Feigenbaum graphs: a complex network perspective of chaos

TL;DR

This paper introduces Feigenbaum graphs, a novel network-based approach to analyze chaos in nonlinear systems, providing universal analytical descriptions and revealing connections between network entropy and dynamical properties.

Contribution

It develops Feigenbaum graphs from horizontal visibility graphs, offering a universal framework for studying chaos and nonlinear dynamics through network theory.

Findings

Exact degree distribution formulas for Feigenbaum graphs

Network entropy correlates with Lyapunov exponent

Fixed points of renormalization group match entropy optimization

Abstract

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.