The Lorenz-Fermi-Pasta-Ulam experiment

Physica D: Nonlinear Phenomena

Volumn 138, Issue 1-2, 2000, Pages 1-43

  Balmforth, N J ^a   Pasquero, C ^a   Provenzale, A ^a  

^a CNR   (Italy)

Author keywords

Coherent behaviour; Coupled dynamical systems; Lorenz system; Synchronization

Indexed keywords

EID: 0347316222     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(99)00193-1     Document Type: Article

References (60)

1. E. Fermi, J.R. Pasta, S. Ulam, Studies of nonlinear problems, Los Alamos Report LA-1940, 1955.
2. N.J. Zabusky, M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965) 240.
3. E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130.
4. E.A. Spiegel, in: W. Malkus, F.J. Mellor (Eds.), Geophysical Fluid Dynamics, vol. WHOI-81-102, Woods Hole Oceanographic Institution, 1981, pp. 1-77.
5. E.N. Lorenz, Dimension of weather and climate attractors, Nature 353 (1991) 241.
6. E.N. Lorenz, in: Proceedings of the Seminar on Predictability, vol. 1, ECMWF, Reading, Berkshire, UK, 1996.
7. E.N. Lorenz, K.A. Emanuel, Optimal sites for supplementary weather observations: simulation with a small model, J. Amtos. Sci. 55 (1998) 399.
8. J. von Hardenberg, A. Provenzale, in: G.C. Castagnoli, A. Provenzale (Eds.), Proceedings of the International School of Physics "Enrico Fermi", vol. 133, IOS Press, Amsterdam, 1997, pp.161-176.
9. R. Burridge, L. Knopoff, Model and theoretical seismology, Bull. Seismol. Soc. Am. 57 (1967) 341.
10. J.M. Carlson, J.S. Langer, Mechanical model of an earthquake fault, Phys. Rev. A 40 (1989) 6470.
11. F. Helot, T. Baumberger, B. Perrin, B. Caroli, C. Caroli, Creep, stick-slip and dry-friction dynamics: experiments and a heuristic model, Phys. Rev. E 49 (1994) 4973.
12. B.N.J. Persson, Theory of friction: stress domains, relaxation and creep, Phys. Rev. B 51 (1995) 13568.
13. R.C. Elson, A.I. Selverston, R. Huerta, N.F. Rulkov, M.I. Rabinovich, H.D.I. Abarbanel, Synchronous behaviour of two coupled biological neurons, Phys. Rev. Lett. 81 (1998) 5692.
14. H. Sompolinksy, D. Golomb, D. Kleinfel, Cooperative dynamics in visual processing, Phys. Rev. A 43 (1991) 6990.
15. A.T. Winfree, The Geometry of Biological Time, Springer, Berlin, 1980.
16. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, New York, 1984.
17. A. Turing, Philos. Trans. R. Soc. London B 237 (1952) 37.
18. L. Brunnet, H. Chaté, P. Manneville, Long-range order with local chaos in lattices of diffusively coupled ODEs, Physica D 78 (1994) 141.
19. J.F. Heagy, T.L. Carroll, L.M. Pecora, Synchronous chaos in coupled oscillator systems, Phys. Rev. E 50 (1994) 1874.
20. J.F. Heagy, T.L. Carroll, L.M. Pecora, Short-wavelength bifurcations and size instabilities in coupled oscillator systems, Phys. Rev. Lett. 74 (1995) 4185.
21. G.V. Osipov, A.S. Pikovsky, M.G. Rosenbaum, J. Kurths, Phase synchronization effects in a lattice of nonidential Rössler oscillators, Phys. Rev. E 55 (1997) 2353.
22. L.M. Pecora, T.L. Carroll, G.A. Johnson, D.J. Mar, J.F. Heagy, Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos 7 (1997) 520.
23. A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization in a population of globally coupled chaotic oscillators, Europhys. Lett. 34 (1996) 165.
24. D.H. Zanette, A.S. Mikhailov, Condensation in globally coupled populations of chaotic dynamical systems, Phys. Rev. E 57 (1998) 276.
25. X.J. Wang, J. Rinzel, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neural Comput. 4 (1992) 84.
26. X.J. Wang, J. Rinzel, Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons, Neuroscience 53 (1993) 899.
27. D. Golomb, J. Rinzel, Dynamics of globally coupled inhibitory neurons with heterogeneity, Phys. Rev. E 48 (1993) 4810.
