论文标题
定期分离和二分法中的可定定ODES解决方案与Hardy
Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy
论文作者
论文摘要
我们为ODES系统的解决方案$ y'= f（x，y）$介绍了定期分离的概念，其中f可以在多项式界限的O-nimal结构中定义，$ y =（y__1，y_2）$。给定一对具有平坦接触的解决方案，我们证明，如果其中一个具有常规分离的属性，则该对交错或生成一个强壮的场。我们将此结果适应具有可定义系数的三维矢量场的轨迹。在真正的分析矢量场的特殊情况下，它改善了F. cano，R。Moussu和第三作者获得的某些积分铅笔的二分法/分离。在这种情况下，我们表明，具有常规分离属性和渐近曲线渐近曲线的一组轨迹永远不会是空的，并且由包含曲线的最小尺寸的亚分析集表示。最后，我们展示了如何使用J.-P。引入的所谓（SAT）属性来构建与亚分析集的正式不变曲线的示例。 Rolin，R。Shaefke和第三作者。
We introduce a notion of regular separation for solutions of systems of ODEs $y'=F(x,y)$, where F is definable in a polynomially bounded o-minimal structure and $y = (y_1,y_2)$. Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property introduced by J.-P. Rolin, R. Shaefke and the third author.
