一类简化的时滞半导体激光方程的Hopf分岔

研究一类简化的时滞半导体激光方程的稳定性和Hopf分岔.以时滞量为参数,分析系统线性化方程零解的稳定性,给出系统产生Hopf分岔临界时滞表达式,最后用数值模拟对结论进行验证.     

一类时滞SIR传染病模型的稳定性与Hopf分岔分析

研究了一类具有时滞及非线性发生率的SIR传染病模型．首先利用特征值理论分析了地方病平衡点的稳定性,并以时滞为分岔参数,给出了Hopf分岔存在的条件．然后,应用规范型和中心流形定理给出了关于Hopf分岔周期解的稳定性及分岔方向的计算公式．最后,用Matlab软件进行了数值模拟．     

时滞速度反馈对强迫自持系统动力学行为的影响

研究强迫自持振动系统因时滞反馈产生的主共振解及其分岔．通过对强迫非自治系统的时滞反馈控制,得到所要研究的数学模型．讨论对应的线性化系统使平凡平衡态失稳出现周期解的稳定性临界条件．特别关注主共振及分岔．结果表明,稳定的主共振解随着时滞的变化周期性地出现在系统中．同时,也给出了不稳定的主共振关于时滞变化的区域,在理论方面给出了系统出现概周期运动的时滞区域．数据模拟证实了理论结果．     

具时滞反馈金融系统的动力学分析与混沌控制

研究了时滞反馈对金融系统的动力学行为的影响.以时滞为分支参数,研究了系统平衡点的局部稳定性,并发现当时滞经过一系列临界值时,系统在平衡点附近经历Hopf分支和Hopf-zero分支.然后,应用规范型方法和中心流形理论得到决定分支周期解性质的详细公式.通过设计合适的反馈增益和时滞,混沌振荡可以控制为稳定的平衡点或周期轨.最后,数值模拟验证了理论结果.     

单时滞类Chen系统Hopf分岔分析

根据非线性动力学理论,以一类新的单时滞Chen系统为分析对象,针对其平衡点的稳定性和Hopf分岔参数等问题进行研究.根据Routh-Hurwitz判据分析了其平衡点的稳定性,通过计算得到单时滞Chen系统特征根的分布,进一步分析得出系统在零平衡点附近是渐进稳定的.结合Hopf分岔理论,运用特征根的分布结果,确定出系统发生Hopf分岔的时滞参数,并给出Hopf分岔条件.通过多组实验仿真验证了理论分析的正确性.     

以滞量为参数的广义Liénard方程的Hopf分支

本文讨论广义Lienard方程的Hopf分支问题首先指出文［3］的错误，并分析了时滞对周期的影响，估计出k＝－f（O）可取多少个不同的值使广义Lienard方程有周期解．然后考虑以时滞r为参数的Hopf分支问题，得到了Hopf分支值及分支方向，并估计出时滞r可取多少个不同的值使方程有周期解，再运用Hassard“规范形”方法，给出了计算以滞量为参数的Lienard方程的Hopf分支公式，利用该公式，能判断周期解的稳定性井得到周期解的近似表达式．     

一类具有时滞的商业周期模型的分支控制

应用一种基于时滞反馈的分支控制方法,对一类具有时滞的商业周期模型的Hopf分支进行了控制.以时滞作为分支参数,当时滞达到某个阈值时,无控系统失去稳定性,Hopf分支产生:将时滞反馈控制器引入无控系统,可以使Hopf分支延迟发生或消失.数值仿真的结果验证了该方法的有效性.     

一类具有时滞的商业周期模型的分支控制北大核心

应用一种基于时滞反馈的分支控制方法,对一类具有时滞的商业周期模型的Hopf分支进行了控制.以时滞作为分支参数,当时滞达到某个阈值时,无控系统失去稳定性,Hopf分支产生:将时滞反馈控制器引入无控系统,可以使Hopf分支延迟发生或消失.数值仿真的结果验证了该方法的有效性.     

