具有三重零奇异的时滞微分方程的分支

主要研究三重零奇异的判定和在R~n上零特征根对应的广义特征空间,利用中心流形简化和规范型计算得到参数时滞微分方程的简化形式,对应于文[A note on the triple zero linear degeneracy:Normal forms,dynamical and bifurcation behaviour of an unfolding.Int J Bifur and Chaos,2002,12:2799-2820]中的结果具体分析具有三重零奇异的参数时滞微分方程的分支行为,并给出一例子来阐述得到的结果.     

一类TCP系统的局部Hopf分支和周期解的全局存在性

研究了一类一阶非线性时滞微分方程描述的TCP系统的动力学行为.通过分析其相应的特征超越方程,得到了当时滞通过一系列临界值时,在正平衡点处Hopf分支产生.利用中心流形和规范型理论,得到了确定Hopf分支方向和稳定性的具体计算表达式.运用Wu[Trans Amer MathSoc,1998,350(12):4799-4838]的方法,得到了全局Hopf分支存在的条件.     

一类三阶时滞微分方程解的渐进稳定性

本文研究了三阶非线性时滞微分方程解的渐近稳定性. 利用Lyapunov 泛函, 得到了微分方程的零解是渐进稳定的, 这一结果推广了文献[2] 的结果.     

一类三阶时滞微分方程解的渐进稳定性

本文研究了三阶非线性时滞微分方程解的渐近稳定性.利用Lyapunov泛函,得到了微分方程的零解是渐进稳定的,这一结果推广了文献[2]的结果.     

奇异摄动中的鸭解问题

鸭解问题是近年来在奇异摄动方程的研究中发现并开始研究的 ,是一种新的分支现象 .用渐近分析方法、微分方程定性理论及不动点方法对一类二维单参数奇异摄动系统进行了研究 ,给出了鸭解和鸭极限环存在的条件及对应的参数估计 .     

具有限时滞一阶线性泛函微分方程的稳定性区域划分

讨论了一阶线性有限时滞泛函微分方程的稳定性区域,用一个超平面把参数空间划分为不同的稳定性区域.这个超平面上的每一点对应于特征方程在纯虚轴上至少存在一个零根(原点除外),所得结论可用于Hofp分枝分析和控制理论.     

一类具有时滞的捕食模型的稳定性和Hopf分支

研究一类具有时滞和Beddington-DeAngelis功能性反应的捕食模型的稳定性和Hopf分支.以滞量为参数,得到了系统正平衡点的稳定性和Hopf分支存在的充分条件.应用一般泛函微分方程的度理论,研究了该系统的全局Hopf分支的存在性.     

多时滞合作系统的稳定性与全局Hopf分支

以时滞为参数,研究了一类多时滞合作系统的正平衡点的稳定性及局部Hopf分支的存在性.在此基础上结合一般泛函微分方程的全局Hopf分支定理,讨论了该系统全局Hopf分支的存在性.     

具有时滞的比率型三种群捕食模型的分支分析

本文研究了一类具有时滞的比率型三种群捕食模型.通过分析该模型的特征方程,证明了该模型在正平衡点的稳定性.通过选择时滞τ为分支参数,得到了当时滞τ通过一系列的临界值时,Hopf分支产生.应用中心流形和规范型理论,得到了关于确定Hopf分支特性的计算公式.最后进行数值模拟验证了我们所得结果的正确性.所得结果是对前人工作的补充.     

一类三阶中立型时滞微分方程的渐近稳定性

讨论了一类三阶中立型时滞微分方程的零解的渐近稳定性,借助于构造函数、推广的Halanay一维时滞微分不等式及泰塔格利亚公式,得到了判定其零解是渐近稳定的且与时滞无关的一个充分条件.     

Second-order normal forms for n-dimensional systems with a nilpotent point

Normal forms theory is one of the most powerful tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Many works paid attention to normal forms associated with nilpotent Jacobian where the critical eigenvalues have algebraic multiplicity k ($k>1$) and geometric multiplicity one, and in particular, the case $k>2$ is more complicated for determining unfolding. Despite a lot of theoretical results on nilpotent normal forms have been obtained, computation developing can not satisfy practical applications. To our knowledge, no results have been reported on the computation of explicit formulas of the nilpotent normal forms for $k>3$ with perturbation parameters. The main difficulty is how to determine the complementary spaces of the Lie transformation. In this paper, we achieve the following results. (1) A simple dimension formula for the complementary space of the Lie transform; (2) a simple direct method to determine a basis of the complementary spaces; (3) a simple direct method to determine the projection of any vector to the complementary spaces. Using this method, the second-order normal forms for any n-dimensional nilpotent systems can be given easily. As an illustrative application, the normal forms for the vector field with triple-zero or four-fold zero singularity and functional differential equation with a triple-zero singularity are presented, and explicit formulas for the normal form coefficients with three or four unfolding parameters are obtained.     

Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity

The paper carries the results on Takens-Bogdanov bifurcation obtained in [T. Faria, L.T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations 122 (1995) 201-224] for scalar delay differential equations over to the case of delay differential systems with parameters. Firstly, we give feasible algorithms for the determination of Takens-Bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in Rn. Next, through center manifold reduction and normal form calculation, a concrete reduced form for the parameterized delay differential systems is obtained. Finally, we describe the bifurcation behavior of the parameterized delay differential systems with T-B singularity in detail and present an example to illustrate the results.     

Ecological complexity and feedback control in a prey–predator system with Holling type III functional response

This article deals with a bioeconomic model of prey–predator system with Holling type III functional response. The dynamical behavior of the system is extensively discussed. Continuous type gestational delay of predators is incorporated in the system to study delay induced instability. It is observed that the system undergoes singularity induced bifurcation at interior equilibrium point when net economic revenue of the system increases through zero. State feedback controller is designed to stabilize the system at positive economic profit. Time delay is considered as a bifurcation parameter to prove the occurrence of Hopf bifurcation phenomenon in the neighborhood of the coexisting equilibrium point. Finally, some numerical simulations are carried out to verify the analytical results and the system is analyzed through graphical illustrations. © 2015 Wiley Periodicals, Inc. Complexity 21: 346–360, 2016     

Multiple stabilities analysis in a magnetic bearing system with time delays

A kind of magnetic bearing system with time delay is considered. Firstly, multiple stabilities of the model is investigated. According to the analysis results, the bifurcation diagram is drawn in the appropriate parameter plane. Then the center manifold reduction and normal form computation for simple zero singularity are performed and detailed bifurcation analysis are carried out. Finally, some numerical simulations are presented to illustrate the results found.     

Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback

In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zero bifurcation is investigated by using center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numerical simulations are given to support the analytic results.     

Local and global Hopf bifurcation in a neutral population model with age structure

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.     

Codimension two bifurcation in a delayed neural network with unidirectional coupling

In this paper, a delayed neural network model with unidirectional coupling is considered. Zero–Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm form at the zero–Hopf singularity and show that the model can exhibit pitchfork, Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Some numerical simulations are given to support the analytic results.     

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.     

一类具有时滞Holling-Ⅲ型捕食-食饵系统的Hopf分支

研究了具有时滞的Holling-Ⅲ型捕食-食饵系统,其中捕食者的数量反应具有leslies形式.采用常微分定性与稳定性方法,推出了当τ=0时,正平衡点全局稳定性的充分条件,并考虑了时滞对于模型稳定性的影响,选取时滞τ作为分支参数,得出了在正平衡点附近产生Hopf分支.