Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a newhyperchaotic circuit system. The stability of the hyperchaotic circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. Animportant feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.     

Hopf Bifurcation in a New Four-Dimensional Hyperchaotic System

In this paper, the Hopf bifurcation in a new hyperchaotic system is studied. Based on the first Lya-punov coefficient theory and symbolic computation, the conditions of supercritical and subcritical bifurcation in the new hyperchaotic system are obtained. Numerical simulations are used to illustrate some main results.     

单周期控制Boost变换器Hopf分岔控制及电路实现

用平均模型分析了单周期控制Boost变换器的运行,分析表明在参考电压变化的情况下,单周期控制Boost变换器会出现Hopf分岔.Hopf分岔使得变换效率下降,器件应力增加.为了消除Hopf分岔,提出了采用washout滤波器的方法.建立了采用washout滤波器的单周期控制Boost变换器平均模型,对于washout滤波器中的两个新参数,可以用Routh-Hurwitz准则来确定.仿真和电路实验验证了所提方法的效果. 关键词： washout滤波器 单周期控制 Boost变换器 Hopf分岔     

忆阻器混沌电路产生的共存吸引子与Hopf分岔

利用荷控忆阻器和一个电感串联设计一种新型浮地忆阻混沌电路.用常规动力学分析方法研究该系统的基本动力学特性,发现系统可以产生一对关于原点对称的"心"型吸引子.将观察混沌吸引子时关注的电压、电流推广到功率和能量信号,观察到蝴蝶结型奇怪吸引子的产生.理论分析Hopf分岔行为并通过数值仿真进行验证,结果表明系统随电路参数变化能产生Hopf分岔、反倍周期分岔两种分岔行为.相对于其它忆阻混沌电路该电路采用的是一个浮地型忆阻器,并且在初始状态改变时,能产生共存吸引子和混沌吸引子与周期极限环共存现象.     

Langford系统Hopf分叉的线性反馈控制

分析Langford系统的Hopf分叉现象，并研究采用线性反馈控制方法控制该系统的Hopf分叉.从理论上推导出受控系统产生Hopf分叉的条件，给出了某些极限环的解析表达式，对该系统进行了分叉点的转移与极限环的稳定性控制.数值模拟说明本文采用的方法对Langford系统的Hopf分叉控制是有效的. 关键词： Langford系统 反馈控制 Hopf分叉 极限环     

一类耦合非线性相对转动系统的Hopf分岔控制

建立一类耦合非线性相对转动系统的动力学方程,研究系统在主共振和1∶1内共振情况下的Hopf分岔行为,设计非线性反馈控制器,控制系统Hopf分岔的发生、极限环的稳定性和幅值,数值模拟证明了该方法的有效性.     

Control of Codimension-2 Bautin Bifurcation in Chaotic Lu System

In this paper, we design a feedback controller, and analytically determine a control criterion so as to control the codimension-2 Bautin bifurcation in the chaotic Lfi system. According to the control criterion, we determine a potential Bautin bifurcation region （denoted by P） of the controlled system. This region contains the Bautin bifurcation region （denoted by Q） of the uncontrolled system as its proper subregion. The controlled system can exhibit Bautin bifurcation in P or its proper subregion provided the control gains are properly chosen. Particularly, we can control the appearance of Bautin bifurcation at any appointed point in the region P. Due to the relationship between Bantin bifurcation and Hopf bifurcation, the control scheme thereby is also viable for controlling the creation and stability of the Hopf bifurcation. In the controller, there are two terms： a linear term and a nonlinear cubic term. We show that the former determines the location of the Hopf bifurcation while the latter regulates its criticality. We also carry out numerical studies, and the simulation results confirm our analyticai predictions.     

Langford系统Hopf分岔极限环幅值控制

系统地研究了Langford 系统Hopf分岔极限环幅值非线性反馈控制问题.根据中心流形定理和规范型降维理论, 推导含控制增益项的非线性曲率系数控制公式和幅值近似解,通过数值仿真验证并绘制极限环幅控关系曲线. 所推导的控制公式为Langford 系统极限环幅值控制提供了方便有效的控制方法.     

Hopf Bifurcation in a New Four-Dimensional Hyperchaotic System

In this paper, the Hopf bifurcation in a new hyperchaotic system is studied. Based on the first Lyapunov coefficient theory and symbolic computation, the conditions of supercritical and subcritical bifurcation in the new hyperchaotic system are obtained. Numerical simulations are used to illustrate some main results.     

