Nonextensive Statistical Mechanics

5.1 Low-Dimensional Dissipative Maps 163Fig. 5.11 Straight lines: qsen av (cycle 3) = 2.5 qav sen (cycle 2) − 0.03 and qav sen (cycle 5) =2.5 qsen av (cycle 2) + 0.03, which suggests qav sen (cycle 5) − qav sen (cycle 3) ≃ 0.06 (from [153]).henceqsen av (cycle 5; z) ≃ 2.5 qav sen (cycle 2; z) + 0.03 , (5.25)qsen av (cycle 5; z) − qav sen (cycle 3; z) ≃ 0.06 . (5.26)The full understanding of all these relations remains an open problem.5.1.1.6 AttractorLet us now focus on an important limiting property, directly related to the CentralLimit Theorem (CLT). It is in fact a dynamical version of the CLT. As an example ofunimodal one-dimensional map, let us consider the z-logistic one for values of thecontrol parameter a such that the Lyapunov exponent λ 1 is positive (i.e., a stronglychaotic map), and start from a given initial condition x 0 . The successive N iteratesx 1 , x 2 , x 3 ..., constitute a time series which associates, with each value of x 0 ,theuniquely defined sum of the first N terms. For fixed N, we may consider a largeset of initial conditions uniformly distributed within the allowed phase-space. Thedistribution of the sums, appropriately centered and scaled, approaches, for N →∞, a Gaussian [154]. See Figs. 5.12, 5.13, and 5.14 (from [370]).The situation changes dramatically if we are at the edge of chaos, where λ 1 = 0(i.e., a weakly chaotic map). The limiting distribution appears to be a q-Gaussianwith q = q stat ≃ 1.7(stat stands for stationary state; this qualification will become