# Chirikov standard map

Boris Chirikov and Dima Shepelyansky (2008), Scholarpedia, 3(3):3550. | doi:10.4249/scholarpedia.3550 | revision #183629 [link to/cite this article] |

The **Chirikov standard map** [1], [2] is an area-preserving map for two canonical dynamical variables, i.e., momentum and coordinate \( (p,x)\).
It is described by the equations:

\[\tag{1} \begin{array}{lcr} \bar{p} = p+K\sin x \\ \bar{x} = x+\bar{p} \end{array} \]

where the bars indicate the new values of variables after one map iteration and \(K\) is a dimensionless parameter that influences the degree of chaos. Due to the periodicity of \( \sin x \) the dynamics can be considered on a cylinder (by taking \( x \!\!\! \mod{2\pi} \)) or on a torus (by taking both \( x,p \!\!\! \mod{2\pi} \)). The map is generated by the time dependent Hamiltonian \( H(p,x,t)= p^2/2 +K \cos(x) \, \delta_1(t) \), where \( \delta_1(t) \) is a periodic \( \delta- \)function with period 1 in time. The dynamics is given by a sequence of free propagations interleaved with periodic kicks.

Examples of the Poincare sections of the standard map on a torus are shown in the following Figs. 1,2,3.

Below the critical parameter \( K < K_c \) (Fig.1) the invariant Kolmogorov-Arnold-Moser (KAM) curves restrict the variation of momentum \( p \) to be bounded. The golden KAM curve with the rotation number \[ r=r_g=(\sqrt{5}-1)/2 =0.618033... \] is destroyed at \(K=K_g=0.971635... \) [3], [4] (Fig.2). This Fig. shows a generic phase space structure typical for various area-preserving maps with smooth generating functions: stability islands are embedded in a chaotic sea, similar structure appears on smaller and smaller scales. In a vicinity of a critical invariant curve, with a golden tail in a continued fraction expansion of \( r\), the phase space structure is universal for all smooth maps [4]. Above the critical value \( K > K_c \) (see Fig.3 showing a chaotic component and visible islands of stability) the variation of \(p\) becomes unbounded and is characterized by a diffusive growth \( p^2 \sim D_0 t\) with number of map iterations \( t \). Here \( D_0 \) is a diffusion rate with \( D_0 \approx (K-K_c)^3/3 \) for \( K_c < K < 4 \) and \( D_0 \approx D_{ql}=K^2/2 \) for \( 4 < K \) [2], [5]. There are strong arguments in favor of the equality \( K_c = K_g \) but rigorously it is only proven that there are no KAM curves for \( K > 63/64 = 0.984375 \) [6]. With the numerical results [3], [4] this implies inequality for the global chaos border, \( K_g \leq K_c < 63/64 \).

A simple analytical criterion proposed in 1959 and now known as the Chirikov resonance-overlap criterion [7] gives the chaos border \( K_c = \pi^2/4\) [1] and after some improvements leads to \( K_c \approx 1.2\) [2],[8]. This accuracy is not so impressive compared to modern numerical methods but still up to now this criterion remains the only simple analytical tool for determining the chaos border in various Hamiltonian dynamical systems.

The Kolmogorov-Sinai entropy of the map is well described by relation \( h \approx \ln(K/2)\) valid for \( K > 4 \)[1], [2].

## Contents |

### Universality and Applications

The map (1) describes a situation when nonlinear resonances are equidistant in phase space
that corresponds to a local description of dynamical chaos. Due to this property various
dynamical systems and maps can be locally reduced to the standard map
and due to this reason the term *standard map* was coined in [2].
Thus, the standard map describes a universal, generic behavior of area-preserving maps
with divided phase space when integrable islands of stability are surrounded
by a chaotic component.
A short list of systems reducible to the standard map is given below:

- chaotic layer around separatrix of a nonlinear resonance induced by a monochromatic force (the whisker map) [2]

- particle dynamics in accelerators [11]

- microwave ionization of Rydberg atoms (linked to the Kepler map) [14] and autoionization of molecular Rydberg states [15]

- electron magnetotransport in a resonant tunneling diode [16]

### Open Problems

- In spite of fundamental advances in ergodic theory [17], a rigorous proof of the existence of a set of positive measure of orbits with positive entropy is still missing, even for specific values of \( K \) (see e.g. [18]).

