Nonextensive Statistical Mechanics

5.1 Low-Dimensional Dissipative Maps 153Fig. 5.1 Left: Absolute values of positions of the first 10 iterations τ for two trajectories of thelogistic map at the edge of chaos, with initial conditions x 0 = 0 (empty circles) and x 0 = δ ≃5×10 −2 (full circles). Right: The same (in log–log plot) for the first 1000 iterations, with δ = 10 −4(from [142]).tangent bifurcations, q sen > 1(weakly insensitive). The case q sen < 1 (with λ qsen >0) yields, in Eq. (5.8), a power-law behavior ξ ∝ t 1/(1−qsen) in the limit t →∞.Thispower-law asymptotics were since long known in the literature [122–126]. The caseq sen < 1 is in fact more complex than indicated in Eq. (5.8). This equation onlyreflects the maximal values of an entire family, fully (and not only asymptotically)described in [150, 155]. See Figs. 5.1 and 5.2 from [142].5.1.1.2 MultifractalityMultifractals are conveniently characterized by the multifractal function f (α) [212].Typically, this function is concave, defined in the interval [α min ,α max ] with f (α min )= f (α max ) = 0; within this interval it attains its maximum d H , d H being the Hausdorffor fractal dimension.It has been proved [129, 142], that, at the edge of chaos, we have 111 − q sen= 1α min− 1α max(q sen < 1). (5.9)1 Virtually all the q-formulae of the present book admit the limit q → 1. This is not the case ofEq. (5.9), since the left member diverges whereas the right member vanishes. Indeed, q = 1 typicallycorresponds to the case of dynamics with positive Lyapunov exponent, hence mixing, henceergodic, hence leading to an Euclidean, nonfractal, geometry. For such a standard one-dimensionalgeometry, it should be α min = α max = f (α min ) = f (α max ) = 1, which clearly makes the rightmember of Eq. (5.9) to vanish. A relation more general than Eq. (5.9) is therefore needed before11taking the q → 1 limit. A relation such as1−q sen=α min− f (α − 1min) α max− f (α mx)for instance. Indeed,it recovers Eq. (5.9) for f (α min ) = f (α max ) = 0, and also admits q → 1, being now possible forboth members to diverge. It should be however noticed that this more general relation is totallyheuristic: we do not yet dispose of numerical indications, and even less of a proof.