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.

We discuss the statistical properties of eigenphases of S matrices in random models simulating quantum systems that exhibit chaotic scattering classically. The energy dependence of the eigenphases is investigated and the corresponding velocity and curvature distributions are obtained both theoretically and numerically. A simple formula describing the velocity distribution ͑and hence the distribution of the Wigner time delay͒ is derived that is capable of explaining the algebraic tail of the time delay distribution observed recently in microwave experiments. A dependence of the eigenphases on other external parameters is also discussed. We show that in the semiclassical limit ͑large number of channels͒ the curvature distribution of S-matrix eigenphases is the same as that corresponding to the curvature distribution of the underlying Hamiltonian and is given by the generalized Cauchy distribution. ͓S1063-651X͑96͒03209-6͔

We consider a Hayden & Preskill like setup for both maximally chaotic and sub-maximally chaotic quantum field theories. We act on the vacuum with an operator in a Rindler like wedge R and transfer a small subregion I of R to the other wedge. The chaotic scrambling dynamics of the QFT Rindler time evolution reveals the information in the other wedge. The holographic dual of this process involves a particle excitation falling into the bulk and crossing into the entanglement wedge of the complement to r = R\I. With the goal of studying the locality of the emergent holographic theory we compute various quantum information measures on the boundary that tell us when the particle has entered this entanglement wedge. In a maximally chaotic theory, these measures indicate a sharp transition where the particle enters the wedge exactly when the insertion is null separated from the quantum extremal surface for r. For sub-maximally chaotic theories, we find a smoothed crossover at a delayed time...

We discuss the statistical properties of eigenphases of S matrices in random models simulating quantum systems that exhibit chaotic scattering classically. The energy dependence of the eigenphases is investigated and the corresponding velocity and curvature distributions are obtained both theoretically and numerically. A simple formula describing the velocity distribution ͑and hence the distribution of the Wigner time delay͒ is derived that is capable of explaining the algebraic tail of the time delay distribution observed recently in microwave experiments. A dependence of the eigenphases on other external parameters is also discussed. We show that in the semiclassical limit ͑large number of channels͒ the curvature distribution of S-matrix eigenphases is the same as that corresponding to the curvature distribution of the underlying Hamiltonian and is given by the generalized Cauchy distribution. ͓S1063-651X͑96͒03209-6͔

In this manuscript we show that a noise-activated escape phenomenon occurs in closed Hamiltonian systems. Due to the energy fluctuations generated by the noise, the isopotential curves open up and the particles can eventually escape in finite times. This drastic change in the dynamical behavior turns the bounded motion into a chaotic scattering problem. We analyze the escape dynamics by means of the average escape time, the probability basins and the average escape time distribution. We obtain that the main characteristics of the scattering are different from the case of noisy open Hamiltonian systems. In particular, the noiseenhanced trapping, which is ubiquitous in Hamiltonian systems, does not play the main role in the escapes. On the other hand, one of our main findings reveals a transition in the evolution of the average escape time insofar the noise is increased. This transition separates two different regimes characterized by different algebraic scaling laws. We provide strong numerical evidence to show that the complete destruction of the stickiness of the KAM islands is the key reason under the change in the scaling law. This research unlocks the possibility of modeling chaotic scattering problems by means of noisy closed Hamiltonian systems. For this reason, we expect potential application to several fields of physics such us celestial mechanics and astrophysics, among others.

In nonlinear dynamics, basins of attraction are defined as the set of points that, taken as initial conditions, lead the system to a specific attractor. This notion appears in a broad range of applications where multistability is present, which is a common situation in neuroscience, economy, astronomy, ecology, and other disciplines. Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. When a single boundary separates three or more different basins of attraction, we call them Wada basins. Usually, Wada basins have been considered even more unpredictable than fractal basins. However, this particular unpredictability has not been fully unveiled until the introduction of the concept of basin entropy. The basin entropy provides a quantitative measure of how unpredictable a basin is. With the help of several paradigmatic dynamical systems, we illustrate how to identify the ingredients that hinder the prediction of the final state. The basin entropy together with two new tests of the Wada property have been applied to some physical systems such as experiments of chaotic scattering of cold atoms, models of shadows of binary black holes, and classical and relativistic chaotic scattering associated to the Hénon-Heiles Hamiltonian system in astrophysics.

