Chaotic Case Research Articles

Transport in Hamiltonian systems with slowly changing phase space structure

Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion of the phase space structures: chaotic, weakly chaotic and regular sets change with time. We show that theoretical predictions of transport rates can be made in the limit where the variations of the stochastic parameters are very slow compared to the Hamiltonian dynamics. Exact asymptotic results can be obtained in the one-dimensional case where the Hamiltonian dynamics is integrable for fixed values of the parameters. For the more interesting chaotic Hamiltonian dynamics case, we show that two mechanisms contribute to the transport. For some range of the parameter variations, one mechanism -called ”transport by migration with the mixing regions” - is dominant. We are then able to model transport in phase space by a Markov model, the local diffusion model, and to give reasonably good transport estimates.

Effect of chaos in a one-channel time-reversal acoustic mirror.

Time-reversal of propagating waves has been intensely studied during the last years and successfully implemented in different experimental contexts. It has been argued that ergodic or chaotic ray dynamics improve the refocusing resolution. In this work we consider this fundamental aspect by studying the reversion of sound waves in two-dimensional reflecting cavities numerically. The boundary of the enclosure is deformed from a rectangle with regular ray dynamics to a completely chaotic hyperbolic billiard. We observed that both the regular and chaotic cases display a prominent refocusing peak, and also that in the first scenario many secondary maxima appear. We developed measures of the spatial and temporal contrasts of the reconstructed signal in order to gain insight on these phenomena and to distinguish between cases. The results obtained point to the necessity for a reconsideration of what is usually understood by successful time-reversal processes.

Phenomenological approach to transport through three-terminal disordered wires.

We study the voltage drop along three-terminal disordered wires in all transport regimes, from the ballistic to the localized regime. This is performed by measuring the voltage drop on one side of a one-dimensional disordered wire in a three-terminal setup as a function of disorder. Two models of disorder in the wire are considered: (i) the one-dimensional Anderson model with diagonal disorder and (ii) finite-width bulk-disordered waveguides. Based on the known β dependence of the voltage drop distribution of the three-terminal chaotic case, β being the Dyson symmetry index (β=1, 2, and 4 for orthogonal, unitary, and symplectic symmetries, respectively), the analysis is extended to a continuous parameter β>0 and uses the corresponding expression as a phenomenological one to reach the disordered phase. We show that our proposal encompasses all the transport regimes with β depending linearly on the disorder strength.

ANALISIS ISI SENSASIONALISME BERITA KRIMINAL (STUDI KASUS PROGRAM INEWS DAN KOMPAS SULSEL)

ABSTRACTThis study discusses the Analysis of the Content of Sensationalism in Criminal News (Case Study of the Inews Program of South Sulawesi and KompasSulsel). The basis of this research began because of the fierce competition between television, newspapers and radio and the limitations of the medium, making television shift from the emphasis on public conversation leading to entertainment. There was a tight competition in the Market News. This condition provides an increase in a phenomenon called "sensationalism". Criminal news is classified into sensational news because the topics and images displayed include sensationalism. In addition, criminal news also has the ability to attract the attention of the public. This study aims to compare the level of sensationalism in news programs, especially criminal news on InewsTv and KompasTv, through the InewsSulsel and KompasSulsel programs. From these comparisons, the research method used is content analysis by comparing messages from different sources. This can be seen in the news of Inews-Sulsel and Kompas-Sulsel which are packaged with the selection of certain words in the title, the use of graphics, interesting drawings and information, the borrowing of expert mouths to convey ideas, angle setting for media presentation, music use, songs and repetitive effects so that they are familiar and determine the agenda agenda [agenda setting]. The results showed that the Inews-sulsel news program was more sensational in presenting news about the chaotic case of the Kingdom of Gowa compared to the Kompas-sulsel program.
 Keywords: Analysi; Sensationalism; Criminal News; Television News

A Method of Untethering by Skipping the Area of Positive Responses on Electrical Stimulation During Surgery of Lumbosacral Lipomatous Malformation: “Hopping on a Stepping Stone”