时滞反馈作用下ENSO充电振子模型的分岔分析

通过数学变换将一类含有时滞反馈机制的ENSO充电振子模型转换成时滞Van der PolDuffing方程,并以此为基础来研究该ENSO系统的零解稳定性、Hopf分岔和极限环等动力学特征.用平均法分析了其零解的稳定性与时滞效应的强度,其和时间都有关系,讨论了时滞负反馈对ENSO振荡的影响并通过简单数值模拟验证理论分析的结果.     

非自治线性时滞微分方程的扰动全局吸引性及一致稳定性

本文研究非自治线性时滞微分方程的扰动全局吸引性及一致稳定性，主要讨论了线性部分为周期系数与非周期系数的两种情形，获得了关于零解全局吸引及一致稳定的充分条件．推广了文献中线性部分为自治情形的结果。     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

具有输入时滞的时滞关联不确定系统的鲁棒分散控制

研究了一类同时具有输入时滞以及不确定参数的时滞关联大系统的稳定性问题.基于所谓的还原法,给出一种新的状态反馈控制器的设计方法,这种方法的不同之处在于利用了时延的大小以及反馈控制的历史信息.根据Lyapunov稳定性理论得到了系统在控制器作用下稳定的充分条件,所有条件都化成可解的标准LMIs(Linear matrix inequalities)形式.文章最后给出了一个数值例子说明本文结果的可行性.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

Robust stabilization for uncertain switched impulsive control systems with state delay: An LMI approach

This paper deals with the problem of robust H_∞ state feedback stabilization for uncertain switched linear systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. In addition, the impulsive behavior is introduced into the given switched system, which results a novel class of hybrid and switched systems called switched impulsive control systems. Using the switched Lyapunov function approach, some sufficient conditions are developed to ensure the globally robust asymptotic stability and robust H_∞ disturbance attenuation performance in terms of certain linear matrix inequalities (LMIs). Not only the robustly stabilizing state feedback H_∞ controller and impulsive controller, but also the stabilizing switching law can be constructed by using the corresponding feasible solution to the LMIs. Finally, the effectiveness of the algorithms is illustrated with an example.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

Parametric resonance in the Rayleigh–Duffing oscillator with time-delayed feedback

We investigate the principal parametric resonance of a Rayleigh–Duffing oscillator with time-delayed feedback position and linear velocity terms. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase of the oscillator. We study the effects of the frequency detuning, the deterministic amplitude, and the time-delay on the dynamical behaviors, such as stability and bifurcation associated with the principal parametric resonance. Moreover, the appropriate choice of the feedback gain and the time-delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time-delay can broaden the stable region of the non-trivial steady-state solutions and enhance the control performance. Theoretical stability analysis is verified through a numerical simulation.     

一类离散不确定线性时滞系统的鲁棒镇定

研究具有时变不确定参数的离散线性时滞系统的鲁棒控制问题,其中不确定性满足匹配条件,利用Ｌｙａｐｕｎｏｖ确定性理论,提出了鲁棒稳定性控制器一种新的设计方法,得到了这类离散不确定线性时滞系统可鲁棒镇定的充分条件。     

State feedback and output feedback control of a class of nonlinear systems with delayed measurements

This paper is concerned with the problem of state feedback and output feedback control of a class of nonlinear systems with delayed measurements. This class of nonlinear systems is made up of continuous-time linear systems with nonlinear perturbations. The nonlinearity is assumed to satisfy a global Lipschitz condition and the time delay is assumed to be time-varying and have no restriction on its derivative. On the basis of the Lyapunov–Krasovskii approach, sufficient conditions for the existence of the state feedback controller and the output feedback controller are derived in terms of linear matrix inequalities. Methods of calculating the controller gain matrices are also presented. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

On a case of resonance for nonlinear systems : PMM vol. 38, n 6, 1974, pp. 986–990

We examine the case of resonance for systems close to nonlinear systems, admitting of a parametric periodic solution. Among the eigenvalues of the matrix of the system's linear part there are zero and pure imaginary ones. We have proved (under certain conditions) the absence of a periodic solution for the original system for which the generating solution is trivial.     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.