Visibility graphs of fractional Wu–Baleanu time series

Abstract

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu in [34], presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.

Applications of Visibility Graphs for the representation of Time Series

“…We have seen that both the scaling factor and the fractional exponent are linked when chaos is present with these results [CLMIR19].…”

“…In the fourth column of Table 3.1 the dierent correlations dened in [INL18] have been computed between the NVG's entropy matrix and the power law exponent tting matrix. As it happened with both visibility graphs matrices, the correlations are high in all the cases, meaning that the chaos-related information encoded by the exponent of the power law tting is qualitatively the same as within the entropy of the NVG's [CLMIR19].…”

“…In this section, we provide a brief overview of some fractional calculus and fractional discrete operators. We also present an alternative way to [WB14] of introducing Wu and Baleanu discrete equation, as we did in our published work [CLMIR19].…”

“…In Chapter 3, we will show a table with dierent correlation metrics for matrices. These correlations have also been computed between the NVGs entropy matrix and the power-law exponent tting matrices to explore if chaosrelated information encoded by the exponent of the power-law tting is qualitatively comparable with the information extracted from the NVGs entropy [CLMIR19]. In this section, we will provide the denitions and properties of several of these metrics.…”

Solutions of abstract integro‐differential equations via Poisson transformation

“…It is remarkable that Volterra difference equations in the form () appears very recently in discrete chaos 16,17 and in the analysis of discrete time neural networks. [ 18 , Formula (10)] In such cases, are the Cesáro numbers 14 .…”