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 $x_{t+1\mathrm{}}=1\mathrm{}-a|x_t{|}^{z_i}$ $(i=1,2\mathrm{}$ correspond to $x_t>0\mathrm{}$ and $x_t<0,$ respectively; $z_i>1,$ $0<a<~2,$ $t=0,1,2,\dots ).\mathrm{}$ The study of the q-generalized second cumulant $C_2^{\mathrm{}\left(q\right)\mathrm{}}$ of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs ${(z}_1{,z}_2).\mathrm{}$ The slope of the $C_2^{\mathrm{}\left(q\right)\mathrm{}}$ versus $C_2^{\mathrm{}\left(1\right)\mathrm{}}$ plot (where $C_2^{\mathrm{}\left(1\right)\mathrm{}}$ 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 ${(z}_1{=z}_2=z)$ by Costa et al. [Phys. Rev. E 56, 245 (1997)]. It appears that as $z_2-z_1\to \pm \infty$ 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 $\overrightarrow{z}\infty ,$ $q^{*}$ approaches 1.