An Interdisciplinary Journal

* 2015, Vol.18, No.3, pp.335-355*

We review our recent works on the dynamical localization in the quantum kicked rotator
(QKR) and related properties of the classical kicked rotator (the standard map, SM).
We introduce the Izrailev *N*-dimensional model of the QKR and analyze the localization
properties of the Floquet eigenstates [*Phys. Rev.* E **87**, 062905 (2013)], and the statistical
properties of the quasienergy spectra. We survey normal and anomalous diffusion in the SM,
and the related accelerator modes [*Phys. Rev.* E **89**, 022905 (2014)]. We analyze the statistical
properties [*Phys. Rev.* E **91**,042904 (2015)] of the localization measure, and show that the
reciprocal localization length has an almost Gaussian distribution which has a finite variance
leven in the limit of the infinitely dimensional model of the QKR, *N* → ∞. This sheds newight on the relation between the QKR and the Anderson localization phenomenon in the
one-dimensional tight-binding model. It explains the so far mysterious strong fluctuations in
the scaling properties of the QKR. The reason is that the finite bandwidth approximation
of the underlying Hamilton dynamical system in the Shepelyansky picture [*Phys. Rev. Lett.*
**56**, 677 (1986)] does not apply rigorously. These results call for a more refined theory of the
localization length in the QKR and in similar Floquet systems where we must predict not only
the mean value of the inverse of the localization length but also its (Gaussian) distribution.
We also numerically analyze the related behavior of finite time Lyapunov exponents in the
SM and of the 2 × 2 transfer matrix formalism.

*Key words: *
quantum and classical chaos, dynamical localization, diffusion, quantum kicked rotator

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Last updated: *October 07, 2015*