Correlation dimension analysis and capillary wave turbulence in Dragon-Wash phenomena

This paper describes the evolution of surface capillary waves of deep water excited by gradually increasing the lateral external force at a single frequency. The vertical velocities of the water surface are measured by using a Polytec Laser Vibrometer with a thin layer of aluminium powder scattering on the surface to reflect the laser beam. Nonlinear interaction processes result in a stationary Fourier spectrum of the vertical surface velocities （the same as the surface elevation）, i.e. Iω -ω^-3-5. The observed spectrum can be interpreted as a wave-turbulent Kolmogorov spectrum for the case of ‘narrowband pumping＇ for a direct cascade of energy. Correlation dimension analysis of the whole development process reveals four distinct stages during the wave structure development and identifies the wave turbulence stage.

Estimation and Prediction of the Condition of the Vehicle Engine Based on the Correlation Dimension

This paper applies the fractal dimension as a characteristic to describe the engine抯 operating condition and its developmental trend. A correlation dimension is one of the quantities that are usually used to characterize a strange attractor. With the operation of the phase space reconstruction, respective correlation dimensions of a series of vibration signals obtained under different conditions are calculated to find the intrinsic relationship between the indicator and the operating condition. The experiment result shows that the correlation dimension is sensitive to the condition evolution and convenient for the identification of abnormal operational states. In advanced prognostic algorithm based on the BP neural network is then applied on the correlation dimensions to predict the short-term running conditions in order to avoid severe faults and realize in-time maintenance. Experimental results are presented to illustrate the proposed methodology.

Correlation dimension based nonlinear analysis of network traffics with different application protocols

This paper uses a correlation dimension based nonlinear analysis approach to analyse the dynamics of network traffics with three different application protocols-HTTP, FTP and SMTP. First, the phase space is reconstructed and the embedding parameters are obtained by the mutual information method. Secondly, the correlation dimensions of three different traffics are calculated and the results of analysis have demonstrated that the dynamics of the three different application protocol traffics is different from each other in nature, i.e. HTTP and FTP traffics are chaotic, furthermore, the former is more complex than the later; on the other hand, SMTP traffic is stochastic. It is shown that correlation dimension approach is an efficient method to understand and to characterize the nonlinear dynamics of HTTP, FTP and SMTP protocol network traffics. This analysis provided insight into and a more accurate understanding of nonlinear dynamics of internet traffics which have a complex mixture of chaotic and stochastic components.

Parallel solutions of correlation dimension calculation

The calculation of correlation dimension is a key problem of the fractals. The standard algorithm requires O（N2） computations. The previous improvement methods endeavor to sequentially reduce redundant computation on condition that there are many different dimensional phase spaces, whose application area and performance improvement degree are limited. This paper presents two fast parallel algorithms： O （N^2/p ＋ logp） time p processors PRAM algo- rithm and O（N^2/p） time p processors LARPBS algorithm. Analysis and results of numeric computation indicate that the speedup of parallel algorithms relative to sequence algorithms is efficient. Compared with the PRAM algorithm, The LARPBS algorithm is practical, optimally scalable and cost optimal.

Evaluation of Surface Roughness of Aluminum Alloy in Burnishing Process Based on Chaos Theory

Burnishing experiments with different burnishing parameters were performed on a computer numerical control milling machine to characterize the surface roughness of an aluminum alloy during burnishing.The chaos theory was employed to investigate the nonlinear features of the burnishing system.The experimental results show that the power spectrum is broadband and continuous,and the Lyapunov exponentλis positive,proving that burnishing has chaotic characteristics.The chaotic characteristic parameter,the correlation dimension D,is sensitive to the time behavior of the system and is used to establish the corresponding relationship with the surface roughness.The correlation dimension was the largest,when the surface roughness was the smallest.Furthermore,when the correlation dimension curve decreases,the roughness curve increases.The correlation dimension and surface roughness exhibit opposite variation trends.The higher the correlation dimension,the lower the surface roughness.The surface roughness of the aluminum alloy can be characterized online by calculating the correlation dimension during burnishing.

Grey Relation between Nonlinear Characteristic and Dynamic Uncertainty of Rolling Bearing Friction Torque

The rolling bearing friction torque which is characterized by its uncertainty and nonlinearity affects heavily the dynamic performance of a system such as missiles, spacecrafts and radars, etc. It is difficult to use the classical statistical theory to evaluate the dynamic evaluation of the rolling bearing friction torque for the lack of prior information about both probability distribution and trends. For this reason, based on the information poor system theory and combined with the correlation dimension in chaos theory, the concepts about the mean of the dynamic fluctuant range （MDFR） and the grey relation are proposed to resolve the problem about evaluating the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque. Friction torque experiments are done for three types of the rolling bearings marked with HKTA, HKTB and HKTC separately; meantime, the correlation dimension and MDFR are calculated to describe the nonlinear characteristic and the dynamic uncertainty of the friction torque, respectively. And the experiments reveal that there is a certain grey relation between the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque, viz. MDFR will become the nonlinear increasing trend with the correlation dimension increasing. Under the condition of fewer characteristic data and the lack of prior information about both probability distribution and trends, the unitive evaluation for the nonlinear characteristic and the dynamic uncertainty of the rolling bearing friction torque is realized with the grey confidence level of 87.7%-96.3%.

Chaotic Characteristic Analysis of Air Traffic System

Chaotic characteristics of traffic flow time series is analyzed to further investigate nonlinear characteristics of air traffic system.Phase space is reconstructed both by time delay which is built through mutual information,and by embedding dimension which is based on false nearest neighbors method.In order to analyze chaotic characteristics of time series,correlation dimensions and the largest Lyapunov exponents are calculated through Grassberger-Procaccia(G-P)algorithm and small-data method.Five-day radar data from the control center in Guangzhou area are analyzed and the results show that saturated correlation dimensions with self-similar structures exist in time series,and the largest Lyapunov exponents are all equal to zero and not sensitive to initial conditions.Air traffic system is affected by multiple factors,containing inherent randomness,which lead to chaos.Only grasping chaotic characteristics can air traffic be predicted and controlled accurately.

Radar Emitter Signal Recognition Based on Complexity Features

Intra-pulse characteristics of different radar emitter signals reflect on signal waveform by way of changing frequency, phase and amplitude. A novel approach was proposed to extract complexity features of radar emitter signals in a wide range of signal-to-noise ratio (SNR), and radial basis probability neural network (RBPNN) was used to recognize different radar emitter signals. Complexity features, including Lempel-Ziv complexity (LZC) and correlation dimension (CD), can measure the complexity and irregularity of signals, which mirrors the intra-pulse modulation laws of radar emitter signals. In an experiment, LZC and CD features of 10 typical radar emitter signals were extracted and RBPNN was applied to identify the 10 radar emitter signals. Simulation results show that the proposed approach is effective and has good application values because average accurate recognition rate is high when SNR varies in a wide range.

Probability Density Function Method for Observing Reconstructed Attractor Structure

Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6-6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system.

Research on a Diagnosis Method Based on a Correlation Integral

For the first time, the diagnosis idea based on a correlation integral isproposed, which regard's the correlation integral as a feature set. The correlation dimension iscontained in the double-log curve of the correlation integral to scale, so extracting featuresdirectly from the correlation integral can avoid the bottleneck problem of determining the range ofnon-scale length. Several features extracted from the correlation integral are better than thesingle feature of the correlation dimension when describing the signal. It is obvious that thismethod utilizes more information of the signal than does the correlation dimension. The diagnosisexamples verify that this method is more accurate and more effective.

A complexity measure approach based on forbidden patterns and correlation degree

Based on forbidden patterns in symbolic dynamics, symbolic subsequences are classified and relations between forbidden patterns, correlation dimensions and complexity measures are studied. A complexity measure approach is proposed in order to separate deterministic （usually chaotic） series from random ones and measure the complexities of different dynamic systems. The complexity is related to the correlation dimensions, and the algorithm is simple and suitable for time series with noise. In the paper, the complexity measure method is used to study dynamic systems of the Logistic map and the Henon map with multi-parameters.

