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

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

2005-01-11

arxiv.org/abs/cond-mat/0501230v2

Complexity, Metastability and Nonextensivity (World Scientific, Singapore, 2005), Eds. C. Beck, A. Rapisarda and C. Tsallis, p

Physics

Condensed Matter

Statistical Mechanics

To appear in "Complexity, Metastability and Nonextensivity" (World Scientific, Singapore, 2005), Eds. C. Beck, A. Rapisarda an

Scientific paper

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.

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