Tamás Tél

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.

Determination of fractal dimensions for geometrical multifractals

Two independent approaches, the box counting and the sand box methods are used for the determination of the generalized dimensions (Dq) associated with the geometrical structure of growing deterministic fractals. We find that the multifractal nature of the geometry results in an unusually slow convergence of the numerically calculated Dqs to their true values. Our study demonstrates that the above-mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres. In this case the latter approach provides better estimates of the generalized dimensions.

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.

Fractals, Multifractals, and Thermodynamics

The basic concept of fractals and multifractals are introduced for pedagogical purposes, and the present status is reviewed. The emphasis is put on illustrative examples with simple mathematical structures rather than on numerical methods or experimental techniques. As a general characterization of fractals and multifractals a thermodynamical formalism is introduced, establishing a connection between fractal properties and the statistical mechanics of spin chains.

Tracer dynamics in open hydrodynamical flows as chaotic scattering

Abstract Methods coming from the theory of chaotic scattering are applied to the advection of passive particles in an open hydrodynamical flow. In a region of parameters where a von Karman vortex street is present with a time periodic velocity field behind a cylinder in a channel, particles can temporarily be trapped in the wake. They exhibit chaotic motion there due to the presence of a nonattracting chaotic set. The experimentally well- known concept of streaklines is interpreted as a structure visualising asymptotically the unstable manifold of the full chaotic set. The evaluation of streaklines can also provide characteristic numbers of this invariant set, e.g. topological entropy, Lyapunov exponent, escape rate. The time delay distributions are also evaluated. We demonstrate these ideas with the aid of both computer simulations of the Navier-Stokes equations and analytical model computations. Properties that could be measured in a laboratory experiment are discussed.

Leaking chaotic systems

There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. A unified treatment of leaking systems is provided and applications to different physical problems, in both the classical and quantum pictures, are reviewed. The treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption and/or transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the limacon family of billiards illustrate the main text. Regarding billiard dynamics, it is emphasized that a correct discrete-time representation can be given only in terms of the so-called true-time maps, while traditional Poincare maps lead to erroneous results. Perron-Frobenius-type operators are generalized so that they describe true-time maps with partial leaks.

Chaotic advection in the velocity field of leapfrogging vortex pairs

The advection problem of passive tracer particles in the time-periodic velocity field of leapfrogging vortex pairs is investigated in the context of chaotic scattering. We numerically determine a few basic unstable periodic orbits of the tracer dynamics, and the non-attracting chaotic set responsible for the motion of particles injected in front of the vortex system. The latter consists of two parts: a hyperbolic component based on strongly unstable periodic orbits, and a non-hyperbolic component that is close to KAM surfaces, The invariant manifolds of the chaotic set are also plotted and their relevance for the particle dynamics is discussed. The tracer dynamics has one single dimensionless parameter: the energy of the vortex system. As a new phenomenon, we point out the existence of stable bounded trajectories between the vortex pairs at sufficiently large energies. A quantitative characterization of the tracer dynamics in terms of the so-called free energy function is given and the multifractal spectrum of Lyapunov exponents, the escape rate and other characteristics of the transient chaotic motion are determined.

chaotic advection, diffusion, and reactions in open flows

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.

Controlling transient chaos

The method of Ott, Grebogi and Yorke (1990) is extended to control transient chaos. The controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor. The time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime. The number of controlled trajectories, however, decreases the maximum perturbation according to a power law. Applicability to experimental situations and comparison with permanent chaotic cases are discussed.

Chemical or biological activity in open chaotic flows.

We investigate the evolution of particle ensembles in open chaotic hydrodynamical flows. Active processes of the type A+B-->2B and A+B-->2C are considered in the limit of weak diffusion. As an illustrative advection dynamics we consider a model of the von Kármán vortex street, a time-periodic two-dimensional flow of a viscous fluid around a cylinder. We show that a fractal unstable manifold acts as a catalyst for the process, and the products cover fattened-up copies of this manifold. This may account for the observed filamental intensification of activity in environmental flows. The reaction equations valid in the wake are derived either in the form of dissipative maps or differential equations depending on the regime under consideration. They contain terms that are not present in the traditional reaction equations of the same active process: the decay of the products is slower while the productivity is much faster than in homogeneous flows. Both effects appear as a consequence of underlying fractal structures. In the long time limit, the system locks itself in a dynamic equilibrium state synchronized to the flow for both types of reactions. For particles of finite size an emptying transition might also occur leading to no products left in the wake.