Asymmetric unimodal maps:  Some results fromq-generalized bit cumulants

In this study, using q-generalized bit cumulants $(q$ is the nonextensivity parameter of the recently introduced Tsallis statistics), we investigate the asymmetric unimodal maps xₜ₊₁=1-a|xₜ|^zᵢ $(i=1,2$ correspond to xₜ>0 and xₜ<0, respectively; zᵢ>1, $0<a<~2,$ t=0,1,2,). The study of the q-generalized second cumulant C₂(q) of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs (z₁,z₂). The slope of the C₂(q) versus C₂⁽¹⁾ plot (where C₂⁽¹⁾ is the standard second cumulant) provides the necessary tool to accomplish this task. The slope behaves exactly the same as the proper q values (say q*) that were obtained for logisticlike maps (z₁=z₂=z) by Costa et al. [Phys. Rev. E 56, 245 (1997)]. It appears that as z₂-z₁→±∞ this slope approaches unity. This behavior is very similar to the behavior of q* as a function of the inflection parameter for some z-dependent maps; namely, as →z∞, q* approaches 1.