AI 中文总结
该研究证明弱法双曲不变环面在小时间周期扰动下的持久性，建立平均原理并引入保多项式性构造，将希尔伯特第16问题的计数推广到高维不变环面，证明相关数量随次数多项式增长且高余维时无界。
AI 中文摘要
我们证明了在小时间周期扰动下吸引型弱法双曲不变环面的持久性定理，该定理将近期关于弱法双曲极限环的延拓结果推广到任意维度的不变环面，为其检测提供了通用分析框架。作为应用，我们建立了平均原理，表明平均系统的吸引型法双曲不变d-环面可生成原非自治系统的吸引型法双曲不变(d+1)-环面。我们还引入了保多项式性的构造，同时提升相空间维度与吸引型法双曲不变环面的维度，对给定次数多项式向量场的余维1法双曲不变环面的最大数量给出递归下界，将希尔伯特第16问题的计数方面推广到高维不变环面。特别地，我们证明该数量随次数至少多项式增长，且对于高余维不变环面会变为无界。
英文摘要
We prove a persistence theorem for attracting weakly normally hyperbolic invariant tori under small time-periodic perturbations. The theorem extends recent continuation results for weakly normally hyperbolic limit cycles to invariant tori of arbitrary dimension, providing a general analytical framework for their detection. As an application, we establish an averaging principle showing that attracting normally hyperbolic invariant $d$-tori of the averaged system give rise to attracting normally hyperbolic invariant $(d+1)$-tori of the original non-autonomous system. We further introduce a polynomiality-preserving construction that simultaneously lifts the dimensions of the phase space and the attracting normally hyperbolic invariant tori, yielding recursive lower bounds for the maximal number of codimension-$1$ normally hyperbolic invariant tori of polynomial vector fields of a given degree, thereby extending the counting aspect of Hilbert's sixteenth problem to higher-dimensional invariant tori. In particular, we prove that this number grows at least polynomially with the degree and becomes unbounded for invariant tori of higher codimension.
