四维超混沌系统Hopf分岔分析与反控制

对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。     

四维超混沌系统Hopf分岔分析与反控制

对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。     

两自由度机械臂λ^2=1下的Hopf分岔与混沌研究

研究了一类周期系数力学系统因周期运动失稳而产生Hopf分岔及混沌问题.首先根据拉格朗日方程给出了该力学系统的运动微分方程,并确定其周期运动的具有周期系数的扰动运动微分方程,再根据Floquet理论建立了其给定周期运动的Poincaré映射,根据该系统的特征矩阵有一对复共轭特征值从-1处穿越单位圆情况,分析该Poincaré映射不动点失稳后将发生次谐分岔、Hopf分岔、倍周期分岔,而多次倍周期分岔将导致混沌.并用数值计算加以验证.结果表明,随着分岔参数的变化,系统的周期运动可通过次谐分岔形成周期2运动,进而发生Hopf分岔形成拟周期运动,并再次经次谐分岔、倍周期分岔形成混沌运动.     

轮对非线性动力学模型稳定性和分岔特性研究

为了探究轮对系统的横向失稳问题,考虑了陀螺效应和一系悬挂阻尼的影响作用,建立非线性轮轨接触关系的轮对动力学模型,研究轮对系统的蛇行稳定性、Hopf分岔特性及迁移转化机理.通过稳定性判据获得了轮对系统失稳临界速度.采用中心流形定理和规范型方法对轮对动力学模型进行化简,得到与轮对系统分岔特性相同的一维复变量方程,理论推导求得轮对系统的第一Lyapunov系数的表达式,根据其符号即可判断轮对系统的Hopf分岔类型.讨论了不同参数对轮对系统Hopf分岔临界速度的影响,探究了轮对系统的超临界、亚临界Hopf分岔域在二维参数空间的分布规律.利用数值模拟得到轮对系统的3种典型Hopf分岔图,验证了轮对系统超临界、亚临界Hopf分岔域分布规律的正确性.结果表明,轮对系统的临界速度随着等效锥度的增大而减小,随着一系悬挂的纵向刚度和纵向阻尼的增大而增大,随着纵向蠕滑系数的增大呈先增大后减小.系统参数变化会引起轮对系统Hopf分岔类型发生改变,即亚临界与超临界Hopf分岔相互迁移转化.轮对系统Hopf分岔域在二维参数空间的分布规律对于轮对系统参数匹配和优化设计具有一定的指导意义.     

轮对系统的Hopf分岔研究

针对轮对系统中的非线性动力学问题, 本文基于Hopf分岔代数判据得到考虑陀螺效应的轮对系统Hopf分岔点解析表达式, 即轮对系统蛇形失稳的线性临界速度解析表达式. 基于分岔理论得到轮对系统的第一、第二Lyapunov系数表达式, 并结合打靶法分别得到不同纵向刚度下, 考虑陀螺效应与不考虑陀螺效应的轮对系统分岔图. 通过对比有无陀螺效应的轮对系统分岔图发现, 在同一纵向刚度下, 考虑陀螺效应的轮对系统线性临界速度和非线性临界速度均大于不考虑陀螺效应的轮对系统, 即陀螺效应可以提高轮对系统的运动稳定性. 基于Bautin分岔理论, 以纵向刚度和纵向速度作为参数, 分别得到考虑陀螺效应和不考虑陀螺效应的轮对系统, 从亚临界Hopf分岔到超临界Hopf分岔, 再从超临界Hopf分岔到亚临界Hopf分岔的迁移机理拓扑图. 通过对比有、无陀螺效应的轮对系统Bautin分岔拓扑图发现, 陀螺效应将改变轮对系统的退化Hopf分岔点, 但对于轮对系统Bautin分岔拓扑图的影响不大.      

