Chaotic Component Research Articles

Quadratic polynomial form of electric arc furnace equation

Purpose – Electric arc furnaces are very often modeled using combined models which cover separately deterministic and stochastic phenomena taking place in the furnace. The deterministic part is expressed by nonlinear differential equations. A closed form of the solution to one of the most popular nonlinear differential equations used for the AC electric arc modeling does not exist for some values of the parameters. The purpose of this paper is to convert electric arc furnace equation for these parameters to the quadratic polynomial form which significantly simplifies solution. Design/methodology/approach – The solution has been obtained in the time domain by a sequence of transformations of the original nonlinear equation which lead to a system of quadratic equations, for which a periodic solution can be found easily using harmonic balance method (HBM). Findings – Quadratic polynomial form of electric arc furnace nonlinear equation in the case for which the solution to the nonlinear differential equation describing electric arc cannot be obtained in a closed form. Research limitations/implications – The complete model of the arc requires extension of the deterministic solution obtained for the quadratic polynomial form using stochastic or chaotic component. Practical implications – The obtained results simplify determination of the arc voltage or radius time waveforms if a closed form solution does not exist. The arc model can be used to evaluate the impact of arc furnaces on power quality during the planning stage of new plants. The proposed approach facilitates calculation of the arc characteristic. Originality/value – In order to avoid problems occurring when a large number of harmonics is required or the system contains strong nonlinearities, a transformation of the original equation has been proposed. Nonlinearities present in the equation have been transformed into purely quadratic polynomial terms. It facilitates application of the classical HBM and allows to follow periodic solutions of the arc equation when its parameters are varied. It also enables better understanding of the phenomenon described by the equation and makes easier the extension of the arc model in order to cover the time-varying character of the arc waveforms.

Quantum chaotic image encryption with one time running key

To improve image encryption mechanism and enhance the security of pixel value transformation, a new image encryption scheme is proposed based on quantum chaos. In the phase of key generation, running key related to plaintext is generated by cipher-text successively disturbing chaotic component. In the process of encryption, polynomial multiplication in Galois field is first introduced to perform pixel encryption and then the cipher-text is encrypted again with cipher-text feedback mechanism. The experiment results show that the introduction of disturbing mechanism implements one time running-key stream, minimization of dynamical degradation of digital chaos, and resistance to reconstruction attack. In addition, polynomial multiplication which is first applied in the encryption system degrades the possibility of breaking our scheme in theory. Finally, some analyses such as correlation, sensitivity, min-entropy, and time complexity further demonstrate the security and efficiency of our scheme.

Functional link artificial neural network applied to active noise control of a mixture of tonal and chaotic noise

Many practical noises emanating from rotating machines with blades generate a mixture of tonal and the chaotic noise. The tonal component is related to the rotational speed of the machine and the chaotic component is related to the interaction of the blades with air. An active noise controller (ANC) with either linear algorithm like filtered-X least mean square (FXLMS) or nonlinear control algorithm like functional link artificial neural network (FLANN) or Volterra filtered-X LMS (VFXLMS) algorithm shows sub-optimal performance when the complete noise is used as reference signal to a single controller. However, if the tonal and the chaotic noise components are separated and separately sent to individual controller with tonal to a linear controller and chaotic to a nonlinear controller, the noise canceling performance is improved. This type of controller is termed as hybrid controller. In this paper, the separation of tonal and the chaotic signal is done by an adaptive waveform synthesis method and the antinoise of tonal component is produced by another waveform synthesizer. The adaptively separated chaotic signal is fed to a nonlinear controller using FLANN or Volterra filter to generate the antinoise of the chaotic part of the noise. Since chaotic noise is a nonlinear deterministic noise, the proposed hybrid algorithm with FLANN based controller shows better performance compared to the recently proposed linear hybrid controller. A number of computer simulation results with single and multitone frequencies and different types of chaotic noise such as logistic and Henon map are presented in the paper. The proposed FLANN based hybrid algorithm was shown to be performing the best among many previously proposed algorithms for all these noise cases including recorded noise signal.

