Chaos Model Research Articles

Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos.

The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well-defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable t in the thermodynamic limit. In particular, we rigorously prove RMT form factor for an odd t, while we formulate a precise conjecture for an even t. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in nonintegrable systems.

Solution of a Minimal Model for Many-Body Quantum Chaos

We solve a minimal model for quantum chaos in a spatially extended many-body system. It consists of a chain of sites with nearest-neighbour coupling under Floquet time evolution. Quantum states at each site span a $q$-dimensional Hilbert space and time evolution for a pair of sites is generated by a $q^2\times q^2$ random unitary matrix. The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbour on one side during the first half of the evolution period, and to its neighbour on the other side during the second half of the period. We show how dynamical behaviour averaged over realisations of the random matrices can be evaluated using diagrammatic techniques, and how this approach leads to exact expressions in the large-$q$ limit. We give results for the spectral form factor, relaxation of local observables, bipartite entanglement growth and operator spreading.

Complexity modeling and analysis of chaos and other fluctuating phenomena

The refined composite multiscale-entropy algorithm was applied to the time-dependent behavior of the Weierstrass functions, colored noise, and Logistic map to provide the fresh insight into the dynamics of these fluctuating phenomena. For the Weierstrass function, the complexity of fluctuations was found to increase with respect to the fractional dimension, D, of the graph. Additionally, the sample-entropy curves increased in an exponential fashion with increasing D. This increase in the complexity was found to correspond to a rising amount of irregularities in the oscillations. In terms of the colored noise, the complexity of the fluctuations was found to be the highest for the 1/f noise (f is the frequency of the generated noise), which is in agreement with findings in the literature. Moreover, the sample-entropy curves exhibited a decreasing trend for noise when the spectral exponent, β, was less than 1 and obeyed an increasing trend when β > 1. Importantly, a direct relationship was observed between the power-law exponents for the curves and the spectral exponents of the noise. For the logistic map, a correspondence was observed between the complexity maps and its bifurcation diagrams. Specifically, the map of the sample-entropy curves was negligible, when the bifurcation parameter, R, varied between 3 and 3.5. Beyond these values, the curves attained non-zero values that increased with increasing R, in general.

Finite dimensional models for random functions

Truncated Karhunen–Loève (KL) representations are used to construct finite dimensional (FD) models for non-Gaussian functions with finite variances. The second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation, polynomial chaos, and translated polynomial chaos models, referred to as T, PC, and PCT models. Following theoretical considerations on KL representations and T, PC, and PCT models, three numerical examples are presented to illustrate the implementation and performance of these models. The PCT models inherit the desirable features of both T and PC models. It approximates accurately all quantities of interest considered in these examples.

Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

Author(s): Boeing, Geoff | Abstract: Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

Chaos theory for clinical manifestations in multiple sclerosis

Multiple sclerosis (MS) is a demyelinating disease which characteristically shows repeated relapses and remissions irregularly in the central nervous system. At present, the pathological mechanism of MS is unknown and we do not have any theories or mathematical models to explain its disseminated patterns in time and space. In this paper, we present a new theoretical model from a viewpoint of complex system with chaos model to reproduce and explain the non-linear clinical and pathological manifestations in MS. First, we adopted a discrete logistic equation with non-linear dynamics to prepare a scalar quantity for the strength of pathogenic factor at a specific location of the central nervous system at a specific time to reflect the negative feedback in immunity. Then, we set distinct minimum thresholds in the above-mentioned scalar quantity for demyelination possibly causing clinical relapses and for cerebral atrophy. With this simple model, we could theoretically reproduce all the subtypes of relapsing-remitting MS, primary progressive MS, and secondary progressive MS. With the sensitivity to initial conditions and sensitivity to minute change in parameters of the chaos theory, we could also reproduce the spatial dissemination. Such chaotic behavior could be reproduced with other similar upward-convex functions with appropriate set of initial conditions and parameters. In conclusion, by applying chaos theory to the three-dimensional scalar field of the central nervous system, we can reproduce the non-linear outcome of the clinical course and explain the unsolved disseminations in time and space of the MS patients.

