Towards global models near homoclinic tangencies of dissipative diffeomorphisms

Nonlinearity

Volumn 11, Issue 3, 1998, Pages 667-770

  Broer, Henk ^a   Simó, Carles ^b   Tatjer, Joan Carles ^b  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b UNIVERSITY OF BARCELONA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0006963160     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/11/3/015     Document Type: Article

References (62)

1. Arnold V I 1965 Small denominators I Transl. Am. Math. Soc 2nd Series 46 213-84
2. Arnold V I 1983 Geometrical Methods in the Theory of Ordinary Differential Equations (New York: Springer)
3. Aronson D G, Chory M A, Hall G R, McGehee R P 1982 Bifurcations from an invariant circle for two-parameter families of maps of the plane Commun. Math. Phys. 83 303-54
4. Bélair J and Glass L 1985 Universality and selfsimilarity in the bifurcations of circle maps Physica D 16 143
5. Benedicks M and Carleson L 1991 The dynamics of the Hénon map Ann. Math. 133 73-169
6. Bosch M and Tatjer J C 1991 On the codimension two bifurcations in families of one- and two-dimensional maps Proc ECIT vol 89 (Singapore: World Scientific) pp 9-18
7. Bosch M 1992 Estudi de sistemes dinàmics mitjançant la reducció de la seva dimensió PhD Thesis Universitat de Barcelona
8. Boyland P L 1986 Bifurcation of circle maps. Arnold tongues, bistability and rotation intervals Commun. Math Phys. 106 353-81
9. Broer H W, Huitema G B and Takens F 1990 Unfoldings of quasi-periodic tori Mem. Am Math. Soc. 421 1-82
10. Broer H W, Roussarie R and Simó C 1993 On the Bogdanov-Takens bifurcation for planar diffeomorphisms Proc. Equadiff 91 (Singapore: World Scientific) pp 81-92
11. Broer H W, Roussane R and Simó C 1996 Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms Ergod. Theor. Dynam. Syst. 16 1147-72
12. Broer H W and Takens F 1989 Formally symmetric normal forms and genericity Dynam. Rep. 2 36-60
13. Broer H W and Tangerman F M 1986 From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems Ergod Theor. Dynam. Syst. 6 345-62
14. Cabré X and Fontich E 1995 Regularity and uniqueness of one dimensional invariant manifolds Preprint Universitat de Barcelona
15. Carcassès J P 1993 Determination of different configurations of fold and flip bifurcation curves of one or two-dimensional maps Int. J. Bifur. Chaos Appl. Sci Eng. 3 869-902
16. Carcassès J P, Mira C, Bosch M, Simó C and Taljer J C 1991 'Crossroad area-spring area' transition (I). Parameter plane representation Int J Bifur. Chaos Appl. Sci. Eng. 1 183-96
17. Carcassès J P, Mira C, Bosch, M, Simó C and Tatjer J C 1991 'Crossroad area-spring area' transition (II). Foliated parametric representation Int. J. Bifur. Chaos Appl. Sci Eng. 1 339-48
18. Devaney R 1989 An Introduction to Chaotic Dynamical Systems 2nd edn (Reading, MA: Addison-Wesley)
19. Díaz L, Rocha J and Viana M 1996 Strange attractors in saddle-node cycles: prevalence and globality Inv. Math 125 37-74
20. Feigenbaum M J, Kadanoff L P and Shenker S J 1982 Quasiperiodicity in dissipative systems: renormalization group analysis Physica D 5 370-86
21. Fontich E and Simó C 1990 The splitting of separatrices for analytic diffeomorphisms Ergod. Theor. Dynam. Syst. 10 295-318
22. Fontich E and Simó C 1990 Invariant manifolds for near identity differentiable maps and splitting of separatrices Ergod. Theor. Dynam. Syst. 10 319-46
23. Gelfreich V G 1997 Reference systems for splitting of separatrices Nonlinearity 10 175-93
24. Giorgilli A, Lazutkin V F and Simó C 1997 Visualization of a hyperbolic structure in area preserving maps Reg. Chaotic Dynam. 2 47-61
25. Guckenheimer J and Holmes P 1983 Nonlinear Oscilations, Dynamical Systems and Bifurcations of Vector Fields (Berlin: Springer)
26. Herman M R 1977 Mesure de Lebesgue et nombre de rotation Lect. Notes Math. 597 271-93
27. Herman M R Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations Publ. Math. IHES 49 1979
