Page 659 - Mathematical Techniques of Fractional Order Systems

630 Mathematical Techniques of Fractional Order Systems

As far as the topological interpretation of the graphic matrix of recur-
rences is concerned, Marwan et al. (2007) distinguish three levels of
analysis: first, homogeneity in the RP graphic representation implies a
homogeneous distribution of the recurring points in the matrix of recur-
rences; which characterizes the stationarity of the system, both in the
case of a uniformly distributed random phenomenon (white noise), and in
the case of a deterministic chaotic structure. On the other hand, the non-
stationarity of the system is at the changes in the distribution of recur-
rence points, visualized graphically by brighten areas. Second, the
periodicity of a system implies the presence of rhythmicity represented
by long parallel diagonal segments at the principal diagonal of the RP
matrix. These long diagonal segments are displayed by shadow areas
related to the occurrence of the recurring points, in a dense manner in
the RP matrix. Third, the drift is characterized by the distance of the
diagonal lines, with respect to the main diagonal. The more the diagonal
lines are distant from the main diagonal, the higher the drift is. On the
contrary, the closer they are to the main diagonal, the lower the drift is.
A weak drift of the recurrence matrix is synonymous with nonstationary
in a time series and vice versa.
As for the scaled topological interpretation of the recursive graph matrix,
four levels of analysis are distinguished: First, the isolated recurring points
imply a stochastic behavior of the system. Then, the small diagonal segments
(small segments parallel to the main diagonal) signify the presence of some
vectors which can be repeated several times during a period of observation.
In the context of attractor dynamics, this implies that the system revisits the
same region of the attractor at different times. Whereas the small segments
perpendicular to the principal diagonal signify the symmetry of the two trian-
gular halves of the matrix of the recurrences, with respect to the principal
diagonal, each half triangle reflects the image of the other. But, the small
horizontal or vertical segments mean the presence of a sequence of vectors
close to each other. Then, the checkered structure is induced by a grouping
of segments of parallel lines, at the main diagonal, in small regions of the
matrix of recurrences. In the context of attractor dynamics, this means that
the system revisits different surfaces of the attractor and switches from one
to the other, quite a few times, during a period of observation. Finally, the
white space band characterizes the nonrecurring points and indicates a
change in structure or a sudden change in the state of the system.
From these topological analyses associated to the matrix of recur-
rences, various statistical indexes are also introduced to transform this
graphical representation into statistical measurements. In this context, a
second statistical technique has developed for measuring complexity,
allowing the analysis of the recurrence quantification of the recurrence
matrix.