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首页> 外文期刊>Journal of dynamics and differential equations >Delayed Feedback Control of a Delay Equation at Hopf Bifurcation
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Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

机译：Hopf分支时滞方程的延迟反馈控制。

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摘要

We embark on a case study for the scalar delay equation with odd nonlinearity f, real nonzero parameters , and three positive time delays . We assume supercritical Hopf bifurcation from in the well-understood single-delay case . Normalizing , branches of constant minimal period are known to bifurcate from eigenvalues at , for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For the branch k of constant period persists as a solution, for any and . Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions of controls and delays such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for in the limit of large k. The only two regions which we were able to detect, in this setting, required delays near 1, controls b near , and were of very small area of order . Our analysis is based on a 2-scale covering lift for the frequencies involved.
机译：我们针对具有奇数非线性f，实数非零参数和三个正时滞的标量延迟方程展开案例研究。我们假设在充分理解的单延迟情况下发生超临界霍普夫分支。对于任何非负整数k，最小化恒定周期不变的分支已知会从的特征值分叉。在局部分支k处，不稳定维数为k。对于任意大的不稳定维k，我们获得了这种分支的稳定性。对于任何和，恒定周期的分支k都作为解持续存在。实际上，由b控制的延迟反馈项在分支k上消失了：反馈控制在那里是无创的。遵循Pyragas的思想，我们寻求控制和延迟的参数区域，以使分支k在Hopf分叉处局部稳定。我们确定在大k的极限内的严格展开。在这种情况下，我们能够检测到的仅有的两个区域要求在1附近有延迟，控制b在附近，并且有很小的有序区域。我们的分析基于所涉及频率的2阶覆盖提升。

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