Singularity analysis of Duffing-van der Pol system with two bifurcation parameters under multi-frequency excitations

Bifurcation properties of a Duffing-van der Pol system with two parameters under multi-frequency excitations are studied. Three cases are discussed： （1） λ 1 is considered as bifurcation parameter, （2） λ 2 is considered as bifurcation parameter, and （3） λ 1 and λ 2 are both considered as bifurcation parameters. According to the definition of transition sets, the whole parametric space is divided into several different persistent regions by the transition sets for different cases. The bifurcation diagrams in different persistent regions are obtained, which provides a theoretical basis for optimal design of the system.

Singularity analysis of a two-dimensional elastic cable with 1:1 internal resonance

A two-degree-of-freedom bifurcation system for an elastic cable with 1：1 internal resonance is investigated in this paper. The transition set of the system is obtained with the singularity theory for three cases. The whole parametric plane is divided into several different persistent regions by the transition set. The bifurcation diagrams in different persistent regions are obtained.

Singularity analysis and avoidance for an SSRMS-type reconfigurable space manipulator with a non-spherical wrist and two lockable passive telescopic links

This study focuses on addressing kinematic singularity analysis and avoidance issues for a space station remote manipulator system(SSRMS)-type reconfigurable space manipulator.The manipulator is equipped with a non-spherical wrist and two lockable passive telescopic links(LPTLs),which enable it to have both active revolute and passive prismatic joints and operate in two distinct modes.To begin with the kinematic singularity analysis,the study derives the differential kinematic equations for the manipulator and identifies the dominant Jacobian matrix that causes singularities.Subsequently,an in-depth analysis of singularities from multiple perspectives is conducted.Firstly,a kinematic singularity map method is proposed to capture the distribution of singularities within the reachable workspace.Then,the influence of the two LPTLs on singularities is thoroughly examined.Finally,a new method based on the matrix rank equivalence principle is introduced to determine singularity conditions,enabling the identification of all the singular configurations for the SSRMS-type reconfigurable manipulator.Notably,this method significantly reduces computational complexity,and the singularity conditions obtained have more concise equations.For the singularity avoidance problem,a novel method is developed,which simultaneously addresses the requirements of real-time performance,high precision,and the avoidance of both kinematic singularities and joint limit constraints.Benefiting from these excellent properties,the proposed method can effectively resolve the singularity issues encountered separately by the SSRMS-type reconfigurable manipulator in its two operational modes.Several typical simulations validate the utility of all the proposed methods.

A novel method for singularity analysis of the 6-SPS parallel mechanisms

Singularity analysis is a basic problem of parallel mechanism, and this problem cannot be avoided in both workspace and motion planning. How to express the singularity locus in an analytical form is the research emphasis for many researchers for a long time. This paper presents a new method for the singularity analysis of the 6-SPS parallel mechanism. The rotation matrix is described by quaternion, and both the rotation matrix and the coordinate vectors have been expanded to four-dimensional forms. Through analyzing the coupling relationship between the position variables and the orientation variables, utilizing properties of the quaternion, eight equivalent equations can be obtained. A new kind of Jacobian matrix is derived from those equations, and the analytical expression of the singularity locus is obtained by calculating the determinant of the new Jacobian matrix. The singularity analysis of parallel mechanisms, whose moving platform actuated by 6 links and the vertices of both the base and the moving platforms has been placed on a circle respectively, can be solved by this analytical expression.

Singularity analysis for single gimbal control moment gyroscope system using space expansion method

Control Moment Gyroscope（CMG） is an effective candidate for agile satellites and large spacecraft attitude control because of its powerful torque amplification capability. The most serious situation, however, in using CMG is the inherent geometric singularity problem, where there＇s no torque output along a particular direction. Space expansion method has been proposed in this work for the singularity analysis. Based on inverse mapping transformation, an expanded Jacobian matrix which is a full rank square matrix is obtained. The singular angle sets of the 3-parallel cluster and pyramid cluster are distinguished using space expansion method. An effective hybrid steering strategy, able to deal with the elliptic singularity, is further proposed. Simulation results demonstrate the excellent performance of the proposed steering logic compared to the generalized singular robust logic and pseudo inverse logic in terms of energy consumption and torque error.

