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
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 df equals unity for all z in contrast with
q which does depend on z, it becomes clear that df plays no major
role in the sensitivity to the initial conditions.
PACS: 05.45.-a – Nonlinear dynamics and nonlinear dynamical systems / 05.20.-y – Statistical mechanics / 05.70.Ce – Thermodynamic functions and equations of state
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 1999