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

We uncover the dynamics at the chaos threshold ${\mu }_{\infty }$ of the logistic map and find that 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 nonextensive generalization of Boltzmann-Gibbs statistical mechanics to critical points of nonlinear maps.