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The generalized dimensions

The dimension of an attractor is an important piece of knowledge of a system, because it is a measure of the information necessary to specify a point on the attractor with a given accuracy. Moreover, the dimension is a lower bound on the number of variables needed to model the system. A strange attractor typically has a non-integer (fractal) dimension. There are many definitions of dimension. For example, the Renyi dimensions are defined as [Renyi, 1970]
 (2.3)

where the -dimensional phase space has been partitioned into boxes of size ; is the probability that the trajectory visits box and is the number of non-empty boxes. For , the sum reduces to the number of non-empty boxes. The is called the topological or fractal dimension. For , the is the ratio between the Shannon information and ; therefore it is called the information'' dimension. The is called correlation'' dimension [Grassberger and Procaccia, 1983b]. The function is monotonically decreasing with and gives information about the inhomogeneity of the attractor. For a homogeneous attractor, is constant.

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