Consider the class QS of all non-degenerate planar quadratic systems and its subclass QSE of all its systems possessing an invariant ellipse. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of \({{\mathbb {R}}}^{12}\) through its coefficients. In this paper we provide necessary and sufficient conditions for a system in QS to have at least one invariant ellipse. We give the global “bifurcation” diagram of the family QS which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or it is not a limit cycle. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for any quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle.

考虑所有非退化平面二次系统的类QS及其所有具有不变椭圆的系统的子类QSE。这是一个有趣的族，因为一方面它是由代数几何性质定义的，另一方面，它是发生极限环的一个族。注意，每个二次微分系统都可以用\（{{\ mathbb {R}}} ^ {12} \通过其系数。在本文中，我们为QS中的系统至少具有一个不变椭圆提供了必要和充分的条件。我们给出了QS族的全局“分叉”图，该图指示了椭圆的存在或不存在的位置，如果椭圆存在，则该图指示椭圆是否为极限环。该图以仿射不变多项式表示，并且在参数的12维空间中完成。该图也是用于确定任何二次系统是否具有不变椭圆以及该椭圆是否为极限环的算法。