Feigenbaum graphs at the onset of chaos

TL;DR

This paper investigates the properties of Feigenbaum graphs derived from unimodal maps at the transition to chaos, revealing universal features, degree fluctuations, and connections to entropy growth and Lyapunov exponents.

Contribution

It introduces a universal network representation of unimodal map dynamics at chaos onset and links graph properties to dynamical measures like Lyapunov exponents and entropy.

Findings

Network degrees fluctuate at all scales with increasing amplitude as network size grows

A graph-theoretical Lyapunov exponent characterizes trajectory expansion in network space

Entropy growth rate in network space aligns with graph-theoretical exponents, forming Pesin-like identities

Abstract

We analyze the properties of the self-similar network obtained from the trajectories of unimodal maps at the transition to chaos via the horizontal visibility (HV) algorithm. We first show that this network is uniquely determined by the encoded sequence of positions in the dynamics within the Feigenbaum attractor and it is universal in that it is independent of the shape and nonlinearity of the maps in this class. We then find that the network degrees fluctuate at all scales with an amplitude that increases as the size of the network grows. This suggests the definition of a graph-theoretical Lyapunov exponent that measures the expansion rate of trajectories in network space. On good agreement with the map's counterpart, while at the onset of chaos this exponent vanishes, the subexponential expansion and contraction of network degrees can be fully described via a Tsallis-type scalar deformation of the expansion rate, that yields a discrete spectrum of non-null generalized exponents. We further explore the possibility of defining an entropy growth rate that describes the amount of information created along the trajectories in network space. Making use of the trajectory distributions in the map's accumulation point and the scaling properties of the associated network, we show that such entropic growth rate coincides with the spectrum of graph-theoretical exponents, what appears as a set of Pesin-like identities in the network.