We study a family of quadratic vector fields in Class I \(\dot{x}=y, \dot{y}=-x-\alpha y+\mu x^2-y^2\), where \((\alpha , \mu ) \in \mathbb {R}^2\). To study the equilibria at infinity on the Poincaré disk of this system completely, we follow the method of generalized normal sectors of Tang and Zhang (Nonlinearity 17:1407–1426, 2004) and give further two new criterions, which allows us to obtain not only the qualitative properties of the equilibria but also asymptotic expressions of these orbits connecting the equilibria at infinity of this system. Further, the complete bifurcation diagram including saddle connection bifurcation curves of this system is given. Moreover, by qualitative properties of the equilibria, the nonexistence of limit cycle and rotated properties about \(\alpha \) and \(\mu \), all global phase portraits on the Poincaré disk of this system are also obtained and the number is 19.

中文翻译：

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一类二次矢量场的分叉图和全局相位肖像

我们研究I 类\（\ dot {x} = y，\ dot {y} =-x- \ alpha y + \ mu x ^ 2-y ^ 2 \）中的二次向量场族，其中\（（\ alpha，\ mu）\ in \ mathbb {R} ^ 2 \）。为了完全研究该系统的庞加莱圆盘上无穷大的平衡，我们采用了Tang和Zhang的广义法向扇形的方法（非线性17：1407–1426，2004），并进一步给出了两个新的准则，这使我们获得了不仅是平衡的定性性质，而且是这些轨道在系统无穷远处的连接的渐近表达。此外，给出了包括该系统的鞍形连接分叉曲线的完整分叉图。此外，通过平衡的定性性质，极限环的不存在和\（\ alpha \）和\（\ mu \）的旋转性质，还获得了该系统的Poincaré圆盘上的所有全局相像，其数量为19