Nonextensive Statistical Mechanics

5.3 High-Dimensional Conservative Maps 181where i, j = 1,...,N. It can be seen that,( ) G T ( ) I B=, (5.42)¯x I (I + B)hence( ) G T (−B IJ =¯x −(I + B) I). (5.43)This quantity, multiplied (from the right side) by the matrix (5.38) yields J.Therefore our system is symplectic. Consequently, the 2N Lyapunov exponentsλ 1 ≡ λ M ,λ 2 ,λ 3 ,...,λ 2N are coupled two by two as follows: λ 1 =−λ 2N ≥ λ 2 =−λ 2N−1 ≥ ... ≥ λ N =−λ N+1 ≥ 0. In other words, as a function of time, aninfinitely small length typically diverges as e λ1t , an infinitely small area divergesas e (λ 1+λ 2 )t , an infinitely small volume diverges as e (λ 1+λ 2 +λ 3 )t , an infinitely smallN-dimensional hypervolume diverges as e (∑ Ni=1 λ i )t( ∑ Ni=1 λ i being in fact equal tothe Kolmogorov–Sinai entropy rate, in agreement with the Pesin identity), an infinitelysmall (N + 1)-hypervolume diverges as e (∑ N−1i=1 λ i )t , and so on. For example,a(2N − 1)-hypervolume diverges as e λ1t , and finally a 2N-hypervolume remainsconstant, thus recovering the conservative nature of the system (of course, this correspondsto the Liouville theorem in classical Hamiltonian dynamics).Typical results are depicted in Figs. 5.31, 5.32, 5.33, 5.34, and 5.35.Fig. 5.31 Lyapunov exponent dependence on system size N in log–log plot, showing that λ M ∼N −κ(α) . Initial conditions correspond to θ 0 = 0.5, δθ = 0.5, p 0 = 0.5, and δp = 0.5. Fixedparameters are a = 0.005 and b = 2. We averaged over 100 realizations. Inset: κ vs. α, exhibitingweak chaos in the limit N →∞when 0 ≤ α 1 (from [359]).