A functional analysis approach to Arnold diffusion
Annales de l'I.H.P. Analyse non linéaire, Volume 19 (2002) no. 4, p. 395-450
@article{AIHPC_2002__19_4_395_0,
     author = {Berti, Massimiliano and Bolle, Philippe},
     title = {A functional analysis approach to Arnold diffusion},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {19},
     number = {4},
     year = {2002},
     pages = {395-450},
     zbl = {1087.37048},
     mrnumber = {1912262},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2002__19_4_395_0}
}
Berti, Massimiliano; Bolle, Philippe. A functional analysis approach to Arnold diffusion. Annales de l'I.H.P. Analyse non linéaire, Volume 19 (2002) no. 4, pp. 395-450. http://www.numdam.org/item/AIHPC_2002__19_4_395_0/

[1] Ambrosetti A., Badiale M., Homoclinics: Poincaré-Melnikov type results via a variational approach, Annales I.H.P., Analyse Nonlin. 15 (2) (1998) 233-252. | Numdam | MR 1614571 | Zbl 1004.37043

[2] Angenent S., A variational interpretation of Melnikov's function and exponentially small separatrix splitting, in: Salamon D. (Ed.), Symplectic Geometry, Lecture Notes of the London Math. Soc., 1993. | Zbl 0810.34037

[3] Arnold V.I., Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl. 6 (1964) 581-585. | Zbl 0135.42602

[4] Berti M., Bolle P., Homoclinics and chaotic behaviour for perturbed second order systems, Annali di Mat. Pura e Applicata, (IV) CLXXVI (1999) 323-378. | MR 1746547 | Zbl 0957.37019

[5] Berti M., Bolle P., Variational construction of homoclinics and chaotic behaviour in presence of a saddle-saddle equilibrium, Annali della Scuola Normale Superiore di Pisa, serie IV XXVII (2) (1998) 331-377. | Numdam | MR 1664692 | Zbl 0938.34039

[6] Berti M., Bolle P., Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems, Rend. Mat. Acc. Naz. Lincei, s. 9 11 (4) (2000) 235-243. | MR 1837581 | Zbl 1009.37044

[7] Berti M., Bolle P., Fast Arnold's diffusion in systems with three time scales, Discrete and Continuous Dynamical Systems, to appear. | MR 1897882 | Zbl 1023.37033

[8] Bessi U., An approach to Arnold diffusion through the calculus of variations, Nonlinear Analysis T.M.A. 26 (1996) 1115-1135. | MR 1375654 | Zbl 0867.70013

[9] Bessi U., Chierchia L., Valdinoci E., Upper bounds on Arnold diffusion times via Mather theory, J. Math. Pures Appl. 80 (1) (2001) 105-129. | MR 1810511 | Zbl 0986.37052

[10] Biasco L., Stime analitiche sui tempi di instabilitá per perturbazioni di sistemi hamiltoniani degeneri, Tesi di Laurea, Univ. Roma 3, 1999.

[11] Bolotin S.V., Homoclinic orbits to invariant tori of Hamiltonian systems, Amer. Math. Soc. Transl., ser 2 168 (1995) 21-90. | MR 1351032 | Zbl 0847.58024

[12] Bolotin S.V., Infinite number of homoclinic orbits to hyperbolic invariant tori of Hamiltonian systems, Regular and Chaotic Dynamics 5 (2) (2000). | MR 1780706 | Zbl 1004.70018

[13] Bourgain J., Golse F., Wennberg B., On the distribution of free path lengths for periodic Lorentz gas, Comm. Math. Phys. 190 (1998) 491-508. | MR 1600299 | Zbl 0910.60082

[14] Chierchia L., Arnold instability for nearly-integrable analytic hamiltonian systems, in: Ambrosetti A., Dell'Antonio G.F. (Eds.), Proceedings of the workshop “Variational and Local Methods in the Study of Hamiltonian Systems”, Trieste, 1994. | Zbl 0951.37036

[15] Chierchia L., Gallavotti G., Drift and diffusion in phase space, Annales de l'IHP, section Physique Théorique 60 (1994) 1-144, see also Erratum in Vol. 68 (1998) 135. | Numdam | MR 1259103 | Zbl 1010.37039

[16] Chierchia L., Zehnder E., Asymptotic expansions of quasi-periodic motions, Annali della Scuola Normale Superiore di Pisa, serie IV XVI (1989) 245-258. | Numdam | MR 1041897 | Zbl 0695.34040

[17] Chirikov B.V., A universal instability of many dimensional oscillator systems, Physics Reports 52 (1979) 263-379. | MR 536429

[18] Cresson J., A λ-lemma for partially hyperbolic tori and the obstruction property, Lett. Math. Phys. 42 (1997) 363-377. | Zbl 0888.58045

[19] Cresson J., Conjecture de Chirikov et optimalité des exposants de stabilité du théorème de Nekhoroshev, preprint Univ. Besançon.

