally in the phase space, and up to a small remainder, the system to a normal form. Then we perform the geometric part of the proof in the normalized variables. As a result, we obtain a proof which is simpler than the usual ones.

    Citation: Dario Bambusi, Beatrice Langella. A $C^\infty$ Nekhoroshev theorem[J]. Mathematics in Engineering, 2021, 3(2): 1-17. doi: 10.3934/mine.2021019

    Related Papers:

  • We prove a $C^\infty$ version of the Nekhoroshev's estimate on the stability times of the actions in close to integrable Hamiltonian systems. The proof we give is a variant of the original Nekhoroshev's proof and it consists in first conjugating, globally in the phase space, and up to a small remainder, the system to a normal form. Then we perform the geometric part of the proof in the normalized variables. As a result, we obtain a proof which is simpler than the usual ones.


    加载中


    [1] Bambusi D (1999) Nekhoroshev theorem for small amplitude solutions in nonlinear Schr?dinger equations. Math Z 230: 345-387.
    [2] Bounemoura A, Féjoz J (2017) Hamiltonian perturbation theory for ultra-differentiable functions. Memoir AMS, arXiv: 1710.01156.
    [3] Benettin G, Gallavotti G (1986) Stability of motions near resonances in quasi-integrable Hamiltonian systems. J Statist Phys 44: 293-338.
    [4] Bambusi D, Giorgilli A (1993) Exponential stability of states close to resonance in infinitedimensional Hamiltonian systems. J Statist Phys 71: 569-606.
    [5] Benettin G, Galgani L, Giorgilli A (1985) A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems. Celestial Mech 37: 1-25.
    [6] Bambusi D, Langella B, Montalto R (2019) On the spectrum of the Schr?dinger operator on Td: A normal form approach. Commun Part Diff Eq 45: 303-320.
    [7] Bambusi D, Maiocchi A, Turri L (2019) A large probability averaging theorem for the defocusing NLS. Nonlinearity 32: 3661-3694.
    [8] Bounemoura A, Niederman L (2012) Generic Nekhoroshev theory without small divisors. Ann Inst Fourier 62: 277-324.
    [9] Bounemoura A (2010) Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians. J Differ Equations 249: 2905-2920.
    [10] Bounemoura A (2011) Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians. Commun Math Phys 307: 157-183.
    [11] De Roeck W, Huveneers F (2015) Asymptotic localization of energy in nondisordered oscillator chains. Commun Pure Appl Math 68: 1532-1568.
    [12] Guzzo M, Chierchia L, Benettin G (2016) The steep Nekhoroshev's theorem. Commun Math Phys 342: 569-601.
    [13] Giorgilli A (2003) Notes on Exponential Stability of Hamiltonian Systems, Pisa: Centro di Ricerca Matematica "Ennio De Giorgi".
    [14] Giorgilli A, Zehnder E (1992) Exponential stability for time dependent potentials. Z Angew Math Phys 43: 827-855.
    [15] Lochak P, Neĭshtadt AI (1992) Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian. Chaos 2: 495-499.
    [16] Lochak P (1992) Canonical perturbation theory: An approach based on joint approximations. Uspekhi Mat Nauk 47: 59-140.
    [17] Marco JP, Sauzin D (2003) Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems. Publ Math Inst Hautes études Sci 96: 199-275.
    [18] Marco JP, Sauzin D (2004) Wandering domains and random walks in Gevrey near-integrable systems. Ergodic Theory Dynam Systems 24: 1619-1666.
    [19] Nekhoroshev NN (1977) An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Uspehi Mat Nauk 32: 5-66.
    [20] Nekhoroshev NN (1979) An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II. Trudy Sem Petrovsk 5: 5-50.
    [21] Niederman L (2004) Exponential stability for small perturbations of steep integrable Hamiltonian systems. Ergodic Theory Dynam Systems 24: 593-608.
    [22] Niederman L (2007) Prevalence of exponential stability among nearly integrable Hamiltonian systems. Ergodic Theory Dynam Systems 27: 905-928.
    [23] Pöschel J (1993) Nekhoroshev estimates for quasi-convex Hamiltonian systems. Math Z 213: 187-216.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(3616) PDF downloads(708) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog