Pattern Selection and Low-Dimensional Chaos in Systems of Coupled Nonlinear Oscillators

  • Alan Bishop
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 30)


The longtime behavior of a number of one- and two-dimensional driven, dissipative, dispersive, many-degree-of-freedom systems is studied. It is shown numerically that the attractors are characterized by strong mode-locking into a small number of (nonlinear) modes. On the basis of the observed profiles, estimates of chaotic attractor dimensions, and projections into nonlinear mode bases, it is argued that the same few modes may (in these extended systems) give a unified picture of spatial pattern selection, low-dimensional chaos, and coexisting coherence and chaos. Analytic approaches to this class of problem are summarized.


Coherent Structure Chaotic Regime Extended Mode Mode Reduction Determine Mode 
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  1. 1.
    See Physica 7D (1983).Google Scholar
  2. 2.
    e.g. M. Cross, Phys. Rev. A 25, 1065 (1982).CrossRefGoogle Scholar
  3. 3.
    e.g. M. Meinhardt, “Models of Biological Pattern Formation” (Academic Press 1982 ).Google Scholar
  4. 4.
    Examples of coesisting coherence and chaos include: clumps and cavitons in turbulent plasmas; filamentation in lasing mediums; large scale structures in turbulent fluids (e.g. modon “blocking” patterns controlling atmospheric flow and weather, or gulf stream “rings” in oceanography); and perhaps even the red spot of Jupiter! In all cases the coherent structures are long-lived and with slower dynamics than the single-particle turbulence - a unifying practical concern is their effect on transport and predictability (in space and time). There are also increasing numbers of controlled laboratory scale observations (e.g. in convection cells, water wave surface solitons) as well as probable applications in biological contexts.Google Scholar
  5. 5.
    In some cases rigorous bounds on the number of determining modes have recently been established (e.g. C. Foias, et al., Phys. Rev. Lett. 50, 1031 (1983)), and it has been possible to bound the attractor (fractal) dimension by the number of determining modes (e.g. O. Manley, et al., preprint (1984), B. Nicolaenko and B. Scheurer, preprint (1984)). In certain cases (e.g. for some reaction-diffusions problems) even a truncated set of linear modes can be accurate. (See J. C. Eilbeck, J. Math. Biol. 16, 233 (1983), B ctN olaenko et al., Proc. Acad. Sc. Paris 298, 23 (1984)).Google Scholar
  6. 6.
    A. R. Bishop, E. Domany and P. S. Lomdahl (unpublished results).Google Scholar
  7. 7.
    e.g., R. K. Dodd, et al. “Solitons and Nonlinear Wave Equations” (Academic Press 1982 ).Google Scholar
  8. 8.
    D. Bennett, A. R. Bishop, S. E. Trullinger, Z. Physik B 47, 265 (1982);CrossRefGoogle Scholar
  9. A. R. Bishop, et al., Phys. Rev. Lett. 50, 1095 (1983);CrossRefGoogle Scholar
  10. A. R. Bishop, et al., Physica 7D, 259 (1983).Google Scholar
  11. 9.
    E. A. Overman, D. W. McLaughlin, A. R. Bishop, preprint (1984)Google Scholar
  12. 10.
    N. Ercolani, et al., preprint (1984).Google Scholar
  13. 11.
    See, e.g., M. Büttiker and R. Landauer, in “Physics in One Dimension”, eds. J. Bernasconi and T. Schneider (Springer 1981 ).Google Scholar
  14. 12.
    See, e.g. J. D. Farmer et al., Physica 7D, 153 (1983).Google Scholar
  15. 13.
    P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983). The calculation of various attractor “dimensions” remains in an early state of development (see Ref. 12). In particular, important questions remain such as sensitivity to the scale of structures and the spatial patterns (similar questions apply to Liapunov exponents - see refs. 21, 22). We emphasize that our error estimates quoted here are realistically conservative. Typically we used an embedding dimension of 5–10 and 80,000 data points.Google Scholar
