可压缩流动的Fourier谱-有限元解法

本文考虑n维(n=2,3)可压缩流动的带有单向周期边值条件问题的数值解.我们在周期方向采用Fourier谱方法,在非周期方向采用有限元方法,从而构造了一类谱-有限元格式.文中严格分析了计算误差,得到了收敛阶的估计.     

蛙卵有丝分裂模型的定性分析

本文对M.T.Borisuk和J.J.Tyson在[1]中所提出的一个有关蛙卵有丝分裂的平面三次系统模型证明了大范围周期解的存在性，给出了周期解所在的空间区域和所对应的参数区域及周期解不存在的空间区域和参数区域，所得结果严格地证明了[1]中给出的数值结果，最后我们证明在[1]的数值结果所用的参数下极限环的唯一性。     

Fitzhugh神经传导方程的张弛振动解

本文用匹配渐近法,计算了Fitzhugh神经传导方程张弛振动解的解析表达式、振动周期,给出了产生张弛振动的参数区域.     

Newton方法在非线性振动理论中的推广与应用

本文提出和证明了,用Newton方法可以求解强(弱)非线性非自治系统的渐近解析周期解,为研究强(弱)非线性系统振动提供了一个新的解析方法.根据本文方法的需要,讨论了二阶线性非齐次周期系统周期解的存在与计算问题.此外,还讨论了Newton方法对于拟线性系统的应用.最后,应用本文方法计算了Duffing方程的周期解.     

周期三对角矩阵逆的一种新算法

给出了一类周期三对角矩阵逆的新的递归算法.新方法充分利用周期三对角矩阵的结构特点,采用递归方法将高阶周期三对角矩阵求逆转化为低阶周期三对角矩阵的求逆.并同时得到简化的计算方法,方法可以有效地减少运算量和存储量,计算精度也有明显的优势.数值实验表明此算法是有效的.     

研究Hénon映射稳定周期轨道的一种方法——代数分析法

本文利用一种简单的代数分析法,研究了Hénon映射的稳定周期轨道存在的参数区域。从理论上精确地确定了,稳定周期轨道产生时的窗口位置和它们发生倍周期分岔的失稳位置。所得的结果不仅包括了Hamouly和Mira的有关数值计算结果,而且包括了最近Hitzl和Zele用离散Fourier变换方法所得的理论计算结果。     

带不完美界面的双曲问题均匀化的一个注记

本文研究了一类二分区域上的具有非周期系数的双曲问题.利用周期Unfolding方法,得到了均匀化及其矫正结果,推广了Donato,Faella和Monsurrò的工作.     

应用F展开法求KdV方程的周期波解

提出了求非线性数学物理演化方程周期波解的F展开法,该方法可看作最近提出的扩展的Jacobi椭圆函数展开方法的浓缩.直接利用F展开法而不计算Jacobi椭圆函数,我们可同时得到著名的KdV方程的多个用Jacobi椭圆函数表示的周期波解.当模数m→1 时,可得到双曲函数解(包括孤立波解).     

扩散方程一种无条件稳定的保正并行有限差分方法

本文针对扩散方程提出了一种保正的并行差分格式,并且这个格式为无条件稳定的.我们在每个时间层将计算区域分成许多个子区域以便于实施并行计算.格式构造中首先我们使用前两个时间层的计算结果在分区界面处通过一种非线性的保正外插来预估子区域界面值.然后在每个子区域内部使用经典的全隐格式进行计算.最后在界面处使用全隐格式进行校正（本质上这一步计算是显式计算）.我们给出了一维与二维情形下的保正并行差分格式,并相应的给出了无条件稳定性证明.数值实验显示此并行格式具有二阶数值精度,而且无条件稳定性与保正性也均在数值实验中得到验证.     

