Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing

Nonlinearity

Volumn 15, Issue 4, 2002, Pages 1205-1267

  Broer, Henk ^a   Simó, Carles ^b   Vitolo, Renato ^a  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b UNIVERSITY OF BARCELONA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042384678     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/15/4/312     Document Type: Article

References (70)

1. Anishchenko V S 1995 Dynamical Chaos - Models and Experiments (Singapore: World Scientific)
2. Anishchenko V S, Safonova M A, Feudel U and Kurths J 1994 Bifurcations and transition to chaos through three-dimensional tori Int. J. Bifur. Chaos 4 595-607
3. Arnol'd V I 1965 Small denominators, I: Mappings of the circumference into itself AMS Transl. Ser. 2 46 213-84
4. Arnol'd V I 1980 Mathematical Methods of Classical Mechanics (Berlin: Springer)
5. Arnol'd V I 1983 Geometrical Methods in the Theory of Ordinary Differential Equations (Berlin: Springer)
6. Benedicks M and Carleson L 1991 The dynamics of the Hénon map Ann. Math. 133 73-169
7. Benettin G, Galgani L, Giorgilli A and Strelcyn J-M 1980 Lyapunov characteristic exponents for smooth dynamical systems: a method for computing all of them Meccanica 15 9
8. Bosch M and Simó C 1993 Attractors in a Shil'nikov-Hopf scenario and arelatedone-dimensional map Physica D 62 217-29
9. Briggs W L and Henson V E The DFT: An Owner's Manual for the Discrete Fourier Transform (Philadelphia: SIAM)
10. Broer H W, Huitema G B and Sevryuk M B 1996 Quasi-periodic motions in families of dynamical systems Springer LNM 1645
11. Broer H W, Huitema G B, Takens F and Braaksma B L J 1990 Unfoldings and bifurcations of quasi-periodic tori Mem. AMS 83 421
12. Broer H W, Osinga H M and Vegter G 1997 Algorithms for computing normally hyperbolic invariant manifolds Z. Angew. Math. Phys. 48 480-524
13. Broer H W and Roussarie R 2001 Exponential confinement of chaos in the bifurcation sets of real analytic diffeomorphisms Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday ed H W Broer, B Krauskopf and G Vegter (Bristol: Institute of Physics Publishing)
14. Broer H W, Roussarie R and Simó C 1996 Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms ETDS 16 1147-72
15. Broer H W and Simó C 1998 Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena Bol. Soc. Bras. Mat. 29 253-93
16. Broer H W, Simó C and Tatjer J C 1998 Towards global models near homoclinic tangendes of dissipative diffeomorphisms Nonlinearity 11 667-770
17. Broer H W and Takens F 1993 Mixed spectrum and rotational symmetry Archive Rat. Mech. An. 124 13-42
18. Broer H W and Takens F 1989 Formally symmetric normal forms and genericity Dynamics Reported 2 36-60
19. Broer H W and Vegter G 1984 Subordinate Shil'nikov bifurcations near some singularities of vector fields having low codimension ETDS 4 509-25
20. Broer H W and Vegter G 1992 Bifurcational aspects of parametric resonance Dynamics Reported 11-53
21. Carcassès J P, Mira C. Bosch M, Simó C and Tatjer J C 1991 'Crossroad area-spring area' transition (II) Parameter plane representation Int. J. Bifur. Chaos Appl. Sci. Eng. 1 183-96
22. Castellà E and Jorba À 2000 On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem Celestial Mechanics 76 35-54
23. Chow S N and Hale J K 1982 Methods of Bifurcation Theory (Berlin: Springer)
24. Cincotta P M and Simó C 2000 Simple tools to study global dynamics in non-axisymmetric galactic potentials-I. Astron. Astrophys. Supp. 147 205-28
25. Devaney R L 1989 An Introduction to Chaotic Dynamical Systems 2nd edn (Reading, MA: Addison-Wesley)
26. Díaz L, Rocha J and Viana M 1996 Strange attractors in saddle cycles: prevalence and globality Inv. Math. 125 37-74
27. Eckmann J-P and Ruelle D 1985 Ergodic theory of chaos and strange attractors Rev. Mod. Phys. 57 617-55
28. Feigenbaum M 1978 Qualitative universality for a class of nonlinear transformations J. Stat. Phys. 21 25-52
29. Flatto M and Levinson N 1955 Periodic solutions of singularly perturbed systems J. Rat. Mech. Anal. 4 943-50
30. Gomez G, Mondelo J M and Simó C 2001 Refined Fourier analysis procedures UB Preprint
31. Gonchenko S V, Shil'nikov L P and Turaev D V 1996 Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits Chaos 6 15-31
32. Grassberger P and Procaccia I 1983 Measuring the strangeness of strange attractors Physica D 9 189-208
33. Grebogi C, Ott E and Yorke J 1982 Chaotic attractors in crisis Phys. Rev. Lett. 48 1507-10
34. Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer)
