Chaotic scattering: An introduction

Relativistic chaotic scattering: Unveiling scaling laws for trapped trajectories

In this paper we study different types of phase space structures which appear in the context of relativistic chaotic scattering. By using the relativistic version of the Hénon-Heiles Hamiltonian, we numerically study the topology of different kind of exit basins and compare it with the case of low velocities in which the Newtonian version of the system is valid. Specifically, we numerically study the escapes in the phase space, in the energy plane, and in the β plane, which richly characterize the dynamics of the system. In all cases, fractal structures are present, and the escaping dynamics is characterized. In every case a scaling law is numerically obtained in which the percentage of the trapped trajectories as a function of the relativistic parameter β and the energy is obtained. Our work could be useful in the context of charged particles which eventually can be trapped in the magnetosphere, where the analysis of these structures can be relevant.

Chaos and resonances in the classical scattering of a positron by a model diatomic molecule

PURPOSE

A recently proposed model potential to study quantum scattering of a positron from a hydrogen molecule is used to solve the Hamilton equations for scattering trajectories. In the present classical description, the positron can transfer energy to the vibrational mode of the molecule, remaining trapped for a while before escaping to infinity. Such vibrational resonances may correspond to trajectories which are embedded in phase-space regions of chaotic scattering.

Wave chaos enhanced light trapping in optically thin solar cells

Enhancing the energy output of solar cells increases their competitiveness as a source of energy. Producing thinner solar cells is attractive, but a thin absorbing layer demands excellent light management in order to keep transmission- and reflection-related losses of incident photons at a minimum. We maximize absorption by trapping light rays to make the mean average path length in the absorber as long as possible. In chaotic scattering systems, there are ray trajectories with very long lifetimes. In this paper, we investigate the scattering dynamics of waves in a model system using principles from the field of quantum chaotic scattering. We quantitatively find that the transition from regular to chaotic scattering dynamics correlates with the enhancement of the absorption cross section and propose the use of an autocorrelation function to assess the average path length of rays as a possible way to verify the light-trapping efficiency experimentally.

Influence of the gravitational radius on asymptotic behavior of the relativistic Sitnikov problem

The Sitnikov problem is a classical problem broadly studied in physics which can represent an illustrative example of chaotic scattering. The relativistic version of this problem can be attacked by using the post-Newtonian formalism. Previous work focused on the role of the gravitational radius λ on the phase space portrait. Here we add two relevant issues on the influence of the gravitational radius in the context of chaotic scattering phenomena. First, we uncover a metamorphosis of the KAM islands for which the escape regions change insofar as λ increases. Second, there are two inflection points in the unpredictability of the final state of the system when λ≃0.02 and λ≃0.028. We analyze these inflection points in a quantitative manner by using the basin entropy. This work can be useful for a better understanding of the Sitnikov problem in the context of relativistic chaotic scattering. In addition, the described techniques can be applied to similar real systems, such as binary stellar systems, among others.

Chaos: A new mechanism for enhancing the optical generation rate in optically thin solar cells

The photogenerated current of solar cells can be enhanced by light management with surface structures. For solar cells with optically thin absorbing layers, it is especially important to take advantage of this fact through light trapping. The general idea behind light trapping is to use structures, either on the front surface or on the back, to scatter light rays to maximize their path length in the absorber. In this paper, we investigate the potential of chaotic scattering for light trapping. It is well known that the trajectories close to the invariant set of a chaotic scatterer spend a very long time inside of the scatterer before they leave. The invariant set, also called the chaotic repeller, contains all rays of infinite length that never enter or leave the region of the scatterer. If chaotic repellers exist in a system, a chaotic dynamics is present in the scatterer. As a model system, we investigate an elliptical dome structure placed on top of an optically thin absorbing film, a system inspired by the chaotic Bunimovich stadium. A classical ray-tracing program has been developed to classify the scattering dynamics and to evaluate the absorption efficiency, modeled with Beer-Lambert's law. We find that there is a strong correlation between the enhancement of absorption efficiency and the onset of chaotic scattering in such systems. The dynamics of the systems was shown to be chaotic by their positive Lyapunov exponents and the noninteger fractal dimension of their scattering fractals.

