Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems

Jean-Pierre Marco; David Sauzin

Publications Mathématiques de l'IHÉS (2003)

  • Volume: 96, page 199-275
  • ISSN: 0073-8301

How to cite


Marco, Jean-Pierre, and Sauzin, David. "Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems." Publications Mathématiques de l'IHÉS 96 (2003): 199-275. <>.

author = {Marco, Jean-Pierre, Sauzin, David},
journal = {Publications Mathématiques de l'IHÉS},
language = {eng},
pages = {199-275},
publisher = {Institut des Hautes Etudes Scientifiques},
title = {Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems},
url = {},
volume = {96},
year = {2003},

AU - Marco, Jean-Pierre
AU - Sauzin, David
TI - Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems
JO - Publications Mathématiques de l'IHÉS
PY - 2003
PB - Institut des Hautes Etudes Scientifiques
VL - 96
SP - 199
EP - 275
LA - eng
UR -
ER -


  1. [Arn64] V. I. ARNOLD, Instability of dynamical systems with several degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9-12; Soviet Math. Dokl. 5 (1964), 581-585. Zbl0135.42602MR163026
  2. [AA67] V. I. ARNOLD and A. AVEZ, Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris (1967). Zbl0149.21704MR209436
  3. [Art91] M. ARTIN, Algebra, Prentice Hall, Englewood Cliffs, New Jersey (1991). Zbl0788.00001MR1129886
  4. [Bae95] C. BAESENS, Gevrey series and dynamical bifurcations for analytic slow-fast mappings, Nonlinearity 8 (1995), 179-201. Zbl0822.58040MR1328593
  5. [Bal94] W. BALSER, From divergent power series to analytic functions, Lecture Notes in Mathematics 1582, Springer-Verlag, Berlin Heidelberg (1994). Zbl0810.34046MR1317343
  6. [BG86] G. BENETTIN and G. GALLAVOTTI, Stability of motions near resonances in quasi-integrable Hamiltonian systems, J. Phys. Stat. 44 (1986), 293-338. Zbl0636.70018MR857061
  7. [BGG85] 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. Zbl0602.58022
  8. [Be96] U. BESSI, An approach to Arnold’s diffusion through the calculus of variations, Nonlinear Anal. TMA 26 (1996), 1115-1135. Zbl0867.70013
  9. [Be97] U. BESSI, Arnold’s example with three rotators, Nonlinearity 10 (1997), 763-781. Zbl0912.70015
  10. [BT99] S. BOLOTIN and D. TRESCHEV, Unbounded growth of energy in nonautonomous Hamiltonian systems, Non-linearity 12 (1999), 365-388. Zbl0989.37050MR1677779
  11. [CD93] M. CANALIS-DURAND, Solution formelle Gevrey d’une équation singulièrement perturbée, Asymptotic Analysis 7 (1993), 1-32. Zbl0814.34045
  12. [Ca59] J. W. S. CASSELS, An introduction to the geometry of numbers, Grundlehren Math. Wiss. 99, Springer-Verlag, Berlin Göttingen Heidelberg (1959). Zbl0086.26203MR157947
  13. [DLS00] A. DELSHAMS, R. DE LA LLAVE and T. M. SEARA, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T 2 , Comm. Math. Phys. 209 (2000), 353-392. Zbl0952.70015MR1737988
  14. [Do86] R. DOUADY, Stabilité ou instabilité des points fixes elliptiques, Ann. Sc. Éc. Norm. Sup. 21 (1988), 1-46. Zbl0656.58020MR944100
  15. [Dy80] E. M. DYN’KIN, Pseudo-analytic extension of smooth functions. The uniform scale, Twelve papers in analysis, Amer. Math. Soc. Transl. 115 (1980), 33-58. Zbl0478.30039
  16. [El97] L. H. ELIASSON, Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum, Acta Math. 179 (1997), 153-196. Zbl0908.34072MR1607554
  17. [Ga86] G. GALLAVOTTI, Quasi-integrable mechanical systems, Phénomènes critiques, systèmes aléatoires, théories de jauge, part II, Les Houches 1984, K. Osterwalder and R. Stora eds., North-Holland, Amsterdam New York (1986), 539-624. Zbl0662.70022MR880535
  18. [Ge18] M. GEVREY, Sur la nature analytique des solutions des équations aux dérivées partielles, Ann. Sc. Éc. Norm. Sup. 35 (1928), 129-190. Zbl46.0721.01JFM46.0721.01
  19. [Gr99] T. GRAMCHEV, Gevrey class, Encyclopaedia of Mathematics, Suppl. II, Kluwer, Doordrecht (1999). 
  20. [GP95] T. GRAMCHEV and G. POPOV, Nekhoroshev type estimates for billiard ball maps, Ann. Inst. Fourier 45 (1995), 859-895. MR1340956
  21. [Ko79] H. KOMATSU, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad. Ser. A 55 (1979), 69-72. Zbl0467.26004MR531445
  22. [Ku93] S. KUKSIN, On the inclusion of an almost integrable analytic symplectomorphism into a Hamiltonian flow, Russian J. Math. Phys. 1 (1993), 191-207. Zbl0931.37030MR1259481
  23. [KP94] S. KUKSIN and J. PÖSCHEL, On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications, Seminar on Dynamical Systems (St. Petersburg, 1991), Progr. Nonlinear Differential Equations Appl. 12, Birkhäuser, Basel (1994), 96-116. Zbl0797.58025MR1279392
