Nonextensive Statistical Mechanics

5.2 Low-Dimensional Conservative Maps 16510 1 N=10 ; n ini= 5x10 610 0N=2x10 6 ; n ini=10 6eq.(11) ; N=2x10 6eq.(11) ; N=1010 –1ρ y(y)10 –210 –310 –410 –5–4 –2 0 2 4yFig. 5.13 Probability density of rescaled sums of iterates of the cubic map (which belongs to thesame universality class of the logistic map) for N = 10 7 and N = 10. The number of initial valuesis n ini = 10 6 n ini = 5 · 10 6 , respectively. The solid lines correspond to the analytical expressionsfor finite N (from [370]).x t+1 = 1 − axt 2 + y t (5.27)y t+1 = bx t . (5.28)The b = 0 particular case corresponds precisely to the z = 2 logistic map. Forb ≥ 0, a line (an infinite number of them, in fact) exists in the (a, b) space onwhich the system is at the edge of chaos, with vanishing Lyapunov exponents. Itis since long well known (see [159] and references therein) that two universalityclasses exist along this line, namely the dissipative (logistic map) universality map∀b ≠ 1, and the conservative universality class for b = 1. One consistently expectsthat the values of q should follow the same classes. In particular, the value of q senfor 0 < b < 1 should be the same as that for b = 0, i.e., q sen = 0.2445 .... Indeed,precisely this, within some numerical precision, has been verified in [156–158].5.2 Low-Dimensional Conservative MapsWe remind that a d-dimensional map has d Lyapunov exponents λ (1)( [286] and references therein). If it is conservative, it satisfies1 ,λ(2)1 , ...,λ(d) 1