Positive Largest Lyapunov Exponent Research Articles

A novel four-wing chaotic system with multiple equilibriums: Dynamical analysis, multistability, circuit simulation and pseudo random number generator (PRNG) based on the voice encryption

Recently, there has been tremendous interest worldwide in the possibility of using chaos in communication systems. Many different chaos-based secure communication schemes have been proposed up until now. However, systems with strong chaoticity are more suitable for chaos-based secure communication. From the viewpoint of Lyapunov exponents, a chaotic system with a larger positive Lyapunov exponent is said to be more complex. This paper constructing a multistable chaotic system that can produce coexisting attractors is an attractive field of research due to its theoretical and practical usefulness. An innovative 3D dynamical system is presented in this research. It can display various coexisting attractors for the same values of parameters. The new system is more suitable for chaos-based applications than recently reported systems since it exhibits strong multistable chaotic behavior, as proved by its large positive Lyapunov exponent. Furthermore, the accuracy of the numerical calculation and the system's physical implementations are confirmed by analog circuit simulation. Finally, implementing the proposed voice encryption is done using a four-wing chaotic system based on the PRNG.

Spatial dynamics of swarmalators’ movements

The simultaneous interactions of synchronization and swarming phenomena are the primary basis for the functioning of many natural systems, such as vinegar eels, sperms, and magnetic particles. Investigating the dual behavior of these particles, which are called swarmalators, has recently attracted the attention of many researchers. However, more needs to be done to analyze the dynamics of swarmalators separately. This paper explores the dynamics of the swarmalators' movements individually over time in two-dimensional plane. This study's findings represent that based on the two rules of high sensitivity to the initial conditions and the positive largest Lyapunov exponent, the swarmalators’ movements in the dynamic states have a chaotic nature. This result can lead to a deeper understanding of the complex dynamics of natural swarmalators. It may provide an opportunity for other researchers to analyze the individual behavior of swarmalators along with their collective behavior.

Semantic and sentiment trajectories of literary masterpieces

The paper deals with semantic and sentiment trajectories of literary masterpieces (we used corpora of 12 languages of various language families), composed of individual embeddings or n-grams. We ascertain that, for all languages, semantic and sentiment trajectories are markedly chaotic: positive largest Lyapunov exponents; ‘entropy-complexity’ pairs belonging to the ‘chaotic’ area of the respective plane; the distinctive ‘chaotic’ drop of the number of false nearest neighbours at a particular value of an embedding dimension. The Russian language turns out to be more ‘chaotic’ than, for example, the English one; we attribute this fact to the free order of words. The Esperanto language, for various ‘approaches’ to different Indo-European languages. The results do not corroborate its claim to be equidistant from all languages. However, it seems to be equidistant from all Indo-European languages. These characteristics are utilised in order to develop a method to compare styles of an original masterpiece and its translations (to automatically assess translation quality). It appears that machine translations are still worse than human ones, however, for example, the Facebook translation is comparable with them.

Nonlinear analysis of biceps surface EMG signals for chaotic approaches

In the present study, the chaos is detected and quantified in the electromyographic (EMG) signal of the biceps muscle using nonlinear features. Biological systems are complex and nonlinear systems composed of a large number of interconnected elements. A thorough understanding of the behavior of these systems, which are sometimes chaotic, will lead to an understanding of the behavior of diseases, finding effective treatments, and using appropriate rehabilitation equipment. Therefore, designing a novel stimulation equation based on Rossler nonlinear dynamics causes chaos in the biceps EMG signal. This stimulation, the sitting and hand positioning is designed in such a way to prevent muscle fatigue and saturated stimulability and also to ensure that with minimal stimulation of the musculocutaneous nerve, there is a maximum activity in the biceps muscle. Then, each signal was divided into chaotic and non-chaotic parts with the positive largest Lyapunov exponent (LLE), and its nonlinear and chaotic criteria, including fractal dimension (FD) using Petrosian method, correlation dimension (CD), for both parts of the EMG signal were calculated as first time. Then, after applying the values of these criteria as features to the support vector machine (SVM) classifier, the separation was carried out using a linear kernel. The results showed that if the classification features include the two features of FD and CD, they are separated from non-chaotic EMG signals with 98.97 % accuracy. Finally, for the first time the extracted features of CD, FD are a good representation of the chaotic behavior of the biceps EMG signal.