Nonlinear Dynamic Analysis of MPEG-4 Video Traffic

The main research motive is to analysis and to veiny the inherent nonlinear character of MPEG-4 video. The power spectral density estimation of the video trafiic describes its 1/f^β and periodic characteristics.The priraeipal compohems analysis of the reconstructed space dimension shows only several principal components can be the representation of all dimensions. The correlation dimension analysis proves its fractal characteristic. To accurately compute the largest Lyapunov exponent, the video traffic is divided into many parts.So the largest Lyapunov exponent spectrum is separately calculated using the small data sets method. The largest Lyapunov exponent spectrum shows there exists abundant nonlinear chaos in MPEG-4 video traffic. The conclusion can be made that MPEG-4 video traffic have complex nonlinear be havior and can be characterized by its power spectral density,principal components, correlation dimension and the largest Lyapunov exponent besides its common statistics.

Determining chaotic invariant properties of short-circuiting gas metal arc welding from an observed time series

The experimental time series of welding current produced by carbon dioxide gas metal arc welding with shortcircuiting transfer were recorded and subsequently evaluated. Based on phase space reconstruction, the correlation dimeusions and Kolmogorov entropies of the corresponding system have been numerically calculated using the Grassberger-Procaccia algorithm at different time delays. It was found out that the time delay has little effect on the estimation of correlation dimension; conversely, it plays a key role in producing precise results on the estimation of Kolmogorov entropy.

Bi-prism-based single-lens three dimensional digital image correlation system with a long working distance: Methodology and application in extreme high temperature deformation test

As a novel three dimensional digital image correlation(3 D DIC) method, the bi-prism-based single lens(BSL) 3 D DIC method has been proposed and developed in recent years. Making use of a bi-prism, this method is able to perform a 3 D DIC measurement using only a single camera. Thus, the integration level of a BSL 3 D DIC system could be much higher than that of the double-camera3 D DIC system. In this paper, using a small-angle bi-prism and a camera with a longer focal length, a special BSL 3 D DIC system with a long working distance is designed for measurements in extreme environments. The principle of the system is first studied,and practical methods are then proposed for the system set-up and the determination of system parameters. Then, feasibility of the measurement system is verified by out-of-plane rigid-body translation tests. Finally, the BSL 3 D DIC system is proven to be capable of combining with a high-temperature testing instrument to perform deformation tests in a high-temperature environment of up to 1500°C.

Fractal Analysis of Rainfall-Induced Landslide and Debris Flow Spread Distribution in the Chenyulan Creek Basin, Taiwan

The Chenyulan Stream in Central Taiwan follows the Chenyulan fault line which is a major boundary fault in Taiwan. In recent years, many destructive landslides have occurred in the Chenyulan Creek Basin after heavy rainfall accompanied by several strong typhoons. Three examples of landslide distributions in the Chenyulan Creek Basin, before and after 1996 and after 2004 are analyzed. The box dimension and two-point correlation dimension are employed to describe the landslide area size distribution and distance distribution between every two landslides, respectively. It is found that the number of landslides increased in this period. However, the average landslide area decreased. The correlation dimension gradually increased from 1.15 to 1.32 during this period（before and after 1996 and after 2004）. This implies that the landslide distribution in the Chenyulan Creek Basin has become diffuse and extensive. The box dimension value shows the degree of the landslide density occupied in a space. The box dimension also increased from 0.3 to 0.69 during this period. The increasing box dimension means that the landslide presented in this creek basin has gradually increased. This indicates that the slopes of this creek basin have become more unstable and susceptible.

Wavelet denoising and nonlinear analysis of solids concentration signal in circulating fluidized bed riser

Particles,particle aggregates,and reactor walls complicate the dynamic microstructures of circulating fluidized beds(CFBs).Using local solids concentration data from a 10-m-high and 76.2-mm-inner-diameter riser with FCC(Fluid Catalytic Cracking)particles(dp=67μm,ρp=1500 kg/m^3),this paper presents an improved denoising process for use before nonlinear chaos analysis.Using the soft-threshold denoising method in the wavelet domain with experimental empty bed signals as base data to estimate the denoising threshold,an efficient denoising algorithm was proposed and used for the dynamic signals in CFBs.Analysis shows that for the local solids concentration time series,high-frequency fluctuations may be one of the system properties,while noise interference can also make a low-frequency contribution.An exact denoising method is needed in such cases.The correlation dimension and Kolmogorov entropy were calculated using denoised data and the results showed that the particle behavior in the CFB is highly complex.Generally,two correlation dimensions coexist in a low-flux CFB.The first correlation dimension is low and corresponds to small-scale fluctuations that reveal a high-frequency pseudo-periodic movement,but the second correlation dimension is high and corresponds to large-scale fluctuations that indicate multi-frequency movements,including particle aggregation and breakage.At the same axial level,the first correlation dimensions change slightly with radial position,and the main tendency is high at the center but slightly lower near the wall.However,the second correlation dimensions show large changes along the radial direction,are again high in the core region,and after r/R≥0.6(r as radial position,R as radius of the riser),the dimensions clearly drop down.This indicates that the particle behavior is more complex and has higher degrees of freedom at the center,but clusters near the wall are restrained to some degree because of wall effects.