In the context of nonhyperbolic chaotic scattering, it has been shown that the evolution of the KAM islands exhibits four abrupt metamorphoses that strongly affect the predictability of Hamiltonian systems. It has been suggested that these metamorphoses are related to significant changes in the structure of the KAM islands. However, previous research has not provided an explanation of the mechanisms underlying the metamorphoses. Here, we show that they occur due to the formation of a homoclinic or heteroclinic tangle that breaks the internal structure of the main KAM island. We obtain similar qualitative results in a two-dimensional Hamiltonian system and a two-dimensional area-preserving map. The equivalence of the results obtained in both systems suggests that the same four metamorphoses play an important role in conservative systems.

The noise-enhanced trapping is a surprising phenomenon that has already been studied in chaotic scattering problems where the noise affects the physical variables but not the parameters of the system. Following this research, in this work we provide strong numerical evidence to show that an additional mechanism that enhances the trapping arises when the noise influences the energy of the system. For this purpose, we have included a source of Gaussian white noise in the Hénon-Heiles system, which is a paradigmatic example of open Hamiltonian system. For a particular value of the noise intensity, some trajectories decrease their energy due to the stochastic fluctuations. This drop in energy allows the particles to spend very long transients in the scattering region, increasing their average escape times. This result, together with the previously studied mechanisms, points out the generality of the noise-enhanced trapping in chaotic scattering problems.

We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare ´map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents.

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few of such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.

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 nume...

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 nume...

We investigate the extent to which the probabilistic properties of a chaotic scattering system with dissipation can be understood from the properties of the dissipation-free system. For large energies E, a fully chaotic scattering leads to an exponential decay of the survival probability P (t) ∼ e -κt with an escape rate κ that decreases with E. Dissipation γ > 0 leads to the appearance of different finite-time regimes in P (t). We show how these different regimes can be understood for small γ ≪ 1 and t ≫ 1/κ0 from the effective escape rate κγ(t) = κ0(E(t)) (including the non-hyperbolic regime) until the energy reaches a critical value Ec at which no escape is possible. More generally, we argue that for small dissipation γ and long times t the surviving trajectories in the dissipative system are distributed according to the conditionally invariant measure of the conservative system at the corresponding energy E(t) < E(0). Quantitative predictions of our general theory are compared with numerical simulations in the Hénon-Heiles model.

Dynamics of scattering in the classical model of o.-cluster nuclei is studied in terms of transport theory. Behavior typical for hyperbolic chaotic scattering is found, which results in an exponential decay of the survival probability. This allows a determination of the unitarity deficit of the S matrix and thus the probability for compound-nucleus formation.

A comparative analysis is performed of the basic methods for calculating fractal dimensions and a new computational technique is proposed. A relationships is derived between the invariant distribution function and the probability distribution function of the distances between points on a strange attractor. The effect of boundaries and singularities in the invariant distribution function on fractal dimensions is investigated. Fractal dimensions are computed for particular systems with chaotic behaviour.

Complex quantum systems consisting of large numbers of strongly coupled states exhibit characteristic level repulsion, leading to a non-Poisson spacing distribution which can be described by Random Matrix Theory. Scattering resonances observed in ultracold atomic and molecular systems exhibit similar features as a consequence of their energy level structure. We study how the overlap between Feshbach resonances affects the distribution of resonance spacings. The spectrum of strongly overlapping resonances turns out to be non-Poisson even when the assumptions of Random Matrix Theory are not fulfilled, but the spectrum is also not completely chaotic and tends towards being semi-Poisson.

