分支问题的有限决定性和万有开折

本文给出了分支问题有限决定的一些充分必要条件，并用横切性条件刻划了分支问题的万有开折，证明了分支问题的万有开折与无穷小稳定是等价的     

常数激励对局部分岔的影响

用奇异性理论讨论了常数激励对1/2内共振系统周期解局部分岔的影响．研究表明,常数激励项能否产生影响关键取决于低频振子中是否存在某些非线性项．常数激励, 一方面起主分岔参数的作用, 另一方面,与系统中某些非线性项的系数一起确定分岔解基本类型、 影响开折参数．在非退化条件下,可不考虑三次非线性项的影响．     

Saddle-node-Hopf bifurcation in a modified Leslie–Gower predator-prey model with time-delay and prey harvesting

From the saddle-node-Hopf bifurcation point of view, this paper considers a modified Leslie–Gower predator-prey model with time delay and the Michaelis–Menten type prey harvesting. Firstly, we discuss the stability of the equilibria, obtain the critical conditions for the saddle-node-Hopf bifurcation, and give the completion bifurcation set by calculating the universal unfoldings near the saddle-node-Hopf bifurcation point by using the normal form theory and center manifold theorem. Then we derive the parameter conditions for the existence of monostable coexistence equilibrium and the parameter regions in which both the prey-extinction and the coexistence equilibrium (or coexistence periodic or quasi-periodic solutions) are simultaneously stabilized. We also investigate the heteroclinic bifurcation, and describe the phenomenon that the periodic behavior disappears as through the heteroclinic bifurcation. Finally, some numerical simulations are performed to support our analytic results.     

周期激励Stuart-Landau方程的某些动力学行为

研究了周期激励Stuart-Landau方程的锁频周期解．利用奇异性理论分别研究了这些解关于外部激励振幅和频率的分岔行为．结果表明:关于外部激励振幅的普适开折具有余维3,在某些条件下,得到了转迁集及分岔图．另外还证明:关于频率的分岔问题具有无穷余维,因此该情形下的动力学分岔行为非常复杂．发现了一些新的动力学现象,它们是孙亮等所获结果的补充．     

含两组状态变量等变分歧问题开折的唯一性和稳定性

基于奇点理论中光滑映射芽的接触等价,讨论多参数等变分歧问题关于接触等价的开折的唯一性和稳定性.将这种等变分歧问题的状态变量分为两组,其中一组的诸状态变量可以独立地变化,而属于另一组的诸状态变量在变化过程中依赖于前一组中的诸状态变量.得出了在接触等价下,满足一定条件的等变分歧问题的万有开折是唯一的,并且给出了一定条件下等变分歧问题开折稳定的一个充要条件.     

Hopf-transcritical bifurcation in toxic phytoplankton–zooplankton model with delay

In this paper, a phytoplankton–zooplankton model with toxic liberation delay is considered. Firstly, the critical values of Hopf bifurcation, transcritical bifurcation and Hopf-transcritical bifurcation are given, and to give more detailed information about the periodic oscillations, the direction and stability of Hopf bifurcation is studied by using the normal-form theory and center manifold theorem. Then, we give the detailed bifurcation set by calculating the universal unfoldings near the Hopf-transcritical bifurcation point. Finally, we show that the plankton system may exhibit quasi-periodic oscillations, which are verified both theoretically and numerically, and explain the experimental observed fluctuation phenomenon of plankton population.     

带有多个分歧参数的等变分歧问题的万有开折

对于含一个分歧参数的分歧问题,已有万有开折定理阻［１,２］．本文考虑带有多个分歧参数的等变分歧问题,并且允许它的状态空间与靶空间可以不同,给出了等变万有开折定理的更一般的形式,［１,２］中相关结果其特殊情形．     

Dynamics in a discrete-time predator-prey system with Allee effect

In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases (within a limited value). Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.     

