关于Guzowska-Luís-Elaydi模型的分岔（英文）

本文考虑了一个离散的Logistic竞争模型。为了讨论分岔,给出了不动点的拓扑类型及非双曲的情况.应用中心流行约化定理,证明了跨临界分岔会在三个不动点上发生.本文还证明了在两个不动点处,跳跃分岔会发生,同时稳定的周期2轨会出现.     

一类双面碰撞振子的对称性尖点分岔与混沌

讨论了一类单自由度双面碰撞振子的对称型周期n-2运动以及非对称型周期n-2运动．把映射不动点的分岔理论运用到该模型,并通过分析对称系统的Poincaré映射的对称性,证明了对称型周期运动只能发生音叉分岔．数值模拟表明:对称系统的对称型周期n-2运动,首先由一条对称周期轨道通过音叉分岔形成具有相同稳定性的两条反对称的周期轨道;随着参数的持续变化,两条反对称的周期轨道经历两个同步的周期倍化序列各自生成一个反对称的混沌吸引子．如果对称系统演变为非对称系统,非对称型周期n-2运动的分岔过程可用一个两参数开折的尖点分岔描述,音叉分岔将会演变为一支没有分岔的分支以及另外一个鞍结分岔的分支．     

基于延时反馈的电力非线性系统Hopf分岔反控制

针对无刷直流电机等效非线性动力系统,设计基于Washout滤波器辅助和延迟反馈相结合的控制器对系统进行Hopf分岔反控制.根据Hopf分岔理论讨论系统在稳定的平衡点处发生Hopf分岔时,延迟参数应满足的条件.讨论结果表明,当延迟参数满足一定条件时,可使系统在所期望的平衡点处发生Hopf分岔,从而实现系统的Hopf分岔反控制.此外,方法也可用于混沌控制.数值仿真证明了控制器的有效性.     

新三维指数系统及其控制系统的Hopf分岔分析

考虑了一个新三维指数系统的Hopf分岔,并且分析了指数系统添加非线性控制器后的Hopf分岔.通过严格的数学推导给出受控系统发生余维一,余维二和余维三的Hopf分岔的参数条件,证明了可以控制系统在指定区域内发生退化分岔和可调控分岔的稳定性,并且通过数值模拟验证了得出的结论.     

一类具稀疏效应的Predator-Prey系统的分岔及混沌

得到了一类稀疏效应下的Predator-Prey系统发生静态分岔和Hopf分岔条件,证明了此类系统存在混沌现象.     

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.     

能源供需随机系统的稳定性研究

研究了江苏省西部能源供需随机系统的稳定性.主要是基于一维扩散过程的奇异边界理论,应用摄动方法研究系统的随机分岔行为.研究结果表明随机因素以及参数的选择会使系统发生分岔行为,从而使系统的稳定性发生质的变化.于是,可以通过调节参数降低发生分岔的概率,使系统处于稳定的发展中.     

一类时变动力系统的高余维分岔及其控制

研究了一类时变动力系统的高余维分岔及其控制问题,首先利用新方法对时变分岔方程的两个方向的分岔转迁和跃迁现象进行分析,分别通过慢变解的线性化近似和量级平衡估计分岔转迁值,然后研究这类时变分岔方程的线性反蚀控制问题,通过分析相应的二维高次自治系统的Hopf分岔,在适当的条件下得到了稳定的动态滞后环,研究揭示出脉冲振动产生的机理是分岔参数随时间周期变化经过定常分岔值时所发生的分岔转迁的滞后和跃迁现象。     

奇点处分岔解支的数目问题*

本文在孤立奇点的假设下证明了带参数非线性方程组解的孤立性,同时指出在分岔点处必有而且仅有有限条解支分岔出来．这一结论是分岔问题数值方法的一个理论基础．     

Asymptotic bifurcation points, and global bifurcation of nonlinear operators and its applications

Using the cone theory and lattice structure, we discuss the existence of asymptotic bifurcation points and the global bifurcation of nonlinear operators which are not assumed to be cone mappings and may not be Frechet differentiable at points at infinity. As an application, the structure of the set of solutions of the superlinear Sturm-Liouville problems is investigated.     

