Chaotic Invariants Research Articles

The entropy of chaotic transitions of EEG phase growth in bipolar disorder with lithium carbonate

The application of chaos measures the association of EEG signals which allows for differentiating pre and post-medicated epochs for bipolar patients. We propose a new approach on chaos necessary for proof of EEG metastability. Shannon entropies of concealed patterns of Schwarzian derivatives from absolute instantaneous frequency transformations of EEG signals after Hilbert transform are compared and found significantly statistically different between pre and post-medication periods when fitted to von Bertalanffy’s functions. Schwarzian dynamics measures was compared at first baseline and then at the end of the first hour of one dose 300 mg lithium carbonate intake for the same subject in depressive patients. With an application of Schwarzian derivative on the prediction of von Bertalanffy’s models, integration and segregation of phase growth orbits of neural oscillations can be understood as an influence of chaos on the mixing of frequencies. A phase growth constant parameter was performed to determine the bifurcation parameter of von Bertalanffy’s model at each given non-overlapped EEG segment. Schwarzian derivative was sometimes very close positive near the origin but stayed negative for most of the number of segments. Lithium carbonate changed the chaotic invariants of the EEG Schwarzian dynamics and removed sharp boundaries in the bipolar spectrum.

Order in chaos: Structure of chaotic invariant sets of square-wave neuron models.

Bursting phenomena and, in particular, square-wave or fold/hom bursting, are found in a wide variety of mathematical neuron models. These systems have different behavior regimes depending on the parameters, whether spiking, bursting, or chaotic. We study the topological structure of chaotic invariant sets present in square-wave bursting neuron models, first detailed using the Hindmarsh-Rose neuron model and later exemplary in the more realistic model of a leech heart neuron. We show that the unstable periodic orbits that form the skeleton of the chaotic invariant sets are deeply related to the spike-adding phenomena, typical from these models, and how there are specific symbolic sequences and a symbolic grammar that organize how and where the periodic orbits appear. Linking this information with the topological template analysis permits us to understand how the internal structure of the chaotic invariants is modified and how more symbolic sequences are allowed. Furthermore, the results allow us to conjecture that, for these systems, the limit template when the small parameter ε, which controls the slow gating variable, tends to zero is the complete Smale topological template.

A Simple Technique for Studying Chaos Using Jerk Equation with Discrete Time Sine Map

Over the past decade, chaotic systems have found their immense application in different fields, which has led to various generalized, novel, and modified chaotic systems. In this paper, the general jerk equation is combined with a scaled sine map, which has been approximated in terms of a polynomial using Taylor series expansion for exhibiting chaotic behavior. The paper is based on numerical simulation and experimental verification of the system with four control parameters. The proposed system’s chaotic behavior is verified by calculating different chaotic invariants using MATLAB, such as bifurcation diagram, 2-D attractor, Fourier spectra, correlation dimension, and Maximum Lyapunov Exponent. Experimental verification of the system was carried out using Op-Amps with analog multipliers.

Network anomaly detection based on logistic regression of nonlinear chaotic invariants

Network anomaly detection has the essential goal of reliably identifying malicious activities within traffic observations collected at specific monitoring points, in order to raise alarms and timely trigger specific reactions and countermeasures. This, ideally, should be done also in presence of previously unknown phenomena, also known as zero-day attacks. However, distinguishing anomalous events due to attacks from normal spikes or sharp variations in traffic flows can become a classic “finding a needle in a haystack” problem, due to the very complex and unpredictable nature of Internet traffic, which is extremely affected by randomness and background noise effects. To face this challenge we leveraged machine learning for developing a novel network anomaly detection solution, based on the exploitation of nonlinear invariant properties of the Internet traffic. These properties, by capturing its chaotic and fractal features, are better suited to represent the more intrinsic and discriminative dynamics within an inductively learned model to be used for effectively classifying, through logistic regression, previously unseen traffic aggregates or individual flows into “normal” or “anomalous” ones. The results of the performance evaluation, obtained within a standard and reproducible experimental validation framework, show that the approach is able to effectively isolate very different kinds of volumetric Denial of Service attacks within the context of complex mixes of traffic flows, with really satisfactory accuracy and precision.

