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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 noninteger (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 nonempty boxes.
For , the sum reduces to the number of nonempty 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.
Next: The generalized entropies
Up: THE CHARACTERIZATION OF CHAOTIC
Previous: Dynamical systems and attractors
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