POWER-LAW SENSITIVITY TO INITIAL CONDITIONS WITHIN A LOGISTICLIKE FAMILY OF MAPS - FRACTALITY AND NONEXTENSIVITY

Power-law sensitivity to initial conditions, characterizing the behavi or of dynamical systems at their critical points (where the standard L iapunov exponent vanishes), is studied in connection with the family o f nonlinear one-dimensional logisticlike maps x(t+1)= 1 - a\x(t)\(z) ( z > 1; 0 < a less than or equal to 2; t = 0,1,2,...). The main ingredi ent of our approach is the generalized deviation law lim (Delta x(0)-- >0)[Delta x(t)/Delta x(0)] = [1 + (1-q)lambda(q)t](1/(l - q)) (equal t o e(lambda lt) for q = 1, and proportional, for large t, to t(1/(1-q)) for q not equal 1; q is an element of R is the entropic index appeari ng in the recently introduced nonextensive generalized statistics). Th e relation between the parameter q and the fractal dimension d(f) of t he onset-to-chaos attractor is revealed: q appears to monotonically de crease from 1 (Boltzmann-Gibbs, extensive, limit) to -infinity when d( f) varies from 1 (nonfractal, ergodiclike, limit) to zero.