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

Chaos thresholds of the z -logistic maps xt+1 =1- 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 limt› S q sen av (t)t= limt› ln q sen av ?(t)t ? q sen av av, where the entropy Sq (1- i piq) (q-1) (S1 =- i pi ln pi), the sensitivity to the initial conditions ? lim?x(0)›0 ?x(t)?x(0), and lnq x (x1-q -1)(1-q) (ln1 x=ln x). The entropic index q sen av <1, and the coefficient ? q sen av av >0 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 1t1 (qrel -1) (qrel >1). These results (i) illustrate the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely, qrel -1 An (1- q sen av) ?n, where the positive numbers (An, ?n) depend on the cycle; (ii) exhibit an unexpected scaling, namely, q sen av (cycle n)= Bn q sen av (cycle 2)+ µ n. © 2006 The American Physical Society.