SIAM Review, Volume 63, Issue 3, Page 625-637, January 2021.
For students in applied mathematics courses, the phenomenon of delay-induced stability and instability offers exciting educational opportunities. Exploration of the onset of instability in delay differential equations (DDEs) invites a blend of analysis (real, complex, and functional), algebra, and computational methods. Moreover, stabilization of unstable but “desirable” equilibria using delayed feedback is of high importance in science and engineering. The primary challenge in classifying the stability of equilibria of DDEs lies in the fact that characteristic equations are transcendental. Here, we survey two approaches for understanding the stability of equilibria of DDEs. The first approach uses a functional analytic framework, a departure from the more familiar textbook methods based upon characteristic equations and completeness-type arguments. The second approach uses Pontryagin's generalization of the Routh--Hurwitz conditions. We apply the latter approach to illustrate how the deliberate introduction of a second time delay in a single-delay differential equation can stabilize an otherwise unstable equilibrium.

中文翻译：

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理解延迟引起的稳定性和不稳定性的方法

SIAM 评论，第 63 卷，第 3 期，第 625-637 页，2021 年 1 月。
对于应用数学课程的学生来说，延迟引起的稳定和不稳定现象提供了令人兴奋的教育机会。探索延迟微分方程 (DDE) 中的不稳定性引发了分析（实数、复数和泛函）、代数和计算方法的混合。此外，使用延迟反馈稳定不稳定但“理想”的平衡在科学和工程中非常重要。对 DDE 的平衡稳定性进行分类的主要挑战在于特征方程是超越的。在这里，我们调查了两种理解 DDE 平衡稳定性的方法。第一种方法使用功能分析框架，与基于特征方程和完整性类型参数的更熟悉的教科书方法不同。第二种方法使用 Pontryagin 对 Routh--Hurwitz 条件的概括。我们应用后一种方法来说明在单延迟微分方程中故意引入第二个时间延迟如何稳定否则不稳定的平衡。