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


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Tirnakli U. , TSALLIS C.

PHYSICAL REVIEW E, cilt.73, 2006 (SCI İndekslerine Giren Dergi) identifier identifier identifier

Özet

Chaos thresholds of the z-logistic maps x(t+1)=1-a parallel to x(t)parallel to(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)< S-qsen(av)>(t)/t=lim(t ->infinity)< ln(qsen)(av) xi >(t)/t equivalent to lambda(avav)(qsen), where the entropy S-q equivalent to(1-Sigma(i)p(i)(q))/(q-1) (S-1=-Sigma(i)p(i)ln p(i)), the sensitivity to the initial conditions xi equivalent to lim(Delta x(0)-> 0)Delta x(t)/Delta x(0), and ln(q)x equivalent to(x(1-q)-1)/(1-q) (ln(1)x=ln x). The entropic index q(sen)(av)< 1, and the coefficient lambda(avav)(qsen)> 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 1/t(rel)(1/(q)-1) (q(rel)> 1). These results (i) illustrate the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely, q(rel)-1 similar or equal to A(n)(1-q(sen)(av))(alpha)(n), where the positive numbers (A(n),alpha(n)) depend on the cycle; (ii) exhibit an unexpected scaling, namely, q(sen)(av)(cycle n)=B(n)q(sen)(av)(cycle 2)+epsilon(n).