The Experts below are selected from a list of 3786 Experts worldwide ranked by ideXlab platform
Haifeng Liu - One of the best experts on this subject based on the ideXlab platform.
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estimating the Largest Lyapunov Exponent and noise level from chaotic time series
Chaos, 2012Co-Authors: Tianliang Yao, Haifeng LiuAbstract:A novel method for estimating simultaneously the Largest Lyapunov Exponent (LLE) and noise level (NL) from a noisy chaotic time series is presented in this paper. We research the influence of noise on the average distance of different pairs of points in an embedding phase space and provide a rescaled formula for calculating the LLE when the time series is contaminated with noise. Our algorithm is proposed based on this formula and the invariant of the LLE in different dimensional embedding phase spaces. With numerical simulation, we find that the proposed method provides a reasonable estimate of the LLE and NL when the NL is less than 10% of the signal content. The comparison with Kantz algorithm shows that our method gives more accurate results of the LLE for the noisy time series. Furthermore, our method is not sensitive to the distribution of the noise.
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noise robust estimates of the Largest Lyapunov Exponent
Physics Letters A, 2005Co-Authors: Haifeng Liu, Zhenghua Dai, Xin GongAbstract:A first-order correction of the Exponential divergence of trajectories in state space of a chaotic time series with noise was proposed. We demonstrated the performance for various examples using data from the Henon map, Ikeda map and logistic map, which were contaminated with noise. It was found that the proposed method provided a reasonable estimate of the Largest Lyapunov Exponent even when the noise level was as high as 30% of the signal content. The new method was not sensitive to the distribution of the noise. Furthermore, the comparison with Wolf et al. algorithm showed that our method is much better when dealing with the time series contaminated with noise. Our algorithm was also valid for more complicated chaotic dynamical systems, such as Lorenz attractor and Rossler-hyperchaos attractor.
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the Largest Lyapunov Exponent of chaotic dynamical system in scale space and its application
Chaos, 2003Co-Authors: Haifeng Liu, Yongzhe Yang, Zhenghua DaiAbstract:The Largest Lyapunov Exponent is an important invariant of detecting and characterizing chaos produced from a dynamical system. We have found analytically that the Largest Lyapunov Exponent of the small-scale wavelet transform modulus of a dynamical system is the same as the system’s Largest Lyapunov Exponent, both discrete map and continuous chaotic attractor with one or two positive Lyapunov Exponents. This property has been used to estimate the Largest Lyapunov Exponent of chaotic time series with several kinds of strong additive noise.
Wahyu Caesarendra - One of the best experts on this subject based on the ideXlab platform.
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application of the Largest Lyapunov Exponent algorithm for feature extraction in low speed slew bearing condition monitoring
Mechanical Systems and Signal Processing, 2015Co-Authors: Wahyu Caesarendra, Buyung Kosasih, A K Tieu, Craig A S MoodieAbstract:Abstract This paper presents a new application of the Largest Lyapunov Exponent (LLE) algorithm for feature extraction method in low speed slew bearing condition monitoring. The LLE algorithm is employed to measure the degree of non-linearity of the vibration signal which is not easily monitored by existing methods. The method is able to detect changes in the condition of the bearing and demonstrates better tracking of the progressive deterioration of the bearing during the 139 measurement days than comparable methods such as the time domain feature methods based on root mean square (RMS), skewness and kurtosis extraction from the raw vibration signal and also better than extracting similar features from selected intrinsic mode functions (IMFs) of the empirical mode decomposition (EMD) result. The application of the method is demonstrated with laboratory slew bearing vibration data and industrial bearing data from a coal bridge reclaimer used in a local steel mill.
