Abstract This article begins with a full description of the quadratic planar vector fields which display two centers. We follow the method proposed by Chengzhi Li and provide more detailed analysis of the different types of double centers using the classification: Hamiltonian, reversible, Lotka-Volterra, Q 4 , currently used for centers of quadratic planar vector fields. We also describe completely the different possible phase portraits and their Poincare compactification. We show that the double center set is a semi-algebraic set for which we give an explicit stratification (see figure 2). Then we initiate a study of the perturbations within quadratic planar vector fields of the most degenerated case which is the double Lotka-Volterra case. The perturbative analysis is made with the method of successive derivatives of return mappings. As usual, this involves relative cohomology of the first integral which is in that case a rational function. In this case, we have to deal with a kind of “relative logarithmic cohomology” already known in singularity theory. We succeed to compute the first bifurcation function by residue techniques around each centers and they differ from one center to the other.

中文翻译：

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二次双中心及其扰动

摘要 本文首先全面描述了显示两个中心的二次平面矢量场。我们遵循李成志提出的方法，并使用以下分类对不同类型的双中心进行更详细的分析：哈密顿量、可逆、Lotka-Volterra、Q 4 ，目前用于二次平面矢量场的中心。我们还完全描述了不同的可能相图及其庞加莱压缩。我们证明双中心集是一个半代数集，我们给出了明确的分层（见图 2）。然后我们开始研究最退化情况（即双 Lotka-Volterra 情况）的二次平面矢量场内的扰动。微扰分析是用返回映射的连续导数的方法进行的。照常，这涉及第一个积分的相对上同调，在这种情况下它是一个有理函数。在这种情况下，我们必须处理奇点理论中已知的一种“相对对数上同调”。我们通过围绕每个中心的残差技术成功地计算了第一个分叉函数，并且它们从一个中心到另一个中心不同。