Chaos edges of z-logistic maps: connection between the relaxation and sensitivity entropic indices.

Chaos thresholds of the z-logistic maps x(t+1)=1-a|xt|(z) (z>1; t=0,1,2,...) are numerically analyzed at accumulation points of cycles 2, 3, and 5 (three different cycles 5). We verify that the nonextensive q-generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More precisely, we computationally verify lim(t-->infinity) [formula-see text], where the entropy S(q) [formula-see text], the sensitivity to the initial conditions xi(triple bond)lim(Deltax(0)-->0)Deltax(t)/Deltax(0), and ln(q)x(triple bond)(x(1-q-1/(1-q)(ln(1)x=ln x). The entropic index [formula-see text], and the coefficient [formula-see text] depend on both z and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases as 1/t(1/q(rel)-1)(q(rel>1). These results (i) illustrate the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely, [formula-see text], where the positive numbers Alpha(n), alpha(n) depend on the cycle; (ii) exhibit an unexpected scaling, namely, [formula-see text].