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Dimension densities for turbulent systems with spatially decaying correlation functions
The introduction of nonlinear, deterministic, and low-dimensional dynamical systems with chaotic solutions led to many conjectures about how these chaotic systems might be related to fluid turbulence. It appears that the time series, produced by a chaotic solution can be as complex as experimentally observed signals from turbulent hydrodynamics. Furthermore, certain transitions from laminar to turbulent flow have their analog in the transition from ordered to chaotic behavior of deterministic chaotic systems. A basic problem in this context is how far methods from nonlinear dynamical systems can be used to describe experimental turbulence. A frequent objection to the approach of using simple dynamical systems as models for turbulence is that these models might reproduce some temporal chaos but would not correspond to real turbulence, for which the spatial structure also is very irregular and chaotic. The dynamics of a turbulent flow are not expected to be spatially coherent and therefore cannot be described by a few global modes. In the following we use the simplest of coupled maps in order to test the applicability of some ideas which should make it possible to estimate the number of degrees of freedom per unit length of a system which is spatially incoherent. This is done by computing the dimension density of the lattice system through a series of two-point measurements at separated lattice points. Then we compare these results with the spatial decay of the correlation function and also of the mutual information content. We find a qualitative agreement with the expected dependence. For precise quantitative measurements the general problem of accuracy and data limitations appear to become dominant