1998/09/10 by Uǧur Tırnaklı, Ugur Tirnakli, Constantino Tsallis +3 · 4 citations
Physics and Astronomy · #Advanced Mathematical Theories and Applications #Statistical Mechanics and Entropy #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1007/s100510050941
15 pages (revtex), 8 figs
arxiv created 1998/09/10 · openalex publication_date 1999/01/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
\n \nDissipative one-dimensional maps may exhibit special points\n(e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently,\nthe sensitivity to the initial conditions has a power-law time dependence,\ninstead of the usual exponential one.\nThe associated exponent can be identified with 1/(1-q), where q\ncharacterizes the nonextensivity of a generalized entropic form currently used\nto extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a\nvariety of anomalous situations.\nIt has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, \n53 (1998)) for such maps the scaling law\n1/(1-q)=1/\α\min - 1/\α\max, where\n\α\min and \α\max are the extreme values appearing in the\nmultifractal f(\α) function. We generalize herein the usual circular map\nby considering inflexions of arbitrary power z, and verify that the scaling\nlaw holds for a large range of z. Since, for this family of maps,\nthe Hausdorff dimension df equals unity for all z in contrast with\nq which does depend on z, it becomes clear that df plays no major\nrole in the sensitivity to the initial conditions.\n\n\n