Effect of a pressure-equalizing film on the trajectory and attitude robustness of an underwater vehicle considering the uncertainty of the platform velocity

ABSTRACTA vertically moving underwater vehicle with an air film attached to its exterior can significantly weaken the hydrodynamic disturbances caused by launch conditions. To quantify the ability of a pressure-equalizing film to improve the trajectory and attitude robustness of vehicles under an uncertain platform velocity input, the response surface model of nonintrusive polynomial chaos was built based on sample spacing constructed with a nested sparse grid-based stochastic collocation method. The results indicate the uncertainty bars of the horizontal displacement exhibit an expanding-contracting-expanding pattern with a ‘spindle’ shape as the vehicle moves. In contrast, the horizontal velocity presents an ‘hourglass’ shape. Additionally, the horizontal velocity and the pitching angular velocity of the vehicle are quite sensitive to the uncertainty of the platform velocity, and the large mean value and uncertainty of these factors will statistically worsen the trajectory and attitude of the vehicle. Implementation of an exhaust during the water-emerging process can weaken the horizontal motion and pitching rotation of a vehicle, as well as the relevant uncertainty. This is attributed to a generated pressure-equalizing region with a low uncertainty. Thus, an exhaust is urgently needed in the water-emerging process of these vehicles.

Analysis of Regulatory of Interrelated Activity of Hepatocyte and Hepatitis B Viruses

In this article, we will present the results of the stability analysis of the equilibrium point of the mathematical model of the regulatory of the interrelated activity of the hepatocyte and hepatitis B viruses. The analysis of this model used the conditions of the Hayes criterion. In this study, the general condition of the Hayes criterion is obtained. If the general condition of the Hayes criterion is satisfied, then the equilibrium point is stable. If the general condition of the Hayes criterion is not fulfilled, then the equilibrium point is not stable, and hence thiscan describe modes limit cycle, chaos and black hole mathematical models of the interrelated activity of the liver cell and hepatitis B viruses. The results of the computational experiment on the quantitative analysis of the regulatory of liver cell and HBV are presented.

Nonintrusive Uncertainty Analysis of Fluid-Structure Interaction with Spatially Adaptive Sparse Grids and Polynomial Chaos Expansion

The propagation of uncertainty in physical parameters of fluid-structure interaction problems is a challenging task---both mathematically and in terms of computational workload. In this paper, we employ nonintrusive polynomial chaos expansion and model the uncertainty in five independent input parameters that characterize both fluid and structure. We propose a novel methodology to compute the expansion's coefficients using spatially adaptive sparse grids and products of one-dimensional integrals, which exploits the tensor structure of both sparse grids and probabilistic space. Furthermore, with spatial adaptivity and modified basis functions, we keep the number of sparse grid points small. We test our approach in two test cases: (i) an elastic vertical flap in a channel flow and (ii) a computationally challenging, well-established benchmark. The outputs of interest are the x-deflection and total force on the structure. In the first test case, we consider six implementations of our methodology and two established methods based on sparse grid quadrature. We evaluate the performance (in terms of number of runs of the numerical solver) and accuracy of all eight methods. The results show that one variant of our approach outperforms all the other implementations. We apply our best variant to the benchmark scenario and address the high computational demands of the resulting problem using three levels of parallelism. Importantly, our approach is not restricted to fluid-structure interaction problems. We can address a broad spectrum of computationally expensive problems, provided that sparse grid approximations can be employed.

Spatiotemporal Chaos in Coupled Logistic Map Lattice With Dynamic Coupling Coefficient and its Application in Image Encryption

This paper proposes a new spatiotemporal chaos model, which is a logistic-dynamic coupled logistic map lattice (LDCML). Through a large number of simulation experiments and theoretical analysis, it can be proved that a chaotic region has the larger parameter space, and the system is more chaotic compared with traditional CML by the introduction of a dynamic coupled method, which is more suitable for chaotic encryption and secure communications. So, this paper presents a bit-level adaptive image encryption algorithm based on this system, and further demonstrates the good chaos of the system from the aspect of encryption performance.

