Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables

Physica D: Nonlinear Phenomena

Volumn 210, Issue 3-4, 2005, Pages 149-179

  Ramírez Ros, Rafael ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Billiards; Exponentially small phenomena; Homoclinic bifurcations; Numeric experiments; Separatrix splitting

Indexed keywords

BIFURCATION (MATHEMATICS); CONVERGENCE OF NUMERICAL METHODS; FUNCTIONS; PARAMETER ESTIMATION;

BILLIARDS; EXPONENTIALLY SMALL PHENOMENA; HOMOCLINIC BIFURCATIONS; SEPARATRIX SPLITTING;

PERTURBATION TECHNIQUES;

EID: 25144476259     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2005.07.012     Document Type: Article

References (23)

1. C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, User's Guide to PARI/GP (freely available from http://www.parigp-home.de/ ).
2. G.D. Birkhoff Dynamical Systems 1927 AMS Providence
3. V.L. Chernov On separatrix splitting of some quadratic area-preserving maps of the plane Regul. Chaotic Dyn. 3 1998 49 65
4. A. Delshams, and R. Ramírez-Ros Poincaré-Melnikov-Arnold method for analytic planar maps Nonlinearity 9 1996 1 26
5. A. Delshams, and R. Ramírez-Ros Exponentially small splitting of separatrices for perturbed integrable standard-like maps J. Nonlinear Sci. 8 1998 317 352
6. A. Delshams, and R. Ramírez-Ros Singular separatrix splitting and the Melnikov method: an experimental study Exp. Math. 8 1999 29 48
7. A. Delshams, Yu. Fedorov, and R. Ramírez-Ros Homoclinic billiard orbits inside symmetrically perturbed ellipsoids Nonlinearity 14 2001 1141 1195
8. E. Fontich, and C. Simó The splitting of separatrices for analytic diffeomorphisms Ergodic Theory Dyn. Syst. 10 1990 295 318
9. 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
10. V.G. Gelfreich A proof of the exponentially small transversality of the separatrices for the standard map Comm. Math. Phys. 201 1999 155 216
11. V.G. Gelfreich, and D. Sauzin Borel summation and splitting of separatrices for the Hénon map Ann. Inst. Fourier (Grenoble) 51 2001 513 567
12. V.G. Gelfreich, and V.F. Lazutkin Splitting of separatrices: perturbation theory and exponential smallness Russian Math. Surveys 56 2001 499 558
13. A. Katok, and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems 1995 Cambridge University Press Cambridge
14. V.V. Kozlov, D.V. Treshchëv, Billiards: a genetic introduction to the dynamics of systems with impacts, Trans. Math. Monographs (AMS, Providence) (1991) 89
15. V.F. Lazutkin, Splitting of separatrices for the standard map. Preprint VINITI (1984).
16. V.F. Lazutkin The splitting of separatrices for a standard family of area-preserving transformations Transl. Am. Math. Soc. Ser. 2 157 1993 27 35
17. R.S. MacKay, J.D. Meiss, and I.C. Percival Transport in Hamiltonian systems Physica D 13 1984 55 81
18. E.M. McMillan A problem in the stability of periodic systems E. Brittin H. Odabasi Topics in Modern Physics 1971 Colorado Association University Press Boulder 219 244
19. C. Simó Analytical and numerical computation of invariant manifolds D. Benest C. Froeschlé Modern Methods in Celestial Mechanics 1990 Editions Frontières Gif-sur-Yvette 285 330
20. C. Simó Analytical and numerical detection of exponentially small phenomena B. Fiedler K. Gröger J. Sprekels Proceedings of the International Conference on Differential Equations World Science Publishing, River Edge 2000 967 976
21. S. Tabachnikov, Billiards, Panor. Synth 1 (SMF, Paris, 1995).
22. M.B. Tabanov Separatrices splitting for Birkhoff's billiard in a symmetric convex domain, close to an ellipse Chaos 4 1994 595 606
23. E.T. Whittaker, and G.N. Watson A Course of Modern Analysis 1927 Cambridge University Press Cambridge