RG universal dynamics at the onset of chaos in unimodal 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 interwined 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 generalized Lyapunov exponent $\lambda_{q}$. 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.
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