CREATION OF PSEUDO-RANDOM SEQUENCES BASED ON CHAOS FOR FORMING OF WIDEBAND SIGNAL

Background. Telecommunication systems with a broadband signal have improved noise immunity, the ability to receive a signal in multipath, as well as electromagnetic compatibility with neighboring radio electronic devices. The use of known pseudo-random sequences to create systems doesn’t ensure their high confidentiality due to the possibility of their selection when receiving a signal. A significant increase in the confidentiality of the system can be achieved by using pseudorandom sequences based on chaos.Objective. The aim of the paper is the development of a technique for creating pseudo-random sequences based on chaos, as well as the analysis of the correlation characteristics of pseudo-random sequences formed on the basis of a chaotic signal.Methods. Chaotic signals are inherently pseudo-random, but they are generated by deterministic systems. All computer models of chaos are approximations of mathematical chaos. Any analysis of these sequences doesn’t allow them to bereproduced and they can’t be intercepted, so they have significant advantages when used for spreading the signal spectrum and creating a pseudo noise broadband signal. Sequence selection with an acceptable level of side lobes of the autocorrelation function is carried out by using the developed graphical interface method.Results. It is shown that, based on the chaos of pseudo-random sequences, it is possible to select sequences with side lobe level up to 0.3, suitable for practical use after analysis of their autocorrelation functions.Conclusions. Using created chaos based pseudo-random sequences is effective for building broadband single-channel telecommunication systems that have a high degree of confidentiality in the information transmission.Keywords: broadband signal; chaos; pseudorandom sequence; autocorrelation function; confidentiality of information transmission.

Zitterbewegung and symmetry switching in Klein’s four-group

Zitterbewegung is the exotic phenomenon associated either with relativistic electron–positron rapid oscillation or to electron–hole transitions in narrow gap semiconductors. In the present work, we enlarge the concept of Zitterbewegung and show that trembling motion may occur due to dramatic changes in the symmetry of the system. In particular, we exploit a paradigmatic model of quantum chaos, the quantum mathematical pendulum (universal Hamiltonian). The symmetry group of this system is Klein’s four-group that possesses three invariant subgroups. The energy spectrum of the system parametrically depends on the height of the potential barrier, and contains degenerate and non-degenerate areas, corresponding to the different symmetry subgroups. Change in the height of the potential barrier switches the symmetry subgroup and leads to trembling motion. We analyzed mean square fluctuations of the velocity operator and observed that trembling is enhanced in highly excited states. We observed a link between the phenomena of trembling motion and the uncertainty relations of noncommutative operators of the system.

Bayesian Inferencing on an Aircraft T-Tail Using Probabilistic Surrogates and Uncertainty Quantification

Uncertainty quantification is a notion that has received much interest over the past decade. It involves the extraction of statistical information from a problem with inherent variability. This variability may stem from a lack of knowledge or through observational uncertainty. Traditionally, uncertainty quantification has been a challenging pursuit owing to the lack of efficient methods available. The archetypal uncertainty quantification method is Monte Carlo theory, however, this method possesses a slow convergence rate and is therefore a computational burden in some scenarios. In contrast to Monte Carlo theory, polynomial chaos theory is an alternative approach that offers the ability to estimate statistical moments efficiently. Because polynomial chaos theory behaves like a surrogate model, it is possible to query this inexpensively for information, which allows it to be useful for Bayesian inferencing. This paper builds upon previous work, because a polynomial chaos model is demonstrated to not only be useful for uncertainty quantification applications by finding inexpensive point estimates for statistical moments, but is also able to form a component of the Bayesian likelihood function. It is thus the aim of this paper to demonstrate and combine the aforementioned notions of uncertainty quantification and Bayesian inferencing on an in-house physical T-tail structure, which is reflective of a realistic scenario. The vibrational modes of the T-tail structure will provide a basis on which the results of the uncertainty quantification analysis may be compared and on which a subsequent Bayesian inferencing procedure may be carried out to infer the correct dimensions, based on comparing the model predictions with real-life vibrational data.

Robust Self-Tuning Control under Probabilistic Uncertainty using Generalized Polynomial Chaos Models

A robust self-tuning controller for a chemical process is developed based on a generalized Polynomial Chaos (gPC) model that accounts for probabilistic time-invariant uncertainty. Using this model, it is possible to calculate analytical expressions of the one-step ahead predicted mean and variances of controlled and manipulated variables. The key idea is to consider these predicted values for performing online robust tuning of the controller through a quadratic optimization procedure. The gPC model is also used to identify overlap between consecutive probability density functions (PDFs) of manipulated variables and to find trade-offs between the aggressiveness of the self-tuning controller and robustness to uncertainty based on this overlap. The proposed methodology is illustrated by a continuous stirred tank reactor (CSTR) system with stochastic variations in the inlet concentration. The efficiency of the proposed algorithm is quantified in terms of control performance and robustness.

