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 μ∞ 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 μ <μ∞. 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 λ 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.
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