Periodic signal extraction of GNSS height time series based on adaptive singular spectrum analysis

Singular spectrum analysis is widely used in geodetic time series analysis.However,when extracting time-varying periodic signals from a large number of Global Navigation Satellite System(GNSS)time series,the selection of appropriate embedding window size and principal components makes this method cumbersome and inefficient.To improve the efficiency and accuracy of singular spectrum analysis,this paper proposes an adaptive singular spectrum analysis method by combining spectrum analysis with a new trace matrix.The running time and correlation analysis indicate that the proposed method can adaptively set the embedding window size to extract the time-varying periodic signals from GNSS time series,and the extraction efficiency of a single time series is six times that of singular spectrum analysis.The method is also accurate and more suitable for time-varying periodic signal analysis of global GNSS sites.

A reweighted damped singular spectrum analysis method for robust seismic noise suppression

(Multichannel)Singular spectrum analysis is considered as one of the most effective methods for seismic incoherent noise suppression.It utilizes the low-rank feature of seismic signal and regards the noise suppression as a low-rank reconstruction problem.However,in some cases the seismic geophones receive some erratic disturbances and the amplitudes are dramatically larger than other receivers.The presence of this kind of noise,called erratic noise,makes singular spectrum analysis(SSA)reconstruction unstable and has undesirable effects on the final results.We robustify the low-rank reconstruction of seismic data by a reweighted damped SSA(RD-SSA)method.It incorporates the damped SSA,an improved version of SSA,into a reweighted framework.The damping operator is used to weaken the artificial disturbance introduced by the low-rank projection of both erratic and random noise.The central idea of the RD-SSA method is to iteratively approximate the observed data with the quadratic norm for the first iteration and the Tukeys bisquare norm for the rest iterations.The RD-SSA method can suppress seismic incoherent noise and keep the reconstruction process robust to the erratic disturbance.The feasibility of RD-SSA is validated via both synthetic and field data examples.

Error analysis and a new steering law design for spacecraft control system using SGCMGs

Based on the singular value decomposition theory,this paper analyzed the mechanism of escaping/avoiding singularity using generalized and weighted singularity-robust steering laws for a spacecraft that uses single gimbal control moment gyros （SGCMGs） as the actuator for the attitude control system.The expression of output-torque error is given at the point of singularity,proving the incompatible relationship between the gimbal rate and the output-torque error.The method of establishing a balance between the gimbal rate and the output-torque error is discussed,and a new steering law is designed.Simulation results show that the proposed steering law can effectively drive SGCMGs to escape away from singularities.

Bifurcation analysis of fan casing under rotating air flow excitation

A fan casing model of cantilever circular thin shell is constructed based on the geometric characteristics of the thin-walled structure of aero-engine fan casing. According to Donnelly＇s shell theory and Hamilton＇s principle, the dynamic equations axe established. The dynamic behaviors are investigated by a multiple-scale method. The effects of casing geometric parameters and motion parameters on the natural frequency of the system are studied. The transition sets and bifurcation diagrams of the system are obtained through a singularity analysis of the bifurcation equation, showing that various modes of the system such as the bifurcation and hysteresis will appear in different parameter regions. In accordance with the multiple relationship of the fan speed and stator vibration frequency, the fan speed interval with the casing vibration sudden jump is calculated. The dynamic reasons of casing cracks are investigated. The possibility of casing cracking hysteresis interval is analyzed. The results show that cracking is more likely to appear in the hysteresis interval. The research of this paper provides a theoretical basis for fan casing design and system parameter optimization.

The Singular Value Decomposition Analysis between Summer Precipitation in the Dongting Lake Region and the Global Sea Surface Temperature

By dint of the summer precipitation data from 21 stations in the Dongting Lake region during 1960-2008 and the sea surface temperature(SST) data from NOAA,the spatial and temporal distributions of summer precipitation and their correlations with SST are analyzed.The coupling relationship between the anomalous distribution in summer precipitation and the variation of SST has between studied with the Singular Value Decomposition(SVD) analysis.The increase or decrease of summer precipitation in the Dongting Lake region is closely associated with the SST anomalies in three key regions.The variation of SST in the three key regions has been proved to be a significant previous signal to anomaly of summer rainfall in Dongting region.

A comparison analysis of multipath extract methods in short-baseline applications of GPS

In order to eliminate the multipath errors existing in static short-baseline applications, a novel de-noising method based on a singular spectrum analysis （named as DSSA） is introduced to extract multipath signals. The multipath error is extracted from the double difference （DD） residuals by DSSA and then applied to the correct multipath error in subsequent measurements based on the correlation among adjacent epochs. Methods based on discrete wavelet transform （DWT） and stationary wavelet transform （SWT） are introduced as comparisons of DSSA based on analysis of a simulated signal. Real baseline residuals are tested to verify different extract methods. Results show that compared with the SWT, the DSSA improves the root mean square （RMS） of the residual by 48.6% and achieves a time reduction of 75.3%.

Singular analysis of bifurcation systems with two parameters

Bifurcation properties of dynamical systems with two parameters are investigated in this paper. The definition of transition set is proposed, and the approach developed is used to investigate the dynamic characteristic of the nonlin- ear forced Duffing system with nonlinear feedback controller. The whole parametric plane is divided into several persistent regions by the transition set, and then the bifurcation dia- grams in different persistent regions are obtained.

Short-range Climate Prediction Experiment of the Southern Oscillation Index Based on the Singular Spectrum Analysis

The Southern Oscillation Index (SOI) time series is analyzed by means of the singular spectrum analysis (SSA) method with 60-month window length. Two major oscillatory pairs are found in the series whose periods are quasi-four and quasi-two years respectively. The auto-regressive model, which is developed on the basis of the Maximum Entropy Spectrum Analysis, is fitted to each of the 9 leading components including the oscillatory pairs. The prediction of SOI with the 36-month lead is obtained from the reconstruction of these extrapolated series. Correlation coefficient between predicted series and 5 months running mean of observed series is up to 0.8. The model can successfully predict the peak and duration of the strong ENSO event from 1997 to 1998. It's also shown that the proper choice of reconstructed components is the key to improve the model prediction.

Dynamic prediction of landslide displacement using singular spectrum analysis and stack long short-term memory network

An accurate landslide displacement prediction is an important part of landslide warning system. Aiming at the dynamic characteristics of landslide evolution and the shortcomings of traditional static prediction models, this paper proposes a dynamic prediction model of landslide displacement based on singular spectrum analysis(SSA) and stack long short-term memory(SLSTM) network. The SSA is used to decompose the landslide accumulated displacement time series data into trend term and periodic term displacement subsequences. A cubic polynomial function is used to predict the trend term displacement subsequence, and the SLSTM neural network is used to predict the periodic term displacement subsequence. At the same time, the Bayesian optimization algorithm is used to determine that the SLSTM network input sequence length is 12 and the number of hidden layer nodes is 18. The SLSTM network is updated by adding predicted values to the training set to achieve dynamic displacement prediction. Finally, the accumulated landslide displacement is obtained by superimposing the predicted value of each displacement subsequence. The proposed model was verified on the Xintan landslide in Hubei Province, China. The results show that when predicting the displacement of the periodic term, the SLSTM network has higher prediction accuracy than the support vector machine(SVM) and auto regressive integrated moving average(ARIMA). The mean relative error(MRE) is reduced by 4.099% and 3.548% respectively, while the root mean square error(RMSE) is reduced by 5.830 mm and 3.854 mm respectively. It is concluded that the SLSTM network model can better simulate the dynamic characteristics of landslides.

Denoising Nonlinear Time Series Using Singular Spectrum Analysis and Fuzzy Entropy

We present a hybrid singular spectrum analysis （SSA） and fuzzy entropy method to filter noisy nonlinear time series. With this approach, SSA decomposes the noisy time series into its constituent components including both the deterministic behavior and noise, while fuzzy entropy automatically differentiates the optimal dominant components from the noise based on the complexity of each component. We demonstrate the effectiveness of the hybrid approach in reconstructing the Lorenz and Mackey--Class attractors, as well as improving the multi-step prediction quality of these two series in noisy environments.

An Intercomparison of Rules for Testing the Significance of Coupled Modes of Singular Value Decomposition Analysis

This paper clarifies the essence of the significance test of singular value decomposition analysis （SVD）, and investigates four rules for testing the significance of coupled modes of SVD, including parallel analysis, nonparametric bootstrap, random-phase test, and a new rule named modified parallel analysis. A numerical experiment is conducted to quantitatively compare the performance of the four rules in judging whether a coupled mode of SVD is significant as parameters such as the sample size, the number of grid points, and the signal-to-noise ratio vary. The results show that the four rules perform better with lower ratio of the number of grid points to sample size. Modified parallel analysis and nonparametric bootstrap perform best to abandon the spurious coupled modes, but the latter is better than the former to retain the significant coupled modes when the sample size is not much larger than the number of grid points. Parallel analysis and random-phase test are robust to abandon the spurious coupled modes only when either （1） the observations at the grid points are spatially uncorrelated, or （2） the coupled signal is very strong for parallel analysis and is not weak for random-phase test. The reasons affecting the accuracy of the test rules are discussed.

