Edge of chaos of the classical kicked top map: Sensitivity to initial conditions

TL;DR

This paper investigates the sensitivity to initial conditions at the edge of chaos in the classical kicked top map, demonstrating a transition in the entropic index q from 0 to 1 as the system becomes fully chaotic.

Contribution

It introduces a q-generalized Lyapunov exponent to characterize the edge of chaos in the classical kicked top map, linking nonextensive statistical mechanics to classical chaos.

Findings

q increases from zero to one as perturbation grows

q remains at one in fully chaotic regime

Sensitivity follows a q-exponential form

Abstract

We focus on the frontier between the chaotic and regular regions for the classical version of the quantum kicked top. We show that the sensitivity to the initial conditions is numerically well characterised by $\xi=e_q^{\lambda_q t}$, where $e_{q}^{x}\equiv [ 1+(1-q) x]^{\frac{1}{1-q}} (e_1^x=e^x)$, and $\lambda_q$ is the $q$-generalization of the Lyapunov coefficient, a result that is consistent with nonextensive statistical mechanics, based on the entropy $S_q=(1- \sum_ip_i^q)/(q-1) (S_1 =-\sum_i p_i \ln p_i$). Our analysis shows that $q$ monotonically increases from zero to unity when the kicked-top perturbation parameter $\alpha $ increases from zero (unperturbed top) to $\alpha_c$, where $\alpha_c \simeq 3.2$. The entropic index $q$ remains equal to unity for $\alpha \ge \alpha_c$, parameter values for which the phase space is fully chaotic.