Geodesic
Hyperbolic tori in Hamiltonian systems with slowly varying parameter
Sbornik. Mathematics, Tome 204 (2013) no. 5, pp. 661-682 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper looks at a Hamiltonian system which depends periodically on a parameter. For each value of the parameter the system is assumed to have a hyperbolic periodic solution. Using the methods in KAM-theory it is proved that if the Hamiltonian is perturbed so that the value of the parameter varies with constant small frequency, then the nonautonomous system will have hyperbolic 2-tori in the extended phase space. Bibliography: 12 titles.
Keywords: KAM-theory, hyperbolic tori, fast-slow systems. 
@article{SM_2013_204_5_a2,
     author = {A. G. Medvedev},
     title = {Hyperbolic tori in {Hamiltonian} systems with slowly varying parameter},
     journal = {Sbornik. Mathematics},
     pages = {661--682},
     year = {2013},
     volume = {204},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_5_a2/}
}
TY  - JOUR
AU  - A. G. Medvedev
TI  - Hyperbolic tori in Hamiltonian systems with slowly varying parameter
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 661
EP  - 682
VL  - 204
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_5_a2/
LA  - en
ID  - SM_2013_204_5_a2
ER  - 
%0 Journal Article
%A A. G. Medvedev
%T Hyperbolic tori in Hamiltonian systems with slowly varying parameter
%J Sbornik. Mathematics
%D 2013
%P 661-682
%V 204
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2013_204_5_a2/
%G en
%F SM_2013_204_5_a2
A. G. Medvedev. Hyperbolic tori in Hamiltonian systems with slowly varying parameter. Sbornik. Mathematics, Tome 204 (2013) no. 5, pp. 661-682. http://geodesic.mathdoc.fr/item/SM_2013_204_5_a2/

[1] V. I. Arnol'd, “On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian”, Soviet Math. Dokl., 3 (1962), 136–140 | MR | Zbl

[2] S. M. Graff, “On the conservation of hyperbolic invariant tori for Hamiltonian systems”, J. Differential Equations, 15 (1974), 1–69 | DOI | MR | Zbl

[3] Yu. N. Bibikov, “A sharpening of a theorem of Moser”, Soviet Math. Dokl., 14 (1973), 1769–1773 | MR | Zbl

[4] G. N. Piftankin, “Skorost diffuzii v zadache Mezera”, Dokl. RAN, 408:6 (2006), 736–737 | MR

[5] S. Bolotin, D. Treschev, “Unbounded growth of energy in nonautonomous Hamiltonian systems”, Nonlinearity, 12:2 (1999), 365–388 | DOI | MR | Zbl

[6] A. Delshams, R. de la Llave, 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 $\mathbb{T}^2$”, Comm. Math. Phys., 209:2 (2000), 353–392 | MR | Zbl

[7] N. Brännström, V. Gelfreich, “Drift of slow variables in slow-fast Hamiltonian systems”, Phys. D, 237:22 (2008), 2913–2921 | DOI | MR | Zbl

[8] V. Gelfreich, D. Turaev, “Unbounded energy growth in hamiltonian systems with a slowly varying parameter”, Comm. Math. Phys., 283:3 (2008), 769–794 | DOI | MR | Zbl

[9] S. V. Bolotin, D. V. Treschev, “Remarks on the definition of hyperbolic tori of Hamiltonian systems”, Regul. Chaotic Dyn., 5:4 (2000), 401–412 | DOI | MR | Zbl

[10] D. Treschev, O. Zubelevich, Introduction to the perturbation theory of Hamiltonian systems, Springer Monogr. Math., Springer-Verlag, Berlin, 2010 | MR | Zbl

[11] V. I. Arnold, A. B. Givental, Simplekticheskaya geometriya, RKhD, Izhevsk, 2000

[12] Symplectic geometry and topology. Lecture notes from the graduate summer school program (Park City, UT, USA, 1997), eds. Ya. Eliashberg, L. Traynor, Amer. Math. Soc., Providence, RI, 1999 | Zbl