On the dynamic prediction of a president's net satisfaction survey: A test of Twitter accuracy

Twitter as a social media platform, contains data that are considered noisy, unstructured, and dynamic in nature. Different studies have been conducted with the use of Twitter in forecasting outcomes such as Box Office, Stock Market, and Electoral Outcomes. In the Philippines, one prominent topic that is always in the news is President Rodrigo Roa Duterte due to his different perspective in serving the country. Surveys that count for the President's net satisfaction are conducted by different organizations across the Philippines, an example of which is the Social Weather Stations (SWS) which conducts quarterly surveys in different social classes in the country. This research aims to test Twitter posts (tweets) as an alternative basis in predicting SWS's result of the president's net satisfaction survey using the Partial Least Squares Regression. To determine whether Twitter sentiment on the President's satisfaction performance can be a basis for predicting outcomes of SWS the dynamics of its data is examined through a chaos analysis using the largest Lyapunov exponent (LLE). The LLE test therefore is used to determine whether the sentiment data obtained from Twitter possess a chaotic or nonchaotic behavior. Chaos test showed positive results for the Twitter variables except for the 2nd quarter of the Number of Negative Tweets, thus determining that for the whole year of 2017, all variables show a chaotic behavior. LLEs in the last quarter of 2017 tend to have a higher exponent during the month of November when the ASEAN summit was held in the Philippines. Evidently, the sentiment Twitter data possess a chaotic behavior as identified by positive LLEs. Partial Least Squares was able to accommodate the dynamic nature of the data and was able to create a forecasting model for the President Net Satisfaction Survey of SWS.

Analysis of rainfall data of some West African countries using wavelet transform and nonlinear time series techniques

ABSTRACT The chaotic behaviour of monthly rainfall data of Benin, Cote d’Ivoire, Cameroon, Ghana, Niger, Nigeria, Senegal and Togo between January 1901 and December 2015 were investigated using wavelet transformation analysis and time series techniques. Wavelet power spectrum was used to split the time series into different scales. Power concentrations between 8 and 16 months were observed for the selected locations. The embedding dimension, delay and largest Lyapunov exponent (LE) were calculated. We observed positive LE ranging from 0.13 to 0.36, indicating the rainfall was chaotic. Ghana had the highest values of LE, while the lowest LE was observed at Niger..

A Novel Chaotic System with Wide Spectrum, its Synchronization, Circuit Design and Application to Secure Communication

Objective: To investigate a novel chaotic system with unique features, its synchronization using nonlinear active control, analog circuit design and application to secure communication. Methods/Analysis: Dynamical tools such as dissipative analysis, instability of equilibrium points, sensitivity to initial conditions, 0-1 test, recurrence plot, Poincare map, Lyapunov exponents, Lyapunov dimension, frequency spectrum and basin of attraction. Synchronization is achieved using modified nonlinear active control technique and analog circuit design, implementation is done in NI Multisim platform. MATLAB and Multisim results are presented to meet the adequate verification of theoretical approach. Findings: A three-dimensional chaotic system with only two nonlinear terms, three parameters and a total of eight terms are proposed. The proposed system has three saddle focus type equilibria. The proposed system is topologically different from Lorenz’s and Rossler’s, Lu’s, Chen’s, and Liu’s families. Such dynamic systems are very few in the literature as per authors best knowledge. The system has basin of chaotic attractors for which first Lyapunov exponent ranges between 2.5 to 3. Frequency spectrum and large positive Lyapunov exponent result comparatively large bandwidth of the proposed systems against some well-known chaotic systems. Chaos, periodic and stable behaviors are obtained by altering the system parameters. Novelty/Application: The proposed three-dimensional chaotic system has significant chaotic behavior and broader spectrum than the six chaotic systems like Lorenz, Rossler, Lu, Chen, BG and Liu systems. Unlike the conventional active control approach, the proposed nonlinear active control does not result decoupled error dynamics. The system has significantly large bandwidth which is helpful in the masking of message signals and enhances the security of transmitted signals during communication. Keywords: Analog circuit design; Chaos; Chaotic system; Chaosbased communication; Chaos synchronization; Nonlinear active control

Determining chaotic characteristics and forecasting tall tower wind speeds in Missouri using empirical dynamical modeling (EDM)

