论文标题
在平面阴影曲线到封闭的逃逸曲线
On Planar Shadowing Curves to Closed Escaping Curves
论文作者
论文摘要
我们介绍了一种名为“阴影问题”的新动力系统模型，在该模型中，一个阴影师通过始终盯着并保持与他的距离来追逐Escaper。当Escaper沿着平面封闭曲线运行时，我们将其与降低的阴影方程相关联，并表明它仅取决于逃逸曲线的几何形状。引入了两个称为临界阴影距离和转弯距离的概念，以表征不同的动态行为。我们表明，根据阴影距离，平面封闭的逃逸曲线可能具有不同类型的阴影曲线，包括周期性，谐波和巨像的曲线。当阴影距离较大时，发现尖峰类型的奇异性。在分析和数值上进行详细研究，对逃逸圆圈的阴影曲线进行了检查。最后，我们猜测，对于典型的逃逸曲线而言，关键的阴影距离和转弯阴影距离是一致的。
We introduce a new dynamical system model called the shadowing problem, where a shadower chases after an escaper by always staring at and keeping the distance from him. When the escaper runs along a planar closed curve, we associate to the reduced shadowing equations the rotation number, and show that it depends only on the geometry of the escaping curve. Two notions called the critical shadowing distance and turning shadowing distance are introduced to characterize different dynamical behaviors. We show that a planar closed escaping curve could have shadowing curves of different types including periodic, subharmonic and ergodic ones, depending on the shadowing distance. Singularities of cusp type are found when the shadowing distance is large. Shadowing curves to an escaping circle are examined in details analytically and numerically. Finally, we conjecture that the critical shadowing distance and turning shadowing distance are coincident for typical escaping curves.
