Nonextensive Statistical Mechanics

5.2 Low-Dimensional Conservative Maps 173p 1central slicewhole phase space(a1) 0 ≤ t ≤ t 1 (b1) t = t 2 (c1) t = t 3111p 1p 2p 2p 1 p 2 p 20.50.50.500 0.5 θ 1 100 0.5 100 0.5 p 1 1(a2) (b2) (c2)1110.50.50.500 0.5 θ 1 100 0.5 p 1 100 0.5 p 1 1Fig. 5.22 Phase-space analysis of the evolution of “water bag” ensembles for two coupled standardmaps for (ã, b) = (0.4, 2). First row: “Water bag” initial conditions 0 ≤ θ 1 ,θ 2 ≤ 1, p 1 , p 2 =0.5±5·10 −3 . Second row: “Water bag” initial conditions 0 ≤ θ 1 ,θ 2 ≤ 1, p 1 , p 2 = 0.25±5·10 −3 .(a) Projectiononthe(θ 1 , p 1 )-plane of the central slice of the phase-space (θ 2 , p 2 = 0.5 ± 10 −2 ),for the orbit 0 ≤ t ≤ t 1 = 10 4 . (c),(c) Projection on the (p 1 , p 2 )-plane of whole phase-space forthe iterate at time t 2 = 15 and t 3 = 2 · 10 4 (from [356]).where a 1 , a 2 , b ∈ R, t = 0, 1,..., and all variables are defined mod 1. If thecoupling constant b vanishes the two standard maps decouple; if b = 2 the points(0, 1/2, 0, 1/2) and (1/2, 1/2, 1/2, 1/2) are a 2-cycle for all (a 1 , a 2 ), hence wepreserve in phase-space the same referential that we had for a single standard map.For a generic value of b, all relevant present results remain qualitatively the same.Also, we set a 1 = a 2 ≡ ã so that the system is invariant under permutation 1 ↔ 2.Since we have two rotors now, the dynamical temperature is naturally given byT ≡ 1 (2 < p21> + < p2 2 > − < p 1 > 2 − < p 2 > 2) , hence the BG temperature remainsT BG = 1/12. The time evolution of the system is depicted in Figs. 5.22, 5.23,and 5.24.5.2.3 Weakly Chaotic Two-Dimensional Conservative MapsIn the previous subsections we have analyzed low-dimensional systems that arestrongly chaotic. We shall dedicate the present subsection to weakly chaotic twodimensionalsystems, namely the Casati–Prosen map (or triangle map) [8–10] andthe Moore map [134–136], the former as focused on in [358], the latter as focusedon in [138].The Casati–Prosen map z n+1 = T (z n ) is defined on a torus z = (x, y) ∈[−1, 1) × [−1, 1)y n+1 = y n + α sgnx n + β (mod 2),x n+1 = x n + y n+1 (mod 2), (5.34)