Lorenz方程不稳定周期轨道的时滞反馈镇定
Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control
摘要：
多年来，人们认为，一个不稳定的周期轨道与奇数个真正的Floquet乘数大于1不能稳定的时间延迟反馈控制机制的皮拉古斯。Fiedler等人最近的一篇论文使用了次临界Hopf分支的规范形式给出了这个定理的一个反例。以Lorenz方程为例，证明了Fiedler等人提出的Hopf规范型稳定化机制同样适用于高维动力系统中次临界Hopf分岔所产生的不稳定周期轨道.我们的分析集中在一个特定的余维两个分支，捕获的稳定机制，在霍普夫正规形的例子，我们表明，相同的余维两个分支是存在于洛伦兹方程与适当选择的皮拉古斯型时间延迟反馈。这个例子提出了一个可能的策略，选择反馈增益矩阵的Pyragus控制不稳定的周期轨道，产生从一个稳定的平衡亚临界Hopf分叉。特别是，我们的反馈增益矩阵的选择是由Fiedler等人的例子，它的工作范围很广的参数，尽管事实上，一个中心流形减少的高维问题不会导致他们的模型问题。
英文摘要：
For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragus. A recent paper by Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragus-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragus control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.
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