标题： 混沌过渡与锥形弹球 显示英文标题

标题： Transition to chaos with conical billiards

摘要： 我们借鉴了几何光学和经典弹道动力学的思想，考虑在圆锥体上以恒定速度运动的粒子轨迹，在倾斜平面与圆锥体相交形成的椭圆边界上发生镜面反射，倾斜角为$\gamma$。 我们研究了动态作为$\gamma$和圆锥体缺陷角$\chi$的函数，该缺陷角控制顶点的尖锐程度，其中正高斯曲率的点源集中于此。 我们在 ($\gamma, \chi$) 平面上发现了区域，根据初始条件，轨迹可能（A）采样整个圆锥底面并避开顶点区域；（B）仅采样底面的一部分，同时再次避开顶点；或（C）更均匀地采样整个圆锥表面，暗示遍历性。 未倾斜圆锥的特殊情况仅显示类型 A 的轨迹，在接近顶点的距离形成环形焦散线。 然而，我们观察到当$\chi$和$\gamma$足够大时，出现了由类型（C）轨迹主导的复杂混沌动力学转变。 一个总结由镜面反射中断的测地线段轨迹的庞加莱映射，为可视化混沌转变提供了一种强有力的手段。 随后，我们分析了圆锥弹道系统通往混沌的路径与其他保面积保守映射之间的相似性和差异性。 显示英文摘要

摘要： We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane, with tilt angle $\gamma$. We explore the dynamics as a function of $\gamma$ and the cone deficit angle $\chi$ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the ($\gamma, \chi$) plane where, depending on the initial conditions, either (A) the trajectories sample the entire cone base and avoid the apex region; (B) sample only a portion of the base region while again avoiding the apex; or (C) sample the entire cone surface much more uniformly, suggestive of ergodicity. The special case of an untilted cone displays only type A trajectories which form a ring caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently large $\chi$ and $\gamma$. A Poincar\'e map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps.