具有饱和接触率和垂直传染的SIRS传染病模型分岔分析

研究了一类具有饱和发生率、脉冲生育、脉冲接种和垂直传染的SIRS传染病模型的复杂动力学行为,首先构造了一个庞卡莱映射,然后利用映射的不动点及其特征值,得到了系统无病周期解的存在和稳定的条件,接着详细讨论了系统的跨临界分岔、超临界分岔和倍周期分岔现象,最后给出了能很好验证理论分析的数值结果.     

一个微生物生态模型分岔的周期轨

本文研究一类与厌氧消化过程微生物生态模型有关的微分方程组在非双曲情形下的扰动。在其一维不变流形上讨论自治扰动下中心附近发生的Pionceare分岔,所涉及的闭轨存在性问题是许多熟知结果^[1-5]不能包含的情形。进而,利用Melnikov函数方法,给出周期扰动下发生次调和分岔的参数条件。     

蛙卵有丝分裂模型的分岔分析

定性分析了Borisuk和Tyson建立的蛙卵有丝分裂模型,讨论了其定态的存在性和稳定性,深入研究了该模型的分岔行为并通过数值实验加以证实。此外,还给出了Tyson数值结果的理论依据。     

指数型一次Logistic迭代方程解的周期倍分岔现象

本文讨论了指数型一次Ｌｏｇｉｓｔｉｃ迭代方程解的周期倍分岔现象的存在性,给出了周期倍分合时的临界点参数应满足的方程,并证明了周期倍分岔的临界点参数序列的极限是存在的,进而证明了当参数越过这个极限时,指数型一次 Ｌｏｇｉｓｔｉｃ迭代方程存在混沌解．     

一个酶催化反应系统的局部分岔分析

讨论一个酶催化反应系统的局部分岔.首先得到该系统只有1个或2个孤立平衡点,或者一条奇线,并给出了所有平衡点的定性性质.进一步分析了孤立平衡点在非双曲情形下发生的分岔,包括跨临界分岔和Hopf分岔,通过计算Lyapunov量得出该系统中细焦点阶数为1.最后利用数值模拟验证了所得结论.     

低维宏观交通流模型的稳定性和分岔分析

交通流模型的分岔点对应临界的交通状态,对研究交通流的稳定性具有重要的理论意义.为了分析宏观交通流模型的分岔特征,通过对低维宏观交通流模型的求解得到两个平衡点,并讨论了其稳定性,发现该模型存在一个跨临界分岔点.数值仿真验证了结论的正确性,并且在一定条件下,通过改变响应时间会影响到最终的平衡状态.     

互联网双时滞TCP-RED模型Hopf分岔和Fold分岔的数值分析

利用时滞动力学分析软件DDE-BIFTOOL研究了时滞互联网TCP-RED拥塞控制模型的动力学.传输时滞τ_1可以使得该系统发生Hopf分岔、Fold分岔,使得平均队列长度和到达速率出现近似恒速运动或周期波动,揭示了时滞对其动力学的重要影响.     

一类SEIS模型的分岔及混沌

建立了一类更为符合实际疫情的种群动态变化下新的SEIS模型,得到了系统的平衡点渐近稳定条件、Hopf分岔以及稳定的极限环,给出了多参数变化对系统混沌的影响和易感种群增减对系统混沌区域伸缩的制约,并附有数值模拟和仿真.     

Lotka-Volterra模型全局指数稳定性和相应方程解的分岔

研究具有变系数的时滞Lottka-Volterra模型,证明该模型在适当的条件下存在正的平衡解,并给出了正平衡解指数稳定的充分条件,进一步讨论了模型(1)的相对退化形式的解产生Hopf分岔现象.     

两种群竞争和合作模型的全局渐近稳定性

其中R=r/K,r为种群的内禀自然增长率,K为环境容纳量,N为种群数量(或密度)。 考虑在周期环境中两种群竞争(合作),此时每一种群的生长由于另一种群的存在而减少(增加),可设这时i种群的环境容纳量为     

Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1 -periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.     

BIFURCATIONS OF ROUGH 3—POINT—LOOP WITH HIGHER DIMENSIONS

The authors study the bifurcation problems of rough heteroclinic loop connecting three saddle points for the case β1 > 1, β2 > 1, β3 < 1 and β1β2β3 < 1. The existence, number, coexistence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied. Meanwhile, the bifurcation surfaces and existence regions are given.     

CODIMENSION 3 BIFURCATIONS OFHOMOCLINIC ORBITS WITH ORBITFLIPS AND INCLINATION FLIPS

The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is non-principal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the twofold periodic orbit and three-fold periodic orbit are also obtained.     

伴随奇点分支的退化异宿轨道的保存和横截性

本文应用指数三分性和不变流形的局部几何表示方法,给出异宿流形上的轨道当两个奇点经历超;临界分支和摄动时保存和横截的条件．     

一类食饵-捕食生物模型的动态分歧

本文运用线性全连续场的谱理论及跃迁理论讨论了一类食饵-捕食生物模型的动态分歧,在一定条件下得到了跃迁类型的判据,并判断了跃迁的类型,同时也给出了分歧解的表达式,最后对获得的结果做了必要的解释.     

高次退化的非线性向量场分支

本文讨论了一个在分支值线性部分具两个零特征根且只有一个Ｊｏｒｄａｎ块，而非线性项为５次的平面向量场，得到了完整的轨线分支图。本文引入的讨论判断函数性质的方法具有一般性，因而也将适用于具更高次退化的非线性情况的研究。     

三维系统余维二分支中周期轨道与不变环面的存在性

本文研究具余维二奇点的三维系统在双参数扰动下周期轨道与不变环面的分支，给出了存在不变环面的最佳条件     

HOPF BIFURCATION OF NONLINEAR ADVERTISING CAPITAL MODEL WITH DISCRETE AND CONTINUOUS TIME DELAYED FEEDBACK

In this paper,stability and Hopf bifurcation of a nonlinear advertising ca- pital model with time delayed are studied.By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties.When delay passes a critical value,Hopf bifurcation may appear.Furthermore,the di- rection and stability of bifurcating periodic solutions are investigated by normal form and center manifold theory.Additionally,we also have some discussion about the model with continuous time delay.     

Bifurcations of rough heteroclinic loop with two saddle points

The bifurcation problems of rough 2-point-loop are studied for the case p11 > λ11, p21 < λ21, P11p21 <λ111λ21. where - pi1 < 0 and λi1 > 0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.     

Dynamical behaviors of a discrete two-dimensional competitive system exactly driven by the large centre

In this paper, a new discrete large-sub-center system is obtained by using the Euler and nonstandard discretization methods for the corresponding continuous system. It is surprised that all dynamic behaviors of the discrete system are exactly driven by the large-center equation, for example, the stabilities, the bifurcations, the period-doubling orbits, and the chaotic dynamics, etc. Additionally, the global asymptotical stability, the existence of exact 2-periodic solutions, the flip bifurcation theorem, and the invariant set of the sub-center equation is also given. These results reveal far richer dynamics of the discrete model compared with the continuous model. Through numerical simulation, we can observe some complex dynamic behaviors, such as period-doubling cascade, periodic windows, chaotic dynamics, etc. Especially, our theoretical results are also showed by those numerical simulations.