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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]
\begin{displaymath}
D_{q} = \lim_{r \rightarrow 0} \frac{1}{(q-1)\ln(r)}
\ln \sum_{i=1}^{N} p_{i}^{q}
\end{displaymath} (2.3)

where the $d$-dimensional phase space has been partitioned into boxes of size $r^{d}$; $p_{i}$ is the probability that the trajectory $\vec{X}$ visits box $i$ and $N$ is the number of non-empty boxes. For $q=0$, the sum reduces to the number of non-empty boxes. The $D_{0}$ is called the topological or fractal dimension. For $q \rightarrow 1$, the $D_1$ is the ratio between the Shannon information $\sum p_i \ln p_i$ and $\ln r$; therefore it is called the ``information'' dimension. The $D_{2}$ is called ``correlation'' dimension [Grassberger and Procaccia, 1983b]. The function $D_q$ is monotonically decreasing with $q$ and gives information about the inhomogeneity of the attractor. For a homogeneous attractor, $D_q$ is constant.


next up previous contents
Next: The generalized entropies Up: THE CHARACTERIZATION OF CHAOTIC Previous: Dynamical systems and attractors   Contents
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