SIAM Journal on Mathematical Analysis, Volume 53, Issue 4, Page 4031-4067, January 2021.
We consider state-dependent delay equations (SDDEs) obtained by adding delays to a planar ODE with a limit cycle. Situations of this type appear in models of several physical processes, where small delay effects are added. Even if the delays are small, they are very singular perturbations since the natural phase space of an SDDE is an infinite-dimensional space. We show that for the SDDE, there are initial values which lead to solutions similar to those of the ODE. That is, there exist a periodic solution and a two parameter family of solutions whose evolution converges to the periodic solution (in the ODE case, these are called the isochrons). The method of proof bypasses the theory of existence, uniqueness, dependence on parameters of SDDE. We consider the class of functions of time that have a well defined behavior (e.g., periodic, or asymptotic to periodic) and derive functional equations which impose that they are solutions of the SDDE. These functional equations are studied using functional analysis methods. We provide a result in “a posteriori” format: given an approximate solution of the functional equation, which has some good condition numbers, we prove that there is a true solution close to the approximate one. Thus, our result can be used to validate the results of numerical computations or formal expansions. The method of proof also leads to practical algorithms. In a companion paper, we present the implementation details and representative results. One feature of the method presented here is that it allows us to obtain smooth dependence on parameters for the periodic solutions and their slow stable manifolds without studying the smoothness of the flow (which seems to be problematic for SDDEs, for now the optimal result on smoothness of the flow is $C^1$).

中文翻译：

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常微分方程状态相关时滞摄动的参数化方法

SIAM 数学分析杂志，第 53 卷，第 4 期，第 4031-4067 页，2021 年 1 月。
我们考虑通过向具有极限环的平面 ODE 添加延迟而获得的状态相关延迟方程 (SDDE)。这种类型的情况出现在几个物理过程的模型中，其中添加了小的延迟效应。即使延迟很小，它们也是非常奇异的扰动，因为 SDDE 的自然相空间是无限维空间。我们表明，对于 SDDE，存在导致类似于 ODE 的解的初始值。也就是说，存在一个周期解和一个演化收敛到周期解的双参数解族（在 ODE 情况下，这些被称为等时线）。证明方法绕过了 SDDE 的存在性、唯一性、依赖参数的理论。我们考虑具有明确定义行为的时间函数类（例如，周期性的，或渐近到周期）并推导出函数方程，这些方程强加于它们是 SDDE 的解。使用函数分析方法研究这些函数方程。我们以“后验”格式提供结果：给定函数方程的近似解，它具有一些良好的条件数，我们证明存在接近近似解的真解。因此，我们的结果可用于验证数值计算或形式扩展的结果。证明方法也导致了实用的算法。在配套论文中，我们介绍了实现细节和代表性结果。