Value Of Largest Lyapunov Exponent Research Articles

Effects of Different Wearable Resistance Placements on Running Stability.

Stability during running has been recognized as a crucial factor contributing to running performance. This study aimed to investigate the effects of wearable equipment containing external loads on different body parts on running stability. Fifteen recreational male runners (20.27 ± 1.23 years, age range 19-22 years) participated in five treadmill running conditions, including running without loads and running with loads equivalent to 10% of individual body weight placed on four different body positions: forearms, lower legs, trunk, and a combination of all three (forearms, lower legs, and trunk). A tri-axial accelerometer-based smartphone sensor was attached to the participants' lumbar spine (L5) to record body accelerations. The largest Lyapunov exponent (LyE) was applied to individual acceleration data as a measure of local dynamic stability, where higher LyE values suggest lower stability. The effects of load distribution appear in the mediolateral (ML) direction. Specifically, running with loads on the lower legs resulted in a lower LyE_ML value compared to running without loads (p = 0.001) and running with loads on the forearms (p < 0.001), trunk (p = 0.001), and combined segments (p = 0.005). These findings suggest that running with loads on the lower legs enhances side-to-side local dynamic stability, providing valuable insights for training.

An analytic estimation for the largest Lyapunov exponent of the Rössler chaotic system based on the synchronization method

&lt;abstract&gt;&lt;p&gt;In this paper, we propose an analytical approach to estimate the largest Lyapunov exponent (LLE) of a Rössler chaotic system, leveraging the synchronization method. This research focuses on establishing an analytical criterion for the synchronization of two identical Rössler chaotic systems through the linear coupling of state variables. This is crucial because the LLE of such systems can be estimated based on the critical coupling required for synchronization. Unlike previous studies, we first transform the synchronization error system between two identical Rössler chaotic systems into a set of Volterra integral equations by using the Laplace transform and convolution theorem. The critical coupling for synchronization is analytically derived using integral equation theory to solve the error system. As compared to the numerical results of the Rössler chaotic system's LLE, our analytical estimates demonstrate high accuracy. Our findings suggest that the challenge of estimating the Rössler chaotic system's LLE can be simplified to solving a cubic algebraic equation, offering a novel perspective on the analysis of how parameters influence the LLE's value in the Rössler chaotic system.&lt;/p&gt;&lt;/abstract&gt;

CFD simulation and experimental analysis of solid-liquid mixing characteristics in a stirred tank with a self-similarity impeller

BackgroundThe nonlinear geometry could create a longer turbulence production region and higher turbulence intensities compared with conventional geometry. A self-similarity impeller based on Hibert curves was introduced to intensify the solid-liquid chaotic mixing process in this work. MethodsThe hydrodynamics of solid-liquid mixing process in a stirred tank with pitched-blade impeller, self-similarity 1 impeller and self-similarity 2 impeller were studied through experimental analyses and numerical simulation. Significant findingsResults showed that self-similarity impeller could consume less power consumption compared with pitched-blade impeller at the same impeller speed. Self-similarity impeller could improve the largest Lyapunov exponent (LLE) value compared with pitched-blade impeller under the constant power consumption, and the LLE value continued to increase with the increase of the number of self-similar iterations. Meanwhile, self-similarity impeller could improve the solid particle suspension quality compared with pitched-blade impeller under the constant power consumption, and increasingly so with the self-similar iteration number. Moreover, self-similarity impeller could destroy the large tailing vortex and recirculation zone, and enhance the turbulent kinetic energy dissipation rate by the jet flows through the concave-convex edges compared with pitched-blade impeller, and this phenomenon became obvious with the increase of self-similar iteration number.

Decreased Postural Complexity in Overweight to Obese Children and Adolescents: A Cross-Sectional Study.