Comparative Study on Chaos Identification of Ionospheric Clutter From HFSWR

In addition to target echoes, high frequency surface wave radar (HFSWR) receives sea and ionospheric clutter. Among these clutters, the ionospheric clutter is dominant and significantly affects the detection performance of HFSWR, particularly when the targets are located 100 kilometers away from the radar, rendering it an unsolved problem for HFSWR. Existing studies concerning HFSWR ionospheric clutter lack empirical research on the nonlinear dynamical characteristics of the ionosphere of HFSWR as existing ionospheric suppression methods are still insufficient to adapt the project application. Therefore, the present study utilized the threshold segmentation method to eliminate the sea clutter in HFSWR Range-Doppler spectrum and extracted ionospheric signals from this spectrum by edge feature extraction. Subsequently, the chaotic invariants, such as correlation dimension and the largest Lyapunov exponent of HFSWR ionospheric clutter, were calculated by phase space reconstruction, whilst the chaotic dynamical characteristics of HFSWR ionospheric clutter were determined by the 0–1 test for chaos and other algorithms. Furthermore, the present study demonstrated, for the first time, the chaotic dynamics of the ionospheric clutter of HFSWR with a low-dimensional attractor by processing and analyzing the experimental data from the Weihai High Frequency Radar Station. The conclusion redefines HFSWR ionospheric clutter based on chaotic dynamics rather than regarding it as a stochastic process, which is conducive to efforts to explore the formation mechanism of ionospheric clutter in essence, which can ultimately improve the detection capability of HFSWR, particularly for long-distance targets.

Attractor-shape descriptors for balance impairment assessment in Parkinson's disease.

In this paper, we propose a computational framework using high-dimensional shape descriptors of reconstructed attractors of center-of-pressure (CoP) tracings collected from subjects with Parkinson's disease while performing dynamical posture shifts, to quantitatively assess balance impairment. Using a dataset collected from 60 subjects, we demonstrated that the proposed method outperforms traditional methods, such as dynamical shift indices and use of chaotic invariants, in assessment of balance impairment.

The Effect of Cathode Pressure Perturbations on the Voltage Stability of a Simulated Fuel Cell Stack

During polymer electrolyte fuel cell (PEFC) operation, proper water management is necessary to optimize performance [1]. It is desirable to have enough water to fully hydrate the electrolyte membrane with no excess liquid water to interfere with the transport of oxygen to the cathode, causing power loss. However, the air flow rate and pressure drop used to remove liquid water from the cathode and maintain power can cause large parasitic power losses. Low cathode pressure drops that are parasitically inexpensive to maintain lead to voltage instability and loss [2], while high flow rates and pressure drops stabilize voltage at the cost of a higher parasitic load on the PEFC and lower system efficiency.In this work, we perform experiments on a simulated fuel cell stack that consists of a constant hydraulic resistance bypass around the cathode, allowing air flow to the cathode to divert through the parallel bypass as the cathode's hydraulic resistance changes in the presence of liquid water. This bypass can be used to simulate a stack of unflooded cells in parallel with the experimental cell [2]. Downstream of the simulated stack we placed a solenoid valve followed by a pressure regulator to apply controlled pressure perturbations to the simulated stack. The perturbations consist of a short (< 1 second) reduction in backpressure followed by backpressure recovery. The PEFC cathode was also equipped with a differential pressure transducer to measure the pressure drop across the flow field. We formulated reduced order nonlinear statistics based on chaotic invariants (the Lyapunov exponent of the voltage return map, λ) to quantify the level of instability in the PEFC and implemented a computer program to monitor the PEFC stability in real time.Using the simulated stack under galvanotstatic operation, we recorded the PEFC voltage, cathode differential pressure (ΔPc), and λ responses to varying magnitudes of pressure perturbations during operation in an unstable regime (low current density, low air flow rate). We found that the pressure perturbations stabilized the voltage, Fig. 1, while allowing for a reduced pressure drop. We then implemented a control scheme that monitors the Lyapunov exponent for values above a threshold and applies a corrective pressure perturbation when this occurs. As Fig. 2 shows, this control scheme recovered lost power, stabilized voltage, and reduced cathode pressure drop with minimal energetic expense.

