Horizontal visibility graphs generated by type-I intermittency

Abstract

The type-I intermittency route to (or out of) chaos is investigated within the horizontal visibility (HV) graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct their associated HV graphs. We show how the alternation of laminar episodes and chaotic bursts imprints a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values for several network parameters. In particular, we predict that the characteristic power-law scaling of the mean length of laminar trend sizes is fully inherited by the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of block entropy functionals defined on the graph. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization-group framework, where the fixed points of its graph-theoretical renormalization-group flow account for the different types of dynamics. We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibits extremal entropic properties.

Generalized Statistical Mechanics at the Onset of Chaos

“…The capability of the method to transfer the most basic properties of different types of time series into their resultant graphs has been demonstrated in recent works. See the references in [50][51][52][53][54]. When the series under study are the trajectories within the attractors generated by unimodal or circle maps, the application of the HV algorithm yield subfamilies of visibility graphs that render the known low-dimensional routes to chaos in a new setting [25,54].…”

Horizontal Visibility graphs generated by type-II intermittency

“…In other words, the operation removes form the graph every node of degree k = 2 along with its two links. Assuming infinitely long series (in order to avoid the rescaling procedure in standard RG), in what follows we argue that the flow induced by iteratively performing this RG operation classifies dynamics coming from above and below transition, although the phase portrait turns to be very different from the one found in type-I intermittency [20].…”

Ordered visibility graph weighted averaging aggregation operator: A methodology based on network analysis

“…VG is investigated increasingly (Donner & Donges, 2012;Donner, Zou, Donges, Marwan, & Kurths, 2010;Donner et al, 2011) and widely used in many applications Ahmadlou, Adeli, & Adeli, 2010;Luque, Lacasa, Ballesteros, & Luque, 2009). The properties of the time series are conserved in the graph topology Lacasa, Nunez, Roldán, Parrondo, & Luque, 2012;Luque, Lacasa, Ballesteros, & Robledo, 2012;Núñez, Luque, Lacasa, Gómez, & Robledo, 2013) and the properties of the visibility graph have been tested (Donges, Donner, & Kurths, 2013;Lacasa, Luque, Luque, & Nuno, 2009;Luque et al, 2009). In the visibility graph, the values of time series are plotted by using vertical bars.…”