- What are the fractal properties of critical chaos parameter \( K_c(r) \) as a function of arithmetic properties of the rotation number \( r \) of KAM curve? do local maxima correspond only to a golden tail of continuous fraction expansion [3], [4] or they may have tails with Markov numbers as it is conjectured in [19]? (see also [20])

- Due to trajectory sticking around stability islands the statistics of Poincare recurrences in Hamiltonian systems with divided phase space (see e.g. Fig.2 with a critical golden KAM curve) is characterized by an algebraic decay \( P(\tau) \propto 1/\tau^\alpha \) with \( \alpha \approx 1.5 \) while a theory based on the universality in a vicinity of critical golden curve gives \( \alpha \approx 3 \ ;\) this difference persists up to 10
^{13}map iterations; as a result correlation functions decay rather slowly \( C(\tau) \sim \tau P(\tau) \propto 1/\tau^{\alpha-1} \) that can lead to a divergence of diffusion rate \( D \sim \tau C(\tau) \) (see [21] and Refs. therein)

## Quantum Map

The quantization of the standard map is obtained by considering variables in (1) as the Heisenberg operators with the commutation relation \( [p,x] = -i \hbar \), where \( \hbar \) is an effective dimensionless Planck constant. In a same way it is possible to use the Schrödinger equation with the Hamiltonian \( H(\hat{p},\hat{x},t) \) given above and \( \hat{p}=-i\hbar \partial/\partial x \). Integration on one period gives the quantum map for the wave function \( \psi \):

\[\tag{2} \bar{\psi} = \hat{U} \psi = e^{-i{\hat p}^2/2\hbar} e^{-i K/\hbar \cos {\hat x}} \psi \]

where bar marks the new value of \( \psi \) after one map iteration. Due to space periodicity of the Hamiltonian the momentum can be presented in the form \( p=\hbar (n + \beta) \), where \( n \) is an integer and \( \beta \) is a quasimomentum preserved by the evolution operator \( \hat U \) . The case with \( \beta =0 \) corresponds to a periodic boundary conditions with \( \psi(x+2\pi) =\psi(x) \) and is known as the kicked rotator introduced in [22].

Other notations with \( \hbar \rightarrow T\), \( K/\hbar \rightarrow k\) are also used to mark the dependence on the period \( T\) between kicks, then \( K = k T\). The diffusion rate over quantum levels \( n \) is \( D=D_0/\hbar^2= n^2/t \approx K^2/2\hbar^2 =k^2/2\), thus the rotator energy \( E = <n^2>/2\) grows linearly with time. Quantum interference effects lead to a suppression of this semiclassical diffusion [22] on the diffusive time scale \( t_D \) so that the quantum probability spreads effectively only on a finite number of states \( \Delta n \sim \sqrt{D t_D} \) (Fig.4). According to the analytical estimates obtained in [23]:

\[\tag{3} t_D \sim \Delta n \sim D \sim k^2 \sim D_0/\hbar^2 . \]

This diffusive time scale is much larger than the Ehrenfest time scale [23], [24] \( t_E \sim \ln(1/\hbar)/2h \) after which a minimal coherent wave packet spreads over the whole phase space due to the exponential instability of classical dynamics. For \( t < t_E \) a quantum wave packet follows the chaotic dynamics of a classical trajectory as it is guaranteed by the Ehrenfest theorem [23]. For the case of Fig.4 the Kolmogorov-Sinai entropy \( h \approx 1 \) and the Ehrenfest time \( t_E \sim 1 \) is extremely short comparing to the diffusive time \( t_D \sim D \sim 200 \). The quantum suppression of chaotic diffusion is similar to the Anderson localization in disordered systems if to consider the level number as an effective site number in a disordered lattice, such an analogy has been established in [25]. However, in contrast to a disordered potential for the case of Anderson localization, in the quantum map (2) diffusion and chaos have a pure deterministic origin appearing as a result of dynamical chaos in the classical limit.

Due to that this phenomenon is called the dynamical localization. The eigenstates of the unitary evolution operator \( \hat U \) are exponentially localized over momentum states \( \psi_m(n) \sim \exp(-|n-m|/l)/\sqrt{l} \) with the localization length \( l \sim \Delta n \sim t_D \) given by the relation [26], [27] \[\tag{4} l=D(K)/2 =D_0(K)/2\hbar^2, \]

where \( D \) is the semiclassical diffusion expressed via a square number of levels per period of perturbation. For \( \hbar = T > 1 \) the chaos parameter \( K \) in the dependence \( D(K) \) should be replaced by its quantum value \( K \rightarrow K_q = 2 k \sin T/2 \) [27]. The quantum localization length \( l \) repeats the characteristic oscillations of the classical diffusion as it is shown in Fig.5. The relation (4) assumes that \( T/4\pi \) is a typical irrational number while for rational values of this ratio the phenomenon of quantum resonance takes place and the energy grows quadratically with time for rational values of quasimomentum [28]. The derivations of the relation (4) based on the field theory methods applied to dynamical systems with chaotic diffusion can be find in [29], [30] (see also Refs. therein).

