Scale-Free Networks Hidden in Chaotic Dynamical Systems

TL;DR

This paper reveals that state-transition networks derived from various chaotic systems are scale-free, uncovering hidden order in chaos and proposing a new network analysis approach to understand complex dynamical phenomena.

Contribution

It introduces the Discretized-State Transition (DST) network method and demonstrates the universal scale-free nature in multiple chaotic maps, revealing hidden order in chaos.

Findings

Scale-free networks are found in several chaotic maps.

The DST network technique helps understand chaotic systems as a whole.

A new network analysis approach to dynamical systems is proposed.

Abstract

In this paper, we show our discovery that state-transition networks in several chaotic dynamical systems are "scale-free networks," with a technique to understand a dynamical system as a whole, which we call the analysis for "Discretized-State Transition" (DST) networks; This scale-free nature is found universally in the logistic map, the sine map, the cubic map, the general symmetric map, the sine-circle map, the Gaussian map, and the delayed logistic map. Our findings prove that there is a hidden order in chaos, which has not detected yet. Furthermore, we anticipate that our study opens up a new way to a "network analysis approach to dynamical systems" for understanding complex phenomena.