35. Hirsch M W, Pugh C C and Shub M 1977 Invariant manifolds Springer LNM 583
36. Homan K 1998 Routes to chaos in the Lorenz-84 atmospheric model Masters Thesis Department of Mathematics, University of Groningen
37. Jorba À and Zou M 2001 On the numerical integration of ODE by means of high-order Taylor methods Preprint
38. Kaplan J and Yorke J 1979 Chaotic behaviour of multidimensional difference equations Functional Differential Equations and Approximations of Fixed Points, Springer LNM 204-27
39. Kock R 1998 Routes to chaos in the periodically driven Lorenz-84 system Masters Thesis Department of mathematics, University of Groningen
40. Kuznetsov Y 1995 Elements of Applied Bifurcation Theory (New York: Springer)
41. Lorenz E N 1984 Irregularity: a fundamental property of the atmosphere Tellus 36A 98-110
42. Lorenz E N 1990 Can chaos and intransitivity lead to interannual variability? Tellus 42A 378-89
43. Masoller C, Sicardi Schifino A C and Romanelli L 1992 Regular and chaotic behaviour in the new Lorenz system Phys. Lett. A 167 185-90
44. Mora L and Viana M 1993 Abundance of strange attractors Acta Math. 171 1-71
45. Newhouse S 1974 Diffeomorphisms with infinitely many sinks Topology 13 9-18
46. m, m ≥ 3 Commun. Math. Phys. 64 35-40
47. Östlund S, Rand D, Sethna J and Siggia E 1983 Universal properties of the transition from quasiperiodicity to chaos in dissipative systems Physica D 8 303-42
48. Palis J and Takens F 1993 Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations Cambridge Studies in Advanced Mathematics vol 35 (Cambridge: Cambridge University Press)
49. Palis J and Viana M 1994 High dimension diffeomorphisms displaying infinitely many sinks Ann. Math. 140 91-136
50. Pomeau Y and Manneville P 1980 Intermittent transition to turbulence in dissipative dynamical systems Commun. Math. Phys. 74 189-97
51. Press W, Teukolsky S, Vetterling W and Flannery B 1992 Numerical Recipies in C: the Art of Scientific Computing 2nd edn (New York: Cambridge University Press)
52. Ruelle D 1988 Elements of Differentiate Dynamics and Bifurcation Theory (New York: Academic)
53. Ruelle D and Takens F 1971 On the nature of turbulence Comm. Math. Phys. 20 167-92
54. Seydel R 1994 Practical Bifurcation and Stability Analysis (New York: Springer)
55. Shil'nikov A, Nicolis G and Nicolis C 1995 Bifurcation and predictability analysis of a low-order atmospheric circulation model Int. J. Bifur. Chaos 5 1701-11
56. Simó C 1979 On the Hénon-Pomeau attractor J. Stat. Phys. 21465-94
57. Simó C 1990 On the analytical and numerical computation of invariant manifolds Modern Methods in Celestial Mechanics ed D Benest and C Froeschlé (Paris: Frontières) pp 285-330
58. Simó C 1996 Effective Computations in Hamiltonian Dynamics Cent ans après les Méthodes Nouvelles de H. Poincaré (Frarvi Société Mathématique de) pp 1-23
59. Simó C, Broer H and Roussarie R 1991 A numerical exploration of the Takens-Bogdanov bifurcation for diffeomorphisms Eur. Conf. Iteration Theory (ECIT 89) (Batschuns, Austria) ed C Mira, N Netzer, C Simó and G Targonski (Singapore: World Scientific) pp 320-34
60. Simó C and Tatjer J C 1994 Basins of attraction near homoclinic tangencies ETDS 14 351-90
61. Swinney H L and Gollub J P 1986 Characterization of hydrodynamic strange attractors Physica D 18 448-54
62. Tatjer J C 2001 Three dimensional dissipative diffeoraorphisms with homoclinic tangencies ETDS 21 249-302
63. van Veen L, Opsteegh T and Verhulst F The dynamics of a low order coupled ocean-atmosphere model J. Nonlin. Sci. submitted; van Veen L, Opsteegh T and Verhulst F Preprint arXiv:chao-dyn/9812024.
64. van Veen L, Opsteegh T and Verhulst F The dynamics of a low order coupled ocean-atmosphere model J. Nonlin. Sci. submitted; van Veen L, Opsteegh T and Verhulst F Preprint arXiv:chao-dyn/9812024
65. Viana M 1993 Strange attractors in higher dimensions Bull. Brat. Math. Soc. 24 13-62
66. Viana M 1997 Multidimensional nonhyperbolic attractors Inst. Hautes Études Sci. Publ. Math. 85 63-96
67. Viana M 1997 Stochastic dynamics or deterministic systems Braz. Math. Colloquium (IMPA); Viana M 1997 Preprint http://www.impa.br/̃viana/
68. Viana M 1997 Stochastic dynamics or deterministic systems Braz. Math. Colloquium (IMPA); Viana M 1997 Preprint http://www.impa.br/̃viana/
69. Wieckzorek S, Krauskopf B and Lenstra D 2002 Mechanisms for multistability in a semiconductor laser with optical injection Opt. Commun. 183 215-26
70. Yorke J and Alligood K 1983 Cascades of period-doubling bifurcations: a prerequisite for horseshoes Bull AMS 9319-22