论文标题
随机向量场中的限制周期枚举
Limit cycle enumeration in random vector fields
论文作者
论文摘要
我们研究平面矢量场的极限循环的数量和分布，其分量函数是随机多项式。当从kostlan-shub-smale集合中采样随机多项式时，我们证明了限制循环的平均数量的下限。调查了Brudnyi [数学Annals（2001）]引入的问题，我们还考虑了在随机扰动的中心焦点附近进行计数限制周期的特殊局部设置，并且当扰动具有I.I.D。系数，我们证明了一个极限定律，表明位于半径磁盘内的极限循环的数量小于统一的磁盘几乎可以肯定地收敛于对数相关的随机单变量功率序列的真实零的数量。我们还考虑无限扰动，在这些扰动中，我们可以在模型家族的全球限制周期中获得精确的渐近造影。这些结果的证据使用了动态系统和随机分析功能的技术组合。
We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-Shub-Smale ensemble. Investigating a problem introduced by Brudnyi [Annals of Mathematics (2001)] we also consider a special local setting of counting limit cycles near a randomly perturbed center focus, and when the perturbation has i.i.d. coefficients, we prove a limit law showing that the number of limit cycles situated within a disk of radius less than unity converges almost surely to the number of real zeros of a logarithmically-correlated random univariate power series. We also consider infinitesimal perturbations where we obtain precise asymptotics on the global average count of limit cycles for a family of models. The proofs of these results use novel combinations of techniques from dynamical systems and random analytic functions.