[20] Cresson J., Guillet C., Periodic orbits and Arnold diffusion, preprint Univ. Besançon. | MR 1952386

[21] Delshams A., Gelfreich V.G., Jorba V.G., Seara T.M., Exponentially small splitting of separatrices under fast quasi-periodic forcing, Comm. Math. Phys. 189 (1997) 35-71. | MR 1478530 | Zbl 0897.34042

[22] DelshamsA., Gutiérrez V., Homoclinic orbits to invariant tori in Hamiltonian systems, in: Jones C., Wiggins S., Khibnik A., Dumortier F., Terman D. (Eds.), Multiple-Time-Dynamical Systems, IMA Vol. in Math and its Applications, Springer-Verlag, to appear. | MR 1846571 | Zbl 01787180

[23] Delshams A., Seara T.M., An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys. 150 (1992) 433-463. | MR 1204314 | Zbl 0765.70016

[24] Gallavotti G., Arnold's diffusion in isochronous systems, Mathematical Physics, Analysis and Geometry 1 (1999) 295-312. | MR 1692234 | Zbl 0936.37031

[25] Gallavotti G., Gentile G., Mastropietro V., Separatrix splitting for systems with three time scales, Comm. Math. Phys. 202 (1999) 197-236. | MR 1686531 | Zbl 0936.37034

[26] Gallavotti G., Gentile G., Mastropietro V., Melnikov approximation dominance. Some examples, Rev. Math. Phys 11 (1999) 451-461. | MR 1682687 | Zbl 0983.37074

[27] Gallavotti G., Gentile G., Mastropietro V., On homoclinic splitting problems, Physica D 137 (2000) 202-204. | MR 1738773 | Zbl 0997.37036

[28] Gallavotti G., Gentile G., Mastropietro V., Hamilton-Jacobi equation and existence of heteroclinic chains in three time scales systems, Nonlinearity 13 (2000) 323-340. | MR 1745371 | Zbl 0989.37059

[29] Gelfreich V., Melnikov method and exponentially small splitting of separatrices, Physica D 101 (1997) 227-248. | MR 1433078 | Zbl 0896.70011

[30] Gelfreich V., A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys. 201 (1999) 155-216. | MR 1669417 | Zbl 1042.37044

[31] Gelfreich V., Separatrix splitting for a high-frequency perturbation of the pendulum, Russian J. Math. Phys. 7 (1) (2000) 48-71. | MR 1832773 | Zbl 1066.34059

[32] Gentile G., A proof of existence of whiskered tori with quasi flat homoclinic intersection in a class of almost integrable systems, Forum Mathematicum 7 (1995) 709-753. | MR 1359423 | Zbl 0841.58038

[33] Holmes P., Marsden J., Scheurle J., Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations, Contemp. Math. 81 (1988) 213-244. | MR 986267 | Zbl 0685.70017

[34] Lazutkin V.F., Splitting of separatrices for the Chirikov's standard map, preprint VINITI 6372/84, 1984. | MR 1993024

[35] Lochak P., Arnold diffusion: a compendium of remarks and questions, in: Proceedings of 3DHAM's Agaro, 1995. | Zbl 0986.37054

[36] Lochak P., Marco J.P., Sauzin D., On the splitting of invariant manifolds in multidimensional Hamiltonian systems, preprint Université Jussieu.

[37] Marco J.P., Transitions le long des chaînes de tores invariants pour les systèmes hamiltoniens analytiques, Annales I.H.P. 64 (1995) 205-252. | Numdam | MR 1386217 | Zbl 0854.70011

[38] Pumarino P., Valls C., Three time scales systems exhibiting persistent Arnold Diffusion, preprint.

[39] Pumarino A., Valls C., Exponentially small splitting of separatrices in open sets of frequencies, preprint.

[40] Range R.M., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, 1986. | MR 847923 | Zbl 0591.32002

[41] Rudnev M., Wiggins S., Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable Hamiltonian systems, Physica D 114 (1998) 3-80, See also Erratum in Physica D 145 (2000) 349-354. | MR 1612043 | Zbl 0997.37038

[42] Rudnev M., Wiggins S., On a homoclinic splitting problem, Regular and Chaotic Dynamics 5 (2) (2000) 227-242. | MR 1780715 | Zbl 0996.37069

[43] Sauzin D., A new method for measuring the splitting of invariant manifolds, Ann. Scient. E.N.S, to appear. | Numdam | MR 1870269 | Zbl 0987.37061

[44] Treschev D., Multidimensional symplectic separatrix maps, preprint. | MR 1888569

[45] Xia Z., Arnold diffusion: a variational construction, Documenta Matematica, extra vol. ICM II (1998) 867-877. | MR 1648133 | Zbl 0910.58015