Control of Chaos in Mechanical System With Time Delay

本文研究了1.一类非线性时滞系统的分支集.非线性时滞系统的等效非线性化。我们构造了等价系统.时滞反馈系统的最优反馈增益，由于时滞反馈系统是由一个微分方程描述的，而微分方程通常是很难求解的，因此很难用解析方法研究这类系统。因此，很难从理论上确定合适的反馈增益。为了分析这类系统，我们已经介绍了泛函微分方程的平均方法。以往的工作表明，这种平均方法是有效的非线性时滞系统。应用这种平均法，研究了这类具有基波响应的系统的分岔集，并构造了其等效系统。并确定了该系统的一个系数，使等效系统的平均系统的行为与原系统的行为一致。通过相图和谐振曲线比较了等效系统和原系统的性能。其中，将运动方程由差分微分方程描述的系统作为原系统，将假设的无时滞系统作为等价系统。当时滞较小时，外力的角频率较大时，该系统具有简谐运动。进一步证明了该系统存在Bogdanov-Takens分岔点，并由此导出了系统的Hopf分岔曲线、鞍结分岔曲线和同宿分岔曲线.等效系统的特性与原系统的特性吻合较好.适当的反馈增益与绕不稳定周期轨道的转速之间的关系。

This paper studies1. bifurcation set of a nonlinear system with time delay.2. equivalent nonlinearization of a nonlinear system with time delay. We construct the equivalent system.3. optimal feedback gains in the time-delayed feedback system.It is known that it is difficult to investigate these systems by analytical methods, because these systems are described by a difference-differential equation which is usually difficult to solve. So, it is difficult to determine an appropriate feedback gain theoretically. To analyze this kind of a system, we have already introduced an averaging method for the functional differential equation. The previous work showed that this averaging method is effective for nonlinear system with time delay. Applying this averaging method, this paper studies bifurcation set of this kind of a system with a fundamental harmonic response, and construction the equivalent system. And, a coefficient of this system is determined that behavior of the averaged system of the equivalent system coincide with that of the original system. We compare the equivalent system with the original system by the phase portrait and resonance curve. Where, we regard the system whose motion equation is described the difference-differential equation as an original system, and an assumed system without time delay as an equivalent system.The result shows1. This system has a simple harmonic motion, when a delay is small and an angular frequency of external force is large. Furthermore, it is shown that this system has a Bogdanov-Takens bifurcation point from which the Hopf bifurcation curve, a saddle-node bifurcation curve and homoclinic bifurcation curve are derived.2. Behavior of the equivalent system is in good agreement with behavior of the original system.3. A relation between the appropriate feedback gain and the revolution around an unstable periodic orbit.