Research article Special Issues

A $C^\infty$ Nekhoroshev theorem

  • 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

Metrics

Article views(2875) PDF downloads(679) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog