可逆二次中心在其周期环外边界的临界性
The criticality of reversible quadratic centers at the outer boundary of its period annulus
摘要：
本文研究可逆二次中心\begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y, \end{equation*}的周期函数，其中$\np=(D,F)\in\R^2.$将向量场紧化为$\Sc^2$，周期环的边界有两个连通分量，中心本身和一个多圈.我们分别称它们为周期环的内边界和外边界。我们感兴趣的是多环$\out_\np$在外边界的临界周期轨道的分支。临界周期是周期函数的一个孤立的临界点。周期函数在外边界的临界性是$X_\np$的临界周期轨道的最大数量，这些轨道在豪斯多夫意义上趋向于$\out_{\np_0}$，作为$\np\to\np_0.$。我们的主要结果（定理A）表明，在外部边界的临界性是最多为2的所有$\np=(D,F)\in\R^2$以外的部分$\{-1\}\times [0,1]$和$\{0\}\times [0,2]$。关于从内边界的分叉，奇科内和雅各布斯在他们的开创性论文中证明了所有$\np\in\R^2.$的上界为2。在本文中，技术是不同的，因为当周期函数解析地延伸到中心时，它没有平滑地延伸到多圈。我们表明，周期函数有一个渐近展开附近的polycycle与其余部分是一致平坦的关于$\np$和其中的主要部分是给定的单式规模包含变形的对数。更准确地说，定理A通过获得四阶渐近展开式并计算其系数而得出，这些系数在$\np$中不是多项式而是超越的。
英文摘要：
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.
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