Nekhoroshev theory and discrete averaging

Vassili Gelfreich; Arturo Vieiro

doi:10.3934/dcds.2026004

Article Contents

2026, Volume 50: 132-160. Doi: 10.3934/dcds.2026004

This volume Previous Article Mass-subcritical Half-Wave equation with mixed nonlinearities: Existence and non-existence of ground states and traveling waves Next Article Regularity of non-stationary stable foliations of toral Anosov maps

Nekhoroshev theory and discrete averaging

Vassili Gelfreich^1, , and
Arturo Vieiro^2,

1.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
2.
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, Barcelona 08007, Spain

^*Corresponding author: Vassili Gelfreich
^*Corresponding author: Vassili Gelfreich

Received: April 2025

Revised: November 2025

Early access: December 2, 2025

Published online: December 2, 2025

Abstract / Introduction Full Text(HTML) Related Papers Cited by

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 a refined version of the theorem on embedding of a near-the-identity symplectic map into an autonomous Hamiltonian flow with exponentially small error that provides explicit constants in the estimates.

Keywords:

Mathematics Subject Classification: Primary: 37J25, 37J40; Secondary: 37J11.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	V. I. Arnol'd, Instability of dynamical systems with several degrees of freedom, Soviet Math. Dokl., 5 (1964), 581-585.
[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 (1963), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.
[3]	D. Bambusi and B. Langella, A $C^\infty$ Nekhoroshev theorem, Math. Eng., 3 (2021), Paper No. 019, 17 pp. doi: 10.3934/mine.2021019.
[4]	G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mech., 37 (1985), 1-25. doi: 10.1007/BF01230338.
[5]	G. Benettin and G. Gallavotti, Stability of motions near resonances in quasi-integrable Hamiltonian systems, J. Statist. Phys., 44 (1986), 293-338. doi: 10.1007/BF01011301.
[6]	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 (1994), 1117-1143. doi: 10.1007/BF02188219.
[7]	L. Biasco and L. Chierchia, Explicit estimates on the measure of primary KAM tori, Ann. Mat. Pura Appl. (4), 197 (2018), 261-281. doi: 10.1007/s10231-017-0678-8.
[8]	A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, J. Differential Equations, 249 (2010), 2905-2920. doi: 10.1016/j.jde.2010.06.004.
[9]	A. Bounemoura and L. Niederman, Generic Nekhoroshev theory without small divisors, Ann. Inst. Fourier (Grenoble), 62 (2012), 277-324. doi: 10.5802/aif.2706.
[10]	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.
[11]	V. Gelfreich and A. Vieiro, Interpolating vector fields for near identity maps and averaging, Nonlinearity, 31 (2018), 4263-4289. doi: 10.1088/1361-6544/aacb8e.
[12]	M. Guzzo, A direct proof of the Nekhoroshev theorem for nearly integrable symplectic maps, Ann. Henri Poincaré, 5 (2004), 1013-1039. doi: 10.1007/s00023-004-0188-2.
[13]	M. Guzzo, L. Chierchia and G. Benettin, The steep Nekhoroshev's theorem, Comm. Math. Phys., 342 (2016), 569-601. doi: 10.1007/s00220-015-2555-x.
[14]	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., Birkhäuser, Basel, 1994, 96-116. doi: 10.1007/978-3-0348-7515-8_8.
[15]	S. B. Kuksin, On the inclusion of an almost integrable analytic symplectomorphism into a Hamiltonian flow, Russian J. Math. Phys., 1 (1993), 191-207.
[16]	P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys, 47 (1992), 57-133. doi: 10.1070/RM1992v047n06ABEH000965.
[17]	P. Lochak and A. I. Neishtadt, Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos, 2 (1992), 495-499. doi: 10.1063/1.165891.
[18]	P. Lochak, A. I. Neishtadt and L. Niederman, 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., Birkhäuser, Basel, 1994, 15-34. doi: 10.1007/978-3-0348-7515-8_2.
[19]	J. P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199-275. doi: 10.1007/s10240-003-0011-5.
[20]	J. K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc., 81 (1968), 60 pp.
[21]	A. I. Neishtadt, Estimates in the Kolmogorov theorem on conservation of conditionally periodic motions, J. Appl. Math. Mech., 45 (1981), 766-772. doi: 10.1016/0021-8928(81)90116-7.
[22]	A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9.
[23]	N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Math. Surveys, 32 (1977), 1-65. doi: 10.1070/RM1977v032n06ABEH003859.
[24]	N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Ⅱ, Trudy Sem. Petrovsk., (1979), 5-50.
[25]	J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., 213 (1993), 187-216. doi: 10.1007/BF03025718.
[26]	J. Pöschel, Integrability of hamiltonian systems on cantor sets, Communications on Pure and Applied Mathematics, 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.
[27]	J.-P. Ramis and R. Schäfke, Gevrey separation of fast and slow variables, Nonlinearity, 9 (1996), 353-384. doi: 10.1088/0951-7715/9/2/004.
[28]	C. Simó, Averaging under fast quasiperiodic forcing, Hamiltonian Mechanics: Integrability and Chaotic Behavior, Springer US, 1994, 13-34. doi: 10.1007/978-1-4899-0964-0_2.
[29]	J. Zhang and K. Zhang, Improved stability for analytic quasi-convex nearly integrable systems and optimal speed of Arnold diffusion, Nonlinearity, 30 (2017), 2918-2929. doi: 10.1088/1361-6544/aa72b7.