一个类Lorenz系统的Hopf分岔分析及分岔控制

对一个新的类Lorenz系统的Hopf分岔行为及分岔控制问题进行研究。首先,通过分岔稳定性指标判定系统的分岔类型。然后,分别对系统施加线性和非线性控制器。在线性控制部分,根据Routh-Hurwitz原理,讨论了线性参数对分岔位置的影响;在非线性控制部分,利用Normal Form(规范形)方法求出系统的Hopf分岔规范式,并通过规范式系数讨论非线性参数对Hopf分岔类型及极限环幅值的影响。结果表明当非线性参数满足一定条件时,原系统的Hopf分岔类型可以被改变,并且在超临界情况下,极限环幅值会随着非线性参数的增加而增加。     

一类比率依赖种群模型在常数收获下的分岔

利用微分方程定性理论以及规范型理论研究了一类具有常数收获项的生物模型.首先,考虑常数收获项对该模型非负平衡点的影响,讨论平衡点的稳定性情况.其次,选取常数收获项作为参数,给出系统存在鞍结点分岔、Hopf分岔以及极限环的充分条件.进一步地,考虑双参数分岔,给出系统存在余维2的Bogdanov-Takens分岔的充分条件.结果表明,由于添加了常数收获项,系统具有丰富的动力学行为.最后,通过数值仿真验证了所得结果的正确性.     

小世界网络模型的非线性动力学研究

基于非线性动力学理论,研究了小世界网络模型的非线性动力学现象.首先,在已有的小世界网络非线性动力学模型基础上,从向量场角度,对其中随参数变化系统的定常状态失稳而出现的Hopf分岔进行了数值分析,并根据Hopf分岔的分析结果,对系统在一定参数条件下定常状态的失稳及周期振荡的产生进行了解释;然后,将向量场控制方程转化为映射,从直观映射的角度,详细分析了其中的定量状态,失稳导致的倍周期分岔、系列倍周期分岔,以及周期-3状态,从而证明系统存在混沌特征.研究表明:该系统蕴含有丰富的非线性动力学行为,通过对该类"非均匀"动力系统深入的理论分析,探索出产生各类复杂非线性动力学现象的机理,从而实现对该类网络系统的有效控制.     

电压型Buck变换器中的低频振荡分析

分析了具有PI控制的Buck变换器中的低频振荡现象。利用离散模型研究了Buck变换器中的低频振荡现象,结果表明系统产生低频振荡的原因是状态变量发生了Hopf分岔。采用状态空间平均的小信号模型分析了变换器的稳定性,它所确定的稳定运行临界点恰好与从离散模型得到的Hopf分岔点的位置相吻合,这表明线性的小信号平均模型可以准确的预测低频振荡的参数稳定域。通过分析发生低频振荡后的频率和幅值,结果表明Buck变换器中的低频振荡现象实际上是一种自激振荡。最后通过数值仿真和电路实验验证了理论分析的合理性。     

独立非交叉传播的分数阶生物竞争网络Hopf分岔

目前绝大多数生态竞争网络是由整数阶系统刻画的,针对系统行为仅受当前时刻影响的问题,提出具有独立非交叉传播特性的分数阶时滞捕食被捕食模型.选取时滞作为分岔参数,通过分析不同阶次影响下系统特征方程根的分布,研究了该模型的稳定性和分岔问题,建立了时滞诱发的稳定性条件和Hopf分岔判据,最后通过数值仿真验证了理论结果的准确性.     

阶段结构广义生物经济模型的分岔及控制

利用微分代数方程理论研究了一类广义生物经济模型.首先,研究参数临界状态下,该模型平衡点的稳定性情况,并进一步给出系统存在跨临界分岔和奇异诱导分岔的充分条件.其次,设计状态反馈控制器,通过控制捕获努力量,消除该系统中存在的奇异诱导分岔及脉冲行为,抑制种群变化,使系统趋于稳定.最后,通过数值仿真说明控制器的有效性.     

广义受迫Van der Pol-Duffing系统的稳定性与分岔

给出了广义受迫VanderPol-Duffing方程,用多初始点分岔分析方法分析了系统外部参数对其稳定性的影响,应用了分岔图、Lyapunov指数图和Poincaré映射图分析了系统的非线性动力学行为,结果很好地解释了该系统中一些复杂的非线性现象,为研究许多模型提供了一定的理论参考和实际意义。     

EEG模型的混沌动力学特性研究

EEG是大脑电生理过程的宏观表现,研究EEG模型的非线性动力学特征可以使大脑的活动规律得到进一步认识。EEG模型中,以兴奋性神经元集群的兴奋性输入作为参数,分析了模型的关联维数和最大Lyapunov指数的变化。数值仿真实验显示该EEG模型具有极限环,倍周期分岔,混沌等复杂的动力学行为,进一步说明混沌存在于EEG模型中。

Abstract：

EEG is the expression of brain electrophysiological activity in macroscopic scales,and the rule of brain activity can be recognized by analyzing nonlinear dynamics of EEG model.Regarding the excitatory inputs to excitatory populations as a parameter,correlation dimension and largest Lyapunov exponent were calculated.Numerical simulation shows complex dynamical behaviors,such as limit cycle,doubling-period bifurcation,chaos and so on,provide the evidence that chaos exists in EEG model.     

