We introduce a class of convex, higher-dimensional billiard models which generalise stadium billiards. These models correspond to the free motion of a point-particle in a region bounded by cylinders cut by planes. They are motivated by models of particles interacting via a string-type mechanism, and confined by hard walls. The combination of these elements may give rise to a defocusing mechanism, similar to that in two dimensions, which allows large chaotic regions in phase space. The remaining part of phase space is associated with marginally stable behaviour. In fact periodic orbits in these systems generically come in continuous parametric families, sociated with a pair of parabolic eigen-directions: the periodic orbits are unstable in the presence of a defocusing mechanism, but marginally stable otherwise. By performing the stability analysis of families of periodic orbits at a nonlinear level, we establish the conditions under which families are nonlinearly stable or unstable. As a result, we identify regions in the parameter space of the models which admit non-linearly stable oscillations in the form of whispering gallery modes. Where no families of periodic orbits are stable, the billiards are completely chaotic, i.e.\ the Lyapunov exponents of the billiard map are non-zero.
Quantifying intermittency in the open drivebelt billiard
一个“传动带”球场台球与边界组成的圆弧不同半径连接其共同的切线与传统的“直”球场，包括双曲性和混合，以及由于边际不稳定的周期轨道（MUPO）的不稳定性的许多属性。有趣的是，直边和曲边的作用是相反的。本文从台球中的洞逃逸的角度讨论了混沌轨道的间歇性，给出了生存概率$P(t)$的精确的首阶系数$\lim_{t\to\infty} t P(t)$，它是固定洞大小的代数.然而，在小孔尺寸与时间成反比的自然缩放极限中，衰减保持指数。直线和传动带视距之间的最大区别是，在传动带的情况下，有多个家庭的MUPO导致定性的新效果。另一个不同之处在于，在这个系统中，大多数边缘周期轨道都倾斜于边界，从而允许利用全内反射的应用，如微激光器。