In complex lumbosacral lipomatous malformation or retethering, "crotch" dissection, an incision on the dura or lipoma at the site lateral to the lipoma-cord fusion line saving underlying roots, is often not easy. Attempts of this dissection in complex and obscured conditions may cause damage to the spinal cord or nerve roots. A method for caudal extension of untethering while positive areas on electrical stimulation were encountered during the crotch dissection of lumbosacral lipomatous malformation is described. After thinning out the negative-response area just caudal to the positive-response area, a small perforation is made at the negative-response area skipping the positive-response area with probing the dissection direction through the subarachnoid space by an inside-out fashion. Subsequently, the procedure is continued caudally from the perforation site. Safe caudal extension of untethering is possible using this method. Illustrative operative photographs are shown. By using this method, safe untethering can be achieved avoiding root injuries in complicated lipomas such as chaotic or retethering cases.

Robust parameter estimation of chaotic systems

Reliable estimation of parameters of chaotic dynamical systems is a long standing problem important in numerous applications. We present a robust method for parameter estimation and uncertainty quantification that requires neither the knowledge of initial values for the system nor good guesses for the unknown model parameters. The method uses a new distance concept recently introduced to characterize the variability of chaotic dynamical systems. We apply it to cases where more traditional methods, such as those based on state space filtering, are no more applicable. Indeed, the approach combines concepts from chaos theory, optimization and statistics in a way that enables solving problems considered as 'intractable and unsolved' in prior literature. We illustrate the results with a large number of chaotic test cases, and extend the method in ways that increase the accuracy of the estimation results.

Scrambling and entanglement spreading in long-range spin chains

We study scrambling in connection to multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time they rise side by side departing only afterwards. Entanglement saturates and becomes extensively multipartite, while scrambling continues its growth up to the recurrence time. Remarkably, the exponential behaviour of scrambling emerges not only in the chaotic case, but also in the regular one, when the dynamics occurs at a dynamical critical point.

Application of chaos theory to the particle dynamics of asymmetry-induced transport

The techniques of chaos theory are employed in an effort to better understand the complex single-particle dynamics of asymmetry-induced transport in non-neutral plasmas. The dynamical equations are re-conceptualized as describing time-independent trajectories in a four-dimensional space consisting of the radius r, rotating frame angle ψ, axial position z, and axial velocity v. Results include the identification of an integral of the motion, fixed-point analysis of the dynamical equations, the construction and interpretation of Poincaré sections to visualize the dynamics, and, for the case of chaotic motion, numerical calculation of the largest Lyapunov exponent. Chaotic cases are shown to be associated with the overlap of resonance islands formed by the applied asymmetry.

Robust Polynomial Observer-Based Chaotic Synchronization for Non-ideal Channel Secure Communication: An SOS Approach

This paper proposes a novel chaotic-based secure communication scheme for non-ideal transmitting public channel. The proposed approach employs a polynomial model and sum-of-squares (SOS) decomposition technique to synchronize chaotic transmitter and receiver systems. Because of high secure communication demands and practical limitations in transforming the signals on the public channel, only limited information of the transmitter system is transformed. Therefore, at the receiver side, a robust polynomial observer-based controller is proposed to estimate the states of the transmitter and synchronize the chaotic systems. Also, since the channel is noisy and non-ideal, H∞ performance criterion is used and the sufficient design conditions of controller and observer are derived in terms of SOS decomposition such that the information is effectively recovered in the presence of external disturbances and noisy environment. In addition, to further remove the effect of the noise on the information recovery, a Savitzky–Golay polynomial filter is utilized. Finally, to show the effectiveness of the proposed approach, two chaotic and hyperchaotic case studies are considered and the obtained results are compared with the existing ones.

The Role of an Implicit Assumption of Causality in the Methodology of Empirical Research

The purpose of the article is to point out the causal ladenness of empirical data in the social sciences. This is a kind of theory ladenness, representing implicit assumptions about the deterministic nature of political processes. The nonlinear and chaotic nature of social phenomena requires the collection of data not only about the current state of the system, but also about the evolution of the system. Using an example, we illustrate that the conclusions made on the basis of information about the final state can be very different from the conclusions made on the basis of monitoring the dynamics of the system. Low-importance factors can have big consequences in a chaotic case and, vice versa, there takes place fading of causality: considerable efforts can lead to more than modest results. For the successful management of political life, it is important to be able to identify the impacts that lead to great consequences.