Uncertainty dimension and basin entropy in relativistic chaotic scattering

Chaotic scattering is an important topic in nonlinear dynamics and chaos with applications in several fields in physics and engineering. The study of this phenomenon in relativistic systems has received little attention as compared to the Newtonian case. Here we focus our work on the study of some relevant characteristics of the exit basin topology in the relativistic Hénon-Heiles system: the uncertainty dimension, the Wada property, and the basin entropy. Our main findings for the uncertainty dimension show two different behaviors insofar as we change the relativistic parameter β, in which a crossover behavior is uncovered. This crossover point is related with the disappearance of KAM islands in phase space, which happens for velocity values above the ultrarelativistic limit, v>0.1c. This result is supported by numerical simulations and by qualitative analysis, which are in good agreement. On the other hand, the computation of the exit basins in the phase space suggests the existence of Wada basins for a range of β<0.625. We also studied the evolution of the exit basins in a quantitative manner by computing the basin entropy, which shows a maximum value for β≈0.2. This last quantity is related to the uncertainty in the prediction of the final fate of the system. Finally, our work is relevant in galactic dynamics, and it also has important implications in other topics in physics such as as in the Störmer problem, among others.

Global relativistic effects in chaotic scattering

The phenomenon of chaotic scattering is very relevant in different fields of science and engineering. It has been mainly studied in the context of Newtonian mechanics, where the velocities of the particles are low in comparison with the speed of light. Here, we analyze global properties such as the escape time distribution and the decay law of the Hénon-Heiles system in the context of special relativity. Our results show that the average escape time decreases with increasing values of the relativistic factor β. As a matter of fact, we have found a crossover point for which the KAM islands in the phase space are destroyed when β≃0.4. On the other hand, the study of the survival probability of particles in the scattering region shows an algebraic decay for values of β≤0.4, and this law becomes exponential for β>0.4. Surprisingly, a scaling law between the exponent of the decay law and the β factor is uncovered where a quadratic fitting between them is found. The results of our numerical simulations agree faithfully with our qualitative arguments. We expect this work to be useful for a better understanding of both chaotic and relativistic systems.

Granular chaos and mixing: Whirled in a grain of sand

In this paper, we overview examples of chaos in granular flows. We begin by reviewing several remarkable behaviors that have intrigued researchers over the past few decades, and we then focus on three areas in which chaos plays an intrinsic role in granular behavior. First, we discuss pattern formation in vibrated beds, which we show is a direct result of chaotic scattering combined with dynamical dissipation. Next, we consider stick-slip motion, which involves chaotic scattering on the micro-scale, and which results in complex and as yet unexplained peculiarities on the macro-scale. Finally, we examine granular mixing, which we show combines micro-scale chaotic scattering and macro-scale stick-slip motion into behaviors that are well described by dynamical systems tools, such as iterative mappings.

Defining chaos

In this paper, we propose, discuss, and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered, including attractors, repellers, and non-periodically forced systems. This definition is based on an entropy-like quantity, which we call "expansion entropy," and we define chaos as occurring when this quantity is positive. We relate and compare expansion entropy to the well-known concept of topological entropy to which it is equivalent under appropriate conditions. We also present example illustrations, discuss computational implementations, and point out issues arising from attempts at giving definitions of chaos that are not entropy-based.

A mechanical analog of the two-bounce resonance of solitary waves: Modeling and experiment

We describe a simple mechanical system, a ball rolling along a specially-designed landscape, which mimics the well-known two-bounce resonance in solitary wave collisions, a phenomenon that has been seen in countless numerical simulations but never in the laboratory. We provide a brief history of the solitary wave problem, stressing the fundamental role collective-coordinate models played in understanding this phenomenon. We derive the equations governing the motion of a point particle confined to such a surface and then design a surface on which to roll the ball, such that its motion will evolve under the same equations that approximately govern solitary wave collisions. We report on physical experiments, carried out in an undergraduate applied mathematics course, that seem to exhibit the two-bounce resonance.

