A dynamical model for hierarchy and modular organization: The trajectories en route to the attractor at the transition to chaos

TL;DR

This paper explores the hierarchical and modular dynamical structure of the transition to chaos in low-dimensional unimodal maps, revealing emergent power-law properties and complex collective behavior.

Contribution

It introduces a detailed hierarchical model of the dynamics leading to chaos, highlighting the emergence of power-law properties in a well-known nonlinear system.

Findings

Hierarchical modular organization in the transition to chaos

Emergence of power-law dynamical properties

Nested structure of the attractor dynamics

Abstract

We show that the full features of the dynamics towards the Feigenbaum attractor, present in all low-dimensional maps with a unimodal leading component, form a hierarchical construction with modular organization that leads to a clear-cut emergent property. This well-known nonlinear model system combines a simple and precise definition, an intricate nested hierarchical dynamical structure, and emergence of a power-law dynamical property absent in the exponential-law that governs the dynamics within the modules. This classic nonlinear system is put forward as a working example for complex collective behavior.