Bifurcation Difference Induced by Different Discrete Methods in a Discrete Predator-prey Model

In this paper, we revisit a discrete predator-prey model with Allee effect and Holling type-I functional response. The most important is for us to find the bifurcation difference: a flip bifurcation occurring at the fixed point $E_3$ in the known results cannot happen in our results. The reason leading to this kind of difference is the different discrete method. In order to demonstrate this, we first simplify corresponding continuous predator-prey model. Then, we apply a different discretization method to this new continuous model to derive a new discrete model. Next, we consider the dynamics of this new discrete model in details. By using a key lemma, the existence and local stability of nonnegative fixed points $E_0$, $E_1$, $E_2$ and $E_3$ are completely studied. By employing the Center Manifold Theorem and bifurcation theory, the conditions for the occurrences of Neimark-Sacker bifurcation and transcritical bifurcation are obtained. Our results complete the corresponding ones in a known literature. Numerical simulations are also given to verify the existence of Neimark-Sacker bifurcation.     

Dynamics of a Discrete Two-species Competitive Model with Michaelies-Menten Type Harvesting in the First Species

In this paper, we use a semidiscretization method to derive a discrete two-species competitive model with Michaelis-Menten type harvesting in the first species. First, the existence and local stability of fixed points of the system are investigated by employing a key lemma. Subsequently, the transcritical bifurcation, period-doubling bifurcation and pitchfork bifurcation of the model are investigated by using the Center Manifold Theorem and bifurcation theory. Finally, numerical simulations are presented to illustrate corresponding theoretical results.     

Complex dynamics of a discrete predator–prey model with the prey subject to the Allee effect

In this paper, complex dynamics of the discrete predator–prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark–Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark–Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.     

Anti-Controlling Codimension-2 Bifurcation of Discrete Dynamical Systems in 1 : 2 Resonance北大核心CSCD

从分岔反控制的角度设计了一套非线性反馈控制策略,来实现离散动力系统1∶2共振情形下余维二分岔的各种分岔解。首先,针对传统分岔准则在确定高余维分岔点时存在的局限性,建立了一个1∶2共振情形下的余维二分岔的新显式准则,基于这个显式准则通过设计线性控制增益来确保此类余维二分岔的存在性。然后,推导了1∶2共振的中心流形,并基于范式方法通过设计非线性控制增益,分析了1∶2共振情形下余维二分岔解的类型和稳定性。最后,以一个Arneodo-Coullet-Tresser映射为例,在指定的参数点处通过控制实现了具有1∶2共振分岔特性的各种分岔解,进一步验证了理论分析。     

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.     

Fundamental solutions and asymptotic numerical methods for bifurcation analysis of nonlinear bi‐harmonic problems

New algorithms, combining asymptotic numerical method (ANM) and method of fundamental solutions, are proposed to compute bifurcation points on branch solutions of a nonlinear bi‐harmonic problem. Three methods, mainly based on asymptotic developments framework, are then proposed. The first one consists in exploiting the ANM step accumulation close to the bifurcation points on a solution branch, the second method allows the introduction of an indicator that vanishes at the bifurcation points, and finally the first real root of the Padé approximant denominator represents the third bifurcation indicator. Two numerical examples are considered to analyze the robustness of these algorithms.     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

Role of transmissible disease in an infected prey-dependent predator – prey system

The role of disease in ecological systems is a very important issue from both mathematical and ecological points of view. This paper deals with the qualitative analysis of a prey-dependent predator – prey system in which a disease is spreading among the prey species only. We have analysed the behaviour of the system around each equilibrium and obtained conditions for global stability of the system around an equilibrium by using suitable Lypunov functions. We have also worked out the region of parametric space under which the system enters a Hopf bifurcation and a transcritical bifurcation but does not experience either saddle-node bifurcations or pitchfork bifurcations around the disease-free equilibrium E ₂. Finally, we have given an example of a real ecological situation with experimental data simulations.