A new method for measuring the splitting of invariant manifolds

 Access to full text

 Full (PDF)

1. [1] Arnol'd V.I, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR156 (1964) 9-12, (in Russian). [English translation: Soviet Math. Dokl.5 (1964) 581–585]. Zbl0135.42602
2. [2] Bott R, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc.7 (2) (1982) 331-358. Zbl0505.58001MR663786
3. [3] Bruno A.D, Local Methods in Nonlinear Differential Equations, Springer-Verlag, Berlin, 1989. Zbl0674.34002MR993771
4. [4] Delshams A, Gelfreich V.G, Jorba À, Seara T.M, Exponentially small splitting of separatrices under fast quasi-periodic forcing, Comm. Math. Phys.189 (1997) 35-71. Zbl0897.34042MR1478530
5. [5] Delshams A, Gutiérrez P, Homoclinic orbits to invariant tori in Hamiltonian systems, in: Jones C, Wiggins S, Khibnik A, Dumortier F, Terman D (Eds.), Multiple-Time-Scale Dynamical Systems, IMA Vol. in Math. and its Appl., Springer-Verlag, Berlin, 1998. Zbl1140.37355MR1846571
6. [6] 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. Zbl0765.70016MR1204314
7. [7] Delshams A, Seara T.M, Splitting of separatrices in Hamiltonian systems with one and half degrees of freedom, Math. Phys. Elec. J.3 (1997), paper 4. Zbl0891.58035MR1474213
8. [8] Écalle J, Six lectures on transseries, analysable functions and the constructive proof of Dulac's conjecture, in: Schlomiuk D (Ed.), Bifurcations and Periodic Orbits of Vector Fields, Kluwer Academic, Dordrecht, 1993, pp. 75-184. Zbl0814.32008MR1258519
9. [9] Eliasson L.H, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Bras. Mat.25 (1) (1994) 57-76. Zbl0799.58026MR1274762
10. [10] Fruchard A, Schäfke R, Exponentially small splitting of separatrices for difference equations with small step size, J. Dynam. Control Syst.2 (2) (1996) 193-238. Zbl0940.39001MR1388695
11. [11] Gallavotti G, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable systems. A review, Rev. Math. Phys.6 (3) (1994) 343-411. Zbl0798.58036MR1305589
12. [12] Gallavotti G, Gentile G, Mastropietro V, Melnikov's approximation dominance. Some examples, Rev. Math. Phys.11 (4) (1999) 451-461. Zbl0983.37074MR1682687
13. [13] Gelfreich V.G, Melnikov method and exponentially small splitting of separatrices, Physica D101 (1997) 227-248. Zbl0896.70011MR1433078
14. [14] Graff S.M, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations15 (1974) 1-69. Zbl0257.34048MR365626
15. [15] Hirsch M.W, Pugh C.C, Shub M, Invariant Manifolds, Lect. Notes in Math., 583, Springer-Verlag, Berlin, 1977. Zbl0355.58009MR501173
16. [16] Lazutkin V.F, Splitting of separatrices for the Chirikov's standard map, Preprint VINITI 6372-84, 1984, (in Russian). MR1993024
17. [17] Lochak P, Effective speed of Arnol'd diffusion and small denominators, Phys. Lett. A143 (1990) 39-42. MR1034234
18. [18] Lochak P, Canonical perturbation theory via simultaneous approximation, Russian Math. Surveys47 (1992) 57-133. Zbl0795.58042MR1209145
19. [19] Lochak P, Hamiltonian perturbation theory: periodic orbits, resonances and intermittency, Nonlinearity6 (1993) 885-904. Zbl0794.70006MR1251248
20. [20] Lochak P, Tores invariants à torsion évanescente dans les systèmes hamiltoniens proches de l'intégrable, C.R. Acad. Sci. Paris, Série I327 (1998) 833-836. Zbl0917.58011MR1663746
21. [21] Lochak P, Marco J.-P, Sauzin D, On the splitting of the invariant manifolds in multidimensional near-integrable Hamiltonian systems, Prépublication 220 de l'Institut de mathématiques de Jussieu, 1999. 
22. [22] Poincaré H, Les méthodes nouvelles de la mécanique céleste, Vol. 2, Gauthier-Villars, Paris, 1893. Zbl25.1847.03JFM25.1847.03
23. [23] Popov G, Invariant tori, effective stability and quasimodes with exponentially small error terms, Preprint, 1999. 
24. [24] Pöschel J, Nekhoroshev estimates for quasi-convex Hamiltonian sytems, Math. Z.213 (1993) 187-216. Zbl0857.70009MR1221713
25. [25] Rudnev M, Wiggins S, Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems, Physica D114 (1998) 3-80. Zbl0941.37021MR1612043
26. [26] Sauzin D, Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé, Ann. Inst. Fourier45 (1995) 453-511. Zbl0826.30004MR1343559
27. [27] Simó C, Averaging under fast quasiperiodic forcing, in: Seimenis J (Ed.), Hamiltonian Mechanics: Integrability and Chaotic Behaviour, NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum Press, New York, 1994, pp. 13-34. MR1316666
28. [28] Treschev D.V, A mechanism for the destruction of resonance tori of Hamiltonian systems, Math. USSR-Sbornik68 (1) (1991) 181-203. Zbl0737.58025MR1025685
29. [29] Treschev D.V, Hyperbolic tori and asymptotic surfaces in Hamiltonian systems, Russian J. Math. Phys.2 (1) (1994) 93-110. Zbl0910.58034MR1297943
30. [30] Yoccoz J.-C, Introduction to hyperbolic dynamics, in: Branner B, Hjorth P (Eds.), Real and Complex Dynamical Systems, NATO Adv. Sci. Inst. Ser. C Math. and Phys. Sciences, 464, Kluwer Academic, Dordrecht, 1995, pp. 265-291. Zbl0834.54023MR1351526

1. José Pedro Gaivão, Analytic invariants for the $1:-1$ resonance
2. Massimiliano Berti, Philippe Bolle, A functional analysis approach to Arnold diffusion