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。     

一类非自治位置时滞反馈控制系统的亚谐共振响应

研究了弹性轨道条件下，控制回路中位置反馈信号存在时滞的磁浮系统在亚谐轨道激励作用下的响应问题. 将动力学模型在平衡点处线性化，以时滞为分岔参数，得到了系统出现Hopf分岔的条件. 用中心流形约化方法得到了包含轨道扰动系统的Poincaré规范型. 用多尺度法从理论上推导了时滞磁浮系统的亚谐共振周期解，得到了自由振动的分岔响应方程，分析了周期解中自由振动项的存在条件，研究了控制参数和激励参数与周期解的关系. 最后用数值仿真的方法分析了时滞参数、控制参数对系统响应的影响，分析结果指出，使系统保持稳定的亚谐响应的时滞边界小于无扰动时的时滞边界，时滞参数不但可以抑制亚谐响应，还能够控制混沌的产生，而控制参数可以控制系统响应中自由振动项的出现和受迫振动的幅值，适当选择这些参数可以有效抑制亚谐振动响应. 关键词： 亚谐共振响应 位置时滞反馈控制 非自治磁浮系统 分岔     

一类准周期参激非线性相对转动动力系统的稳定性与时滞反馈控制

建立了一类含准周期参数激励和时滞反馈的相对转动非线性系统的动力学方程. 采用多尺度法求解1/2亚谐波主参数共振下的分岔响应方程,并分析了系统的稳定性. 在求解非受控系统的定常解的基础上,通过讨论系统的动力学特性,研究了准周期参数激励对系统响应的影响. 采用时滞反馈控制的方法对系统分岔和极限环(域)进行控制,数值模拟的结果表明通过改变时滞参数可以实现对系统分岔的控制,并能有效地控制极限环(域)的幅值和稳定性. 关键词： 相对转动 准周期参激 时滞反馈 极限环     

有界噪声激励下带有时滞反馈的随机Mathieu-Duffing系统的响应

研究了参数激励下带有时滞反馈的随机Mathieu-Duffing方程的主参数共振响应问题.运用多尺度方法分离了系统的快慢变量.分析了系统的分岔性质,发现调谐参数、时滞、时滞项的系数以及非线性项的强度等都可以影响系统的分岔行为,适当选择这些参数可以改变系统的分岔响应.同时,还讨论了非零解的稳定性,得到了非零解稳定的充要条件,而且发现在随机激励的带宽较小时,系统的多解现象仍然存在,分岔和跳跃现象仍会发生,数值模拟验证了理论推导的有效性. 关键词： 随机Mathieu-Duffing系统 多尺度 稳定性 分岔     

基于状态反馈参数激励系统的超谐共振分岔控制

设计了非线性参数控制器，改变了参数激励系统2 倍超谐共振时的稳态响应，减小了系统的响应幅值和消除了超谐共振时的鞍结分岔，从而消 除了跳跃和滞后现象.首先由多尺度法得到参数系统的近似频响方程，再进行分岔分析，从 而实现非线性控制的目标.通过数值模拟，说明状态反馈控制是可行的和有效的. 关键词： 参数激励 鞍结分岔 分岔控制 2倍超谐共振     

具有对应分段系统和指数系统的新混沌系统的Hopf分岔控制研究

为进一步了解一个复杂的有不稳定奇点的三维动力系统在Hopf分岔点附近的非线性特性,采用非线性控制器,提出了相应的控制系统,使得受控系统可能发生余维一、余维二和余维三的Hopf分岔.通过严格的数学推导给出了受控系统发生分岔的参数条件,证明了可控制系统在指定区域内发生退化分岔和可调控分岔的稳定性.     

非线性传送带系统的复杂分岔

讨论了一类单自由度非线性传送带系统. 首先通过分段光滑动力系统理论得出系统滑动区域的解析分析和平衡点存在性条件; 其次利用数值方法, 对系统几种类型的周期轨道进行单参数和双参数延拓, 得到系统的余维一滑动分岔曲线和若干余维二滑动分岔点, 以及系统在参数空间中的全局分岔图. 通过对系统分岔行为的研究, 反映出传送带速度和摩擦力振幅对系统动力学行为有较大影响, 揭示了非线性传送带系统的复杂动力学现象. 关键词： 传送带系统 滑动分岔 周期运动     

耦合相对转动非线性动力系统的稳定性与近似解

研究了一类含三次非线性耦合项的相对转动非线性动力系统的动力学行为. 建立了具有非线性弹性力、广义摩阻力耦合项的系统动力学方程. 运用多尺度法求解谐波激励下耦合非自治系统的近似解,通过讨论系统的主共振和内共振特性,分析了耦合项对系统响应的影响. 应用奇异性理论研究了主振稳态响应分岔方程的稳定性,得到了系统的转迁集和分岔曲线的拓扑结构. 关键词： 相对转动 非线性耦合动力系统 奇异性理论 稳定性     

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.     

