On the criterion for the absolute stability of the control system

In this paper, firstly we give the criterion for the absolute stability of the second canonical form for the control system, including the equation of the longitudinal motions of a plane as a particular example. The corresponding result in [8], [9] is a particular example given in this paper. Secondly, we give the criteria for the absolute stability of the first canonical form in the usual case and in the critical case. Finally, we give some criteria for the absolute stability of the general form for the direct control system.All the results in this paper merely depend upon the relations between the parameters of the system itself to give an explicit algebraic discriminant.     

The state space reconstruction technology of different kinds of chaotic data obtained from dynamical system

Certain deterministic nonlinear systems may show chaotic behavior. We consider the motion of qualitative information and the practicalities of extracting a part from chaotic experimental data. Our approach based on a theorem of Takens draws on the ideas from the generalized theory of information known as singular system analysis. We illustrate this technique by numerical data from the chaotic region of the chaotic experimental data. The method of the singular-value decomposition is used to calculate the eigenvalues of embedding space matrix. The corresponding concrete algorithm to calculate eigenvectors and to obtain the basis of embedding vector space is proposed in this paper. The projection on the orthogonal basis generated by eigenvectors of timeseries data and concrete paradigm are also provided here. Meanwhile the state space reconstruction technology of different kinds of chaotic data obtained from dynamical system has also been discussed in detail. The project supported by the National Natural Science Foundation of China (19672043)     

STUDY ON PREDICTION METHODS FOR DYNAMIC SYSTEMS OF NONLINEAR CHAOTIC TIME SERIES＊

The prediction methods for nonlinear dynamic systems which are decided by chaotic time series are mainly studied as well as structures of nonlinear self-related chaotic models and their dimensions. By combining neural networks and wavelet theories, the structures of wavelet transform neural networks were studied and also a wavelet neural networks learning method was given. Based on wavelet networks, a new method for parameter identification was suggested, which can be used selectively to extract different scales of frequency and time in time series in order to realize prediction of tendencies or details of original time series. Through pre-treatment and comparison of results before and after the treatment, several useful conclusions are reached:High accurate identification can be guaranteed by applying wavelet networks to identify parameters of self-related chaotic models and more valid prediction of the chaotic time series including noise can be achieved accordingly.     

PREDICTION TECHNIQUES OF CHAOTIC TIME SERIES AND ITS APPLICATIONS AT LOW NOISE LEVEL

The paper not only studies the noise reduction methods of chaotic time series with noise and its reconstruction techniques,but also discusses prediction techniques of chaotic time series and its applications based on chaotic data noise reduction.In the paper,we first decompose the phase space of chaotic time series to range space and null noise space.Secondly we restructure original chaotic time series in range space.Lastly on the basis of the above,we establish order of the nonlinear model and make use of the nonlinear model to predict some research.The result indicates that the nonlinear model has very strong ability of approximation function,and Chaos predict method has certain tutorial significance to the practical problems.     

非等距时间序列的GM（1，N）模型及其在地下工程中的应用

文献〔１〕给出了等距时间序列的灰色预测ＧＭ（１，Ｎ）模型，本文把它推广到非等距时间序列，得到了相应模型，并应用于地下工程的几个实例     

On the predictability of chaotic systems with respect to maximally effective computation time

The round-off error introduces uncertainty in the numerical solution. A computational uncertainty principle is explained and validated by using chaotic systems, such as the climatic model, the Rossler and super chaos system. Maximally effective computation time (MECT) and optimal stepsize (OS) are discussed and obtained via an optimal searching method. Under OS in solving nonlinear ordinary differential equations, the self-memorization equations of chaotic systems are set up, thus a new approach to numerical weather forecast is described. The project supported by the National Natural Science Foundation of China (40275031 and 40231006), the National Key Program for Developing Basic Sciences (G1999043408) and the Key Innovation Project of Chinese Academy of Sciences (K2CX1-10-07)     

An eigenvector‐based linear reconstruction approach for time stepping in discontinuous Galerkin scheme used to solve shallow water equations

Discontinuous Galerkin (DG) methods have shown promising results for solving the two‐dimensional shallow water equations. In this paper, the classical Runge–Kutta (RK) time discretisation is replaced by the eigenvector‐based reconstruction (EVR) that allows the second‐order time accuracy to be achieved within a single time‐stepping procedure. Moreover, the EVRDG approach yields stable solutions near drying and wetting fronts, whereas the classical RKDG approach yields instabilities. The proposed EVRDG technique is compared with the original RKDG approach on various test cases with analytical solutions. The EVRDG solutions are shown to be as accurate as those obtained with the RKDG scheme. Besides, the EVRDG scheme is 1.6 times faster than the RKDG method. Simulating dambreaks involving dry beds confirms that EVRDG scheme gives correct solutions, whereas the RKDG method yields instabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation

An approach based on a Discontinuous Galerkin discretisation is proposed for the Bhatnagar–Gross–Krook model kinetic equation. This approach allows for a high-order polynomial approximation of molecular velocity distribution function both in spatial and velocity variables. It is applied to model one-dimensional normal shock wave and heat transfer problems. Convergence of solutions with respect to the number of spatial cells and velocity bins is studied, with the degree of polynomial approximation ranging from zero to four in the physical space variable and from zero to eight in the velocity variable. This approach is found to conserve mass, momentum and energy when high-degree polynomial approximations are used in the velocity space. For the shock wave problem, the solution is shown to exhibit accelerated convergence with respect to the velocity variable. Convergence with respect to the spatial variable is in agreement with the order of the polynomial approximation used. For the heat transfer problem, it was observed that convergence of solutions obtained by high-degree polynomial approximations is only second order with respect to the resolution in the spatial variable. This is attributed to the temperature jump at the wall in the solutions. The shock wave and heat transfer solutions are in excellent agreement with the solutions obtained by a conservative finite volume scheme.     

A novel multidimensional solution reconstruction and edge‐based limiting procedure for unstructured cell‐centered finite volumes with application to shallow water dynamics

On unstructured meshes, the cell‐centered finite volume (CCFV) formulation, where the finite control volumes are the mesh elements themselves, is probably the most used formulation for numerically solving the two‐dimensional nonlinear shallow water equations and hyperbolic conservation laws in general. Within this CCFV framework, second‐order spatial accuracy is achieved with a Monotone Upstream‐centered Schemes for Conservation Laws‐type (MUSCL) linear reconstruction technique, where a novel edge‐based multidimensional limiting procedure is derived for the control of the total variation of the reconstructed field. To this end, a relatively simple, but very effective modification to a reconstruction procedure for CCFV schemes, is introduced, which takes into account geometrical characteristics of computational triangular meshes. The proposed strategy is shown not to suffer from loss of accuracy on grids with poor connectivity. We apply this reconstruction in the development of a second‐order well‐balanced Godunov‐type scheme for the simulation of unsteady two‐dimensional flows over arbitrary topography with wetting and drying on triangular meshes. Although the proposed limited reconstruction is independent from the Riemann solver used, the well‐known approximate Riemann solver of Roe is utilized to compute the numerical fluxes, whereas the Green–Gauss divergence formulation for gradient computations is implemented. Two different stencils for the Green–Gauss gradient computations are implemented and critically tested, in conjunction with the proposed limiting strategy, on various grid types, for smooth and nonsmooth flow conditions. Copyright © 2012 John Wiley & Sons, Ltd.