Nonlinear dynamics of atoms in a crossed optical dipole trap

We explore the classical dynamics of atoms in an optical dipole trap formed by two identical Gaussian beams propagating in perpendicular directions. The phase space is a mixture of regular and chaotic orbits, the latter becoming dominant as the energy of the atoms increases. The trapping capabilities of these perpendicular Gaussian beams are investigated by considering an atomic ensemble in free motion. After a sudden turn on of the dipole trap, a certain fraction of atoms in the ensemble remains trapped. The majority of these trapped atoms has energies larger than the escape channels, which can be explained by the existence of regular and chaotic orbits with very long escape times.

Weakly noisy chaotic scattering

The effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest. We investigate how a weak source of additive uncorrelated Gaussian noise affects both the dynamics and the topology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regime of the Hénon-Heiles Hamiltonian system. We have found long transients for the time escape distributions for critical values of the noise intensity for which the particles escape slower as compared with the noiseless case. An analysis of the survival probability of the scattering function versus the Gaussian noise intensity shows a smooth curve with one local maximum and with one local minimum which are related to those long transients and with the basin structure in phase space. On the other hand, the computation of the exit basins in phase space shows a quadratic curve for which the basin boundaries lose their fractal-like structure as noise turned on.

Distribution of scattering matrix elements in quantum chaotic scattering

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory, the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.

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.

Stable regimes for hard disks in a channel with twisting walls

We study a gas of N hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N=2 and expected to be true for all N≥2). We study various perturbations by twisting the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations, and however small they are.

Billiards: a singular perturbation limit of smooth Hamiltonian flows

Nonlinear multi-dimensional Hamiltonian systems that are not near integrable typically have mixed phase space and a plethora of instabilities. Hence, it is difficult to analyze them, to visualize them, or even to interpret their numerical simulations. We survey an emerging methodology for analyzing a class of such systems: Hamiltonians with steep potentials that limit to billiards.

Model for ultraintense laser-plasma interaction at normal incidence

An analytical study of the relativistic interaction of a linearly polarized laser field of ω frequency with highly overdense plasma is presented. In agreement with one-dimensional particle-in-cell simulations, the model self-consistently explains the transition between the sheath inverse bremsstrahlung absorption regime and the J×B heating (responsible for the 2ω electron bunches), as well as the high harmonic radiations and the mean electron energy.

Transient chaos in optical metamaterials

We investigate the dynamics of light rays in two classes of optical metamaterial systems: (1) time-dependent system with a volcano-shaped, inhomogeneous and isotropic refractive-index distribution, subject to external electromagnetic perturbations and (2) time-independent system consisting of three overlapping or non-overlapping refractive-index distributions. Utilizing a mechanical-optical analogy and coordinate transformation, the wave-propagation problem governed by the Maxwell's equations can be modeled by a set of ordinary differential equations for light rays. We find that transient chaotic dynamics, hyperbolic or nonhyperbolic, are common in optical metamaterial systems. Due to the analogy between light-ray dynamics in metamaterials and the motion of light in matter as described by general relativity, our results reinforce the recent idea that chaos in gravitational systems can be observed and studied in laboratory experiments.

Newtonian and special-relativistic predictions for the trajectories of a low-speed scattering system

Newtonian and special-relativistic predictions, based on the same model parameters and initial conditions for the trajectory of a low-speed scattering system are compared. When the scattering is chaotic, the two predictions for the trajectory can rapidly diverge completely, not only quantitatively but also qualitatively, due to an exponentially growing separation taking place in the scattering region. In contrast, when the scattering is nonchaotic, the breakdown of agreement between predictions takes a very long time, since the difference between the predictions grows only linearly. More importantly, in the case of low-speed chaotic scattering, the rapid loss of agreement between the Newtonian and special-relativistic trajectory predictions implies that special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly describe the scattering dynamics.

An automated algorithm for the generation of dynamically reconstructed trajectories

The lack of long enough data sets is a major problem in the study of many real world systems. As it has been recently shown [C. Komalapriya, M. Thiel, M. C. Romano, N. Marwan, U. Schwarz, and J. Kurths, Phys. Rev. E 78, 066217 (2008)], this problem can be overcome in the case of ergodic systems if an ensemble of short trajectories is available, from which dynamically reconstructed trajectories can be generated. However, this method has some disadvantages which hinder its applicability, such as the need for estimation of optimal parameters. Here, we propose a substantially improved algorithm that overcomes the problems encountered by the former one, allowing its automatic application. Furthermore, we show that the new algorithm not only reproduces the short term but also the long term dynamics of the system under study, in contrast to the former algorithm. To exemplify the potential of the new algorithm, we apply it to experimental data from electrochemical oscillators and also to analyze the well-known problem of transient chaotic trajectories.

