Roberto Artuso

Spectral properties and anomalous transport in a polygonal billiard.

We analyze a class of polygonal billiards, whose behavior is conjectured to exhibit a variety of interesting dynamical features. Correlation functions are numerically investigated, and in a subclass of billiard tables they give indications about a singular continuous spectral measure. By lifting billiard dynamics we are also able to study transport properties: the (normal or anomalous) diffusive behavior is theoretically connected to a scaling index of the spectral measure; the proposed identity is shown to agree with numerical simulations. (c) 2000 American Institute of Physics.

Numerical experiments on billiards

We investigate decay properties of correlation functions in a class of chaotic billiards. First we consider the statistics of Poincaré recurrences (induced by a partition of the billiard): the results are in agreement with theoretical bounds by Bunimovich, Sinai, and Bleher, and are consistent with a purely exponential decay of correlations out of marginality. We then turn to the analysis of the velocity-velocity correlation function: except for intermittent situations, the decay is purely exponential, and the decay rates scale in a simple way with the (uniform) curvature of the dispersing arcs. A power-law decay is instead observed when the system is equivalent to an infinite-horizon Lorentz gas. Comments are given on the behaviour of other types of correlation functions, whose decay, during the observed time scale, appears slower than exponential.

Periodic orbit theory of strongly anomalous transport

We establish a deterministic technique to investigate transport moments of an arbitrary order. The theory is applied to the analysis of different kinds of intermittent one-dimensional maps and the Lorentz gas with infinite horizon: the typical appearance of phase transitions in the spectrum of transport exponents is explained.

Effects of atomic interactions on quantum accelerator modes

We consider the influence of the inclusion of interatomic interactions on the

ESTIMATING HYPERBOLICITY OF CHAOTIC BIDIMENSIONAL MAPS

\ensuremath{\delta}

Delocalized and resonant quantum transport in nonlinear generalizations of the kicked rotor model.

-kicked accelerator model. Our analysis concerns in particular quantum accelerator modes, namely quantum ballistic transport near quantal resonances. The atomic interaction is modeled by a Gross-Pitaevskii cubic nonlinearity, and we address both attractive (focusing) and repulsive (defocusing) cases. The most remarkable effect is enhancement or damping of the accelerator modes, depending on the sign of the nonlinear parameter. We provide arguments showing that the effect persists beyond mean-field description, and lies within the experimentally accessible parameter range.

Periodic orbit theory of deterministic diffusion

We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. [2007] to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so-called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for nonperiodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, such splitting being noninvariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov–Taylor standard map, we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.

Weak chaos and anomalous transport: a deterministic approach

We analyze the effects of a nonlinear cubic perturbation on the delta-kicked rotor. We consider two different models, in which the nonlinear term acts either in the position or in the momentum representation. We numerically investigate the modifications induced by the nonlinearity in the quantum transport in both localized and resonant regimes and a comparison between the results for the two models is presented. Analyzing the momentum distributions and the increase of the mean square momentum, we find that the quantum resonances asymptotically are very stable with respect to nonlinear perturbation of the rotors phase evolution. For an intermittent time regime, the nonlinearity even enhances the resonant quantum transport, leading to superballistic motion.

Anomalous dynamics and the choice of Poincaré recurrence set

We review recent results on deterministic diffusion, obtained via periodic orbit expansions. The general formalism is applied to a number of one-dimensional examples. We also discuss how anomalous diffusion can be analyzed by similar techniques.

Recycling Parrondo games

Abstract We review how transport properties for chaotic dynamical systems may be studied through cycle expansions, and show how anomalies can be quantitatively described by hierarchical sequences of periodic orbits.