The chaotic characteristics of the tall tower wind speed data within Missouri was investigated using both quantitative and qualitative methodologies. The phase space diagrams were constructed using the method of time delay. The two parameters needed in the construction of the attractor are the embedding dimension and the time delay. The former was determined using the Cao Algorithm and the latter by Average Mutual Information (AMI). Qualitatively, the phase portraits display chaos for all the wind speed time series for the various stations and height levels. They did not illustrate periodicity nor were they random motions, rather, they depicted a single attractor representative of chaos. Quantitatively the Largest Lyapunov Exponent (LLE) was evaluated. It was determined that for the Columbia station the wind speeds display chaotic characteristics representative of the positive LLEs. However, the increasing level of chaos characteristics did not coincide with the increasing height levels of the tall tower. Thereafter, a simple non-linear prediction algorithm was used to forecast wind speeds using a moving window. The attractor was constructed using the first 56 days and the subsequent 6 h or 36 (10 min) time steps were predicted. The preceding forecast was done when the attractor was reconstructed using the training data of 56 days starting from a 6-h delay from the previous run. The RMSE, MAE and Correlation were investigated for the model with the errors evaluated cumulatively for all of the 1st through 36st predictions. It was determined that the errors increase as the forecasting steps increased for all stations and height levels. The RMSE plateaus at higher wind speeds for increasing height levels with the exception of the station, Neosho, where it plateaued at all height levels at approximately 3.0 ms−1. For Columbia at all height levels, after the 20th time step or 3.33 h, the model's normalized errors exceeds 1 or 100%. However, using a 50% normalized error cap, it was noted that these values occurred for Columbia's height levels after the 1st, 2nd and 3rd time steps respectively. For Blanchard, this value was given by the 2nd time step for both heights whilst for Neosho, at all heights this percentage occurred after at most, 2 time steps. From the Lyapunov exponent, the prediction horizons or the time limits to obtain accurate predictions from the chaotic system were determined to be 6 time steps for all the height levels in the Columbia station using a 95% confidence band. When a range of confidence bands was used, it was shown that for the 90% confidence, this value was decreased to 4 time steps. This model was compared to the benchmark model of persistence where it was determined that the EDM is comparable to persistence and it beats it in the very short-term range of one time step for Columbia and Blanchard. Seasonality and diurnal cycle analyses were also accomplished. Seasonality was investigated by slicing the results every 6 h or extracting every 36th forecast error. It was shown that four of the eight stations' height levels had the season of summer incurring the lowest magnitude of average errors and standard deviations. The diurnal cycle was examined by extracting every four of the 6 time slices done previously. The time of day was analysed by lagging these slices by 6, 12 and 18 h. It was determined that there was no evident trend where a particular time of day the model incurred more errors and had greater standard deviations for all stations and heights.

On the Correlation Dimension of Discrete Fractional Chaotic Systems

This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann–Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger–Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.

Analysis, Control and FPGA Implementation of a Fractional-Order Modified Shinriki Circuit

In this paper, we introduce a novel integer-order memristor-modified Shinriki circuit (MMSC). We investigate the dynamic properties of the MMSC system and the existence of chaos is proved with positive largest Lyapunov exponent. Bifurcation plots are derived to analyze the parameter dependence of the MMSC system. The fractional-order model of the MMSC system (FOMMSC) is derived and the bifurcation analysis of the FOMMSC system with the fractional orders is carried out. Fractional-order adaptive sliding-mode controllers (FOASMCs) and genetically optimized PID controllers are designed to synchronize identical FOMMSC systems with unknown parameters. Numerical simulations are conducted to validate the theoretical results. FPGA implementation of the FOASMC controllers is presented to show that the proposed control algorithm is hardware realizable. MMSC has trigonometric functions which make the system more complex and the optimization and synchronization of such systems in the integer order itself are harder, so the paper does the same in fractional order. The proposed system is a memristive circuit which can show special features such as multistability, hyperchaos, and multiscroll attractor. Such a system with these features is very rare in the literature.

