Largest Lyapunov Exponent Method Research Articles

Combining forecast of landslide displacement based on chaos theory

This study aimed to propose a new forecast combination method for landslide displacement based on the monitoring dataset. Considering the existence of outlier data, the 3σ method was used to eliminate them firstly. In order to diminish the effect of different length units, Z-score normalization was employed in this paper. Space reconstruction was performed to calculate the Lyapunov exponent, which was to verify whether the time series were chaotic. Three models were further studied here, namely, adding-weight one-rank local-region prediction method, adding-weight zero-rank local-region prediction method, and the largest Lyapunov exponent method. Furthermore, they are compared with respect to accuracy with each other and measured data so as to establish a forecast combination method. In order to check its validity, this method was applied to the analysis of the dataset of a 434-day displacement monitoring project performed on 12 monitoring points distributed on 4 profile maps on Qinghuasi Landslide Treatment Project in Yan’an, China. This model, which was not only implemented on a single point but also extended some results to a profile map, effectively, predicted the slope deformation in Qinghuasi landslide site. Prediction results indicate that the displacement of A-A profile map is relatively minimal, while slope deformation has been significant on the whole profile map of C-C. Situations at B-B and D-D are better than C-C profile map, though local failures have occurred at these two profile maps. This new model achieved a higher accuracy. Engineering measures based on these predicting results work effectively, which shows that it would be helpful in practical applications. This paper provides new idea and calculating method that can be used to develop deep data-mining approaches for landslide displacement prediction.

EEG Control of a Bionic Hand With Imagination Based on Chaotic Approximation of Largest Lyapunov Exponent: A Single Trial BCI Application Study

This paper investigates a method for imaginary hand fisting pattern recognition based on the electroencephalography (EEG). The proposed method estimate the largest Lyapunov exponent (LLE) chaotic feature that is based on approximation of mutual information (MI) and false nearest neighbor (FNN) methods for reconstructing a phase space. The selected method for MI and FNN approximation approaches is a new version of Tug of War optimization algorithm. The new algorithm utilizes chaotic maps to update candidate solutions. The chaotic approximation of the LLE (CALLE) is the utilized method for extracting the chaotic features and then categorizing features by means of soft margin support vector machine with a generalized radial basis function kernel classifier. Accuracy and paired t-test values are obtained and compared with the traditional LLE method; 18 candidates were participated to record the EEG for imaginary right-hand fisting task. The results show improvements for the CALLE algorithm in comparison with the traditional LLE by achieving a higher accuracy of 68.25%. Feature changes between two imaginary statuses were significant for 17 subjects, and the paired t-test values were (p <; 0.05). From the results, it is concluded that the Tug of War optimization method finds different values to reach a higher accuracy than the traditional LLE method, and the traditional methods for the LLE are not optimum.

Dynamics of solar wind speed: Cycle 23

A statistical signal processing approach has been made to study the dynamics of the speed of steady flow of hot plasma from the corona of sun known as solar wind generated in Solar Cycle 23. A long time series of solar wind speed of length 2492days from 1st Jan, 1997 to 28th October, 2003 collected from Coordinated Heliospheric Observations (COHO) data base at NASA’s National Space Science Data Center (NSSDC) is investigated for this purpose. Detection of nonlinearity and chaos in dynamics of solar wind speed is the prime objective of this work. In the present analysis delay vector variance (DVV) method is used to detect the existence of nonlinearity within the dynamics of solar wind speed. To explore the signature of the chaos in it multiple statistical methodologies like ‘0–1’ test, the correlation dimension analysis, computation of Information Entropy of the time series and Largest Lyapunov Exponent method have been applied. It has been observed that though the coronal plasma i.e. solar wind flow rate has a nonlinear dynamics but without any chaos. The absence of chaos indicates a probable regular behaviour of the series. The unit magnitude of the Correlation dimension indicates the presence of the deterministic component of the series. Embedding Dimension obtained argues that the deterministic component has dimension of six. The nearly zero value of the Lyapunov exponent claims that the system is conservative and exhibits Lyapunov stability. These revelations establish that not only the solar wind speed alone but the solar wind-magnetosphere coupling is also contributing towards the complexity of the magnetospheric plasma dynamics.

Lyapunov function method for analyzing stability of quasi-Hamiltonian systems under combined Gaussian and Poisson white noise excitations

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi-Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied by using Lyapunov function method. According to the integrability and resonance, quasi-Hamiltonian systems can be divided into five classes, namely quasi-non-integrable, quasi-completely integrable and non-resonant, quasi-completely integrable and resonant, quasi-partially integrable and non-resonant, and quasi-partially integrable and resonant. Lyapunov functions for these five classes of systems are constructed. The derivatives for these Lyapunov functions with respect to time are obtained by using the stochastic averaging method. The approximately sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian system under parametric excitations of combined Gaussian and Poisson white noises is determined based on a theorem due to Khasminskii. Four examples are given to illustrate the application and efficiency of the proposed method. And the results are compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent method.

