N阶分歧问题解的数值分析

本文构造了Ｎ阶分歧问题解分支扩充系统的有限维逼近形式最其解的存在性,并给出了解的误差估计。     

简单高阶奇异性的数值模拟

本文对高阶奇异性的分歧数值计算问题进行了研究，构造了计算简单高阶分歧点的扩充系统，并数值模拟了相应的实际问题。     

简单高阶分歧点的正则化

构造了计算简单高阶分歧点的扩充系统．     

Ⅳ阶分歧问题的数值分析

构造了求解一类非退化分歧点及相关参数的扩充系统，给出了拟牛顿迭代法并证明了收敛性。     

一类非线性方程分歧点的摄动定理

本文给出了（Ｔ，Ｋ１，Ｋ２，．．．Ｋｎ）－单本征值的摄动结果，进而证明了一类非线性方程分歧点的摄动定理。     

四阶非线性微分方程周期解的存在性

应用构造Ｌｉａｐｕｎｏｖ函数的方法，给出了四阶非线性微分方程存在周期解的一组充分条件．     

一类非线性方程分歧点的存在性

讨论多参数非线性方程F（λ，x）=0的分歧问题，给出了（λ0，0）是F（λ，x）=0的分歧点的一个充分条件．     

一种计算二重X0—对称破缺分歧点的有效方法

本文将构造一种直接的方法来计算二重Ｘ０－对称破缺分歧点,其构造的扩张系统没有将零向量引入当成变量,因此节约了计算的工作量和计算机内存,数值例子的实际显示了该方法是有效的。     

一类典型分歧问题的初等识别方法

给出了一类典型分歧问题的初等识别方法．     

Numerical analysis of bifurcation problem with corank-n

In this paper, numerical approximation of solution branches of bifurcation problem with corank-n is studied. This project is supported by the Natural Science Foundation of Liaoning Province     

Bifurcation problems in nonlinear parametric programming

The nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters. This system, which is motivated by the Fritz John first-order necessary conditions, contains all Fritz John and all Karush-Kuhn-Tucker points as well as local minima and maxima, saddle points, feasible and nonfeasible critical points. Necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition, the linear dependence of the gradients of the active constraints, and the singularity of the Hessian of the Lagrangian on a tangent space. Any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution. For problems with one parameter, we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition, and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity. The research of this author was supported by National Science Foundation through NSF Grant DMS-85-10201 and by the Air Force Office of Scientific Research through instrument number AFOSR-ISSA-85-00079.     

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps.     

Bifurcation of a three-unit neural network

Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. Discussing the change of the number of eigenvalues with zero real part, we locate the boundary of the stability region and finally determine the largest stability region of trivial solution. We investigate the existence of bifurcation phenomena of codimension one/two of the trivial equilibrium by considering the intersections of some parameter curves, which, in the aτ-half parameter plane, correspond to zero root or pure imaginary roots. In particular, the equivariant bifurcation is studied because of the equivariance of the system. We also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant Pitchfork-Hopf bifurcation points, Hopf-Hopf bifurcation points, and some higher codimension bifurcation points.     

Bifurcation of periodic solutions with large periods for a delay differential equation

We consider a one-parameter family of delay differential equations which has been proposed as a model for a prize and prove that at a critical parameter where the linearization at equilibrium has a double zero eigenvalue periodic solutions bifurcate off with periods descending from infinity. AMS Subject Classification. 34K18,34K13,37G15     

具有暂时免疫传染病模型同宿分支

讨论了具有暂时免疫传染病模型同宿轨道分支的存在性,利用Melnikov函数确定了系统双曲不动点的稳定和不稳定流形的相对位置,从而给出存在极限环的参数范围.     

具有分布时滞的互惠系统Hopf分歧分析

考虑了一类具有扩散项和分布时滞的互惠系统,研究了系统Hopf分歧的出现,用中心流行定理证明了周期解的稳定性.     

Numerical Analysis of Bifurcation Problems of Nonlinear Equations

The paper presents some essential results of branch solutions of nonlinear problems and their numerical approximation. The general theory is applied to the bifurcation problems of the Navier-Stokes equations.