Nonextensive Statistical Mechanics

152 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicsy t+1 = μ y t (1 − y t ) (0 ≤ μ ≤ 4; 0 ≤ y t ≤ 1) . (5.2)The z-periodic map is defined as follows [129]:x t+1 = d cos(π|x t − 1/2| z/2 ) (z > 1; d > 0; |x t |≤d). (5.3)It belongs to the same universality class of the z-logistic map since they both sharean extremum with inflexion of order z. The standard case is recovered for z = 2,and its primary edge to chaos occurs at d c (2) = 0.8655 ...The z-circular map is defined as follows [132]:θ t+1 = + [θ t − 12π sin(2πθ t)] z/3 (z > 0). (5.4)The case z = 3 recovers the standard case, and its primary edge to chaos occurs at c (3) = 0.6066 .... Various interesting properties and analytical results can be seenin [145].The z-exponential map is defined as follows [146]:x t+1 = 1 − ae −1/|x t | z (z > 0; a ∈ [0, a ∗ (z)]; |x t |≤1), (5.5)where a ∗ (z) depends slowly from z (e.g., a ∗ (0.5) ≃ 5.43). This map was introduced[146] in order to have an extremum flatter than any power, which is the case ofthe z-logistic and the z-periodic ones. It shares with the z-logistic and z-periodicmaps the same topological properties, although they differ in the metric ones. Thecase z = 1/2 is a typical one, and its primary edge to chaos occurs at a c (1/2) =3.32169594 ...5.1.1.1 Sensitivity to the Initial ConditionsThe sensitivity to the initial conditions ξ for a one-dimensional dynamical systemis, as previously addressed, defined as follows:x(t)ξ ≡ limx(0)→0 x(0) , (5.6)where x denotes the phase-space variable. The sensitivity ξ is quite generically expectedto satisfyhence [127, 141, 142, 150]dξdt= λ qsen ξ q sen, (5.7)ξ = e λ qsen tq sen, (5.8)where q sen = 1 if the Lyapunov exponent λ 1 ≠ 0(strongly sensitive if λ 1 > 0, andstrongly insensitive if λ 1 < 0), and q sen ≠ 1 otherwise; sen stands for sensitivity.Atthe edge of chaos, q sen < 1(weakly sensitive), and at both the period-doubling and