Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time-reversal symmetry, and such high accuracy has not been obtained in more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying chaotic orbits, island chains, and rotational invariant circles that does not rely on these symmetries. We use Chirikov’s standard map as our test case, and also demonstrate that our methods apply to three other, well-studied cases.

保区图的旋转不变圆是KAM tori的重要且经过充分研究的例子。约翰·格林（John Greene）推测，局部最鲁棒的旋转圆具有高贵的旋转数，即具有连续分数的尾部为1的连续分数，而在这些圆中，最鲁棒的旋转圆具有金色的平均旋转数。这些猜想的精确数值确认依赖于具有时间反转对称性的地图，而在更一般的地图中尚未获得如此高的准确性。在本文中，我们开发了一种基于加权Birkhoff平均数的方法，该方法可以识别不依赖于这些对称性的混沌轨道，岛链和旋转不变圆。我们使用Chirikov的标准图作为测试用例，并且还证明了我们的方法适用于其他三个经过充分研究的案例。