Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics

Abstract

We uncover the dynamics at the chaos threshold ${\mu }_{\infty }$ of the logistic map and find it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for $\mu <{\mu }_{\infty }$. We corroborate this structure analytically via the Feigenbaum renormalization group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a $q$-exponential, of which we determine the $q$-index and the $q$-generalized Lyapunov coefficient ${\lambda }_q$. Our results are an unequivocal validation of the applicability of the non-extensive generalization of Boltzmann-Gibbs (BG) statistical mechanics to critical points of nonlinear maps.

Revtex, 3 figures. Updated references and some general presentation improvements. To appear published as a Rapid communication of PRE

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