We study the kicked rotator in the classically fully chaotic regime using Izrailev's $N$-dimensional model for various $N\le 4000$, which in the limit $N\to \infty$ tends to the quantized kicked rotator. We do treat not only the case $K=5$, as studied previously, but also many different values of the classical kick parameter $5\le K\le 35$ and many different values of the quantum parameter $k\in [5,60]$. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable $\Lambda =l_{\infty }/N$ to the case of anomalous diffusion in the classical phase space by deriving the localization length $l_{\infty }$ for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents ${\beta }_{\mathrm{BR}}$. (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by ${\beta }_{\mathrm{loc}}$ in the interval $[0,1]$. The level repulsion parameters ${\beta }_{\mathrm{BR}}$ and ${\beta }_{\mathrm{loc}}$ are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between ${\beta }_{\mathrm{loc}}$ and the relative localization length $\Lambda$, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).

• Received 17 January 2013

DOI:https://doi.org/10.1103/PhysRevE.87.062905