Syndrome de Goujerot-Sjögren associé à une cryoglobulinémie de type I avec atteinte multisystémique

More efficient water management techniques are required to decrease the cost of polymer electrolyte fuel cell (PEFC) systems while maintaining robust performance. In this study, we use nonlinear statistical analysis of experimental data to characterize PEFC dynamics under conditions where water accumulation in the cathode air-delivery microchannels causes decreases in performance accompanied by chaotic fluctuations. Using experimental PEFC voltage signals, we estimate chaotic invariants indicative of the degrees of freedom of the dynamics (the correlation dimension) and the instability of the dynamics (the Kolmogorov entropy). We find that these invariants decrease with increasing gas flow commensurate with greater fuel cell current and air stoichiometric ratio, and that they are indicative of the channel two-phase flow regime. We correlate the Lyapunov exponents of the one-dimensional voltage return map and the Hurst exponents of the voltage time series with the chaotic invariants for use in future PEFC water management and control strategies. In addition, we examine the relationships between the invariants estimated from the voltage signal and the two-phase friction multiplier calculated from measured cathode pressure drop in order to distinguish the distinct dynamics of two-phase channel flow.

Nonlinear analysis of voltage dynamics in a polymer electrolyte fuel cell due to two-phase channel flow

More efficient water management techniques are required to decrease the cost of polymer electrolyte fuel cell (PEFC) systems while maintaining robust performance. In this study, we use nonlinear statistical analysis of experimental data to characterize PEFC dynamics under conditions where water accumulation in the cathode air-delivery microchannels causes decreases in performance accompanied by chaotic fluctuations. Using experimental PEFC voltage signals, we estimate chaotic invariants indicative of the degrees of freedom of the dynamics (the correlation dimension) and the instability of the dynamics (the Kolmogorov entropy). We find that these invariants decrease with increasing gas flow commensurate with greater fuel cell current and air stoichiometric ratio, and that they are indicative of the channel two-phase flow regime. We correlate the Lyapunov exponents of the one-dimensional voltage return map and the Hurst exponents of the voltage time series with the chaotic invariants for use in future PEFC water management and control strategies. In addition, we examine the relationships between the invariants estimated from the voltage signal and the two-phase friction multiplier calculated from measured cathode pressure drop in order to distinguish the distinct dynamics of two-phase channel flow.

Fractal dimension algorithms and their application to time series associated with natural phenomena

Chaotic invariants like the fractal dimensions are used to characterize non-linear time series. The fractal dimension is an important characteristic of systems, because it contains information about their geometrical structure at multiple scales. In this work, three algorithms are applied to non-linear time series: spectral analysis, rescaled range analysis and Higuchi's algorithm. The analyzed time series are associated with natural phenomena. The disturbance storm time (Dst) is a global indicator of the state of the Earth's geomagnetic activity. The time series used in this work show a self-similar behavior, which depends on the time scale of measurements. It is also observed that fractal dimensions, D, calculated with Higuchi's method may not be constant over-all time scales. This work shows that during 2001, D reaches its lowest values in March and November. The possibility that D recovers a change pattern arising from self-organized critical phenomena is also discussed.

Human action recognition based on chaotic invariants

A new human action recognition approach was presented based on chaotic invariants and relevance vector machines (RVM). The trajectories of reference joints estimated by skeleton graph matching were adopted for representing the nonlinear dynamical system of human action. The C-C method was used for estimating delay time and embedding dimension of a phase space which was reconstructed by each trajectory. Then, some chaotic invariants representing action can be captured in the reconstructed phase space. Finally, RVM was used to recognize action. Experiments were performed on the KTH, Weizmann and Ballet human action datasets to test and evaluate the proposed method. The experiment results show that the average recognition accuracy is over 91.2%, which validates its effectiveness.

Characterizing gas–solid fluidization by nonlinear tools: Chaotic invariants and dynamic moments

The chaotic behavior, that gas–solid fluidization exhibits, has been characterized using the state space analysis. The space trajectories (attractor) of the dynamic system, reconstructed from the experimental time series of pressure fluctuations in the bed, have been described by the commonly used invariants: correlation dimension (CD), largest Lyapunov exponent (Ly) and the Kolmogorov entropy (K). The evolution of these invariants has been evaluated for bubbling, slugging and turbulent regime by analyzing the dependence on the flow regimes investigated. A new correlation has been proposed to predict the K for slugging regime. Due to some uncertainty involved in the delay determination for the attractor reconstruction, an attractor shape descriptor has been introduced. As shape descriptor, the dynamic moments of the attractor have been chosen and the morphological differences on the shape have been analyzed by varying the delay. The fluidization regimes have been characterized by the dynamic moments.

