具有时迟的Volterra方程的周期解分歧现象

本文考虑具有时迟的Ｖｏｌｔｅｒｒａ方程其中α,β,δ,γ为正常数．给出方程（Ｅ）出现周期解分歧现象的条件并给出重要参数μ（ε）,Ｔ（ε）,β（ε）的计算方法．     

FitzHugh-Nagumo方程的分歧

运用非线性分歧理论,研究FitzHugh-Nagumo方程的定态分歧和Hopf分歧.证明了FitzHugh-Nagumo方程在适当条件下有定态分歧发生,此时FitzHugh-Nagumo方程的定态方程有非平凡解存在.另外还证明了FitzHugh-Nagumo方程在适当的条件下有Hopf分歧发生,此时该方程从平凡解分歧出非平凡的周期解.最后分析得出影响FitzHugh-Nagumo方程分歧发生的主要因素是离子电压门控通道打开与关闭的延迟反应的快慢.理论分析所得结果与实验现象是相一致的.     

FitzHugh-Nagumo方程的分歧北大核心

运用非线性分歧理论,研究FitzHugh-Nagumo方程的定态分歧和Hopf分歧.证明了FitzHugh-Nagumo方程在适当条件下有定态分歧发生,此时FitzHugh-Nagumo方程的定态方程有非平凡解存在.另外还证明了FitzHugh-Nagumo方程在适当的条件下有Hopf分歧发生,此时该方程从平凡解分歧出非平凡的周期解.最后分析得出影响FitzHugh-Nagumo方程分歧发生的主要因素是离子电压门控通道打开与关闭的延迟反应的快慢.理论分析所得结果与实验现象是相一致的.     

Brusselator行波解的Hopf分歧

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一类多分子反应模型的Hopf分歧解

本文用Ｌｉａｐｕｎｏｖ函数和Ｈｏｐｆ分歧定理讨论一类多分子反应模型ｄｘ／ｄｔ＝δ－ａｘ－ｘ＾ｐｙ＾ｑ，ｄｙ／ｄｔ＝ｘ＾ｐｙ＾ｑ－ｂｙ对ａ＝０，ｐ＝２，ｑ≥２，得到Ｈｏｐｆ分歧解的存在性，唯一性，稳定性及分歧解的渐近表达式。     

一类三维Lotka-Volterra方程和Poincaré标准型及有关参数的讨论

讨论了一类三维Lotka-Volterra方程和Poincaré标准型问题.在给出方程出现周期解分歧现象的条件和确定重要参数中有广泛应用的一个级数的基础上,进一步讨论了与之相关的Poincaré标准型及确定若干个重要参数的表达式问题,并给出了参数的计算方法.     

时滞偏害系统的稳定性和分歧分析

本文讨论了具离散和分布时滞的偏害系统.以时滞作为分歧参数,通过分析原系统在正平衡点处线性化系统的特征方程,获得了正平衡点渐近稳定以及在它周围分歧出周期解的条件.另外,通过使用规范形和中心流形定理,我们获得了Hopf分歧的方向和分歧周期解稳定性的显式算法.最后,数值模拟支持了我们的理论分析.     

Riccati方程周期解的分支

证明带参数λ的Riccati方程x′=x~2+(λ+Q(t))存在周期解的分支点λ_0,当λλ_0时有且仅有两个周期解,当λ=λ_0时有且仅有一个周期解,当λλ_0时所有解无界.     

具有扩散的广义Brusselator系统的Hopf分歧

在齐次Neumann边界条件下,考虑广义Brusselator系统．首先讨论常微分系统Hopf分歧的存在性,得到渐近稳定的周期解．其次讨论具有扩散的偏微分系统,在扩散系数满足一定的条件下,得到超临界的Hopf分歧,并利用规范形理论和中心流形定理给出空间齐次周期解的渐近稳定性．最后,借助Matlab软件进行数值模拟,证明了定理的结论．同时,正平衡态解和空间非齐次周期解的描绘补充了理论分析结果．     

一类具有时滞的病菌-免疫力模型的Hopf分歧

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性及方向.运用计算机数值模拟验证所得理论结果,为传染病的控制和预防提供了理论基础和数值依据.     

ON A PERIODIC VOLTERRA EQUATION WITH SEVERAL FINITE DELAYS AND AN INFINITE DELAY

The existence of positive periodic solutions for a periodic Volterra equation with several finite delays and an infinite delay is established. Sufficient conditions for the periodic solutions having global attractivity are obtained.     

人口动力学中一类含时滞周期反应—扩散方程的正周期解分歧

本文研究人口动力学中一类含时滞周期反应-扩散方程的正周期解问题;利用紧算子的全局分歧结果给出了这个方程正周期解存在的充分必要条件。     

退化时滞微分方程的周期解问题

本文讨论退化时滞微分方程的周期解问题．特别地，我们给出二维退化滞后微分方程的周期解的存在性问题，并在最后举例说明其应用。     

POSITIVE PERIODIC SOLUTION TO A CLASS OF NONLINEAR PERIODIC DIFFERENTIAL EQUATION WITH IMPULSES AND DELAY

In this paper, by using a fixed point theorem of Krasnoselskii, we study the positive periodic solution for a class of nonlinear periodic differential equation with impulses and delay. Firstly, definition of periodic solution and some lemmas are stated. Then some results of the existence of positive periodic solution about the equation are obtained.     

一类时滞积分方程概周期解的存在性

本文讨论一类具广泛应用背景的概周期时滞积分方程,利用极限方程理论和不动点定理等方法,给出了关于概周期解的几个存在性定理.     

一类非线性中立型时滞微分方程周期解的存在性

本文利用抽象连续定理，研究了一类非线性时滞微分方程x′(t)=f(t，x(t)，x(t-τ)，x′(t-τ)) p(t)的周期解问题，并在滞量τ的不同范围内分别得到了周期解存在的充要条件。     

PERIODIC SOLUTIONS OF SCALAR NEUTRAL VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

This paper deals with the existence and uniqueness of periodic solutions of the following scalar neutral Volterra integro-differential equation with infinite delaywhere a, C, D, f are continuous functions, also a(t + T) = a(t), C(t + T,s + T) = C(t, s), D(t + T,s + T) = D(t, s), f(t + T) = f(t). Sufficient conditions on the existence and uniqueness of periodic solution to this equation are obtained by the contraction mapping theorem.     

POSITIVE PERIODIC SOLUTION TO SECOND-ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION WITH MIXED DELAY

We discuss two types of second-order neutral functional differential equations with mixed delay in this paper. By applying Krasnoselskii's fixed-point theorem, some results about the existence of positive periodic solution to the systems are obtained, which complement the previous known results.     

NUMERICAL ANALYSIS FOR VOLTERRA INTEGRAL EQUATION WITH TWO KINDS OF DELAY

In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1,1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L~∞ and L_(ω~c)~2-norm. In the end, we give numerical test to confirm the conclusion.