Interval dimension of strange attractors
I.V. Pentegov and V.N. Sidorets
E.O. Paton Institute of Electric Welding, Kiev, Ukraine
Existing kinds of dimension, such as the Hausdorf dimension, the correlational and informational dimensions, the Liapunov dimension and a number of others are not always convenient in the analysis of dot sets and are labour-consuming in calculation, which has resulted in extremely rare use of the dimension for identification of chaotic processes. In paper [1] the interval dimension of dot sets, distinguished by simplicity of calculation and convenience for automated methods of computation was proposed.
The substantiation of interval dimension consists in following. We shall define an interval measure of the size of a dot set as the limit for the quantity:
where n - the number of points of the set, 
-
the least interval between points (in relative units) at given
n. In general case measure 
equals zero, infinite or
finite corresponding to the choice of the exponent d - dimension
of the measure, the numerical invariant depending on the space
metrics and characterizing it. The interval dimension 
of
a dot set - is the critical dimension, at which measure has
finite nonzero value. For smooth growth of d the measure
jumps from 0 up to 
in the close vicinity of
. By definition
>From this, having in mind that 
for
, we shall find
The interval dimension is the local characteristic of a dot
set. We shall note, that 
in the only case when
. If the dot set represents points of the
Poincare section of some attractor, this condition corresponds to
the onset of the periodic regime. Here lies the main difference
and advantage of the interval dimension compared to other
dimensions -- it signals at once the approach of periodicity.
This property combined with the simplicity of calculations makes
interval dimension irreplaceable for identification of chaos: in
case of chaos is nonintegral number.