The Phase-space Reconstruction, Bifurcation And Attractor Analysis Of Economy System

Among a large number of non-linear systems existing in the nature and thesociety, a lot of inside structure of system is not clear, its external characteristic is onlya certain form time series of variable usually. Therefore, it is important to deal withsingle variable time series and analyze the characteristic of the original dynamics ofsystem. Chaos is an irregular behaviour which is a wide existent phenomenon. Chaos isa complicated behaviour which is from a deterministic nonlinear dynamical system.Phase space reconstruction is the base of using dynamical methods to analysenonlinear time series. The key of phase space reconstruction is the estimation of itsparameter. It has far-reaching meanings to study the non-linear time series, especiallymacro-economy time series. The meaning of using chaos theory to analyze theeconomic data is: though this process some economic systems can be understoodclearly like the operation laws of the securities business, strengthen the various kindsof economic regulation and control management of financial market better.Chaotic time series are extremely sensitive to its initial conditions. The change ofthe initial value will be expanded in the rate of exponent. So it is very difficult topredict chaotic time series. But chaos is a deterministic system determined by thenonlinear dynamical mechanism. There is a deterministic rule in the interior of thechaotic system which is seemed as a random move. So chaotic time series can bepredicted in the short term.And the main contributions of this thesis are:1. On the base of the introduction of the conceptions such as non-linear timeseries analysis, phase-space reconstruction and chaos, the delay-coordinate method isimproved by using the K-L transformation, the classical chaos time series such as theLorenz system etc and economic time series are analyzed deeply, at the same time, the chaotic attractors implied in the economic systems are acquired.In this paper we analyse the existing prediction methods. And simulation resultsshow the prediction performance of the existing prediction methods.In this paper we discuss the theory and methods of phase space reconstruction. Inthis paper we use the matrix constructed by high-order statistics function to estimatethe embedding dimension. We study methods using high-order statistics function toconstruct matrixes and findthe best two methods. Simulation results proved the validity and the stability of thismethod.2. There are many methods which can analyze the single time series in thenon-linear time series and resume the nonlinear kinetics characteristics for examplechaotic attractor etc. In this paper, the delay-coordinate method is adopted toreconstructed the space phase. The key of this method is to select fixed embeddingdimension and delay time. In this paper, a large number of theory analysis which canchose the two parameters are made and research, the cardinal principle and one's ownpluses and minuses of these methods are expounded, the programs that can realizethese method are made, and have been proved.Single variable time series are reconstructed by the delay-coordinate method torebuild phase space, on this base the method that K-L transformation which is appliedin the phase space is put forward. By this method the most of energy of the originalsignal is kept, the correlations in between all the vectors are eliminated absolutely .Atthe same time, the characters of the original system can be better resumed, moreinformation can be acquired and the computation can also be reduced.3. On the base of the space-phase reconstruction, the two important parametersthat can characterize non-linear system attractors included Lyapunov exponent andcorrection dimension are studied, the programs that can calculate these two parametersare made by myself. For proving the dependability and the accuracy of theseprograms, some classical chaotic systems are used, the result indicate the values calculated by the programs accord with the theory values. These classical chaoticsystems are Lorenz system, Henon map and Logistic map.Noise is an omnipresent phenomenon. All experimental date are to some extentcontaminated by noise. The sensitivity to noise of chaotic system is a consequence ofthe inherent instability. And the inside structure of chaotic time series are destroyed bynoise. So it is very important to reduce the noise in chaotic time series. At last wediscuss the noise reduction methods.On the basis of the research work of domestic and foreign scholars, the paperstudies noise effects to largest Lyapunov exponent of the chaotic time series, accordingto the small data sets method which developed by Rosenstein et al. We demonstratedfor various examples using data from Logistic map, Henon map and Lorenz map,which were contaminated with noise. In this paper, the time series were produced by asuperposition of Gauss white noise generated by random number and noise-free data.We analysed effects of length of time series and embedding dimension to the largestLyapunov exponent under the same signal-to-noise ratio(SNR), as well as weresearched the change tendency of largest Lyapunov exponent under different SNR.The conclusion implies that the effects of noise isnot obvious when SNR is in givenscope, which has the important theory and practice significance to diagnosis ofnonlinear chaotic characteristic of the economy or finance time serics contaminatedwith noise in certain degree in actual problem.4. Based on the short-term predictability of chaotic time series and the adaptivetracking chaotic trajectory of adaptive algorithm, a new multi-step-prediction methodis proposed in this paper, and this method is used to improve the adaptive predictionmethod and the local adaptive prediction method. Simulation results show that themulti-step prediction performance of the improved method is rapidly improved. Thisresult is very important to newly understand the predictability of chaotic time series.In addition, in the paper a few of methods are used to set up predicting models ofthe time array, analyze the time series in order to study characteristic of non-linear The design and manufacture methods of high-strength spiral bevel gears andhypoid gears are researched in this thesis. Due to the transmitting smoothing and higherstrength, the spiral bevel gears and hypoid gears have widely been used in automobiles,aeronautical engineering, astronautical engineering, engineering mechanics, miningmachinery', machine tool, instrument and so on. With the development of modernscience and technology, the spiral bevel gears and hypoid gears are required with higherstrength and lower noise; at the same time, the new research platform of the spiral bevelgears and hypoid gears is constructed. Thus the research on the strength and noise of thespiral bevel gears and hypoid gears is hot topics and paid attention by a number ofexperts. By using the local synthesis method, tooth contact analysis, loaded toothcontact analysis, finite element method and with the aid of the Natural ScienceFoundation of China and, the Science Foundation of Chinese Aeronautical Engineering,the method of improving the strength of spiral bevel gears and hypoid gears is studied.Some conclusions and creative contents are listed as follows:(1) With the aid of the local synthesis method, the tooth cutting of non-zeromodified spiral bevel gears is researched. By using the finite element method, thebending stress, tooth profile and tooth thickness are analyzed. The computer simulationand computational analysis show that the maximum tensile bending stress and themaximum tooth surface contact stress of the non-zero modified spiral bevel gears withpositive modification coefficients are markedly reduced. On the condition of invariantouter diameter of gear and the geometric parameters of the cutters, tooth root thicknessof the non-zero modified spiral bevel gears with positive modification coefficients isthicker than the spiral bevel gears designed by Gleason method.(2) The modified pitch cone method of the hypoid gear manufactured by HFT isintroduced. The tooth-cutting design of the hypoid gears designed by the modified pitchcone method is studied with the aid of the local synthesis method. The computersimulation and computational analysis show that the root chordal tooth thickness of thepinion designed by the modified pitch cone method becomes thicker, and pinion outerdiameter becomes longer; the tooth surface contact stress, the maximum tensile bendingstress and the maximum compressive bending stress of the pinion designed by newmethod can be markedly reduced. The tooth machining experiment and the comparativeexperiment for strength of hypoid gears designed by the modified pitch cone methodand Gleason method respectively show that the hypoid gears designed by the modifiedpitch cone method are manufactured successfully by using the general machine-tool andcutter, and the changing of bending stress, tooth profile and tooth thickness areconsistent with theoretical design and analysis. As the application of the modified pitchcone method, the hypoid gears of high-speed ratio (tooth number of pinion is 3 or 4) are up already are adopted to carried on short-term prediction of the macro-economy timeseries.