Nonextensive Statistical Mechanics

5.2 Low-Dimensional Conservative Maps 167aP (y) / P (0)10 0 a = 1.40209a = 1.401354a = 1.401198a = 1.401164410 –1 a = 1.40115716Eq.(4); q = 1.3; β = 6.210 –210 –3N = 2 2nn ini = 10 81.00.80.60.40.20.0–3 –2 –1 0 1 2 310 –410 –5 –15 –10 –5 0 5 10 15y P(0)bP (y) / P (0)a = 1.401588a = 1.401248a = 1.401175a = 1.4011594510 –1 Eq.(4); q = 1.7; β = 6.210 –210 –3N = 2 2nn ini = 10 81.00.80.60.40.20.0–3 –2 –1 0 1 2 310 0 y P(0)10 –4–15 –10 –5 0 5 10 15Fig. 5.15 Probability density of the quantity y/σ at the critical point a c for z = 2, N = 2 14 ,and N = 2 15 (from [371]).(x t+1 , y t+1 ) ={(2x t , y t /2) (0 ≤ x t < 1/2)(2x t − 1, (y t + 1)/2) (1/2 ≤ x t ≤ 1) .(5.30)We verify that |(x t+1 , y t+1 )/(x t , y t )| = 1, and λ (1)1= −λ (2)1= ln 2. SeeFig. 5.18. The time dependence of the entropy S q (t) is depicted in Figs. 5.19and 5.20.The generalized cat map is defined as follows [85]p t+1 = p t + kx t (mod 1)x t+1 = p t + (1 + k) x t (mod 1) (k ≥ 0) . (5.31)We verify that |(p t+1 , x t+1 )/(p t , x t )|=1, and λ (1)1=−λ (2)1= ln 2+k+√ k 2 +4kFor typical values of k, it has been numerically verified in [85] that2.