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condition monitoring of slow speed slewing bearing based on Largest Lyapunov Exponent algorithm and circular domain feature extractions
2013Co-Authors: Wahyu Caesarendra, P B Kosasih, Kiet A Tieu, Craig A S MoodieAbstract:This paper presents a combined nonlinear and circular features extraction-based condition monitoring method for low speed slewing bearing. The proposed method employs the Largest Lyapunov Exponent (LLE) algorithm as a signal processing method based on vibration data. LLE is used to detect chaos existence in vibration data in discrete angular positions of the shaft. From the processed data, circular features such as mean, skewness and kurtosis are calculated and monitored. It is shown that the onset and the progressively deteriorating bearing condition can be detected more clearly in circular-domain features compared to timedomain features. The application of the method is demonstrated with laboratory run slewing bearing data.
Xingyuan Wang - One of the best experts on this subject based on the ideXlab platform.
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simple estimation method for the Largest Lyapunov Exponent of continuous fractional order differential equations
Physica A-statistical Mechanics and Its Applications, 2021Co-Authors: Shuang Zhou, Xingyuan WangAbstract:Abstract In this paper, a simple method based on the perturbation of the initial value is presented to directly estimate the Largest Lyapunov Exponent (LLE) from continuous fractional-order differential equations. Two nearby trajectories are used to directly compute the LLE and reduce parameter errors. Another initial value is obtained by perturbing the given initial value. Two solutions are then developed from a fractional-order chaotic system by using the two initial values. The evolutionary distance between the two solutions is calculated, and the LLE is determined from the curve of the track distance. Some continuous fractional-order chaotic and nonchaotic differential equations are applied to verify the effectiveness of our method. Experimental results indicate that the proposed method is feasible and easy to implement instead of computing the Jacobian matrix and phase space.
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simple estimation method for the second Largest Lyapunov Exponent of chaotic differential equations
Chaos Solitons & Fractals, 2020Co-Authors: Shuang Zhou, Xingyuan WangAbstract:Abstract The Largest Lyapunov Exponent (LLE) is an important tool used to identify systems. However, it cannot easily distinguish between chaos and hyperchaos. Therefore, the second-Largest Lyapunov Exponent (SLLE) must be calculated. Some methods to calculate this index have already been proposed, but these require massive computation times and data points. Thus, to reduce the calculating complexity, we propose a simple novel method based on three nearby orbits to directly calculate the finite-time local SLLE from continuous chaotic systems with improved accuracy. First, we obtain a solution orbit of chaotic equations. Then, two selected points from the solution are used as the initial condition to solve the same equation and obtain two solution orbits. Next, we calculate the evolution area of three solution orbits and the distance between the two solution trajectories. The LLE and area Exponent are then obtained from the logarithmic curve of the track distance and the area. Finally, the SLLE is obtained from the difference computed between the two indices. Some chaotic differential equations and non-chaotic systems are presented to demonstrate the efficiency of the proposed method. Calculating the finite-time local Exponents does not require the calculation of all the evolution lengths. Thus, the proposed method is simple, very fast and robust without the need to reconstruct the phase space. For now, the findings of this research provide new ideas related to the SLLE calculation theory.
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a novel method based on the pseudo orbits to calculate the Largest Lyapunov Exponent from chaotic equations
Chaos, 2019Co-Authors: Shuang Zhou, Xingyuan Wang, Zhen Wang, Chuan ZhangAbstract:To reduce parameter error caused by human factors and ensure the accuracy of the Largest Lyapunov Exponent (LLE) obtained from chaotic equations, this paper proposes a simple method based on two nearby pseudo-orbits. First, a point is selected from a solution trajectory of chaotic equation by the roundoff error. Second, the selected point is used as an initial condition to solve the same equation to obtain another solution trajectory. Third, the evolution distance of the two solution trajectories is calculated. Finally, the LLE is the slope of the linear region in the curve of the track distance of the natural algorithm. Our method has been successfully applied to simulate five well-known chaotic systems and some non-chaotic systems. The results show that, compared with other traditional methods, the proposed method is efficient, simple, and robust without reconstructing phase space and computing the Jacobian matrix.