If the quantum map (2) is taken on a torus with \( N \) levels then the level spacing statistics is described by the Poisson law for \( N \gg l \) and by the Wigner-Dyson law of the random matrix theory for \( N \ll l \) [24],[31]. In the later case the quantum eigenstates are ergodic on a torus in agreement with the Shnirelman theorem and the level spacing statistics agrees with the Bohigas-Giannoni-Schmit conjecture (see books on quantum chaos in Recommended Reading).

The quantum map (2) was built up experimentally with cold atoms in a kicked optical lattice by the group of M.Raizen [32]. Such a case corresponds to a particle in an infinite periodic lattice with averaging over many various \( \beta \). The quantum resonances at \( \beta \approx 0 \) were also experimentally observed with the Bose-Einstein condensate (BEC) in [33]. Quantum accelerator modes for kicked atoms falling in the gravitational field were found and analyzed in [34].

### Extensions and Related Quantum Systems

Due to universal properties of the standard map its quantum version also finds applications for various systems and various physical effects:

- dynamical localization for ionization of excited hydrogen atoms in a microwave field

was theoretically predicted in [35] and was experimentally observed by the group of P.Koch [36] (see more details in [14],[37],[38])

- quantum particle in a triangular well and monochromatic field with a quantum delocalization transition [39]

- the kicked Harper model where in contrast to the relation (4) the quantum delocalization can take place due to quasi-periodicity of unperturbed spectrum (see [40], [41] and Refs. therein)

- 3D Anderson transition in kicked rotator with modulated kick strength and quantum transport in mesoscopic conductors (see [42] and Refs. therein)

- dissipative quantum chaos [43]

- fractal Weyl law for the quantum standard map with absorption (see [44] and Refs. therein)

## Time Reversibility and Boltzmann - Loschmidt Dispute

The statistical theory of gases developed by Boltzmann leads to macroscopic irreversibility and entropy growth even if dynamical equations of motion are time reversible. This contradiction was pointed out by Loschmidt and is now known as the Loschmidt paradox. The reply of Boltzmann relied on the technical difficulty of velocity reversal for material particles: a story tells that he simply said "then go and do it" [45]. The modern resolution of this famous dispute, which took place around 1876 in Wien, came with the development of the theory of dynamical chaos (see e.g. [8], [17]). Indeed, for chaotic dynamics the Kolmogorov-Sinai entropy is positive and small perturbations grow exponentially with time, making the motion practically irreversible. This fact is convenient to illustrate on the example of the standard map which dynamics is time reversible, e.g. by inverting all velocities at the middle of free propagation between two kicks (see Fig.6). This explanation is valid for classical dynamics, while the case of quantum dynamics requires special consideration. Indeed, in the quantum case the exponential growth takes place only during the rather short Ehrenfest time, and the quantum evolution remains stable and reversible in presence of small perturbations [46] (see Fig.7). Quantum reversibility in presence of various perturbations has been actively studied in recent years and is now described through the Loschmidt echo (see [47] and Refs. therein). A method of approximate time reversal of matter waves for ultracold atoms in the regime of quantum chaos, like those in [32], [33], is proposed in [48]. In this method a large fraction of the atoms returns back even if the time reversal is not perfect. This fraction of the atoms exhibits Loschmidt cooling which can decrease their temperature by several orders of magnitude. At the same time a kicked BEC of attractive atoms (soliton) described by the Gross-Pitaevskii equation demonstrates a truly chaotic dynamics for which the exponential instability breaks the time reversibility [49]. However, since a number of atoms in BEC is finite and since BEC is a really quantum object one should expect that the Ehrenfest time is still very short and hence the time reversibility should be preserved in presence of small errors if the second quantization is taken into account.