相对转动非线性动力系统的稳定性与强迫激励下的近似解

研究相对转动非线性动力系统的运动稳定性.建立具有一般广义阻尼力和外扰激励的一类两质量相对转动非线性动力系统的动力学方程.研究相对转动非线性动力自治系统的稳定性，证明系统在一定条件下可发生闭轨分岔.应用多尺度法得到强迫激励下非自治系统的近似解. 关键词： 相对转动 非线性动力系统 运动稳定性 近似解     

Generalized Ginzburg-Landau equation for self-pulsing instability in a two-photon laser

A nonlinear analysis is made for a degenerate two-photon ring laser near its critical point corresponding to a self-pulsing instability by using the slaving principle and normal form theory. It turns out that the system undergoes two kinds of transitions, a usual Hopf bifurcation to a stable or unstable limit cycle and a co-dimension two Hopf bifurcation where the limit cycles disappear. An analytical criterion is given to distinguish the super-from the sub-critical bifurcation. We have also solved the equations numerically to confirm and to supplement our analytical results. In the case of super-critical bifurcation, a period-doubling bifurcation sequence to chaos is also observed with the decrease in pumping.     

A time delay control for a nonlinear dynamic beam under moving load

The bifurcation resulted from moving force may lead to instability for the system. Based on time delay feedback controller, a nonlinear beam under moving load is discussed in the case of the primary resonance and the 1/3 subharmonic resonance. The bifurcation may be eliminated or the bifurcation point's position may be changed. The perturbation method is used to obtain the bifurcation equation of the nonlinear dynamic system. The result indicates time delay feedback controller may work well on this system, but the selection of a proper time delay and its coefficient may depend on the engineering condition. This paper presents some theoretical results.     

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.     

The instability of chaotic synchronization in coupled Lorenz systems: from the Hopf to the Co-dimension two bifurcation

We investigate the Hopf bifurcation of the synchronous chaos in coupled Lorenz oscillators. We find that the system undergoes a phase transition along the Hopf instability of the synchronous chaos. The phase transition makes the traveling wave component with the phase difference φ(i)-φ(i+1)=2π/N between adjacent sites unstable. The phase transition also plays a role to relate the Hopf bifurcation with the co-dimension two bifurcation of the synchronous chaos.     

Helical solitons in vector modified Korteweg–de Vries equations

We study existence of helical solitons in the vector modified Korteweg–de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.     

一类非线性相对转动系统的组合共振分岔与混沌

研究一类具有异宿轨道的非线性相对转动系统的分岔与混沌运动. 应用耗散系统的拉格朗日方程建立一类组合谐波激励作用下非线性相对转动系统的动力学方程. 利用多尺度法求解相对转动系统发生组合共振时满足的分岔响应方程并进行奇异性分析, 得到了系统稳态响应的转迁集. 根据相对转动系统异宿轨道参数方程, 求解了异宿轨道的Melnikov函数, 并给出了系统发生Smale马蹄变换意义下混沌的临界条件. 最后采用数值方法, 通过分岔图, 最大Lyapunov指数图, 相轨迹图和庞加莱截面图研究系统参数对混沌运动的影响.     

RESONANCES OF A NON-LINEAR s.d.o.f. SYSTEM WITH TWO TIME-DELAYS IN LINEAR FEEDBACK CONTROL

The primary, superharmonic, and subharmonic resonances of a harmonically excited non-linear s.d.o.f. system with two distinct time-delays in the linear state feeback are studied. The two different time-delays are presented in the proportional feedback and the derivative feedback respectively. The method of multiple scales is utilized to obtain the first order approximation of response. The effect of the feedback gains and time-delays on the steady state responses of three types of resonances is investigated. It is found that a proper selection of the feedback gains and time-delays can enhance the control performance.