Dynamic reconstruction of chaotic system based on exponential weighted online sequential extreme learning machine with kernel

For the dynamic reconstruction of the chaotic dynamical system, a method of identifying an exponential weighted online sequential extreme learning machine with kernel(EW-KOSELM) is proposed. The kernel recursive least square (KRLS) algorithm is directly extended to an online sequential ELM framework, and weakens the effect of old data by introducing a forgetting factor. Meanwhile, the proposed algorithm can deal with the ever-increasing computational difficulties inherent in online kernel learning algorithms based on the ‘fixed-budget’ memory technique. The employed EW-KOSELM identification method is firstly applied to the numerical example of Duffing-Ueda oscillator for chaotic dynamical system based on simulated data, the qualitative and quantitative analysis for various validation tests of the dynamical properties of the original system as well as the identification model are carried out. A set of qualitative validation criteria is implemented by comparing chaotic attractors i.e. embedding trajectories, computing the corresponding Poincare mapping, plotting the bifurcation diagram, and plotting the steady-state trajectory i.e. the limit cycle between the original system and the identification model. Simultaneously, the quantitative validation criterion which includes computing the largest positive Lyapunov exponent and the correlation dimension of the chaotic attractors is also calculated to measure the closeness i.e. the approximation error between the original system and the identification model. The employed method is further applied to a practical implementation example of Chua's circuit based on the experimental data which are generated by sampling and recording the measured voltage across a capacitor, the inductor current from the double-scroll attractor, the measured voltage across a capacitor from the Chua's spiral attractor and an experimental time series from a chaotic circuit. The digital filtering technique is then used as a preprocessing approach, on the basis of wavelet denoising the measured data with lower signal-noise ratio (SNR) which can produce the double-scroll attractor or the spiral attractor, the reconstruction attractor of the identification model is compared with the reconstruction attractor from the experimental data for original system. The above experimental results confirm that the EW-FB-KOSELM identification method has a better performance of dynamic reconstruction, which can produce an accurate nonlinear model of process exhibiting chaotic dynamics. The identification model is dynamically equivalent or system approximation to the original system.

Modelings and mechanism analysis underlying both the 4D Euler equations and Hamiltonian conservative chaotic systems

Four sub-Euler equations for four sub-rigid bodies are generalized by extending the dimension of the state space from 3D to 4D. Six integrated 4D Euler equations are proposed by combining any two of the four sub-Euler equations with two common axes. These 4D equations are essential in providing the symplectic structures for the dynamics of rigid body and fluid mechanics and generalized Hamiltonian systems. The conservation of both the Hamiltonian and Casimir energies is proved for the six 4D Euler equations. Conservative chaos is more advantageous than dissipative chaos regarding ergodicity, the distribution of probability, and fractional dimensions in the application of chaos-based secure communications and generation of pseudo-random numbers. Six 4D Hamiltonian chaotic systems are proposed through breaking of the conservation of Casimir energies and preserving of the Hamiltonian energies, one of which is analyzed in detail. This system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 80 K), a large state amplitude and energy, and power spectral density with a wide bandwidth. The system passed the NIST tests performed on it. Therefore, strong pseudo-randomness of this Hamiltonian conservative chaotic system is confirmed. The Casimir power method is verified as an alternative analytical measuring index of orbital mode to the Lyapunov exponent. The force interaction and exchange in Casimir energy are the causes of chaos production. The mechanism underlying the transition from regular orbits to irregular orbits to stronger irregular orbits is studied using the Casimir power and the variability of physical parameters of the chaotic system. The supremum is also found using the property of Hamiltonian conservation.

Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm

It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

Lévy noise improves the electrical activity in a neuron under electromagnetic radiation.

As the fluctuations of the internal bioelectricity of nervous system is various and complex, the external electromagnetic radiation induced by magnet flux on membrane can be described by the non-Gaussian type distribution of Lévy noise. Thus, the electrical activities in an improved Hindmarsh-Rose model excited by the external electromagnetic radiation of Lévy noise are investigated and some interesting modes of the electrical activities are exhibited. The external electromagnetic radiation of Lévy noise leads to the mode transition of the electrical activities and spatial phase, such as from the rest state to the firing state, from the spiking state to the spiking state with more spikes, and from the spiking state to the bursting state. Then the time points of the firing state versus Lévy noise intensity are depicted. The increasing of Lévy noise intensity heightens the neuron firing. Also the stationary probability distribution functions of the membrane potential of the neuron induced by the external electromagnetic radiation of Lévy noise with different intensity, stability index and skewness papremeters are analyzed. Moreover, through the positive largest Lyapunov exponent, the parameter regions of chaotic electrical mode of the neuron induced by the external electromagnetic radiation of Lévy noise distribution are detected.

Are continuum predictions of clustering chaotic?