IntroductionAlthough a few studies suggest that young overweight to obese children and adolescents (YO) may have impaired postural control compared to young normal-weight (YN) peers, little information exists about how these two groups differ in the quality of the underlying balance strategies employed. Hence, the aim of the present study was a first comprehensive examination of the structural complexity of postural sways in these two cohorts during quiet bilateral standing.MethodsNineteen YO secondary school students (13.0 ± 1.4 years; male = 10, female = 9) were carefully matched to YN controls (13.0 ± 1.5 years) for age, sex, height, and school. Mediolateral (ML) and anteriorposterior (AP) acceleration signals were recorded with an inertial measurement unit (IMU) positioned at the trunk while standing barefoot in two conditions: firm and foam support surface. The magnitude of postural fluctuations was obtained using the root mean square (RMS). The temporal structure of the signals was analyzed via sample entropy (SEn), largest Lyapunov exponent (LyE), and detrended fluctuation analysis (α-DFA) algorithm. Reliability was assessed using a test–retest design.ResultsIn both groups, foam standing caused higher postural fluctuations (higher RMS values) and reduced structural complexity (lower SEn values, higher LyE values, higher α-DFA values). In comparison to YN, YO exhibited a higher RMSAP. Especially in ML direction, the acceleration signals of the YO had higher repeatability (smaller SEn values), greater long-range correlations (higher α-DFA values), and lower local stability (higher LyE values). However, these observations were largely independent of the task difficulty. Except for α-DFAAP, the IMU approach proved reliable to characterize posture control.DiscussionOur outcomes confirm postural control deficits in YO compared to their YN peers and indicate impaired regulatory mechanisms reflected as rigidity. Such less complex patterns usually reflect diverse pathologies, are detrimental to compensate for internal or external perturbations, and are attributed to lower adaptability and task performance. Without targeted balance stimuli, YO likely end in a lifelong vicious circle of mutually dependent poor balance regulation and low physical activity.

Nonlinear Dynamics Suppression of Bending Deflection in a Composite Laminated Plate Using a Beam Stiffener Due to Critical Buckling Load and Shear Force

In this paper, the nonlinear dynamics behavior of the bending deflection of a stiffened composite laminated plate is suppressed using beam stiffeners at different fiber volume fractions and different aspect ratios. The non-periodic motion and chaos in a stiffened composite laminated plate is detected using the largest Lyapunov exponent parameter and power density function of a fast Fourier transform (FFT). The critical buckling load is calculated at different thickness ratios, numbers of stiffeners, lamination angles and stiffener–depth ratios based on different boundary conditions. The nonlinear response of the bending deflection is analyzed analytically, numerically and experimentally. The analytic solution has been derived using Levy and Navier solutions of classical laminate plate theory at different boundary conditions (CLPT). The numerical simulation was conducted using the ANSYS program while the experiment test was carried out using a strain gauge through a strain meter device. Experimentally, a Southwell plot is used to investigate the value of the critical buckling load. The combined loading are the in-plane compression mechanical load and shear force. All the values of the largest Lyapunov exponent are positive, which gives indication to non-periodic motion and chaos. The nonlinear dynamics behavior of the bending deflection is decreased with the increasing of number of stiffeners in which the value of largest Lyapunov exponent has been decreased. The nonlinear dynamics behavior is increased with the increasing of aspect ratios and fiber volume fractions. The system with an aspect ratio (2.5) and fiber volume fraction (υf = 80%) for an un-stiffened plate is more chaotic than the other systems.

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 hybrid wavelet-Lyapunov exponent model for river water quality forecast

Abstract The use of spectral theory and chaos theory on river water quality modeling is reported in a very limited way. This study proposes a wavelet-maximum Lyapunov exponent (WMLE) hybrid model for river water quality dynamics, combining spectral theory and chaos theory. The methodology involves the following major steps: (1) use of wavelet transformation to filter the noisy signal in the water quality time series; (2) reconstruction of phase space to embed the water quality time series and determine the trajectory of the underlying dynamics; and (3) identification of the presence/absence of chaos and prediction using the largest Lyapunov exponent value. Case studies on the Huaihe River in China and the Potomac River in the United States, as representatives of low-frequency and high-frequency forecast, show average relative errors on weekly dissolved oxygen (DO), chemical oxygen demand (COD), and ammonia nitrogen (NH3-N) data are 2.35%, 4.53%, and 18.85%, and on 15-minute based DO data are 1.185%. It also indicates that the hybrid model performs better to some extent when compared to the purely Lyapunov exponent model, ARMA model, and ANN model. This study is a proof that the combination of spectral theory and chaos theory is promising to describe and predict fluctuation of particular water quality indicators in rivers.

ACL injury and reconstruction affect control of ground reaction forces produced during a novel task that simulates cutting movements.