28. D. Golomb, J. Rinzel, Clustering in globally coupled inhibitory neurons, Physica D 72 (1994) 259.
29. R. Huerta, M. Bazhenov, M.I. Rabinovich, Clusters of synchronization and bistability in lattices of chaotic neurons, Europhys. Lett. 43 (1998) 719.
30. K. Kaneko, Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements, Physica D 41 (1990) 137.
31. K. Kaneko, Partition complexity in a network of chaotic elements, J. Phys. A 24 (1991) 2107.
32. N.J. Balmforth, A. Jacobson, A. Provenzale, Synchronized family dynamics in globally coupled maps, Chaos 9 (1999) 738.
33. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators, Phys. Rev. Lett. 76 (1996) 1804.
34. Y.-H. Qian, D.-H. Feng, Diffusive Lorenz dynamics: coherent structures and spatio-temporal chaos, Preprint, 1998.
35. P. Coullet, C. Tresser, C.-W. Wu, The diffusive Lorenz equations, Preprint, 1998.
36. P. Ashwin, J. Buescu, I. Stewart, From attractor to chaotic saddle: a tale of transverse stability, Nonlinearity 9 (1994) 703.
37. H. Fujisaka, T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Prog. Theoret. Phys. 69 (1983) 32.
38. V.S. Afraimovich, W.-W. Lin, Synchronization in lattices of coupled oscillators with Neumann/periodic boundary conditions, Dyn. Stability Syst. 13 (1998) 237.
39. M. Peyrard, M.D. Kruskal, Kink dynamics in the highly discrete sine-Gordon system, Physica D 14 (1984) 88.
40. L.P. Shil'nikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl. 6 (1965) 163.
41. N.J. Balmforth, Solitary waves and homoclinic orbits, Ann. Rev. Fluid Mech. 27 (1993) 335.
42. A.R. Champneys, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Physica D 112 (1998) 158.
43. C. Elphick, G.R. Ierley, O. Regev, E.A. Spiegel, Interacting localized structures with Galilean invariance, Phys. Rev. A 44 (1991) 1110.
44. N.J. Balmforth, G.R. Ierley, R. Worthing, Pulse dynamics in an unstable medium, SIAM J. Appl. Math. 57 (1997) 205.
45. R.L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc. 218 (1976) 89.
46. A.R. Champneys, J.F. Toland, Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems, Nonlinearity 6 (1993) 655.
47. J. Härterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Physica D 112 (1998) 187.
48. K. Weisenfeld, P. Hadley, Attractor crowding in oscillator arrays, Phys. Rev. Lett. 62 (1989) 1335.
49. D. Golomb, D. Hansel, B. Shraiman, H. Sompolinsky, Clustering in globally coupled phase oscillators, Phys. Rev. A 45 (1992) 3516.
50. N.J. Balmforth, E.A. Spiegel, C. Tresser, Checkerboard maps, Chaos 5 (1995) 216.
51. N. Platt, E.A. Spiegel, C. Tresser, On-off intermittency: a mechanism for bursting, Phys. Rev. Lett. 70 (1993) 279.
52. H. Kantz, T. Shreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, 1997.
53. N.J. Balmforth, A. Provenzale, E.A. Spiegel, M. Martens, C. Tresser, C.-W. Wu, Red spectra from white and blue noise, Proc. Roy. Soc. London B 266 (1999) 311.
54. A. White, M. Begon, R.G. Bowers, Explaining the colour of power spectra in chaotic ecological models, Proc. R. Soc. London B 263 (1996) 1731.
55. T. Bohr, M.H. Jensen, G. Paladin, A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, Cambridge, 1998.
56. F. Takens, in: D.A. Rand, L.S. Young (Eds.), Lecture Notes in Mathematics, vol. 898, Springer, New York, 1981, p. 366.
57. P. Cessi, W.R. Young, Some unexpected consequences of the interaction between convective adjustment and horizontal diffusion, Physica D 98 (1996) 287.
58. H.J. Xu, L. Knopoff, Periodicity and chaos in a one-dimensional dynamical model of earthquakes, Phys. Rev. E 50 (1994) 3577.
59. S.C. Venkataramani, B.R. Hunt, E. Ott, D.J. Gauthier, J.C. Bienfang, Transitions to bubbling of chaotic systems, Phys. Rev. Lett. 77 (1996) 5361.
60. S.C. Venkataramani, B.R. Hunt, E. Ott, Bubbling transition, Phys. Rev. E 54 (1996) 1346.