Remarkable statistical behavior for truncated Burgers-Hopf dynamics

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.