Gauss白噪声激励下的永磁同步电动机模型的分岔分析北大核心CSCD

针对永磁同步电动机(PMSM)模型引入Gauss白噪声,根据极坐标变换和随机平均法得到系统It8随机微分方程,并计算出系统概率密度函数,通过数值模拟揭示了系统P-分岔的机理.此外,探讨了系统在双参数空间中的复杂动力学,仿真结果表明在参数空间中出现了大量的“鱼”形周期区域,并且这些“鱼”形周期区域不可避免地受到噪声的影响变得紊乱.值得注意的是,从数值模拟结果中发现了一个新的现象,一定的噪声强度下,可以诱导系统在周期振荡区域内的收敛行为,这也表明了噪声对系统影响的双面性.     

Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems

We present a new parallel algorithm for time-periodic problems by combining the waveform relaxation method and the parareal algorithm, which performs the parallelism both in sub-systems and in time. In the new algorithm, the waveform relaxation propagator is chosen as a new fine propagator instead of the classical fine propagator. And because of the characteristic of time-periodic problems, the new parareal waveform relaxation algorithm needs to solve a periodic coarse problem at the coarse level in each iteration. The new algorithm is proved to converge linearly at most. Then the theoretic parallel efficiency of the new algorithm is also considered. Numerical experiments confirm our analysis finally.     

Non-existence of small-amplitude doubly periodic waves for dispersive equations

We formulate the question of the existence of spatially periodic, time-periodic solutions for evolution equations as a fixed point problem, for certain temporal periods. We prove that if a certain estimate applies for the Duhamel integral, then time-periodic solutions cannot be arbitrarily small. This provides a partial analogue in the spatially periodic case of scattering results for dispersive equations on the real line, as scattering implies the non-existence of small-amplitude traveling waves. Furthermore, it also complements small-divisor methods (e.g., the Craig–Wayne–Bourgain method) for proving the existence of small-amplitude time-periodic solutions (again, for frequencies in certain set).     

Multiplicity and stability of time-periodic solutions of Ginzburg-Landau equations of superconductivity

In this article we shall show that the Ginzburg-Landau equations admit at least three time-periodic solutions. One of the time-periodic solutions describes the non-superconductive (or normal) state and the other one describes the superconductivity state. We will also show that the time-periodic solutions are exponentially stable. Furthermore, the method we use in this article can be used to find numerical approximations to the time-periodic solutions.     

CONCERNING TIME-PERIODIC SOLUTIONS OF THE NAVIER-STOKES EQUATIONS IN CYLINDRICAL DOMAINS UNDER NAVIER BOUNDARY CONDITIONS

The problem of the existence of time-periodic flows in infinite cylindrical pipes in correspondence to any given, time-periodic, total flux, was solved only quite recently in [1]. In this last reference we solved the above problem for flows under the non-slip boundary condition as a corollary of a more general result. Here we want to show that the abstract theorem proved in [1] applies as well to the solutions of the well known slip (or Navier) boundary condition (1.7) or to the mixed boundary condition (1.14). Actually, the argument applies for solutions of many other boundary value problems. This paper is a continuation of reference [1], to which the reader is referred for some notation and results.     

Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.     

Existence of periodic solutions and their asymptotic stability to the Navier–Stokes equations with the Coriolis force

We consider the time-periodic problem for the Navier–Stokes equations in the rotational framework. We prove the unique existence of time-periodic solutions for the prescribed external force. Furthermore, we also show the asymptotic stability of small time-periodic solutions provided the initial disturbance is sufficiently small.     

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka-Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Asymptotics of Small Exterior Navier-Stokes Flows with Non-Decaying Boundary Data

We prove the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time. As a corollary, the existence of unique small time-periodic solutions is shown. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically self-similar, then the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity.     

Vortical and self-similar flows of 2D compressible Euler equations

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino’s solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.     

Time-periodic Solutions to Systems of Conservation Laws with Ellipticity

We study the time-periodic solutions to systems of conservation laws with ellipticity. It is shown that under time-periodic boundary condition, the system admits at least one global time-periodic solution bounded uniformly with the same period.