论文标题
希尔伯特（Hilbert
A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori
论文作者
论文摘要
希尔伯特（Hilbert）的第16个问题，关于给定程度$ m $的平面多项式矢量场的最大限制循环数量，一直是矢量场定性理论中最重要的驱动力之一。增加尺寸，人们不能期望存在有限的上限，例如，给定度量$ m $的$ 3 $ d多项式矢量字段。在这里，作为在$ 3 $ d空间中此类问题的扩展，我们调查了$ 3 $ d多项式矢量字段中隔离的不变托里的数量。在这种情况下，鉴于天然数量$ m $，我们用$ n（m）$表示的上限，用于$ 3 $ d多项式矢量$ m $ $ m $的隔离不变托里的数量。基于最近开发的平均方法来检测不变的Tori，我们的第一个主要结果提供了一种机制，可以从给定平面差异矢量字段中使用$ H $ H $双曲线限制限制的给定平面差分矢量字段的数字$ h $ $ h $ h $ h $ h $ h $ h $ h $ h $ bundiale vor fibs提供机制。我们研究数字$ n（m）$的机制的强度在于以下事实：构建的$ 3 $ d差分矢量场是多项式的，只要给定的平面差异矢量场是多项式的。因此，我们的第二个主要结果在$ n（m）$的下界建立了一个下限，用于平面多项式矢量的双曲线极限周期，度$ [m/2] -1 $。基于最后一个结果，我们应用了由于Christopher＆Lloyd而导致的方法，以表明$ n（m）$的生长速度高达$ m^3/128 $。最后，上述问题还针对较高维度多项式矢量场进行了制定。
Hilbert's 16th Problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree $m$, has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, $3$D polynomial vector fields of a given degree $m$. Here, as an extension of such a problem in the $3$D space, we investigate the number of isolated invariant tori in $3$D polynomial vector fields. In this context, given a natural number $m$, we denote by $N(m)$ the upper bound for the number of isolated invariant tori of $3$D polynomial vector fields of degree $m$. Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing $3$D differential vector fields with a number $H$ of normally hyperbolic invariant tori from a given planar differential vector field with $H$ hyperbolic limit cycles. The strength of our mechanism in studying the number $N(m)$ lies in the fact that the constructed $3$D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for $N(m)$ in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree $[m/2]-1$. Based on this last result, we apply a methodology due to Christopher & Lloyd to show that $N(m)$ grows as fast as $m^3/128$. Finally, the above-mentioned problem is also formulated for higher dimensional polynomial vector fields.
