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Entropy of weighted recurrence plots

Thomas Peron, Juergen Kurths
First conceived to visualize the time-dependent behavior of complex dynamical systems, recurrence plots (RPs) have
been shown to be a powerful technique to uncover statistically many characteristic properties of such systems.
A crucial issue in the study of time series originating from complex systems is the detection of dynamical transitions,
a task that RPs have been accomplishing due to a set of RP-based measures of complexity. Examples of their
successful application in real-world systems can be found in neuroscience, earth science, astrophysics , and other areas
of research. The so-called recurrence quantification analysis (RQA) provides measurements based on the density
and the length of diagonal and vertical line patterns in RPs, which turn out to be an alternative way to quantify
the complexity of physical systems. The time-dependent behavior of nonlinear time series can then be uncovered by
setting sliding time windows in order to identify dynamical transitions, such as periodic to chaos transitions and
even chaos-chaos transitions. The great merit of this approach resides in the fact that the calculation of Lyapunov
exponents is often impracticable when the equations of motion are unknown. In this way, many measurements
based on RQA have been proposed in order to better quantify the properties of dynamical systems. For instance,
one of the most employed quantifiers able to detect bifurcation points are entropy-based measurements, e.g., the
normalized entropy of recurrence times or the Shannon entropy of the distribution of length of diagonal line
segments. However, the entropy of the diagonal line segments is known to present in some cases a counterintuitive
anticorrelation with Lyapunov exponents, leading to high values for periodic dynamics and low values in chaotic
regimes. In this work we tackle this problem of the apparent contradiction by proposing an alternative RP-based
entropy of weighted Recurrence Plots (wRPs). In other words, instead of considering binary RPs, we allow them to
have weights proportional to the euclidean distances between the points in the phase space
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