Chaos and crises in a model for cooperative hunting: a symbolic dynamics approach

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K,C(0)) and (K,sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K(c) decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

Effect of noise on chaotic scattering

When noise is present in a scattering system, particles tend to escape faster from the scattering region as compared with the noiseless case. For chaotic scattering, noise can render particle-decay exponential, and the decay rate typically increases with the noise intensity. We uncover a scaling law between the exponential decay rate and the noise intensity. The finding is substantiated by a heuristic argument and numerical results from both discrete-time and continuous-time models.

The nonequilibrium Ehrenfest gas: a chaotic model with flat obstacles?

It is known that the nonequilibrium version of the Lorentz gas (a billiard with dispersing obstacles [Ya. G. Sinai, Russ. Math. Surv. 25, 137 (1970)], electric field, and Gaussian thermostat) is hyperbolic if the field is small [N. I. Chernov, Ann. Henri Poincare 2, 197 (2001)]. Differently the hyperbolicity of the nonequilibrium Ehrenfest gas constitutes an open problem since its obstacles are rhombi and the techniques so far developed rely on the dispersing nature of the obstacles [M. P. Wojtkowski, J. Math. Pures Appl. 79, 953 (2000)]. We have developed analytical and numerical investigations that support the idea that this model of transport of matter has both chaotic (positive Lyapunov exponent) and nonchaotic steady states with a quite peculiar sensitive dependence on the field and on the geometry, not observed before. The associated transport behavior is correspondingly highly irregular, with features whose understanding is of both theoretical and technological interests.

Scattering off an oscillating target: Basic mechanisms and their impact on cross sections

We investigate classical scattering off a harmonically oscillating target in two spatial dimensions. The shape of the scatterer is assumed to have a boundary which is locally convex at any point and does not support the presence of any periodic orbits in the corresponding dynamics. As a simple example we consider the scattering of a beam of noninteracting particles off a circular hard scatterer. The performed analysis is focused on experimentally accessible quantities, characterizing the system, like the differential cross sections in the outgoing angle and velocity. Despite the absence of periodic orbits and their manifolds in the dynamics, we show that the cross sections acquire rich and multiple structure when the velocity of the particles in the beam becomes of the same order of magnitude as the maximum velocity of the oscillating target. The underlying dynamical pattern is uniquely determined by the phase of the first collision between the beam particles and the scatterer and possesses a universal profile, dictated by the manifolds of the parabolic orbits, which can be understood both qualitatively as well as quantitatively in terms of scattering off a hard wall. We discuss also the inverse problem concerning the possibility to extract properties of the oscillating target from the differential cross sections.

Radiationless energy exchange in three-soliton collisions

We revisit the problem of the three-soliton collisions in the weakly perturbed sine-Gordon equation and develop an effective three-particle model allowing us to explain many interesting features observed in numerical simulations of the soliton collisions. In particular, we explain why collisions between two kinks and one antikink are observed to be practically elastic or strongly inelastic depending on relative initial positions of the kinks. The fact that the three-soliton collisions become more elastic with an increase in the collision velocity also becomes clear in the framework of the three-particle model. The three-particle model does not involve internal modes of the kinks, but it gives a qualitative description to all the effects observed in the three-soliton collisions, including the fractal scattering and the existence of short-lived three-soliton bound states. The radiationless energy exchange between the colliding solitons in weakly perturbed integrable systems takes place in the vicinity of the separatrix multi-soliton solutions of the corresponding integrable equations, where even small perturbations can result in a considerable change in the collision outcome. This conclusion is illustrated through the use of the reduced three-particle model.