碰撞振动系统强共振下的两参数动力学分析

建立了两自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在一对共轭复特征值在单位圆上并满足强共振(λ40=1)条件时,通过中心流型-范式方法将四维映射转变为二维范式映射。理论分析了系统两参数开折的局部动力学行为,扩展了单参数分岔理论,给出了n-1周期运动产生Hopf分岔和次谐分岔的条件。数值仿真验证了所得出的理论,证明系统在共振点附近存在稳定的Hopf分岔不变环面和次谐分岔4-4周期运动。     

Hopf分岔的代数判据及其在车辆动力学中的应用

利用Hurwitz行列式,给出平衡点失稳而发生Hopf分岔的代数判定准则和计算方法,这一方法将Hopf分岔点的求解转化为一个非线性方程的求解问题,从而克服了以前方法在计算Hopf分岔点时,对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量。应用这一方法,我们进行了非线性车辆系统蛇行运动稳定性的研究,得到了轮对系统发生蛇行运动的临界速度的解析表达式。     

三自由度齿轮系统的双参数分岔与全局特性研究

以三自由度齿轮系统为研究对象,通过构造参数平面内不同运动类型的边界线算法,得到了系统在参数平面内的分岔曲线。为了判断分岔曲线的分岔类型,构造了三自由度齿轮系统Poincaré映射的Jacobi矩阵及Floquet乘子算法。结合系统的分岔图、最大Lyapunov指数图(TLE)、相图、Poincaré映射图和Floquet理论,讨论了双参数平面上系统的分岔特性以及参数平面内系统动力学特性的演变,并利用胞映射法对系统随啮合频率变化下的全局动力学特性进行了研究。结果表明:系统在参数平面k-ξ33内存在倍化分岔曲线、鞍结分岔曲线、Hopf分岔曲线等;阻尼系数越大,综合误差越小,系统运动越稳定;鞍结分岔对系统的全局稳定性影响较大,而Hopf分岔对系统的全局稳定性影响较小。研究结果可为齿轮系统设计和参数选择提供理论依据,研究方法也适用于其它非线性系统的双参数分岔分析。     

快慢耦合振子的张驰簇发及其非光滑分岔机制

通过引入适当的参数值, 得到了两时间尺度下的快慢耦合振子, 分析了耦合系统及子系统的平衡点及其性质, 进而利用微分包含理论, 探讨了非光滑分界面上的奇异性, 指出在适当的参数条件下, 系统轨迹在穿越分界面时会产生由Hopf分岔和Fold分岔组合的非常规分岔. 给出了不同参数条件下的周期簇发行为, 分析了簇发过程的振荡特性, 指出激发态的频率取决于快子系统在非光滑分界面上的Hopf分岔频率, 而慢子系统的固有频率影响了簇发行为的振荡周期, 并进一步揭示了由非光滑分岔引起的不同周期簇发的分岔机制.     

时滞Rucklidge系统Hopf分岔分析与电路实现

以一类新的单时滞Rucklidge系统为分析对象,通过计算时滞系统的平衡点,分析该系统在各平衡点的稳定性和Hopf分岔的存在性,得到其发生Hopf分岔的条件。Matlab多组数值仿真验证了理论分析的正确性。基于此设计了一种可切换时滞与非时滞的混沌电路,并运用Multisim14.0进行仿真,实验结果表明,该电路可行且有效。     

神经网络动力系统吸引子对称性的变化

用余维２的双Ｈｏｐｆ分叉的规范形方程设计了期望存储振荡型记忆模式的模拟四阶关系神经网络，所设计的网络向量场具有中心对称性。研究表明该网络发生二阶双Ｈｏｐｆ分叉后可以出现不变二环面。观察二环面上的销相运动，发现了系统出现对称破裂的规律，邓把双Ｈｏｐｆ分叉的两个频率比表示成既约分数的形式，当该分数的分子和分母均为奇数时，网络的吸引子保持对称性，而当分子主分母中任一个为偶数时，就会发生对称破裂。     

Local stability and Hopf bifurcations analysis of the Muthuswamy-Chua-Ginoux system

The three-dimensional Muthuswamy–Chua–Ginoux (MCG, for short) circuit system based on a thermistor is a generalization of the classical Muthuswamy–Chua circuit differential system. At present, there are only partial numerical simulations for the qualitative analysis of the MCG circuit system. In this work, we study local stability and Hopf bifurcations of the MCG circuit system depending on 8 parameters. The emerging of limit cycles under zero-Hopf bifurcation and Hopf bifurcation is investigated in detail by using the averaging method and the center manifolds theory, respectively. We provide sufficient conditions for a class of the circuit systems to have a prescribed number of limit cycles bifurcating from the zero-Hopf equilibria by making use of the third-order averaging method, as well as the methods of Gröbner basis and real solution classification from symbolic computation. Such algebraic analysis allows one to study the zero-Hopf bifurcation for any other differential system in dimension 3 or higher. After, the classical Hopf bifurcation of the circuit system is analyzed by computing the first three focus quantities near the Hopf equilibria. Some examples and numerical simulations are presented to verify the established theoretical results.