After anterior cruciate ligament (ACL) injury and reconstruction, biomechanical and neuromuscular control deficits persist and 25% of those who have experienced an ACL injury will experience a second ACL rupture in the first year after returning to sports. There remains a need for improved rehabilitation and the ability to detect an individual's risk of secondary ACL rupture. Nonlinear analysis metrics, such as the largest Lyapunov exponent (LyE) can provide new biomechanical insight in this population by identifying how movement patterns evolve over time. The purpose of this study was to determine how ACL injury, ACL reconstruction (ACLR), and participation in high-performance athletics affect control strategies, evaluated through nonlinear analysis, produced during a novel task that simulates forces generated during cutting movements. Uninjured recreational athletes, those with ACL injury who have not undergone reconstruction (ACLD [ACL deficient]), those who have undergone ACL reconstruction, and high-performance athletes completed a task that simulates cutting forces. The LyE calculated from forces generated during this novel task was greater (ie, force control was diminished) in the involved limb of ACLD and ACLR groups when compared with healthy uninjured controls and high-performance athletes. These data suggest that those who have experienced an ACL injury and subsequent reconstructive surgery exhibit poor force control when compared with both uninjured controls and high-performance athletes. Clinical significance: significantly larger LyE values after ACL injury and reconstruction when compared with healthy athletes suggest a continuing deficit in force control not addressed by current rehabilitation protocols and evaluation metrics that could contribute to secondary ACL rupture.

Fluid flow signals processing based on fractional Fourier transform in a stirred tank reactor

The analysis of fluid flow signals and the characterization of fluid flow behavior are of great importance for two-phase flow studies. In this work, the fractional Fourier transform (FRFT), which was based on the optimum order calculated by stepping search method, was proposed to extract the characteristics of fluid flow signals. Meanwhile, the largest Lyapunov exponent (LLE), which is an indication of the chaotic degree of mixing process, was adopted to quantify fluid flow behavior. The maximum amplitude (MA) and LLE value were taken together to inquire into the relationship between the characteristics of fluid flow signals and the characterization of fluid flow behavior. In addition, differences between the two adjacent values (AD) and the maximum differences (MD) are employed to further analyze the differences in behavioral characterization with MA and LLE. The results show that the MA value performs the same increasing trend as the LLE value when the gas flow rate and agitation speed increase. AD and MD values of the MA are one to two orders of magnitude greater than those of the LLE. The eigenvalues (MA) solved by the FRFT method is facilitates capturing small changes owing to changes in external conditions. These findings can provide new ideas for the extraction and characterization of fluid flow behavior.

A robust method to estimate the largest Lyapunov exponent of noisy signals: A revision to the Rosenstein’s algorithm

This study proposed a revision to the Rosenstein’s method of numerical calculation of the largest Lyapunov exponent (LyE) to make it more robust to noise. To this aim, the effect of increasing number of initial neighboring points on the LyE value was investigated and compared to values obtained by filtering the time series. Both simulated (Lorenz and passive dynamic walker) and experimental (human walking) time series were used to calculate the LyE. The number of initial neighbors used to calculate LyE for all time series was 1 (the original Rosenstein’s method), 2, 3, 4, 5, 10, 15, 20, 25, and 30 data points. The results demonstrated that the LyE graph reached a plateau at the 15-point neighboring condition implying that the LyE values calculated using at least 15 neighboring points were consistent. The proposed method could be used to calculate more consistent LyE values in experimental time series acquired from biological systems where noise is omnipresent.

Design of Impeller Blades for Intensification of Gas-Liquid Dispersion Process in a Stirred Tank

Abstract Gas-liquid dispersion characteristics were experimentally investigated by measuring largest Lyapunov exponent (LLE), relative power demand (RPD), and local gas holdup in a stirred tank with rigid impellers, rigid-flexible impellers, and punched rigid-flexible impellers. Results showed that punched rigid-flexible impeller could enhance the value of LLE, namely, the chaotic mixing degree of gas-liquid system compared with rigid impeller and rigid-flexible impeller. RPD for punched rigid-flexible impeller was higher than that for rigid impeller and rigid-flexible impeller. The local gas holdup of punched rigid-flexible impeller system was higher than those of rigid impeller system and rigid-flexible impeller system at the same Pg,m. In addition, a long flexible connection piece length could improve the chaotic mixing degree, RPD, and local gas holdup. The aperture diameter of 8 mm and free area ratio of 12 % of punched rigid-flexible impeller were particularly suitable for the gas-liquid dispersion process in this work.

Nonlinear analysis of heart rate variability of healthy subjects and patients with chronic obstructive pulmonary disease

Chronic obstructive pulmonary disease (COPD) is one of the causes of mortality worldwide with an increasing prevalence. Heart rate variability (HRV) reflects the regulation mechanism of the cardiac activity by the autonomic nervous system. The assessment of HRV by using nonlinear methods is more sensitive for the detection of complexity when compared to linear methods. This study aims to get information about the autonomic dysfunction occurred in patients with COPD by analysing the complexity of HRV. Electrocardiogram signals recorded from healthy subjects, patients with moderate COPD and severe COPD (eight subjects per group) were analysed. The HRV signals were acquired from ECG signals. Signals were reconstructed in the phase space and largest Lyapunov exponent (LLE), correlation dimension, Hurst exponent and approximate entropy (ApEn) values were calculated. It has seen that for the patients with COPD LLE, correlation dimension, Hurst exponent and ApEn values were less than control group. According to this, HRV complexity decreases in the presence of COPD. However, there is no significant difference between COPD groups and the severity of COPD has no effect on the chaoticity of the system. The results revealed that autonomic dysfunction occurred in patients with COPD is associated with reduced HRV complexity.