## Links to Other Physical Topics

### Frenkel-Kontorova Model

The Frenkel-Kontorova model describes a one-dimensional chain of atoms/particles with harmonic couplings placed in a periodic potential [50]. This model was introduced with the aim to study crystal dislocations but it also successfully applies for the description of commensurate-incommensurate phase transitions, epitaxial monolayers on the crystal surface, ionic conductors, glassy materials, charge-density waves and dry friction [51]. The Hamiltonian of the model is \( H= \sum_i \left({P_i^2 \over 2} + {(x_i -x_{i-1})^2 \over 2}- K \cos x_i \right)\), where \( P_i, x_i \) are momentum and position of atom \( i \). At the equilibrium the momenta \( P_i =0\) and \( \partial H/\partial x_i =0\) so that the positions of atoms are described by the map (1) with \( p_{i+1} = x_{i+1}- x_i , \; p_{i+1}= p_i +K\sin x_i\). The density of atoms corresponds to the rotation number \( r \) of an invariant KAM curve. For the golden density with \( r =r_g\) the chain slides in the periodic potential for \( K < K_g \) (KAM curve regime) while for \( K > K_g \) the transition by the breaking of analyticity, or Aubry transition, takes place, the chain becomes pinned and atoms form an invariant Cantor set called cantorus (see [52] and Aubry-Mather theory). In this regime the phonon spectrum has a gap so that the phonon excitations are suppressed at low temperature. The mathematical Aubry-Mather theory guarantees that the ground state of the chain exists and is unique. However there exist exponentially many static equilibrium configurations which are exponentially close to the energy of the ground state. The energies of these configurations form a fractal quasi-degenerate band structure and become mixed at any physically realistic temperature. Thus, such configurations can be viewed as a dynamical spin glass. For a case of Coulomb interactions between particles (e.g. ions or electrons) one obtains a problem of Wigner crystal in a periodic potential which again is locally described by the Frenkel-Kontorova model since the map (1) gives the local description of the dynamics. For the quantum Frenkel-Kontorova model the dynamics of atoms (ions) in the chain is quantum. In this case the quantum vacuum fluctuations and instanton tunneling lead to a quantum melting of pinned phase: above a certain effective Planck constant a quantum phase transition takes place from pinned instanton glass to sliding phonon gas (see [53] and Refs. therein).

### Quantum Computing

One iteration of maps (1) and (2) can be simulated on a quantum computer in a polynomial number of quantum gates for an exponentially large vector representing a Liouville density distribution or a quantum state. The quantum algorithm of such a quantum computation is described in [54], effects of quantum errors are analyzed in [55] (see also Refs. therein).

## Historical Notes

The standard map (1) in a form of recursive relation for atoms in a periodic potential appears already in the works of Kontorova and Frenkel [50]. As a dynamical map it first appeared as a description of electron dynamics in a new relativistic accelerator proposed by V.I.Veksler (Dokl. Akad. Nauk SSSR 43: 346 (1944)). The regime of a stable regular acceleration was studied later also by A.A.Kolomensky (Zh. Tekh. Fiz. 30: 1347 (1960)) and S.P.Kapitsa, V.N.Melekhin ("Microtron", Nauka, Moscow (1969) in Russian). Among the early researchers of model (1) was also British physicist J.B.Taylor (unpublished reports). The description of chaos in map (1) and its main properties, including chaos border, diffusion rate and positive entropy, was given in [1]. The term "standard map" appeared in [2], "Chirikov-Taylor map" [8] and "Chirikov standard map" [16] are also used, the quantum standard map or kicked rotator was first considered in [22]. Appearance of other terms: Kolmogorov-Arnold-Moser theory [1], Arnold diffusion [1], Kolmogorov-Sinai entropy [2], Ehrenfest time [24].

## Recommended Reading

B.V.Chirikov, "Research concerning the theory of nonlinear resonance and stochasticity", Preprint N 267, Institute of Nuclear Physics, Novosibirsk (1969), (Engl. Trans., CERN Trans. 71-40 (1971))

B.V.Chirikov, "A universal instability of many-dimensional oscillator systems", Phys. Rep. 52: 263 (1979).

B.V.Chirikov, "Time-dependent quantum systems" in "Chaos and quantum mechanics", Les Houches Lecture Series, Vol. 52, pp.443-545, Eds. M.-J.Giannoni, A.Voros, J.Zinn-Justin, Elsevier Sci. Publ., Amsterdam (1991)

A.J.Lichtenberg, M.A.Lieberman, "Regular and chaotic dynamics", Springer, Berlin (1992).

F.Haake, "Quantum signatures of chaos", Springer, Berlin (2001).

L.E.Reichl, "The Transition to chaos in conservative classical systems and quantum manifestations", Springer, Berlin (2004).

**Internal references**

- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.