Gas-solid multiphase flows are prone to develop an instability known as clustering. Two-fluid models, which treat the particulate phase as a continuum, are known to reproduce the qualitative features of this instability, producing highly-dynamic, spatiotemporal patterns. However, it is unknown whether such simulations are truly aperiodic or a type of complex periodic behavior. By showing that the system possesses a sensitive dependence on initial conditions and a positive largest Lyapunov exponent, λ1≈1/τ, we provide a tentative answer: continuum predictions of clustering are chaotic. We further demonstrate that the chaotic behavior is dimensionally dependent, a conclusion which unifies previous results and strongly suggests that the chaotic behavior is not a direct consequence of the fundamental kinematic instability, but requires a secondary (inherently multidimensional) instability.

Nonlinear dynamics analysis of the human balance control subjected to physical and sensory perturbations.

Postural control after applying perturbation involves neural and muscular efforts to limit the center of mass (CoM) motion. Linear dynamical approaches may not unveil all complexities of body efforts. This study was aimed at determining two nonlinear dynamics parameters (fractal dimension (FD) and largest Lyapunov exponent (LLE)) in addition to the linear standing metrics of balance in perturbed stance. Sixteen healthy young males were subjected to sudden rotations of the standing platform. The vision and cognition during the standing were also interfered. Motion capturing was used to measure the lower limb joints and the CoM displacements. The CoM path length as a linear parameter was increased by elimination of vision (p<0.01) and adding a cognitive load (p<0.01). The CoM nonlinear metric FD was decreased due to the cognitive loads (p<0.001). The visual interference increased the FD of all joints when the task included the cognitive loads (p<0.01). The slightly positive LLE values showed weakly-chaotic behavior of the whole body. The local joint rotations indicated higher LLEs. Results indicated weakly chaotic response of the whole body. Increase in the task difficulty by adding sensory interference had difference effects on parameters. Linear and nonlinear metrics of the perturbed stance showed that a combination of them may properly represent the body behavior.

Quaternions and joint angles in an analysis of local stability of gait for different variants of walking speed and treadmill slope

The authors describe an example of application of a nonlinear time series analysis directed at identifying the presence of deterministic chaos in human gait kinematic data by means of the largest Lyapunov exponent (LLE). A positive LLE value is interpreted as an indicator of local instability. The research was aimed at assessment of the influence of both walking speed and ground slope on the resilience of gait control to infinitesimally small perturbations that occur naturally during walking. The analysis of treadmill gait data was carried out twofold: 1) for the time series representing the following joint angles: hip flexion/extension, knee flexion/extension and dorsiflexion/plantarflexion of the ankle, and 2) for the time series representing rotations of foot, tibia and femur segments through Euler angles converted to a quaternion representation. A comparison between both approaches as well as a dependency between treadmill inclination and LLE values constitute the original aspects of this study. The LLE value was estimated threefold for every time series: as the short-term LLE for both the first step and the first stride and as the long-term LLE over a fixed interval between the fourth and the tenth stride. It was confirmed that all considered movements are characterized by positive LLE values which quantify a local instability. Moreover, a tendency to attenuate the perturbation consequences is evident in all variants of walking speed and treadmill slope.

Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations.In this work, we have studied the chaoticity properties of FIGARCH (p,d,q) processes by computing mutual information, correlation dimensions, FNNs (False Nearest Neighbour), the largest Lyapunov exponents (LLE) for both the stochastic difference equation and for the financial time series by applying Wolf’s algorithm, Kant’z algorithm and Jacobian algorithm. Although Wolf’s algorithm produced positive LLE’s, Kantz’s algorithm and Jacobian algorithm which are subsequently developed methods due to insufficiency of Wolf’s algorithm generated negative LLE’s constantly.So, as well as experimenting Wolf’s methods’ inefficiency formerly pointed out by Rosenstein (1993) and later Dechert and Gencay (2000), based on Kantz’s and Jacobian algorithm’s negative LLE outcomes, we concluded that it can be suggested that FIGARCH (p,d,q) is not deterministic chaotic process.

Chaos in a Fractional-Order Single-Machine Infinite-Bus Power System and Its Adaptive Backstepping Control

This paper has numerically studied the dynamical behaviors of a fractional-order single-machine infinite-bus (FOSMIB) power system. Periodic motions, period- doubling bifurcations and chaotic attractors are observed in the FOSMIB power system. The existence of chaotic behavior is affirmed by the positive largest Lyapunov exponent (LLE). Based on the fractional-order backstepping method, an adaptive controller is proposed to suppress chaos in the FOSMIB power system. Numerical simulation results demonstrate the validity of the proposed controller.