Chaotic scattering in solitary wave interactions: a singular iterated-map description

We derive a family of singular iterated maps--closely related to Poincare maps--that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary-wave collisions depends on the transfer of energy to a secondary mode of oscillation, often an internal mode of the pulse. This map allows us to go beyond previous analyses and to understand the interactions in the case when this mode is excited prior to the first collision. The map is derived using Melnikov integrals and matched asymptotic expansions and generalizes a "multipulse" Melnikov integral. It allows one to find not only multipulse heteroclinic orbits, but exotic periodic orbits. The maps exhibit singular behavior, including regions of infinite winding. These maps are shown to be singular versions of the conservative Ikeda map from laser physics and connections are made with problems from celestial mechanics and fluid mechanics.

Chaotic scattering by steep repelling potentials

Consider a classical two-dimensional scattering problem: a ray is scattered by a potential composed of several tall, repelling, steep mountains of arbitrary shape. We study when the traditional approximation of this nonlinear far-from-integrable problem by the corresponding simpler billiard problem, of scattering by hard-wall obstacles of similar shape, is justified. For one class of chaotic scatterers, named here regular Sinai scatterers, the scattering properties of the smooth system indeed limit to those of the billiards. For another class, the singular Sinai scatterers, these two scattering problems have essential differences: though the invariant set of such singular scatterers is hyperbolic (possibly with singularities), that of the smooth flow may have stable periodic orbits, even when the potential is arbitrarily steep. It follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards' deviation from a singular scatterer. Thus, even in this singular case, the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow. Finally, we see that corners have nontrivial and significant impact on the scattering functions.

Dynamics of scattering on a classical two-dimensional artificial atom

A classical two-dimensional (2D) model for an artificial atom is used to make a numerical "exact" study of elastic and nonelastic scattering. Interesting differences in the scattering angle distribution between this model and the well-known Rutherford scattering are found in the small energy and/or small impact parameter scattering regime. For scattering off a classical 2D hydrogen atom different phenomena such as ionization, exchange of particles, and inelastic scattering can occur. A scattering regime diagram is constructed as function of the impact parameter (b) and the initial velocity (v) of the incoming particle. In a small regime of the (b,v) space the system exhibits chaos, which is studied in more detail. Analytic expressions for the scattering angle are given in the high impact parameter asymptotic limit.

Generalization of the deflection angle in the classical scattering of particles

Current conventions define the deflection angle associated with the classical elastic scattering of particles in terms of the system's position vector. This is not consistent with the definition of the scattering angle, a function of the momentum vector. A definition of the deflection angle which resolves this inconsistency is introduced and developed for the case of an arbitrary potential in two dimensions. It is shown that the generalized deflection angle reduces to that of Cross [J. Chem. Phys. 49, 609 (1967)] when angular momentum is conserved. An efficient algorithm for the calculation of the generalized deflection angle is given and its utility in the analysis of collision dynamics is demonstrated with a numerical example.

Systems with Escapes

There are two types of escapes in conservative dynamical systems with two degrees of freedom: escapes to infinity and escapes to certain singular points at a finite distance. In both cases the areas on a surface of section are not preserved. We consider the basins of escape to infinity in simple Hamiltonian systems. The initial conditions of orbits escaping after 1, 2, ... intersections with a surface of section form in general spiral fractal sets. Then we consider the sets of escapes into two fixed black holes for various values of the energy. The forms of these sets depend on the unstable periodic orbits and their asymptotic curves. We find the characteristics of the simple periodic orbits and their changes for various values of the energy.