- Martin Gutzwiller (2007) Quantum chaos. Scholarpedia, 2(12):3146.

## External Links

Selected publications of Boris Chirikov [1]

Sputnik of Chaos [2]

Google query for "standard map" [3]

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## See also

Hamiltonian systems, Mapping, Chaos, Kolmogorov-Arnold-Moser Theory, Kolmogorov-Sinai entropy, Aubry-Mather theory, Quantum chaos

## Updates added after 2008

Here some additional features of the standard map, which appeared after 2008 or were omitted in the 2008 article edition, are presented.

### Mathematical aspects

Mathematical results are presented here for the classical (UM1-UM3) and quantum (UM4) standard map.

- UM1) A rigorous proof is given in the standard map for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant \( K > K_0 \), where \( K_0 \) depends on a coding representation of a trajectory. The obtained chaotic trajectories correspond to stationary configurations of the Frenkel-Kontorowa model with a finite (non-zero) photon gap. The concept of anti-integrability emerges from the theorems presented in [UM1].

- UM2) The large basic sets, which fill in the torus as the parameter runs to infinity, are constructed. It is proven that, for a residual set of large parameters, these basic sets accumulated by elliptic periodic islands. It is shown there exists a \( K_0 > 0 \) and a dense set of parameters \( K_0 \leq K < \infty \) for which the standard map exhibits homoclinic tangencies [UM2].

- UM3) It is proven that stochastic sea of the standard map has full Hausdorff dimension

for sufficiently large topologically generic parameters [UM3].

- UM4) For the quantum standard map with a generic quadratic rotational spectrum the localization is proven for small kick amplitudes [UM4].

### Physical aspects and numerical results

Results for the Ulam method for the standard map are presented in [UP1], Poincare recurrences in [UP2] and the fractal Weyl law for Perron-Frobenius operators in [UP3]. The results of the field theory for the quantum standard map are discussed in [UP4].

- UP1) In 1960 Ulam proposed a method, known now as the Ulam method, for construction a finite size matrix approximate for the Perron-Frobenius operator of a dynamical system in a continuous phase space [UP1a]. The method allows to construct numerically a matrix of Markov transitions between cells in a discretized phase space with fully chaotic dynamics. The method is knwon to be converging to the continuous limit of Perron-Frobenius operator when the phase space is fully chaotic. However, for systems with a divided phase space an effective noise induced by a finite cell size breaks the convergence leading to a destruction of the invariant KAM curve. This problem was resolved in [UP1b] by a generalized Ulam method in which the Markov transitions between cells are generated by one chaotic trajectory starting inside a chaotic component. The extensive numerical studies based on the Arnoldi method show that the Ulam approximate of the Perron-Frobenius operator \( S \) (UPFO) on a chaotic component converges to a continuous limit. Typically, in this regime the spectrum of relaxation modes is characterized by a power law decay for small relaxation rates. The numerical results show that the exponent of this decay is approximately equal to the exponent of Poincare recurrences in such systems. The eigenmodes, or eigenstates, show links with trajectories sticking around stability islands. An example of such an eigenstate is shown in Fig.8U. The spectrum of UFPO \( S \) is shown in Fig.9U.

- UP2) The numerical studies of the Poincare recurrences in the standard map with the critical golden curve have been performed in [UP2a] with a new survival Monte Carlo method which allows to study recurrences on times changing by ten orders of magnitude (see Fig.10U). The comparison is done with the results of generalized Ulam method and localization properties of eigenstates of the Ulam matrix are analyzed. The recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. On the investigated scales the Poincare decay exponent, in <math P \propto 1/t^{\alpha} </math> is found to be approximately </math> \alpha \approx 1.58 </math> in a satisfactory agreement with early and more recent studies of various symplectic 2D maps [19], [20], [UP2c], [UP2d] thus indicating the universality of the exponent. The detailed theoretical explanation of the algebraic decay of Poincare recurrences and the exponent value is still lucking. It is interesting to add a few notes: A) in the standard map with dissipation (e.g. the right hand side of upper line Eq.(1) is multiplied by a coefficient less than unity) in the regime of a strange attractor the decay of Poincare recurrences is exponential [UP2c]; B) in the symplectic case additional noise leads to a diffusive type decay \( P \propto 1/\sqrt{t} \) on a long time scale due to diffusion inside stability islands, when they are present; C) in the quantum standard map the quantum Poincare recurrences are characterized by a decay \( P \propto 1/t \) due to quantum tunneling inside the stability islands [UP2e].