Chaos and chaotic transients in an aeroelastic system

Chaotic motions of a two dimensional airfoil with coupled structural nonlinearities, both in pitch as well as plunge degrees of freedom, are investigated via a numerical integration method. The original system of coupled integro-differential equations governing the motion of the present aeroelastic model is transformed into a simple system of six ordinary differential equations (ODEs), rather than the previously frequently used eight ODEs. Complex dynamical behaviors are revealed and identified through the means of bifurcation diagrams, the phase portraits, the amplitude spectra and the Poincare maps. Besides, a more quantitative method, namely that of observing the evolution of the largest Lyapunov exponent (LLE) is also applied to diagnose the motions. Two peculiar phenomena, namely, long (perhaps super-persistent) chaotic transients, and fluctuating Lyapunov exponents, are observed; in the two such cases the LLE method fails to work. In addition, the effects of various system parameters, namely, the position of the elastic axis, the frequency ratio, the airfoil/air mass ratio, the viscous damping ratios, and the location of the center of mass, on the response of the aeroelastic system, are investigated.

The Application of Random Noise Reduction By Nearest Neighbor Method To Forecasting of Economic Time Series

Abstract Since the deterministic chaos appeared in the literature, we have observed a huge increase in interest in nonlinear dynamic systems theory among researchers, which has led to the creation of new methods of time series prediction, e.g. the largest Lyapunov exponent method and the nearest neighbor method. Real time series are usually disturbed by random noise, which can complicate the problem of forecasting of time series. Since the presence of noise in the data can significantly affect the quality of forecasts, the aim of the paper will be to evaluate the accuracy of predicting the time series filtered using the nearest neighbor method. The test will be conducted on the basis of selected financial time series.

Lyapunov function construction for nonlinear stochastic dynamical systems

Though the Lyapunov function method is more efficient than the largest Lyapunov exponent method in evaluating the stochastic stability of multi-degree-of-freedom (MDOF) systems, the construction of Lyapunov function is a challenging task. In this paper, a specific linear combination of subsystems’ energies is proposed as Lyapunov function for MDOF nonlinear stochastic dynamical systems, and the corresponding sufficient condition for the asymptotic Lyapunov stability with probability one is then determined. The proposed procedure to construct Lyapunov function is illustrated and validated with several representative examples, where the influence of coupled/uncoupled dampings and excitation intensities on stochastic stability is also investigated.

NONLINEARITY AND CHAOS IN 8B SOLAR NEUTRINO FLUX SIGNALS FROM SUDBURY NEUTRINO OBSERVATORY

The Sudbury neutrino observatory (SNO) detects 8 B solar neutrino fluxes from both the D2O and Salt detector. In the present analysis we have taken into consideration the flux data from 2nd November, 1999 to 27th May, 2001 from the D2O detector and that from 26th July, 2001 to 28th August, 2003 from the Salt detector. We have applied Delay Vector Variance analysis, 0-1 test, correlation dimension analysis, largest Lyapunov exponent method, recurrence plot and recurrence quantification analysis to explore the complexity and chaosity in these two time series. Present study reveals deterministic chaotic behaviour of these two signals which in turn suggests that long-term forecasting is not possible for these two signals but short-term forecasting can be made provided the model for the process dynamics is known to us.

Chaotic characteristics analysis and prediction model study on wind power time series

In order to reveal the internal dynamic property of wind power time series the nonlinear analysis method is used to identify the chaotic property of wind power set which is the basis for the prediction of the wind power time series. Firstly day correlation property on wind power time series of a certain wind farmer is analyzed. Secondly the largest Lyapunov exponent of wind power set is calculated on the basis of phase space construction to verify the presence of chaos in wind power time series. The ultra-short-term predicted of wind power would produce larger errors by using the Volterra filter multi-step prediction so the predicted results of Volterra filter are corrected by combining the results predicted by Local-region Multi-steps Method and the largest Lyapunov exponent method with weighted Markov chain and ordered operator. Finally the prediction on wind power of a certain wind farmer is presented and the simulation results illustrate that the correction forecasting model improves high predictive accuracy effectively, which provides a useful reference for wind power prediction by the Volterra filter multi-step method.

CHAOS SYNCHRONIZATION VIA UNIDIRECTIONAL COUPLING AND ITS APPLICATION TO SECURE COMMUNICATION

This paper studies chaos synchronization via unidirectional coupling. The self-synchronization of Lorenz systems, modified coupled dynamos systems and hyperchaotic Chen systems is studied by three methods: the Lyapunov function method, the global synchronization method and the numerical calculation of the largest Lyapunov exponent method. In regard to application to communication, we show that via transmitting single signal the synchronization of the drive system and the response system can be achieved. An example of applying self-synchronization of hyperchaotic Chen systems to chaotic masking secure communication is presented in this paper. Simulation results show the effectiveness of the method.