Extracting predictable components and forecasting techniques in extended-range numerical weather prediction

This paper refers to the CNOP-related algorithms and formulates the practical method and forecast techniques of extracting predictable components in a numerical model for predictable components on extended-range scales. Model variables are divided into predictable components and unpredictable chaotic components from the angle of model prediction error growth. The predictable components are defined as those with a slow error growth at a given range. A targeted numerical model for predictable components is established based on the operational dynamical extended-range forecast (DERF) model of the National Climate Center. At the same time, useful information in historical data are combined to find the fields for predictable components in the numerical model that are similar to those for the predictable components in historical data, reducing the variable dimensions in a similar judgment process and further correcting prediction errors of predictable components. Historical data is used to obtain the expected value and variance of the chaotic components through the ensemble forecast method. The numerical experiment results show that this method can effectively improve the forecast skill of the atmospheric circulation field in the 10–30 days extended-range numerical model and has good prospects for operational applications.

Measuring Instantaneous and Spectral Information Entropies by Shannon Entropy of Choi-Williams Distribution in the Context of Electroencephalography

The theory of Shannon entropy was applied to the Choi-Williams time-frequency distribution (CWD) of time series in order to extract entropy information in both time and frequency domains. In this way, four novel indexes were defined: (1) partial instantaneous entropy, calculated as the entropy of the CWD with respect to time by using the probability mass function at each time instant taken independently; (2) partial spectral information entropy, calculated as the entropy of the CWD with respect to frequency by using the probability mass function of each frequency value taken independently; (3) complete instantaneous entropy, calculated as the entropy of the CWD with respect to time by using the probability mass function of the entire CWD; (4) complete spectral information entropy, calculated as the entropy of the CWD with respect to frequency by using the probability mass function of the entire CWD. These indexes were tested on synthetic time series with different behavior (periodic, chaotic and random) and on a dataset of electroencephalographic (EEG) signals recorded in different states (eyes-open, eyes-closed, ictal and non-ictal activity). The results have shown that the values of these indexes tend to decrease, with different proportion, when the behavior of the synthetic signals evolved from chaos or randomness to periodicity. Statistical differences (p-value < 0.0005) were found between values of these measures comparing eyes-open and eyes-closed states and between ictal and non-ictal states in the traditional EEG frequency bands. Finally, this paper has demonstrated that the proposed measures can be useful tools to quantify the different periodic, chaotic and random components in EEG signals.

Forecasting of chaotic dynamical processes based on the trend component extraction

In the present work the problem of chaotic dynamical processes forecasting are considered if methods of a preliminary filtration are used. Schemes of chaotic dynamical processes forecasting by means of preliminary extraction abnormal, chaotic and trend components and the further application to trend component for forecasting are considered. Extraction trend component on a basis of robust schemes, in particular, new one-sided robust estimation scheme can be carried out by means of robust orthogonal polynomials. The examples showing efficiency of offered schemes of forecasting are presented.

Time Series Forecasting Using Nonlinear Dynamic Methods and Identification of Deterministic Chaos

The study is devoted to the application of nonlinear dynamic methods to explore and model chaotic processes. The criteria of deterministic chaos and the general stages of modeling time series are presented in the work. It is proposed to improve the forecast accuracy by the identification of the chaotic component of the time process using deterministic nonlinear dynamic systems with chaotic solutions in terms of small number of available observations and one process implementation. Decomposition in the system of chaotic processes described by the logistic map is used as a model of chaotic signal. Moreover the parameter of the logistic map and the state of the system of each previous step are known inaccurately and are estimated using the unscented Kalman filter (UKF). Divergence process due to rounding estimations of the parameters of the systems is analyzed in the research.

Entropy and the fourth moment phenomenon

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijnʼs formula of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery–Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Steinʼs method for normal approximations.

Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems

In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.

Literature survey on nonlinear components and chaotic nonlinear components of block ciphers

In the modern era of secure communication, it is important to create uncertainty in the original data in order to avoid unauthorized entities to extract or manipulate information. From simple methods such as permutations of original data to different mapping algorithms, the security of the ciphers rely on the substitution process. There are many types of components proposed in literature that are evolved by different methodologies and ideas. The prevailing ciphers use substitution boxes (S-boxes) to do this transformation process. In this work, we present a literature review of the design, construction, and analysis of the S-boxes used in block ciphers.

