Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity

Power-law sensitivity to initial conditions, characterizing the behavior of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear one-dimensional logisticlike maps xₜ₊₁=1-a|xₜ|ᶻ ($z>1;$ $0<a<~2;$ t=0,1,2,). The main ingredient of our approach is the generalized deviation law limop_Δx(0)→0[Δx(t)/Δx(0)]=[1+(1-q)λqt]^1/(1-q) (equal to e^λ₁t for $q=1,$ and proportional, for large $t,$ to t^1/(1-q) for q≠1; q∈R is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension df of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to -∞ when df varies from 1 (nonfractal, ergodiclike, limit) to zero.