On the organisation of transient chaos-application to irregular scattering

New developments in classical chaotic scattering

Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described.

Fractals and dynamical chaos in a two-dimensional Lorentz gas with sinks

We consider a two-dimensional periodic reactive Lorentz gas, in which a moving point particle undergoes elastic collisions on fixed hard disks and annihilates on absorbing disks, called sinks. We present clear evidence of the existence of a fractal repeller in this open system. Moreover, we establish a relation between the reaction rate, describing the macroscopic evolution of the system, and two characteristic quantities of the microscopic chaos: the average Lyapunov exponent and the Hausdorff codimension of the fractal repeller.

Dynamical ensembles in nonequilibrium statistical mechanics and their representations

The stationary states of driven systems of particles are considered from the point of view of the invariant probability distributions in the phase space which characterize them. The main features of various representations of such distributions are reviewed, and a brief derivation of the one based on orbital measures is given. We mention the limits of the mathematical derivations, and discuss the expected range of applicability beyond such limits. (c) 1998 American Institute of Physics.

Chaotic scattering: An introduction

In recent years chaotic behavior in scattering problems has been found to be important in a host of physical situations. Concurrently, a fundamental understanding of the dynamics in these situations has been developed, and such issues as symbolic dynamics, fractal dimension, entropy, and bifurcations have been studied. The quantum manifestations of classical chaotic scattering is also an extremely active field, with new analytical techniques being developed and with experiments being carried out. This issue of Chaos provides an up-to-date survey of the range of work in this important field of study.

Conditions for the abrupt bifurcation to chaotic scattering

One of the generic ways in which chaotic scattering can come about as a system parameter is varied is the so-called "abrupt bifurcation" in which the scattering is nonchaotic on one side of the bifurcation and is chaotic and hyperbolic on the other side. Previous work demonstrating the abrupt bifurcation [S. Bleher et al., Phys. Rev. Lett. 63, 919 (1989); Physica D 46, 87 (1990)] was primarily for the case where the scattering potential had maxima ("hilltops") which had locally circular isopotential contours. Here we extend these considerations to the more general case of locally elliptically shaped isopotential contours at the hilltops. It turns out that the conditions for the abrupt bifurcation change drastically as soon as even a small amount of noncircularity is included (i.e., the circular case is singular). The illustrative case of scattering from three isolated potential hills is dealt with in detail. One interesting result is a simple geometrical sufficient condition for an abrupt bifurcation in the case of large enough ellipticity of the hill with lowest potential at its hilltop.

Application of scattering chaos to particle transport in a hydrodynamical flow

The dynamics of a passive particle in a hydrodynamical flow behind a cylinder is investigated. The velocity field has been determined both by a numerical simulation of the Navier-Stokes flow and by an analytically defined model flow. To analyze the Lagrangian dynamics, we apply methods coming from chaotic scattering: periodic orbits, time delay function, decay statistics. The asymptotic delay time statistics are dominated by the influence of the boundary conditions on the wall and exhibit algebraic decay. The short time behavior is exponential and represents hyperbolic effects.

Statistical properties of chaos demonstrated in a class of one-dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.