一类时滞非线性相对转动系统的Hopf分岔与周期解的稳定性

讨论了一类时滞非线性相对转动系统的Hopf分岔现象,给出了系统产生Hopf分岔的临界时滞条件,利用多尺度法研究了系统在主共振情况下时滞参数对分岔方向和周期解稳定性的影响,最后,用数值模拟对所得结论进行验证.     

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     

相位噪声诱发神经放电的单次或两次相干共振

神经元电活动可以从静息通过Hopf分岔到放电,放电频率有固定周期;也可以从静息通过鞍-结分岔到放电,放电频率接近零.在具有周期性的相位噪声作用下的Hopf分岔和鞍-结分岔点附近,都会产生相干共振.噪声的周期小于Hopf分岔点附近的放电的周期时,相位噪声可以引起神经系统产生一次相干共振,位于系统内在的固有频率附近;噪声的周期大于系统的固有周期时,相位噪声可以引起双共振,对应低噪声强度的共振产生在噪声频率附近,对应高噪声强度的共振产生在系统的固有频率附近;并对双共振的产生原因进行了解释.在鞍-结分岔点附近,无论噪声的周期是大是小,都只会引起一次共振,研究结果不仅揭示了相位噪声作用下平衡点分岔点相干共振的动力学特性和对应于两类分岔的两类神经兴奋性的差别,还对近期的相位噪声诱发产生单或双共振的不同研究结果给出了解释.     

四维Qi系统零平衡点的Hopf分岔反控制

通过线性与非线性状态反馈, 实现了对四维Qi系统零平衡点的Hopf分岔反控制.首先确定产生Hopf分岔的线性控制项,得到线性控制增益的选取原则.然后,利用稳定性分析,借助于对线性受控Qi系统的Jordan标准型的直接控制以及适当的变换,确定影响Hopf分岔稳定性的非线性控制项,得到非线性控制增益的选取原则.针对所考虑分岔参数的不同,给出不同的控制方案.最后通过数值模拟验证了理论分析结果的正确性. 关键词： Qi系统 Hopf分岔 反控制 稳定性     

非磁化等离子体Hopf分岔处随机共振实验

在非磁化的直流放电等离子体中,利用系统内禀的周期驱动力与外加的高斯白噪声的协同作用,成功地在系统动力学轨道Hopf分岔的不动点一侧,观察到了周期信号的信噪比被噪声改善的随机共振现象.一维过阻尼系统的随机共振模型可以解释所观察的实验结果.比较实验结果与理论模型可知,系统动力学轨道发生Hopf分岔的原因是系统双稳态之间势垒的变化 在Hopf分岔的极限环一侧观察不到随机共振现象,其原因是此时系统已经处于超阈值周期驱动的状态     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响。     

线性延时反馈Josephson结的Hopf分岔和混沌化

考虑线性延时反馈控制下电阻-电容分路的Josephson结,运用非线性动力学理论分析了受控系统平凡解的稳定性.理论分析表明,随着控制参数的改变,系统的稳定平凡解将会通过Hopf分岔失稳,并推导了发生Hopf分岔的临界参数条件.对不同参数条件下受控系统的动力学进行了数值分析.结果显示,系统由Hopf分岔产生的稳定周期解,将进一步通过对称破缺分岔和倍周期分岔通向混沌. 关键词： 约瑟夫森结 线性延时反馈 Hopf分岔 混沌     

基于Washout滤波器的Rössler系统Hopf分岔控制

针对Rössler系统平衡点的Hopf分岔,以Washout滤波器为控制器,详细讨论了控制器参数对Hopf分岔点位置、分岔类型以及周期解振幅的控制问题.首先根据Routh-Hurwitz判据计算了受控系统的参数空间稳定域,找出了对应的Hopf分岔边界,并由此分析了滤波器时间常数、线性控制增益对分岔点位置的影响.然后,引入Normal Form直接法方便地求出系统Hopf分岔Normal Form系数,由此确定出改变分岔类型和周期解振幅的控制器非线性增益选择原则.最后用数值计算验证了本文的结论. 关键词： Rö ssler系统 Washout滤波器 Hopf分岔 Normal Form     

