Exponentially Small Splitting of Separatrices for Perturbed Integrable Standard-Like Maps

Journal of Nonlinear Science

Volumn 8, Issue 3, 1998, Pages 317-352

  Delshams, A ^a   Ramirez Ros R ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000467096     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003329900054     Document Type: Article

References (53)

1. C. Amick, S. C. E. Ching, L. P. Kadanoff, and V. Rom-Kedar. Beyond all orders: Singular perturbations in a mapping. J. Nonlinear Sci., 2:9-67, 1992.
2. S. Angenent. A variational interpretation of Melnikov's function and exponentially small separatrix splitting. In D. Salamon, editor, Symplectic Geometry, vol. 192 of London Math. Soc. Lecture Note Ser., pages 5-35. Cambridge Univ. Press, Cambridge, Eng., 1993.
3. M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover, New York, 1972.
4. V. L. Chernov. Resurgence of separatrices of the quadratic map. Math. Preprints Series 185, Univ. Barcelona, November 1995.
5. R. Cushman. Examples of nonintegrable analytic Hamiltonian vectorfields with no small divisors. Trans. Am. Math. Soc., 238(1):45-55, 1978.
6. D. Delatte. On normal forms in Hamiltonian dynamics, a new approach to convergence questions. Ergodic Theory Dynamical Systems, 15:49-66, 1995.
7. A. Delshams, V. G. Gelfreich, A. Jorba, and T. M. Seara. Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys, 189:35-71, 1997.
8. A. Delshams, A. Mir, and T. M. Seara. Psi-series and splitting of separatrices. Preprint in progress, 1997.
9. A. Delshams and R. Ramírez-Ros. Poincaré-Melnikov-Arnold method for analytic planar maps. Nonlinearity, 9(1):1-26, 1996.
10. A. Delshams and R. Ramírez-Ros. Melnikov potential for exact symplectic maps. Commun. Math. Phys. 190:213-245, 1997.
11. A. Delshams and R. Ramírez-Ros. Singular separatrix splitting and Melnikov method: An experimental study. Submitted, 1997.
12. A. Delshams and T. M. Seara. An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Commun. Math. Phys., 150:433-463, 1992.
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):1-40 1997.
14. E. Fontich. On the existence of homoclinic points for conservative diffeomorphisms. In C. Alsina, J. Llibre, Ch. Mira, C. Simó, G. Targonski, and R. Thibault, eds., European Conference on Iteration Theory (ECIT 87), pp. 189-197. Caldes de Malavella, Spain, 20-26 September, 1987. World Scientific, Singapore, 1989.
15. E. Fontich. Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations. Nonlinear Anal., 20(6):733-744, 1993.
16. E. Fontich. Rapidly forced planar vector fields and splitting of separatrices. J. Diff. Eq., 119(2):310-335, 1995.
17. E. Fontich and C. Simó. The splitting of separatrices for analytic diffeomorphisms. Ergodic Theory Dynamical Systems, 10:295-318, 1990.
18. B. Fiedler and J. Scheurle. Discretization of homoclinic orbits, rapid forcing and "invisible" chaos. Mem. Amer. Math. Soc., 119(570), 1996. viii + 79 pp.
19. 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.
20. V. G. Gelfreich. Separatrices splitting for the rapidly forced pendulum. In S. Kuksin, V. F. Lazutkin, and J. Pöschel, eds., Proc. Dynam. Syst. Semester, pp. 47-67. St. Petersburg, Russia, 17-30 November, 1991. Birkhäuser, Basel-Boston-Stuttgart, 1993.
21. V. G. Gelfreich. Melnikov method and exponentially small splitting of separatrices. Physica D, 101:227-248, 1997.
22. V. G. Gelfreich. Conjugation to a shift and the splitting of invariant manifolds. Appl. Math., 24(2):127-140, 1996.
23. V. G. Gelfreich, V. F. Lazutkin, and N. V. Svanidze. A refined formula for the separatrix splitting for the standard map. Physica D, 71:82-101, 1994.
24. V. G. Gelfreich, V. F. Lazutkin, and M. B. Tabanov. Exponentially small splitting in Hamiltonian systems. Chaos, 1(2):137-142, 1991.
25. M. L. Glasser, V. G. Papageorgiou, and T. C. Bountis. Melnikov's function for two-dimensional mappings. SIAM J. Appl. Math., 49(3):692-703, 1989.
26. V. Hakim and K. Mallick. Exponentially small splitting of separatrices, matching in the complex plane and Borel summation. Nonlinearity, 6:57-70, 1993.
