We consider several models of state dependent delay differential equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure in celestial mechanics.Truncations of these power series can be taken as input for a posteriori theorems, that show that near the approximate solutions there are true solutions. In this way, we hope that one can construct a catalogue of solutions for SDDEs, bypassing the need of a systematic theory of existence and uniqueness for all initial conditions.

中文翻译：

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状态相关延迟方程的准周期解的延迟展开

我们考虑状态依赖的延迟微分方程（SDDE）的几种模型，其中延迟受一个小的参数影响。由于方程的性质发生变化，因此这是非常奇异的扰动。在某些情况下，我们构造形式幂级数，以逐级求解SDDE。这些序列是时间的准周期函数。这与天体力学中的Lindstedt过程非常相似。这些幂级数的截断可以作为后验定理的输入，表明后验定理附近存在真解。通过这种方式，我们希望人们可以绕开所有初始条件下关于存在性和唯一性的系统性理论的需求，为SDDE构造一套解决方案。