Circular-like Maps: Sensitivity to the Initial Conditions, Multifractality and Nonextensivity

TL;DR

This paper extends the circular map by introducing arbitrary power inflexions, analyzing the scaling law's validity across different parameters, and exploring the relationship between Hausdorff dimension and nonextensivity.

Contribution

It generalizes the circular map with arbitrary power inflexions and examines the scaling law's applicability and the role of Hausdorff dimension and nonextensivity.

Findings

Scaling law holds for a wide range of the parameter z.

Hausdorff dimension remains unity regardless of z.

Nonextensivity parameter q varies with z, unlike d_f.

Abstract

We generalize herein the usual circular map by considering inflexions of arbitrary power $z$, and verify that the scaling law which has been recently proposed [Lyra and Tsallis, Phys.Rev.Lett. 80 (1998) 53] holds for a large range of $z$. Since, for this family of maps, the Hausdorff dimension $d_f$ equals unity for all $z$ values in contrast with the nonextensivity parameter $q$ which does depend on $z$, it becomes clear that $d_f$ plays no major role in the sensitivity to the initial conditions.