Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


A note on micro-instability for Hamiltonian systems close to integrable

Authors: Abed Bounemoura and Vadim Kaloshin
Journal: Proc. Amer. Math. Soc. 144 (2016), 1553-1560
MSC (2010): Primary 37J25, 37J40
Published electronically: December 21, 2015
MathSciNet review: 3451232
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of ``micro-diffusion'': under generic assumptions on $ h$ and $ f$, there exists an orbit of the system for which the drift of its action variables is at least of order $ \sqrt {\varepsilon }$, after a time of order $ \sqrt {\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $ h$ and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.

References [Enhancements On Off] (What's this?)

  • [Arn63] V. I. Arnol′d, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 13–40 (Russian). MR 0163025
  • [Arn64] V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady 5 (1964), 581-585.
  • [Arn94] V. I. Arnol′d, Mathematical problems in classical physics, Trends and perspectives in applied mathematics, Appl. Math. Sci., vol. 100, Springer, New York, 1994, pp. 1–20. MR 1277190,
  • [BF13] Abed Bounemoura and Stéphane Fischler, A Diophantine duality applied to the KAM and Nekhoroshev theorems, Math. Z. 275 (2013), no. 3-4, 1135–1167. MR 3127051,
  • [BK14] Abed Bounemoura and Vadim Kaloshin, Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom, Mosc. Math. J. 14 (2014), no. 2, 181–203, 426 (English, with English and Russian summaries). MR 3236491
  • [BKZ11] P. Bernard, V. Kaloshin, and K. Zhang, Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders, Acta Mathematica (2011), conditionally accepted, arXiv:1112.2773.
  • [Bou10] Abed Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, J. Differential Equations 249 (2010), no. 11, 2905–2920. MR 2718671,
  • [Bou13] Abed Bounemoura, Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians, Regul. Chaotic Dyn. 18 (2013), no. 3, 261–276. MR 3061809,
  • [Che13] C.-Q. Cheng, Arnold diffusion in nearly integrable Hamiltonian systems, Preprint (2013), arXiv:1207.4016v2.
  • [Kol54] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527–530 (Russian). MR 0068687
  • [KZ12] V. Kaloshin and K. Zhang, A strong form of Arnold diffusion for two and a half degrees of freedom, preprint (2012), arXiv:1212.1150.
  • [KZ14a] V. Kaloshin and K. Zhang, Partial averaging and dynamics of the dominant Hamiltonian, with applications to Arnold diffusion, preprint (2014), arXiv:1410.1844.
  • [KZ14b] V. Kaloshin and K. Zhang, A strong form of Arnold diffusion for three and a half degrees of freedom, preprint (2014), vkaloshi/papers/announce-three-and-half.pdf.
  • [Loc92] P. Loshak, Canonical perturbation theory: an approach based on joint approximations, Uspekhi Mat. Nauk 47 (1992), no. 6(288), 59–140 (Russian); English transl., Russian Math. Surveys 47 (1992), no. 6, 57–133. MR 1209145,
  • [Mos60] Jürgen Moser, On the elimination of the irrationality condition and Birkhoff’s concept of complete stability, Bol. Soc. Mat. Mexicana (2) 5 (1960), 167–175. MR 0133544
  • [Mos62] Jürgen Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen II (1962), 1-20.
  • [Nek77] N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk 32 (1977), no. 6(198), 5–66, 287 (Russian). MR 0501140
  • [Nek79] N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk. 5 (1979), 5–50 (Russian). MR 549621

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37J25, 37J40

Retrieve articles in all journals with MSC (2010): 37J25, 37J40

Additional Information

Abed Bounemoura
Affiliation: CNRS-IMPA UMI, Rio de Janeiro AC 22460-320, Brazil

Vadim Kaloshin
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Received by editor(s): December 19, 2014
Received by editor(s) in revised form: March 30, 2015
Published electronically: December 21, 2015
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society