A D-M Chaotic Model ATPG in Mixed Circuits BIST

In order to realize BIST (Built-In Self-Test) of mixed circuits, the D-M chaos ATPG (Auto Test Pattern Generator) method is proposed in this paper. The features of time series generated by D-M chaos model are analyzed to prove that it can be used in ATPG. The ATPG is a wide spectrum of signal sequences, and its correlation is small and close to the white noise, so it can be used in digital circuits BIST. Furthermore, in order to make it can be used in the test of analog circuits, the chaotic time series is reconstructed. FPGA is used as BIST controller of digital-analog mixed circuits test system to validate this method. The experimental results show that the proposed method can effectively identify the fault in mixed-signal circuits, and it can be used in mixed circuits BIST as a general method.

Bayesian inference of earthquake parameters from buoy data using a polynomial chaos-based surrogate

This work addresses the estimation of the parameters of an earthquake model by the consequent tsunami, with an application to the Chile 2010 event. We are particularly interested in the Bayesian inference of the location, the orientation, and the slip of an Okada-based model of the earthquake ocean floor displacement. The tsunami numerical model is based on the GeoClaw software while the observational data is provided by a single DARTⓇ buoy. We propose in this paper a methodology based on polynomial chaos expansion to construct a surrogate model of the wave height at the buoy location. A correlated noise model is first proposed in order to represent the discrepancy between the computational model and the data. This step is necessary, as a classical independent Gaussian noise is shown to be unsuitable for modeling the error, and to prevent convergence of the Markov Chain Monte Carlo sampler. Second, the polynomial chaos model is subsequently improved to handle the variability of the arrival time of the wave, using a preconditioned non-intrusive spectral method. Finally, the construction of a reduced model dedicated to Bayesian inference is proposed. Numerical results are presented and discussed.

Quantum chaos and holographic tensor models

A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp-plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.

Chaotic cellular automaton for generating measurement matrix used in CS coding

Exact compressed sensing (CS) recovery theoretically depends on a large number of random measurements. In this study, the authors present a novel CS measurement technique based on the cellular automata chaos (CAC) model. The proposed method selects original signal thresholding (OST) as its initial seed to realise CS signal coding. The benefits of CS coding with CAC-OST are that: (i) the signal compression ratio of this coding method can be far below the signal sparsity level and (ii) the signal can be recovered perfectly, even with slow CS measurements. This study reports some experiments that demonstrate the excellent performance of CAC-OST in CS coding.

Building Better Ecological Machines: Complexity Theory and Alternative Economic Models

Computer models of the economy are regularly used to predict economic phenomena and set financial policy. However, the conventional macroeconomic models are currently being reimagined after they failed to foresee the current economic crisis, the outlines of which began to be understood only in 2007-2008. In this article we analyze the most prominent of this reimagining: Agent-Based models (ABMs). ABMs are an influential alternative to standard economic models, and they are one focus of complexity theory, a discipline that is a more open successor to the conventional chaos and fractal modeling of the 1990s. The modelers who create ABMs claim that their models depict markets as ecologies, and that they are more responsive than conventional models that depict markets as machines. We challenge this presentation, arguing instead that recent modeling efforts amount to the creation of models as ecological machines. Our paper aims to contribute to an understanding of the organizing metaphors of macroeconomic models, which we argue is relevant conceptually and politically, e.g., when models are used for regulatory purposes.

On CFT and quantum chaos

We make three observations that help clarify the relation between CFT and quantum chaos. We show that any 1+1-D system in which conformal symmetry is non-linearly realized exhibits two main characteristics of chaos: maximal Lyapunov behavior and a spectrum of Ruelle resonances. We use this insight to identify a lattice model for quantum chaos, built from parafermionic spin variables with an equation of motion given by a Y-system. Finally we point to a relation between the spectrum of Ruelle resonances of a CFT and the analytic properties of OPE coefficients between light and heavy operators. In our model, this spectrum agrees with the quasi-normal modes of the BTZ black hole.