Chaos-induced intensification of wave scattering

Sound-wave propagation in a strongly idealized model of the deep-water acoustic waveguide with a periodic range dependence is considered. It is investigated how the phenomenon of ray and wave chaos affects the sound scattering at a strong mesoscale inhomogeneity of the refractive index caused by the synoptic eddy. Methods derived in the theory of dynamical and quantum chaos are applied. When studying the properties of wave chaos we decompose the wave field into a sum of Floquet modes analogous to quantum states with fixed quasi-energies. It is demonstrated numerically that the "stable islands" from the phase portrait of the ray system reveal themselves in the coarse-grained Wigner functions of individual Floquet modes. A perturbation theory has been derived which gives an insight into the role of the mode-medium resonance in the formation of Floquet modes. It is shown that the presence of a weak internal-wave-induced perturbation giving rise to ray and wave chaos strongly increases the sensitivity of the monochromatic wave field to an appearance of the eddy. To investigate the sensitivity of the transient wave field we have considered variations of the ray travel times--arrival times of sound pulses coming to the receiver through individual ray paths--caused by the eddy. It turns out that even under conditions of ray chaos these variations are relatively predictable. This result suggests that the influence of chaotic-ray motion may be partially suppressed by using pulse signals. However, the relative predictability of travel time variations caused by a large-scale inhomogeneity is not a general property of the ray chaos. This statement is illustrated numerically by considering an inhomogeneity in the form of a perfectly reflecting bar.

Scattering off two oscillating disks: dilute chaos

We investigate the role of the unstable periodic orbits and their manifolds in the dynamics of a time-dependent two-dimensional scattering system. As a prototype we use two oscillating disks on the plane with the oscillation axes forming an angle theta. The phase space of the system is five dimensional and it possesses a variety of families of unstable periodic orbits (UPOs) with intersecting manifolds. We perform numerical experiments to probe the structure of distinct scattering functions, in one and two dimensions, near the location of the UPOs. We find that the corresponding manifolds occur only in a very particular and localized way in the high-dimensional phase space. As a consequence the underlying fractal structure is ubiquitous only in higher-dimensional, e.g., two-dimensional, scattering functions. Both two-dimensional and one-dimensional scattering functions are dominated by seemingly infinite sequences of discontinuities characterized by small values of the magnitude of the projectile's outgoing velocity. These peaks accumulate toward the phase-space locations of the UPOs, with a rate which monotonically depends on the corresponding instability exponent. They represent the intersections of the set of the initial conditions with invariant sets of larger dimensionality embedded in the phase space of the system, which are not directly related with the UPOs. We adopt the term "dilute chaos" to characterize these phenomenological aspects of the scattering dynamics.

Classical scattering for a driven inverted Gaussian potential in terms of the chaotic invariant set

We study the classical electron scattering from a driven inverted Gaussian potential, an open system, in terms of its chaotic invariant set. This chaotic invariant set is described by a ternary horseshoe construction on an appropriate Poincaré surface of section. We find the development parameters that describe the hyperbolic component of the chaotic invariant set. In addition, we show that the hierarchical structure of the fractal set of singularities of the scattering functions is the same as the structure of the chaotic invariant set. Finally, we construct a symbolic encoding of the hierarchical structure of the set of singularities of the scattering functions and use concepts from the thermodynamical formalism to obtain one of the measures of chaos of the fractal set of singularities, the topological entropy.

Resurgence in quasiclassical scattering

In quasiclassical spectral theory, "resurgence" means that long periodic orbits can be expressed by short ones in such a way that the spectral determinant is real. The question has thus long been posed whether long scattering orbits can be expressed by short orbits in such a way as to make the quasiclassical scattering matrix unitary. We here find a resurgent and manifestly Hermitean expression for Wigner's R matrix, implying a unitary scattering matrix. The result is particularly important if the average resonance width is comparable with the average resonance spacing.

Chaotic character of two-soliton collisions in the weakly perturbed nonlinear Schrödinger equation

We analyze the exact two-soliton solution to the unperturbed nonlinear Schrödinger equation and predict that in a weakly perturbed system (i) soliton collisions can be strongly inelastic, (ii) inelastic collisions are of almost nonradiating type, (iii) results of a collision are extremely sensitive to the relative phase of solitons, and (iv) the effect is independent on the particular type of perturbation. In the numerical study we consider two different types of perturbation and confirm the predictions. We also show that this effect is a reason for chaotic soliton scattering. For applications, where the inelasticity of collision, induced by a weak perturbation, is undesirable, we propose a method of compensating it by perturbation of another type.