一类相对转动系统Hopf分岔的非线性反馈控制

研究一类非线性摩擦阻尼力作用下的相对转动系统的Hopf分岔现象,给出系统产生Hopf分岔的充要条件,提出一种非线性反馈控制方法对系统的Hopf分岔点进行转移,并控制极限环的稳定性和幅值,数值模拟说明该方法对一类相对转动系统的Hopf分岔控制是有效的. 关键词： 相对转动 非线性反馈控制 Hopf分岔 极限环     

Stochastic analysis of a Hopf bifurcation: Master equation approach

The effect of inhomogeneous fluctuations in a reaction-diffusion system exhibiting a Hopf bifurcation is analyzed using the master equation approach. A Taylor expansion of the logarithm of the stationary probability, known as the stochastic potential, is calculated. This procedure displays marked analogies with the theory of normal forms. The critical potential, reduced to its local expansion around an arbitrary point of the limit cycle, brings out the essential role played by the phase of the oscillating variables. A comparison with the Langevin analysis of Walgraefet al. [J. Chem. Phys. 78(6):3043 (1983)] is performed.     

Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

This paper is concerned with the Hopf bifurcation analysis of tumor-immune system competition model with two delays. First, we discuss the stability of state points with different kinds of delays. Then, a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points. Furthermore, under this condition, the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory. Finally, a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically. At last, some numerical experiments are given to match well with the main conclusion of this paper.     

基于状态反馈控制的无线网络拥塞控制流体流模型的Hopf分岔

针对控制无线网络拥塞控制系统中流体流模型的Hopf分岔的问题,提出一种状态反馈控制器.通过选择通信时延作为分岔参数,验证模型在加入状态反馈控制器后,①增加了分岔参数的临界值,扩大了稳定性区域,使系统的Hopf分岔延迟;②通过选择合适的参数,可以容易地改变分岔周期解的稳定性及其分岔方向.理论分析和数据仿真验证了该方法能够有效地控制系统的Hopf分岔.     

Modified Projective Synchronization of a New Hyperchaotic System via Nonlinear Control

In this paper, a nonlinear control scheme of two identical hyperchaotic Chert systems is developed to realize their modified projective synchronization. We achieve modified projective synchronization between the two identical hyperchaotic systems by directing the scaling factor onto the desired value. With symbolic computation system Maple and Lyapunov stability theory, numerical simulations are given to perform the process of the synchronization.     

Modified Projective Synchronization of a New Hyperchaotic System via Nonlinear Control

In this paper, a nonlinear control scheme of two identical hyperchaotic Chensystems is developed to realize their modified projective synchronization.We achieve modified projective synchronization between the two identicalhyperchaotic systems by directing the scaling factor onto the desired value. With symbolic computation system Maple and Lyapunov stability theory, numerical simulations are given to perform the process of the synchronization.     

Amplitude control of limit cycle in a van der Pol Duffing system

This paper applies washout filter technology to amplitude control of limit cycles emerging from Hopf bifurcation of the van der Pol--Duffing system. The controlling parameters for the appearance of Hopf bifurcation are given by the Routh--Hurwitz criteria. Noticeably, numerical simulation indicates that the controllers control the amplitude of limit cycles not only of the weakly nonlinear van der Pol--Duffing system but also of the strongly nonlinear van der Pol--Duffing system. In particular, the emergence of Hopf bifurcation can be controlled by a suitable choice of controlling parameters. Gain-amplitude curves of controlled systems are also drawn.     

Hopf bifurcation analysis and circuit implementation for a novelfour-wing hyper-chaotic system

In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare′ maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.     

Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [K. Ikeda, K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D 29 (1987) 223–235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental “period-2” mode found in Ikeda-type systems.     

Nearly Vertical Hopf Bifurcation for a Passively Q-Switched Microchip Laser

Passively Q-switched microchip lasers generate strongly pulsating intensity oscillations that emerge from a Hopf bifurcation point. We show that this bifurcation is nearly vertical and explain why strongly pulsating oscillations are immediately observed as we pass the Hopf bifurcation point. The laser dynamical problem is mathematically a singular perturbation problem which we investigate. The leading order problem is conservative and corresponds to Lotka–Volterra equations.