Singular separatrix splitting and the melnikov method: An experimental study

Experimental Mathematics

Volumn 8, Issue 1, 1999, Pages 29-48

  Delshams, Amadeu ^a   Ramírez Ros, Rafael ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033245665     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.1999.10504385     Document Type: Article

References (42)

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 (1992), 9-67.
2. A. Benseny and C. Olivé, “High precision angles between invariant manifolds for rapidly forced Hamiltonian systems”, pp. 315-319 in International Conference on Differential Equations (Formula Presented) (Barcelona, 1991), edited by C. Perelló et al., World Scientific, Singapore, 1993.
3. W.-J. Beyn and J.-M. Kleinkauf, “The numerical computation of homoclinic orbits for maps”, SIAM J. Numer. Anal. 34:3 (1997), 1207-1236.
4. V. L. Chernov, “Separatrix splitting for the Hénon map: a resurgence approach”, Math. Preprints Series 188, Univ. Barcelona, November 1995.
5. A. Delshams and R. Ramírez-Ros, “Poincaré-Melnikov Arnold method for analytic planar maps”, Nonlinearity 9:1 (1996), 1-26.
6. A. Delshams and R. Ramfrez-Ros, “Melnikov potential for exact symplectic maps”, Comm. Math. Phys. 189:1 (1997), 213-245.
7. A. Delshams and R. Ramfrez-Ros, “Exponentially small splitting of separatrices for perturbed integrable standard-like maps”, J. Nonlinear Sci. 8:3 (1998), 317-352.
8. 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 (1997), 1-40.
9. A. Delshams, R. Ramfrez- Ros, and T. M. Seara, “Splitting of separatrices in Hamiltonian systems and symplectic maps”, in Hamiltonian systems with three or more degrees of freedom (S’Agaro, Spain, 1995), edited by C. Simo, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer, Dordrecht, 1999.
10. R. W. Easton, “Transport through chaos”, Nonlinearity 4 (1991), 583-590.
11. B. Fiedler and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and “invisible” chaos, Mem. Amer. Math. Soc. 570, Amer. Math. Soc., Providence, 1996.
12. E. Fontich, “On the existence of homoclinic points for conservative diffeomorphisms”, pp. 189-197 in European Conference on Iteration Theory (ECIT 87) (Caldes de Malavella, Spain, 1987), edited by C. Alsina et al., World Scientific, Singapore, 1989.
13. E. Fontich, “Rapidly forced planar vector fields and splitting of separatrices”, J. Differential Equations 119:2 (1995), 310-335.
14. E. Fontich and C. Simó, “The splitting of separatrices for analytic diffeomorphisms”, Ergodic Theory Dynam. Systems 10 (1990), 295-318.
15. V. G. Gelfreich, “Conjugation to a shift and the splitting of invariant manifolds”, Appl. Math. (Warsaw) 24:2 (1996), 127-140.
16. V. G. Gelfreich, V. F. Lazutkin, and M. B. Tabanov, “Exponentially small splitting in Hamiltonian systems”, Chaos 1:2 (1991), 137-142.
17. V. G. Gelfreich, V. F. Lazutkin, and N. V. Svanidze, “A refined formula for the separatrix splitting for the standard map”, Phys. D 71 (1994), 82-101.
18. V. Hakim and K. Mallick, “Exponentially small splitting of separatrices, matching in the complex plane and Borel summation”, Nonlinearity 6 (1993), 57-70.
19. D. E. Knuth, The art of computer programming, 2: Seminumerical algorithms, Addison- Wesley, Reading, MA, 1969.
20. V. F. Lazutkin, “Splitting of separatrices for Chirikov’s standard map”, Preprint 6372-84, VINITI, Moscow, 1984. In Russian.
21. V. F. Lazutkin, I. G. Schach- mannski, and M. B. Tabanov, “Splitting of separatrices for standard and semistandard mappings”, Phys. D 40 (1989), 235-248.
22. P. Levallois, “Calcul d’une fonction de Melnikov et de ses zeros pour une perturbation algebrique du billard elliptique”, Ergodic Theory Dynam. Systems 17 (1997), 435-444.
23. P. Levallois and M. B. Tabanov, “Séparation des séparatrices du billard elliptique pour une perturbation algébrique et symetrique de l’ellipse”, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 589-592.
24. H. E. Lomelí, “Perturbations of elliptic billiards”, Phys. D 99 (1996), 59-80.
25. H. E. Lomelf and J. D. Meiss, “Heteroclinic orbits, and transport in a perturbed integrable standard map”, Preprint 96-77, February 1996. Send e-mail to mp_arc@math.utexas.edu.
26. R. S. MacKay, J. D. Meiss, and I. C. Percival, “Transport in Hamiltonian systems”, Phys. D 13 (1984), 55-81.
27. R. S. MacKay, J. D. Meiss, and I. C. Percival, “Resonances in area preserving maps”, Phys. D 27 (1987), 1-20.
28. E. M. McMillan, “A problem in the stability of periodic systems”, pp. 219-244 in Topics in modem physics: a tribute to E. V. Condon, edited by E. Brittin and H. Odabasi, Colorado Assoc. Univ. Press, Boulder, CO, 1971.
29. J. D. Meiss, “Symplectic maps, variational principles, and transport”, Rev. Mod. Phys. 64:3 (1992), 795-848.
30. A. I. Neishtadt, “The separation of motions in systems with rapidly rotating phase”, J. Appl. Math. Mech. 48:2 (1984), 133-139.
31. A. Nikitin, “Separatrix splitting via resurgence for the ‘twist map’: A method for the calculation of the coefficients of the splitting formula”, Math. Preprints Series 184, Univ. Barcelona, November 1995.
32. J.-P. Ramis and R. Schafke, “Gevrey separation of fast and slow variables”, Nonlinearity 9 (1996), 353-384.
33. V. Rom-Kedar, L. P. Kadanoff, S. C. Ching, and C. Amick, “The break-up of a heteroclinic connection in a volume preserving mapping”, Phys. D 62 (1993), 51-65.
34. D. Sauzin, “Résurgence parametrique et exponentielle petitesse de l’ecart des separatrices du pendule rapidament force”, Ann. Inst. Fourier (Grenoble) 45:2 (1995), 453-511.
35. J. Scheurle, J. Marsden, and P. J. Holmes, “Exponentially small estimate for separatrix splittings”, pp. 187-195 in Asymptotics beyond all orders, edited by H. Segur et al., NATO Adv. Sci. Inst. Ser. B Phys. 284, Plenum, New York, 1991.
36. C. Simó, “On the analytical and numerical approximation of invariant manifolds”, pp. 285-329 in Les méthodes modemes de la mécanique celeste (Goutelas, 1989), edited by D. Benest and C. Froeschlé, Éditions Frontières, Gif-sur-Yvette, 1990.
37. Y. B. Suris, “Integrable mappings of the standard type”, Functional Anal. Appl. 23 (1989), 74-76.
38. Y. B. Suris, “On the complex separatrices of some standard-like maps”, Nonlinearity 7 (1994), 1225-1236.
39. E. Tabacman, “Variational computation of homoclinic orbits for twist maps”, Phys. D 85, 548-562.
40. M. B. Tabanov, “Separatrices splitting for Birkhoff’s billiard in a symmetric convex domain, close to an ellipse”, Chaos 4:4 (1994), 595-606.
41. D. V. Treschev, “An averaging method for Hamiltonian systems, exponentially close to integrable ones”, Chaos 6:1 (1996), 6-14.
42. A. Weinstein, “Lagrangian submanifolds and Hamiltonian systems”, Ann. of Math. (2) 98 (1973), 377-410.