ABSTRACT Quasi-periodic high order degenerate bifurcation theories have been well established, but works which are related to almost-periodic bifurcations seem to be very few. In this paper, we consider the almost-periodic time-dependent perturbations of one-dimensional degenerate vector field With the KAM theory and singularity theory, we show that the universal unfolding of the vector field can persist under small almost-periodic perturbation if some appropriate non-resonant conditions are satisfied, which implies strongly non-resonant invariant tori in the integrable part and all bifurcation scenario can survive under any small almost-periodic perturbation.

中文翻译：

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一维退化矢量场的几乎周期性分岔*

摘要 准周期高阶退化分岔理论已经很成熟，但与准周期分岔相关的工作似乎很少。在本文中，我们考虑了一维退化向量场的几乎周期性时间相关扰动 结合 KAM 理论和奇点理论，我们证明了矢量场的普遍展开可以在小的几乎周期性扰动下持续存在，如果一些适当的满足非共振条件，这意味着可积部分具有强非共振不变环面，并且所有分叉场景都可以在任何小的几乎周期性的扰动下幸存下来。