Semiclassical prediction of large spectral fluctuations in interacting kicked spin chains

While plenty of results have been obtained for single-particle quantum systems with chaotic dynamics through a semiclassical theory, much less is known about quantum chaos in the many-body setting. We contribute to recent efforts to make a semiclassical analysis of many-body systems feasible. This is nontrivial due to both the enormous density of states and the exponential proliferation of periodic orbits with the number of particles. As a model system we study kicked interacting spin chains employing semiclassical methods supplemented by a newly developed duality approach. We show that for this model the line between integrability and chaos becomes blurred. Due to the interaction structure the system features (non-isolated) manifolds of periodic orbits possessing highly correlated, collective dynamics. As with the invariant tori in integrable systems, their presence lead to significantly enhanced spectral fluctuations, which by order of magnitude lie in-between integrable and chaotic cases.

Consequences of a wave-correction extended ray dynamics for integrable and chaotic optical microcavities

Ray optics have proven to be an effcient and versatile tool to describe dielectric optical microcavities and their far-field emission based on the principle of ray-wave correspondence. Whereas often the well-known ray-optics at planar interfaces yields reasonable results, semiclassically and boundary-curvature induced corrections will become more important as the cavity size is further reduced. In this paper, we summarize the various ray optics descriptions of optical microresonators, in addition paying special attention to the differences that arise between chaotic versus non-chaotic (integrable or nearly integrable) resonator geometries, respectively. Whereas the far-field pattern in the chaotic case is known to be determined by the overlap of the unstable manifold with the leaky region, it results from the emission of trajectories with the smallest nonzero decay rates in the non-chaotic situation. We present an enhanced ray optical description, extended by wave-inspired (semiclassical) corrections, and discuss their consequences for the ray dynamics. In particular, we find clear indications for the presence of attractors resulting from the non-Hamiltonian character of the extended ray dynamics in phase space. We illustrate their impact on the far-field emission and show that it can considerably differ from the conventional ray description result.

Quantum Ising model in transverse and longitudinal fields: chaotic wave functions

The construction of a statistical model for eigenfunctions of the Ising model in transverse and longitudinal fields is discussed in detail for the chaotic case. When the number of spins is large, each wave function coefficient has the Gaussian distribution with zero mean and variance calculated from the first two moments of the Hamiltonian. The main part of the paper is devoted to the discussion of various corrections to the asymptotic result. One type of correction is related to higher order moments of the Hamiltonian, and can be taken into account by Gibbs-like formulae. Other corrections are due to symmetry contributions, which manifest as different numbers of non-zero real and complex coefficients. The statistical model with these corrections included agrees well with numerical calculations of wave function moments.

Determination of scale invariance in random-matrix spectral fluctuations without unfolding.

We apply the singular value decomposition (SVD) method, based on normal-mode analysis, to decompose the spectra of finite random matrices of standard Gaussian ensembles in trend and fluctuation modes. We use the fact that the fluctuation modes are scale invariant and follow a power law, to characterize the transition between the extreme regular and chaotic cases. Thereby, we quantify the quantum chaos in systems described by random matrix theory in a direct way, without performing any previous unfolding procedure, and therefore, avoiding possible artifacts.

Temporal information partitioning: Characterizing synergy, uniqueness, and redundancy in interacting environmental variables

AbstractInformation theoretic measures can be used to identify nonlinear interactions between source and target variables through reductions in uncertainty. In information partitioning, multivariate mutual information is decomposed into synergistic, unique, and redundant components. Synergy is information shared only when sources influence a target together, uniqueness is information only provided by one source, and redundancy is overlapping shared information from multiple sources. While this partitioning has been applied to provide insights into complex dependencies, several proposed partitioning methods overestimate redundant information and omit a component of unique information because they do not account for source dependencies. Additionally, information partitioning has only been applied to time‐series data in a limited context, using basic pdf estimation techniques or a Gaussian assumption. We develop a Rescaled Redundancy measure (Rs) to solve the source dependency issue, and present Gaussian, autoregressive, and chaotic test cases to demonstrate its advantages over existing techniques in the presence of noise, various source correlations, and different types of interactions. This study constitutes the first rigorous application of information partitioning to environmental time‐series data, and addresses how noise, pdf estimation technique, or source dependencies can influence detected measures. We illustrate how our techniques can unravel the complex nature of forcing and feedback within an ecohydrologic system with an application to 1 min environmental signals of air temperature, relative humidity, and windspeed. The methods presented here are applicable to the study of a broad range of complex systems composed of interacting variables.

