Papers
Topics
Authors
Recent
Search
2000 character limit reached

Nekhoroshev theory and discrete averaging

Published 4 Nov 2024 in math.DS | (2411.02190v1)

Abstract: This paper contains a proof of the Nekhoroshev theorem for quasi-integrable symplectic maps. In contrast to the classical methods, our proof is based on the discrete averaging method and does not rely on transformations to normal forms. At the centre of our arguments lies the theorem on embedding of a near-the-identity symplectic map into an autonomous Hamiltonian flow with exponentially small error.

Authors (2)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (25)
  1. V.I. Arnol’d. Instability of dynamical systems with several degrees of freedom. Soviet Math. Dokl., 5:581–585, 1964.
  2. V.I. Arnol’d. Proof of a theorem of a. n. kolmogorov on the invariance of quasi-periodic motions under small perturbations of the hamiltonian. Russian Math. Surveys, 18(5):9–36, 1963.
  3. D. Bambusi and B. Langella. A C∞superscript𝐶C^{\infty}italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT Nekhoroshev theorem. Math. Eng., 3(2):Paper No. 019, 17, 2021.
  4. A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems. Celestial Mech., 37:1–25, 1985.
  5. G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms. J. Statist. Phys., 74(5-6):1117–1143, 1994.
  6. L. Biasco and L. Chierchia. Explicit estimates on the measure of primary KAM tori. Ann. Mat. Pura Appl. (4), 197(1):261–281, 2018.
  7. A. Bounemoura. Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians. J. Differential Equations, 249(11):2905–2920, 2010.
  8. A. Bounemoura and L. Niederman. Generic Nekhoroshev theory without small divisors. Ann. Inst. Fourier (Grenoble), 62(1):277–324, 2012.
  9. J.W.S. Cassels. An introduction to Diophantine approximation, volume No. 45 of Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, New York, 1957.
  10. V. Gelfreich and A. Vieiro. Interpolating vector fields for near identity maps and averaging. Nonlinearity, 31(9):4263–4289, 2018.
  11. M. Guzzo. A direct proof of the Nekhoroshev theorem for nearly integrable symplectic maps. Ann. Henri Poincaré, 5(6):1013–1039, 2004.
  12. The steep Nekhoroshev’s theorem. Comm. Math. Phys., 342(2):569–601, 2016.
  13. S. Kuksin and J. Pöschel. On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications. In Seminar on Dynamical Systems (St. Petersburg, 1991), volume 12 of Progr. Nonlinear Differential Equations Appl., pages 96–116. Birkhäuser, Basel, 1994.
  14. P. Lochak. Canonical perturbation theory via simultaneous approximation. Russian Math. Surveys, 47(6):57–133, 1992.
  15. P. Lochak and A.I. Neishtadt. Estimates of stability time for nearly integrable systems with a quasiconvex hamiltonian. Chaos, 2:495, 1992.
  16. Stability of nearly integrable convex Hamiltonian systems over exponentially long times. In Seminar on Dynamical Systems (St. Petersburg, 1991), volume 12 of Progr. Nonlinear Differential Equations Appl., pages 15–34. Birkhäuser, Basel, 1994.
  17. J.P. Marco and D. Sauzin. Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems. Publ. Math. Inst. Hautes Études Sci., (96):199–275, 2002.
  18. J.K. Moser. Lectures on Hamiltonian systems. Mem. Amer. Math. Soc., 81:60, 1968.
  19. A.I. Neishtadt. Estimates in the Kolmogorov theorem on conservation of conditionally periodic motions. J. Appl. Math. Mech., 45(6):766–772, 1981.
  20. A.I. Neishtadt. The separation of motions in systems with rapidly rotating phase. J. Appl. Math. Mech., 48:133 – 139, 1984.
  21. N.N. Nekhoroshev. An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russian Math. Surveys, 32:1–65, 1977.
  22. N.N. Nekhoroshev. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II. Trudy Sem. Petrovsk., No. 5:5–50, 1979.
  23. J. Pöschel. Nekhoroshev estimates for quasi-convex Hamiltonian systems. Math. Z., 213(2):187–216, 1993.
  24. J. Pöschel. Integrability of hamiltonian systems on cantor sets. Communications on Pure and Applied Mathematics, 35(5):653–696, 1982.
  25. C. Simó. Averaging under fast quasiperiodic forcing. Hamiltonian Mechanics: Integrability and Chaotic Behavior, Springer US:13–34, 1994.
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free