论文标题
可逆二次中心在其周期环的外边界的关键性
The criticality of reversible quadratic centers at the outer boundary of its period annulus
论文作者
论文摘要
本文介绍了可逆二次中心的期间函数\ begin {equation*} x _ {\ np} = - y（1-x）\ partial_x+（x+dx^2+dx^2+fy^2）\ partial_y，\ partial_y，\ end end end end eNd {qore*} $ \ sc^2 $，周期环的边界具有两个连接的组件，中心本身和一个多循环。我们将它们称为周期环的内部和外边界。我们对外界的polycycle $ \ out_ \ np $的关键周期轨道分叉感兴趣。关键时期是周期函数的孤立关键点。外边界的周期功能的关键性是$ x_ \ np $的关键周期性轨道数量的最大数量是$ \ out _ {\ np_0} $在hausdorff sense中的最大数量，即$ \ np \ to \ np_0。我们的主要结果（定理A）表明，对于所有$ \ np =（d，f）\ in \ r^2 $，外部边界的关键性最多是2 $ in \ r^2 $外部的$ \ { - { - 1 \} \ times [0,1] $和$ \ \ \ \ {0 \} \} \} \ times [0,2] $。关于内部边界的分叉，Chicone和Jacobs在开创性的论文中证明了一个问题，即所有$ \ np \ in \ r^2 in \ r^2的上限为2。我们表明，周期函数在多囊附近具有渐近膨胀，其余的相对于〜$ \ np $均匀平坦，而主要部分则以对数变形为单一尺度给出。更确切地说，定理〜a遵循的是，将渐近扩展到第四顺序并计算其系数，而该系数在〜$ \ np $但超越性的情况下不是多项式。
This paper deals with the period function of the reversible quadratic centers \begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y, \end{equation*} where $\np=(D,F)\in\R^2.$ Compactifying the vector field to $\Sc^2$, the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle $\out_\np$ at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of $X_\np$ that tend to $\out_{\np_0}$ in the Hausdorff sense as $\np\to\np_0.$ This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all $\np=(D,F)\in\R^2$ outside the segments $\{-1\}\times [0,1]$ and $\{0\}\times [0,2]$. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all $\np\in\R^2.$ In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to~$\np$ and where the principal part is given in a monomial scale containing a deformation of the logarithm. More precisely, Theorem~A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in~$\np$ but transcendental.
