School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom.
Chaos. 2012 Jun;22(2):026113. doi: 10.1063/1.3685522.
A "drivebelt" stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional "straight" stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient lim(t→∞)tP(t) of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.
带有由其公切线连接的不同半径的圆弧边界的“驱动带”体育场台球与传统的“直”体育场具有许多相同的性质，包括双曲性和混合性，以及由于边界不稳定周期轨道（MUPO）而产生的间歇性。有趣的是，直边和曲边的作用是相反的。在这里，我们从通过台球中的孔逃逸的角度讨论混沌轨迹的间歇性特性，给出生存概率 P(t)的精确领头阶系数 lim(t→∞)tP(t)，对于固定孔大小，该系数是代数的。然而，在小孔尺寸与时间成反比的自然标度极限下，衰减仍然是指数级的。直边和驱动带体育场之间的主要区别在于，在驱动带情况下，存在多个导致定性新效应的 MUPO 族。另一个区别是，该系统中的大多数边界周期性轨道都与边界成斜交，从而允许利用全内反射的应用，例如微激光器。
