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

F. Baldovin·A. Robledo
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Abstract

We uncover the dynamics at the chaos threshold $μ_{\infty}$ 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 $μ<μ_{\infty}$. 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 $λ_{q}$. 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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F. Baldovin, A. Robledo. 2002-09-11. Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. https://doi.org/10.1103/physreve.66.045104

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