Chaotic Dynamics

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain non-periodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a non-periodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts.

The dynamics of Rydberg states of atomic hydrogen perturbed simultaneously by a static electric field and a resonant microwave field of elliptical polarization is analysed in the quantum perturbative limit of small amplitudes. For some configurations, the secular motion (i.e. evolution in time of the elliptical electronic trajectory) may be classically predominantly chaotic. By changing the orientation of the static field with respect to the polarization of the microwave field, one can modify the global symmetries of the system and break any generalized time-reversal invariance. This has a dramatic effect on the statistical properties of the energy levels.

Quantized, compact graphs were shown to be excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity we show that they display all the features which characterize scattering systems with an underlying classical chaotic dynamics. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of Random Matrix Theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems, and their semiclassical description.

We address the problem of the classical deterministic dynamics of a particle in a periodic asymmetric potential of the ratchet type. We take into account the inertial term in order to understand the role of the chaotic dynamics in the transport properties. By a comparison between the bifurcation diagram and the current, we identify the origin of the current reversal as a bifurcation from a chaotic to a periodic regime. Close to this bifurcation, we observed trajectories revealing intermittent chaos and anomalous deterministic diffusion.

Lagrangian chaos is experimentally investigated in a convective flow by means of Particle Tracking Velocimetry. The Finite Size Lyapunov Exponent analysis is applied to quantify dispersion properties at different scales. In the range of parameters of the experiment, Lagrangian motion is found to be chaotic. Moreover, the Lyapunov depends on the Rayleigh number as Ra^(1/2). A simple dimensional argument for explaining the observed power law scaling is proposed.

We study the spatial patterns formed by interacting populations or reacting chemicals under the influence of chaotic flows. In particular, we have considered a three-component model of plankton dynamics advected by a meandering jet. We report general results, stressing the existence of a smooth-filamental transition in the concentration patterns depending on the relative strength of the stirring by the chaotic flow and the relaxation properties of planktonic dynamical system. Patterns obtained in open and closed flows are compared.

The statistical properties of a classical electromagnetic field in interaction with matter are numerically investigated on a one-dimensional model of a radiant cavity, conservative and with finite total energy. Our results suggest a trend towards equipartition of energy, with the relaxation times of the normal modes of the cavity increasing with the mode frequency according to a law, the form of which depends on the shape of the charge distribution.

We study eigenstates of chaotic billiards in the momentum representation and propose the radially integrated momentum distribution as useful measure to detect localization effects. For the momentum distribution, the radially integrated momentum distribution, and the angular integrated momentum distribution explicit formulae in terms of the normal derivative along the billiard boundary are derived. We present a detailed numerical study for the stadium and the cardioid billiard, which shows in several cases that the radially integrated momentum distribution is a good indicator of localized eigenstates, such as scars, or bouncing ball modes. We also find examples, where the localization is more strongly pronounced in position space than in momentum space, which we discuss in detail. Finally applications and generalizations are discussed.

We investigate the decay process from a time dependent potential well in the semiclassical regime. The classical dynamics is chaotic and the decay rate shows an irregular behavior as a function of the system parameters. By studying the weak-chaos regime we are able to connect the decay irregularities to the presence of nonlinear resonances in the classical phase space. A quantitative analytical prediction which accounts for the numerical results is obtained.

Chaotization of supercritical (Z>137) hydrogenlike atom in the monochromatic field is investigated. A theoretical analysis of chaotic dynamics of the relativistic electron based on Chirikov criterion is given. Critical value of the external field at which chaotization will occur is evaluated analytically. The diffusion coefficient is also calculated.