Lyapunov Exponent Of Time Series Research Articles

Study on the Precursors of Coal-Rock Fracture Based on the Maximum Lyapunov Exponent of Acoustic Emission Parameters

In this paper, the relationship between acoustic emission (AE) parameters and coal-rock fracture is investigated by using the discrete element particle flow simulation software PFC. PFC is used to simulate the coal-rock uniaxial compression test for extracting the time series of AE event number, calculate the maximum Lyapunov exponent of the time series, and analyze the chaotic characteristics of AE event number. Combined with the relationship between the maximum point of the Lyapunov exponent and the peak point of the stress-strain curve, the coal-rock fracture precursor model was developed. The results indicate that during the uniaxial compression of coal-rock, the number of AE events first increases and then decreases, while the maximum point appears after the peak point of the stress-strain curve. The lower the coal-rock homogeneity, the earlier is the initial occurrence time of AE events. The AE event number has chaotic characteristics. In most cases, the chaotic characteristics of the number of AE events are the most evident before the specimen is completely destroyed. When the maximum Lyapunov exponent of AE event number time series mutates, it indicates that the specimen is about to be destroyed entirely, which can be used as a precursor criterion for coal-rock fracture.

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.

A new method for the study of two-phase flow patterns based on the chaotic characteristic method of image fields

The phenomenon of two-phase flow is widespread in industrial production. Different flow regimes have different mechanisms for heat-transfer and flowing. Changes of flow regimes in a channel can lead to flow resistance, a change of stability and heat transfer problems. Therefore, understanding the dynamics and identification of the flow regime in a gas–liquid two-phase flow is vital in industrial production. Because of complex interfacial effects and relative motion between the phases, it is difficult to accurately identify the two-phase flow patterns. This paper emphasizes the study of flow dynamics, and flow pattern identification. This paper also proposes a new method for extracting time series. Each frame of the video signal is divided into smaller areas. The gray scale difference of two adjacent frames is calculated to obtain the maximum points in the smaller areas and form a time series. The maximal Lyapunov exponents of time series are respectively extracted, and its matrix is composed. The videos of gas–liquid two-flow patterns are divided into different chaotic areas by the characteristics of the Lyapunov exponent. Then the overall and detailed analyses are conducted respectively by zero and one distribution map and contour map. The mechanism of gas–liquid two-phase flow is analyzed by combining the fractal box dimension, Shannon entropy, and the average value of the maximal Lyapunov exponent matrix. As there are chaotic characteristics with different intensities in the background of the gas–liquid two-phase flow video and changed phase interfaces, the results show that the extracting method of the maximum distance series of small areas combined with the maximal Lyapunov exponent can be used to distinguish the characteristics of different flow patterns, which is proven to be an effective method for analyzing the gas–liquid two-phase flow signals.

Positive Lyapunov exponents calculated from time series of strange nonchaotic attractors.

Time-series methods for estimating Lyapunov exponents may give a positive exponent when they are applied to the time series of strange nonchaotic systems. Strange nonchaotic systems are characterized by expanding and contracting regions in phase space that result in repeatedly expanding or contracting trajectories. Using time-series methods, the maximum time-series Lyapunov exponent is calculated as an average of the locally most expanding exponents that characterize the divergence of nearby trajectories following a reconstructed attractor over time. A positive exponent is reported by time-series methods for trajectories in an expanding region. While in a converging region, the most expanding dynamics are related to the quasiperiodic driving force. Statistically, a zero exponent related to the quasiperiodic force is obtained through time-series methods within converging regions. As a result, the calculated maximum Lyapunov exponent is positive.