Asymmetric unimodal maps: some results from q-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)=1-a|x(t)|(z(i)) (i=1,2 correspond to x(t)>0 and x(t)<0, respectively; z(i)>1, 0<a</=2, t=0,1,2,ellipsis). The study of the q-generalized second cumulant C((q))(2) of these maps allows us to determine the dependence of the nonextensivity parameter q on the inflection parameter pairs (z(1),z(2)). The slope of the C((q))(2) versus C((1))(2) plot (where C((1))(2) 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)-->+/-infinity 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-->infinity, q(*) approaches 1.
Tirnakli
U
Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil and Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey.