The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential pr...The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.展开更多
The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a ...The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.展开更多
A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of part...A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.展开更多
We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev...We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.展开更多
双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Ku...双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Kutta进行求解。将该法得到的数值结果与精确解进行比较,验证了方法的有效性,数据结果的误差与其他方法相比有较高的精确度。Hyperbolic partial differential equation is one of the important partial differential equations. The Chebyshev spectral method is proposed to solve the telegraph equation. Chebyshev-gauss-lobatto is used to assign points, the derivative matrix is constructed by Chebyshev polynomial, and the telegraph equation is approximated as an ordinary differential equation. The error estimation of the discrete Chebyshev spectral method for the telegraph equation was proved. Runge-Kutta was used to solve the problem. The numerical results obtained by the method are compared with the exact solution, and the effectiveness of the method is verified. The error of the data results is more accurate than that of other methods.展开更多
在求解奇异摄动两点边值问题时,本文构造了基于Chebyshev点的B样条配置法。该方法采用三次B样条函数作为基函数,利用Chebyshev点作为配置点直接对方程进行求解。文中探讨了该方法在实施时的具体步骤及需要注意的若干细节。通过奇异摄动...在求解奇异摄动两点边值问题时,本文构造了基于Chebyshev点的B样条配置法。该方法采用三次B样条函数作为基函数,利用Chebyshev点作为配置点直接对方程进行求解。文中探讨了该方法在实施时的具体步骤及需要注意的若干细节。通过奇异摄动扩散反应问题、奇异摄动对流扩散反应问题这两个算例的研究,表明基于Chebyshev点的B样条配置法与等距节点下的B样条配置法相比,前者具有高精度和高效率的优势。In solving the singular perturbation two-point boundary value problems, this paper constructs a Chebyshev B-spline collocation method. This method uses cubic B-spline functions as basis functions and utilizes the Chebyshev point as the configuration point to solve the equation directly. The specific steps in the implementation of the method and several details that need to be noted are discussed in the paper. Through the study of two arithmetic cases, namely, the singular regent diffusion response problem and the singular regent convection diffusion response problem, it is shown that the Chebyshev B-spline collocation method has the advantages of high accuracy and high efficiency as compared with the B-spline configuration method under equidistant nodes.展开更多
基金Project supported by the National Natural Science Foundation of China （Grants Nos 10472091 and 10332030）.
文摘The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.
基金supported by the National Basic Research Program of China (Grant 2013CB733004)
文摘The dynamic characteristics of a beam-cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam-cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.
文摘A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.
文摘We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.
文摘双曲偏微分方程是重要的偏微分方程之一。提出求解电报方程的Chebyshev谱法,采用Chebyshev-Gauss-Lobatto配点,利用Chebyshev多项式构造导数矩阵,将电报方程近似为常微分方程,证明了电报方程的离散Chebyshev谱法的误差估计,采用Runge-Kutta进行求解。将该法得到的数值结果与精确解进行比较,验证了方法的有效性,数据结果的误差与其他方法相比有较高的精确度。Hyperbolic partial differential equation is one of the important partial differential equations. The Chebyshev spectral method is proposed to solve the telegraph equation. Chebyshev-gauss-lobatto is used to assign points, the derivative matrix is constructed by Chebyshev polynomial, and the telegraph equation is approximated as an ordinary differential equation. The error estimation of the discrete Chebyshev spectral method for the telegraph equation was proved. Runge-Kutta was used to solve the problem. The numerical results obtained by the method are compared with the exact solution, and the effectiveness of the method is verified. The error of the data results is more accurate than that of other methods.
文摘在求解奇异摄动两点边值问题时,本文构造了基于Chebyshev点的B样条配置法。该方法采用三次B样条函数作为基函数,利用Chebyshev点作为配置点直接对方程进行求解。文中探讨了该方法在实施时的具体步骤及需要注意的若干细节。通过奇异摄动扩散反应问题、奇异摄动对流扩散反应问题这两个算例的研究,表明基于Chebyshev点的B样条配置法与等距节点下的B样条配置法相比,前者具有高精度和高效率的优势。In solving the singular perturbation two-point boundary value problems, this paper constructs a Chebyshev B-spline collocation method. This method uses cubic B-spline functions as basis functions and utilizes the Chebyshev point as the configuration point to solve the equation directly. The specific steps in the implementation of the method and several details that need to be noted are discussed in the paper. Through the study of two arithmetic cases, namely, the singular regent diffusion response problem and the singular regent convection diffusion response problem, it is shown that the Chebyshev B-spline collocation method has the advantages of high accuracy and high efficiency as compared with the B-spline configuration method under equidistant nodes.