Virtual Chimera States for Delayed-Feedback Systems

Time-delayed systems are found to display remarkable temporal patterns the dynamics of which split into regular and chaotic components repeating at the interval of a delay. This novel long-term behavior for delay dynamics results from strongly asymmetric nonlinear delayed feedback driving a highly damped harmonic oscillator dynamics. In the corresponding virtual space-time representation, the behavior is found to develop as a chimeralike state, a new paradigmatic object from the network theory characterized by the coexistence of synchronous and incoherent oscillations. Numerous virtual chimera states are obtained and analyzed, through experiment, theory, and simulations.

Dynamical localization of chaotic eigenstates in the mixed-type systems: spectral statistics in a billiard system after separation of regular and chaotic eigenstates

We study the quantum mechanics of a billiard (Robnik 1983 J. Phys. A: Math. Gen. 16 3971) in the regime of mixed-type classical phase space (the shape parameter λ = 0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré–Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all to be dynamically localized, meaning that they do not uniformly occupy the entire available chaotic classical phase space component, but are localized on a proper subset. We find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the Poisson statistics, and the chaotic ones obey the Brody statistics, as anticipated in a recent paper by Batistić and Robnik (2010 J. Phys. A: Math. Theor. 43 215101), where the entire spectrum was found to obey the Berry–Robnik–Brody statistics. There are no effects of dynamical tunneling in this regime, due to the high energies, as they decay exponentially with the inverse effective Planck constant which is proportional to the square root of the energy.

Relativistic quantum tunneling of a Dirac fermion in nonhyperbolic chaotic systems

Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking ``clustering'' phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the ``mixed'' role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.

Individuality of breathing patterns in patients under noninvasive mechanical ventilation evidenced by chaotic global models

Autonomous global models based on radial basis functions were obtained from data measured from patients under noninvasive mechanical ventilation. Some of these models, which are discussed in the paper, turn out to have chaotic or quasi-periodic solutions, thus providing a first piece of evidence that the underlying dynamics of the data used to estimate the global models are likely to be chaotic or, at least, have a chaotic component. It is explicitly shown that one of such global models produces attractors characterized by a Horseshoe map, two models produce toroidal chaos, and one model produces a quasi-periodic regime. These topologically inequivalent attractors evidence the individuality of breathing profiles observed in patient under noninvasive ventilation.

Recognition of the deep anesthesia stage from parameters of the approximated entropy of EEG signal

The paper considers theoretical problems and results of experiments on recognition of anesthesia stages from electroencephalograms (EEGs) by methods based on analyzing the parameters of approximated entropy. It is shown that a discrete sequence of point entropy estimates can be efficiently used for describing regular and chaotic components of the EEG signal and for identifying a patient's functional conditions during surgical operations. The minimum length of the time interval is determined for which the obtained entropy characteristics of the signal are reliable and stable. The computational resources required for obtaining entropy parameters of the signal are estimated in online EEG analysis. On this basis, restrictions on the signal sampling frequency are introduced and the number of point estimates of the approximated entropy is selected for which the proposed analyzing algorithms can be implemented in medical systems with midlevel computers.

Wavelet-scalogram based study of non-periodicity in speech signals as a complementary measure of chaotic content

In recent studies, the chaotic behavior of a signal is confirmed using scalogram analysis of continuous-wavelet transform. Chaotic component of a speech signal can be verified through scalogram analysis, since, it investigates the periodicity content of a signal. The periodicity analysis helps in proving that a signal is not periodic, which is an essential condition on chaotic activity. In this work, a scale-index based on scalogram-analysis is calculated for a set of recordings of Arabic vowels. Also, Largest-Lyapunov Exponents (LLE) are computed for these recordings. The obtained measures are, then, compared. The comparison proves the efficacy of scale index for confirming chaotic behavior even for highly-periodic waveforms which is the case in speech vowels. Additionally, it is noted that both LLE and scale-index exhibit classification ability for Arabic vowels.

