Abstract
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.
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References
See Physica 7D (1983).
e.g. M. Cross, Phys. Rev. A 25, 1065 (1982).
e.g. M. Meinhardt, “Models of Biological Pattern Formation” (Academic Press 1982 ).
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.
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)).
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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.
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.
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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.
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Bishop, A. (1985). Pattern Selection and Low-Dimensional Chaos in Systems of Coupled Nonlinear Oscillators. In: Takeno, S. (eds) Dynamical Problems in Soliton Systems. Springer Series in Synergetics, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02449-2_36
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