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We review our recent works on the dynamical localization in the quantumkicked rotator (QKR) and the related properties of the classical kicked rotator(the standard map, SM). We introduce the Izrailev $N$-dimensional model of theQKR and analyze the localization properties of the Floquet eigenstates [{\emPhys. Rev. E} {\bf 87}, 062905 (2013)], and the statistical properties of thequasienergy spectra. We survey normal and anomalous diffusion in the SM, andthe related accelerator modes [{\em Phys. Rev. E} {\bf 89}, 022905 (2014)]. Weanalyze the statistical properties [{\em Phys. Rev. E} {\bf 91},042904 (2015)]of the localization measure, and show that the reciprocal localization lengthhas an almost Gaussian distribution which has a finite variance even in thelimit of the infinitely dimensional model of the QKR, $N\rightarrow \infty$.This sheds new light on the relation between the QKR and the Andersonlocalization phenomenon in the one-dimensional tight-binding model. It explainsthe so far mysterious strong fluctuations in the scaling properties of the QKR.The reason is that the finite bandwidth approximation of the underlyingHamilton dynamical system in the Shepelyansky picture [{\em Phys. Rev. Lett.}{\bf 56}, 677 (1986)] does not apply rigorously. These results call for a morerefined theory of the localization length in the QKR and in similar Floquetsystems, where we must predict not only the mean value of the inverse of thelocalization length but also its (Gaussian) distribution. We also numericallyanalyze the related behavior of finite time Lyapunov exponents in the SM and ofthe $2\times2$ transfer matrix formalism.

Studies of dynamical localization in a finite-dimensional model of the quantum kicked rotator