28. van der Heijden G H M 1994 Nonlinear drillstring dynamics, a quest for the origin of chaotic vibrations PhD Thesis Utrecht University
29. Hirsch M W, Pugh C C and Shub M 1977 Invariant Manifolds (Springer Lecture Notes in Mathematics 583) (Berlin: Springer)
30. Holmes P and Whitley D 1984 Bifurcations of one- and two-dimensional maps Phil. Trans. R. Soc. A 311 43-102
31. Lazutkin V F 1989 Splitting of separatrices for standard and semistandard mappings Physica D 40 235-48
32. 3, in preparation
33. Mañé R 1978 Persistent manifolds are normally hyperbolic Trans. Am. Math. Soc. 246 261-83
34. Mira C 1987 Chaotic Dynamics (Singapore: World Scientific)
35. Mira C, Kawakami H and Allam R 1993 The dovetail bifurcation structure and its qualitative changes Int. J. Bifur. Chaos Appl. Sci. Eng. 3 903-14
36. Mora L and Viana M 1993 Abundance of strange attractors Acta Math. 171 1-71
37. Neishtadt A I 1984 The separation of motions in systems with rapidly rotating phase Prikl. Mat. Mekh. 48 133-9
38. Newhouse S 1979 The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms Publ. Math. IHES 50 101-51
39. Newhouse S, Palis J and Takens F Stable families of diffeomorphisms Publ. Math. IHES 57 1983
40. Nikitin A 1995 Separatrix splitting for the 'twist map'. A method of calculation of the coefficients of the splitting formula Math. Preprint Series 184 Universitat de Barcelona
41. Nitecki Z 1971 Differentiable Dynamics (Cambridge, MA: MIT)
42. Olvera A and Simó C 1987 An obstruction method for the destruction of invariant curves Physica D 28 181-92
43. Ostlund S, Rand D, Sethna J and Siggia E 1983 Universal properties of the transition from quasi-periodicity to chaos in dissipative systems Physica D 8 303-42
44. Palis J and de Melo W C 1982 Geometric Theory of Dynamical Systems (Berlin: Springer)
45. Palis J and Takens F 1993 Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Cambridge Studies in Advanced Mathematics 35) (Cambridge: Cambridge University Press)
46. Pumariño A and Rodríguez J A 1997 Coexistence and Persistence of Strange Attractors (Lecture Notes in Mathematics 1658) (Berlin: Springer)
47. Rand D 1988 Universality and renormalization New Directions in Dynamical Systems (London Math. Soc. L. N. Ser. 127) ed T Bedford and J Swift
48. Roussarie R and Simó C Universal models near homoclinic tangencies for planar maps, in preparation
49. Shub M 1978 Stabilité globale des systèmes dynamiques Astérisque 56
50. Simó C 1979 On the Hénon-Pomeau attractor J. Stat Phys. 21 (1979), 465-494
51. Simó C 1990 Analytical and numerical computation of invariant manifolds Modern Methods in Celestial Mechanics ed D Benest and C Froeschlé (Paris: Editions Frontières) pp 285-330
52. Simó C Direct numerical computation of coefficients in refined splitting formulas, in preparation
53. Simó C 1997 Invariant curves of perturbations of non twist integrable area preserving maps Preprint
54. Simó C, Broer H W and Roussarie R 1991 A numerical survey on the Takens-Bogdanov bifurcation for diffeomorphisms ECIT 89 (Singapore: World Scientific) pp 320-34
55. Simó C and Treschev D 1997 Evolution of the 'last' invariant curve in a family of area preserving maps: the case of the separatrix map Preprint
56. Tatjer J C 1989 On the strongly dissipative Hénon map ECIT 87 (Singapore: World Scientific) pp 331-7
57. Tatjer J C 1990 Invariant manifolds and bifurcations for one and two dimensional maps PhD Thesis Universitat de Barcelona
58. Tatjer J C 1992 Codimension two bifurcations near a cubic homoclinic tangency ECIT 91 (Singapore: World Scientific) pp 308-18
59. Tatjer J C and Simó C 1994 Basins of attraction near homoclinic tangencies Ergod. Theor. Dynam. Syst. 14 351-90
60. Viana M 1992 Saddle-node critical cycles and prevalence of strange attractors, course given at Groningen University
61. 1-conjugaison des difféomorphismes du cercle Lect. Notes Math. 1007 814-27
62. Zaslavsky G, Zakharov M, Sagdeev R, Usikov D and Chernikov A 1986 Stochastic web and diffusion of particles in a magnetic field Zh. Eksp. Teor. Fiz. 91 500-15