- UP3) For the standard map with absorption or dissipation in a chaotic regime it is show that the Ulam approximate of Perron-Frobenius operator is characterized by the fractal Weyl law with the exponent given by a half of the fractal dimension of related chaotic repeller or strange attractor [UP3] (see also the section Ulam networks in Google matrix). The fractal Weyl law for the quantum standard map with absorption is analyzed in [44].

- UP4) The field theory methods for the quantum standard map have been developed in [UP4]. It is shown that the effective theory describing the long wave length physics of the system is precisely the supersymmetric nonlinear sigma-model for quasi one-dimensional metallic wires. It is shown that the localization length is given by Eq.(4). This proves that the analogy between chaotic systems with dynamical localization and disordered metals can indeed be exact, as claimed by the authors. However, this approach misses certain properties of quantum evolution, thus it gives the finite localization length for the quantum kicked Harper model in the chaotic regime while the numerical results show the existence of delocalized quantum phase.

### Related models and systems

- UR1) The properties of quantum kicked rotator with rotational phases depending linearly on level number in Eq.(2) and generalized to any number of dimensions are considered in [UR1a],[UR1b], [UR1c]. The mathematical prove of localization of all eigenstates is given for small [UR1b] and arbitrary kick amplitudes [UR1c]. This result is rather clear from the view point of classical dynamics where linear dependence of Hamiltonian on actions (linear spectrum) leads to complete integrability of motion.

### References added after 2008

[UM1] |
S.Aubry, G.Abramovici, "Chaotic trajectories in the standard map. The concept of anti-integrability", Physica D 43: 199 (1990). |

[UM2] |
P.Duarte, "Plenty of elliptic islands for the standard family of area preserving maps", Annales de l'I.H.P. Sec. C 11(4): 359 (1994). |

[UM3] |
A.Gorodentski, "On stochastic sea of the standard map", Comm. Math. Phys. 309(1): 155 (2012). |

[UM4] |
J.Bourgain, "Estimates on Green's functions, localization and the quantum kicked rotor model", Ann. Math. 156: 249 (2002). |

[UP1a] |
S.M.Ulam, "A Collection of mathematical problems, Interscience tracs in pure and applied mathematics", Interscience, New York 8: 73 (1960). |

[UP1b] |
K.M.Frahm, D.L.Shepelyansky, "Ulam method for the Chirikov standard map", Eur. Phys. J. B 76: 57 (2010). |

[UP2a] |
K.M.Frahm, D.L.Shepelyansky, "Poincare recurrences and Ulam method for the Chirikov standard map", Eur. Phys. J. B 86: 322 (2013). |

[UP2b] |
M.Weissm L.Hufnagel, R.Ketzmerick, "Universal power-law decay in Hamiltonian systems?", Phys. Rev. Lett. 100: 184101 (2008). |

[UP2c] |
B.V.Chirikov, D.L.Shepelyansky, "Correlation properties of dynamical chaos in Hamiltonian systems", Physica D 13: 395 (1984), [24], detailed BINP preprint in RUssian [25] |

[UP2d] |
G.Cristadoro, R.Ketzmerick, "Universality of algebraic decays in Hamiltonian systems", Phys. Rev. Lett. 100: 184191 (2008). |

[UP2e] |
G.Casati, G.Maspero, D.L.Shepelyansky, "Quantum poincare recurrences", Phys. Rev. Lett. 82: 524 (1999). |

[UP3] |
L.Ermann, D.L.Shepelyansky, "Ulam method and fractal Weyl law for Perron-Frobenius operators", Eur. J. B 75: 299 (2010). |

[UP4] |
A.Altland, M.Zinbauer, "Field theory of the quantum kicked rotor", Phys. Rev. Lett. 77: 4536 (1996). |

[UR1a] |
S.Fishman, D.R.Grenpel, R.E.Prange, "Localization in an incommensurate potential: an exactly solvable model", Phys. Rev. Lett. 49: 833 (1982). |

[UR1b] |
J.Bellissard, R.Lima, E.Scoppola, "Localization in v-dimensional incommensurate structures", Comm. Math. Phys. 88: 465 (1983). |

[UR1c] |
A.L.Figotin, L.A.Pastur, "An exactly solvable model of a multidimensional incommensurate structure localization for linear spectrum", Comm. Math. Phys. 95: 401 (1984). |

### See also added after 2008

Anderson localization and quantum chaos maps, Cold atom experiments in quantum chaos, Google matrix, kicked Harper model, Microwave ionization of hydrogen atoms