基于O-D网络的交通流模型的分岔与混沌

以交通流O-D网络系统为背景,通过分岔图和相图讨论了在无交通阻塞下的交通费用和交通容量变化下系统非线性动力学行为的变化和运动复杂性.交通费用和交通容量作为交通网络的固有属性,对定性分析交通网络的动力学行为具有至关重要的作用.同时,数值验证了两种计算交通流O-D网络系统分岔图的方法的等价性.     

保险市场中三寡头垄断的动态博弈模型研究

考虑三寡头垄断的保险市场动态价格博弈模型.在模型中,一个寡头采取自适应决策,其余两个寡头采取有限理性决策,由此建立三寡头价格博弈微分方程模型.系统有唯一的Nash均衡点.并对该系统的稳定性和Hopf分岔的存在性进行研究.数值模拟结果证实了理论的准确性,并且展示了系统的动态行为.保险公司在考虑延迟的价格博弈的过程中,必须控制好延迟参数的值,并适当地降低自身价格调整速度,以使系统尽快稳定到平衡状态.     

Hopf bifurcation analysis and bifurcation control of a lorenz-like system北大核心CSCD

Hopf bifurcation behavior and bifurcation control of a new Lorenz-like system are studied in this paper. Firstly, Hopf bifurcation type is determined by bifurcation stability norm. Then the linear controller and the non-linear controller are applied to control the original system respectively. In the section of linear control, the effect of linear parameter on the position of Hopf bifurcation is discussed by Routh-Hurwitz criterion; In the section of non-linear control, the Hopf bifurcation Normal Form of controlled system is obtained by using direct Normal Form method, and the effects of nonlinear parameter on amplitude of limit cycle and Hopf bifurcation type are discussed by coefficient of Normal Form. Discussions show that if non-linear parameter satisfies certain condition, bifurcation type of original system will be changed, and the periodic solution amplitude will increase with the parameter increasing. ©, 2015, The Journal Agency of Complex Systems and Complexity Science. All right reserved.     

EXISTENCE AND ANALYTICAL APPROXIMATIONS OF LIMIT CYCLES IN A THREE-DIMENSIONAL NONLINEAR AUTONOMOUS FEEDBACK CONTROL SYSTEM

This paper is concerned with the existence and the analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system. Based on three-dimensional Hopf bifurcation theorem, the existence of limit cycles is first proved. Then the homotopy analysis method （HAM） is applied to obtain the analytical approximations of the limit cycle and its frequency. In deriving the higher-order approximations, the authors utilized the idea of a perturbation procedure proposed for limit cycles＇ approximation in van der Pol equation. By comparing with the numerical integration solutions, it is shown that the accuracy of the analytical results obtained in this paper is very high, even when the amplitude of the limit cycle is large.     

LOCAL AND GLOBAL BIFURCATIONS AND APPLICATIONS IN A PREDATOR-PREY SYSTEM WITH SEVERAL PARAMETERS

A predator-prey system,depending on several parameters,is investigated forbifurcation of equilibria,Hopf bifurcation,global bifurcation occurring saddle connection,and global existence and nonexistence of limit cycles,and changes of the topological structureof trajectory as parameters are varied.     

Singularly perturbed bifurcation subsystem and its application in power systems

The singularly perturbed bifurcation subsystem is described, and the test conditions of subsystem persistence are deduced. By use of fast and slow reduced subsystem model, the result does not require performing nonlinear transformation. Moreover, it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold. Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.     

Singularly perturbed bifurcation subsystem and its application in power systems

The singularly perturbed bifurcation subsystem is described,and the test conditions of subsystem persistence are deduced.By use of fast and slow reduced subsystem model,the result does not require performing nonlinear transformation.Moreover,it is shown and proved that the persistence of the periodic orbits for Hopf bifurcation in the reduced model through center manifold.Van der Pol oscillator circuit is given to illustrate the persistence of bifurcation subsystems with the full dynamic system.     

Codimension-two bifurcations analysis and tracking control on a discrete epidemic model

In this paper, the dynamic behaviors of a discrete epidemic model with a nonlinear incidence rate obtained by Euler method are discussed, which can exhibit the periodic motions and chaotic behaviors under the suitable system parameter conditions. Codimension-two bifurcations of the discrete epidemic model, associated with 1:1 strong resonance, 1:2 strong resonance, 1:3 strong resonance and 1:4 strong resonance, are analyzed by using the bifurcation theorem and the normal form method of maps. Moreover, in order to eliminate the chaotic behavior of the discrete epidemic model, a tracking controller is designed such that the disease disappears gradually. Finally, numerical simulations are obtained by the phase portraits, the maximum Lyapunov exponents diagrams for two different varying parameters in 3-dimension space, the bifurcation diagrams, the computations of Lyapunov exponents and the dynamic response. They not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.     

QUALITATIVE ANALYSIS FOR A THIRD-ORDER DIFFERENTIAL EQUATION IN A MODEL OF CHEMICAL SYSTEMS

In this paper a mathematical model of chemical systems is investigated.We present the conditions for the existence and local stability of the steady statesand the periodic solution of the Hopf type.Specifically,we show by using an ana-lytical method that there may exist two or four Hopf bifurcation points separatedat a finite distance from each other;at the same time,a technique for studying theHopf bifurcation value is given.