Chaotic scattering in open Hamiltonian systems is a topic of fundamental interest in physics, which has been mainly studied in the purely conservative case. However, the effect of weak perturbations in this kind of systems has been an important focus of interest in the last decade. In a previous work, the authors studied the effects of a periodic forcing in the decay law of the survival probability, and they characterized the global properties of escape dynamics. In the present paper, we add two important issues in the effects of periodic forcing: the fractal dimension of the set of singularities in the scattering function, and the unpredictability of the exit basins, which is estimated by using the concept of basin entropy. Both the fractal dimension and the basin entropy exhibit a resonant-like decrease as the forcing frequency increases. We provide a theoretical reasoning, which could justify this decreasing in the fractality near the main resonant frequency, that appears for ω ≈ 1. We attribute the decrease in the basin entropy to the reduction of the area occupied by the KAM islands and the basin boundaries when the frequency is close to the resonance. Finally, the decay rate of the exponential decay law shows a minimum value of the amplitude, A c , which reflects the complete destruction of the KAM islands in the resonance. We expect that this work could be potentially useful in research fields related to chaotic Hamiltonian pumps, oscillations in chemical reactions and companion galaxies, among others.

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases the particles escape faster. For this reason the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been uncovered. The reason is that when increasing the energy, both the size and geometry of the KAM islands undergo abrupt changes. These fluctuations do not appear within the hyperbolic regime. Hence, the fluctuations in the basin entropy allow us to ascertain the hyperbolic or nonhyperbolic nature of a system. In this manuscript we have used continuous and discrete open Hamiltonian systems in order to show the relevant role of the KAM islands on the unpredictability of the exit basins, and the utility of the basin entropy to analyze this kind of systems.

Chaotic scattering in three dimensions has not received as much attention as in two dimensions so far. In this paper, we deal with a three-dimensional open Hamiltonian system whose Wada basin boundaries become non Wada when the critical energy value is surpassed in the absence of dissipation. In particular, we study here the dissipation effects on this topological change, which has no analogy in two dimensions. Hence, we find that non-Wada basins, expected in the absence of dissipation, transform themselves into partially Wada basins when a weak dissipation reduces the system energy below the critical energy. We provide numerical evidence of the emergence of the Wada points on the basin boundaries under weak dissipation. According to the paper findings, Wada basins are typically driven, enhanced and, consequently, structurally stable under weak dissipation in three-dimensional open Hamiltonian systems.

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases the particles escape faster. For this reason the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been uncovered. The reason is that when increasing the energy, both the size and geometry of the KAM islands undergo abrupt changes. These fluctuations do not appear within the hyperbolic regime. Hence, the fluctuations in the basin entropy allow us to ascertain the hyperbolic or nonhyperbolic nature of a system. In this manuscript we have used continuous and discrete open Hamiltonian systems in order to show the relevant role of the KAM islands on the unpredictability of the exit basins, and the utility of the basin entropy to analyze this kind of systems.

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&gt;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 exis...

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.

Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon–Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases the particles escape faster. For this reason the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been uncovered. The reason is that when increasing the energy, both the size and geometry of the KAM islands undergo abrupt changes. These fluctuations do not appear within the hyperbolic regime. Hence, the fluctuations in the basin entropy allow us to ascertain the hyperbolic or nonhyperbolic nature of a system. In this manuscript we have used continuous and discrete open Hamiltonian systems in order to show the relevant role of the KAM islands on the unpredictability of the exit basins, and the utility of the basin entropy to analyze this kind of systems.

The noise-enhanced trapping is a surprising phenomenon that has already been studied in chaotic scattering problems where the noise affects the physical variables but not the parameters of the system. Following this research, in this work we provide strong numerical evidence to show that an additional mechanism that enhances the trapping arises when the noise influences the energy of the system. For this purpose, we have included a source of Gaussian white noise in the Hénon-Heiles system, which is a paradigmatic example of open Hamiltonian system. For a particular value of the noise intensity, some trajectories decrease their energy due to the stochastic fluctuations. This drop in energy allows the particles to spend very long transients in the scattering region, increasing their average escape times. This result, together with the previously studied mechanisms, points out the generality of the noise-enhanced trapping in chaotic scattering problems.

An important effect of dynamical localization of light waves in liquid crystal electro-hydrodynamic instabilities is reported by investigating coherent backscattering effects. Recurrent multiple scattering in dynamic and chaotic complex fluids lead to a cone of enhanced backscattered light. The cone width and the related mean free path dependence on the dynamic scattering regimes emphasize the diverse light localization scales related to the internal structures present in the sample. The systems investigated up to now were mainly nano-powdered solutions or biological tissues, without any external control on the disorder. Here, an anisotropic complex fluid is "driven" throughout chaotic regimes by an external electric field, giving rise to dynamics that evolve through several spatio-temporal patterns.