ANALYSIS OF THE CHAOTIC DYNAMICS OF A HIGH-FLUX CFB RISER USING SOLIDS CONCENTRATION MEASUREMENTS

A high-flux circulating fluidized bed （CFB） riser （0.076-m I.D. and 10-m high） was operated in a wide range of operating conditions to study its chaotic dynamics, using FCC catalyst particles （dp= 67μm, ρp = 1500 kg·m^-3）. Local solids concentration fluctuations measured using a reflective-type fiber optic probe were processed to determine chaotic invariants （Kolmogorov entropy and correlation dimension）, Radial and axial profiles of the chaotic invariants at different operating conditions show that the core region exhibits higher values of the chaotic invariants than the wall region. Both invariants vary strongly with local mean solids concentration. The transition section of the riser exhibits more complex dynamics while the bottom and top sections exhibit a more uniform macroscopic and less-complex microscopic flow structure. Increasing gas velocity leads to more complex and less predictable solids concentration fluctuations, while increasing solids flux generally lowers complexity and increases predictability. Very high solids flux, however, was observed to increase the entropy.

Dynamic Reconstruction-Based Fuzzy Neural Network Method for Fault Detection in Chaotic System

This paper presents a method for detecting weak fault signals in chaotic systems based on the chaotic dynamics reconstruction technique and the fuzzy neural system （FNS）. The Grassberger-Procaccia algorithm and least squares regression were used to calculate the correlation dimension for the model order estimate. Based on the model order, an appropriately structured FNS model was designed to predict system faults. Through reasonable analysis of predicted errors, the disturbed signal can be extracted efficiently and correctly from the chaotic background. Satisfactory results were obtained by using several kinds of simulative faults which were extracted from the practical chaotic fault systems. Experimental results demonstrate that the proposed approach has good prediction accuracy and can deal with data having a -40 dB signal to noise ratio （SNR）. The low SNR requirement makes the approach a powerful tool for early fault detection.

CHAOTIC HYDRODYNAMICS IN HORIZONTAL GAS-LIQUID BUBBLY FLOW

To investigate the non-linear hydrodynamics in horizontal gas-liquid bubblyflow, time-series data of local void fraction were obtained using a double sensor probe. Bothdeterministic chaos and stochastic analyses were used to diagnose the dynamic behaviors in thesystem. The power spectra of the local void fraction exhibited no single distinct peak, indicatingthe aperiodic nature of the variation of the local void fraction. The kolmogorov entropy andcorrelation dimension of the attractor reconstructed from the time-series data by the embeddingmethod were to be positive and very large, the Hurst exponent computed from the rescaled rangeanalysis was found to be larger than 0. 5, indicating the non-linear chaotic feature of the system.The Kolmogorov entropy and the correlation dimension decrease with increasing liquid superficialvelocity, and the correlation dimension changes less if the gas superficial velocity is high. On theother hand, the fluctuation of local void fraction in horizontal gas-liquid bubbly flow could beconsidered to take place due to the zigzag motion of bubbles with different shape, size andvelocity.

STUDY OF THE RANDOMNESS INHERENT IN THE TYPHOON MOVEMENT

A difference scheme is adopted to perform numerical integrations of a deterministic barotropic vorticity equation for four designed experiments,and the 1-D and 2-D sequences are obtained for the velocity of the typhoon center with the aid of model output,together with the correlation dimension and K entropy investigated.Results show that there ex- ists randomness inherent in the center movement in the model atmosphere.Eventually the influence of localized heat sources and basic flow upon the stochasticity and predictability of the track of typhoon is discussed.