Some more universal scaling laws for critical mappings

Abstract

We study such nonlinear mappingsx _n +1=F(x _n ;b _cr) of an intervalI into itself for which the Feigenbaum scaling laws hold (i.e., for which b_cr is an accumulation point of bifurcation points). Letx ₀ be a random variable with some absolutely continuous distribution inI. We show in particular that (i) the geometric average distance ofx _n from the nearest point of the attractor decreases liken ^−1.93387; (ii) the geometric average of ¦∂x _n /∂x ₀¦ increases liken ^0.60; (iii) the geometric mean distance ¦x _n −y _n ¦ between the iterates of two close-by pointsx ₀,y ₀ asymptotically tends towards a value ∼¦x ₀−y ₀¦^0.77. These-and other-properties are also borne out from a simple probabilistic model which depicts the evolution as a random walklike process.