Page 658 - Mathematical Techniques of Fractional Order Systems

Fractional Order Chaotic Systems Chapter | 21 629

the matrix elements correspond to those times at which a state of a dynam-
ical system recurs (columns and rows correspond then to a certain pair of
times). Techniqually, the RP reveals all the times when the phase space tra-
jectory of the dynamical system visits roughly the same area in the phase
space.

The Recurrence Plot concept has been introduced by (Eckmann et al.,
1987) in order to quantify the recurrence properties of chaotic dynamics
systems. The aim of this work is to represent whether an attractor pos-
sesses, in its temporal evolution, states which are repeated. The RPs
denoted R ij are considered as an associated diagram to a square matrix
constructed around the indicated assumption as follows. If each point y i of

is near to or not of another point y j ,thena
the phase portrait y i
i51; 2;...;N
recurrence between point’s y i and y j of the trajectory will take place. Thus,
if the distance between two points is less than a predetermined threshold
ε, the points are said to be recurrent and are associated with a black point;
otherwise, they are not recurring and are associated with a white point.
Mathematically, this is written by an order N square matrix such that
(2.1):

R ij 5 Θ ε2:y i 2y j : ; y i ; y j Aℜ ð21:1Þ
i;j51; 2;...;N
where N is the number of considered states, Θ :ðÞ is the Heaviside function,ε
is a threshold distance, ::: is the norm, and y i and y j are two points of the
trajectory. The principle of this technique consists in the visual analysis and
inspection of the graph matrix of recurrence constructed with vector dis-
tances when each portion of the curve is compared to all the others and
represented on a recurrence map, as well as the quantitative characteristics
are necessary for the evaluation of the distances, between the points in the
temporal space and in the phase space.
Several methodological advantages, of which not all are allowed in the
analysis of nonlinear dynamic systems by other less adapted tools, justify the
need to use this relevant technique since it does not require some constraints
on the stability over time of the statistical or stationary properties of systems,
or on the form of the statistical distribution of the associated measures.
Influenced by the characteristic behavior of the trajectory of the phase
space, the structure of RPs is composed of elements of minimal sizes, such
as single points, diagonal lines, and vertical or horizontal lines, or a mixture
of them. The macroscopic structure, known as texture, can be visually domi-
nated by homogeneity, periodicity, drift, or perturbation. The visual appear-
ance of an RP can provide the necessary information about the dynamics of
the system. (Marwan and Kurths, 2002) proposed measurements based on
vertical and/or horizontal structures capable of highlighting, in particular,
chaos chaos transitions. Thus, RP allows two broad and small-scale topo-
logical approaches. For a historical review of recurrence plots, it is appropri-
ate to see Marwan (2008).