Multichannel singular spectrum analysis of the axial atmospheric angular momentum

Earth＇s variable rotation is mainly produced by the variability of the AAM（atmospheric angular momentum）. In particular, the axial AAM component X_3, which undergoes especially strong variations,induces changes in the Earth＇s rotation rate. In this study we analysed maps of regional input into the effective axial AAM from 1948 through 2011 from NCEP/NCAR reanalysis. Global zonal circulation patterns related to the LOD（length of day） were described. We applied MSSA（Multichannel Singular Spectrum Analysis） jointly to the mass and motion components of AAM, which allowed us to extract annual, semiannual, 4-mo nth, quasi-biennial, 5-year, and low-frequency oscillations. PCs（Principal components） strongly related to ENSO（El Nino southern oscillation） were released. They can be used to study ENSO-induced changes in pressure and wind fields and their coupling to LOD. The PCs describing the trends have captured slow atmospheric circulation changes possibly related to climate variability.

A singular spectrum analysis on Holocene climatic oscillation from lake sedimentary record in Minqin Basin, China

The total organic carbon (TOC) content series from the lake sediment of Minqin Basin (100°57′–104°57′E, 37°48′–39°17′N) in northwestern China, which has a 10 000-year-long paleo-climatic proxy record, was used to analyze the Holocene climate changes in the local region. The proxy record was established in the Sanjiaocheng (SJC), Triangle Town in Chinese, Section (103°20′25″E, 39°00′38″N), which is located at the northwestern boundary of the present Asian summer monsoon in China, and is sensitive to global environmental and climate changes. Applying singular spectrum analysis (SSA) to the TOC series, principal climatic oscillations and periodical changes were studied. The results reveal 3 major patterns of climate change regulated by reconstructed components (RCs). The first pattern is natural long-term trend of climatic change in the local area (Minqin Basin), indicating a relatively wetter stage in early Holocene (starting at 9.5 kaBP), and a relatively dryer stage with a strong lake desiccation and a declined vegetation cover in mid-Holocene (during 7–6 kaBP). From 4.0 kaBP to the present, there has been a gradually decreasing trend in the third reconstructed component (RC3) showing that the local climate changed again into a dryer stage. The second pattern shows millennial-centennial scale oscillations containing cycles of 1 600 and 800 years that have been present throughout almost the entire Holocene period of the last 10 000 years. The third pattern is a millennial-centennial scale variation with a relatively smaller amplitude and unclear cycles showing a nonlinear interaction within the earth’s climate systems.

DENOISING METHOD BASED ON SINGULAR SPECTRUM ANALYSIS AND ITS APPLICATIONS IN CALCULATION OF MAXIMAL LIAPUNOV EXPONENT

An algorithm based on the data-adaptive filtering characteristics of singular spectrum analysis (SSA) is proposed to denoise chaotic data. Firstly, the empirical orthogonal functions (EOFs) and principal components (PCs) of the signal were calculated, reconstruct the signal using the EOFs and PCs, and choose the optimal reconstructing order based on sigular spectrum to obtain the denoised signal. The noise of the signal can influence the calculating precision of maximal Liapunov exponents. The proposed denoising algorithm was applied to the maximal Liapunov exponents calculations of two chaotic system, Henon map and Logistic map. Some numerical results show that this denoising algorithm could improve the calculating precision of maximal Liapunov exponent.

Improved interpolation method based on singular spectrum analysis iteration and its application to missing data recovery

A novel interval quartering algorithm （IQA） is proposed to overcome insufficiency of the conventional singular spectrum analysis （SSA） iterative interpolation for selecting parameters including the number of the principal components and the embedding dimension. Based on the improved SSA iterative interpolation, interpolated test and comparative analysis are carried out to the outgoing longwave radiation daily data. The results show that IQA can find globally optimal parameters to the error curve with local oscillation, and has advantage of fast computing speed. The improved interpolation method is effective in the interpolation of missing data.