Fiedler B, Flunkert V, Georgi M, Hövel P, Schöll E
Institut für Mathematik I, FU Berlin, Arnimallee 2-6, D-14195 Berlin, Germany.
Phys Rev Lett. 2007 Mar 16;98(11):114101. doi: 10.1103/PhysRevLett.98.114101. Epub 2007 Mar 14.
We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.
我们反驳了一个经常被引用的定理，该定理声称，具有奇数个大于1的实弗洛凯乘数的周期轨道永远无法通过皮拉加斯提出的那种形式的时滞反馈控制来稳定。使用一般范式，我们证明了由亚临界霍普夫分岔产生的具有单个实不稳定弗洛凯乘数的不稳定周期轨道实际上可以被稳定。我们根据反馈控制增益的幅度和相位推导出控制矩阵的明确解析条件，并给出一个数值例子。我们的结果与物理、化学、技术和生命科学中发生亚临界霍普夫分岔的广泛系统相关。