We have studied the rich dynamics of a damped particle inside an external double-well potential under the influence of state-dependent time-delayed feedback. In certain regions of the parameter space, we observe multistability with the existence of two different attractors (limit cycle or strange attractor) with well separated mean Lyapunov energies forming a two-level system. Bifurcation analysis reveals that, as the effects of the time-delay feedback are enhanced, chaotic transitions emerge between the two wells of the double-well potential for the attractor corresponding to the fundamental energy level. By computing the residence time distributions and the scaling laws near the onset of chaotic transitions, we rationalize this apparent tunneling-like effect in terms of the crisis-induced intermittency phenomenon. Further, we investigate the first passage times in this regime and observe the appearance of a Cantor-like fractal set in the initial history space, a characteristic feature of hyperbolic chaotic scattering. The non-integer value of the uncertainty dimension indicates that the residence time inside each well is unpredictable. Finally, we demonstrate the robustness of this tunneling intermittency as a function of the memory parameter by calculating the largest Lyapunov exponent.

Reflection of microwaves from a cavity is measured in a frequency domain where the underlying classical chaotic scattering leaves a clear mark on the wave dynamics. We check the hypothesis that the fluctuations of the S matrix can be described in terms of parameters characterizing the chaotic classical scattering. Absorption of energy in the cavity walls is shown to significantly affect the results, and is linked to time-domain properties of the scattering in a general way. We also show that features whose origin is entirely due to wave dynamics (e.g. , the enhancement of the Wigner time delay due to timereversal symmetry) coexist with other features which characterize the underlying classical dynamics.

We discuss the phenomenon of chaotic scattering and its application in the study of transmission of electrons in mesoscopic devices as well as the transmission of microwaves through junctions. We show that the fact that the ray optics (classical dynamics) is chaotic, implies fluctuations in the observed transmission coefficients, whose statistics is determined by the theory of random matrices. We also show how the classical distribution functions which reflect the chaotic nature of the classical dynamics, determine the dependence of the correlations observed in the fluctuating transmission coefficients on external parameters. The time domain properties of chaotic scattering systems are also examined, and are shown to depend on the chaotic nature of the classical dynamics, together with a wave mechanical enhancement in time reversal invariant systems. Finally, we study the role of absorption and discuss its effects on the transmission fluctuations and their statistics.

This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e. , not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of "fully developed" chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horseshoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed. Recently, some progress has been made on the general problem we address in this paper: understanding how and why scattering can become chaotic as a parameter is varied. ' In Ref.

We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph-its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances 'topological' to emphasize their origin in the non-trivial connectivity. Topological resonances have a clear and unique signature which is apparent in the statistics of the resonance parameters (such as e.g., the width, the delay time or the wavefunction intensity in the graph). We discuss this phenomenon by providing analytical arguments supported by numerical simulation, and identify the features of the above distributions which depend on genuine topological quantities such as the length of the shortest cycle (girth). These signatures cannot be explained using any of the other paradigms for narrow resonances. Finally, we propose an experimental setting where the topological resonances could be demonstrated, and study the stability of the relevant distribution functions to moderate dissipation.

We consider a quasi one-dimensional chain of N chaotic scattering elements with periodic boundary conditions. The classical dynamics of this system is dominated by diffusion. The quantum theory, on the other hand, depends crucially on whether the chain is disordered or invariant under lattice translations. In the disordered case, the spectrum is dominated by Anderson localization whereas in the periodic case, the spectrum is arranged in bands. We investigate the special features in the spectral statistics for a periodic chain. For finite N , we define spectral form factors involving correlations both for identical and non-identical Bloch numbers. The short-time regime is treated within the semiclassical approximation, where the spectral form factor can be expressed in terms of a coarse-grained classical propagator which obeys a diffusion equation with periodic boundary conditions. In the long-time regime, the form factor decays algebraically towards an asymptotic constant. In the limit N → ∞, we derive a universal scaling function for the form factor. The theory is supported by numerical results for quasi one-dimensional periodic chains of coupled Sinai billiards. 05.45.+b,03.65.Sq

