Hilbert第16问题与初等多圈的圈性估计
The Hilbert 16-th problem and an estimate for cyclicity of an elementary polycycle
摘要：
Hilbert-Arnold（HA）问题是由Hilbert第16问题启发而来的，它证明了对于一般的k-参数光滑向量场族{ \dot x=v（x，\eps）}_{ \eps \in B^k}在二维球面S ^2上有一致有界数目的极限环（孤立周期解），用LC（\eps）表示，在参数\eps上，即对于某个K，max_{\eps \in B^k} LC（\eps）<= K < \infty。HA问题可以归结为所谓的局部Hilbert-Arnold（LHA）问题。设一般k参数族{ \dot x=v（x，\eps）}_{ \eps \in B^k}，x \in S^2对于某个参数\eps^* \in B^k有一个多环（分界线多边形）gamma，它由作为顶点的平衡点和作为边的连接分界线组成。LHA问题是对域\dot x=v（x，\eps），估计在伽玛邻域内可能产生的极限环的最大个数B（k），其中\eps接近于\eps^* .
英文摘要：
Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of limit cycles (isolated periodic solutions), denoted by LC(\eps), over the parameter \eps, i.e. max_{\eps \in B^k} LC(\eps) <= K < \infty for some K. The HA problem can be reduced to so-called Local Hilbert-Arnold (LHA) problem. Suppose that a generic k-parameter family {\dot x=v(x,\eps)}_{\eps \in B^k}, x\in S^2 for some parameter \eps^*\in B^k has a polycycle (separatrix polygon) gamma consisting of equilibrium points as vertices and connecting separatrices as sides. LHA problem is to estimate B(k)--- the maximal number of limit cycles that can be born in a neighbourhood of gamma for a field \dot x=v(x,\eps), where \eps is close to \eps^*.
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