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 [Formula Presented] ([Formula Presented] [Formula Presented] [Formula Presented]). The main ingredient of our approach is the generalized deviation law [Formula Presented] (equal to [Formula Presented] for [Formula Presented] and proportional, for large [Formula Presented] to [Formula Presented] for [Formula Presented] [Formula Presented] is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter [Formula Presented] and the fractal dimension [Formula Presented] of the onset-to-chaos attractor is revealed: [Formula Presented] appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to [Formula Presented] when [Formula Presented] varies from 1 (nonfractal, ergodiclike, limit) to zero. © 1997 The American Physical Society.