Multifractality and nonextensivity at the edge of chaos of unimodal maps

E. Mayoral; A. Robledo

doi:10.1016/j.physa.2004.04.010

paper

Multifractality and nonextensivity at the edge of chaos of unimodal maps

arXiv:cond-mat/0401128 · doi:10.1016/j.physa.2004.04.010

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 $ξ_{t}$ . 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 $2^{- η}$ in terms of the generalized dimensions $D_{β}$ . There exists a gap $Δ η$ in the values of $η$ equal to $λ_{q} = 1/ (1 - q) = D_{\infty}^{- 1} - D_{- \infty}^{- 1}$ where $λ_{q}$ 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

Contribution to the proceedings of 2nd International Conference on News and Expectations in Thermostatistics (NEXT03), Cagliari, Italy, 21-28/09/2003. Submitted to Physica A

Multifractality and nonextensivity at the edge of chaos of unimodal maps

Abstract

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