Wide-Area Measurement—Based Model-Free Approach for Online Power System Transient Stability Assessment

Online assessment of transient stability of power systems is critical for avoiding blackout. The increasing installation of phasor measurement units in power systems and the advancement of the wide-area measurement systems make it possible to develop methods for online assessment of power systems by using a large amount of real-time synchronous data. In this paper, we propose an approach based on the largest Lyapunov exponent (LLE) for online transient rotor angle stability assessment, using only wide-area measurement systems data. Through establishing a mathematical model that accounts for the LLE value and the rotor speed, as well as analyzing the post-fault phase-plane trajectory, we present a fast stability criterion to determine whether transient instability occurs. The proposed approach is model-free and does not need long-time LLE data to identify the final sign of the LLE. Furthermore, to reduce computational cost and realize online assessment for large-scale power systems, the critical generator pair (CGP) is recognized as the observation for the rotor angle stability assessment, and an effective algorithm for identifying the CGP is presented. Several case studies on the IEEE-39 bus test system and East China power grid are reported to demonstrate the accuracy and effectiveness of the proposed approach.

Effect of noise and filtering on largest Lyapunov exponent of time series associated with human walking

This study aimed to determine the effect of added noise, filtering and time series length on the largest Lyapunov exponent (LyE) value calculated for time series obtained from a passive dynamic walker. The simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series. The generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples. The LyE was calculated as the exponential rate of divergence of neighboring trajectories of the state space using Rosenstein’s algorithm. To determine the effect of noise on LyE values, seven levels of Gaussian white noise (SNR=55–25dB with 5dB steps) were added to the time series. In addition, the filtering was performed using a range of cutoff frequencies from 3Hz to 19Hz with 2Hz steps. The LyE was calculated for both noise-free and noisy time series with different lengths of 6, 50, 100 and 150 strides. Results demonstrated a high percent error in the presence of noise for LyE. Therefore, these observations suggest that Rosenstein’s algorithm might not perform well in the presence of added experimental noise. Furthermore, findings indicated that at least 50 walking strides are required to calculate LyE to account for the effect of noise. Finally, observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating LyE.

Intensification of chaotic mixing in a stirred tank with a punched rigid-flexible impeller and a chaotic motor

Laminar mixing of non-Newtonian CMC solution was investigated experimentally in a stirred tank with a rigid impeller, a rigid impeller coupled with a chaotic motor, a rigid-flexible impeller, a rigid-flexible impeller coupled with a chaotic motor, a punched rigid-flexible impeller, and a punched rigid-flexible impeller coupled with a chaotic motor. Visualization experiment was used to reveal the presence of segregated regions, and observe the structure changes of segregated regions. The mixing performance was characterized by the largest Lyapunov exponent (LLE) and dimensionless mixing time (NTm). Results showed that punched rigid-flexible impeller could further enhance the value of LLE, namely, the chaotic extent of mixing system compared with rigid impeller and rigid-flexible impeller, and the NTm for punched rigid-flexible impeller was shorter than that for rigid impeller and rigid-flexible impeller at the same consumption per unit volume (Pv). The chaotic motor could further increase LLE value, and reduce the NTm on the basis of the normal motor. Moreover, the optimum aperture diameter and aperture ratio of punched rigid-flexible impeller were 8mm and 12%, respectively, and the variable speed period of 10s and constant speed period of 5s of chaotic motor were particularly suitable for the laminar mixing process.

Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor

The mixing performance of solid-liquid system was investigated in a stirred tank with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor. The effects of the impeller types, the flexible connection piece width/length, the aperture size/ratio, and the variable/constant speed periods were investigated. The mixing performance was characterized by the largest Lyapunov exponent (LLE) and axial local solid concentration profile. The results indicated that the double punched rigid-flexible impeller and chaotic motor could enhance the LLE value of solid-liquid mixing system on the basis of the double rigid-flexible impeller and traditional motor under the same power consumption. The variation in solid homogeneous degree was in good agreement with that of LLE. It was found that the double punched rigid-flexible impeller coupled with a chaotic motor could further improve the solid suspension quality and energy efficiency compared with the double rigid impeller or double rigid-flexible impeller coupled with a traditional motor in the solid-liquid mixing process.

