Figure 1.
Horizontal Visibility Graph method applied to the time series .
denotes the node degree distribution of the obtained graph (HVG-PDF).
Figure 2.
Confidence intervals for values for Noninvertible, Dissipative, Conservative and Schuster Chaotic Maps.
The values were obtained following the methodology proposed by Lacasa et al.. Symmetric confidence intervals at the confidence level were obtained for the
parameter assuming the Gaussian model, a linear structure for the regression and independent zero-mean errors. The horizontal line represents the value of
corresponding to white noise (uncorrelated stochastic dynamics). The list of names for each map is the same given in Sec.. Full circles (blue) are in agreement with Lacassa and Toral [36] proposal rule. Empty circles (red) not.
Figure 3.
Parameter values of HVG-PDF
for fBm, fGn and noise with
power spectrum time series with total length of
data.
The values were obtained following the methodology proposed by Lacasa et al.: from the graph
versus
, the
parameter was computed by adjusting using the least square method, a straight line being
its slope. The linear scaling region considered in all cases is
, or
(if
). Symmetric confidence intervals at the
confidence level were obtained for the
parameter assuming the Gaussian model, a linear structure for the regression and independent zero-mean errors. The horizontal line represents the value of
corresponding to white noise (uncorrelated stochastic dynamics) Full circles (blue) are in agreement with Lacassa and Toral [36] proposal rule. Empty circles (red) not.
Figure 4.
-value determination: examples of analyzed dynamical systems where a good linear scaling region was found.
For the Holmes cubic map (), however, even having a good fitting, the
-value obtained is greater than
which not satisfied the chaotic distinction suggested by Lacasa and Toral [36]. In all cases, time series with
are considered, and linear scaling regions are defined by
for chaotic and
for stochastic time series.
Figure 5.
-value determination in the case of time series generated by stochastic dynamics with
power spectrum with
.
Time series with data. Two different linear scaling zones: a)
given
; and b)
given
. Note that the slope of the straight line change significantly.
Figure 6.
Cases with bad -value determination: a) Cusp map and Schuster map with
, the associated HVG-PDF present heavy tail making difficult to define an unique linear scaling zone representative of all the data.
b) Tinkerbell map (Y) and the Burger’s map (X) for which it is impossible to define an unique linear scaling zone, and in consequence the hypothesis of an exponential behavior cannot be confirmed. Time series with data are considered.
Table 1.
Dynamical systems and their statistical quantifiers skewness (), kurtosis (
evaluated for
and
.
Figure 7.
Examples of HVG-PDF for some chaotic and stochastic systems.
Only are displayed. Note that the corresponding cut-offs (
) are also shown. The length of the time series is
.
Figure 8.
Study of the effect of the series length on the Information Theory quantifiers.
The dynamical systems here considered are the Logistic map and noises with power spectrum, for
and
.
Figure 9.
Representation on the Shannon-Fisher plane, , for all dynamical systems.
The quantifiers were evaluated with the HVG-PDF from time series length . The stars (
) represent the obtained values for chaotic flows (RS: Rössler system (X-coordinate), and LS: Lorenz system (X-coordinate)).
Figure 10.
Shannon-Fisher plane, zoom, see Fig. 9.
Figure 11.
Shannon-Fisher plane, zoom, see Fig. 9.
Figure 12.
Shannon-Fisher plane, , for the logistic map (
) contaminated with additive noise with uniform PDF and amplitude
.
Time series with data are considered.