Short-lived two-soliton bound states in weakly perturbed nonlinear Schrodinger equation

Resonant soliton collisions in the weakly discrete nonlinear Schrodinger equation are studied numerically. The fractal nature of the soliton scattering, described in our previous works, is investigated in detail. We demonstrate that the fractal scattering pattern is related to the existence of the short-lived two-soliton bound states. The bound state can be regarded as a two-soliton quasiparticle of a new type, different from the breather. We establish that the probability P of a bound state with the lifetime L follows the law P approximately L(-3). In the frame of a simple two-particle model, we derive the nonlinear map, which generates the fractal pattern similar to that observed in the numerical study of soliton collisions. (c) 2002 American Institute of Physics.

Autocatalytic reactions of phase distributed active particles

We investigate the effect of asynchronism of autocatalytic reactions taking place in open hydrodynamical flows, by assigning a phase to each particle in the system to differentiate the timing of the reaction, while the reaction rate (periodicity) is kept unchanged. The chaotic saddle in the flow dynamics acts as a catalyst and enhances the reaction in the same fashion as in the case of a synchronous reaction that was studied previously, proving that the same type of nonlinear reaction kinetics is valid in the phase-distributed situation. More importantly, we show that, in a certain range of a parameter, the phenomenon of phase selection can occur, when a group of particles with a particular phase is favored over the others, thus occupying a larger fraction of the available space, or eventually leading to the extinction of the unfavored phases. We discuss the biological relevance of this latter phenomenon. (c) 2002 American Institute of Physics.

Cusp-scaling behavior in fractal dimension of chaotic scattering

A topological bifurcation in chaotic scattering is characterized by a sudden change in the topology of the infinite set of unstable periodic orbits embedded in the underlying chaotic invariant set. We uncover a scaling law for the fractal dimension of the chaotic set for such a bifurcation. Our analysis and numerical computations in both two- and three-degrees-of-freedom systems suggest a striking feature associated with these subtle bifurcations: the dimension typically exhibits a sharp, cusplike local minimum at the bifurcation.

Chaotic scattering in deformed optical microlasing cavities

We consider a common class of dielectric optical microlasing cavities with quadrupolar deformations and address the question of the maximally allowed amount of deformation for both high-Q operation and a high degree of directionality of light emission. Our approach is to compute the probability for light rays to be trapped in the cavity by examining chaotic scattering dynamics in the classical phase space. We develop a dynamical criterion for high-Q operation and introduce a measure to quantify the directionality of the light emission. Our results suggest that high-Q and directionality can be achieved simultaneously in a wide range of the deformation parameter.

Driving trajectories in chaotic scattering

In this work we introduce a general approach for targeting in chaotic scattering that can be used to find a transfer trajectory between any two points located inside the scattering region. We show that this method can be used in association with a control of chaos strategy to drive around and keep a particle inside the scattering region. As an illustration of how powerful this approach is, we use it in a case of practical interest in celestial mechanics in which it is desired to control the evolution of two satellites that evolve around a large central body.

Critical number in scattering and escaping problems in classical mechanics

Scattering and escaping problems for Hamiltonian systems with two degrees of freedom of the type kinetic plus potential energy arise in many applications. Under some discrete symmetry assumptions, it is shown that important quantities in these problems are determined by a relation between two canonical invariant numbers that can be explicitly computed.

Nonperiodic delay mechanism and fractallike behavior in classical time-dependent scattering

We study the occurrence of delay mechanisms other than periodic orbits in scattering systems with time-dependent potentials. By using as model system two harmonically oscillating disks on a plane, we have found the existence of a mechanism not related to the periodic orbits of the system, that delays trajectories in the scattering region. This mechanism creates a fractallike structure in the scattering functions and can possibly occur in several time-dependent scattering systems.

Topological aspects of chaotic scattering in higher dimensions

We investigate the topological properties of the chaotic invariant set associated with the dynamics of scattering systems with three or more degrees of freedom. We show that the separation of one degree of freedom from the rest in the asymptotic regime, a common property in a large class of scattering models, defines a gate which is a dynamical object with phase space separating invariant manifolds. The manifolds form an invariant set causing singularities in the scattering process. The codimension one property of the manifolds ensures that the fractal structure of the invariant set can be studied by scattering functions defined over simple one-dimensional families of initial conditions as usually done in two-degree-of-freedom scattering problems. It is found that the fractal dimension of the invariant set is not due to the gates but to interior hyperbolic periodic orbits.

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.