     

Hopf bifurcation of nonlinear system with multisource stochastic factors

The article mainly explores the Hopf bifurcation of a kind of nonlinear system with Gaussian white noise excitation and bounded random parameter. Firstly, the nonlinear system with multisource stochastic factors is reduced to an equivalent deterministic nonlinear system by the sequential orthogonal decomposition method and the Karhunen–Loeve (K-L) decomposition theory. Secondly, the critical conditions about the Hopf bifurcation of the equivalent deterministic system are obtained. At the same time the influence of multisource stochastic factors on the Hopf bifurcation for the proposed system is explored. Finally, the theorical results are verified by the numerical simulations.     

Hopf bifurcation analysis and numerical simulation in a 4D-hyoerchaotic system

This paper investigates the Hopf bifurcation of a four-dimensional hyperchaotic system with only one equilibrium. A detailed set of conditions is derived which guarantees the existence of the Hopf bifurcation. Furthermore, the standard normal form theory is applied to determine the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods. In addition, numerical simulations are used to justify theoretical results.     

时变小扰动Hamilton系统的Hopf分岔

运用Melnikov方法研究了时变小扰动Ｈamilton系统周期轨道发生Ｈopf分岔的条件,并将这些条件应用到一类三维时变小扰动非自治系统,数值结果验证了本文理论的正确性,进一步数值积分表明,所研究的系统还存在复杂而有规律的环面分岔行为。     

Global stability and bifurcation analysis of a delay induced prey-predator system with stage structure

This paper describes a delay induced prey–predator system with stage structure for prey. The dynamical characteristics of the system are rigorously studied using mathematical tools. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Sufficient conditions are derived for the global stability of the system. The optimal harvesting problem is formulated and solved in order to achieve the sustainability of the system, keeping the ecological balance, and maximize the monetary social benefit. Maturation time delay of prey is incorporated and the existence of Hopf bifurcation phenomenon is examined at the coexistence equilibria. It is shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Moreover, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.     

Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays

In this paper, we consider the effect of distributed delays in a three-neuron unidirectional ring. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. Taking the average delay as a bifurcation parameter, we find two critical values at which the system undergoes Hopf bifurcations. The orbital asymptotic stability of the Hopf bifurcating periodic solutions is investigated by using the method of multiple scales. The global Hopf bifurcation is also studied. Finally, the theoretical results are illustrated by some numerical simulations.     

Stability and bifurcation analysis of delay coupled Van der Pol–Duffing oscillators

In this paper, we aim to investigate the dynamics of a system of Van der Pol–Duffing oscillators with delay coupling. First, taking the time delay as a bifurcation parameter, the stability of the equilibrium, and the existence of Hopf bifurcation are investigated. Then using the center manifold reduction technique and normal form theory, we give the direction of the Hopf bifurcation. And then by means of the symmetric bifurcation theory for delay differential equations and the representation theory of groups, we claim the bifurcation periodic solution induced by time delay is antiphase locked oscillation. Finally, at the end of the paper, numerical simulations are carried out to support our theoretical analysis.     

Dynamical analysis in a 4D hyperchaotic system

Inspirited by Li and Jin (Nonlinear Dyn. 67:2857?C2864 2012), this paper investigates the Hopf bifurcation of a four-dimensional hyperchaotic system with only one equilibrium. A detailed set of conditions are derived, which guarantee the existence of the Hopf bifurcation. Furthermore, the standard normal form theory is applied to determine the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods. In addition, numerical simulations are used to justify theoretical results.     

Modelling and analysis of a multiple delayed exploited ecosystem towards coexistence perspective

This paper describes a multiple delayed modified Leslie–Gower type predator–prey system with a strong Allee effect in prey population growth. Non-selective effort is used to harvest the population. The dynamical characteristics of the delay induced system are rigorously studied using mathematical tools. The existence of coexistence equilibria is ensured, and the dynamic behavior of the system is investigated around coexistence equilibria. Uniform strong persistence and permanence of the system are discussed in order to ensure long-term survival of the species. The stability of the delay preserved system is investigated. Sufficient conditions are derived for local and global stability of the system. The existence of Hopf bifurcation phenomenon is examined around interior equilibria of the system. Subsequently, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, numerical simulations are carried out to validate the analytical findings.