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identifying the linear region based on machine learning to calculate the Largest Lyapunov Exponent from chaotic time series
Chaos, 2018Co-Authors: Shuang Zhou, Xingyuan WangAbstract:To reduce the error caused by the human factor, this paper proposes a modification of a well-known small data method to obtain the Largest Lyapunov Exponent more accurately, which is based on machine learning for better identification of linear region. Firstly, we use the k-d tree neighborhood search algorithm to improve the computational efficiency of the average divergence index data. Secondly, the unsaturated data are obtained by the density peak based clustering algorithm from the average divergence index data. Thirdly, we use the density peak based clustering algorithm to identify the linear region from the first-order difference curve of the retained data. Finally, the Largest Lyapunov Exponent is obtained by using the least squares method to fit the linear region. Our method is applied to simulate five famous theoretical chaotic systems, the results show that the proposed method can automatically identify the linear region, which is more accurate than the small data method for the Largest Lyapunov Exponent calculation and the effectiveness of our method is verified through the simulation of two real-world time series.
Sina Mehdizadeh - One of the best experts on this subject based on the ideXlab platform.
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a robust method to estimate the Largest Lyapunov Exponent of noisy signals a revision to the rosenstein s algorithm
Journal of Biomechanics, 2019Co-Authors: Sina MehdizadehAbstract:Abstract 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.
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a robust method to estimate the Largest Lyapunov Exponent of noisy signals a revision to the rosensteins algorithm
bioRxiv, 2018Co-Authors: Sina MehdizadehAbstract:Aim. This study proposed a revision to the Rosenstein9s method of numerical calculation of Largest Lyapunov Exponent (LyE) to make it more robust to noise. Methods. To this aim, the effect of increasing number of initial neighboring points on the LyE value was investigated and compared to the values obtained by filtering the time series. Both simulated (Lorenz and passive dynamic walker) and experimental (human walking) time series were used to calculate 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. Results. The results demonstrated that the LyE graph reached a plateau at the 15-point neighboring condition inferring that the LyE values calculated using at least 15 neighboring points were consistent and reliable. Conclusion. The proposed method could be used to calculate LyE more reliably in experimental time series acquired from biological systems where noise is omnipresent.
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the Largest Lyapunov Exponent of gait in young and elderly individuals a systematic review
Gait & Posture, 2018Co-Authors: Sina MehdizadehAbstract:The Largest Lyapunov Exponent (LyE) is an accepted method to quantify gait stability in young and old adults. However, a range of LyE values has been reported in the literature for healthy young and elderly adults in normal walking. Therefore, it has been impractical to use the LyE as a clinical measure of gait stability. The aims of this systematic review were to summarize different methodological approaches of quantifying LyE, as well as to classify LyE values of different body segments and joints in young and elderly individuals during normal walking. The Pubmed, Ovid Medline, Scopus and ISI Web of Knowledge databases were searched using keywords related to gait, stability, variability, and LyE. Only English language articles using the Lyapunov Exponent to quantify the stability of healthy normal young and old subjects walking on a level surface were considered. 102 papers were included for full-text review and data extraction. Data associated with the walking surface, data recording method, sampling rate, walking speed, body segments and joints, number of strides/steps, variable type, filtering, time-normalizing, state space dimension, time delay, LyE algorithm, and the LyE values were extracted. The disparity in implementation and calculation of the LyE was from, (i) experiment design, (ii) data pre-processing, and (iii) LyE calculation method. For practical implementation of LyE as a measure of gait stability in clinical settings, a standard and universally accepted approach of calculating LyE is required. Therefore, future studies should look for a standard and generalized procedure to apply and calculate LyE.
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effect of noise and filtering on Largest Lyapunov Exponent of time series associated with human walking
Journal of Biomechanics, 2017Co-Authors: Sina Mehdizadeh, Mohammad Ali SanjariAbstract: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.
Shuang Zhou - One of the best experts on this subject based on the ideXlab platform.
