周期分岔解的鲁棒控制

根据C-L方法,可以得到非线性动力系统的分岔方程和拓扑分岔图．根据得到的分岔图,结合控制理论,提出了周期解的鲁棒控制方法．该方法将运动模式控制到目标模式．由于该方法对控制器的参数没有严格的控制,所以在设计和制造控制器方面是很方便的．数值研究验证了该方法的有效性．     

Stability analysis on steady-state bifurcation for arbitrary order autocatalytic reaction model

This paper deals with an autocatalytic reaction–diffusion model with arbitrary order functional response subject to no-flux boundary conditions. We mainly discuss the stability of the steady-state bifurcation which emanates from the unique positive constant equilibrium. On the stability of the bifurcation solution, the conventional way is to consider the sign of the first derivative of a certain function. However, sometimes, the first derivative may be equal to zero. This leads to the uncertainty of the stability. In such case, it needs to break through the common idea. We present an approach which determines the stability of the bifurcation solution.     

Oscillations in three-reaction quadratic mass-action systems

It is known that rank-two bimolecular mass-action systems do not admit limit cycles. With a view to understanding which small mass-action systems admit oscillation, in this paper we study rank-two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greater generality. One key finding is that an isolated periodic orbit cannot occur in a three-reaction, trimolecular, mass-action system with bimolecular sources. In fact, we characterize all networks in this class that admit a periodic orbit; in every case, all nearby orbits are periodic too. Apart from the well-known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov–Hopf bifurcation. Furthermore, we characterize all two-species, three-reaction, bimolecular-sourced networks that admit an Andronov–Hopf bifurcation with mass-action kinetics. These include two families of networks that admit a supercritical Andronov–Hopf bifurcation and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that are tetramolecular.     

PERIODIC REACTION AND TURBULENCE OF CHEMICAL BRUSSELATOR

In this paper,we discuss the reaction model of tri-molecules and the exi- stence of periodic reaction under certain original density conditions of two ob- jects without diffusion phenomena.We prove that the periodic reaction exhibit some chemical phenomena that violate the second law of thermo-mechanics. Finally,we research similar properties and phenomena concerning Belousov- Zhabotinsky reaction model.     

一类反应扩散系统的分歧与斑图

本文研究了一类发生在密闭容器中的不可激活的高次自催化反应扩散系统．在适当的条件下,用渐进近似的方法讨论了系统平衡态的稳定范围;用多重尺度的方法证明了当扩散系数λ充分小时,系统出现两种类型的斑图,一类是由Hopf分歧引出的驻波斑图;另一类是由 Pitchfork分歧引出的定波斑图．进一步还讨论了,在分歧点附近,对于大于空间或等于空间波数的小扰动,斑图是局部稳定的,而小于自身空间波数的小扰动,斑图是不稳定的．     

一类三维神经元模型的分支研究

考虑一类三维神经元模型的分支问题.利用常微分方程的定性与分支理论的知识,讨论了模型的平衡点个数及其稳定性,主要分析了平衡点的Hopf分支和Bogdanov-Takens分支,并得到了相应的鞍结点分支曲线,Hopf分支曲线与同宿分支曲线.     

Degasperis-Procesi方程的孤立尖波解

利用动力系统的定性分析理论对D egasperis-P rocesi方程的孤立尖波解进行了研究.给出了D e-gasperis-P rocesi方程对应行波系统的相图分支,利用相图获得了孤立尖波解和周期尖波解的解析表达式,通过数值模拟给出了部分解的图像.     

Peaked wave solutions of CH-r equation

In this paper the qualitative analysis methods of planar autonomous systems and numerica simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained,and some of their relationships arerevealed.Why peaked points are generated is discussed.     

A Consumption Behavior Model with Advertising And Word-of-Mouth Effect

It is widely-held belief that people’s consumption behavior is partly determined by advertising and word-of-mouth effect especially in monopolistic competitive market. Owing to this, we propose a mathematical model to interpret consumer behavior under the advertising and word-of-mouth effects, which is divided into continuous and discrete types for dynamic behavior analysis. Our research indicates the continuous model undergoes fold bifurcation, Hopf bifurcation, and degenerate fold-Hopf bifurcation; the discrete model undergoes flip bifurcation and Neimark-Sacker bifurcation. Moreover, bifurcation diagrams are given by using MATLAB to illustrate the model. Based on the theory of Hopf bifurcation or Flip bifurcation, the system undergoes supercritical Hopf bifurcation or Flip bifurcation under certain conditions. This lead to both advertising impacts periodically on consumer behavior, and a short-term reduction cause no effect the public¡s recognition of the brand, which can guide precision advertising investment.     

非线性算子的歧点

本文研究了Banach空间中非线性全连续算子在不假设Fréchet可微的条件下,歧点的存在性和不存在性.利用齐次算子,获得了这类算子歧点的存在和不存在的充分条件,推广和改进了已有文献中的结果.     

Feigenbaum刚性和界于简单与混沌之间的动力系统

本文主要综述了关于一类光滑动力系统族周期倍分支现象中通有比例规则的最新研究进展，而这一规则是由Ｆｅｉｇｅｎｂａｕｍ在美国、Ｃｏｕｌｌｅｔ与Ｔｒｅｓｓｅｒ在法国分别独立发现的，文中还阐述了一类界于简单与混沌之间的动力系统的几何性质。     

Existence and bifurcation of positive solutions for fractional p$$ p $$-Kirchhoff problems

In this paper, we are interested in the existence and bifurcation of positive solutions for Kirchhoff-type eigenvalue problems involving the fractional $p$-Laplacian. First, we investigate the properties of the first eigenvalue for fractional $p$-Laplacian equations with weighted functions. Furthermore, by using fixed-point argument and modified global bifurcation theorem of Rabinowitz, together with topological degree theory, we obtain the existence of unbounded continuum of positive weak solutions to Kirchhoff-type equations with subcritical and critical nonlinearities, where the bifurcation emanates from $(\mathrm{0,0})$. It is worth mentioning that our main results fill in some gaps of the available results.     

竞争不可逆化学反应过程工艺特征数学模型的建立

讨论提出竞争不可逆反应过程工艺方案和建立过程工艺数学模型,包括反应物完全转化条件下主反应的选择率数学模型,过程稳定态数学模型,特定极限分离边界数学模型和极值边际数学模型.     

Logistic阻滞增长模型的稳定性与混沌

将Logistic阻滞增长模型的差分形式简化，讨论了它的稳定性，用计算机进行迭代求解。模拟了这一简单差分方程从收敛，分叉，2^n倍周期收敛进入混沌现象的过程。直观地展示了序列{yk}收敛，2倍周期，4倍周期……直至混沌的现象，这对Logistic阻滞增长模型的应用和混沌现象的模拟有很好的参考价值。     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

One-Dimensional Topologically Nontrivial Solutions in the Skyrme Model

We consider the Skyrme model using the explicit parameterization of the rotation group through elements of its algebra. Topologically nontrivial solutions already arise in the one-dimensional case because the fundamental group of is . We explicitly find and analyze one-dimensional static solutions. Among them, there are topologically nontrivial solutions with finite energy. We propose a new class of projective models whose target spaces are arbitrary real projective spaces .