Temporal complexity of daily precipitation records from different atmospheric environments: Chaotic and Lévy stable parameters

Rainfall events are very erratic at short and large temporal and spatial scales. The main objectives of the present study were (i) to describe different time series of daily precipitation records using both chaos theory and stable distribution methods and (ii) to search for potential relationships between chaotic and Lévy-stable parameters. We studied eight time series of daily rainfall from different latitudes around the world. Each rainfall signal spanned nine years (1997–2005). We used methods derived from chaos theory (embedding delays, spectrum of Lyapunov exponents, determinism tests and others) and parameters computed after fitting a stable distribution model to each differenced time series of rainfall data. All the rainfall signals showed chaotic structures with two positive Lyapunov exponents. The stability index was α < 2 which accounts for the scale-free behavior of rainfall dynamics. There were found latent statistical relationships between chaotic and Lévy stable parameters. That represents a potential connection between chaotic behavior, sub-Gaussian statistics and rainfall dynamics. Future research should deal with the connection between chaotic invariants, stable parameters and rainfall phenomenology.

EMOTIONAL STRESS RECOGNITION SYSTEM FOR AFFECTIVE COMPUTING BASED ON BIO-SIGNALS

In this paper, we propose a new approach to classify emotional stress in the two main areas of the valance-arousal space by using bio-signals. Since electroencephalogram (EEG) is widely used in biomedical research, it is used as the main signal. We designed an efficient acquisition protocol to acquire the EEG and psychophysiological. Two specific areas of the valence-arousal emotional stress space are defined, corresponding to negatively excited and calm-neutral states. Qualitative and quantitative evaluation of psychophysiological signals have been used to select suitable segments of EEG signal for improving efficiency and performance of emotional stress recognition system. After pre-processing the EEG signals, wavelet coefficients and chaotic invariants like fractal dimension, correlation dimension and wavelet entropy were used to extract the features of the signal. So, by using independent-sample T-Test and Linear Discriminate Analysis (LDA), effective features are selected. The results show that, the average classification accuracy were 80.1% and 84.9% for two categories of emotional stress states using the LDA and Support Vector Machine (SVM) classifiers respectively. We achieved an improvement in accuracy, in compared to our previous studies in the similar field. Therefore, this new fusion link between EEG and psychophysiological signals are more robust in comparison to the separate signals.

Nonlinear dynamics of meteorological variables: multifractality and chaotic invariants in daily records from Pastaza, Ecuador

Weather represents the daily state of the atmosphere. It is usually considered as a chaotic nonlinear dynamical system. The objectives of the present study were (1) to investigate multifractal meteorological trends and rhythms at the Amazonian area of Ecuador and (2) to estimate some nonlinear invariants for describing the meteorological dynamics. Six meteorological variables were considered in the study. Datasets were collected on a daily basis from January 1st 2001 to January 1st 2005 (1,460 observations). Based on a new multifractal method, we found interesting fractal rhythms and trends of antipersistence patterns (Fractal Dimension >1.5). Nonlinear time series analyses rendered Lyapunov exponent spectra containing more than one positive Lyapunov exponent in some cases. This sort of hyperchaotic structures could explain, to some extent, larger fractal dimension values as the Kaplan–Yorke dimension was also in most cases larger than two. The maximum prediction time ranged from ξ = 1.69 days (approximately 41 h) for E/P ratio to ξ = 14.71 days for evaporation. Nonlinear dynamics analyses could be combined with multifractal studies for describing the time evolution of meteorological variables.

Prediction of flow regimes transitions in bubble columns using passive acoustic measurements

Passive acoustic sound, measured by a hydrophone in an air–water bubble column, is used to study the hydrodynamics of the unit. The recorded measurements taken at different superficial gas velocities are processed using both spectral and chaos-based techniques in order to characterize the column flow regimes and to predict the transitions points. These processing tools were supported by digital video imaging of bubbles motion inside the column. The results of data analysis indicate the applicability of passive sound measurements to identify flow regimes in the bubble column. In this regard the analysis of sound spectra gives a useful qualitative comparison of flow regimes. Chaos-based techniques, on the other hand, are more successful in predicting the transition points between the homogenous and the churn-turbulent flow regimes in the column. The critical gas velocities of the transition are associated with a marked change in some of the calculated chaotic invariants of sound pressures. The calculated superficial gas velocities of the critical points are also found to be consistent with experimental observations. Moreover, a useful visualization of the dynamics induced in the column by the alternation of small and large bubbles is possible through the inspection of phase-space trajectories reconstructed from time-series measurements. The shape and size of the trajectories are closely linked to the size distribution of bubbles, and they change as the flow moves from one regime to an other.