  24. [Lo92] P. LOCHAK, Canonical perturbation theory via simultaneous approximation, Usp. Mat. Nauk 47 (1992), 59- 140; Russian Math. Surveys 47 (1992), 57-133. Zbl0795.58042MR1209145
  25. [Lo93] P. LOCHAK, Hamiltonian perturbation theory: periodic orbits, resonances and intermittency, Nonlinearity 6 (1993), 885-904. Zbl0794.70006MR1251248
  26. [LN92] P. LOCHAK and A. I. NEISHTADT, Estimates in the theorem of N. N. Nekhorocheff for systems with a quasi- convex Hamiltonian, Chaos 2 (1992), 495-499. Zbl1055.37573MR1195881
  27. [LNN93] P. LOCHAK, A. I. NEISHTADT and L. NIEDERMAN, Stability of nearly integrable convex Hamiltonian systems over exponentially long times, Proc. 1991 Euler Institute Conf. on Dynamical Systems, Birkhäuser, Boston (1993). Zbl0807.70020MR1279386
  28. [LMS03] P. LOCHAK, J.-P. MARCO and D. SAUZIN, On the splitting of the invariant manifolds in multidimensional near-integrable Hamiltonian systems, Memoirs of the Amer. Math. Soc. 163 (2003). Zbl1038.70001MR1964346
  29. [Mat93] J. MATHER, Variational construction of connecting orbits, Ann. Inst. Fourier 43 (1993), 1349-1386. Zbl0803.58019MR1275203
  30. [Mal95] B. MALGRANGE, Resommation des séries divergentes, Expo. Math. 13 (1995), 163-222. Zbl0836.40004MR1346201
  31. [Nei84] A. I. NEISHTADT, The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh. 48, 2 (1984), 197-204; J. Appl. Math. Mech. 48, 2 (1984), 133-139. Zbl0571.70022MR802878
  32. [Nekh77] N. N. NEKHOROSHEV, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Usp. Mat. Nauk 32 (1977), 5-66; Russian Math. Surveys 32 (1977), 1-65. Zbl0389.70028MR501140
  33. [Ni00] L. NIEDERMAN, Exponential stability for small perturbations of steep integrable Hamiltonian systems, Preprint University of Paris-Sud 2000-73 (2000), to appear in Ergod. Th. and Dyn. Syst. Zbl1071.37038MR2054052
  34. [Po00] G. POPOV, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms, Ann. Henri Poincaré 1 (2000), 223-248. Zbl0970.37050MR1770799
  35. [Po93] J. PÖSCHEL, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z. 213 (1993), 187-216. Zbl0857.70009MR1221713
  36. [PT97] A. V. PRONIN and D. V. TRESCHEV, On the inclusion of analytic maps into analytic flows, Reg. Chaot. Dyn. 2, 2 (1997), 14-24. Zbl0921.58024MR1652125
  37. [Ra80] J.-P. RAMIS, Les séries k-sommables et leurs applications, Proc. 1979 ‘Les Houches’ (Analysis, Microlocal Calculus and Relativistic Quantum Theory), Lecture Notes in Physics 126, Springer-Verlag, Berlin Heidelberg (1980), 178-199. 
  38. [Ra84] J.-P. RAMIS, Théorèmes d’indices Gevrey pour les équations différentielles ordinaires, Mem. Am. Math. Soc. 296 (1984). Zbl0555.47020
  39. [Ra93] J.-P. RAMIS, Séries divergentes et théories asymptotiques, Panoramas et synthèses 1, Soc. Math. de France (1993). Zbl0830.34045MR1272100
  40. [Ro62] C. ROUMIEU, Ultradistributions définies sur R n et sur certaines classes de variétés différentiables, J. Analyse Math. 10 (1962-1963), 153-192. Zbl0122.34802MR158261
  41. [RS96] J.-P. RAMIS and R. SCHÄFKE, Gevrey separation of slow and fast variables, Nonlinearity 9 (1996), 353-384. Zbl0925.70161MR1384480
  42. [Sa92] D. SAUZIN, Caractère Gevrey des solutions formelles d’un problème de moyennisation, C. R. Acad. Sci. Paris 315 (1992), Série I, 991-995. Zbl0794.35094
  43. [Th96] V. THILLIEZ, Quelques propriétés de quasi-analyticité, Gazette des Mathématiciens 70 (1996), 49-68. Zbl0918.26015MR1423692
  44. [Th97] V. THILLIEZ, Sur les fonctions composées ultradifférentiables, J. Math. Pures Appl. 76 (1997), 499-524. Zbl0878.58008MR1465608
  45. [To90] J.-C. TOUGERON, An introduction to the theory of Gevrey expansions and to the Borel-Laplace transform with some applications, Cours à Rennes et Toronto (1989-90). 
  46. [Wa00] G. WALLET, La variété des équations surstables, Bull. Soc. math. France 128 (2000), 497-528. Zbl0969.34077MR1815396

Citations in EuDML Documents

  1. Claire Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles
  2. Giuseppe De Donno, Alessandro Oliaro, Hypoellipticity and local solvability of anisotropic PDEs with Gevrey nonlinearity
  3. Abed Bounemoura, Laurent Niederman, Generic Nekhoroshev theory without small divisors
  4. Pierre Lochak, Jean-Pierre Marco, Diffusion times and stability exponents for nearly integrable analytic systems
  5. Meysam Nassiri, Enrique R. Pujals, Robust transitivity in hamiltonian dynamics
  6. Laurent Stolovitch, Smooth Gevrey normal forms of vector fields near a fixed point
  7. Laurent Niederman, Hamiltonian stability and subanalytic geometry

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.