  It is well known that, for chaotic systems, the production of relevant
entropy (Boltzmann-Gibbs) is always linear and the system has strong
(exponential) sensitivity to initial conditions. In recent years, various
numerical results indicate that basically the same type of behavior emerges at
the edge of chaos if a specific generalization of the entropy and the
exponential are used. In this work, we contribute to this scenario by
numerically analysing some generalized nonextensive entropies and their related
exponential definitions using $z$-logistic map family. We also corroborate our
findings by testing them at accumulation points of different cycles.