Flow Regime Transitions in an Internal‐Loop Airlift Reactor

AbstractThe flow regime transitions of the riser and downcomer in an internal‐loop airlift reactor are investigated. Analysis of the effects of mean value, standard deviation and chaotic time series on pressure fluctuation signals is recorded to determine the transition of the hydrodynamics in the riser and downcomer of the internal‐loop airlift reactor. Two major chaotic invariants, the correlation dimension and the largest Lyapunov exponent, are employed to indicate the regime transitions. The regime transitions are determined by the sudden increase and decrease of the chaotic invariants, which are computed by the pressure fluctuation signals obtained by varying the gas flow velocity. The determination of chaotic invariants predicts that there is no heterogeneous phase existing in the downcomer of the internal‐loop airlift reactor. The experimental observations agree well with the predicted results.

Data-Driven Dynamic Modeling and Control of a Surface Aeration System

In this study we have developed a support vector regression (SVR) based data-driven model for predicting two important design parameters of surface aerators, namely, the volumetric mass transfer coefficient (kLa) and fractional gas hold-up (eG). The dynamical state of the surface aerator system was captured by acquiring pressure fluctuation signals (PFSs) at various design and operating conditions. The most informative features from PFS were extracted using the chaos analysis technique, which includes estimation of Lyapunov exponent, correlation dimensions, and Kolmogorov entropy. At similar conditions the values of kLa and eG were also measured. Two different SVR models for predicting the volumetric mass transfer coefficient (kLa) and overall gas hold-up (eG) as a function of chaotic invariants, design parameters, and operating parameters were developed showing test accuracies of 98.8% and 97.1%, respectively. Such SVM based models for the surface aerator can be potentially useful on a commercial scale f...

Analysis of the chaotic dynamics of a high-flux CFB riser using solids concentration measurements

A high-flux circulating fluidized bed (CFB) riser (0.076-m I.D. and 10-m high) was operated in a wide range of operating conditions to study its chaotic dynamics, using FCC catalyst particles ( d p = 67 μm, ρ p = 1500 kg·m −3). Local solids concentration fluctuations measured using a reflective-type fiber optic probe were processed to determine chaotic invariants (Kolmogorov entropy and correlation dimension). Radial and axial profiles of the chaotic invariants at different operating conditions show that the core region exhibits higher values of the chaotic invariants than the wall region. Both invariants vary strongly with local mean solids concentration. The transition section of the riser exhibits more complex dynamics while the bottom and top sections exhibit a more uniform macroscopic and less-complex microscopic flow structure. Increasing gas velocity leads to more complex and less predictable solids concentration fluctuations, while increasing solids flux generally lowers complexity and increases predictability. Very high solids flux, however, was observed to increase the entropy.

Chaotic characteristics in an evaporator with a vapor–liquid–solid boiling flow

In this paper, quantitative and qualitative investigations on the unstable fluctuations of wall temperature signals in a multiple-tube evaporator with a vapor–liquid–solid boiling flow were carried out by employing the nonlinear analysis tools including reconstructed phase space analysis, correlation dimension analysis and Kolmogorov entropy analysis besides the traditional analysis methods such as power spectral analysis and autocorrelation analysis. The values of Kolmogorov entropy are positive and finite, and the values of correlation dimension are from 1.5 to 2.0. The analysis results indicate that the physical parameter fluctuations of the system with a vapor–liquid–solid three-phase boiling flow are chaotic. Based on the estimates of correlation dimension, it can be found that at least two independent variables are needed in order to describe the flow boiling behavior of such system. The curve shapes of correlation integral versus radius of hypersphere vary with the variations of multi-phase flow regimes or states. So do the curve shapes of slope or Kolmogorov entropy versus radius of hypersphere. And the multi-phase flow regimes or states at given operation conditions in such system may be identified or characterized by the shape variations of these curves. In the estimations of chaotic invariants including correlation dimension and Kolmogorov entropy at the same given operation condition, the multi-value phenomena are found. The phenomena reflect the multi-scale behavior occurred in such multi-phase flow boiling systems.