We investigate the two-point correlations in the band spectra of periodic systems that exhibit chaotic diffusion in the classical limit, in terms of form factors with the winding number as a spatial argument. For times below the Heisenberg time, they contain the full space-time dependence of the classical propagator. They approach constant asymptotes via a regime, reflecting quantal ballistic motion, where they decay by a factor proportional to the number of unit cells. We derive a universal scaling function for the long-time behaviour. In the limit of long chains, our results are consistent with expressions obtained by field-theoretical methods. They are substantiated by numerical studies of the kicked rotor and a billiard chain.

We present an in-depth study of the universal correlations of scattering-matrix entries required in the framework of non-stationary many-body scattering where the incoming states are localized wavepackets. Contrary to the stationary case the emergence of universal signatures of chaotic dynamics in dynamical observables manifests itself in the emergence of universal correlations of the scattering matrix at different energies. We use a semiclassical theory based on interfering paths, numerical wave function based simulations and numerical averaging over random-matrix ensembles to calculate such correlations and compare with experimental measurements in microwave graphs, finding excellent agreement. Our calculations show that the universality of the correlators survives the extreme limit of few open channels relevant for electron quantum optics, albeit at the price of dealing with large-cancellation effects requiring the computation of a large class of semiclassical diagrams.

We report on the experimental investigation of the dependence of the elastic enhancement, i.e., enhancement of scattering in backward direction over scattering in other directions of a wave-chaotic system with partially violated time-reversal (T) invariance on its openness. The elastic enhancement factor is a characteristic of quantum chaotic scattering which is of particular importance in experiments, like compound-nuclear reactions, where only cross sections, i.e., the moduli of the associated scattering matrix elements are accessible. In the experiment a quantum billiard with the shape of a quarter bow-tie, which generates a chaotic dynamics, is emulated by a flat microwave cavity. Partial T-invariance violation of varying strength 0 ≤ ξ 1 is induced by two magnetized ferrites. The openness is controlled by increasing the number M of open channels, 2 ≤ M ≤ 9, while keeping the internal absorption unchanged. We investigate the elastic enhancement as function of ξ and find that for a fixed M it decreases with increasing time-reversal invariance violation, whereas it increases with increasing openness beyond a certain value of ξ 0.2. The latter result is surprising because it is opposite to that observed in systems with preserved T invariance (ξ = 0). We come to the conclusion that the effect of T-invariance violation on the elastic enhancement then dominates over the openness, which is crucial for experiments which rely on enhanced backscattering, since, generally, a decrease of the openness is unfeasible. Motivated by these experimental results we, furthermore, performed theoretical investigations based on random matrix theory which confirm our findings.

In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic Hénon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dynamics is non-ergodic independently of the existence of KAM tori, since the chaotic saddle, in whose vicinity trajectories are more likely to spend a finite amount of time evolving before escaping forever, is not utterly spread over the energy shell. Taking this into consideration, we provide a clarifying discussion about ergodicity in open Hamiltonian systems and explore the limitations of ergodic decay laws when describing escapes in this kind of systems. Finally, we generalize our claims by deriving a new decay law in the relativistic regime for an inertial and a non-inertial reference frames under the assumption of ergodicity, and suggest another approach to the description of escape laws in open Hamiltonian systems.

The Hénon-Heiles Hamiltonian is investigated in the context of chaotic scattering, in the range of energies where escaping from the scattering region is possible. Special attention is paid to the analysis of the different nature of the orbits, and the the invariant sets, such as the stable and unstable manifolds and the chaotic saddle. Furthermore, a discussion on the average decay time associated to the typical chaotic transients, which are present in this problem, is presented. The main goal of this paper is to show, by using various computational methods, that the corresponding exit basins of this open Hamiltonian are not only fractal, but they also verify the more restrictive property of Wada. We argue that this property is verified by typical open Hamiltonian systems with three or more escapes.

