Pattern selection in extended periodically forced systems: A continuum coupled map approach

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 63, Issue 4, 2001, Pages

  Ott, Edward ^b  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITY OF MARYLAND   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; FAST FOURIER TRANSFORMS; GRANULAR MATERIALS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS;

CONTINUUM COUPLED MAPS (CCM); COUPLED MAP LATTICES (CML);

CONTINUUM MECHANICS;

EID: 0035305595     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.63.046202     Document Type: Article

References (65)

1. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
2. J. P. Gollub and J. S. Langer, Rev. Mod. Phys. 71, S396 (1999).
3. A. M. Turing, Philos. Trans. R. Soc. London, Ser. B 327, 37 (1952);
4. G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Chemical Systems (Wiley, New York, 1977);
5. K. J. Lee, W. D. McCormick, H. L. Swinney, and J. E. Pearson, Nature (London) 369, 215 (1994).
6. M. Faraday, Philos. Trans. R. Soc. London 121, 319 (1831);
7. B. Christiansen, P. Alstrom, and M. T. Levinson, Phys. Rev. Lett. 68, 2157 (1992);
8. A. Kudrolli and J. P. Gollub, Physica D 97, 133 (1996);
9. D. Binks and W. van de Water, Phys. Rev. Lett. 78, 4043 (1997).
10. H. Bénard, Ann. Chim. Phys. 23, 62 (1901);
11. Lord Rayleigh, Philos. Mag. 32, 529 (1916);
12. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, London, 1961);
13. J. B. Swift and P. C. Hohenberg, Phys. Rev. A 15, 319 (1977);
14. A. V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics (World Scientific, Singapore, 1998);
15. E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. 32, 709 (2000).
16. A. T. Winfree, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7, 487 (1997). This article also contains an extensive list of references.
17. T. Shinbrot and J. M. Ottino, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 955 (1995).
18. C. Diks, F. Takens, and J. DeGoede, Physica D 104, 269 (1997).
19. A. Lemaître and H. Chaté, Phys. Rev. Lett. 82, 1140 (1999);
20. A. LemaîtreH. ChatéJ. Stat. Phys. 96, 915 (1999).
21. F. Melo, P. B. Umbanhowar, and H. L. Swinney, Phys. Rev. Lett. 75, 3838 (1995).
22. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Nature (London) 382, 793 (1996).
23. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Physica A 249, 1 (1998).
24. For other experiments exhibiting phenomena in vertically vibrated granular layers, see T. Metcalf, J. B. Knight, and H. M. Jaeger, Physica A 236, 202 (1997);
25. F. Melo, P. B. Umbanhowar and H. L. Swinney, Phys. Rev. Lett. 72, 172 (1994);
26. H. K. Pak and R. P. Behringer, Phys. Rev. Lett. 71, 1832 (1993);
27. C. Larouche, S. Douady, and S. Fauve, J. Phys. (France) 50, 699 (1989).
28. T. Benjamin and F. Ursell, Proc. R. Soc. London, Ser. A 225, 505 (1954).
29. K. Kumar and L. Tuckerman, J. Fluid Mech. 279, 49 (1994);
30. K. Kumar, Proc. R. Soc. London, Ser. A 452, 1113 (1996);
31. E. Cerda and E. Tirapegui, Phys. Rev. Lett. 78, 859 (1997).
32. H. Müller, Phys. Rev. E 49, 1273 (1994);
33. W. Zhang and J. Viñals, J. Fluid Mech. 336, 301 (1997);
34. P. Chen and J. Viñals, Phys. Rev. E 60, 559 (1999).
35. H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod. Phys. 68, 1259 (1996).
36. L. P. Kadanoff, Rev. Mod. Phys. 71, 435 (1999).
37. C. Bizon, Phys. Rev. Lett. 80, 57 (1998);
38. C. BizonJ. Stat. Phys. 93, 449 (1998).
39. C. K. K. Lun, S. B. Savage, D. J. Jeffery, and N. Chepurniy, J. Fluid Mech. 140, 223 (1983);
40. J. T. Jenkins and M. W. Richman, Arch. Ration. Mech. Anal. 87, 355 (1985);
41. J. T. Jenkins and M. W. Richman, Phys. Fluids 28, 3485 (1985).
42. J. R. de Bruyn, C. Bizon, M. D. Shattuck, D. Goldman, J. B. Swift, and H. L. Swinney, Phys. Rev. Lett. 81, 1421 (1998).
43. C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Phys. Rev. E 60, 4340 (1999);
44. M. D. Shattuck, C. Bizon, J. B. Swift, and H. L. Swinney, Physica A 274, 158 (1999);
45. C. Bizon, M. D. Shattuck, and J. B. Swift, Phys. Rev. E 60, 7210 (1999).
46. L. S. Tsimring and I. S. Aronson, Phys. Rev. Lett. 79, 213 (1997).
47. T. Shinbrot, Nature (London) 389, 574 (1997).
48. E. Cerda, F. Melo, and S. Rica, Phys. Rev. Lett. 79, 4570 (1997).
49. D. Rothman, Phys. Rev. E 57, R1239 (1998).
50. J. Eggers and H. Riecke, Phys. Rev. E 59, 4476 (1999).
51. C. Crawford and H. Riecke, Physica D 129, 83 (1999).
52. S. C. Venkataramani and E. Ott, Phys. Rev. Lett. 80, 3495 (1998).
53. A. Mehta and J. M. Luck, Phys. Rev. Lett. 65, 393 (1990).
54. R. Lifshitz and D. M. Petrich, Phys. Rev. Lett. 79, 1261 (1997). The authors modified the Swift-Hohenberg equation by introducing a second length scale, and this led to patterns besides stripes.
55. M. Golubitsky, I. N. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory (Springer-Verlag, New York, 1988), Vol. II.
56. S. C. Venkataramani and E. Ott (unpublished).
57. E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991); R. E. O’Malley Jr., Introduction to Singular Perturbations (Academic Press, New York, 1974).
58. We do not have to restrict (Formula presented) to be time periodic. In particular, we can let (Formula presented) be a chaotic sequence or a random sequence. In certain parameter regimes varying (Formula presented) irregularly gives states that intermittently switch between stripe patterns and hexagon patterns [Markus Löcher (private communication)].
59. Since the time index n is a discrete variable, it is incorrect to say that (Formula presented) continuously. However, (Formula presented) varies on a scale (Formula presented) and by appropriate coarse graining on a scale T we can define a function (Formula presented) for a continuous variable (Formula presented) such that (Formula presented) By the statement (Formula presented) in a continuous fashion, we mean that the coarse graining yields a continuous, slowly varying, function (Formula presented) such that (Formula presented) as (Formula presented)
60. See, e.g., K. Kaneko, Chaos 2, 279 (1992). This issue of Chaos focuses on the subject of coupled map lattices.
61. Another in between case is provided by coupled ordinary differential equations on a spatial lattice; see, e.g., D. K. Umberger, C. Grebogi, E. Ott, and B. Afeyan, Phys. Rev. A 39, 4835 (1989).
62. In our numerical solutions of the model [Eqs. (12345678)], a spatial grid is used, but the spacing between the grid points is small compared to the characteristic scale (Formula presented)
63. O. Lioubashevski, H. Arbell, and J. Fineberg, Phys. Rev. Lett. 76, 3959 (1996);
64. J. Fineberg and O. Lioubashevski, Physica A 249, 10 (1998);
65. O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, and J. Fineberg, Phys. Rev. Lett. 83, 3190 (1999).