Eur. Phys. J. B 11, 309-315 (1999)
https://doi.org/10.1007/s100510050941

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

¹ Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey
² Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro - RJ, Brazil
³ Departamento de Fisica, Universidade Federal de Alagoas, 57072-970 Maceio-AL, Brazil

Corresponding author: ^a tirnakli@sci.ege.edu.tr

Received: 5 February 1999
Published online: 15 September 1999

Abstract

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with , where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions.