Biomedical Engineering Reference
In-Depth Information
the phase space trajectory visits roughly the same area in the phase space. In this
way distinct recurrent behavior, e.g., periodicities and also irregular cyclicities can
be detected. The recurrence of states, in the meaning that states become arbitrarily
close after some time, is a fundamental property of deterministic dynamical systems
and is typical for non-linear or chaotic systems.
Recurrence of a state at time i against a different state at time j is marked within
a two-dimensional squared matrix with ones and zeros (black and white dots in the
plot), where both axes are time axes. More formally RP can be expressed as:
R m
R i , j
=
Θ
(
ε i
−||
x i
x j
|| ) ,
x i
,
i
,
j
=
1
...,
N
(2.137)
where N is the number of considered states x i , ε i is a threshold distance,
||·||
a norm,
and Θ
( · )
the Heaviside function. The visual appearance of an RP (Figure 2.23) gives
FIGURE 2.23: Examples of the recurrence plots (lower panels) for different sig-
nals shown above: a) a periodic signal obtained from the logistic equation for the
control parameter r = 3.829, b) a chaotic signal obtained from the logistic equation
for r = 3.9999, c) white noise. Note that in the RP for the deterministic chaotic sig-
nal short lines parallel to the diagonal are present. Adapted from [Klimaszewska and
Zebrowski, 2009].
hints about the dynamics of a system. Uncorrelated white noise results in uniformly
distributed dots, periodic patterns are connected with cyclicities in the process—time
distance between patterns (e.g., lines) corresponds to period, diagonal lines mean that
the evolution of states is similar at different times—the process could be determin-
istic; if these diagonal lines occur beside single isolated points, the process could be
chaotic.
The visual interpretation of RPs requires some experience. Their quantification
offers a more objective way of investigating the considered system. With this quan-
tification, the RPs have become a tool for analysis of certain biomedical processes,
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