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 analyzing 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. (c) 2006 Elsevier B.V. All rights reserved.