Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localize all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations.

确定微分方程的平面多项式系统可以有多少个极限环是一个非常困难的问题。主要困难之一是极限环可能存在于不同尺度的区域内。这使得数值探索非常难以执行，需要高精度计算，而必要的精度是事先未知的。使用严格的计算，我们可以动态确定所需的精度，并定位给定系统的所有极限环。我们证明松岭平面二次多项式微分方程组正好有四个极限环。此外，我们使用基于区间算法的严格计算方法为这些极限环的位置给出了精确的界限。