Determination of Kolmogorov Entropy of Chaotic Attractor Included in One-Dimensional Time Series of Meteorological Data

The 1970-1985 day to day averaged pressure dataset of Shanghai and the extension method in phase space are used to calculate the correlation dimension D and the second-order Renyi entropy K2 of the approximation of Kolmogorov's entropy, the fractional dimension D = 7.7-7.9 and the positive value K2 - 0.1 are obtained. This shows that the attractor for the short-term weather evolution in the monsoon region of China exhibits a chaotic motion. The estimate of K2 yields a predictable time scale of about ten days. This result is in agreement with that obtained earlier by the dynamic-statistical approach.The effects of the lag time i on the estimate of D and K2 are investigated. The results show that D and K2 are convergent with respect to i. The day to day averaged pressure series used in this paper are treated for the extensive phase space with T = 5, the coordinate components are independent of each other; therefore, the dynamical character quantities of the system are stable and reliable.

Uniform attractors for non-autonomous Klein-Gordon-Schrdinger lattice systems

The existence of a compact uniform attractor for a family of processes corre- sponding to the dissipative non-autonomous Klein-Gordon-SchrSdinger lattice dynamical system is proved. An upper bound of the Kolmogorov entropy of the compact uniform attractor is obtained, and an upper semicontinuity of the compact uniform attractor is established.

Comparison between the new mechanistic and the chaos scale-up methods for gas-solid fluidized beds

The chaotic scale-up approach by matching the Kolmogorov entropy(E_K) proposed by Schouten et al.(1996) was assessed in two geometrically similar gas–solid fluidized bed columns of 0.14 and 0.44 m diameter.We used four conditions of our validated new mechanistic scale-up method based on matching the radial profiles of gas holdup where the local dimensionless hydrodynamic parameters were similar as measured by advanced measurement techniques.These experimental conditions were used to evaluate the validity of the chaotic scale-up method,which were selected based on our new mechanistic scale-up methodology.Pressure gauge transducer measurements at the wall and inside the bed at various local radial locations and at three axial heights were used to estimate KE.It was found that the experimental conditions with similar or close radial profiles of the Kolmogorov entropy and with similar or close radial profiles of the gas holdup achieve the similarity in local dimensionless hydrodynamic parameters,and vice versa.

THE FRACTAL RESEARCH AND PREDICATING ON THE TIME SERIES OF SUNSPOT RELATIVE NUMBER

In this paper, with the theory of nonlinear dynamic systems, It is analyzed that the dynamic behavior and the predictability for the monthly mean variations of the sunspot relative number recorded from January 1891 to December 1996. In the progress, the fractal dimension (D = 3.3 +/- 0.2) for the variation process rt as computed. This helped us to determine the embedded dimension [2 x D + 1] = 7. By computing the Lyapunov index (lambda(1) = 0.863), it was indicated that the variation process is a chaotic system. The Kolmogorov entropy (K = 0.0260) was also computed, which provides, theoretically, the predicable time scale. And at the end, according to the result of the analysis above, an experimental predication is made, whose data was a part cut from the sample data.

Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler

A nonlinear and non-averaged model of a two-beam free-electron laser （FEL） wiggler that is tapered nonlinearly in the absence of slippage is presented. The two beams are assumed to have different energies, and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam. By using Maxwell＇s equations and the full Lorentz force equation of motion for the electron beams, coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method. The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs. By simulation, the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found. This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions. The effect of tapering on the dynamical stability of the fast electron beam is also studied.

Research on Chaotic Time Series Prediction Based on K-entropy and RBF Neural Networks

In this paper, a method of direct multi-step prediction of chaotic time series is proposed, which is based on Kolmogorov entropy and radial basis functions neural networks. This is done first by reconstructing a phase space using chaotic time series, then using K-entropy as a quantitative parameter to obtain the maximum predictability time of chaotic time series, finally the predicted chaotic time series data can be acquired by using RBFNN. The application considered is Lorenz system. Simulation results for direct multi-step prediction method are compared with recurrence multi-step prediction method. The results indicate that the direct multi-step prediction is more accurate and rapid than the recurrence multi-step prediction within the maximum predictability time of chaotic time series. So, it is convenient to forecast and control with real time using the method of direct multi-step prediction.

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.

Effect of bed size on the gas-solid flow characterized by pressure fluctuations in bubbling fluidized beds

Pressure fluctuations in four bubbling fluidized beds having different bed sizes (three square cross-sections of 5, 10, and 15 cm in side length, and one rectangular cross-section of 2 × 10 cm2) were measured at four axial positions (P1, P2, P3, and P4). Several characteristic indicators of the flow specifically of the pressure were calculated. In terms of these characteristic indicators, the effect of bed size on flow behavior was investigated. The results show that in the fully fluidized state, the pressure drop is slightly higher in smaller beds, but the pressure drops in the 10- and 15-cm beds are close. The 15-cm bed has the lowest pressure-fluctuation amplitude. The amplitudes at P1 and P2 in the lower part of the bed are very close for bed sizes below 10 cm, but the amplitude at P3 near the bed surface increases with decreasing bed size. No general trend was observed regarding the effect of bed size on skewness and kurtosis of the pressure for all four axial heights. For the average, standard deviation, skewness, and kurtosis of the pressure at P4, the values are close for the two small beds (2 × 10 and 5 × 5 cm2) and the two large beds (10 × 10 and 15 × 15 cm2), and hence the effect of bed size separates the beds into two groups. In the fully fluidized state, for P1, P2, and P3, the Kolmogorov entropy and the dominant frequency both increase with increasing bed size, but in the pseudo-2D bed both are between the values for the 5- and 10-cm beds.

Quantifying Dynamical Predictability:the Pseudo-Ensemble Approach

The ensemble technique has been widely used in numerical weather prediction and extended-range forecasting.Current approaches to evaluate the predictability using the ensemble technique can be divided into two major groups.One is dynamical,including generating Lyapunov vectors,bred vectors,and singular vectors,sampling the fastest error-growing directions of the phase space,and examining the dependence of prediction efficiency on ensemble size.The other is statistical,including distributional analysis and quantifying prediction utility by the Shannon entropy and the relative entropy.Currently,with simple models,one could choose as many ensembles as possible,with each ensemble containing a large number of members.When the forecast models become increasingly complicated,however,one would only be able to afford a small number of ensembles,each with limited number of members,thus sacrificing estimation accuracy of the forecast errors.To uncover connections between different information theoretic approaches and between dynamical and statistical approaches,we propose an (∈;T)-entropy and scale-dependent Lyapunov exponent——based general theoretical framework to quantify information loss in ensemble forecasting.More importantly,to tremendously expedite computations,reduce data storage,and improve forecasting accuracy,we propose a technique for constructing a large number of "pseudo" ensembles from one single solution or scalar dataset.This pseudo-ensemble technique appears to be applicable under rather general conditions,one important situation being that observational data are available but the exact dynamical model is unknown.

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.

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.