Multifractality and nonextensivity at the edge of chaos of unimodal maps

TL;DR

This paper explores the multifractal and dynamical properties at the chaos threshold of logistic maps with nonlinearity, revealing a relationship between generalized dimensions, Lyapunov exponents, and nonextensive entropy.

Contribution

It analytically links the multifractal structure, sensitivity to initial conditions, and nonextensive statistics at the edge of chaos in unimodal maps.

Findings

Derived an analytical expression for sensitivity to initial conditions.

Established a relationship between multifractal dimensions and nonextensive Lyapunov exponents.

Demonstrated invariance of the partition function under a renormalization group operation.

Abstract

We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity $z>1$. First we determine analytically the sensitivity to initial conditions $\xi_{t}$. Then we consider a renormalization group (RG) operation on the partition function $Z$ of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of $Z$ fixes a length-scale transformation factor $2^{-\eta}$ in terms of the generalized dimensions $D_{\beta}$. There exists a gap $\Delta \eta $ in the values of $\eta $ equal to $\lambda _{q}=1/(1-q)=D_{\infty}^{-1}-D_{-\infty}^{-1}$ where $\lambda_{q}$ is the $q$-generalized Lyapunov exponent and $q$ is the nonextensive entropic index. We provide an interpretation for this relationship - previously derived by Lyra and Tsallis - between dynamical and geometrical properties. Key Words: Edge of chaos, multifractal attractor, nonextensivity