A low-mode dynamo with meridional circulation and dipolar symmetry of the magnetic field

A dynamic system for the Parker dynamo including meridional circulation that is applicable to astrophysical objects is constructed. The meridional circulation is able to control the regimes for the generation of magnetic fields. If the meridional flows are weak, regimes with steady oscillations, dynamo bursts, fluctuations, and chaotic components are possible. When the meridional circulation is strengthened, the range of dynamo numbers required for fluctuations and dynamo bursts is reduced, and gradually vanishes; at the same time, the range required for oscillations is increased and raised to higher dynamo numbers. The latitude-time distributions of the toroidal and poloidal magnetic fields for steady oscillations are presented.

A broad-band laser-driven double Fano system—photoelectron spectra

Fano profiles, with their asymmetric character, have many potential applications in technology. The design of Fano profiles into optical systems may create new nonlinear and switchable metamaterials, high-quality optical waveguides, ultrasensitive media for chemical or biosensing etc. In this paper, we consider an external field-driven double Fano model, in which, instead of one autoionizing state, there are two discrete states embedded in one continuum. We assume further that the external electromagnetic field can be decomposed into two parts: a deterministic (or coherent) part and a randomly fluctuating chaotic component, which is a δ -correlated, Gaussian, Markov and stationary process (white noise). This assumption corresponds to the case of the real multimode laser operating without any correlation between the modes. We solve a set of coupled stochastic integro-differential equations involving the Fano model with two discrete levels. We derive an exact formula by determining the photoelectron spectrum and compare it with the results obtained in our previous papers.

The 2009 summer low temperature in northeast china and its association with prophase changes of the air-sea system

Under the background of global warming, summer (JJA) low temperature events in Northeast China had not occurred for about 15 yr since 1994, but one such event took place in 2009. By using the NCEP/NCAR reanalysis data, the 100-yr station temperature data at Harbin and Changchun, and the Hadley Center sea surface temperature (SST) data, this paper intends to reveal the cause, circulation background, and influencing factors of this event. Analysis of both horizontal and vertical circulations of a low-value system over Northeast China in summer 2009 during the low temperature event shows that anomalous activities of the Northeast China cold vortex (NECV) played the most direct role. A decadal cooling trend of — 0.8°C (10 yr)−1 over 1999–2008 at Changchun and Harbin was found, which is obviously out-of-phase with the linear warming trend (0.2°C (10 yr)−1) over 1961–2000 for Northeast China in response to the global warming. The previous winter North Pacific polar vortex (NPPV) area index, significantly positively related to the observed summer temperatures of Harbin and Changchun, was also in a significantly declining tendency. These provide favorable decadal backgrounds for the 2009 low temperature event. Different from the average anomaly field of 500-hPa height for summer 1994–2008 in Northeast China, in the summer of 2009, the Arctic Oscillation (AO) showed a strong negative phase distribution, and significant negative height anomalies dominated Northeast Asia, Aleutian Islands, and North Atlantic. Furthermore, the negative phase of North Pacific Oscillation (NPO) in the winter of 2008 was obviously strong, and it maintained in the spring of 2009. Meanwhile, the SSTA in the equatorial eastern-central Pacific Ocean in the winter of 2008 showed a La Nina phase, but the strength of the La Nina weakened obviously in the spring of 2009. The abnormally strong activities of NECV in June and July of 2009 were related to the disturbances of stationary waves that replaced the original ultra-long waves over the North Pacific region in April and May 2009. The singular value decomposition (SVD) and harmonic analysis results suggest that the anomalous phase of NPO is an important precursor for summer temperature variations over Northeast China, and also a stable planetary-scale component that can be extracted from the atmospheric circulation in addition to the chaotic components on the synoptic scale.

Noisy Chaotic Time Series Prediction Based on Wavelet Echo State Network

As a research focus of intelligence algorithm, the prediction of classic noiseless chaotic time series has a great development in recent years. However, the existing prediction models cannot get good performance for real-world chaotic time series because of the interference of noise components. In order to take full advantage of the property of real-world chaotic time series, the paper proposes a novel prediction model based on wavelet transform and echo state network (WESN). The basic idea of WESN is that firstly wavelet decomposition is used to separate the chaotic dynamics component and noise components, then the gotten components can be predicted by echo state network (ESN) independently, and finally the prediction results of time series are obtained by assembling the prediction values of all components. By using real-world sunspot time series for verification, the prediction results show that the proposed model has higher prediction accuracy by comparing with the models of direct echo state network (DESN), and direct echo state network (DSVM).