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Remarkable Statistical Behavior for Truncated Burgers-Hopf Dynamics
Andrew J. Majda and Ilya Timofeyev
Proceedings of the National Academy of Sciences of the United States of America
Vol. 97, No. 23 (Nov. 7, 2000), pp. 12413-12417
Published by: National Academy of Sciences
Stable URL: http://www.jstor.org/stable/123862
Page Count: 5
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A simplified one-dimensional model system is introduced and studied here that exhibits intrinsic chaos with many degrees of freedom as well as increased predictability and slower decay of correlations for the large-scale features of the system. These are important features in common with vastly more complex problems involving climate modeling or molecular biological systems. This model is a suitable approximation of the Burgers-Hopf equation involving Galerkin projection on Fourier modes. The model has a detailed mathematical structure that leads to a well-defined equilibrium statistical theory as well as a simple scaling theory for correlations. The numerical evidence presented here strongly supports the behavior predicted from these statistical theories. Unlike the celebrated dissipative and dispersive approximations of the Burgers-Hopf equation, which exhibit exactly solvable and/or completely integrable behavior, these model approximations have strong intrinsic chaos with ergodic behavior.
Proceedings of the National Academy of Sciences of the United States of America © 2000 National Academy of Sciences