Horizontal Visibility graphs generated by type-I intermittency

TL;DR

This paper explores how Horizontal Visibility graphs can characterize type-I intermittency in dynamical systems, revealing scaling laws, bifurcation signatures, and a renormalization group framework for understanding intermittent chaos.

Contribution

It introduces a phenomenological theory linking graph properties to intermittency features and develops a graph-theoretical RG approach to classify dynamical regimes.

Findings

Power-law scaling of laminar lengths reflected in degree distribution variance

Scaling of Lyapunov exponent inherited in graph block entropy

RG fixed points correspond to different dynamical behaviors

Abstract

The type-I intermittency route to (or out of) chaos is investigated within the Horizontal Visibility graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct, according to the Horizontal Visibility algorithm, their associated graphs. We show how the alternation of laminar episodes and chaotic bursts has a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values of several network parameters. In particular, we predict that the characteristic power law scaling of the mean length of laminar trend sizes is fully inherited in 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 the block entropy over the degree distribution. 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 RG 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 exhibit extremal entropic properties.