We classify the phase portraits in the Poincaré disc of the differential equations of the form \(x^{\prime } = -y + x f(x,y)\), \(\dot y =x + y f(x,y)\) where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2,3,4,5, and f has only simple zeroes. We also provide some general results on these uniform isochronous centers for all n ≥ 2. All our results have been revised by the program P4; see Chaps. 9 and 10 of Dumortier et al. (UniversiText, Springer-Verlag, New York, 2006).

中文翻译：

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具有均匀非线性的等时同步中心的相图

我们在Poincaré圆盘中以\（x ^ {\ prime} = -y + xf（x，y）\）形式，\（\ dot y = x + yf（x，y ）\）其中˚F（X，ÿ）是度的均匀多项式ñ - 1时ñ = 2,3,4,5，和˚F仅具有简单零。我们还针对所有n≥2在这些统一的等时中心上提供了一些一般结果。参见章节。Dumortier等人的第9和10页。（UniversiText，Springer-Verlag，纽约，2006年）。