A proof of the exponentially small transversality of the separatrices for the standard map

Communications in Mathematical Physics

Volumn 201, Issue 1, 1999, Pages 155-216

  Gelfreich, V G ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033241455     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050553     Document Type: Article

References (19)

1. [Chi79] Chirikov, B.V: A universal instability of many-dimensional oscillator systems. Phys.Rep. 52, 263-379 (1979)
2. [DGJS97] Delshams, A., Gelfreich, V., Jorba, À., and Seara, T.M.: Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys. 189, 35-71 (1997)
3. [DS92] Delshams, A. and Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly force pendulum. Commun. Math. Phys. 150, 433-463 (1992)
4. [Fon95] Fontich, E.: Rapidly forced planar vector fields and splitting of separatrices. J. Diff. Eq. 119(2), 310-335 (1995)
5. [FS90] Fontich, E. and Simó, C.: Invariant manifolds for near identity differentiable maps and splitting of separatrices. Ergod. Th. and Dynam. Sys. 10, 319-346 (1990)
6. [Gel96] Gelfreich, V.G.: Conjugation to a shift and the splitting of invariant manifolds. Applicationes Mathematicae 24, 2, 127-140 (1996)
7. [Gel97a] Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Physica D 101, 227-248 (1997)
8. [Gel97b] Gelfreich, V.G.: Reference systems for splitting of separatrices. Nonlinearity 10, 175-193 (1997)
9. [GLS94] Gelfreich, V.G., Lazutkin, V.F. and Svanidze, N.V.: A refined formula for the separatrix splitting for the standard map. Physica D 71(2), 82-101 (1994)
10. [GP81] Greene, J.M. and Percival, I.C.: Hamiltonian maps in the complex plane. Physica D 3, 530-548 (1981)
11. [Laz84] Lazutkin, V.F.: Splitting of separatrices for the Chirikov's standard map. VINITI no. 6372/84, 1984, (Russian)
12. [Laz90] Lazutkin, V.F.: Analytic integrals of the semistandard map, and splitting of separatrices. Leningrad Math. J. 1(2), 427-445 (1990)
13. [Laz91] Lazutkin, V.F.: Exponential splitting of separatrices and an analytical integral for the semistandard map. Preprint, Université Paris VII, 1991
14. [Laz92] Lazutkin, V.F.: An analytic integral along the separatrix of the semistandard map: Existence and an exponential estimate for the distance between the stable and unstable separatrices. Algebra i Analiz 4(4), 110-142 (1992) (Russian edition). St. Petersburg Math. J. 4(4), 721-748 (1993) (English edition)
15. [Laz92] Lazutkin, V.F.: An analytic integral along the separatrix of the semistandard map: Existence and an exponential estimate for the distance between the stable and unstable separatrices. Algebra i Analiz 4(4), 110-142 (1992) (Russian edition). St. Petersburg Math. J. 4(4), 721-748 (1993) (English edition)
16. [LST89] Lazutkin, V.F., Schakhmanski, I.G. and Tabanov, M.B.: Splitting of separatrices for standard and semistandard mappings. Physica D 40, 235-348 (1989)
17. [Sin94] Sinai, Ya.G.: Topics in ergodic theory. Princeton Math. Ser. 44, Princeton, NJ: Princeton University Press, 1994, pp. 218
18. [Sur94] Suris, Yu.B.: On the complex separatrices of some standard-like maps. Nonlinearity 7(4), 1225-1236 (1994)
19. [Tre96] Treshchev, D.V: An averaging method for Hamiltonian systems, exponentially close to integrable ones. Chaos 6(1), 6-14 (1996)