Conduction at the onset of chaos

After a general discussion of the thermodynamics of conductive processes, we introduce specific observables enabling the connection of the diffusive transport properties with the microscopic dynamics. We solve the case of Brownian particles, both analytically and numerically, and address then whether aspects of the classic Onsager's picture generalize to the non-local non-reversible dynamics described by logistic map iterates. While in the chaotic case numerical evidence of a monotonic relaxation is found, at the onset of chaos complex relaxation patterns emerge.

EXPLICIT SYMPLECTIC-LIKE INTEGRATORS WITH MIDPOINT PERMUTATIONS FOR SPINNING COMPACT BINARIES

ABSTRACT We refine the recently developed fourth-order extended phase space explicit symplectic-like methods for inseparable Hamiltonians using Yoshida’s triple product combined with a midpoint permuted map. The midpoint between the original variables and their corresponding extended variables at every integration step is readjusted as the initial values of the original variables and their corresponding extended ones at the next step integration. The triple-product construction is apparently superior to the composition of two triple products in computational efficiency. Above all, the new midpoint permutations are more effective in restraining the equality of the original variables and their corresponding extended ones at each integration step than the existing sequent permutations of momenta and coordinates. As a result, our new construction shares the benefit of implicit symplectic integrators in the conservation of the second post-Newtonian Hamiltonian of spinning compact binaries. Especially for the chaotic case, it can work well, but the existing sequent permuted algorithm cannot. When dissipative effects from the gravitational radiation reaction are included, the new symplectic-like method has a secular drift in the energy error of the dissipative system for the orbits that are regular in the absence of radiation, as an implicit symplectic integrator does. In spite of this, it is superior to the same-order implicit symplectic integrator in accuracy and efficiency. The new method is particularly useful in discussing the long-term evolution of inseparable Hamiltonian problems.

Wave functions of channeling electrons in regular and chaotic cases

The spectral method applicability to the problem of the charged particles axial channeling is studied. The quantum state characteristics for the electron motion near the atomic string [110] direction of the silicon crystal are obtained using this method of Schrödinger equation numerical solution. The full set of eigenenergies and eigenfunctions for the electron with longitudinal energy of 5 MeV is computed. The results obtained illustrate the difference in the wave function morphology for regular and chaotic cases.

Wave functions of channeling electrons in regular and chaotic cases

The spectral method applicability to the problem of the charged particles axial channeling is studied. The quantum state characteristics for the electron motion near the atomic string [110] direction of the silicon crystal are obtained using this method of Schrödinger equation numerical solution. The full set of eigenenergies and eigenfunctions for the electron with longitudinal energy of 5 MeV is computed. The results obtained illustrate the difference in the wave function morphology for regular and chaotic cases.

Numerical methods for calculating the response of a deterministic and stochastically excited Duffing oscillator

When compared to independent harmonic or stochastic excitation, there exist relatively few methods to model the response of non-linear systems to a combination of deterministic and stochastic vibration despite the likelihood of harmonic oscillations containing noise in realistic applications. This paper uses the Duffing oscillator to illustrate how the joint probability density function (JPDF) of the displacement and velocity responds to this form of excitation. Monte Carlo simulations were performed to generate the JPDF which was observed, in general, to spread around the trajectory that would be observed if only deterministic excitation was present. In the deterministic chaotic case, the JPDF is known to be a diffuse chaotic attractor when noise is present. This paper assesses the ability of a useful class of methods, global weighted residual methods, to produce the geometrically complex JPDF responses produced from harmonic and white noise excitation. A technique using a JPDF in the form of a Gram–Charlier type C series was found to produce accurate results, although the method fails due to ill-conditioning as the shape of the JPDF required by the dynamics becomes too complex.