耗散孤立波方程的吸引子

本文研究耗散孤立波方程的长期动力学行为：吸引子的存在性、吸引子的几何结构、耗散系统参数扰动下动力行为、吸引子的分形维估计．     

(4+1)维Fokas方程的怪波解和呼吸子解

本文研究了(4+1)维Fokas方程的多阶怪波解,呼吸子解和高阶呼吸子解.利用Hirota双线性形式和简化的Hirota双线性形式,丰富了(4+1)维Fokas方程的解的多样性.最后分析了精确解的动力学行为.     

两自由度非对称三次系统非奇异时的非线性模态及叠加性

本文利用非线性模态子空间的不变性研究两自由度非对称三次系统在非奇异条件下的非线性模态及其模态叠加解有效性，重点考虑这种有效性与模态动力学方程静态分岔之间的关系·大量的数值结果表明，非线性模态解的有效性不仅与其局部性的限制有关，而且与模态动力学方程静态解分岔有关·     

高阶各向异性Cahn-Hilliard-Navier-Stokes系统的弱解

该文主要研究二维高阶各向异性的Cahn-Hilliard-Navier-Stokes系统的弱解.该系统由高阶各向异性的Cahn-Hilliard方程与不可压缩的Navier-Stokes方程耦合而成.首先引入泛函空间,给出弱解的定义.其次,给出了解的能量估计,由Galerkin方法得到该系统的弱解,最后得到解的唯一性.     

Extended Fisher-Kolmogorov系统的渐近吸引子

考虑了ExtendedFisher-Kolmogorov系统的解的长时间行为,构造了一个有限维解序列即该系统的渐近吸引子,证明了它在长时间后无限趋于方程的整体吸引子,并给出了渐近吸引子的维数估计.     

两个模态耦合的Ginzburg-Landau方程的时空混沌同步化

根据数值计算的结果提出了模态耦合的条件,两个方程在高频模态上是耦合的,而在低频模态上是不耦合的．利用了无穷维动力系统理论,证明了两个高频模态耦合的Ginzburg-Landau方程在函数空间中存在吸引域,因而存在连通的、有限维的紧的整体吸引子．驱动方程存在时空混沌．将方程组联系一个截断形式,得到的修正方程组将保持原方程组的动力学行为．高频模态耦合的两个方程在一定的条件下具有挤压性质,证明了可达到完全的时空混沌同步化．在数学上定性解释了无穷维动力系统的同步化现象．研究方法不同于有限维动力系统中通常使用的Liapunov函数方法与近似线性方法．     

KGS格点系统的全局吸引子

考虑了对应于Klein-Gordon-Schrdinger方程的格点系统(KGS格点系统)的解的长时间行为．首先通过引入一个加权范数与采用解的“切尾”法,证明了全局吸引子的存在性．在此基础上,采用元素分解法与多面体的球覆盖性质, 得到了此吸引子的Kolmogorov δ-熵的上界的一个估计．最后,我们用有限维的常微分方程的全局吸引子逼近它．     

稳态Poisson-Nernst-Planck方程的残量型后验误差估计

针对稳态的Poisson-Nernst-Planck方程研究了一种残量型的后验误差估计子,对方程的两个解-浓度和电势,都分别给出了上界和下界估计.数值实验表明,基于这种后验误差估计子构造的自适应有限元算法对于稳态的Poisson-Nernst-Planck方程是有效的.     

非定常Stokes方程一种基于POD方法的简化有限差分格式

特征正交分解(proper orthogonal decomposition,简记为POD)方法是一种可对偏微分方程的物理模型(如流体流动)做简化的技术．这种方法已经成功地用于对复杂系统模型降阶．推广应用POD方法,将POD方法应用于具有实际应用背景的非定常Stokes方程经典的有限差分格式,建立一种维数较低而精度足够高的简化差分格式,并给出简化差分格式解与经典差分格式解的误差估计．数值例子说明数值计算结果与理论结果相吻合．进一步表明基于POD方法的简化差分格式对求解非定常Stokes方程数值解是可行和有效的．     

非自治Ginzburg-Landau方程的有限维行为

本文研究Ω R~n(n=1,2,3)上具有几乎周期外力的非自治Ginzburg-Landau方程的有限维行为。证明了非自治Ginzburg-Landau系统存在紧的一致吸引子A_1。当外力是时间拟周期时,得到了吸引子A_1的Hausdorff维数的上界估计,当外力是时间周期时,证明了吸引子里一定含有周期解,而且当耗散系数λ满足适当条件时,系统在空间H=L~2(Q)上存在唯一周期解,该周期解指数吸引H中的任何有界集。     

Expansion of solution of an inverted pendulum system with time delay

In this paper, the solution expansion for an inverted pendulum system with time delay is studied. The linearized model of this nonlinear system near its equilibrium is derived on the assumption that a unique equilibrium exists in it. Then the asymptotic expressions of its eigenvalues and the eigenvalues’ corresponding eigenvectors are obtained. Moreover, although the set of these eigenvectors does not form a Schauder basis for the state space, the solution of this model still can be expressed by these eigenvectors in the form of infinite series under certain conditions. Finally, a simulation is provided to support these results.     

具有强结构阻尼的非线性梁方程的整体动力学和控制

在本文中，我们考虑了高维具有强结构阻尼和全指数Ｂａｌａｋｒｉｓｈｎａｎ－Ｔａｙｌｏｒ阻尼的非线性固定边界可伸展的弹性梁方程，得到它的吸收集和平坦惯性流形的存在性．基于无控制方程的惯性流形的存在性，得到了相应的溢出问题的有限维反馈镇定控制．进而，此结果关于结构参数的不确定性是鲁棒的．     

On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators

Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character.     

Global dynamics and control of a nonlinear body equation with strong structural damping

A nonlinear hinged extensible elastic body equation with strong structural damping and Balakrishnan-Taylor damping of full exponent is studied as a general model for large space structures of higher dimensions. In this paper, the absorbing sets and flat inertial manifold are obtained for this nonlinear body equation. The control spillover problem associated with the stabilization of this equation is resolved by constructing a linear finite dimensional feedback, control based on the existence of inertial manifolds of the uncontrolled equation. Moreover, the results obtained are robust with respect to the uncertainty in structural parameters. Supported by the National Natural Science Foundation of China (No. 19701023)     

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

Approximate reduction of non-linear discrete models with two time scales

The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system.     

Error bounds for computed eigenvalues and eigenvectors

Summary On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.Dedicated to Professor Yoshikazu Nakai on his sixtieth birthday     

Preconditioned gradient flows for nonlinear eigenvalue problems and application to the Hartree‐Fock functional

In this article, we propose to solve numerically the problem of finding the smallest eigenvalues of a Hermitian operator (and the space spanned by the corresponding eigenvectors) by a gradient flow technique. This method is then applied to the Hartree‐Fock problem. Improvements are also proposed in two directions: preconditioning of the dynamical system and development of a specific flow that enables to compute directly the eigenvectors. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

线张力作用下微纳米尺度液滴的非线性粘附

三相接触线上的线张力对微纳米尺度液滴的粘附行为有至关重要的影响．首次在由表面张力、线张力和液滴尺度组成的全参数空间中研究了液滴粘附的非线性行为．研究表明:液滴粘附解空间可以被归纳为4个特征区;在每个特征区内,液滴的粘附行为是相同的;在具有高度非线性的特征区中,给定材料体系,存在着多重粘附态;液滴粘附解空间中存在着两个公共不动点．这些新结果,与数值计算和实验观测相符合．     

Stabilization of a nonlinear flow-plate interaction via component-wise decomposition

Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [8] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [6, 7]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology. We show a decomposition of the dynamics into “smooth” component and global-in-timeHadamard continuous component, thus permitting approximation by smooth data. That the flows are subsonic is critical for our approach. Our result implies that flutter (a periodic or chaotic end behavior) is not present in subsonic flows with sufficient viscous damping in the structure.