基于Washout滤波器的Rssler系统Hopf分岔控制

针对Rssler系统平衡点的Hopf分岔,以Washout滤波器为控制器,详细讨论了控制器参数对Hopf分岔点位置、分岔类型以及周期解振幅的控制问题.首先根据Routh-Hurwitz判据计算了受控系统的参数空间稳定域,找出了对应的Hopf分岔边界,并由此分析了滤波器时间常数、线性控制增益对分岔点位置的影响.然后,引入NormalForm直接法方便地求出系统Hopf分岔Normal Form系数,由此确定出改变分岔类型和周期解振幅的控制器非线性增益选择原则.最后用数值计算验证了本文的结论.     

基于Washout滤波器的R(o)ssler系统Hopf分岔控制

针对R(o)ssler系统平衡点的Hopf分岔,以Washout滤波器为控制器,详细讨论了控制器参数对Hopf分岔点位置、分岔类型以及周期解振幅的控制问题.首先根据Routh-Hurwitz判据计算了受控系统的参数空间稳定域,找出了对应的Hopf分岔边界,并由此分析了滤波器时间常数、线性控制增益对分岔点位置的影响.然后,引入Normal Form直接法方便地求出系统Hopf分岔Normal Form系数,由此确定出改变分岔类型和周期解振幅的控制器非线性增益选择原则.最后用数值计算验证了本文的结论.     

Hopf bifurcation control for a coupled nonlinear relative rotation system with time-delay feedbacks

This paper investigates the Hopf bifurcations resulting from time delay in a coupled relative-rotation system with time- delay feedbacks. Firstly, considering external excitation, the dynamical equation of relative rotation nonlinear dynamical system with primary resonance and 1：1 internal resonance under time-delay feedbacks is deduced. Secondly, the averaging equation is obtained by the multiple scales method. The periodic solution in a closed form is presented by a perturbation approach. At last, numerical simulations confirm that time-delay theoretical analyses have influence on the Hopf bifurcation point and the stability of periodic solution.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

Hopf bifurcation control of a Pan-like chaotic system

This paper is concerned with the Hopf bifurcation control of a modified Pan-like chaotic system. Based on the Routh-Hurwtiz theory and high-dimensional Hopf bifurcation theory, the existence and stability of the Hopf bifurcation depending on selected values of the system parameters are studied. The region of the stability for the Hopf bifurcation is investigated.By the hybrid control method, a nonlinear controller is designed for changing the Hopf bifurcation point and expanding the range of the stability. Discussions show that with the change of parameters of the controller, the Hopf bifurcation emerges at an expected location with predicted properties and the range of the Hopf bifurcation stability is expanded. Finally,numerical simulation is provided to confirm the analytic results.     

负能模式在驱动漂移波非线性不稳定性中的作用(Ⅱ)——与正能模式交换,“回避交叉”和Hopf分岔

在正弦波驱动的非线性漂移波中,当驱动波频率和强度改变时,一个负能模式与另一个正能模式的复本征值可能出现交叉和“回避交叉”,在出现“回避交叉”的同时,这两个模式的地位发生了某种交换。正负能量两个模式之间的这种非线性共振在弱耗散下可能引起Hopf分岔。     

Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance

This study investigates the two-to-one internal resonance of the shallow arch with both ends elastically constraining, and the primary resonance case is considered. The full-basis Galerkin method and the multi-scale method are applied to obtain the modulation equations. It is shown that the natural frequencies of the first two modes cross/avoid to each other when the stiffness of elastic supports at two ends is the same/different. Moreover, the nonlinear modal interactions between these two modes may not/may be activated. The force/frequency-response curves are employed to explore the nonlinear response of the elastically supported shallow arch. The saddle-node bifurcation points and Hopf bifurcation points are observed in these cases. Moreover, the dynamic solutions, i.e., the periodic solution, quasi-periodic solution and chaotic solution are discussed. The numerical simulations are used to illustrate the route to chaos via period-doubling bifurcation.