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.

Feature Extraction and Classification of Breast Tumors Using Chaos and Fractal Analysis on Dynamic Magnetic Resonance Imaging

Background: Breast cancer is one of the leading causes of death in the world. Early diagnosis of breast cancer can reduce the rate of mortality of this type of cancer. An increasing number of reports have confirmed the excellent sensitivity of dynamic contrast enhanced magnetic resonance imaging (DCE-MRI). Despite the excellent sensitivity of DCE-MRI, there is still some difficulty in the prediction of malignancy in these patients because of the lack of the optimum guidelines for the interpretation of breast magnetic resonance (MR) studies as well as the reported overlap in T1 and T2 relaxation times. Objectives: The aim of this study was to extract significant features from MRI images of the breast using chaos, fractal and time series analysis and to classify breast tumors into malignant and benign using the calculated features. Methods: In this research, we utilized the chaos theory and fractal analysis in the interpretation of breast tumors on DCE-MRI. This cross-sectional study was done at Pardisnoor imaging center during years 2015 and 2016 in Iran. Our sample size was 18 mass lesions, which were randomly selected among patients with BIRAD 3 and BIRAD 4 classification by the expert radiologist. The analysis was performed after injecting patients with a contrast agent and 18 mass lesions were extracted from dynamic MR images. After pre-processing and segmentation stages, time series of the tumor was generated for each MR image. The largest Lyapunov exponent (LLE) and statistical parameters for each mass lesion were extracted. Also, fractal analysis was utilized to extract meaningful features from mass contour to evaluate the roughness of tumor margin. Results: We found that the value of LLE in malignant tumors was higher than benign mass lesions. The obtained results demonstrated that chaos and time series features, such as LLE and non-circularity of the tumor, were the best parameters among all features. Conclusions: The extracted descriptors can improve the performance of classifiers in the early detection of breast cancer. Significant shape features can also help radiologists increase diagnosis accuracy in classification of suspicious breast masses.

Pulsatile flow in a compliant stenosed asymmetric model

Time-varying velocity field in an asymmetric constricted tube is experimentally studied using a two-dimensional particle image velocimetry system. The geometry resembles a vascular disease which is characterized by arterial narrowing due to plaque deposition. The present study compares the nature of flow patterns in rigid and compliant asymmetric constricted tubes for a range of dimensionless parameters appearing in a human artery. A blood analogue fluid is employed along with a pump that mimics cardioflow conditions. The peak Reynolds number range is Re ~ 300–800, while the Womersley number range considered in experiments is Wo ~ 6–8. These values are based on the peak velocity in a straight rigid tube connected to the model, over a pulsation frequency range of 1.2–2.4 Hz. The medial-plane velocity distribution is used to investigate the nature of flow patterns. Temporal distribution of stream traces and hemodynamic factors including WSS, TAWSS and OSI at important phases of the pulsation cycle are discussed. The flow patterns obtained from PIV are compared to a limited extent against numerical simulation. Results show that the region downstream of the constriction is characterized by a high-velocity jet at the throat, while a recirculation zone, attached to the wall, evolves in time. Compliant models reveal large flow disturbances upstream during the retrograde flow. Wall shear stress values are lower in a compliant model as compared to the rigid. Cross-plane flow structures normal to the main flow direction are visible at select phases of the cycle. Positive values of largest Lyapunov exponent are realized for wall movement and are indicative of chaotic motion transferred from the flow to the wall. These exponents increase with Reynolds number as well as compliance. Period doubling is observed in wall displacement of highly compliant models, indicating possible triggering of hemodynamic events in a real artery that may cause fissure in the plaque deposits.

Thermal instability-driven multiple solutions in a grooved channel

ABSTRACTWe present rich non-linear characteristics like multiple-stable states, bifurcation, and transition to chaos as a result of thermal instability-driven multiple solutions in a bottom-heated grooved channel undergoing natural convection with Rayleigh number varying from 1 × 105 to 9 × 105. From our analysis, we identify two different routes in which the solution of the system evolves along with the critical Rayleigh numbers associated with such transitions. In one route, the solution transforms from steady symmetric to chaotic solutions, while for the second route it first undergoes pitchfork bifurcation, where steady symmetric solution transforms to steady asymmetric, followed by Hopf bifurcation to periodic solutions. The two distinct solution routes clearly bring out the dependence of system solutions on initial conditions. The route to chaos in the grooved channel system has been successfully identified, which is supported by the results of time-series, power-spectra density and three-dimensional phase plots, apart from the positive values of largest Lyapunov exponent.