Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems

Journal of Nonlinear Science

Volumn 10, Issue 4, 2000, Pages 433-476

  Delshams, A ^a   Gutierrez P ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0013256095     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003329910016     Document Type: Article

References (38)

1. R. Abraham and J. E. Marsden. Foundations of mechanics. Benjamin/Cummings, Reading, Mass., 2nd edition, 1978.
2. V. I. Arnold. Proof of a theorem of A. N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian. Russ. Math. Surveys, 18(5):9-36, 1963.
3. V. I. Arnold. Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl., 5(3):581-585, 1964.
4. G. Benettin, L. Galgani, A. N. Giorgilli, and J.-M. Strelcyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method. Nuovo Cimenta B, 79(2):201-223, 1984.
5. S. V. Bolotin, Homoclinic orbits in invariant tori of Hamiltonian systems. In V. V. Kozlov, editor, Dynamical systems in classical mechanics, volume 168 of Amer. Math. Soc. Trans. Ser. 2, pages 21-90. Amer. Math. Soc., Providence. RI, 1995. (Adv. Math. Sci., 25).
6. S. V. Bolotin, Homoclinic orbits in invariant tori of Hamiltonian systems. In V. V. Kozlov, editor, Dynamical systems in classical mechanics, volume 168 of Amer. Math. Soc. Trans. Ser. 2, pages 21-90. Amer. Math. Soc., Providence. RI, 1995. (Adv. Math. Sci., 25).
7. L. Chierchia and G. Gallavotti. Drift and diffusion in phase space. Ann. Inst. H. Poincaré Phys. Théor., 60(1): 1-144, 1994.
8. S.-N. Chow and J. K. Hale. Methods of bifurcation theory, volume 251 of Grundlehren Math. Wiss. Springer-Verlag, New York, 1982.
9. B. V. Chirikov. A universal instability of many-dimensional oscillator systems. Phys. Rep., 52(5):263-379, 1979.
10. A. Delshams and P. Gutiérrez. Homoclinic orbits to invariant tori in Hamiltonian systems. In C. Jones, S. Wiggins, A. Khibnik, F. Dumortier, and D. Terman, editors, Multiple-time-scale dynamical systems, IMA Volumes in Mathematics and Its Applications. Held in Minneapolis, 27-31 October 1997. Springer-Verlag, New York (to appear).
11. A. Delshams, V. G. Gelfreich, À. Jorba, and T. M. Seara. Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys., 189:35-71, 1997.
12. A. Delshams and R. Ramírez-Ros. Melnikov potential for exact symplectic maps. Commun. Math. Phys., 190:213-245, 1997.
13. A. Delshams and T. M. Seara. Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3(4):1-10, 1997.
14. L. H. Eliasson. Biasymptotic solutions of perturbed integrable Hamiltonian systems. Bol. Soc. Bras. Mat, 25(1):57-76, 1994.
15. G. Gallavotti. Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review. Rev. Math. Phys., 6(3):343-411, 1994.
16. G. Gallavotti, G. Gentile, and V. Mastropietro. Separatrix splitting for systems with three time scales. Commun. Math. Phys., 202(1): 197-236, 1999.
17. S.M. Graff. On the conservation of hyperbolic invariant tori for Hamiltonian systems. J. Diff. Eq., 15:1-69, 1974.
18. P. Holmes and J. E. Marsden. Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys., 23:669-675, 1982.
19. A. N. Kolmogorov. The general theory of dynamical systems and classical mechanics. In Foundations of mechanics [AM78]. pages 741-757 (appendix). The original version appeared in 1954.
20. P. Lochak and C. Meunier. Multiphase averaging for classical systems: With applications to adiabatic theorems, volume 72 of Appl. Math. Sci. Springer-Verlag, New York, 1988.
21. P. Lochak, J.-P. Marco, and D. Sauzin. On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Preprint, 1999.
22. P. Lochak. Canonical perturbation theory via simultaneous approximation. Russ. Math Surveys, 47(6):57-133. 1992.
23. J. Moser. The analytic invariants of an area-preserving mapping near a hyperbolic fixed point. Commun. Pure Appl. Math., 9:673-692. 1956.
24. L. Niederman. Dynamics around simple resonant tori in nearly integrable Hamiltonian systems (to appear in J. Diff. Eq.). Preprint, 1999.
25. J. Pöschel. Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math., 35:653-696, 1982.
26. A. Pronin and D. Treschev. Continuous averaging in multi-frequency slow-fast systems. Preprint. 1999.
27. C. Robinson. Horseshoes for autonomous Hamiltonian systems using the Melnikov integral. Ergodic Theory Dynam. Systems, 8:395-409, 1988.
28. H. Rüssmann. On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. In J. Moser, editor, Dynamical systems, theory and applications, volume 38 of Lect. Notes in Phys., pages 598-624. Springer-Verlag, New York, 1975.
29. M. Rudnev and S. Wiggins. KAM theory near multiplicity one resonant surfaces in perturbations of a-priori stable Hamiltonian systems. J. Nonlinear Sci., 7:177-209, 1997.
30. M. Rudnev and S. Wiggins. Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems. Phys. D, 114:3-80, 1998.
31. M. Rudnev and S. Wiggins. On a homoclinic splitting problem. Preprint, 1999.
32. D. Sauzin. A new method for measuring the splitting of invariant manifolds. Preprint, 1999.
33. C. Simó. Averaging under fast quasiperiodic forcing. In J. Seimenis, editor, Hamiltonian mechanics: Integrability and chaotic behavior, volume 331 of NATO ASI Ser. B: Phys., pages 13-34. Held in Toruń. Poland, 28 June-2 July 1993. Plenum, New York, 1994.
34. D. Treschev. The mechanism of destruction of resonance tori of Hamiltonian systems. Math. USSR Sb., 68(1): 181-203, 1991.
35. D. Treschev. Hyperbolic tori and asymptotic surfaces in Hamiltonian systems. Russ. J. Math. Phys., 2(1):93-110, 1994.
36. E. Valdinoci. Whiskered transition tori for a priori stable and unstable Hamiltonian systems. Preprint 98-560, mp_arc@math.utexas.edu, 1998.
37. S. Wiggins. Global bifurcations and chaos: Analytical methods, volume 73 of Appl. Math. Sci. Springer-Verlag, New York, 1988.
38. E. Zehnder. Generalized implicit function theorems with applications to some small divisor problems/II. Commun. Pure Appl. Math., 29:49-111, 1976.