Nonextensivity and multifractality in low-dimensional dissipative systems

TL;DR

This paper explores the connection between nonextensive statistics and multifractal properties in low-dimensional dissipative systems, revealing a relation between the entropic index and the multifractal spectrum at the edge of chaos.

Contribution

It establishes a theoretical relation between the entropic index q and the multifractal spectrum of attractors, supported by numerical validation in dissipative maps.

Findings

Derived a formula linking q to multifractal spectrum bounds.

Numerically confirmed the relation in standard dissipative maps.

Provides insight into the microscopic dynamics underlying nonextensivity.

Abstract

Power-law sensitivity to initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity as prescribed in the generalized statistics recently introduced by one of us (C.T.) and characterized by the entropic index $q$. We show that general scaling arguments imply that $1/(1-q) = 1/\alpha_{min}-1/\alpha_{max}$, where $\alpha_{min}$ and $\alpha_{max}$ are the extremes of the multifractal singularity spectrum $f(\alpha)$ of the attractor. This relation is numerically checked to hold in standard one-dimensional dissipative maps. The above result sheds light on a long-standing puzzle concerning the relation between the entropic index $q$ and the underlying microscopic dynamics.