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simple estimation method for the Largest Lyapunov Exponent of continuous fractional order differential equations
Physica A-statistical Mechanics and Its Applications, 2021Co-Authors: Shuang Zhou, Xingyuan WangAbstract:Abstract In this paper, a simple method based on the perturbation of the initial value is presented to directly estimate the Largest Lyapunov Exponent (LLE) from continuous fractional-order differential equations. Two nearby trajectories are used to directly compute the LLE and reduce parameter errors. Another initial value is obtained by perturbing the given initial value. Two solutions are then developed from a fractional-order chaotic system by using the two initial values. The evolutionary distance between the two solutions is calculated, and the LLE is determined from the curve of the track distance. Some continuous fractional-order chaotic and nonchaotic differential equations are applied to verify the effectiveness of our method. Experimental results indicate that the proposed method is feasible and easy to implement instead of computing the Jacobian matrix and phase space.
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simple estimation method for the second Largest Lyapunov Exponent of chaotic differential equations
Chaos Solitons & Fractals, 2020Co-Authors: Shuang Zhou, Xingyuan WangAbstract:Abstract The Largest Lyapunov Exponent (LLE) is an important tool used to identify systems. However, it cannot easily distinguish between chaos and hyperchaos. Therefore, the second-Largest Lyapunov Exponent (SLLE) must be calculated. Some methods to calculate this index have already been proposed, but these require massive computation times and data points. Thus, to reduce the calculating complexity, we propose a simple novel method based on three nearby orbits to directly calculate the finite-time local SLLE from continuous chaotic systems with improved accuracy. First, we obtain a solution orbit of chaotic equations. Then, two selected points from the solution are used as the initial condition to solve the same equation and obtain two solution orbits. Next, we calculate the evolution area of three solution orbits and the distance between the two solution trajectories. The LLE and area Exponent are then obtained from the logarithmic curve of the track distance and the area. Finally, the SLLE is obtained from the difference computed between the two indices. Some chaotic differential equations and non-chaotic systems are presented to demonstrate the efficiency of the proposed method. Calculating the finite-time local Exponents does not require the calculation of all the evolution lengths. Thus, the proposed method is simple, very fast and robust without the need to reconstruct the phase space. For now, the findings of this research provide new ideas related to the SLLE calculation theory.
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a novel method based on the pseudo orbits to calculate the Largest Lyapunov Exponent from chaotic equations
Chaos, 2019Co-Authors: Shuang Zhou, Xingyuan Wang, Zhen Wang, Chuan ZhangAbstract:To reduce parameter error caused by human factors and ensure the accuracy of the Largest Lyapunov Exponent (LLE) obtained from chaotic equations, this paper proposes a simple method based on two nearby pseudo-orbits. First, a point is selected from a solution trajectory of chaotic equation by the roundoff error. Second, the selected point is used as an initial condition to solve the same equation to obtain another solution trajectory. Third, the evolution distance of the two solution trajectories is calculated. Finally, the LLE is the slope of the linear region in the curve of the track distance of the natural algorithm. Our method has been successfully applied to simulate five well-known chaotic systems and some non-chaotic systems. The results show that, compared with other traditional methods, the proposed method is efficient, simple, and robust without reconstructing phase space and computing the Jacobian matrix.
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identifying the linear region based on machine learning to calculate the Largest Lyapunov Exponent from chaotic time series
Chaos, 2018Co-Authors: Shuang Zhou, Xingyuan WangAbstract:To reduce the error caused by the human factor, this paper proposes a modification of a well-known small data method to obtain the Largest Lyapunov Exponent more accurately, which is based on machine learning for better identification of linear region. Firstly, we use the k-d tree neighborhood search algorithm to improve the computational efficiency of the average divergence index data. Secondly, the unsaturated data are obtained by the density peak based clustering algorithm from the average divergence index data. Thirdly, we use the density peak based clustering algorithm to identify the linear region from the first-order difference curve of the retained data. Finally, the Largest Lyapunov Exponent is obtained by using the least squares method to fit the linear region. Our method is applied to simulate five famous theoretical chaotic systems, the results show that the proposed method can automatically identify the linear region, which is more accurate than the small data method for the Largest Lyapunov Exponent calculation and the effectiveness of our method is verified through the simulation of two real-world time series.