The action of a set of ergodic magnetic limiters in tokamaks is investigated from the Hamiltonian chaotic scattering point of view. Special attention is paid to the influence of invariant sets, such as stable and unstable manifolds, as well as the strange saddle, on the formation of the chaotic layer at the plasma edge. The nonuniform escape process associated to chaotic field lines is also analyzed. It is shown that the ergodic layer produced by the limiters has not only a fractal structure, but it possesses the even more restrictive Wada property.

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.

The effect of weak dissipation on chaotic scattering is relevant to situations of physical interest. We investigate how the fractal dimension of the set of singularities in a scattering function varies as the system becomes progressively more dissipative. A crossover phenomenon is uncovered where the dimension decreases relatively more rapidly as a dissipation parameter is increased from zero and then exhibits a much slower rate of decrease. We provide a heuristic theory and numerical support from both discrete-time and continuous-time scattering systems to establish the generality of this phenomenon. Our result is expected to be important for physical phenomena such as the advection of inertial particles in open chaotic flows, among others.

The creation of an outer layer of chaotic magnetic field lines in a tokamak is useful to control plasma-wall interactions. Chaotic field lines (in the Lagrangian sense) in this region eventually hit the tokamak wall and are considered lost. Due to the underlying dynamical structure of this chaotic region, namely a chaotic saddle formed by intersections of invariant stable and unstable manifolds, the exit patterns are far from being uniform, rather presenting an involved fractal structure. If three or more exit basins are considered, the respective basins exhibit an even stronger Wada property, for which a boundary point is arbitrarily close to points belonging to all exit basins. We describe such a structure for a tokamak with an ergodic limiter by means of an analytical Poincaré field line mapping.

Noisy scattering dynamics in the randomly driven Hénon-Heiles system is investigated in the range of initial energies where the motion is unbounded. In this paper we study, with the help of the exit basins and the escape time distributions, how an external perturbation, be it dissipation or periodic forcing with a random phase, can enhance or mitigate the unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have the Wada property, predictability becomes very complicated, since the basin boundaries start to intermingle, what means that there are points of different basins close to each other. The main responsible of this unpredictability is the external forcing with random phase, while the dissipation can recompose the basin boundaries and turn the system more predictable. Therefore, we do the necessary simulations to find out the values of dissipation and external forcing for which the exit basins present the Wada property. Through these numerical simulations, we show that the presence of the Wada basins have a specific relation with the damping, the forcing amplitude and the energy value. Our approach consists on investigating the dynamics of the system in order to gain knowledge able to control the unpredictability due to the Wada basins.

Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Hénon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical e...

We use the Hénon-Heiles system as a paradigmatic model for chaotic scattering to study the Lorentz factor effects on its transient chaotic dynamics. In particular, we focus on how time dilation occurs within the scattering region by measuring the time in a non-inertial clock co-moving with the particle. We observe that the several events of time dilation that the particle undergoes exhibit sensitivity to initial conditions. However, the structure of the singularities appearing in the escape time function remains invariant under coordinate transformations. This occurs because the singularities are closely related to the chaotic saddle. We then demonstrate using a Cantor-like set approach that the fractal dimension of the escape time function is relativistic invariant. In order to verify this result, we compute the fractal dimensions of the escape time functions as measured with inertial and non-inertial clocks by means of the uncertainty dimension algorithm. We conclude that, from a mathematical point of view, chaotic transient phenomena are equally predictable in any reference frame and that transient chaos is coordinate invariant.

We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincaré map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents.

Relaxation in correlated systems such as interacting ions, entangled polymer chains or viscous liquids requires a time-dependent relaxation rate for a full accounting of the observed phenomena. This has been demonstrated by the coupling scheme for relaxation in correlated systems. A fundamental theory of these systems necessarily involves non-integrable interactions or constraints which are known to produce chaos in Hamiltonian models. Van Kampen has shown that the transition rates in the master equation for a relaxing Hamiltonian system can be interpreted in terms of diffusion in phase space. In this paper, this approach is generalized to include chaotic Hamiltonians. It is demonstrated that general results from anomalous diffusion in fractals produce all the properties of the time-dependent relaxation rate required to explain the experiments. This leads to a conjecture on the non-Euclidean and possibly fractal phase space structure of relaxing chaotic Hamiltonians which would allow a basic understanding of the central results of the coupling scheme.