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Multifractality and nonextensivity at the edge of chaos of unimodal maps

Author

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  • Mayoral, E
  • Robledo, A

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 ξt. Then we consider a renormalization group 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 Dβ. There exists a gap Δη in the values of η equal to λq=1/(1−q)=D∞−1−D−∞−1 where λq 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.

Suggested Citation

  • Mayoral, E & Robledo, A, 2004. "Multifractality and nonextensivity at the edge of chaos of unimodal maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 219-226.
    • Handle: RePEc:eee:phsmap:v:340:y:2004:i:1:p:219-226
    DOI: 10.1016/j.physa.2004.04.010
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