Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2002-05-17
Phys. Rev. E {\bf 66}, 045104(R) (2002).
Physics
Condensed Matter
Statistical Mechanics
Revtex, 3 figures. Updated references and some general presentation improvements. To appear published as a Rapid communication
Scientific paper
10.1103/PhysRevE.66.045104
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.
Baldovin Fulvio
Robledo Alberto
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