Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems

Massimiliano Berti; Philippe Bolle

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: 4, page 235-243
  • ISSN: 1120-6330

Abstract

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We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincaré-Melnikov function.

How to cite

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Berti, Massimiliano, and Bolle, Philippe. "Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.4 (2000): 235-243. <http://eudml.org/doc/252375>.

@article{Berti2000,
abstract = {We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincaré-Melnikov function.},
author = {Berti, Massimiliano, Bolle, Philippe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Arnold’s diffusion; Shadowing; Splitting of separatrices; Heteroclinic orbits; Variational methods; Arnold diffusion; shadowing; splitting of separatrices; heteroclinic orbits; isochronous oscillators},
language = {eng},
month = {12},
number = {4},
pages = {235-243},
publisher = {Accademia Nazionale dei Lincei},
title = {Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems},
url = {http://eudml.org/doc/252375},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Berti, Massimiliano
AU - Bolle, Philippe
TI - Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/12//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 4
SP - 235
EP - 243
AB - We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincaré-Melnikov function.
LA - eng
KW - Arnold’s diffusion; Shadowing; Splitting of separatrices; Heteroclinic orbits; Variational methods; Arnold diffusion; shadowing; splitting of separatrices; heteroclinic orbits; isochronous oscillators
UR - http://eudml.org/doc/252375
ER -

References

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  2. Angenent, S., A variational interpretation of Melnikov’s function and exponentially small separatrix splitting. In: D.A. Salamon (ed.), Symplectic geometry. London Math. Soc., Lecture Notes Series, vol. 192, Cambridge University Press1993. Zbl0810.34037MR1297127
  3. Arnold, V.I., Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl., 6, 1964, 581-585. Zbl0135.42602
  4. Berti, M. - Bolle, P., Homoclinics and Chaotic Behaviour for Perturbed Second order Systems. Annali di Mat. Pura e Applicata, (IV), vol. CLXXVI, 1999, 323-378. Zbl0957.37019MR1746547DOI10.1007/BF02506001
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  6. Berti, M. - Bolle, P., Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. Preprint SISSA 98/2000/M, october 2000. MR1837581
  7. Bessi, U. - Chierchia, L. - Valdinoci, E., Upper Bounds on Arnold Diffusion Time via Mather theory. Journal de Mathématiques Pures et Appliquées, neuvième série, to appear. Zbl0986.37052MR1810511DOI10.1016/S0021-7824(00)01188-0
  8. Chierchia, L. - Gallavotti, G., Drift and diffusion in phase space. Ann. Inst. Henri Poincaré, Phys. Théor., 60, 1994, 1-144; see also Erratum in vol. 68, 1998, 135. Zbl1010.37039MR1259103
  9. Cresson, J., Conjecture de Chirikov et Optimalité des exposants de stabilité du théorème de Nekhoroshev. Dép. de Mathématiques de Besançon, 1998, preprint. 
  10. Delshams, A. - Gelfreich, V.G. - Jorba, V. G. - Seara, T. M., Exponentially small splitting of separatrices under fast quasi-periodic forcing. Comm. Math. Ph., 189, 1997, 35-71. Zbl0897.34042MR1478530DOI10.1007/s002200050190
  11. Gallavotti, G., Arnold’s Diffusion in Isochronous Systems. Mathematical Physics, Analysis and Geometry, 1, 1999, 295-312. Zbl0936.37031MR1692234DOI10.1023/A:1009893118532
  12. Gallavotti, G. - Gentile, G. - Mastropietro, V., Separatrix splitting for systems with three time scales. Commun. Math. Phys., 202, 1999, 197-236. Zbl0936.37034MR1686531DOI10.1007/s002200050579
  13. Gallavotti, G. - Gentile, G. - Mastropietro, V., A possible counter example to a paper by Rudnew and Wiggins. Physica D, 137, 2000, 202-204. Zbl0997.37036MR1738773DOI10.1016/S0167-2789(99)00071-8
  14. Lochak, P., Arnold diffusion: a compendium of remarks and questions. Proceedings of 3DHAM’s Agaro, 1995. Zbl0986.37054
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  17. Pumarino, A. - Valls, C., Three time scales systems exhibiting persisent Arnold’s diffusion. Preprint; www.ma.utexas.edu/mp arc. 

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