  16. 14.
    Assuming only 2 breathers (or 4 kinks) as a truncated modal set, the maximum dimension of the space containing the attractor is 8. Our initial data symmetry reduces this to 4. The presence of dissipation will typically further reduce the “active” dimension. Our estimates of y are generally in the range 2–2.5. This is entirely reasonalbe in view of our estimates (unpublished) of y for a chaotic single particle with similar damping and driving strengths: there the maximum dimension is 2 but we generally find v = 1. 1–1. 3.Google Scholar
  17. 15.
    A. R. Bishop, J. C. Eilbeck, G. Wysin, APS March Meeting Bulletin (1984), and preprint; See also M. P. Soerensen, et al., Phys. Rev. Lett. 51, 1919 (1983).Google Scholar
  18. 16.
    P. S. Lomdahl, et al., Phys. Rev. B 25, 5737 (1982).CrossRefGoogle Scholar
  19. 17.
    J. C. Eilbeck, P. S. Lomdahl, A. C. Newell, Phys. Lett. 87A, 1 (1981).CrossRefGoogle Scholar
  20. 18.
    G. Wysin and A. R. Bishop, APS March Meeting Bulletin (1984), and preprint.Google Scholar
  21. 19.
    O. H. Olsen, P. S. Lomdahl, A. R. Bishop, J. C. Eilbeck, preprints (1984).Google Scholar
  22. 20.
    D. W. McLaughlin, J. V. Moloney, A. C. Newell, Phys. Rev. Lett. 51, 75 (1983); and preprint (1984).Google Scholar
  23. 21.
    M. Imada, J. Phys. Soc. Jpn. 52, 1946 (1983).CrossRefGoogle Scholar
  24. 22.
    G. D. Doolen, et al., Phys. Rev. Lett. 51, 335 (1983).CrossRefGoogle Scholar
  25. 23.
    Of course there are many different mechanisms for pattern selection and self-organization depending on the context - see, e.g., “Fronts, Interfaces and Patterns,” eds. A. R. Bishop, L. J. Campbell, P. J. Channell, Physica D (1984). In particular, our cases should be contrasted with those where diffusion or reaction - diffusion dominates, even though physical questions (such as mode reduction) can be quite similar.Google Scholar
  26. 24.
    Some degree of hysteresis and coexistence of attractors persists in these manydegree-of-freedom systems, although it is typically much less pronounced than for a single oscillator.Google Scholar
  27. 25.
    N. Ercolani and D. W. McLaughlin, unpublished.Google Scholar
  28. 26.
    S. E. Trullinger et al., unpublished.Google Scholar
  29. 27.
    D. J. Kaup and A. C. Newell, Proc. Roy. Soc. (London) A361, 413 (1978).CrossRefGoogle Scholar
  30. 28.
    See, e.g., P. J. Holmes and J. E. Marsden, Archive Rational Mechanics and Analysis, 76, 135 (1981).Google Scholar
  31. 29.
    See, e.g., E. Domany, Phys. Rev. Lett. 52, 871 (1984).CrossRefGoogle Scholar
  32. 30.
    A. R. Bishop, and E. Domany, unpublished.Google Scholar
  33. 31.
    See, e.g., P. Bak, Rep. Prog. Phys. 45, 587 (1982).CrossRefGoogle Scholar
  34. 32.
    J. Oitmaa and A. R. Bishop, preprint (1984). The effect of a discrete lattice is to produce many metastable states becuase of the Peierls-Nabarro pinning forces. The situation is similar to that of large scale dynamics in a discrete discommensurate model (c.f. Refs. 1,31), and all combinations of “order” and “chaos” in both space and time and possible.Google Scholar
  35. 33.
    e.g. N. Bekki and K. Nozaki, these proceedings, and references therein.Google Scholar
  36. 34.
    e.g. H. T. Moon et al., Physica 7D, 135 (1983);Google Scholar
  37. K. Nozaki and N. Bekki, Phys. Rev. Lett. 51, 2171 (1983).CrossRefGoogle Scholar
  38. 35.
    K. Fesser, et al., preprint (1984); B. Paulus, et al., Phys. Lett. 102A, 89 (1984), and preprint.Google Scholar
  39. 36.
    K. Kaneko, these proceedings.Google Scholar
  40. 37.
    See, e.g., “Cellular Automata,” eds. J. D. Farmer, T. Toffoli, S. Wolfram (North-Holland Amsterdam 1984 ).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Alan Bishop
    • 1
  1. 1.Los Alamos National Lab.Center for Nonlinear Studies and Theoretical DivisionLos AlamosUSA

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