具有初始缺陷弹性板的稳定性分析

本文以Marguerre方程为基础,用奇异性理论研究了初始挠度缺陷以及横向载荷对弹性板屈曲后分叉解的影响。借助于普适开折的原理,在单特征值局部邻域内将该问题的失稳分析转化为三次代数方程的讨论,从而确定出分叉解的性态。同时绘出了在不同参数下的分叉解文,讨论了几何缺陷和横向载荷对特征值的影响。     

Zn—等变的奇点理论

本详细讨论了Ｚｎ－等变奇点理论的基本概念，范式和万有开折的求法，并具有给出了非退化情况下Ｚ３－等变点的范式和万有开折。它们为研究周期参数激励系统的亚谐分叉提供了一种方法。     

DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS

§1. IntroductionA key problem in the study of problems in mathematical physics and mechanics is tounderstand and predict patterns and their transitions/evolutions. In ?uid mechanics, forinstance, it is important to study the periodic, quasi-periodic, ape…     

Bifurcation analysis on a diffusive Holling–Tanner predator–prey model

This paper is concerned with a diffusive Holling–Tanner predator–prey model subject to homogeneous Neumann boundary condition. By choosing the ratio of intrinsic growth rates of predator to prey λ as the bifurcation parameter, we find that spatially homogeneous and non-homogeneous Hopf bifurcation occur at the positive constant steady state as λ varies. The steady state bifurcation of simple and double eigenvalues are intensively investigated. The techniques of space decomposition and the implicit function theorem are adopted to deal with the case of double eigenvalues. Our results show that this model can exhibit spatially non-homogeneous periodic and stationary patterns induced by the parameter λ. Numerical simulations are presented to illustrate our theoretical results.     

Bifurcations of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and prey harvesting

The dynamics of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle-node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results.     

约束含参分岔问题的分类

约束边界与分岔参数有关的约束分岔问题,称为约束含参分岔问题．通过引入适当的变换,将约束含参分岔问题转化为新变量的非约束分岔问题,推导出了约束含参分岔问题转迁集的一般形式,结果表明只有约束分岔集受约束含参的影响,其它转迁集与不含参约束分岔的转迁集相同．以含参约束树枝分岔为例分析了此类问题的分岔分类,讨论了约束含参对分岔分类的影响．     

Analysis of a Shil’nikov Type Homoclinic Bifurcation

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.     

Effects of the killing rate on global bifurcation in an oncolytic-virus system with tumors

Oncologists and virologist are quite concerned about many kinds of issues related to tumor-virus dynamics in different virus models. Since the virus invasive behavior emerges from combined effects of tumor cell proliferation, migration and cell-microenvironment interactions, it has been recognized as a complex process and usually simulated by nonlinear differential systems. In this paper, a nonlinear differential model for tumor-virus dynamics is investigated mathematically. We first give a priori estimates for positive steady-states and analyze the stability of the positive constant solution. And then, based on these, we mainly discuss effects of the rate of killing infected cells on the bifurcation solution emanating from the positive constant solution by taking the killing rate as the bifurcation parameter.     

Numerical simulation of the transition to chaos in a dissipative Duffing oscillator with two-frequency excitation

A mathematical modeling technique is proposed for oscillation chaotization in an essentially nonlinear dissipative Duffing oscillator with two-frequency excitation on an invariant torus in ?². The technique is based on the joint application of the parameter continuation method, Floquet stability criteria, bifurcation theory, and the Everhart high-accuracy numerical integration method. This approach is used for the numerical construction of subharmonic solutions in the case when the oscillator passes to chaos through a sequence of period-multiplying bifurcations. The value of a universal constant obtained earlier by the author while investigating oscillation chaotization in dissipative oscillators with single-frequency periodic excitation is confirmed.     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

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.     

投资过程中具有时滞的一类动态宏观经济模型

根据Klecki思想,对投资过程中具有常数时滞的一类动态宏观经济模型进行了讨论.通过选择时滞作为Hopf分歧参数,当时滞经过临界值时周期行为产生.并用数值例子解释了理论结果.