Page 666 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 637

If the dimension is greater than or equal to 2 and integer, the system is
quasi-periodic.
If the dimension is not an integer (and greater than 2), the system is
chaotic.

The attractors associated with chaotic behaviors are called strange attrac-
tors and they have a fractal structure if their individual dimension isn’t a nat-
ural number. The fractal geometry is a priori notion completely independent
of the theory of chaos (due to the fractal character of chaotic attractors) that
was developed by Mandelbrot (1982). The starting idea of Mandelbrot is that
in nature many objects are self-similar, i.e., a small part resembles to the
whole and it is replicated. It is a phenomenon of morphology of endogenous
structures by memorization using the relatively homogeneous and synchro-
nized invariance recomposition. Then, it will be possible to account for the
invariance measures of dynamic structures by using the topological synchro-
nization of fractional order chaotic systems thanks to this theory of self-
similar fractal geometry. The basic assumption implies that the knowledge of
attractors and their previous history characteristics determines the dynamic
systems in phase space. Thus, the characterization of the topological proper-
ties of the chaotic attractor consists in calculating its invariant measures,
such as important fractal dimensions and the maximum Lyapunov exponent.

21.4.1 The Hausdorff Besicovitch Dimension
According to Vladimirsky and Ismailov (2015a) the Hausdorff Besicovitch
dimension of fractional order chaotic system is defined as:

Definition: For any compact metric space X,

2 log N ε; d ðXÞ
d H ðXÞ 5 inf lim ð21:16Þ
ε-0 logðεÞ
where ε is the sphere of radius ε, d is a metric of X, (X, d f ) is a compact
fractional metric space with dimension d f ,N ε; d ðXÞ 5 min U jj and U is a finite
open covering of X with mesh less than ε.

21.4.2 The Correlation Dimension

This measure is based on the idea that a chaotic process with dimension m
does not fill a space of dimension m 1 1. According to Kantz and Schreiber
(1997) such indicator is defined as:

Definition: For two-dimensional square matrices with order N and any com-
pact metric space X,