27. P. Holmes, J. Marsden, and J. Scheurle. Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. Contemp. Math., 81:213-244, 1988.
28. L. J. Laslett. Some illustrations of stochasticity. In S. Jorna, ed., Topics in Nonlinear Dynamics, A Tribute to Sir Edward Bullard, no. 46 in AIP Conference Proceedings, pages 221-243. Am. Inst. Phys., New York, 1978.
29. V. F. Lazutkin. Splitting of separatrices for the Chirikov standard map. Preprint 6372-84, VINITI, 1984. (In Russian).
30. V. F. Lazutkin. On the width of the instability zone near the separatrices of a standard mapping. Soviet Math. Dokl., 42(1):5-9, 1991.
31. V. F. Lazutkin. Resurgent approach to the separatrices splitting. In C. Perelló, C. Simó, and J. Solà-Morales, eds., International Conference on Differential Equations (Equa∂i/∂1ff 91), pp. 163-176. Barcelona, 26-31 August 1991. World Scientific, Singapore, 1993.
32. P. Levallois. Calcul d'une fonction de Melnikov et de ses zeros pour une perturbation algébrique du billiard elliptique. Ergodic Theory Dyn. Syst., 17:435-444, 1997.
33. H. E. Lomelí and J. D. Meiss. Heteroclinic orbits, and transport in a perturbed integrable standard map. Preprint 96-77, mp_arc@math.utexas.edu, February 1996.
34. H. E. Lomelí. Perturbations of elliptic billiards. Physica D, 99:59-80, 1996.
35. V. F. Lazutkin, I. G. Schachmannski, and M. B. Tabanov. Splitting of separatrices for standard and semistandard mappings. Physica D, 40:235-248, 1989.
36. P. Levallois and M. B. Tabanov. Séparation des séparatrices du billard elliptique pour une perturbation algébrique et symétrique de l'ellipse. C. R. Acad. Sci. Paris Sér. I Math., 316:589-592, 1993.
37. E. M. McMillan. A problem in the stability of periodic systems. In E. Brittin and H. Odabasi, eds., Topics in Modern Physics, a Tribute to E. V. Condon, pp. 219-244. Colorado Assoc. Univ. Press, Boulder, CO, 1971.
38. J. D. Meiss. Symplectic maps, variational principles, and transport. Rev. Mod. Phys., 64(3):795-848, 1992.
39. J. Moser. The analytic invariants of an area-preserving mapping near a hyperbolic fixed point. Commun. Pure Appl. Math., 9:673-692, 1956.
40. A. I. Neishtadt. The separation of motions in systems with rapidly rotating phase. J. Appl. Math. Mech., 48(2):133-139, 1984.
41. H. Poincaré. Les méthodes nouvelles de la mécanique céleste, volumes 1, 2, 3. Gauthier-Villars, Paris, 1892-1899.
42. V. Rom-Kedar and S. Wiggins. Transport in two-dimensional maps. Arch. Rat. Mech. Anal., 109:239-298, 1988.
43. M. Rudnev and S. Wiggins. Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems. Preprint 97-4, mp_arc@math.utexas.edu, January 1997. To appear in Phys. D.
44. D. Sauzin. Résurgence paramétrique et exponentielle petitese de l'ecart des séparatrices du pendule rapidament forcé. Ann. Inst. Fourier (Grenoble), 45(2):453-511, 1995.
45. C. Simó. Averaging under fast quasiperiodic forcing. In J. Seimenis, editor, Hamiltonian Mechanics: Integrability and Chaotic Behaviour, vol. 331 of NATO Adv. Sci. Inst. Ser. B Phys., pp. 13-34. Toruń, Poland, 28 June-2 July 1993. Plenum, New York, 1994.
46. B. Y. Sternin and V. E. Shatalov. Borel-Laplace transform and asymptotic theory: Introduction to resurgent analysis. CRC Press, Boca Raton, Florida, 1996.
47. Y. B. Suris. Integrable mappings of the standard type. Funct. Anal. Appl., 23:74-76, 1989.
48. Y. B. Suris. On the complex separatrices of some standard-like maps. Nonlinearity, 7:1225-1236, 1994.
49. M. B. Tabanov. Separatrices splitting for Birkhoffs billiard in a symmetric convex domain, close to an ellipse. Chaos, 4(4):595-606, 1994.
50. D. V. Treschev. Separatrix splitting for a pendulum with rapidly oscillating suspension point. Russ. J. Math. Phys., 5:63-98, 1997.
51. D. V. Treschev. An averaging method for Hamiltonian systems, exponentially close to integrable ones. Chaos, 6(1):6-14, 1996.
52. D. V. Treschev. Width of stochastic layers in near-integrable two-dimensional symplectic maps. Preprint, 1996.
53. E. T. Whittaker and G. N. Watson. A course of modern analysis. Cambridge Univ. Press, Cambridge, 1927.