随机向量场中的极限环计数
Limit cycle enumeration in random vector fields
AI总结：
该研究针对随机多项式平面向量场的极限环问题，证明了Kostlan-Shub-Smale系综下平均极限环数的下界，得到扰动中心焦点附近极限环数的极限定律与无穷小扰动下的精确渐近，结合动力系统与随机解析函数技术推进了相关计数问题研究。
摘要：
我们研究分量函数为随机多项式的平面向量场的极限环的数量与分布。我们证明了当随机多项式从Kostlan-Shub-Smale系综（科斯特兰-舒布-斯梅尔系综）中采样时，平均极限环数量的一个下界。针对Brudnyi在《数学年刊》（2001）中提出的问题，我们还考虑了一种特殊的局部情形：计数随机扰动的中心焦点附近的极限环；当扰动具有独立同分布（i.i.d.）系数时，我们证明了一条极限定律，表明位于半径小于1的圆盘内的极限环数量几乎必然收敛于一个对数相关随机一元幂级数的实零点数量。我们还研究了无穷小扰动情形，针对一族模型得到了全局平均极限环数量的精确渐近结果。这些结果的证明创新性地结合了动力系统与随机解析函数领域的技术。
英文摘要：
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.
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